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2,869,038,154,828 | arxiv | \section{Introduction}
As the nearest large spiral galaxy to the Milky Way, M31 has been the subject of intense observational scrutiny, including recent detections at gamma-ray energies: While early gamma-ray telescopes were able to only set upper limits on the gamma-ray signal, the Fermi Large Area Telescope (LAT) \citep{Atwood:2009ez} was the first instrument to obtain a significant positive detection \citep{M31first}. The LAT detection was found to be compatible both with a point source and with an extended source emission tracing an infrared map at 100$\mu$m intended to indicate star-forming regions, with the extended emission preferred non-significantly at the confidence level of 1.8$\sigma$ \citep{M31first}. Subsequent studies of LAT data including longer exposure have added to the evidence for gamma-ray emission in M31, including the tentative, and controversial, detection of potential ``bubble-like'' features analogous to the Milky Way Fermi bubbles \citep{Pshirkov:2016qhu}. In a recent study, the Fermi-LAT collaboration reported a $10\sigma$ detection of M31 with a strong detection of spatially extended emission out to $\sim 5$ kpc at the $4\sigma$ significance level \citep{ackermann}.
The nature and origin of the emission from the central regions of M31 remain somewhat controversial: on the one hand, M31's observed gamma-ray luminosity does not significantly deviate from the expectation from the known scaling relationship between infrared and gamma-ray luminosity \citep{Ajello:2020zna, Storm:2012gn}; this, in turn, would hint at cosmic rays, accelerated in supernova explosions, as the physical counterpart to the observed emission. This possibility was quantitatively explored in \cite{McDaniel:2019niq}, which found, however, that the required input power from supernova explosion would imply, in the case of a leptonic or hadronic, or even of a mixed scenario, a supernova rate around two orders of magnitude larger than expected. \cite{McDaniel:2017ppt} and \cite{McDaniel:2018vam} explored, instead, a dark matter annihilation scenario, where gamma rays originate as a result of the pair-annihilation of dark matter particles. This possibility was recently also considered in \cite{M31HaloDM}. While in principle consistent with the so-called, controversial, ``Galactic Center Excess'' in the Milky Way \citep{fermiGCE}, and marginally in tension with the non-observation of gamma-ray emission from local dwarf galaxies by Fermi-LAT \citep{albert2017, fermi_dwarfs2015}, this is an intriguing possibility. Finally, unresolved emission from point sources such as millisecond pulsars or other compact objects, which has been considered in \citet{fragione} as well as in \citet{eckner}, remains an unavoidable component of the observed signal, albeit with uncertain relative importance.
Other recent observations of gamma-ray emission in M31 have searched for emission at large radii in the outer halo of the galaxy. As part of a detailed study of the gamma-ray emission in M31 using roughly 8 years of Fermi-LAT data in a $60^{\circ}$ region of interest centered at $(l,b) = (121.17^{\circ}, -21.57^{\circ})$, \citet{M31Halo} reports evidence for an extended gamma-ray excess separate from the Milky Way foreground. This purported emission extends out to roughly 100-200 kpc above the plane of the galaxy, although the authors acknowledge that the emission from the ``far outer halo'' (at angles from M31's center $8.5^\circ<r<21^\circ$) is likely related to mis-modeling of the significant foreground emission from the Milky Way and thus less robust than the emission from the ``Spherical Halo'' region at angles between $0.4^\circ<r<8.5^\circ$ and than the robustly-detected inner galaxy emission at $r<0.4^\circ$. \citet{M31Halo}, while not ruling it out, argues against an extended cosmic-ray halo \citep{Feldmann:2012rx, Pshirkov:2015oqu} based on the radial extent, spectral shape, and intensity of the observed large-radii signal. However, as we argue below, the radial extent and intensity depend critically on assumptions on cosmic-ray diffusion outside the Galactic plane and in the halo; and the spectral shape is strongly affected by foreground Galactic emission and from the intrinsic weakness and limited statistics of the signal.
In this study, we study cosmic-ray electron and proton transport in M31 under a variety of assumptions on the nature of diffusion within and beyond the traditional cylindrical ``diffusion box'', used in Milky Way cosmic-ray studies, around M31's galactic plane. Since we relax the simplifying assumption of homogeneity for the diffusion coefficient, we solve the transport equation via a stochastic approach in the standard way (namely turning the Fokker-Planck partial differential equation describing cosmic-ray transport into a stochastic differential equation solved by means of a Monte Carlo method). We consider both a sharp discontinuity and a gradual transition from the inside to the outside of the inner diffusion region; in addition, we also consider a model, that is becoming increasingly well-motivated by observations of TeV halos \citep{Abeysekara:2017old}, where diffusion within the sites of cosmic-ray acceleration is inefficient. Finally, we also consider a variety of possible injection sites for the cosmic rays.
Cosmic-ray electrons and protons both produce gamma rays as they propagate through the galaxy. However, while electrons radiate highly efficiently and lose energy quickly, protons' energy losses are significantly less efficient, with time scales much longer than those associated with propagation. At the gamma-ray energies of interest, and in the outer regions we are concerned with, cosmic-ray electron emission proceeds through inverse-Compton scattering, primarily by up-scattering photons in the cosmic microwave background (CMB). Cosmic-ray protons instead produce gamma rays as a result of inelastic collisions with the interstellar and circumgalactic medium, producing neutral pions eventually decaying to gamma-ray pairs.
In the study below we also re-assess the contribution of millisecond pulsars and of younger pulsars to the gamma-ray emission, making use of dedicated pulsar population synthesis modeling and of observationally-motivated predictions for gamma-ray emission from pulsars.
Our results indicate that it is quite plausible that {\em (i) most of the spherical halo gamma-ray emission originate from a cosmic-ray halo possibly extending out to M31's virial radius, and well beyond the galactic disk}; we find that this interpretation is {\em (ii) possible both within hadronic and leptonic cosmic-ray scenarios}, albeit in the latter case only a fraction of the spherical-halo gamma-ray emission can be explained; finally, we find that the {\em (iii) inner-galaxy emission is most likely a combination of pulsar gamma-ray emission and of hadronic and leptonic cosmic-ray-induced gamma rays}, somewhat attenuating the tension with the expected supernova rate found in \cite{McDaniel:2019niq}.
The remainder of this study is structured as follows: in the next section \ref{sec:diff} we outline our approach to solving the relevant transport equation, and give details about the diffusion and cosmic-ray models we consider; the following sections \ref{sec:results} and \ref{sec:psr_gamma} detail our results on cosmic-ray driven gamma-ray emission and on pulsar gamma-ray emission, respectively; finally, sec.~\ref{sec:conclusions} presents a final discussion of our results and our conclusions.
\section{Solution to the transport equation and diffusion models}\label{sec:diff}
The transport of cosmic rays on galactic scales, describing the particles' flux and energy spectrum $n(\vec x,\vec p,t)$, with $\vec x$ position and $\vec p$ momentum, is customarily described through diffusive processes in phase space via equations with a structure of the type \citep{Strong:1998pw}
\begin{multline}
\frac{\partial n}{\partial t}+\vec u\cdot\nabla n=\nabla\cdot(\hat \kappa\nabla n)+\frac{1}{p^2}\frac{\partial}{\partial p}\left(p^2\kappa_{pp}\frac{\partial n}{\partial p}\right)\\
+\frac{1}{3}(\nabla\cdot \vec u)\frac{\partial n}{\partial \ln p}+S(\vec x,p,t).
\end{multline}
In the equation above, $\vec u$ is the advection speed, $\hat\kappa$ the spatial diffusion tensor, $p=|\vec p|$, $\kappa_{pp}$ is the momentum-space diffusion coefficient which effectively describes re-acceleration, and $S(\vec x,p,t)$ describes the cosmic-ray sources.
In the context of cosmic-ray (CR) transport, the diffusion equation has been solved through a variety of methods. In \citet{Colafrancesco:2005ji,Colafrancesco:2006he}, a Green's function method, with image ``charges'' suitably accounting for the boundary conditions of the problem, was developed and employed to solve for the steady-state solution to a diffusion problem with spherical symmetry, for arbitrary injection spectra, but with spatially constant energy loss term ad diffusion coefficient \citep[see also][]{2020arXiv201111947V}. The method was generalized in \cite{McDaniel:2017ppt} for the calculation of the radio and inverse-Compton emission, with the possibility to also include a spatially-dependent magnetic field and target radiation field energy density.
An alternate approach is to solve for the differential equation on a lattice by discretizing the problem in a standard fashion. This method is employed by popular codes that solve for the diffusion problem in cylindrical coordinates such as GALPROP \citep{Vladimirov:2010aq} or DRAGON \citep{Tomasik:2008fq}. While this method can in principle be adapted to different geometries and to different assumptions on the spatial dependence of the various transport coefficients, the method is not easily adapted to complex diffusion setups such as the ones we are interested in here.
Finally, using the well-known connection between the Fokker-Planck equation and Stochastic differential equations, other codes model CR transport by means of stochastic processes, see for example the CRPropa code \citep{Merten:2017mgk, Batista:2019nbw}. Here, we utilize precisely this approach, as it is the most flexible to study largely inhomogeneous diffusion setups and complex CR injection morphologies. We refer the reader to classic literature on the equivalence between Fokker-Planck partial differential equations and stochastic differential equations, see e.g. \cite{23677}.
We model diffusion as a stochastic process in space (we model energy losses separately, and neglect reacceleration). We assume diffusion to be isotropic, thus each pseudo-particle's step is taken to occur in a random direction in space. The step size is taken to correspond to the mean free path $\lambda$, which for a diffusive process in 3 dimensions is $\lambda\simeq 3D/v$, with $D$ the (energy-dependent) diffusion coefficient at the particle's location, and $v$ the pseudo-particle velocity; for instance, for typical values of $D\sim3\times 10^{28}\ {\rm cm}^2/s$ and $v\simeq c$, we get $\lambda\simeq 1$ pc. Since the diffusion coefficient is taken to be a function of energy, so is the corresponding mean free path. To reduce the computational complexity of our simulations, we occasionally needed to resort to extrapolations of the results of simulations with larger step sizes. The extrapolation procedure involves running several simulations with the same parameters, except with different step sizes. At each step size, there is a density of particles for a given radius. The densities are extrapolated as a function of step size and a best fit line is produced from the results. We then extrapolate the density to a step size of .002 using the best fit line. A visual of this procedure is shown in fig.~\ref{fig:d2d1}, left.
\begin{figure*}
\centering
\includegraphics[width=0.44\linewidth]{Inset_figure.png}
\includegraphics[width=0.48\linewidth]{optD2D1delta.png}
\caption{Left: Illustration of an extrapolation procedure used in some of our simulations: The inner step size refers to the step size within the diffusion region. This image represents the CRP1 diffusion setup. Right: Averaged $\chi^2$ for a fit to the observed gamma-ray emission morphology from cosmic-ray protons with an ensemble of simulation at various values of the width of the transition region $\delta$ and of the ratio $D_2/D_1$. The optimal choice corresponds to $D_2/D_1=25$ and $\delta=1000$ kpc.}
\label{fig:d2d1}
\end{figure*}
To model the CR spatial distribution in steady state (the case of interest here, since we assume all injection sources to be in steady state), we run our simulations with a limited number of pseudo-particles and assess the CR residence time-scale $\tau$ for pseudo-particle loss outside the region of interest (which is typically taken to be 200 kpc, the same region of interest used in \cite{M31Halo}, and around 2/3 of M31's virial radius) by fitting, after an initial transient, for the exponential decay behavior of the number $N(t)$ of pseudo-particles still within the diffusion region versus the total initial number of particles $N(t=0)=N_0$,
\begin{equation}
N(t)\simeq N_0\exp(-t/\tau);
\label{eq:particle number}
\end{equation}
In practice, we fit the exponential form to the interval $0.1<N(t)/N_0<0.9$ to prevent both fitting for the initial transient corresponding to the drift to the boundary of the diffusion region, and for the noisy tail of the distribution at the end of the simulation. Once the residence time-scale $\tau$ is found, we run extensive simulations with a large number of pseudo-particles (on the order of $10^6$ to $10^8$) for a time $t=\tau$.
We validated the procedure outlined above and the code by comparing our results with a simplified version of the diffusion equation's Green function, which solves
\begin{equation}
\frac{\partial n}{\partial t}=\nabla\cdot (\hat\kappa\nabla n)+\delta(\vec r)\delta(t)
\end{equation}
and which reads
\begin{equation}
n(r_i,t)=\frac{\exp\left(-\frac{r_i^2}{4\kappa_{ii}t}\right)}{\left(4\pi \kappa_{ii}t\right)^{d/2}},
\end{equation}
with $d$ the dimension and $\kappa_{ii}$ the diffusion coefficient in the direction $r_i$. We cross-checked our code for $d=1,\dots,3$ and for isotropic and non-isotropic diffusion tensor. We completely developed the code we employed in-house.
We neglect energy losses for the case of cosmic-ray protons, while in the case of cosmic-ray electrons we assume the standard quadratic dependence on energy for energy losses for high-energy electrons,
\begin{equation}
\frac{dE}{dt}\simeq -b_0 E^2,
\end{equation}
with $b_0\sim 10^{-16}\ {\rm GeV}^{-1}{\rm sec}^{-1}$ \citep{Colafrancesco:2005ji}. One can integrate the equation above by separation of variables between an initial and a final time/energy to get (for initial time $t=0$)
\begin{equation}
-\frac{1}{E_f}+\frac{1}{E_i}=-b_0t=-b_0x/c,
\end{equation}
where in the last equation we assumed that the propagation step has length $x=ct$ because the electrons are ultra-relativistic. Solving for $E_f$, one finds:
\begin{equation}\label{eq:energyloss}
E_f=\frac{E_i}{1+\left(\frac{b_0}{c}\right)x\ E_i}.
\end{equation}
Expressing $x$ in kpc and $E_i$ in GeV, the multiplicative constant gives
\begin{equation}
E_f=\frac{E_i}{1+10^{-5}\frac{x}{\rm kpc}\ \frac{E_i}{\rm GeV}}.
\end{equation}
Thus, in the case of the electrons, we run simulations that track not only position, but also energy. We assume an injection spectrum inspired by Fermi second-order acceleration, $dN/dE\sim 1/E^2$, and
spawn electrons with energy in proportion to that spectrum; we then track both where the final position of the electrons is and which energy they have, with the proviso that, for every step where the electron has moved by a distance $x$, the energy is reduced according to the equation (\ref{eq:energyloss}) above.
Notice that while we diffuse electrons in the same way as protons, i.e. simulating diffusion via a stochastic processed as described above, the relevant time scale for the simulation in this case is the {\em energy-loss} time scale rather than the diffusion of the particles outside the diffusive region of interest. The energy-loss time scale is defined by the number of steps it takes particles to fall below an energy such that emission of gamma rays in the energy range to which the Fermi LAT is sensitive to is no longer viable. Since we are interested here primarily in emission far away from the stellar population in M31, the main photon field we are concerned with is the cosmic microwave background (CMB) (the inverse-Compton emission scales with the energy density in a given radiation field, and in the spherical halo this is dominated by the CMB). The average energy of a CMB photon is $E_\gamma\simeq 6.4\times 10^{-4}$ eV; the typical energy of the up-scattered photon is $E_f\sim \gamma_e^2 E_\gamma$, where $\gamma_e$ is the Lorentz factor of the incoming electron. Requiring $E_f\gtrsim 0.1$ GeV, which is at the lowest energy detectable by the LAT, we get that $\gamma_e=E_e/m_e\gtrsim \sqrt{E_f/E_\gamma} \simeq 4\times 10^5$, thus $E^{\rm min}_e\simeq 200$ GeV. Notice that the energy dependence of the diffusion coefficient is quite critical in the case of high-energy electrons, $D(E)=D_0(E/{\rm GeV})^\delta$. We assume $\delta\simeq 0.3$.
We can estimate the average path length $\lambda_e$ the electrons take to lose their initial energy $E_i$ from Eq.~(\ref{eq:energyloss}) above; being a diffusive process, we first calculate the time $T_{i\to f}$ for the electrons to lose energy from $E_i$ to $E_f$
\begin{equation}
T_{i\to f}=\frac{1}{b_0}\left(\frac{1}{E_f}-\frac{1}{E_i}\right)\simeq 10^{16}\frac{\rm GeV}{E_f}\ {\rm sec};
\end{equation}
The typical distance traveled by an electron from its injection point is then given by
\begin{equation}
\lambda_e\simeq\sqrt{D_0(\bar E/{\rm GeV})^\delta T_{i\to f}}\sim 5.8\ {\rm kpc}\ \left(\frac{E_f}{\rm GeV}\right)^{-0.35}\approx 0.9\ {\rm kpc},
\end{equation}
where in the last equation we assumed $E_f\simeq 200$ GeV. Thus, electrons contributing IC emission off of CMB are not expected to diffuse further than approximately 1 kpc from the source location. Our simulations are found to be consistent with the simple estimate above.
While in the case of electrons up-scattering CMB photons there is a one-to-one correspondence between the final electron locations and the gamma-ray emission along a given line of sight, in the case of protons the structure of the interstellar and ciscumstellar target gas density is crucial, as the gamma-ray emissivity is proportional to the line-of-sight integral of the product of the cosmic-ray proton density and the target gas density. Given the lack of detailed information about the gas density along the line of sight especially between the MW and M31 centers, we choose to adopt the results of the simulations in \cite{2014MNRAS.441.2593N}. Specifically, we utilize their results on the gas density along the line of sight between M31 and the MW (thick black line in fig.~16), up to a distance of approximately 30 kpc from M31; at that point, we match the gas density in their simulation results for the gaseous halo of M31, fig.~5, left panel; finally, we use the results of the simulations in \cite{2013ApJ...763...21F}, their fig.~1, summing upon all components, in the innermost 1.5 kpc.
In producing the gamma-ray morphology plots, we simply count the number of pseudo-particle corresponding to cosmic-ray electrons, given the homogeneity of the background radiation field. In the case of cosmic-ray protons we instead weigh each pseudo-particle with the corresponding target gas density at its location, and then integrate (i.e. sum) along the given line of sight.
In order to compare our results with observations, we digitized the map corresponding to the tentative signal (i.e. residual) intensity map in \cite{M31Halo}, fig. 34, top-left, using the numpy, matplotlib, and OpenCv libraries in python to digitally input the image and the associated color-bar, which was then converted from RGB values back into physical values using the nearest color index of the color-bar. This was then scaled down to the source's original 20x20 pixel resolution. We note that the source image contained overlays for spatial reference, which resulted in artifacts upon digitizing and re-scaling; in order to resolve this, pixels were manually corrected by sampling the nearest unaffected pixels from the full-scale digitized image.
\begin{figure*}
\centering
\mbox{\includegraphics[width=0.3\linewidth]{diff1.pdf}\qquad\qquad \includegraphics[width=0.4\linewidth]{diff2.pdf}}
\caption{Spatially varying diffusion coefficient, with the functional form in Eq.~(\ref{eq:diff}) and $D_2/D_1=25$ and $\delta=1000$ kpc. The left panel shows curves of constant $D_2/D_1$ on the $(r,z)$ plane of cylindrical coordinates across the entire region of interest, while the right panel shows a three dimensional plot of the same quantity, on the same plane. The inner diffusion region, with constant $D=D_1$, is bordered by white lines.}
\label{fig:diffcoeff}
\end{figure*}
\section{Results}\label{sec:results}
In this analysis we test the impact of both the source morphology and the diffusion setup on the gamma-ray emission from (i) inelastic collisions of cosmic-ray protons with the ISM and (ii) inverse Compton emission of high-energy cosmic-ray electrons off of CMB photons, using the simulation techniques described in the previous section.
As far as the source morphology is concerned, we entertain three scenarios:
\begin{enumerate}
\item Cosmic rays are produced near the central region of M31; this scenario assumes that the main acceleration mechanism for cosmic rays in M31 is physics associated with the innermost region of the galaxy, such as for instance accretion around, and jets emanating from, the central supermassive black hole of M31;
\item Cosmic rays are produced in star-forming regions; this possibility physically relies on the notion that the main cosmic-ray acceleration sites are likely supernova shocks, whose locations trace star-forming regions. Observationally and theoretically, this possibility was explored in \cite{Carlson:2015daa, Carlson:2015ona} and in \cite{Carlson:2016iis}, which found that a significant fraction of cosmic rays in the Milky Way are likely injected from star-forming regions. We use as a tracer of star-forming regions in M31 the IR emission map from \cite{Gordon_2006}.
\item Finally, we use the {\tt PrsPopPy} code (\cite{psrpoppy}) to produce a synthetic population of pulsars (see sec.~\ref{sec:psr_gamma} for details on the population synthesis procedure we adopt) as a proxy for a scenario where cosmic-ray electrons and positrons are produced in the magnetosphere of rotating neutron stars \citep[see e.g.][]{Grasso:2009ma, Profumo:2008ms}.
\end{enumerate}
We also entertain a variety of diffusion models, taking advantage of the flexibility provided by the stochastic solution to the diffusion equation. In particular, we assume:
\begin{enumerate}
\item[(a)] A traditional ``leaky box'' diffusion scenario (hereafter referred to as our ``benchmark'' model) inspired by similar setups for the Milky Way that successfully reproduce the measured abundance of cosmic-ray species \citep[see e.g.][]{Vladimirov:2010aq, Tomasik:2008fq}. Here, we assume that cosmic rays diffuse primarily inside a cylindrical diffusion region of radius 20 kpc and half-height 10 kpc (we have also considered variations of these parameters, with marginal impact on our results described below), effectively free-streaming outside the diffusion region; we model this latter effect by a sudden, step-like jump by a factor of 100 in the diffusion coefficient outside the cylindrical box;
\item[(b)] A ``constant'' diffusion scenario, where cosmic rays diffuse in an isotropic and homogeneous medium with a constant diffusion coefficient. Albeit physically unrealistic, this scenario aims at assuming that the circumgalactic medium in the Local Group continues to support cosmic-ray diffusion well outside M31 and the Milky Way and out to much larger radii than the galaxies' size;
\item[(c)] A ``gradual'' spatially-dependent diffusion coefficient defined so that the diffusion coefficient inside a cylindrical box of height $z_t$ and radius $r_t$ is $D_1$ and, after a transition region of size $\delta$, it asymptotes to an outer value $D_2$. The distance of a point $(r,z)$ in cylindrical coordinates from the diffusion box is
\begin{equation}
{\rm dist}(r,z)=\sqrt{(r-{\rm min}[r,r_t])^2+(z-{\rm min}[z,z_t])^2};
\end{equation}
The diffusion coefficient at a point of cylindrical coordinates $(r,z)$ is then calculated as
\begin{equation}\label{eq:diff}
D(r,z)=D_1+\left(D_2-D_1\right)\frac{{\rm ArcTan}\Big[{\rm dist}(r,z)/\delta\Big]}{\pi/2}.
\end{equation}
We searched for the values of $D_2/D_1$ and $\delta$ producing the gamma-ray emission morphology most closely resembling (based quantitatively on a pixel-by-pixel $\chi^2$ procedure) the diffuse gamma-ray emission measured in the inner halo and spherical halo of M31 \citep{M31Halo}. For this calculation, we utilized the gamma-ray emission from cosmic-ray protons, with the procedure explained above. The $\chi^2$ was computed by comparing the predicted and measured emission pixel by pixel, after normalizing both maps to the same average emission. We show in fig.~\ref{fig:d2d1}, right, the results for the $\chi^2$ for different values of $\delta$ and $D_2/D_1$.
For every combination of $D_2/D_1$ and $\delta$ we ran a set of 10 independent simulations, with the inferred standard deviation shown in the figure. Our results indicate a preference for {\em large} values of $\delta$, implying, in turn, a preference for a mild ``gradient'' in transitioning to the larger outer value of the diffusion coefficient. Similarly, we observe a preference for smaller ratios of the outer to inner diffusion coefficient. However, for large $\delta\sim1$ Mpc, we find that ratios $5\lesssim D_2/D_1\lesssim 50$ give equally good fits to the observed morphology. While there is no strong statistical preference, we adopted as our benchmark choice the lowest $\chi^2$ central value which corresponded to $D_2/D_1=25$. We show in fig.~\ref{fig:diffcoeff} with an iso-level contour plot in the left and with a three-dimensional rendering of a 100$\times$100 kpc region on the right the resulting diffusion coefficient (normalized to the value inside the inner diffusion box) in cylindrical coordinates $(r,z)$, with parameters corresponding to the optimal choices $D_2/D_1=25$ and $\delta=1000$ kpc.
\begin{figure*}
\centering
\begin{subfigure}[b]{1\textwidth}
\includegraphics[width=\linewidth]{CRE_Morphology.png}
\end{subfigure}
\caption{Morphology of each of the six CRE cases. For each case, a contour plot of the observed gamma-ray emission from the Fermi-LAT observations \citep{M31Halo} is mapped over the CRE configurations for comparison.}
\label{fig:cre_morphology}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{Final_CRE_Rad_Intensity_Plot.png}
\caption{Radial intensity profile for cosmic-ray electron simulations, for the six diffusion and injection source profile combinations discussed in the text.}
\label{fig:cre_radial_intensity}
\end{figure}
\item[(d)] As a second example of a spatially-varying diffusion coefficient, we utilize a model (which we dub ``Swiss cheese'' diffusion coefficient) where inside the diffusion region there exist spherical sub-regions of inefficient cosmic-ray transport associated with the turbulent medium inside pulsar wind nebulae (PWNe). This scenario reflects the recent findings of \cite{Abeysekara:2017old} that high-energy cosmic-ray electrons diffuse much less efficiently (around a factor 100 smaller effective diffusion coefficient) inside PWNe than outside. Following \cite{Profumo:2018fmz}, we use the model of \cite{Abdalla:2017vci} to relate the pulsar age to the radial size of the corresponding PWN, and we assume a sudden transition to a diffusion coefficient $D_0/100$ inside the PWN; outside the cylindrical box, we assume, as for the benchmark model, a large diffusion coefficient $100\times D_0$.
\end{enumerate}
We employ slightly different sets of source distribution and diffusion models for cosmic-ray electrons (CRE) and protons (CRP), based on different expected injection sources (protons are not thought to be produced by pulsars' magnetospheres). We describe below our choices and results.
\subsection{Cosmic-Ray Electrons}
Here we present results for gamma rays from inverse Compton (IC) up-scattering of CMB photons by high-energy cosmic ray electrons. We consider six different cases:
\begin{enumerate}
\item[(CRE1):] Benchmark diffusion scenario (a), with CRE injected at the very center of M31, i.e. scenario (i)
\item[(CRE2):] Benchmark diffusion scenario (a), with CRE injected in star-forming regions, i.e. scenario (ii)
\item[(CRE3):] Benchmark diffusion scenario (a), with CRE injected at the location of mature synthetic pulsar locations, i.e. scenario (iii)
\item[(CRE4):] Constant diffusion scenario (b), with CRE injected at the very center of M31, i.e. scenario (i)
\item[(CRE5):] Gradual diffusion scenario (c), with $D_2/D_1=25$ and $\delta=1,000$ kpc, with CRE injected at the very center of M31, i.e. scenario (i)
\item[(CRE6):] ``Swiss cheese" diffusion scenario (d), with CRE injected at the location of mature synthetic pulsar locations, i.e. scenario (iii)
\end{enumerate}
\begin{figure*}
\centering
\begin{subfigure}[b]{1\textwidth}
\includegraphics[width=\linewidth]{CRP_Morphology.png}
\end{subfigure}
\caption{Morphology of each of the five CRP cases. For each case, a contour plot of the observed gamma-ray emission from Fermi-LAT \protect\citet{M31Halo} is mapped over the CRP configurations for comparison.}
\label{fig:crp_morphology}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{Final_CRP_Rad_Intensity_Plot.png}
\caption{Radial intensity profile for cosmic-ray proton simulations, for the five diffusion and injection source profile combinations discussed in the text.}
\label{fig:crp_radial_intensity}
\end{figure}
Fig.~\ref{fig:cre_morphology} shows the emission morphology from IC of CMB photons by CRE. We notice that virtually in all cases the emission is mostly circumscribed to the inner regions of M31, albeit with different morphology for the different assumptions on diffusion and source location. The radially-averaged intensity profiles of the six cases is shown in fig.~\ref{fig:cre_radial_intensity}. First, we note that CRE1 and CRE5 are very similar, indicating that the transport conditions beyond the inner regions play a relatively mild role. In these cases, there is also very marginal emission in the spherical halo region. CRE2 and CRE3 also look remarkably similar, as is somewhat expected since in both cases the CR injection sites trace star formation, in CRE2 via the IR emission map we utilize as a proxy for the star formation rate, and in CRE3 via the synthetic pulsar population model we constructed (described in detail in the following section). CRE2 and CRE3 exhibit a more extended morphology compared to CRE1 and CRE5, and are found to contribute somewhat to the emission in the spherical halo region, which is around $5\times 10^{-8}$ ph cm$^{-2}$ s$^{-1}$ sr$^{-1}$, out to around 15 kpc. Finally, CRE6 is the model that has the most pronounced emission in the innermost few kpc, driven by CR electrons being ``trapped'' in bubbles of inefficient transport associated with PWNe.
\subsection{Cosmic-Ray Protons}
\begin{figure*}
\centering
\includegraphics[width=0.4\linewidth]{PulsarSpitzer.png}\qquad \includegraphics[width=0.4\linewidth]{303psrs.pdf}
\caption{Left: Model pulsar population locations in relation to the galactic plane of M31. The colors of the pulsars are a generated by a standard 3-D kernel density estimation. The plane image is the de-projected 24 $\mu$m image from \protect\citet{Gordon_2006}. Right: example sample of 303 PWNe (enlarged by a factor 20).}
\label{fig:psr_locations}
\end{figure*}
In the case of protons, which are not produced in pulsars' magnetospheres, we consider a different set of cases (although, to ease the comparison with the CRE case, we follow a similar numbering convention), specifically:
\begin{enumerate}
\item[(CRP1):] Benchmark diffusion scenario (a), with CRP injected at the very center of M31, i.e. scenario (i)
\item[(CRP2):] Benchmark diffusion scenario (a), with CRP injected in star-forming regions, i.e. scenario (ii)
\item[(CRP4):] Constant diffusion scenario (b), with CRP injected at the very center of M31, i.e. scenario (i)
\item[(CRP5):] Inhomogeneous ``gradual'' diffusion scenario (c), with $D_2/D_1=25$ and $\delta=1,000$ kpc, with CRP injected at the very center of M31, i.e. scenario (i)
\item[(CRP6$^\prime$):] ``Swiss cheese" diffusion scenario (d), but with CRP injected in star-forming regions, i.e. scenario (ii)
\end{enumerate}
As above, we show in two separate figures the results for the morphology of the innermost 25$\times$25 kpc region in fig.~\ref{fig:crp_morphology} and the radial intensity profile in fig.~\ref{fig:crp_radial_intensity}. Our results indicate that CRP1, CRP2, and CRP6$^\prime$ all exhibit a relatively similar morphology, likely due to the fact that in those cases the emission tracks quite closely the residence time, in turn related to the diffusion coefficient. Since protons diffuse for much longer times than electrons, the source injection site is less critical, and information thereof is asymptotically lost.
Larger values of the diffusion coefficient in the spherical halo and outer halo regions, as in CRP4 and CRP5, yields, as expected, a much brighter emission at large radii; because CRP5 has a gradual ramp up to a larger diffusion coefficient, its relative brightness at large radii is lower than the constant diffusion coefficient case of CRP4. This is also clearly shown in the radial intensity profile of fig.~\ref{fig:crp_radial_intensity}.
Our results for cosmic-ray protons indicate, as somewhat expected, that in order to support significant emission beyond the inner region, a comparatively small diffusion coefficient needs to be present in the outer regions of M31, as in CRP4 and CRP5. In either case, including when, as shown in fig.~\ref{fig:diffcoeff}, the diffusion coefficient is almost five times larger at the outskirts than around the inner M31 regions, the gamma-ray emission in the spherical halo and that in the inner region are quite accurately reproduced. Only a constant, suppressed diffusion coefficient would explain the outermost gamma-ray emission; in this case the emission in the inner region would also additionally be self-consistently explained (see the pink dashed line in fig.~\ref{fig:crp_radial_intensity}).
\section{Pulsar Emission}\label{sec:psr_gamma}
The pulsar population in M31 is modeled using the population synthesis code PsrPopPy \citep{psrpoppy}.
In total we generate $10,000$ pulsars using default parameters for a Milky Way-type galaxy, appropriate in the present case. In particular, the radial distribution is based on the analysis of \cite{lorimer06}, while the vertical distribution assumes a two-sided exponential with scale height of $z_{scale}=0.33$ kpc. The pulsar spin period also is given in \cite{lorimer06} as a log-normal distribution with $\mu=2.7$, and standard deviation $\sigma=-0.34$. The total pulsar number was chosen as a reasonable estimate based on estimates from \cite{Lorimer2009}, where the theorized pulsar birth rate is estimated between 2.8 pulsars per century, which would give a population of approximately 3000 pulsars, and a larger possible number, which \cite{Lorimer2009} suggests could be up to a factor 5 larger.
We sample the PSR ages homogeneously and linearly between the ages of $10^3$ yrs and $10^5$ yrs, to bracket the observationally-motivated age range of pulsars exhibiting a wind nebula (PWN) \cite{HESS2018}.
As far as the size of the PWN, we implement the functional dependence between age and radius of the nebula as in \cite{HESS2018, Profumo:2018fmz}. In figure \ref{fig:psr_locations} we show for illustrative purposes an image of the pulsar locations relative to the plane of the galaxy compared with a de-projected 24 $\mu$m image of M31 from \cite{Gordon_2006}. Notice that in the figure we use different scales for the $z$ axis and for the galactic plane. The right panel of fig.~\ref{fig:psr_locations} shows the size of the region of inefficient diffusion for a down-selection of 303 random pulsars, enhanced for visibility by a factor 20 in size.
\begin{figure*}
\centering
\includegraphics[width=0.85\linewidth]{PWNmorphology.png}
\caption{The morphology of the gamma-ray emission from unresolved gamma-ray pulsars as predicted in the population synthesis model we constructed, for the case where each pulsar features the same gamma-ray luminosity (PWN0), for $L_\gamma\propto \dot E\propto 1/(\tau P^2)$ (PWN1), $L_\gamma\propto \sqrt{\dot E}\propto 1/(\sqrt{\tau} P)$ (PWN2).}
\label{fig:psr_morphology}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{radprof_log.png}
\caption{The radial intensity profile, normalized to the measured gamma-ray intensity from the inner regions of M31, for the gamma-ray emission from unresolved gamma-ray pulsars as predicted in the population synthesis model we constructed, for the case PWN0 of identical emission from each pulsar (black triangles), of $L_\gamma\propto \dot E\propto 1/(\tau P^2)$ (PWN1, purple circles), $L_\gamma\propto \sqrt{\dot E}\propto 1/(\sqrt{\tau} P)$ (PWN2, green squares).}
\label{fig:psr_radial}
\end{figure}
We utilize the synthetic population constructed as described above to simulate the sources of high-energy CRE in the case of CRE6, and we adopt for diffusion scenario (iii), again from CRE6 and for CRP6$^\prime$ a suppressed diffusion coefficient 100 times smaller than in the rest of the diffusive cylindrical inner region. We also use the synthetic pulsar population to model a further possible (and plausible) source of gamma radiation.
As detailed in \cite{TheFermi-LAT:2013ssa}, the emission from gamma-ray pulsars is found to correlate with the pulsar's spin-down luminosity $\dot E$. Specifically, \cite{TheFermi-LAT:2013ssa} finds that, while several pulsars over a wide range of spin-down luminosities exhibit gamma-ray luminosities $L_\gamma\propto \dot E$, numerous other follow a phenomenological behavior where $L_\gamma\propto \sqrt{\dot E}$. Noting that the spin-down luminosity $\dot E\propto \dot P/P^3$, where $P$ is the pulsar period and $\dot P$ the period derivative, and that the characteristic pulsar age $\tau=P/(2\dot P)$, given our synthetic pulsar catalog, we simulated pulsar emission from the following two observationally motivated \cite{TheFermi-LAT:2013ssa} prescriptions:
\begin{itemize}
\item [(1)] $L_\gamma\propto \dot E\propto 1/(\tau P^2)$;
\item [(2)] $L_\gamma\propto \sqrt{\dot E}\propto 1/(\sqrt{\tau} P)$.
\end{itemize}
We also considered a model (0) where all pulsars produce the {\em same} gamma-ray emission. This latter case (0) can be considered a proxy for the emission from older, ``recycled'' millisecond pulsars, as considered for instance in \cite{eckner} and in \cite{fragione}.
We show in fig.~\ref{fig:psr_morphology} the expected gamma-ray emission from unresolved gamma-ray pulsars from the synthetic population we built as described above, for the three cases PWN1 where $L_\gamma\propto \dot E\propto 1/(\tau P^2)$, PWN2 where $L_\gamma\propto \sqrt{\dot E}\propto 1/(\sqrt{\tau} P)$ and in the case PWN0 where the same gamma-ray luminosity is associated with every pulsar in the catalog. We observe a slight increase to the extension of the emission from PWN1 to PWN2 to the same-luminosity case. Overall, however, the pulsar emission appears to be relevant only for the innermost few kpc, as also reproduced in the detailed radial intensity profile shown in fig.~\ref{fig:psr_radial}, where we normalize the emission to the inner galaxy data point, shown in red.
We estimate that in order to account for 100\% of the inner galaxy emission, the typical gamma-ray luminosity of each pulsar in our catalog would need to be around $4\times 10^{40}$ ph sec$^{-1}$. Since the gamma-ray luminosity of both gamma-ray pulsars from the Fermi-LAT pulsar catalog \citep{TheFermi-LAT:2013ssa} and for millisecond pulsar \citep[see e.g.][]{Hooper:2015jlu} is $10^{33}-10^{37}$ erg/sec, and given that the observed emission peaks around 1 GeV, we find that the energetics of the gamma-ray emission is compatible with being primarily or in significant part fuelled by emission from gamma-ray pulsars. In summary, based on both morphological arguments and energetics, we find that emission from unresolved gamma-ray pulsars in M31 is likely to be a significant contributor in the inner regions of M31.
\section{Discussion and Conclusions}\label{sec:conclusions}
We studied the gamma-ray emission expected from the Andromeda galaxy (M31) due to high-energy cosmic-ray electrons and protons and by unresolved pulsar emission. The key motivation for the present study is the detection of gamma-ray emission in the spherical halo and far outer halo of M31 \citep{M31Halo}, and the possibility that such emission be in part or entirely associated with a cosmic-ray ``halo'' extending significantly beyond the disk and bulge of M31.
We considered a broad ensemble of diffusion scenarios, including ones where diffusion is relatively efficient out to large radii, and ones where diffusion is significantly inhomogeneous. We found that cosmic-ray electrons up-scattering cosmic microwave background photons is likely responsible for a significant portion of the inner region gamma-ray emission and, possibly, of the spherical halo, especially if diffusion is highly inefficient near the sites of cosmic-ray electron acceleration. Cosmic-ray protons also definitely contribute to the inner-region emission, and possibly to the emission in the outer-most region, if the increase in the diffusion coefficient from the inner regions out to the virial radius is limited to within a factor 5-10. Finally, we studied the possible contribution of unresolved point-like sources associated with pulsars, and found that this should only contribute to the inner region, with limited impact on the spherical halo emission.
In this study we did not consider spectral information in investigating the nature of the spherical halo and outer halo emission from M31. The reason for this choice is that, as shown explicitly in \cite{McDaniel:2019niq}, in both the hadronic and the leptonic case the gamma-ray spectrum is largely dependent on the assumed cosmic-ray injection spectrum. The latter, in turn, depends on the acceleration sites and mechanism. As a result, while some information on the expected cosmic-ray spectrum is inferred from direct measurements at Earth, and from Galactic gamma rays, spectral information provides only a highly model-dependent input to the origin of the M31 gamma-ray emission.
In conclusion, our results suggest a possible direct evidence for an extended cosmic-ray halo around M31, and thus possibly around our own Galaxy, as first entertained for instance in \cite{Feldmann:2012rx} and \cite{Pshirkov:2015oqu}. The composition of the halo in terms of its leptonic and hadronic components is at present unclear, as is the detailed structure of the diffusion scenario, although our results give some boundaries to the nature of the latter. Given the possible new-physics interpretation of the gamma-ray emission from the outer regions of M31 \citep{M31HaloDM}, the cosmic-ray halo scenario should be carefully explored. Future observations both at gamma-ray and at other frequencies in conjunction with additional detailed cosmic-ray simulations and predictions at other wavelengths (including e.g. radio and X-ray, where detailed data exist) will help further elucidate the origin of the gamma-ray emission from M31. For instance, \cite{McDaniel:2019niq} showed that the inner galaxy M31 gamma-ray emission can only be explained exclusively by cosmic-ray electrons as long as the magnetic field in the inner regions is highly suppressed compared to expected values in the several micro-gauss range. Finally, given the similarities between M31 and the Milky Way, our results warrant establishing whether
our own Galaxy possesses an extended cosmic-ray halo and, if so, how it would manifest observationally \citep[see e.g.][]{Tibaldo:2015ooa}.
\section*{Acknowledgements}
SP is partly supported by the U.S. Department of Energy grant numbers DE-SC0010107 and A00-1465-001. We gratefully acknowledge early contributions by Henri Geneste, Lili Manzo, Lilianne Callahan, and helpful conversations with Tesla Jeltema.
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
2,869,038,154,829 | arxiv | \section{Introduction}
Many objects, especially these made by humans, have intrinsic symmetry \cite{Rosen12,Hong04} and Manhattan properties (meaning that 3 perpendicular axes are inferable on the object \cite{Coughlan99,Coughlan03,Furukawa09}), such as cars, aeroplanes, see Fig~\ref{fig:symmetry_manhattan}. The purpose of this paper is to investigate the benefits of using symmetry and/or Manhattan constraints to estimate the 3D structures of objects from one or more images. As a key task in computer vision, numerous studies have been conducted on estimating the 3D shapes of objects from multiple images \cite{Hartley2004,Torresani03,Xiao04,Torresani08,Akhter2011,Gotardo11,Hamsici2012,Dai12,Dai14,Agudo14}. There is also a long history of research on the use of symmetry \cite{Gordon90,kontsevich93,Vetter94,Mukherjee95,Hong04,Thrun05,Li07,jiang2015polyhedral} and a growing body of work on Manhattan world \cite{Coughlan99,Coughlan03,Furukawa09}. There is, however, little work that combines these cues.
\begin{figure}[t]
\centering
\includegraphics[width=0.3\linewidth]{car.pdf}
\hspace{15mm}
\includegraphics[width=0.3\linewidth]{Fig1.pdf}
\caption{Left Panel: Illustration of symmetry and Manhattan structure. The car has a bilateral symmetry with respect to the plane in red. There are three Manhattan axes. The first is normal to the symmetry plane of the car (\eg, from {\it left wheels} to {\it right wheels}). The second is from the front to the back of the car (\eg, from {\it back wheels} to {\it front wheels}) while the third is in the vertical direction. Right Panel: Illustration of the 3 Manhattan directions on a real aeroplane image, shown by Red, Green, Blue lines. These 3 Manhattan directions can be obtained directly from the labeled keypoints.}
\label{fig:symmetry_manhattan}
\end{figure}
This paper aims at estimating the 3D structure of an object class, taking a single or multiple intra-class instances as input, \eg, different cars from various viewpoints. Following \cite{Vicente14,Kar15,Gao16}, we use 2D positions of keypoints as input to estimate the 3D structure and the camera projection, leaving the detection of the 2D keypoints to methods such as \cite{chen2014articulated}. In this paper, different combinations of the three cues, \ie, symmetry, Manhattan and multiple images, are investigated and two algorithms/derivations are proposed (assuming orthographic projection), \ie, single image reconstruction using both symmetry and Manhattan constraints, and multiple-image reconstruction using symmetry\footnote{We experimented with using Manhattan for the multiple-image case, but found it gave negligible improvement. Please see the derivations of using Manhattan for the multiple-image case in the supplementary material.}.
Specifically, we start with the single image reconstruction case, using the symmetry and Manhattan constraints, see Fig.~\ref{fig:symmetry_manhattan}. Our derivation is inspiring that a single image is sufficient for reconstruction when the symmetry and Manhattan axes can be inferred. Specifically, our analysis shows that using Manhattan alone is sufficient to recover the camera projection (up to several axis-sign ambiguities) by a single input image, then the 3D structure can be reconstructed uniquely by symmetry thereafter. We show these results on {\it aeroplane} in Pascal3D+. But we note that all the 3 Manhattan axes are hard to observe from a single image sometimes, particularly if the keypoints we rely on are occluded.
Hence, we extend the use of symmetry to the multiple-image case using structure from motion (SfM). The input is different intra-class object instances with various viewpoints. We formulate the problem in terms of energy minimization (\ie, MLE) of a probabilistic model, in which symmetry constraints are included. The energy also involves missing/latent variables for unobserved keypoints due to occlusion. These complications, and in particular symmetry, implies that we cannot directly apply singular value decomposition (SVD) (as in the previous SfM based work \cite{kontsevich87,Tomasi92}) to directly minimize the energy function. Instead we must rely on coordinate descent (or hard EM methods if we treat the missing points are latent variable) which risk getting stuck in local minimum. To address this issue we define a {\it surrogate energy function} which exploits symmetry, by grouping the keypoints into symmetric keypoint pairs, and assumes that the missing data are known (\eg, initialized by another process). We show that the surrogate energy can be decomposed into the sum of two independent energies, each of which can be minimized directly using SVD. This leads to a two-stage strategy where we first minimize the surrogate energy function as initialization for coordinate descent on the original energy function.
Recall that the classic SfM has a ``gauge freedom" \cite{morris2001gauge} because we can rotate the object and camera pose by equivalent amounts without altering the observations in the 2D images. This gauge freedom can be thought of as the freedom to {\it choose our coordinate system}. In this paper, we exploit this freedom to choose the symmetry axis to be in the $x$ axis (and the other two Manhattan axes to be in the $y$ and $z$ directions).
In the following, we group keypoints into keypoint pairs and use a superscript $\dag$ to denote symmetry, \eg, $Y$ and $Y^{\dag}$ are symmetric keypoint pairs.
The rest of the paper is organized as follows: firstly, we review related work in Section 2. In Section 3, we describe the common experiment design for all our methods. Then the mathematical details and evaluations on the single image reconstruction are given in Section 4, followed by the derivation and experiments on the multiple-image case, \ie, the symmetric rigid structure from motion (Sym-RSfM), in Section 5. Finally, we give our conclusions in Section 6.
\section{Related Works}
Symmetry has been studied in computer vision for several decades. For example, symmetry has been used as a cue in depth recovery \cite{Gordon90,kontsevich93,Mukherjee95} as well as for recognizing symmetric objects \cite{Vetter94}. Grossmann and Santos-Victor utilized various geometric clues, such as planarity, orthogonality, parallelism and symmetry, for 3D scene reconstruction \cite{grossmann2002maximum,grossmann2005least}, where the camera rotation matrix was pre-computed by vanishing points \cite{grossmann2002single}. Recently, researchers applied symmetry to scene reconstruction \cite{Hong04}, and 3D mesh reconstruction with occlusion \cite{Thrun05}. In addition, symmetry, incorporated with planarity and compactness priors, has also been studied to reconstruct structures defined by 3D keypoints \cite{Li07}. By contrast, the Manhattan world assumption was developed originally for scenes \cite{Coughlan99,Coughlan03,Furukawa09}, where the authors assumed visual scenes were based on a Manhattan 3D grid which provided 3 perpendicular axis constaints. Both symmetry and Manhattan can be straightforwardly combined, and adapted to 3D object reconstruction, particularly for man made objects.
The estimation of 3D structure from multiple images is one of the most active research areas in computer vision.
Classic SfM for rigid objects built on matrix factorization methods \cite{kontsevich87,Tomasi92}. Then, more general non-rigid deformation was considered, and the rigid SfM in \cite{kontsevich87,Tomasi92} was extended to non-rigid case by Bregler \etal \cite{Bregler00}. Non-rigid SfM was shown to have ambiguities \cite{Xiao04} and various non-rigid SfM methods were proposed using priors on the non-rigid deformations \cite{Xiao04,Torresani08,Olsen08,Akhter08,Gotardo11,Akhter2011}. Gotardo and Martinez proposed a Column Space Fitting (CSF) method for rank-$r$ matrix factorization and applied it to SfM with smooth time-trajectories assumption \cite{Gotardo11}. A more general framework for rank-$r$ matrix factorization was proposed in \cite{hongsecrets}, containing the CSF method as a special case\footnote{However, the general framework in \cite{hongsecrets} cannot be used to SfM directly, because it did not constrain that all the keypoints within the same frame should have the same translation. Instead, \cite{hongsecrets} focused on better optimization of rank-$r$ matrix factorization and pursuing better runtime.}. More recently, it has been proved that the ambiguities in non-rigid SfM do not affect the estimated 3D structure, \cite{Akhter09} which leaded to prior free matrix factorization methods \cite{Dai12,Dai14}.
SfM methods have been used for category-specific object reconstruction, \eg, estimating the structure of \emph{cars} from images of different \emph{cars} under various viewing conditions \cite{Kar15,Vicente14}, but these did not exploit symmetry or Manhattan. We point out that in \cite{Ceylan14}, the repetition patterns have been incorporated into SfM for urban facades reconstruction, but \cite{Ceylan14} focused mainly on repetition detection and registration.
\section{Experimental Design}
This paper discusses 2 different scenarios to reconstruct the 3D structure: (i) reconstruction from a single image using symmetry and the Manhattan assumptions, (ii) reconstruction from multiple images using symmetric rigid SfM. The experiments are performed on Pascal3D+ dataset. This contains object categories such as \emph{aeroplane} and \emph{car}. These object categories are sub-divided into subtypes, such as \emph{sedan} car. For each object subtype, we estimate an 3D structure and the viewpoints of all the within-subtype instances. The 3D structure is specified by the 3D keypoints in Pascal3D+ \cite{Xiang14} and the corresponding keypoints in the 2D images are from Berkeley \cite{Bourdev10}. These are the same experimental settings as used in \cite{Kar15,Gao16}.
For evaluation we report the rotation error $e_R$ and the shape error $e_S$, as in \cite{Dai12,Dai14,Akhter08,Gotardo11,Gao16}. The 3D groundtruth and our estimates may have different scales, so we normalize them before evaluation. For each shape $S_n$ we use its standard deviations in $X, Y, Z$ coordinates $\sigma_n^x, \sigma_n^y, \sigma_n^z$ for normalization: $S_n^{\text{norm}} = 3 S_n/(\sigma_n^x + \sigma_n^y + \sigma_n^z)$. To deal with the rotation ambiguity between the 3D groundtruth and our result, we use the Procrustes method \cite{Schonemann66} to align them. Assuming we have $2P$ keypoints, \ie, $P$ keypoint pairs, the rotation error $e_R$ and the shape error $e_S$ are calculated as:
\begin{align}
&e_R = \frac{1}{N} \sum_{n=1}^{N} ||R_n^{\text{aligned}} - R_n^* ||_F, \nonumber \\
&e_S = \frac{1}{2NP} \sum_{n=1}^{N} \sum_{p = 1}^{2P} ||S_{n,p}^{\text{norm aligned}} - S_{n,p}^{\text{norm}*}||_F,
\label{eq:Error}
\end{align}
where $R_n^{\text{aligned}}$ and $R_n^*$ are the recovered and the groundtruth camera projection matrix for image $n$. $S_{n,p}^{\text{norm aligned}}$ and $S_{n,p}^{\text{norm}*}$ are the normalized estimated structure and the normalized groundtruth structure for the $p$'th point of image $n$. These are aligned by the Procrustes method \cite{Schonemann66}.
\section{3D Reconstruction of A Single Image}
In this section, we describe how to reconstruct the 3D structure of an object from a single image using its symmetry and Manhattan properties. Theoretical analysis shows that this can be done with a bilateral symmetry and three Manhattan axes, but the estimation is ambiguous if less than three Manhattan axes are visible. Specifically, the three Manhattan constraints alone are sufficient to determine the camera projection up to sign ambiguities (\eg, we cannot distinguish between front-to-back and back-to-front directions). Then, the symmetry property is sufficient to estimate the 3D structure uniquely thereafter.
Let $Y, Y^{\dag} \in \mathbb{R}^{2 \times P}$ be the observed 2D coordinates of the $P$ symmetric pairs, then the orthographic projection implies:
\begin{equation}
Y = RS, \qquad Y^{\dag} = RS^{\dag}, \label{SingleImg}
\end{equation}
where $S, S^{\dag} \in \mathbb{R}^{3 \times P}$ are the 3D structure and $R \in \mathbb{R}^{2 \times 3}$ is the camera projection matrix. We have eliminated translation by centralizing the 2D keypoints.
\begin{remark}
We first estimate the camera projection matrix using the Manhattan constraints. Each Manhattan axis gives us one constraint on the camera projection. Hence, three axes give us the well defined linear equations to estimate the camera projection. We now describe this in detail.
\end{remark}
Consider a single Manhattan axis specified by 3D points $S_a$ and $S_b$. Without loss of generality, assume that these points are along the $x$-axis, \ie, $S_a - S_b = [x, 0, 0]^T$. It follows from the orthographic projection that:
\begin{equation}
Y_a - Y_b = R(S_a - S_b) =
\begin{bmatrix}
r_{11}, & r_{12}, & r_{13} \\
r_{21}, & r_{22}, & r_{23}
\end{bmatrix}
\begin{bmatrix}
x \\ 0 \\ 0
\end{bmatrix}
=
\begin{bmatrix}
r_{11} x \\
r_{21} x
\end{bmatrix}. \label{x-axis}
\end{equation}
where $R = \begin{bmatrix} r_{11}, & r_{12}, & r_{13} \\ r_{21}, & r_{22}, & r_{23} \end{bmatrix}$. Setting $Y_a = [y_a^1, y_a^2]^T$, $Y_b = [y_b^1, y_b^2]^T$, yields
$r_{21} / r_{11} = (y_a^2 - y_b^2) / (y_a^1 - y_b^1)$. With other Manhattan axes, \eg, if $S_c, S_d$ are along the $y$-axis, $S_e, S_f$ are along the $z$-axis, we can get similar constraints.
Let $\mu_1 = r_{21}/r_{11}$, $\mu_2 = r_{22}/r_{12}$, $\mu_3 = r_{23}/r_{13}$, we have:
\begin{align} \mu _1 = (y_a^2 - y_b^2)/(y_a^1 - y_b^1), \nonumber \\ \mu _2 = (y_c^2 - y_d^2)/(y_c^1 - y_d^1), \nonumber \\ \mu _3 = (y_e^2 - y_f^2)/(y_e^1 - y_f^1),\label{eq:alan}\end{align}
Now, consider the orthogonality constraint on $R$, \ie, $RR^T = I$, which implies:
\begin{align}
& r_{11}^2 + r_{12}^2 + r_{13}^2 = 1, \nonumber \\ & r_{21}^2 + r_{22}^2 + r_{23}^2 = 1, \nonumber \\
& r_{11}r_{21} + r_{12}r_{22} + r_{13}r_{23} = 0.
\end{align}
Replacing $r_{21}, r_{22}, r_{23}$ by the known values $\mu_1, \mu_2, \mu_3$ of Eq. \eqref{eq:alan} indicates the following linear equations:
\begin{equation}
\begin{bmatrix}
1, & 1, & 1 \\
\mu_1^2, & \mu_2^2, & \mu_3^2 \\
\mu_1, & \mu_2, & \mu_3
\end{bmatrix}
\begin{bmatrix}
r_{11}^2 \\ r_{12}^2 \\ r_{13}^2
\end{bmatrix}
=
\begin{bmatrix}
1 \\ 1 \\ 0
\end{bmatrix}
\end{equation}
These equations can be solved for the unknowns $r_{11}^2$, $r_{12}^2$ and $r_{13}^2$ provided the coefficient matrix (above) is invertible (\ie, has full rank). This requires that
$(\mu_1 - \mu_2)(\mu_2 - \mu_3)(\mu_3 - \mu_1) \neq 0$. Because $\mu_1, \mu_2, \mu_3$ are the slopes of the projected Manhattan axes in 3D space, this constraint is violated only in the very special case when the camera principal axis and two Manhattan axes are in the same plane.
Note that there are sign ambiguities for solving $r_{11}, r_{12}, r_{13}$ from $r_{11}^2, r_{12}^2, r_{13}^2$. But these ambiguities do not affect the estimation of the 3D shape, because they are just choices of the coordinate system. Next we can calculate $r_{21}, r_{22}, r_{23}$ directly based on $r_{11}, r_{12}, r_{13}$ and $\mu_1, \mu_2, \mu_3$. This recovers the projection matrix.
\begin{remark}
We have shown that the camera projection matrix $R$ can be recovered if the three Manhattan axes are known. Next we show that the 3D structure can be estimated using symmetry (provided the projection $R$ is recovered).
\end{remark}
Assume, without loss of generality, the object is along the $x$-axis. Let $Y \in \mathbb{R}^{2 \times P}$ and $Y^{\dag} \in \mathbb{R}^{2 \times P}$ be the $P$ symmetric pairs, $S \in \mathbb{R}^{3 \times P}$ and $S^{\dag} \in \mathbb{R}^{3 \times P}$ be the corresponding $P$ symmetric pairs in the 3D space. Note that for the $p$'th point pair in the 3D space, we have $S_p = [x_p, y_p, z_p]^T$ and $S_p^{\dag} = [-x_p, y_p, z_p]$. Thus, we can re-express the camera projection in Eq.~\ref{SingleImg} by:
\begin{align}
L &= \frac{Y - Y^{\dag}}{2} = R \begin{bmatrix} x_1, \ ..., \ x_P \\ 0, \ ..., \ 0 \\ 0, \ ..., \ 0 \end{bmatrix} =
\begin{bmatrix}
r_{11} x_1, \ ..., \ r_{11} x_P \\
r_{21} x_1, \ ..., \ r_{21} x_P
\end{bmatrix},
\label{SS1}\\
M &= \frac{Y + Y^{\dag}}{2} = R \begin{bmatrix} 0, \ ..., \ 0 \\ y_1, \ ..., \ y_P \\ z_1, \ ..., \ z_P \end{bmatrix} \nonumber \\
& = \begin{bmatrix}
r_{12} y_1 + r_{13} z_1, \ ..., \ r_{12} y_P + r_{13} z_P \\
r_{22} y_1 + r_{23} z_1, \ ..., \ r_{22} y_P + r_{23} z_P
\end{bmatrix} \label{SS2}.
\end{align}
Finally, we can solve Eqs. \eqref{SS1} and \eqref{SS2} to estimate the components $(x_p,y_p,z_p)$ of all the points $S_p$ (since $R$ is known), and hence, recover the 3D structure. Observe that we have only just enough equations to solve $(y_p,z_p)$ uniquely. On the other hand, the $x_p$ is over-determined due to symmetry. We also note that the problem is ill-posed if we do not exploit symmetry, \ie, it involves inverting a $2 \times 3$ projection matrix $R$ if not exploiting symmetry.
\subsection{Experiments on 3D Reconstruction Using A Single Image}
We use \emph{aeroplane}s for this experiment, because the 3 Manhattan directions (\eg, left wing $\rightarrow$ right wing, nose $\rightarrow$ tail and top rudder $\rightarrow$ bottom rudder) can be obtained directly on aeroplanes, see Fig. \ref{fig:symmetry_manhattan}. Also aeroplanes are generally far away from the camera, implying orthographic projection is a good approximation.
We selected 42 images with clear 3 Manhattan directions and with no occluded keypoints from the \emph{aeroplane} category of Pascal3D+ dataset, and evaluated the results by the Rotation Error and Shape Error (Eq. \eqref{eq:Error}). The shape error is obtained by comparing the reconstructed structure with their subtype groundtruth model of Pascal3D+ \cite{Xiang14}.
The \emph{average rotation and shape errors} for aeroplane using the Manhattan and symmetry constraints on the single image case are \emph{0.3210} and \emph{0.6047}, respectively. These results show that using the symmetry and Manhattan properties alone can give good results for the single image reconstruction. Indeed, the performance is better than some of the structure from motion (SfM) methods which use multiple images, see Tables \ref{table:Rot_Results} and \ref{table:Shp_Results} (on Page 8). But this is not a fair comparison, because these 42 images are selected to ensure that all the Manhattan axes are visible, while the SfM methods have been evaluated on all the aeroplane images. Some reconstruction results are illustrated in Fig. \ref{single_illust}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{Fig2.pdf}
\caption{Illustration of the reconstruction results for \emph{aeroplane} using the symmetry and Manhattan constraints on a single image. For each subfigure triplet, the first subfigure is the 2D image with input keypoints, the second and third subfigures are the 3D structure from the original and rectified viewpoints. The gauge freedom of sign ambiguities can also be observed by comparing the rectified 3D reconstructions, \ie, the third subfigures. The Red, Green, Blue lines represent the three Manhattan directions we used.}
\label{single_illust}
\end{figure}
\section{Symmetric Rigid Structure from Motion}
This section describes symmetric rigid structure from motion (Sym-RSfM). We start by defining the a full energy function for the problem, see Section~\ref{sec:original}, and the coordinate descent algorithm to minimize it, see Section~\ref{sec:coordinatedescent}. Then, the missing points are initialized in Section~\ref{sec:missingData}. After that, we describe the surrogate energy and the algorithm to minimize it, see Section~\ref{sec:surrogate}, which serves to initialize coordinate descent on the full energy function.
For consistency with our baseline methods \cite{Tomasi92}, we assume orthographic projection and the keypoints are centralized without translation for the Sym-RSfM. Note that due to the iterative estimation and recovery of the occluded data, the translation has to be re-estimated to re-centralize the data during each iteration. The update of the translation is straightforward and will be given in Section~\ref{sec:coordinatedescent}.
\subsection{Problem Formulation: The Full Energy \label{sec:original}}
We extend the image formation model from a single image to multiple images indexed by $n=1,...,N$. The keypoints are grouped into $P$ keypoint pairs, which are symmetric across the $x$-axis. We introduce noise in the image formation, and the translation is removed as discussed above.
Then, by assuming that the 3D structure $S$ is symmetric along the $x$-axis, the 2D keypoint pairs for image $n$, \ie, $Y_n \in \mathbb{R}^{2 \times P}$ and $Y^{\dag}_n \in \mathbb{R}^{2 \times P}$, are given by:
\begin{align}
Y_n = R_n S + N_n \ &\Rightarrow \ P(Y_n| R_n, S) \sim \mathcal{N}(R_n S, 1) \nonumber \\
Y^{\dag}_n = R_n \a S + N_n \ &\Rightarrow \ P(Y_n^{\dag}| R_n, S) \sim \mathcal{N}(R_n \a S, 1) \label{rigid_model}
\end{align}
where $R_n \in \mathbb{R}^{2 \times 3}$ is the projection matrix, $S \in \mathbb{R}^{3 \times P}$ is the 3D structure, and $\mathcal{A}$ is a matrix operator $\mathcal{A} = \text{diag}([-1, 1, 1])$, by which $\a S$ changes the sign of the first row of $S$, making $S^{\dag} = \a S$. $N_n$ is zero mean Gaussian noise with unit variance $N_n \sim \mathcal{N}(0,1)$\footnote{Note that we experimented with another model which treated the variance as an unknown parameter $\sigma^2_n$ and estimated it during optimization, but found this made negligible difference to the experimental results.}.
Observe that the noise is independent for all keypoints and for all images. Hence, the 2D keypoints are independent when conditioned on the 3D structure $S$ and the camera projections $R_n$. Therefore, the problem is formulated in terms of
minimizing the following energy function with unknown $R_n, S$:
\begin{align}
\mathcal{Q}(R_n, S) &= -\sum_n \ln P(Y_n, Y^{\dag}_n | R_n, S) \nonumber \\ & = -\sum_n \left( \ln P(Y_n| R_n, S) - \ln P(Y^{\dag}_n | R_n, S) \right) \nonumber \\
&\sim \sum_n ||Y_n - R_n S||^2_2 + \sum_n||Y^{\dag}_n - R_n \a S ||^2_2. \label{RigidEner0}
\end{align}
This formulation relates to the classic structure from motion problems \cite{kontsevich87,Tomasi92}, but those classic methods do not impose symmetry, and therefore, they have only the first term in Eq. \eqref{RigidEner0}, \ie, $\mathcal{Q}(R_n, S) = \sum_n||Y_n - R_n S||^2_2$. If all the data is fully observed then $S$ is of rank 3 (assuming the 3D points do not lie in a plane or a line) and so the energy can be minimized by first stacking the keypoints for all the images together, \ie, $\mathbf{Y} = [Y_1^T, ..., Y_N^T]^T \in \mathbb{R}^{2N \times P}$, then applying SVD to $\mathbf{Y}$. The solution is unique up to some coordinate transformations, \ie, the rotation ambiguities \cite{kontsevich87,Tomasi92}.
\begin{remark}
Our problem is more complex than the classic rigid SfM because of two confounding issues: (i) some keypoints will be unobserved in each image, (ii) we cannot directly solve Eq. \eqref{RigidEner0} by SVD because it consists of two energy terms which are not independent (even if all data is observed). Hence, we first formulate a full energy function with missing points, where the missing points are initialized in Section~\ref{sec:missingData}. After that, in Section~\ref{sec:surrogate}, a surrogate energy is defined, which exploits symmetry and can be minimized by SVD, and therefore, can be used for initializing camera projection and the 3D structure.
\end{remark}
To deal with unobserved keypoints we divide them into {\it visible sets} ${\it VS}, {\it VS}^{\dag}$and {\it invisible sets} ${\it IVS}, {\it IVS}^{\dag}$. Then the {\it full energy function} can be formulated as:
\vspace{-1mm}
{\small
\begin{align}
&\mathcal{Q}(\mathbf{R}, S, \{Y_{n,p}, (n,p) \in {\it IVS}\}, \{Y_{n,p}^{\dag}, (n,p) \in {\it IVS}^{\dag}\}) \nonumber \\
=& \sum_{(n,p) \in {\it VS}} ||Y_{n,p} - R_n S_{n, p}||^2_2 + \sum_{(n,p) \in {\it VS}^{\dag}}||Y_{n,p}^{\dag} - R_n \a S_{n, p} ||^2_2 + \nonumber \\
& \sum_{(n,p) \in I{\it VS}} ||Y_{n,p} - R_n S_{n, p}||^2_2 + \sum_{(n,p) \in {\it IVS}^{\dag}}||Y_{n,p}^{\dag} - R_n \a S_{n, p} ||^2_2. \label{RigidEner2}
\end{align}
}Here $\{Y_{n,p}, (n,p) \in {\it IVS}\}$, $\{Y_{n,p}^{\dag}, (n,p) \in {\it IVS}^{\dag}\}$ are the missing points.
Note that it is easy to add the Manhattan constraints in Eq. \eqref{RigidEner2} as a regularization term for the Sym-RSfM on multiple images. But we found no significant improvement in our experiments when we used Manhattan, perhaps because it was not needed due to the extra images available as input. Please see the supplementary materials.
\subsection{Optimization of The Full Energy Function \label{sec:coordinatedescent}}
We now define a \emph{coordinate descent} algorithm to estimate the 3D structure, the camera projection, and the missing data. This algorithm is not guaranteed to converge to the global minimum, so it will be initialized using the surrogate energy function, described in Section~\ref{sec:surrogate}.
We use a \emph{coordinate descent} method to optimize Eq. \eqref{RigidEner2} by updating $R_n, S$ and the missing points $\{Y_{n,p}, (n,p) \in IVS\}, \{Y_{n,p}^{\dag}, (n,p) \in IVS^{\dag}\}$ iteratively. Note that the energy in Eq. \eqref{RigidEner2} \emph{w.r.t} $R_n, S$, \ie, when the missing points are fixed, is given in Eq. \eqref{RigidEner0}.
Firstly, we vectorize Eq. \eqref{RigidEner0} and $S$ to update it in matrix form by $\mathbb{S}$:
\begin{equation}
\scalebox{0.8}{$
\mathbb{S} = \left( \sum_{n=1}^{N} (G_n^TG_n + \a_P^T G_n^T G_n \a_P) \right)^{-1} \left(\sum_{n=1}^{N} ( G_n^T \mathbb{Y}_n + \a_P^T G_n^T \mathbb{Y}_n^{\dag} ) \right). \label{Rigid-S1}
$}
\end{equation}
where $\mathbb{S} \in \mathbb{R}^{3P \times 1}, \mathbb{Y}_n \in \mathbb{R}^{2P \times 1}, \mathbb{Y}_n^{\dag} \in \mathbb{R}^{2P \times 1}$ are vectorized $S, Y_n, Y_n^{\dag}$, respectively. $G_n = I_{P} \otimes R_n$ and $\a_P = I_{P} \otimes \a$. $I_{P} \in \mathbb{R}^{P \times P}$ is an identity matrix.
Each $R_n$ is updated under the nonlinear orthogonality constraints $R_n R_n^T = I$ similar to the idea in EM-PPCA \cite{Torresani08}: we first parameterize $R_n$ to a full $3 \times 3$ rotation matrix $Q$ and update $Q$ by its rotation increment. Please refer to the supplementary materials for the details.
\begin{algorithm}[t]
\SetNlSkip{0.5em}
\SetInd{0.5em}{1em}
\caption{Optimization of the full energy Eq. \eqref{RigidEner2}.}
\label{algo1}
\KwIn{The stacked keypoint sets (for all the $N$ images) $\mathbf{Y}$ and $\mathbf{Y}^{\dag}$ with occluded points, in which each occluded point is set to $\mathbf{0}$ initially.
}
\KwOut{The camera projection matrix $R_n$ for each image, the 3D structure $S$, and the keypoints with recovered occlusions $(\mathbf{Y})^t$ and $(\mathbf{Y}^{\dag})^t$.}
Initialize the occluded points by Algorithm \ref{algo2}. \\
Initialize the each camera projection $R_n$ and the 3D structure $S$ by Algorithm \ref{algo3}. \\
\Repeat{Eq. \eqref{RigidEner2} converge}{
Update $S$ by Eq. \eqref{Rigid-S1} and update each $R_n$ (see the supplementary materials).
\\
Calculate the occluded points by Eq. \eqref{Rigid-OP}, and update them in $Y_n, Y_n^{\dag}$. \\
Centralize the $Y_n, Y_n^{\dag}$ by Eq. \eqref{Rigid-centralize}.
}
\end{algorithm}
From Eq. \eqref{RigidEner2}, the occluded points of $\mathbf{Y}$ and $\mathbf{Y}^{\dag}$ (\ie, the $p$-th point $Y_{n,p}$ and $Y_{n,p}^{\dag}$) can be updated by minimizing: $\mathcal{Q}(Y_{n,p}, Y_{n,p}^{\dag}) = \sum_{(n,p) \in IVS} ||Y_{n,p} - R_n S_p||^2_2 + \sum_{(n,p) \in IVS^{\dag}}||Y_{n,p}^{\dag} - R_n \a S_p ||^2_2$, which implies the update rule for the missing points:
\begin{equation}
Y_{n,p} = R_n S_p, \qquad Y_{n,p}^{\dag} = R_n \a S_p, \label{Rigid-OP}
\end{equation}
where $(n,p) \in IVS$.
Note that we do not model the translation explicitly for the sake of consistency with the baseline method \cite{Tomasi92}, where the translation is assumed to be eliminated by centralizing the data. However, since the occluded points have been updated iteratively in our method, we have to re-estimate the translation and re-centralize the data during each iteration. This can be done by:
\begin{align}
&Y_n \gets Y_n - \mathbf{1}_{2P}^T \otimes t_n, \qquad Y_n^{\dag} \gets Y_n^{\dag} - \mathbf{1}_{2P}^T \otimes t_n, \nonumber \\
&t_n = \sum_p (Y_{n,p} - R_n S_p + Y_{n,p}^{\dag} - R_n \mathcal{A} S_p). \label{Rigid-centralize}
\end{align}
The algorithm to optimize Eq. \eqref{RigidEner2} is summarized in Algorithm \ref{algo1}, in which the initialization of the missing points, the 3D structure and the camera projection, \ie, Algorithms \ref{algo2} and \ref{algo3}, will be discussed in the following sections.
\subsection{Initialization of The Missing Data \label{sec:missingData}}
In this section, the missing data is initialized by the whole input data ignoring symmetry. This will be used both for coordinate descent of the full energy and for applying singular value decomposition to the surrogate energy.
Let $\mathbf{Y} = [Y_1^T, ..., Y_N^T]^T, \mathbf{Y}^{\dag} = [(Y_1^{\dag})^T, ..., (Y_N^{\dag})^T]^T \in \mathbb{R}^{2N \times P}$ are the stacked keypoints for all the images, and $\mathbf{R} = [R_1^T, ...R_N^T]^T \in \mathbb{R}^{2N \times 3}$ are the stacked camera projection. Thus, we have $\mathbf{Y}^{\text{All}} = [\mathbf{Y}$, $\mathbf{Y}^{\dag}] = \mathbf{R} [S, \a S]$. It implies that $\mathbf{Y}^{\text{All}}$ has the same rank, namely 3, with $\mathbf{R} [S, \a S]$ given all the points of $[S, \a S]$ do not lie on a plane or a line. Therefore, rank 3 recovery can be used to initialize the missing points. Also, the same centralization as in the previous section has to be done after each iteration of the missing points, so as to eliminate the translations.
The occlusions initialization is shown in Algorithm \ref{algo2}.
\subsection{The Surrogate Energy: Initialization of Camera Projection and 3D Structure
\label{sec:surrogate}}
\begin{remark}
We now define a surrogate energy function that exploits the symmetry constraints, which enables us to decompose the energy into two independent terms and leads to an efficient minimization algorithm using SVD.
\end{remark}
To construct the surrogate energy, we first change the coordinates to exploit symmetry, so that the problem breaks down into two independent energy terms. Since $S$ and $S^{\dag}$ are symmetric along x-axis, we can decompose $S$ by:
{\small
\begin{align}
& \mathbf{L} = \frac{\mathbf{Y} - \mathbf{Y}^{\dag}}{2} = \mathbf{R} \begin{bmatrix} x_1, &..., &x_P \\ 0, &..., &0 \\ 0, &..., &0 \end{bmatrix} = \mathbf{R}^1 S_x, \quad \nonumber \\
& \mathbf{M} = \frac{\mathbf{Y} + \mathbf{Y}^{\dag}}{2} = \mathbf{R} \begin{bmatrix} 0, &..., &0 \\ y_1, &..., &y_P \\ z_1, &..., &z_P \end{bmatrix} = \mathbf{R}^2 S_{yz} \label{SRSfM2},
\end{align}
}where $\mathbf{R}^1 \in \mathbb{R}^{2N \times 1}, \mathbf{R}^2 \in \mathbb{R}^{2N \times 2}$ are the first single column and second-third double columns of $\mathbf{R}$, $S_x \in \mathbb{R}^{1 \times P}, S_{yz} \in \mathbb{R}^{2 \times P}$ are the first single row and second-third double rows of $S$, respectively. Equation \eqref{SRSfM2} gives us the energy function on $\mathbf{R}, S$ to replace Eq. \eqref{RigidEner0} into:
\begin{equation}
Q(\mathbf{R}, S) = ||\mathbf{L} - \mathbf{R}^1 S_x||_2^2 + ||\mathbf{M} - \mathbf{R}^2 S_{yz}||_2^2. \label{SVD}
\end{equation}
This is essentially changing the coordinate system by rotating $\mathbf{Y}, \mathbf{Y}^T$ with 45$^{\circ}$ (except a scale factor of $\sqrt{2}$).
\begin{algorithm}[t]
\SetNlSkip{0.5em}
\SetInd{0.5em}{1em}
\caption{The initialization of the occluded points.}
\label{algo2}
\KwIn{The stacked keypoint sets (for all the $N$ images) $\mathbf{Y}$ and $\mathbf{Y}^{\dag}$ with occluded points, in which each occluded point is set to $\mathbf{0}$ initially. The number of iterations $T$ (default 10).}
\KwOut{The keypoints with initially recovered occlusions $(\mathbf{Y})^t$ and $(\mathbf{Y}^{\dag})^t$.}
Set $t = 0$, initialize the occluded points ignoring symmetry by: \\
\While{$t < T$}{
Centralize $\mathbf{Y}^{\text{All}} = [(\mathbf{Y})^t$, $(\mathbf{Y}^{\dag})^t]$ by Eq. \eqref{Rigid-centralize}. \\
Do SVD on $\mathbf{Y}^{\text{All}}$ ignoring the symmetry, \ie, $[\mathbf{A}, \Sigma, \mathbf{B}] = \text{SVD}\left(\mathbf{Y}^{\text{All}}\right)$. \\
Use the first 3 component of $\Sigma$ to reconstruct the keypoints $(\mathbf{Y}^{\text{All}})^{\text{new}}$. \\
Replace the occluded points in $(\mathbf{Y})^t$, $(\mathbf{Y}^{\dag})^t$ by these in $(\mathbf{Y}^{\text{All}})^{\text{new}}$ and set $t \leftarrow t+1$.
}
\end{algorithm}
\begin{remark}
We have decomposed the energy into two independent terms, and therefore, they can be solved separately by SVD up to some ambiguities. Then we will combine them to study and resolve the ambiguities. Note that we assume the occluded keypoints are replaced by the initialization described in the previous section.
\end{remark}
Equation \eqref{SVD} implies that we can estimate $\mathbf{R}^1, S_x$ and $\mathbf{R}^2, S_{yz}$ by matrix factorization on $\mathbf{L}$ and $\mathbf{M}$ independently up to ambiguities. Then we combine them to remove this ambiguity by exploiting the orthogonality constraints on each $R_n$: \ie, $R_n R_n^T = I$. Applying SVD to $\mathbf{L}$ and $\mathbf{M}$ gives us estimates, \ie, $(\mathbf{\hat{R}}^1, \hat{S}_x)$ of $(\mathbf{R}^1, S_x)$, and $(\mathbf{\hat{R}}^2, \hat{S}_{yz})$ of $(\mathbf{R}^2, S_{yz})$, up to ambiguities $\lambda$ and $B$:
\begin{equation} \mathbf{L} = \mathbf{R}^1 S_x = \mathbf{\hat{R}}^1 \lambda \lambda^{-1} \hat{S}_x, \quad \mathbf{M} = \mathbf{R}^2 S_{yz} = \mathbf{\hat{R}}^2 B B^{-1} \hat{S}_{yz}, \label{ambi}\end{equation}
here $\mathbf{R}^1$ and $\mathbf{R}^2$ are the decomposition of the true projection matrix $\mathbf{R}$, \ie, $\mathbf{R} = [\mathbf{R}^1, \mathbf{R}^2]$, and $\mathbf{\hat{R}}^1$ and $\mathbf{\hat{R}}^2$ are the output estimates from SVD. Equation \eqref{ambi} shows that there is a scale ambiguity $\lambda$ between $\mathbf{\hat{R}}^1$ and $\mathbf{R}^1$, and a 2-by-2 matrix ambiguity $B \in \mathbb{R}^{2 \times 2}$ between $\mathbf{\hat{R}}^2$ and $\mathbf{R}^2$.
\begin{remark}
Next we show how to resolve the ambiguities $\lambda$ and $B$. This is done by using the \emph{orthogonality constraints}, namely $R_n R_n^T = I$.
\end{remark}
Observe from Eqs. \eqref{ambi} that the ambiguities (\ie, $\lambda$ and $B$) are the same for the projection matrices of all the images. In the following derivation, we analyze the ambiguity for the $n$'th image, \ie, projection matrix $R_n$.
Using Eqs. \eqref{ambi}, the true $R_n$ can be represented by:
\begin{equation}
R_n = [R_n^1, R_n^2] = [\hat{R}^1_n, \hat{R}^2_n]
\begin{bmatrix}
\lambda, & \mathbf{0} \\
\mathbf{0}, & B
\end{bmatrix} \label{Rotation} \\
=
\hat{R}_n
\begin{bmatrix}
\lambda, & \mathbf{0} \\
\mathbf{0}, & B
\end{bmatrix}\end{equation}
where $R_n^1 \in \mathbb{R}^{2 \times 1}$ and $R_n^2 \in \mathbb{R}^{2 \times 2}$ are the first single
column and second-third double columns of the true projection matrix $R_n$. $\hat{R}^1_n \in \mathbb{R}^{2 \times 1}$ and $\hat{R}_n^2 \in \mathbb{R}^{2 \times 2}$ are the initial estimation of $R_n^1$ and $R_n^2$ from the matrix factorization.
Let $\hat{R}_n = [\hat{R}^1_n, \hat{R}^2_n] = \begin{bmatrix} \hat{r}^{1,1}_n, & \hat{r}^{1,2:3}_n \\ \hat{r}^{2,1}_n, & \hat{r}^{2,2:3}_n \end{bmatrix} \in \mathbb{R}^{2 \times 3}$, imposing the orthogonality constraints $R_n R_n^T = I$ using Eq. \eqref{Rotation} gives:
{\small
\begin{align}
& R_n R_n^T = \hat{R}_n \begin{bmatrix}
\lambda^2, & \mathbf{0} \\
\mathbf{0}, & BB^T
\end{bmatrix}
\hat{R}_n^T \nonumber \\
=& \begin{bmatrix}
\hat{r}^{1,1}_n, & \hat{r}^{1,2:3}_n \\
\hat{r}^{2,1}_n, & \hat{r}^{2,2:3}_n
\end{bmatrix}
\begin{bmatrix}
\lambda^2, & \mathbf{0} \\
\mathbf{0}, & BB^T
\end{bmatrix}
\begin{bmatrix}
\hat{r}^{1,1}_n, & \hat{r}^{1,2:3}_n \\
\hat{r}^{2,1}_n, & \hat{r}^{2,2:3}_n
\end{bmatrix}^T = I \label{R_orth}
\end{align}
}
Vectorizing $BB^T$ of Eq. \eqref{R_orth} using $\text{vec}(AXB^T) = (B \otimes A)\text{vec}(X)$, we can get the following linear equations:
\begin{equation}
\begin{bmatrix}
(\hat{r}^{1,1}_n)^2, & \hat{r}^{1,2:3}_n \otimes \hat{r}^{1,2:3}_n \\
(\hat{r}^{2,1}_n)^2, & \hat{r}^{2,2:3}_n \otimes \hat{r}^{2,2:3}_n \\
\hat{r}^{1,1}_n \hat{r}^{2,1}_n, & \hat{r}^{1,2:3}_n \otimes \hat{r}^{2,2:3}_n \\
\end{bmatrix}
\begin{bmatrix}
\lambda^2 \\
\text{vec}(BB^T)
\end{bmatrix}
=
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix} .
\label{Rigid_orth}
\end{equation}
Note that $BB^T$ is a symmetric matrix, the second and third elements of $\text{vec}(BB^T)$ are the same. Let $\text{vec}(BB^T) = [bb_1, bb_2, bb_2, bb_3]^T$, we can enforce the symmetriy of $B B^T$ by rewriting Eq. \eqref{Rigid_orth}:
\begin{equation}
\begin{bmatrix}
(\hat{r}^{1,1}_n)^2, & \hat{r}^{1,2:3}_n \otimes \hat{r}^{1,2:3}_n \\
(\hat{r}^{2,1}_n)^2, & \hat{r}^{2,2:3}_n \otimes \hat{r}^{2,2:3}_n \\
\hat{r}^{1,1}_n \hat{r}^{2,1}_n, & \hat{r}^{1,2:3}_n \otimes \hat{r}^{2,2:3}_n \\
\end{bmatrix}
\begin{bmatrix}
1 \ 0 \ 0 \ 0 \\
0 \ 1 \ 0 \ 0 \\
0 \ 0 \ 1 \ 0 \\
0 \ 0 \ 1 \ 0 \\
0 \ 0 \ 0 \ 1
\end{bmatrix}
\begin{bmatrix}
\lambda^2 \\
bb_1 \\
bb_2 \\
bb_3
\end{bmatrix}
= A_i \mathbf{x} =
\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix} ,
\label{Rigid_orth2}
\end{equation}
where the constant matrix of the left term is a matrix operator to sum the third and forth columns of the coefficient matrix with $r$'s, $\mathbf{x}$ is the unknown variables $[\lambda^2, bb_1, bb_2, bb_3]^T$, and $A_i \in \mathbb{R}^{3 \times 4}$ is the corresponding coefficients.
\begin{algorithm}[t]
\SetNlSkip{0.5em}
\SetInd{0.5em}{1em}
\caption{The initialization of the camera projection and structure.}
\label{algo3}
\KwIn{The keypoint sets $\mathbf{Y}$ and $\mathbf{Y}^{\dag}$ with initially recovered occluded points by Algorithm \ref{algo2}.}
\KwOut{The initialized camera projection $\mathbf{R}$ and the 3D structure $S$.}
Change the coordinates to decouple the symmetry constraints by Eq. \eqref{SRSfM2}. \\
Get $\mathbf{\hat{R}}^1, \mathbf{\hat{R}}^2, \hat{S}_x, \hat{S}_{yz}$ by SVD on $\mathbf{L}, \mathbf{M}$, \ie, Eq. \eqref{ambi}. \\
Solve the squared ambiguities $\lambda ^2$, $BB^T$ by Eq. \eqref{Rigid_orth2}. \\
Solve for $\lambda$ from $\lambda^2$, and $B$ from $BB^T$, up to sign and rotation ambiguities. \\
Obtain the initialized $\mathbf{R}$ and $S$ by Eq. \eqref{initial_RS}.
\end{algorithm}
Stacking all the $\hat{\mathbf{R}}$'s together, \ie, let $\mathbf{A} = [A_1^T, ..., A_N^T]^T \in \mathbb{R}^{3N \times 4}$ and $\mathbf{b} = \mathbf{1}_{N} \otimes [1,1,0]^T$, we have a over-determined equations for the unknown $\mathbf{x}$: $\mathbf{A}\mathbf{x} = \mathbf{b}$ (\ie, $3N$ equations for 4 unknowns), which can be solved efficiently by LSE: $\mathbf{x} = (\mathbf{A}^T\mathbf{A})^{-1}\mathbf{A}^T\mathbf{b}$.
\begin{remark}
The ambiguity of $\mathbf{R}^1$ and $\hat{\mathbf{R}}^1$, (\ie, in the symmetry direction), is just a sign change, which cause by calculating $\lambda$ from $\lambda^2$. In other words, the symmetry direction can be fixed as $x$-axis in our coordinate system using the decomposition Eq. \eqref{SRSfM2}.
\end{remark}
After obtained $BB^T$, $B$ can be recovered up to a rotation ambiguity on $yz$-plane, which does not affect the reconstructed 3D structure (See the supplementary materials).
Given $\lambda, B, \hat{\mathbf{R}}, \hat{S}$, we can get the true $\mathbf{R}$ and $S$ by:
\begin{equation}
\mathbf{R} = \mathbf{\hat{R}}
\begin{bmatrix}
\lambda, & \mathbf{0} \\
\mathbf{0}, & B
\end{bmatrix}, \qquad
S =
\begin{bmatrix}
\lambda, & \mathbf{0} \\
\mathbf{0}, & B
\end{bmatrix}^{-1} \hat{S}. \label{initial_RS}
\end{equation}
\begin{table*}[t!]
\fontsize{9pt}{0.9\baselineskip}\selectfont
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c | c | c | c || c | c | c | c | c | c ||}
\hline
& \multicolumn{7}{c||}{\textbf{aeroplane}} & \multicolumn{6}{c||}{\textbf{bus}} \\
\hline
& I & II & III & IV & V & VI & VII & I & II & III & IV & V & VI \\
\hline
RSfM & 0.52 & 1.17 & 1.99 & 0.28 & 0.94 & 1.77 & 1.74 & 0.32 & 1.05 & 0.27 & 0.27 & 1.13 & 0.51 \\
CSF (S) & 0.92 & 0.70 & 0.87 & 0.83 & \underline{\textbf{0.89}} & 0.96 & \underline{\textbf{1.02}} & 0.72 & 0.86 & 0.68 & 0.92 & 0.94 & 1.04 \\
CSF (R) & 0.93 & 0.82 & \underline{\textbf{0.80}} & 0.91 & 0.99 & 1.02 & 1.37 & 0.69 & 0.88 & 0.81 & 0.93 & 0.88 & 1.08 \\
Sym-RSfM & \underline{\textbf{0.13}} & \underline{\textbf{0.46}} & 2.00 & \underline{\textbf{0.17}} & 1.81 & \underline{\textbf{0.89}} & 1.69 & \underline{\textbf{0.19}} & \underline{\textbf{0.33}} & \underline{\textbf{0.03}} & \underline{\textbf{0.22}} & \underline{\textbf{0.58}} & \underline{\textbf{0.50}} \\
\hline
\end{tabular*}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c | c | c | c | c | c | c || c | c ||}
\hline
& \multicolumn{10}{c||}{\textbf{car}} & \multicolumn{2}{c||}{\textbf{sofa}} \\
\hline
& I & II & III & IV & V & VI & VII & VIII & IX & X & I & II \\
\hline
RSfM & 0.58 & 0.71 & 0.54 & 1.10 & 0.67 & 1.51 & 0.67 & 1.41 & 0.97 & 0.37 & 1.18 & 0.75 \\
CSF (S) & 0.95 & 1.22 & 1.06 & 1.12 & 1.00 & 1.03 & 1.13 & \underline{\textbf{1.04}} & 1.33 & 1.03 & 1.02 & 0.75 \\
CSF (R) & 0.95 & 1.32 & 1.08 & 1.06 & 1.09 & 0.98 & 1.22 & 1.05 & 1.29 & 1.17 & 0.85 & 0.76 \\
Sym-RSfM & \underline{\textbf{0.36}} & \underline{\textbf{0.43}} & \underline{\textbf{0.30}} & \underline{\textbf{0.43}} & \underline{\textbf{0.31}} & \underline{\textbf{0.26}} & \underline{\textbf{0.32}} & \underline{\textbf{1.04}} & \underline{\textbf{0.25}} & \underline{\textbf{0.16}} & \underline{\textbf{0.67}} & \underline{\textbf{0.26}} \\
\hline
\end{tabular*}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c || c | c | c | c || c | c | c | c ||}
\hline
& \multicolumn{4}{c||}{\textbf{sofa}} & \multicolumn{4}{c||}{\textbf{train}} & \multicolumn{4}{c||}{\textbf{tv}} \\
\hline
& III & IV & V & VI & I & II & III & IV & I & II & III & IV \\
\hline
RSfM & 1.90 & 1.00 & 1.99 & 1.90 & 1.95 & 1.44 & 1.33 & 1.01 & 0.86 & 0.38 & 0.39 & 1.38 \\
CSF (S) & 1.16 & 0.99 & 1.66 & 1.21 & 0.84 & 0.69 & 0.86 & 0.85 & 0.99 & 0.79 & 0.95 & 0.73 \\
CSF (R) & 0.87 & 0.86 & \underline{\textbf{0.98}} & 1.70 & 0.92 & \underline{\textbf{0.67}} & \underline{\textbf{0.84}} & \underline{\textbf{0.82}} & 1.00 & 0.82 & 0.91 & 0.84 \\
Sym-RSfM & \underline{\textbf{0.11}} & \underline{\textbf{0.69}} & 1.57 & \underline{\textbf{0.97}} & \underline{\textbf{0.18}} & 0.68 & 0.88 & 0.97 & \underline{\textbf{0.23}} & \underline{\textbf{0.14}} & \underline{\textbf{0.26}} & \underline{\textbf{0.44}} \\
\hline
\end{tabular*}
\caption{The mean \emph{rotation} errors for \emph{aeroplane, bus, car, sofa, train, tv}, calculated using the images from the same subtype (denoted by the Roman numerals) as input.
}
\label{table:Rot_Results}
\end{table*}
\begin{table*}[t!]
\fontsize{9pt}{0.9\baselineskip}\selectfont
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c | c | c | c || c | c | c | c | c | c ||}
\hline
& \multicolumn{7}{c||}{\textbf{aeroplane}} & \multicolumn{6}{c||}{\textbf{bus}} \\
\hline
& I & II & III & IV & V & VI & VII & I & II & III & IV & V & VI \\
\hline
RSfM & 0.44 & 1.17 & 0.52 & \underline{\textbf{0.29}} & 0.76 & 0.54 & 0.61 & 1.29 & 1.27 & 1.16 & 0.97 & 1.52 & 1.21 \\
CSF (S) & 1.25 & \underline{\textbf{0.36}} & 1.42 & 0.84 & 0.33 & 0.47 & \underline{\textbf{0.59}} & 1.11 & \underline{\textbf{0.39}} & 0.56 & \underline{\textbf{0.16}} & 3.02 & \underline{\textbf{0.44}} \\
CSF (R) & 0.25 & 0.44 & 0.34 & 1.40 & 0.58 & 1.73 & 0.69 & 0.99 & 1.06 & 1.33 & 0.88 & 1.94 & 2.00 \\
Sym-RSfM & \underline{\textbf{0.19}} & 0.88 & \underline{\textbf{0.27}} & 0.34 & \underline{\textbf{0.33}} & \underline{\textbf{0.30}} & 0.62 & \underline{\textbf{0.68}} & 0.58 & \underline{\textbf{0.35}} & 0.24 & \underline{\textbf{0.76}} & 0.47 \\
\hline
\end{tabular*}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c | c | c | c | c | c | c || c | c ||}
\hline
& \multicolumn{10}{c||}{\textbf{car}} & \multicolumn{2}{c||}{\textbf{sofa}} \\
\hline
& I & II & III & IV & V & VI & VII & VIII & IX & X & I & II \\
\hline
RSfM & 1.48 & 1.49 & 1.33 & 1.38 & 1.45 & 1.39 & 1.21 & 1.81 & 1.22 & 1.07 & 2.50 & 1.09 \\
CSF (S) & 1.06 & 2.33 & 1.15 & 1.17 & 1.36 & 1.17 & 1.03 & 1.10 & 2.03 & 0.99 & 1.78 & 0.24 \\
CSF (R) & 1.34 & 1.07 & 1.03 & 1.16 & 1.18 & 1.26 & 0.88 & \underline{\textbf{0.90}} & 1.65 & 1.13 & \underline{\textbf{0.76}} & 0.25 \\
Sym-RSfM & \underline{\textbf{1.03}} & \underline{\textbf{0.96}} & \underline{\textbf{0.95}} & \underline{\textbf{1.07}} & \underline{\textbf{0.89}} & \underline{\textbf{1.00}} & \underline{\textbf{0.81}} & 1.66 & \underline{\textbf{0.88}} & \underline{\textbf{0.71}} & 2.27 & \underline{\textbf{0.22}} \\
\hline
\end{tabular*}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}} || l || c | c | c | c || c | c | c | c || c | c | c | c ||}
\hline
& \multicolumn{4}{c||}{\textbf{sofa}} & \multicolumn{4}{c||}{\textbf{train}} & \multicolumn{4}{c||}{\textbf{tv}} \\
\hline
& III & IV & V & VI & I & II & III & IV & I & II & III & IV \\
\hline
RSfM & 1.49 & 1.60 & 3.44 & 2.56 & 1.68 & 0.39 & 0.28 & 0.22 & 0.23 & 0.88 & 0.64 & 1.77 \\
CSF (S) & 3.14 & 1.54 & 2.74 & 1.55 & 0.83 & 0.85 & 0.25 & 0.26 & 0.66 & 0.77 & 0.34 & 0.34 \\
CSF (R) & 1.82 & 1.19 & 1.42 & 1.20 & 1.05 & \underline{\textbf{0.37}} & 0.24 & \underline{\textbf{0.17}} & 0.22 & 0.97 & 0.55 & 0.36 \\
Sym-RSfM & \underline{\textbf{0.40}} & \underline{\textbf{1.07}} & \underline{\textbf{0.87}} & \underline{\textbf{1.14}} & \underline{\textbf{0.73}} & 0.61 & \underline{\textbf{0.13}} & 0.24 & \underline{\textbf{0.09}} & \underline{\textbf{0.29}} & \underline{\textbf{0.32}} & \underline{\textbf{0.14}} \\
\hline
\end{tabular*}
\caption{The mean \emph{shape} errors for \emph{aeroplane, bus, car, sofa, train, tv}, calculated using the images from the same subtype (denoted by the Roman numerals) as input.}
\label{table:Shp_Results}
\end{table*}
\vspace{-2mm}
\subsection{Experiments on The Symmetric Rigid Structure from Motion} \label{Rigid_Exp}
\vspace{-1mm}
We estimate the 3D structures of each subtype and the orientations of all the images within that subtype for \emph{aeroplane, bus, car, sofa, train, tv} in Pascal3D+ \cite{Xiang14}. Note that Pascal3D+ provides a single 3D shape for each subtype rather than for each object. For example, it provides 10 subtypes for the \emph{car} category, such as \emph{sedan, truck}, but ignores the within-subtype variation \cite{gao2016semi}. Thus, we divide the images of the same category into subtypes, and then input the images of each subtype for the experiments.
Following \cite{Kar15,Gao16}, images with more than 5 visible keypoints are used. The rotation and shape errors are calculated by Eq. \eqref{eq:Error}. The rigid SfM (RSfM) \cite{Tomasi92} and a more recent CSF method \cite{Gotardo11}, which both do not exploit symmetry, are used for comparison. Note that the CSF method \cite{Gotardo11} utilized smooth time-trajectories as initialization, which does not always
hold in our application, as the input images here are not from a continuous video. Thus, we also investigate the results from CSF method with random initialization. We report the CSF results with smooth prior as \emph{CSF (S)} and the \emph{best} results with 10 random initialization as \emph{CSF (R)}.
The results (mean rotation and shape errors) are shown in Tables \ref{table:Rot_Results} and \ref{table:Shp_Results}, which indicate that our method outperforms the baseline methods for most cases. The cases that our method does not perform as the best may be caused by that Pascal3D+ assumes the shapes from objects within the same subtype are very similar to each other, but this might be violated sometimes. Moreover, our method is robust to imperfect annotations (\ie, result in imperfect symmetric pairs) for practical use. This was simulated by adding Gaussian noise to the 2D annotations in the supplementary material.
\vspace{-2mm}
\section{Conclusions}
\vspace{-2mm}
We show that symmetry, Manhattan and multiple-image cues can be utilized to achieve good quality performance on object 3D structure reconstruction. For the single image case, symmetry and Manhattan together are sufficient if we can identify suitable keypoints. For the multiple-image case, we formulate the problem in terms of energy minimization exploiting symmetry, and optimize it by a coordinate descent algorithm. To initialize this algorithm, we define a surrogate energy function exploiting symmetry, and decompose it into a sum of two independent terms that can be solved by SVD separately. We further study the ambiguities of the surrogate energy and show that they can be resolved assuming the orthographic projection. Our results outperform the baselines on most object classes in Pascal3D+. Future works involve using richer camera models, like perspective \cite{park20153d,hartley2008perspective}, and keypoints extracted and matched automatically from images \cite{chen2014articulated,ma2015robust,ma2013regularized} with outliers \cite{ma2015non,ma2014robust,ma2015robust2} and occlusions \cite{jacobs2001linear,jacobs1997linear,Lin2010} handled.
~\\
~\\
\noindent \textbf{Acknowledgments.} We would like to thank Ehsan Jahangiri, Wei Liu, Cihang Xie, Weichao Qiu, Xuan Dong, and Siyuan Qiao for giving feedbacks on this paper. This work is partially supported by ONR N00014-15-1-2356 and the NSF award CCF-1317376.
\bibliographystyle{ieee}
|
2,869,038,154,830 | arxiv | \section{Introduction and notation}
One of the most pressing questions in particle physics today is to determine whether or not the discovered Higgs-like particle \cite{:2012gk,:2012gu}
is compatible with the Standard-Model expectations \cite{Djouadi:2005gi}, or whether it belongs to an enlarged scalar sector.
The most popular extension is certainly the supersymmetric one \cite{Djouadi:2005gj}, but while waiting for hints of supersymmetry, it may be worthwhile to entertain also other possibilities \cite{Gunion:1989we}.
We shall here review and update the interpretation of the 125~GeV Higgs particle in the Two-Higgs-Doublet model \cite{Basso:2012st}, with Type~II Yukawa couplings, like in the MSSM.
In the spirit of the original motivation for the model, we allow CP violation \cite{Lee:1973iz} (see also Ref.~\cite{Shu:2013uua}), and take the potential to be
\begin{align}
\label{Eq:pot_7}
V&=\frac{\lambda_1}{2}(\Phi_1^\dagger\Phi_1)^2
+\frac{\lambda_2}{2}(\Phi_2^\dagger\Phi_2)^2
+\lambda_3(\Phi_1^\dagger\Phi_1) (\Phi_2^\dagger\Phi_2) \nonumber \\
&+\lambda_4(\Phi_1^\dagger\Phi_2) (\Phi_2^\dagger\Phi_1)
+\frac{1}{2}\left[\lambda_5(\Phi_1^\dagger\Phi_2)^2+{\rm h.c.}\right] \\
&-\frac{1}{2}\left\{m_{11}^2(\Phi_1^\dagger\Phi_1)
\!+\!\left[m_{12}^2 (\Phi_1^\dagger\Phi_2)\!+\!{\rm h.c.}\right]
\!+\!m_{22}^2(\Phi_2^\dagger\Phi_2)\right\}. \nonumber
\end{align}
with $m_{12}^2$ and $\lambda_5$ complex.
The potential uniquely determines the mass spectrum. There are three neutral states, $H_1$, $H_2$ and $H_3$, with masses $M_1\leq M_2\leq M_3$, and a charged pair, $H^\pm$, with mass $M^\pm$.
With the field decomposition (ghosts are removed):
\begin{equation}\label{Eq:Higgs_goldstones}
\Phi_1=
\left(
\begin{array}{c}
- s_\beta H^+ \\
\frac{1}{\sqrt{2}} [v_1 + \eta_1 - i s_\beta \eta_3]
\end{array}
\right), \qquad
\Phi_2 =
\left(
\begin{array}{c}
c_\beta H^+ \\
\frac{1}{\sqrt{2}} [v_2 + \eta_2 + i c_\beta \eta_3 ]
\end{array}
\right).
\end{equation}
where $c_\beta=\cos\beta$ and $s_\beta=\sin\beta$, and the ratio defines
$\tan{\beta}=v_2/v_1$,
the neutral Higgs states can be expressed via a rotation matrix $R$ as
\begin{equation} \label{Eq:R-def}
\begin{pmatrix}
H_1 \\ H_2 \\ H_3
\end{pmatrix}
=R
\begin{pmatrix}
\eta_1 \\ \eta_2 \\ \eta_3
\end{pmatrix}.
\end{equation}
The rotation matrix can be parametrized in terms of 3 rotation angles, $\alpha_j$ \cite{Accomando:2006ga}.
It is instructive to take the input parameters in terms of quantities that have a more direct physical interpretation.
A convenient choice is to specify the masses of the two lightest Higgs bosons ($M_1$, $M_2$), as well as that of the charged one, $M^\pm$. Supplementing these data by $\tan\beta$, the three angles $\alpha_j$, as well as $\mu^2=\Re m_{12}^2/(2\cos\beta\sin\beta)$, the potential can be trivially reconstructed \cite{Khater:2003wq,ElKaffas:2007rq}. This is useful, since some of the theoretical constraints (see below) are more simply expressed in terms of the potential parameters.
\section{Constraints}
The parameter space is constrained both by theoretical considerations, and by experimental data.
\subsection{Theory constraints}
We impose the familiar theory constraints: positivity, tree-level unitarity, and perturbativity. In addition, we impose the less familiar constraint of requiring that the potential minimum be a global one. This is computationally rather expensive, and therefore checked only if all other constraints (including experimental ones) are satisfied. For details, see Ref.~\cite{Basso:2012st}.
\subsection{Experimental constraints}
We impose the constraints from flavor physics (in particular, $b\to s\gamma$), $\Gamma(Z\to b\bar b)$, and electroweak precision observables $T$ and $S$.
While allowing for CP violation opens up a larger parameter space than CP-conserving models, one has to make sure that excessive CP violation is not induced. A representative observable that is easily checked, is the electron electric dipole moment. For details, see Ref.~\cite{Basso:2012st}.
As compared with our original work, the LHC constraints have tightened: ATLAS has presented the new (preliminary) result $R_{\gamma\gamma}=1.65\pm0.3$ \cite{ATLAS:2013oma}, where
\begin{equation} \label{Eq:R_gammagamma}
R_{\gamma\gamma}=\frac{\sigma(pp\to H_1X){\rm BR}(H_1\to\gamma\gamma)}
{\sigma(pp\to H_\text{SM}X){\rm BR}(H_\text{SM}\to\gamma\gamma)}.
\end{equation}
As a 2-$\sigma$ interval, we allow $1.05\leq R_{\gamma\gamma}\leq2.33$. The exclusion of values below unity has significant implications for the allowed parameter range.\footnote{After this scan was performed, also CMS released their updated results for $R_{\gamma\gamma}$ \cite{CMS-2013}. They find the 2-$\sigma$ range $0.26\leq R_{\gamma\gamma}\leq1.34$, significantly lower than that obtained by ATLAS, and adopted here. Since the ATLAS and CMS results barely overlap, our scans should not be taken as definitive, but rather as an illustration of how improved data can further constrain the model.}
Also, CMS has presented more tight exclusions of a SM-like Higgs particle in the high-mass region \cite{Guillelmo:2013cca},
\begin{equation} \label{Eq:R_ZZ}
R_{ZZ}=\frac{\sigma(pp\to H_jX)){\rm BR}(H_j\to ZZ)}
{\sigma(pp\to H_\text{SM}X){\rm BR}(H_\text{SM}\to ZZ)},
\end{equation}
having implications for how strongly the heavier partners, $H_2$ and $H_3$ can couple to $W$ and $Z$. Also, both ATLAS \cite{ATLAS:2013oma} and CMS \cite{Guillelmo:2013cca} presented new results on $R_{ZZ}$ and $R_{WW}$, relevant for $H_1$. As a 2-$\sigma$ envelope covering both the $ZZ$ and $WW$ channels, we adopt the range $0.3\leq R_{VV}\leq2.7$. Finally, we also take into account new preliminary results from a $H\to b\bar b$ search \cite{ATLAS:2013nma}, yielding the 2-$\sigma$ range $0.49\leq R_{b\bar b}\leq 1.69$, with $R_{b\bar b}$ defined in analogy with (\ref{Eq:R_gammagamma}) and (\ref{Eq:R_ZZ}).
In both (\ref{Eq:R_gammagamma}) and (\ref{Eq:R_ZZ}), we approximate the cross section ratio by the corresponding ratio of gluon-gluon branching ratios,
\begin{equation}
\frac{\sigma(pp\to H_jX)}{\sigma(pp\to H_\text{SM}X)}
\simeq\frac{\Gamma(H_j\to gg)}{\Gamma(H_\text{SM}\to gg)},
\end{equation}
i.e., we consider only production via the dominant gluon-gluon fusion.
It is interesting to see how the new data have led to a shrinking of the allowed parameter space. We shall refer to the constraints described above as ``Moriond 2013'' and compare the still allowed parameter space to that allowed by the constraints considered in Ref.~\cite{Basso:2012st}, which we refer to as ``2012''. Since the constraints on $R_{VV}$ and $R_{b\bar b}$ are still rather loose, the main impact of the new data are in the following two sectors: (1) tighter constraints on how $H_2$ and $H_3$ couple to $ZZ$ (or $W^+W^-$), and (2) tighter constraints on how $H_1$ couples to photons, $R_{\gamma\gamma}$.
Key features of these LHC constraints can be summarized as follows:
\begin{alignat}{4} \label{Eq:constraint-2012}
&\text{``2012'' \cite{Basso:2012st}:} &\qquad
H_1\to\gamma\gamma: 0.5\leq &R_{\gamma\gamma}\leq2, &\qquad
&H_{2,3}\to VV:&\quad
&\text{Ref.}~\cite{ATLAS:2012ae,Chatrchyan:2012tx}\\
&\text{``Moriond 2013'':} &\qquad
H_1\to\gamma\gamma: 1.03\leq &R_{\gamma\gamma}\leq2.33, &\qquad
&H_{2,3}\to VV:&\quad
&\text{Ref.}~\cite{ATLAS:2013oma,Guillelmo:2013cca}
\label{Eq:constraints-2013}
\end{alignat}
\section{Allowed parameter space}
We shall here illustrate how the allowed parameter space has shrunk as a result of the recent LHC results, and determine remaining regions.
\subsection{The general CP-violating case}
The $H_1t\bar t$ coupling is essential to the production of $H_1$ via gluon-gluon fusion.
In the general case, the $H_it\bar t$ coupling differs from that of the SM by the factor
\begin{equation} \label{Eq:H_j_Yuk}
H_j t\bar t \sim
\frac{1}{\sin\beta}\, [R_{j2}-i\gamma_5\cos\beta R_{j3}],
\end{equation}
where $R_{jk}$ refers to the matrix defined by Eq.~(\ref{Eq:R-def}).
Two points are worth noting: (1) At low $\tan\beta$, a reduced value of $|R_{12}|=|\sin\alpha_1\cos\alpha_2|$ can be compensated for by the factor $\sin\beta$ in the denominator, and (2) there is an additional contribution from the pseudoscalar coupling, proportional to $R_{13}/\tan\beta$ (and a different loop function).
For the $H_1\to\gamma\gamma$ rate, the $H_1W^+W^-$ coupling is likewise essential. The $H_jZZ$ (and $H_jW^+W^-$) coupling is, relative to that of the SM, given by
\begin{equation} \label{Eq:ZZH}
H_j ZZ\ (H_jW^+W^-)\sim
[\cos\beta R_{j1}+\sin\beta R_{j2}].
\end{equation}
This coupling, for $H_2$ and $H_3$, is also important for the constraint from the high-mass exclusion of a Higgs particle, since the most sensitive channels are various sub-channels of the $H\to ZZ$ and $H\to W^+W^-$ ones.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas12-01-400-400.png}
\caption{Allowed regions in the $\alpha_1$--$\alpha_2$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=1$, $M_2=400~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas12-01-400-400}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas23-01-400-400.png}
\caption{Allowed regions in the $\alpha_2$--$\alpha_3$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=1$, $M_2=400~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas23-01-400-400}
\end{center}
\end{figure}
We shall show a few scans over the $\alpha$-space, identifying allowed regions. We start with the case
\begin{equation} \label{Eq:parameters-01-400-400}
\tan\beta=1, \quad M_2=400~\text{GeV}, \quad M_{H^\pm}=400~\text{GeV},
\end{equation}
and two values of $\mu$, namely $\mu=250~\text{GeV}$ and $\mu=350~\text{GeV}$.
In Fig.~\ref{Fig:alphas12-01-400-400} we show allowed regions in the $\alpha_1$--$\alpha_2$ plane, whereas in Fig.~\ref{Fig:alphas23-01-400-400} we show allowed regions in the $\alpha_2$--$\alpha_3$ plane.
We see that the new constraints permit solution for both values of $\mu$ (for these values of $\tan\beta$, $M_2$ and $M_{H^\pm}$). However, the case of $\mu=200~\text{GeV}$, studied in Ref.~\cite{Basso:2012st}, is no longer allowed. This is due to the new constraint on $R_{\gamma\gamma}$.
Already within the ``2012'' constraints, the parameter space is very constrained, as shown in green (in some cases, this is practically covered by the blue and black regions). When we impose the constraints from the high-mass exclusion (reduced couplings of $H_2$ and $H_3$), we obtain the blue regions.
When we also impose the $R_{\gamma\gamma}$ constraint, we are left with the black regions.
In these examples, there are three regions that are almost or fully allowed (see Fig.~\ref{Fig:alphas23-01-400-400}). There are two regions at small values of $\alpha_2$, one around $\alpha_3=0$, and the other around $\alpha_3=\pi/2$. These both have $\alpha_1\sim\pi/4$. The third region has $\alpha_2<0$, somewhat larger values of $\alpha_1$, and $\alpha_3\sim\pi/4$.
The two first-mentioned regions are close to CP-conserving limits. We recall that $H_3=A$ corresponds to $(\alpha_2,\alpha_3)=(0,0)$, whereas $H_2=A$ corresponds to $(\alpha_2,\alpha_3)=(0,\pi/2)$ \cite{ElKaffas:2007rq}, $A$ being the CP-odd boson.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas12-01-500-400.png}
\caption{Allowed regions in the $\alpha_1$--$\alpha_2$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=1$, $M_2=500~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas12-01-500-400}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas23-01-500-400.png}
\caption{Allowed regions in the $\alpha_2$--$\alpha_3$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=1$, $M_2=450~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas23-01-500-400}
\end{center}
\end{figure}
As a second example, we consider
\begin{equation} \label{Eq:parameters-01-500-400}
\tan\beta=1, \quad M_2=500~\text{GeV}, \quad M_{H^\pm}=400~\text{GeV},
\end{equation}
and show allowed regions in Figs.~\ref{Fig:alphas12-01-500-400} and \ref{Fig:alphas23-01-500-400} for two values of $\mu$, this time 350 and 400~GeV.
For the lower value of $\mu$, two of the three regions described above, are connected. In fact, a large region of CP-violating parameter sets is allowed. For the higher value of $\mu$, all three regions are allowed, but distinct.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas12-02-400-400.png}
\caption{Allowed regions in the $\alpha_1$--$\alpha_2$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=2$, $M_2=400~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas12-02-400-400}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas23-02-400-400.png}
\caption{Allowed regions in the $\alpha_2$--$\alpha_3$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=2$, $M_2=400~\text{GeV}$ and $M_{H^\pm}=400~\text{GeV}$.}
\label{Fig:alphas23-02-400-400}
\end{center}
\end{figure}
As a third example, we consider $\tan\beta=2$, with $M_2=M_{H^\pm}=400~\text{GeV}$, in Figs.~\ref{Fig:alphas12-02-400-400} and \ref{Fig:alphas23-02-400-400}. Here, we take $\mu=250~\text{GeV}$ and $\mu=400~\text{GeV}$.
In this case, the blue regions practically cover the green regions, meaning that the ``Moriond 2013'' constraints in the high-mass region have little impact. Instead, the new $R_{\gamma\gamma}$ constraint significantly reduces the allowed regions (black). Indeed, the low-$\mu$ case is fully excluded. For $\mu=400~\text{GeV}$, there is a major difference with respect to the $\tan\beta=1$ case: the whole range of $\alpha_3$-values is allowed, i.e., in addition to the CP-conserving limits also a band in the CP-violating interior of the $\alpha_2$--$\alpha_3$ space is allowed.
In Tables~\ref{Table:tanbeta01-m2-mch-mu} and \ref{Table:tanbeta02-m2-mch-mu} we summarize the results of very coarse scans over $\mu$, for selected grids in $M_2$ and $M_{H^\pm}$. Centered roughly around the average of these values, 50~GeV increments in $\mu$ are explored and reported in these tables. For example, the notation ``[250,350]'' means that $\mu$ values of 250, 300 and 350 give allowed solutions, whereas the next values below (200) and above (400) do not.
\newcommand{\twolines}[2]{\genfrac{}{}{0pt}{}{\text{#1}}{\text{#2}}}
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\diagbox{$M_{H^\pm}$}{$M_2$} & 300 & 350 & 400 & 450 & 500 \\
\hline
500 & none & none & none & none & [450,500] \\
\hline
450 & none & [250,350] & [300,400] & [350,450] & [400,450] \\
\hline
400 & none & [250,350] & [250,400] & [300,400] & [350,400] \\
\hline
\end{tabular}
\end{center}
\caption{Some allowed values of $\mu$ for selected values of $M_2$ and $M_{H^\pm}$ [all in GeV], for $\tan\beta=1$. \label{Table:tanbeta01-m2-mch-mu}}
\end{table}
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\diagbox{$M_{H^\pm}$}{$M_2$} & 300 & 350 & 400 & 450 & 500 \\
\hline
500 & none & none & none & none & [500] \\
\hline
450 & none & none & [400] & [400,450] & [450,500] \\
\hline
400 & none & [350] & [350,450] & [350,450] & [400] \\
\hline
\end{tabular}
\end{center}
\caption{Some allowed values of $\mu$ for selected values of $M_2$ and $M_{H^\pm}$ [all in GeV], for $\tan\beta=2$. \label{Table:tanbeta02-m2-mch-mu}}
\end{table}
We see that the allowed range of $\mu$ is rather narrow, and constrained to lie around ${\cal O}(M_2,M_{H^\pm})$. See also the discussion in sect.~\ref{Sect:high-M-tanbeta}.
Related aspects of the model were presented recently \cite{Barroso:2013zxa}. We agree with the findings of those authors that it is difficult for this model to yield high values of $R_{\gamma\gamma}$ consistent with the other constraints.
\subsection{The CP-conserving case}
There are three CP-conserving limits, any one of $H_1$, $H_2$ or $H_3$ (with $M_1\leq M_2\leq M_3$) could be CP-odd, usually referred to as $A$. We shall here discuss the case $H_3=A$, which in the above terminology corresponds to $\alpha_2=0$, $\alpha_3=0$, namely the origin in Figs.~\ref{Fig:alphas23-01-400-400}, \ref{Fig:alphas23-01-500-400} and \ref{Fig:alphas23-02-400-400}.
In Figs.~\ref{Fig:alphas1-mh3-01} and \ref{Fig:alphas1-mh3-03} we show allowed regions in the $\alpha_1$-$M_3$ plane, in the same color code as above. Two values of $\tan\beta$ are studied (1 and 3), and two values of $M_2$ (400 and 600~GeV).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas1-mh3-01.png}
\caption{Allowed regions in the $\alpha_1$--$M_3$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=1$, $M_2=400~\text{GeV}$ and 600~GeV.}
\label{Fig:alphas1-mh3-01}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.85\textwidth]{alphas1-mh3-03.png}
\caption{Allowed regions in the $\alpha_1$--$M_3$
parameter space, with ``2012'' (green) and ``Moriond 2013'' constraints (blue and black), for
$\tan\beta=3$, $M_2=400~\text{GeV}$ and 600~GeV.}
\label{Fig:alphas1-mh3-03}
\end{center}
\end{figure}
In this limit, the $H_iZZ$ (or $H_iW^+W^-$) couplings are proportional to
\begin{equation}
H_1ZZ\sim\cos(\beta-\alpha_1), \quad
H_2ZZ\sim\sin(\beta-\alpha_1), \quad
H_3ZZ=0,
\end{equation}
(in the familiar notation of the CP-conserving model, $\cos(\beta-\alpha_1)\to\sin(\beta-\alpha)$ and $\sin(\beta-\alpha_1)\to-\cos(\beta-\alpha)$).
Thus, maximizing the $H_1W^+W^-$ coupling (in order to obtain an acceptable $H_1\to\gamma\gamma$ rate), simultaneously makes the $H_2ZZ$ coupling small, and the tightened ``Moriond 2013'' high-mass exclusion has little effect.
In fact, for the CP-conserving case, it is only at low $\tan\beta$ and low $M_2$ (see left panel of Fig.~\ref{Fig:alphas1-mh3-01}) that they have any impact. For higher values of $\tan\beta$ and $M_2$, even rather loose constraints (\ref{Eq:constraint-2012}) on $H_1\to\gamma\gamma$ are more relevant than the high-mass ``Moriond 2013'' exclusion applied to $H_2$ and $H_3$.
For low values of $M_2$, the new $R_{\gamma\gamma}$ constraints significantly reduce the allowed range in $\alpha_1$, whereas for higher values of $M_2$ everything is excluded.
The CP-conserving case was recently studied in \cite{Grinstein:2013npa,Coleppa:2013dya,Chen:2013rba,Eberhardt:2013uba}. Our results are in qualitative agreement with one notable exception: Eberhardt {\it et al.} \cite{Eberhardt:2013uba} find that high masses (of order 1~TeV) are allowed, whereas we do not. We comment on that limit in Sect.~\ref{Sect:high-M-tanbeta}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{tanb-mhch}
\caption{Allowed regions in the $\tan\beta$--$M_{H^\pm}$
parameter space, without (green) and with (black) the recent LHC constraints, for $M_1=125~\text{GeV}$
and four values of $M_2$,
as indicated.
The dashed lines show the recent lower bounds at 360 and 380~GeV \cite{Hermann:2012fc}.
\label{Fig:tanbeta-mh_ch}}
\end{center}
\end{figure}
\subsection{Overview vs $\tan\beta$ and $M_{H^\pm}$}
In Fig.~\ref{Fig:tanbeta-mh_ch} we present an overview, in the $\tan\beta$-$M_{H^\pm}$ plane, of allowed regions. Two colors are used: green refers to regions considered allowed in our 2012 paper \cite{Basso:2012st}, whereas black refers to regions compatible with the recent ``Moriond 2013'' constraints (adopting the ATLAS range for $R_{\gamma\gamma}$). The horizontal dashed lines represent the recent constraint from $b\to s\gamma$ transitions \cite{Hermann:2012fc}, according to which, $M_{H^\pm}<360$ or $380~\text{GeV}$ is excluded for all $\tan\beta$. (The published version of the paper quotes both these values, depending on the choice of input.)
The new constraints from LHC are seen to significantly reduce the allowed region in parameter space, in particular at high values of $M_{H^\pm}$. However, we note that they depend on adopting the ATLAS result for $R_{\gamma\gamma}$ \cite{ATLAS:2013oma}.
\section{Charged Higgs Benchmarks}
We consider the production of the charged Higgs boson in association with a $W$ boson, and its decay to a neutral Higgs boson and a second $W$:
\begin{equation}
pp\to W^\mp H^\pm \to W^\mp W^\pm H_1 \to W^\mp W^\pm b \bar{b} \to 2j+2b+1\ell + \mbox{MET}.
\end{equation}
i.e., we let one $W$ decay hadronically, and the other leptonically.
There is a considerable $t\bar t$ background, but it was found \cite{Basso:2012st} that a certain combination of cuts can reduce this background to a tolerable level (see also Ref.~\cite{Basso:2013hs}).
In Ref.~\cite{Basso:2012st} we proposed a set of benchmarks, which, in
addition to being compatible with the constraints, also yielded
acceptable signal rates for the above channel. In the face of the new
LHC constraints, they are {\it all excluded}. Then, in the spirit of our previous work, we have adopted the same procedure and picked new benchmarks which could replace the old ones with no qualitative modification of our previous strategy to clean the signal from the background. In Table~\ref{Table:points} we present a new set of candidate points: the value of $\tan{\beta}$ is chosen to be $\mathcal{O}(1)$ and the free mass parameters are $\sim 400-500$ GeV.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\ & $\alpha_1/\pi$ & $\alpha_2/\pi$ & $\alpha_3/\pi$ & $\tan\beta$ & $M_2$ & $M_{H^\pm}^\text{min} ,M_{H^\pm}^\text{max}$ \\
\hline
$P_1^\prime$ & $0.27$ & $-0.002$ & $0.045$ & $1$ & $400$ & $\sim400$ \\
$P_2^\prime$ & $0.27$ & $-0.02$ & $0.02$ & $1$ & $400$ & $\sim400$ \\
$P_3^\prime$ & $0.29$ & $-0.03$ & $0.03$ & $1$ & $450$ & $\sim400$ \\
$P_4^\prime$ & $0.38$ & $-0.22$ & $0.26$ & $1$ & $450$ & $\sim450$ \\
$P_5^\prime$ & $0.37$ & $-0.15$ & $0.23$ & $1$ & $500$ & $\sim400$ \\
$P_6^\prime$ & $0.38$ & $-0.02$ & $0.04$ & $2$ & $400$ &$\sim400$ \\
$P_7^\prime$ & $0.39$ & $-0.03$ & $0.13$ & $2$ & $400$ & $\sim400$ \\
$P_8^\prime$ & $0.38$ & $-0.0005$ & $0.49$ & $2$ & $400$ & $\sim400$ \\
\hline
\end{tabular}
\end{center}
\caption{New benchmark points selected from the allowed parameter space. Masses $M_2$ and allowed range of $M_{H^\pm}$ are in GeV, and $\mu\simeq\min(M_2,M_{H^\pm})$. \label{Table:points}}
\end{table}
All the candidate points are allowed by the theoretical and experimental contraints that we have described in the previous sections, but the open question is whether at these benchmarks our selection strategy is still allowed. In order to establish this, we must profile the charged Higgs at each point of the parameter space and check if there is room for an observation at the LHC. In Table~\ref{Table:BRHc} we show the branching ratios for the charged Higgs boson main decay channels: $WH_1$, $tb$ and $ts$. Since we are interested in decay and production associated with bosons, we focus on the $WH_1$ decay mode. From the table is clear that the points $P'_4$, $P'_5$ and $P'_7$ provide interesting branching ratio values.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\ & $P_1^\prime$ & $P_2^\prime$ & $P_3^\prime$ & $P_4^\prime$ & $P_5^\prime$ & $P_6^\prime$ & $P_7^\prime$ & $P_8^\prime$ \\
\hline
$W^+ H_1$ & $0.0034$ & $0.0067$ & $0.020$ & $0.35$ &
$0.21$ & $0.037$ & $0.071$ & $0.025$ \\
$t\bar{b}$ & $0.99$ & $0.99$ & $0.98$ & $0.64$ &
$0.79$ & $0.96$ & $0.93$ & $0.97$ \\
$t\bar{s}$ & $0.0016$ & $0.0016$ & $0.0016$ & $0.0010$ &
$0.0012$ & $0.0015$ & $0.0015$ & $0.0015$ \\
\hline
\end{tabular}
\end{center}
\caption{Branching ratios of the main charged Higgs decay channels. \label{Table:BRHc}}
\end{table}
Thereafter, in Table~\ref{Table:CSHc} we present the cross sections for the main charged Higgs production channels at the LHC with $\sqrt{s}=8$ TeV and $\sqrt{s}=14$ TeV. Again, we are interested in the production associated with bosons, in particular the most sizeable one, which is $H^\pm W^\mp$. By direct comparison, we immediately see that the most promising points are represented by the choice $P'_1$, $P'_2$, $P'_3$, $P'_5$ and $P'_6$.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\ & $P_1^\prime$ & $P_2^\prime$ & $P_3^\prime$ & $P_4^\prime$ & $P_5^\prime$ & $P_6^\prime$ & $P_7^\prime$ & $P_8^\prime$ \\
\hline
$H^\pm H_i (8)$ & $0.0010$ & $0.00099$ & $0.00073$ & $0.0013$ &
$0.0013$ & $0.00075$ & $0.00092$ & $0.0010$ \\
$H^\pm H_i (14)$ & $0.0043$ & $0.0042$ & $0.0033$ & $0.0049$ &
$0.0049$ & $0.0033$ & $0.0039$ & $0.0043$ \\
$H^\pm W (8)$ & $0.18$ & $0.18$ & $0.19$ & $0.059$ &
$0.20$ & $0.13$ & $0.041$ & $0.042$ \\
$H^\pm W (14)$ & $1.24$ & $1.19$ & $1.15$ & $0.40$ &
$1.02$ & $0.62$ & $0.26$ & $0.28$ \\
$H^\pm t (8)$ & $0.31$ & $0.31$ & $0.31$ & $0.21$ &
$0.26$ & $0.078$ & $0.078$ & $0.078$ \\
$H^\pm t (14)$ & $1.97$ & $1.97$ & $1.97$ & $1.39$ &
$1.69$ & $0.49$ & $0.49$ & $0.49$ \\
\hline
\end{tabular}
\end{center}
\caption{Cross sections (pb) for the charged Higgs production channels at the LHC with $\sqrt{s}=8-14$ TeV. \label{Table:CSHc}}
\end{table}
If we combine the two results, it is clear that the best candidate is $P'_5$, which shows the best features for both the production and the decay. Such strong production is triggered by the high value of the $H_2$ mass ($500$ GeV), which produces a charged Higgs $H^\pm$ ($400$ GeV) together with a vector boson $W^\pm$ ($\sim 80$ GeV) almost resonantly. This point has a very similar behaviour to the benchmarks $P_5$ and $P_6$ of Refs.~\cite{Basso:2012st,Basso:2013hs}, for which our proposed strategy can be easily applied. We can conclude that the portion of the parameter space in the neighborhood of $P'_5$ is a good candidate for a charged Higgs analysis in association with bosons.
\section{High masses and high $\tan\beta$}
\label{Sect:high-M-tanbeta}
The parameter space is severely constrained at high values of $M_{H^\pm}$ and $\tan\beta$. There is actually an interplay of three constraints that operate in this region. Below, we shall comment on these.
First of all, the electroweak precision data, in particular $T$ (or $\Delta\rho$), severely constrain the {\it splitting} of the second doublet. However, in the exact limit
\begin{equation}
M_2=M_3=M_{H^\pm}\equiv M,
\end{equation}
the additional contributions to $T$ cancel. So this particular constraint can be evaded by such tuning of the heavy masses, which we will explore in the following.
There is a price to pay, the ``soft'' mass parameter must be carefully tuned, $\mu\simeq M$, due to the positivity and unitarity constraints.
In the limit when $M$ and $\mu$ both are large (compared to $M_1$) and $\tan\beta\gg1$, we find \cite{ElKaffas:2007rq}
\begin{subequations}
\begin{align}
\lambda_1&\simeq\frac{\tan^2\beta}{v^2}[c_1^2c_2^2M_1^2+(1-c_1^2c_2^2)M^2-\mu^2], \label{Eq:lambda1}\\
\lambda_2&\simeq\frac{1}{v^2}[s_1^2c_2^2M_1^2+(1-s_1^2c_2^2)M^2], \label{Eq:lambda2} \\
\lambda_3&\simeq\frac{\tan\beta}{v^2}[c_1s_1c_2^2(M_1^2-M^2)]
+\frac{1}{v^2}[2M^2-\mu^2], \label{Eq:lambda3} \\
\lambda_4&\simeq\frac{1}{v^2}[s_2^2M_1^2+(c_2^2-2)M^2+\mu^2], \label{Eq:lambda4} \\
\Re\lambda_5&\simeq\frac{1}{v^2}[-s_2^2M_1^2+(\mu^2-c_2^2M^2)], \label{Eq:lambda5} \\
\Im\lambda_5&\simeq\frac{1}{v^2}c_2s_2(c_1+\tan\beta s_1)(M^2-M_1^2).
\end{align}
\end{subequations}
In the CP-conserving limit with $c_2=1$ and $H_3=A$, we have
\begin{subequations}
\begin{align}
\lambda_1&\simeq\frac{\tan^2\beta}{v^2}[c_1^2M_1^2+s_1^2M^2-\mu^2], \label{Eq:lambda1bis}\\
\lambda_2&\simeq\frac{1}{v^2}[s_1^2M_1^2+c_1^2M^2], \label{Eq:lambda2bis} \\
\lambda_3&\simeq\frac{\tan\beta}{v^2}c_1s_1(M_1^2-M^2)
+\frac{1}{v^2}[2M^2-\mu^2], \label{Eq:lambda3bis} \\
\lambda_4&\simeq\frac{1}{v^2}(\mu^2-M^2), \label{Eq:lambda4bis} \\
\Re\lambda_5&\simeq\frac{1}{v^2}(\mu^2-M^2), \label{Eq:lambda5bis} \\
\Im\lambda_5&=0.
\end{align}
\end{subequations}
Tree-level unitarity roughly requires $|\lambda_i|<8\pi$ \cite{Ginzburg:2003fe}.
From Eqs.~(\ref{Eq:lambda4}) and (\ref{Eq:lambda5}) it follows that we must have $c_2\simeq1$ and $\mu\simeq M$.
Eqs.~(\ref{Eq:lambda1}) and (\ref{Eq:lambda2}) further require $|s_1|\simeq1$. The latter statement is made more precise by considering Eq.~(\ref{Eq:lambda3}), from which it follows that $c_1 \tan\beta M^2/v^2\leq{\cal O}(1)$, meaning that when $\tan\beta$ is large, $c_1$ must be very small, or $\alpha_1$ close to $\pm\pi/2$.
The positivity constraints can be written as \cite{Deshpande:1977rw}
\begin{subequations}
\begin{align} \label{Eq:positivity}
\lambda_1&>0, \quad \lambda_2>0, \\
\lambda_3&+\min[0,\lambda_4-|\lambda_5|]>-\sqrt{\lambda_1\lambda_2}.
\label{Eq:pos:lambda3}
\end{align}
\end{subequations}
The first of these equations, together with Eq.~(\ref{Eq:lambda1}), leads to $\mu\leq M$.
Then, $\min[0,\lambda_4-|\lambda_5|]=-2(M^2-\mu^2)/v^2$. By unitarity, the RHS of Eq.~(\ref{Eq:pos:lambda3}) can not be lower than $-8\pi$. Thus, we have a bound on $\tan\beta$, unless $c_1s_1$ vanishes:
\begin{equation}
c_1s_1 \tan\beta <\frac{M^2+8\pi v^2}{M^2-M_1^2},
\end{equation}
where we have approximated $\mu\simeq M$. On the other hand, if we set $s_1=1$ (but keep $\mu< M$), we find
\begin{equation}
\tan\beta<\sqrt{\frac{8\pi v^2}{M^2-\mu^2}}.
\end{equation}
Clearly, large values of $\tan\beta$ require either $c_1s_1\to0$ or $\mu\to M$.
For $\mu=0$, the cut-off is around $\tan\beta=5$--7, depending on the value of $M_{H^\pm}$ considered \cite{WahabElKaffas:2007xd}.
In addition, for a fixed value of $M_{H^\pm}$ there is a cut-off on $\tan\beta$ from $B\to\tau\nu$ and $B\to\tau\nu X$ decays \cite{Hou:1992sy}.
As stressed by Ref.~\cite{Eberhardt:2013uba}, there is an $M\gg v$ region, but our higher-dimensional parameter scan has some difficulty finding it. See, however, Fig.~\ref{Fig:alphas1-mh3-03}, where the case $M_2=M_{H^\pm}=600~\text{GeV}$ is excluded only by the ATLAS result on $R_{\gamma\gamma}$.
\section{Summary}
In this note, we have reviewed the status of the experimental and theoretical limits on a CP-violating version of the 2HDM type-II in view of the Moriond 2013 updates from the LHC. From the surviving parameter space, we have chosen some candidate points and we have checked the possibility of applying the selection strategy previously explained in \cite{Basso:2012st}. It turns out that, despite the considerable shrinking of the parameter space, there is still room for our proposed analysis. In practice, among our proposed benchmarks, a choice like $P'_5$ and similar configurations with $M_{H_2}\sim M_{H^\pm}+M_W$ give raise to a very good production cross section and decay rate. Hence, we persist with our suggestion for the next era of charged Higgs searches: after the discovery of the Higgs-like boson, the production and decay charged Higgs channels associated with bosons at hadron colliders deserve a special attention, because while the fermionic-associated channels are accompained by a huge background ($t\bar{t}$), the latter can be suppressed by a proper cutting strategy in the case of bosonic-associated channels.
\medskip
\noindent
{\bf Acknowledgement:}
We are grateful to the authors of Ref.~\cite{Eberhardt:2013uba} for clarifying discussions.
|
2,869,038,154,831 | arxiv | \section{Introduction}
Multiple choice questions provide the examinee with the ability to compare the answers, in order to eliminate some choices, or even guess the correct one.
Even when validating the choices one by one, the examinee can benefit from comparing each choice with the query and infer patterns that would have been missed otherwise. Indeed, it is the ability to synthesize de-novo answers from the space of correct answers that is the ultimate test for the understanding of the question.
In this work, we consider the task of generating a correct answer to a Raven Progressive Matrix (RPM) type of intelligence test~\cite{raven2003raven,carpenter1990one}. Each query (a single problem) consists of eight images placed on a grid of size $3 \times 3$. The task is to generate the missing ninth image, which is on the third row of the third column, such that it matches the patterns of the rows and columns of the grid.
The method we developed has some similarities to previous methods that recognize the correct answer out of eight possible choices. These include encoding each image and aggregating these encodings along rows and columns. However, the synthesis problem demands a new set of solutions.
Our architecture combines three different pathways: reconstruction, recognition, and generation. The reconstruction pathway provides supervision, that is more accessible to the network when starting to train, than the other two pathways, which are much more semantic. The recognition pathway shapes the representation in a way that makes the semantic information more explicit. The generation pathway, which is the most challenging one, relies on the embedding of the visual representation from the first task, and on the semantic embedding obtained with the assistance of the second, and maps the semantic representation of a given query to an image.
In the intersection of the reconstruction and the generation pathways, two embedding distributions need to become compatible. This is done through variational loss terms on the two paths. Since there are many different answers for every query, the variational formulation is also used to introduce randomness. However, since the representation obtained from the generation task is conditioned on a complex pattern, uniform randomness can be detrimental. For this reason, we present a new form of a variational loss, which varies dynamically for each condition, to support partial randomness.
Due to the non-deterministic nature of the problem and to the high-level reasoning required, the generation pathway necessitates a semantic loss. For this reason, we employ a perceptual loss that is based on the learned embedding networks. Since the suitability of the generated image is only exposed in the context of the row and column to which it belongs, the perceptual representation requires a hierarchical encoding.
Given the perceptual representation, a contrastive loss is used to compare the generated image to both the correct, ground truth, choice images, as well as to the other seven distractors. Since the networks that encode the images and subsequently the rows and columns also define the context embedding for the image generation, backpropogration needs to be applied with care. Otherwise, the networks that define the perceptual loss would adapt in order to reduce the loss, instead of the generating pathway that we wish to improve.
Our method presents very convincing generation results. The state of the art recognition methods regard the generated answer as the right one in a probability that approaches that of the ground truth answer. This is despite the non-deterministic nature of the problem, which means that the generated answer is often completely different pixel-wise from the ground truth image. In addition, we demonstrate that the generation capability captures most rules, with little neglect of specific ones. Finally, the recognition network, which is employed to provide an auxiliary loss, is almost as effective as the state of the art recognition methods.
\begin{figure*}[t]
\centering
\begin{minipage}[t]{0.5\textwidth}
\includegraphics[width=.6\textwidth, trim={0 0 0 2}, clip]{images/query.png}\qua
\includegraphics[width=.3\textwidth, trim={0 0 0 0}, clip]{images/chioces2.png
\caption{Example of a PGM problem with a $3\times 3$ grid and 8 choices.}\label{fig:dataset}
\end{minipage} \hfil
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=.6\textwidth]{images/condition2.png}
\caption{Each row has equivalent answers for some attribute (from top: shape type, shape number, and line color)}\label{fig:condition}
\end{minipage}
\end{figure*}
\section{Related work}\label{sec:related}
In RPM, the participant is given the first eight images of a $3 \times 3$ grid (context images) and has to select the missing ninth image out of a set of eight options (choice images). The correct image
fits the most patterns over the rows and columns of the grid. In this work, we utilize two datasets: (i) the Procedurally Generated Matrices (PGM)~\cite{santoro2018measuring} dataset, which depicts various lines and shapes of different types, colors, sizes and positions. Each grid has between one and four rules on both rows and columns, where each rule applies to either the shapes or the lines, see Fig.~\ref{fig:dataset} for a sample, and (ii) the recently proposed RAVEN-FAIR~\cite{MRNet} dataset, which is a RPM dataset that is based on the RAVEN~\cite{zhang2019raven} dataset and is aimed to remove a bias in the process of creating the seven distractors.
In parallel to presenting PGM, Santoro et al. presented the Wild Relation Network (WReN)~\cite{santoro2018measuring}, which considers the choices one by one.
Considering all possible answers at once, both CoPINet~\cite{zhang2019learning} and LEN~\cite{zheng2019abstract} were able to infer more context about the question. CoPINet and LEN also introduced row-wise and column-wise relation operations. MRNet~\cite{MRNet} introduced multi-scale processing of the eight query and eight challenge images. Another contribution of MRNet is a new pooling technique for aggregating the information along the rows and the columns of the query's grid. Variational autoencoders~\cite{kingma2013auto} were considered as a way to disentangle the representations for improving on held-out rules~\cite{steenbrugge2018improving}.
While our architecture uses a similar pooling operator as MRNet, unlike~\cite{MRNet,zhang2019learning,zheng2019abstract}, we cannot perform pooling that involves the third row and the third column. Otherwise, information from the choices would leak to our generator.
Despite having this limitation, the ability of our method to select the correct answer does not fall short of any of the previous methods, except for MRNet.
Our work is related to supervised image-to-image translation~\cite{pix2pix,pix2pixHD}, since the input takes the form of a stack of images. However, the semantic nature of the problem requires a much more abstract representation. For capturing high-level information, the perceptual loss is often used following Johnson et al.~\cite{johnson2016perceptual}. Closely related to it is the usage of feature matching loss terms when training GANs~\cite{salimans2016improved}. In this case, one uses the trained discriminator to provide a semantic feature map for matching a distribution, or, in the conditional generation case, to compare the generated image to the desired output. In our work, we use an elaborate type of a perceptual loss, using some of the trained networks, in order to provide a training signal based on the query's rows and columns.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth, trim={40 610 40 20}, clip]{images/architecture.pdf}
(a) \\
\begin{tabular}{c@{~}c}
\includegraphics[width=0.48\textwidth, trim={25 405 300 285}, clip]{images/architecture.pdf} &
\includegraphics[width=0.48\textwidth, trim={300 405 25 285}, clip]{images/architecture.pdf} \\
(b)&(c) \\
\end{tabular}
\caption{(a) Our architecture. (b) The CEN part. (c) Application of the relation-wise loss.}
\label{fig:architecture}
\end{figure*}
\section{Method}
Our method consists of the following main components: (i) an encoder $E$ and (ii) a generator $G$, which are trained together as a variational autoencoder (VAE) on the images; (iii) a Context Embedding Network (CEN), which encodes the context images and produces the embedding for the generated answer, and (iv) a discriminator $D$, which provides an adversarial training signal for the generator. An illustration of the model can be found in Fig.~\ref{fig:architecture}(a). In this section, we describe the functionality of each component. The exact architecture of each one is listed in the supplementary.
{\bf Variational autoencoder\quad }
The VAE pathway contains the encoder $E$ and the generator $G$, and it autoencodes the choice images $\{I_{a_i} | i \in [1,8]\}$ one image at a time. The latent vector produced by the encoder is sampled with the reparameterization trick~\cite{kingma2013auto} with a random vector $z$ sampled from a unit gaussian distribution.
\begin{equation}
\mu^{a_i}_v, \sigma^{a_i}_v = E(I_{a_i})\quad h_{a_i} = \mu^{a_i}_v + \sigma^{a_i}_v \circ z\quad \hat{I}_{a_i} = G(h_{a_i})\,,
\end{equation}
where $\mu^{a_i}_v, \sigma^{a_i}_v,h_{a_i},z \in \mathbb{R}^{64}$, and $\circ$ marks the element-wise multiplication. (The symbol $v$ is used to distinguish from the second reparameterization trick employed by the translation pathway below, with the symbol $g$). The VAE is trained with the following loss $\mathcal{L}_{VAE} = \frac{1}{8} \sum_{i=1}^8\left(\lambda_{KL_1} \cdot \mathcal{D}_{KL}(\mu^{a_i}_v, \sigma^{a_i}_v) + MSE(\hat{I}_{a_i}, I_{a_i})\right)$,
where $\mathcal{D}_{KL}$ is the KL-divergence between $h_{a_i}$ and a unit gaussian distribution, $MSE$ is the mean squared error on the image reconstruction and $\lambda_{KL_1}=4$ is a tradeoff parameter between the KL-Divergence and the reconstruction terms.
{\bf Context embedding network\quad}
Our CEN is comprised of multiple sub-modules, which are trained together to provide input to the generator, see Fig.~\ref{fig:architecture}(b) for illustration. The network has an additional auxiliary task of predicting if a choice image is a correct answer conditioned on the context images and predicting an auxiliary 12-bit ``meta data'' ($\psi$) that define the applied rules.
Our embedding process has some similarity to previous work, MRNet~\cite{MRNet}. Our method follows the same encoding process, where it relies on a multi-stage encoder to encode the images into multiple resolutions and relies on a similar Reasoning Module (RM). The key difference from MRNet is that the generation pathway cannot be allowed to observe the choice images $I_{a_i}$, since it will then learn to retrieve the correct image instead of fully generating it.
To generate the answer image, we first encode each context image $I_{i}$, $i=1,2,..,8$ with the mutli-scale context encoder ($E^C_h, E^C_m, E^C_l$) in order to extract encodings in three different scales $e^i_h \in \mathbb{R}^{64,20,20}, e^i_m \in \mathbb{R}^{128,5,5}, e^i_l \in \mathbb{R}^{256,1,1}$. These are applied sequentially, such that the tensor output of each scale is passed to the encoder of the following scale.
\begin{equation}
e^i_h = E^C_h(I_{i}), \quad e^i_m = E^C_m(e^i_h), \quad e^i_l = E^C_l(e^i_m)
\end{equation}
Since the context embedding is optimized with an auxiliary classifier $C$ that predicts the correctness of each choice, the encoder also encodes the choice images $\{I_{a_i} | i \in [1,8]\}$, obtaining $\{e^{a_i}_t\}$ for $t\in\{h,m,l\}$.
The embeddings of the context images are then passed to a Reasoning Module (RM) in order to detect a pattern between them that will be used to define the properties of the generated image. Since the rules are applied in a row-wise and column-wise orientation, the RM aligns the context embeddings in rows and column, similar to~\cite{zheng2019abstract,zhang2019learning,MRNet}. Following~\cite{MRNet}, this is done for all three scales, and is denoted by the scale index $t\in\{h,m,l\}$.
The row representations are the concatenated triplets $(e^1_t, e^2_t, e^3_t),(e^4_t,e^5_t,e^6_t)$ and the column representations are $(e^1_t, e^4_t, e^7_t),(e^2_t,e^5_t,e^8_t)$, where the images are arranges as in Fig.~\ref{fig:architecture}(a) "Context Images". Each representation is passed through the scale-approriate RM to produce a single representation of the triplet. There is a single RM per scale that encodes both rows and columns.
\begin{equation}\label{eq:RM}
r^1_t = RM_t(e^1_t, e^2_t, e^3_t), ~ r^2_t = RM_t(e^4_t,e^5_t,e^6_t), ~ c^1_t = RM_t(e^1_t, e^4_t, e^7_t), ~ c^2_t = RM_t(e^2_t,e^5_t,e^8_t)
\end{equation}
Note that unlike~\cite{MRNet}, only two rows are used and we do not use the triplets $(e^7_t, e^8_t, e^{a_i}_t), (e^3_t, e^6_t, e^{a_i}_t)$, since they contain the embeddings of the choices and using the choices at this stage will reveal the potential correct image to the generation path.
The two row-representations and two column-representations are then joined to form the intermediate context embedding $q_t$. Following~\cite{MRNet}, this combination is based on the element-wise differences between the vectors, which are squared and summed element-wise. Unlike~\cite{MRNet}, in our case there are only two rows and two columns:
\begin{equation}
q_t = (r^1_t - r^2_t).^2 + (c^1_t - c^2_t).^2
\end{equation}
While the first steps of the CEN shared some of the architecture of previous work, the rest of it is entirely novel. The next step considers the representation of the third row and column in each scale, where the embedding $q_t$ replaces the role of the missing element.
\begin{equation}\label{eq:P}
x_t = P_t(e^7_t, e^8_t, q_t) + P_t(e^3_t, e^6_t, q_t),
\end{equation}
where $\{P_t | t\in\{h,m,l\}\}$ is another set of learned sub-networks of the CEN module.
This is the point where the model splits into generation path and recognition path. The merged representations $x_h, x_m, x_l$, are used for two purposes. First, they are used in the auxiliary task that predicts the correctness of each image and the rule type. Second, they are merged to produce a context embedding for the downstream generation.
For the auxiliary tasks, two classifier networks are used, $C_O$ for predicting the correctness of each choice image $I_{a_i}$, and $C_M$ for predicting the rule described in the metadata. The first classifier, unlike the second, is, therefore, conditioned on the three embeddings $e^{a_i}_t$ of $I_{a_i}$.
\begin{equation}
\hat{y}_i = Sigmoid(C_O(x_h, x_m, x_l, e^{a_i}_h, e^{a_i}_m, e^{a_i}_l)), \quad
\hat{\psi} = Sigmoid(C_M(x_h,x_m,x_l)),
\end{equation}
where $\hat{y}_i\in[0,1]$ and $\hat{\psi} \in [0,1]^K$, with $K=12$.
The classifiers apply a binary cross-entropy on the eight choices separately, with $y_i \in \{0,1\}$ and on the meta target $\psi \in \{0,1\}^K$.
\begin{equation}
\mathcal{L}_C = \frac{1}{8} \sum^8_{i=1} BCE(\hat{y}_i, y_i) + \frac{1}{K} \sum^K_{k=1} BCE(\hat{\psi}[k], \psi[k])
\end{equation}
For generation purposes, the 3 embeddings $x_h,x_m,x_l$ are combined to a single context embedding $x$
\begin{equation}\label{eq:R}
x = R(x_h, x_m, x_l)\,,
\end{equation}
where R is a learned network.
{\bf Generating a plausible answer\quad} Network $T$ needs to transform the context embedding vector $x$ to a vector in the latent space of the VAE.
It maps $x$ to the mean and the standard deviation of a diagonal multivariate Gaussian distribution with parameters $\mu_g, \sigma_g \in \mathbb{R}^{64}$. These are then used together with a random vector $z' \sim \mathcal{N}(0,1)^{64}$ to sample a new non-deterministic representation $h_g$ in the latent space of the VAE.
\begin{equation}\label{eq:T1}
\mu_g, \sigma_g = T(x) \quad\quad h_g = \mu_g + \sigma_g \cdot z'\,,
\end{equation}
Instead of regularizing the reparameterization with the standard KL-divergence loss, we use a novel loss for the reparameterization we call Dynamic Selective KLD (DS-KLD).
The loss applies the regularization on a subset of indices, and have this subset change for each case.
This way, the model is allowed to reduce the noise on some indices, while other elements of $x$ maintain their information. In other words, we use this novel loss to encourage the model to add noise only for those indices that affect the distracting attributes. This way, it adds variability to the generation process, while not harming the correctness of the generated image.
The KL-divergence loss between some i.i.d Gaussian distribution and the normal distribution
is defined as two unrelated terms: the mean and the variance:
\begin{equation}
\mathcal{L}_{KL} = - \frac{1}{2}
\underbrace{\sum\mu^2}_{\mathcal{L}_{KL_\mu}} -\frac{1}{2}
\underbrace{\sum (\log(\sigma^2) - \sigma^2}_{\mathcal{L}_{KL_\sigma}})
\end{equation}
Our method applied the mean term as usual, to densely pack the latent space, but applies the variance term only on the subset of indices with variance above the median.
We then use the generator $G$ to synthesize the image.
\begin{equation}\label{eq:G}
I_g = G(h_g)
\end{equation}
Two loss terms are applied to the generated image. An unconditional adversarial loss $\mathcal{L}_G$, which trains the generation to produce images that look real, and a conditioned perceptual loss $\mathcal{L}_{COND}$, which trains the generation to produce images with attribute that match the correct answer.
The adversarial loss is optimized with an unconditioned discriminator $D$, which is trained with the standard GAN loss $\min_G \max_D \mathbb{E}_x[\log(D(x))] + \mathbb{E}_z[\log(1 - D(G(z)))]$.
The loss on the discriminator is: $ \mathcal{L}_D = \log(D(I_{a^{*}})) + \log(1 - D(I_g))$,
where ${a^{*}}$ is the index of the correct target image for the context the generation is conditioned on. The adversarial loss on the Generator (and upstream computations) is $\mathcal{L}_G = \log(D(I_g))$.
In order to {enforce the generation to be conditioned on the context}, we apply a contrastive loss between the generated image and the choices $I_{a}^i$. For two vectors $x_0, x_1$, $y \in \{0,1\}$, and a margin hyper-parameter $\alpha$, the contrastive loss is defined as:
$ \text{Contrast}(x_0, x_1, y) := y \cdot \|x_0 - x_1\|^2_2 + (1 - y) \cdot \max \left(0, \alpha - \|x_0 - x_1 \|^2_2\right)$.
This loss learns a metric between the two vectors given that they are of the same type ($y=1$) or not ($y=0$).
Measuring similarity in this setting is highly nontrivial. The images can be compared on the pixel-level (directly comparing the images) or with a perceptual loss on some semantic level (comparing the image encodings $e^i_t$). However, two images can be very different pixel-wise and semantic-wise and still both be correct with respect to the relational rules. This is shown in Fig.~\ref{fig:condition}, where each of the three images would be considered correct under the specified rule, but would not be correct under any other rule.
For this reason, we do not follow any of these approaches. Our approach is to apply the perceptual loss, conditioned on the context, by computing the third row and column representations $(e^7_t, e^8_t, e^9_t),(e^3_t, e^6_t, e^9_t)$, where $e^9_t\in \{e^g_t\} \cup \{e^{a_i}_t | i\in [1,8]\}$, compute their relation-wise encodings, and compare those of the generated image to those of the choice images. Here, the index $a_i$ is used to denote the embedding that arises when $I_a=I_{a_i}$, and $e^g_t$ is the encoding of the generated image $I_g$.
The relation-wise encodings are formulated as: $\dot{e}^g_h = \dot{E}^C_h(I_g)$, $\dot{e}^g_m = \dot{E}^C_m(\dot{e}^g_h)$, $\dot{e}^g_l = \dot{E}^C_l(\dot{e}^g_m)$, $r^{3,g}_t = \dot{RM}_t(\ddot{e}^7_t$, $\ddot{e}^8_t, \dot{e}^g_t)$,
$c^{3,g}_t = \dot{RM}_t(\ddot{e}^3_t, \ddot{e}^6_t, \dot{e}^g_t)$,
$r^{3,{a_i}}_t = \dot{RM}_t(\ddot{e}^7_t, \ddot{e}^8_t, \ddot{e}_t^{a_i})$,
$c^{3,{a_i}}_t = \dot{RM}_t(\ddot{e}^3_t, \ddot{e}^6_t, \ddot{e}_t^{a_i})$. Here,
$\ddot{e}$ means that the variable does not backpropagate (a detached copy). $\dot{RM}_t$ specifies that this network is frozen as well. The single dot $\dot{e}$ means that the encoders $E^C_h,E^C_m,E^C_l$ were frozen for this encoding. The rest of the modules ,$G,T,R,P_t$, along with $E^C_t$,$RM_t$ (through their first paths only), which are part of the generation of $I_g$, are all trained through this loss.
This selective freezing of the model is done, since the frozen model is used as a critic in this instance and one cannot optimize $E_t,RM_t$ to artificially try to reduce this loss.
The total contrastive loss $\mathcal{L}_T$ is applied by computing the contrastive loss between the generated image and the choice images $I_{a_i}, i \in [1,8]$.
\begin{equation}
\begin{gathered}
\mathcal{L}^{1^t}_{T,r} = \text{Contrast}(r^{3,g}_t, r^{3,a^{*}}_t, 1), \quad
\mathcal{L}^{0^t}_{T,r} = \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( \text{Contrast}(r^{3,g}_t, r^{3,a_i}_t, 0) \right) \\
\mathcal{L}^{1^t}_{T,c} = \text{Contrast}(c^{3,g}_t, c^{3,a^{*}}_t, 1), \quad
\mathcal{L}^{0^t}_{T,c} = \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( \text{Contrast}(c^{3,g}_t, c^{3,a_i}_t, 0) \right) \\
%
\mathcal{L}^{{1}^t}_T = \mathcal{L}^{{1}^t}_{T,r} + \mathcal{L}^{{1}^t}_{T,c}, \quad
\mathcal{L}^{{0}^t}_T = \mathcal{L}^{{0}^t}_{T,r} + \mathcal{L}^{{0}^t}_{T,c} \\
\mathcal{L}^{1}_T = \mathcal{L}^{{1}^h}_T + \mathcal{L}^{{1}^m}_T + \mathcal{L}^{{1}^l}_T, \quad
\mathcal{L}^{0}_T = \mathcal{L}^{{0}^h}_T + \mathcal{L}^{{0}^m}_T + \mathcal{L}^{{0}^l}_T \\
\mathcal{L}_{COND} = \mathcal{L}^{1}_T + \mathcal{L}^{0}_T \\
\end{gathered}
\end{equation}
In the ablation, we apply other variants of this loss.
(1) Contrastive pixel-wise comparison to $I_a$:
$\mathcal{L}_{COND_1} = Contrast(I_g, I_{a^{*}}, 1) + \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( \text{Contrast}(I_g, I_{a_i}, 0) \right)$, (2) Contrastivefeature-wise comparison to $e^a$:
$\mathcal{L}_{COND_2} = \text{Contrast}(\dot{e}^g, \ddot{e}^{a^{*}}, 1) + \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( Contrast(\dot{e}^g, \ddot{e}^{a_i}, 0) \right)$, and (3) Non-contrastive, without $\mathcal{L}^0_T$ (MSE): $\mathcal{L}_{COND_3} = \mathcal{L}^{1}_T$.
\begin{figure}
\centering
\begin{tabular}{c@{~}c@{~}c@{~}c}
\includegraphics[width=0.238\textwidth]{images/line_type_half_colored.png} &
\includegraphics[width=0.238\textwidth]{images/shape_position_half_colored.png} &
\includegraphics[width=0.238\textwidth]{images/shape_number_half_colored.png} &
\includegraphics[width=0.238\textwidth]{images/shape_type_half_colored.png} \\
(a)&(b)&(c)&(d) \\
\end{tabular}
\caption{Generation results for selected rules in PGM, each with 5 problems. The top row is the ground truth answer. The bottom is the generated.
The good (bad) results are highlighted in green (red) respectively.
(a) line type. (b) shape position. (c) shape number. (d) shape type.}
\label{fig:generation}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.4\textwidth]{images/variability_of_shape_position.png} &
\includegraphics[width=0.4\textwidth]{images/variability_of_line_type.png} \\
(a)&(b) \\
\end{tabular}
\caption{Generation variability of distracting attributes by sampling $z'$ (Eq.~\ref{eq:T1}). A collection of five different PGM problems. the first row contains the real answers and the other rows contain two different generated answers from the same context and different $z'$. (a) shape position. (b) line type.}
\label{fig:generation2}
\end{figure}
\section{Experiments}
\label{sec:exp}
The Adam optimizer is used with a learning rate of $10^{-4}$. {The margin hyper-parameter $\alpha$ (for the contrastive loss) is updated every 1000 iterations to be the mean measured distance between the choices images and the generated.} The contrastive loss with respect to the target choice image was multiplied by $3\cdot10^{-3}$, and the contrastive loss with respect to the negative choice image was multiplied by $10^{-4}$. The VAE losses were multiplied by 0.1 (with $\beta$ of 4), and the auxiliary $C_m$ loss was multiplied by 10. all other losses were not weighted. The CEN was trained for 5 epochs for the recognition pathway only, after which all subnetworks were trained for ten additional epochs. We train on the train set and evaluate on the test set for all datasets.
{The experiments were conducted on the two regimes of the PGM dataset~\cite{santoro2018measuring}, ``neutral'' and ``interpolation'' as well as on the recently proposed RAVEN-FAIR dataset~\cite{MRNet}. In ``neutral'' and RAVEN-FAIR, train and test sets are of the same distribution, and in ``interpolation'', ordered attributes (colour and size) differ in test and train sets. In order to evaluate the generated results, two different approaches were used: machine evaluation (using other recognition models), and human evaluation.}
{\bf Machine evaluation in the ``neutral'' regime of PGM\quad} A successful generation would present two properties: (i) image quality would be high and the generated images would resemble the ground truth images of the test set. (ii) the generated answers would be correct. The first property is evaluated with FID~\cite{heusel2017gans} that is based on an PGM classification networks (see supplementary). To evaluate generation correctness, we employ the same automatic recognition networks. While these networks are not perfectly accurate they support reproducibility. To minimize bias, both evaluations are repeated with three pretrained models of largely different architectures: WReN~\cite{santoro2018measuring}, LEN~\cite{zheng2019abstract} and MRNet~\cite{MRNet}.
The accuracy evaluation is performed by measuring the fraction of times, in which the generated answer is chosen over the seven distractors (false choices) of each challenge. This number is compared to the recognition results of each network given the ground truth target $I_{a^*}$. Ideally, our generation method would obtain the same accuracy. However, just by applying reconstruction to $I_{a^*}$, there is a degradation in quality that reduces the reported accuracy. Therefore, to quantify this domain gap between synthetic and real images, we also compare to two other versions of the ground truth: in one it is reconstructed without any randomness added and in the other, no reparameterization is applied. These two are denoted by $G(\mu_{a^*}^{v})$ and $G(h_{a^*})$, respectively.
As can be seen in Table \ref{tab:generation}, our method, denoted `Full', performs somewhat lower in terms of accuracy in comparison to the real targets. However, most of this gap arises from the synthetic to real domain gap. It is also evident that this gap is larger when randomness is added to the encoding before reconstruction takes place. However, for our method (comparing it to `Full, w/o reparam in test') this gap is smaller, suggesting it was adapted for this randomization.
Considering the FID scores, we can observe that while the FID of the generated answer is somewhat larger than that of the reconstructed versions of the ground truth answer, it is still relatively low. This is despite the VAE itself lacking as a generator for random seeds, as is evident when considering the `random VAE image' row in the table. This row is obtained by sampling from the normal distribution in the latent space of the VAE and performing reconstruction. Autoencoders, unlike GANs, usually do not provide good reconstruction for random seeds.
{\bf Generated examples\quad} The visual quality of the output can be observed by considering the examples in Fig.~\ref{fig:generation}. It is evident that the generated images mimic the domain images, yet, maybe with a tendency for fainter colors. More images are shown in the supplementary.
The generations in Fig.~\ref{fig:generation} are in the context of a specific query that demonstrates a selected rule. The top row shows the ground truth answer, and the bottom row shows the generated one. The correct answers (validated manually) are marked in green. As can be seen, the correct solution generated greatly differs from the ground truth one, demonstrating the variability in the space of correct answers. This variability is also demonstrated in Fig.~\ref{fig:generation2}, in which we present two solutions (out of many) for a given query. The obtained solutions are different. However, in some cases, they tend to be more similar to one another than to the ground truth answer.
{\bf Ablation study in the “neutral” regime of PGM\quad} To validate the contribution of each of the major components of our method, an ablation analysis was conducted. The following variations of our full method are trained from scratch and tested:
(1) {\bf W/o reparameterization in train:} generate answers without the reparameterization trick, but use a random vector $z' \sim \mathcal{N}(0,1)^{128}$ concatenated to the context embedding.
(2) {\bf Standard KLD:} reparameterization trick employing the standard KLD loss.
(3) {\bf Static half KLD:} applying the KLD loss to a fixed half of the latent space and do not apply it to the other indices.
(4) {\bf W/o $VAE$:} without autoencoding, using a discriminator to train a GAN on the generated images and the real answer images.
(5) {\bf W/o auxiliary $C_M$:} without the auxiliary task of predicting the correct rule type.
(6) {\bf W/o $\mathcal{L}^{\cdot}_T$:} instead of using contrastive loss $\mathcal{L}^{\cdot}_T$, we trained using the MSE loss just for minimizing the relation distance between the generated image to the target image.
(7) {\bf $\mathcal{L}_T$ on $e^a$:} instead of using contrastive perceptual loss with the relation module, we train by using contrastive feature-wise loss with the features encoded vectors $e_a$:
$\mathcal{L}_{COND_2} = \text{Contrast}(\dot{e}^g, \ddot{e}^{a^{*}}, 1) + \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( Contrast(\dot{e}^g, \ddot{e}^{a_i}, 0) \right)$.
(8) {\bf $\mathcal{L}_T$ on $I_a$:} instead of using contrastive perceptual loss with the relation module, we train by using contrastive pixel-wise comparison loss on the images $I_a$:
$\mathcal{L}_{COND_1} = Contrast(I_g, I_{a^{*}}, 1) + \frac{1}{7} \sum_{i: a_i \neq a^{*}} \left( \text{Contrast}(I_g, I_{a_i}, 0) \right)$. Finally,
(9) {\bf W/o freeze:} without selectively freezing of the model. Variants 1--4 test the selective reparameterization and the VAE, variants 6-9 test the perceptual loss and its application.
As can be seen in Table~\ref{tab:generation}, each of these variants leads to a decrease in the accuracy of the generated answer performance. For FID the effect is less conclusive and there is often a trade-off between it and the accuracy. Interestingly, removing randomness altogether is better than using the conventional KLD term, in which there is no selection of half of the vector elements.
\begin{table*}[t]
\caption{Performance on each evaluator. Acc is $I_g$ vs. the seven $I_a$ for $a\neq a*$.}\label{tab:generation}
\centering
\begin{tabular}{lcccccc}
\toprule
& \multicolumn{2}{c}{WReN} & \multicolumn{2}{c}{LEN} & \multicolumn{2}{c}{MRNet} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
& Acc & FID & Acc & FID & Acc & FID \\
\midrule
Real Target ($I_{a^*}$) & 76.9 & - & 79.6 & - & 93.3 & - \\
Recon. Target ($G(\mu_{a^*}^{v})$) & 62.9 & 1.9 & 66.2 & 12.2 & 80.6 & 2.9 \\
Recon. Target with reparam ($G(h_{a^*})$) & 58.4 & 2.2 & 62.3 & 14.2 & 76.5 & 3.6 \\
\midrule
Full & 58.7 & 5.9 & 60.1 & 38.6 & 65.4 & 8.1 \\
Full, w/o reparam. in test & {\bf59.0} & 4.9 & {\bf60.4} & 37.3 & {\bf65.5} & 7.5 \\
\midrule
(1) W/o reparam. in train & 54.7 & {\bf4.7} & 56.3 & {\bf37.2} & 60.1& {\bf7.1} \\
(2) Standard KLD & 47.4 & 5.7 & 49.3 & 38.1 & 53.3 & 7.6 \\
(3) Static half KLD & 50.6 & 6.0 & 52.9 & 40.2 & 55.8 & 8.8 \\
(4) W/o VAE & 50.5 & 6.2 & 52.7 & 44.0 & 55.6 & 12.7 \\
(5) W/o auxiliary $C_M$ & 53.6 & 6.2 & 54.7 & 38.8 & 57.2 & 8.2\\
(6) W/o $\mathcal{L}^{\cdot}_T$ & 51.5 & 5.2 & 50.4 & 40.9 & 52.7 & 8.5 \\
(7) $\mathcal{L}_T$ on $e^a$ & 47.2 & 6.8 & 46.6 & 41.3 & 48.4 & 8.4 \\
(8) $\mathcal{L}_T$ on $I_a$ & 46.0 & 8.0 & 46.7 & 613.9 & 48.1 & 65.0 \\
(9) W/o freeze & 40.3 & 7.5 & 44.8 & 869.0 & 44.6 & 58.1 \\
\midrule
Random VAE image & 22.3 & 8.1 & 29.6 & 1637.0 & 21.5 & 94.4 \\
\bottomrule
\end{tabular}
\smallskip
\begin{minipage}[c]{0.67\linewidth}
\caption{Accuracy and FID per rule (MRNet)}\label{tab:generation-WReN}
\smallskip
\centering
\begin{tabular}{@{}l@{~}c@{~}c@{~}c@{~}c@{~}c@{~}c@{~}c@{}}
\toprule
& \multicolumn{2}{c}{Line} & \multicolumn{5}{c}{Shape} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-8}
\parbox[t]{15mm}{Model} & Type & Color & Type & Color & Pos. & Num. & Size\\
\midrule
Acc. on Real Target & 96.3 & 96.3 & 88.3 & 76.5 & 99.0 & 98.5 & 89.4 \\
Acc. on recon. Target & 89.8 & 83.1 & 75.6 & 60.4 & 90.1 & 81.2 & 73.4 \\
Acc. of our method & 86.6 & 57.2 & 41.0 & 37.8 & 88.0 & 55.9 & 45.7 \\
\midrule
FID of our method & 4.9 & 6.52 & 5.3 & 6.1 & 5.0 & 6.4 & 6.1 \\
\bottomrule
\end{tabular}
\end{minipage}%
\hfill
\begin{minipage}[c]{0.32\linewidth}
\caption{The accuracy obtained by the auxiliary classifier $C_o$}\label{tab:classification}
\centering
\begin{tabular}{@{}l@{~}c@{~}c@{}}
\toprule
\parbox[t]{15mm}{Model} & PGM & PGM\_aux\\
\midrule
WReN~\cite{santoro2018measuring} & 62.6 & 76.9 \\
CoPINet~\cite{zhang2019learning} & 56.4 & - \\
LEN~\cite{zheng2019abstract} & 68.1 & 82.3 \\
MRNet~\cite{MRNet} & 93.3 & 92.6 \\
Our~$C_o$ & 68.2 & 82.9 \\
\bottomrule
\end{tabular}
\end{minipage}
\end{table*}
{\bf Performance on each task in the “neutral” regime of PGM\quad} Generative models often suffer from mode collapse and even if, on average, the generation achieves high performance, it is important to evaluate the success on each rule. For this purpose, we employ MRNet, which is currently the most accurate classification model. In Tab.~\ref{tab:generation-WReN} we present the breakdown of the accuracy per type of rule. As can be seen, the performance of our generated result is not uniformly high. There are rules such as Line-Type and Shape-Pos that work very well, and rules such as Shape-Type and Shape-Color that our method struggles with.
{\bf Human evaluation in the “neutral” regime of PGM\quad} Two user studies were conducted. The first is a user study that follows the same scheme as the machine evaluation. Three participants were extensively trained on the task of PGM questions. After training, each got 30 random questions with the correct target image, reconstructed by VAE to match the quality, and 30 with the generated target instead. Human performance on the correct image was 72.2\%, and on the generated image was 63.3\%. {We note that due to the extensive training required, the number of participants leads to a small sample size.}
To circumvent the training requirement, we conducted a second user that is suitable for untrained individuals. The study is motivated by the qualitative image analysis in Fig.~\ref{fig:generation}. In PGM, an image is correct if and only if it contains the right instance of the object attribute which the rule is applied on (this information is in the metadata). By comparing the generated object attribute to the correct answer object attribute, one can be easily determined if the generation is correct. This study had $n=22$ participants, 140 random image comparing instances for the generated answers, and 140 for a random choice image (reconstructed by VAE) as a baseline. The results show that 70.1\% of the generations were found to be correct and only 6.4\% of the random choice images (baseline).
{\bf Recognition performance in the “neutral” regime of PGM\quad} We evaluate the recognition pathway, i.e., the accuracy obtained by the classifier $C_O$. This classifier was trained as an auxiliary classifier, and was designed with the specific constraint of not using the relational module RM on the third row and column. It is therefore at a disadvantage in comparison to the literature methods. Tab.~\ref{tab:classification} presents results for two versions of $C_O$. One was trained without the metadata (this is the `W/o auxiliary $C_M$ ablation`) and one with. These are evaluated in comparison to classifiers that were trained with and without this auxiliary information. As can be seen, our method is highly competitive and is second only to MRNet~\cite{MRNet}.
{\bf The ``interpolation'' regime of PGM\quad} Out-of-distribution generalization was demonstrated by training on this regime of PGM, in which the ordered attributes (colour and size) differ in test and train sets. Evaluation was done using the MRNet model that was trained on the ``neutral'' regime. The generation accuracy was 61.7\%, which is very close to the 68.1\% (MRNet) and 64.4\% (WReN) accuracy in the much easier recognition task in this regime.
{\bf ``RAVEN-FAIR''\quad} Further experiments were done on this recent variant of the RAVEN dataset, see Fig.~\ref{fig:raven_generation} for some typical examples. machine evaluation was performed using MRNet, which is the state of the art network for this dataset, with 86.8\% accuracy. The generation accuracy was 60.7\%, this is to be compared to 69.5\% on the target image reconstructed by VAE, and only 8.9\% on a random generated image. We also evaluate the recognition pathway (auxiliary classifier $C_O$ performance). Tab.~\ref{tab:classification_raven} presents those results in comparison to other classifiers. It seems that for RAVEN-FAIR, our method achieve far greater recognition results then most classifiers, and is second only to MRNet~\cite{MRNet}.
\begin{figure}[t]
\centering
\begin{minipage}[c]{0.74\linewidth}
\includegraphics[width=1\textwidth]{images/raven_half_colored2.jpg}
\captionof{figure}{A collection of ten different RAVEN-FAIR problems. Real target images on the top, and generated images on the bottom. some attributes are allowed to change when no rules are applied on them (correct in green, incorrect in red). }
\label{fig:raven_generation}
\end{minipage}
\hfill
\begin{minipage}[c]{0.24\linewidth}
\captionof{table}{ Recognition results on RAVEN-FAIR}
\label{tab:classification_raven}
\begin{tabular}{@{}l@{~}c@{~}}
\toprule
\parbox[t]{15mm}{Model} & Accuracy \\
\midrule
WReN~\cite{santoro2018measuring} & 30.3 \\
CoPINet~\cite{zhang2019learning} & 50.6\\
LEN~\cite{zheng2019abstract} & 51.0 \\
MRNet~\cite{MRNet} & 86.8\\
Our~$C_o$ & 60.8 \\
\bottomrule
\end{tabular}
\end{minipage}
\end{figure}
\section{Conclusions}
In problems in which the solution space is complex enough, the ability to generate a correct answer is the ultimate test of understanding the question, since one cannot extract hints from any of the potential answers. Our work is the first to address this task in the context of RPMs. The success in this highly semantic task relies on a large number of crucial technologies: applying the reparameterization trick selectively and multiple times, reusing the same networks for encoding and to provide a loss signal, selective backpropagation, and an adaptive variational loss.
\clearpage
\section*{Broader Impact}
The shift from selecting an answer from a closed set to generating an answer could lead to more interpretable methods, since the generated output may reveal information about the underlying inference process. Such networks are, therefore, more useful for validating cognitive models through the implementation of computer models.
The field of answer generation may play a crucial part in automatic tutoring. Ideally, the generated answer would fit the level of the student and allow for automated personalized teaching. Such technologies would play a role in making high-level education accessible to all populations.
\section*{Acknowledgements}
This project has received funding from the European Research Council (ERC) under the European Unions Horizon
2020 research and innovation programme (grant ERC CoG 725974).
\nocite{stgan}
\bibliographystyle{plain}
\section{Links for datasets}
The Procedurally Generated Matrices (PGM) dataset can be found in {\url{https://github.com/deepmind/abstract-reasoning-matrices}} and the RAVEN-FAIR dataset can be found in {\url{https://github.com/yanivbenny/RAVEN_FAIR}}.\\
\section{Calculating FID}
FID calculation was performed on the real target image and the generated image.
For WReN we used the output of the CNN encoder of [32, 4,4] vector, flattened to a 512 size vector.
For MRnet we used the outputs of the perception encoders – low, mid and high features sizes – [256, 1, 1], [128, 5, 5] and [64, 20, 20], and average pooled them to 256, 128 and 64 sized vectors. Than concatenated them to 448 size vector.
And for LEN we used the output of the CNN encoder of [32, 4,4] vector, and average pooled to a 32 size vector.\\
\section{Generated images}
In Fig.~\ref{fig:generation_supp} we present some examples of our method's generated images.
For analyzing the generation results in PGM we must understand for each of the rules, what counts as a good generated image. For 'line type' (a) rules, the lines must stay at the same place, but all other attributes may change, here we get great results. For 'shape position' (b), the shape's position must always be the same, and all other attributes may change, including the shape's size, type, color, and the lines. Here we can see that almost all of the results are great. For 'shape number' (c), the shape's number must always be the same, and all other attributes may change, including the shape's size, type, color, position, and the lines. Here also, we get great results. For 'shape type' (d), the shape's type and number must always be the same, and all other attributes may change, including the shape's size, color, position, and the lines. Here we can see that the results are average. For 'shape color' (e), the shape's color and number must always be the same, and all other attributes may change, including the shape's size, type, position, and the lines. Here we can see that the results are also average. For 'shape size' (f), the shape's size and number must always be the same, and all other attributes may change, including the shape's type, position, color, and the lines. Results are average. For 'line color' (g), the line's color must always be the same, and all other attributes may change, including the shape's attributes and the line's type. Here results look good.
\begin{figure}[t]
\centering
\begin{tabular}{c@{~}c@{~}c@{~}c@{~}c@{~}c@{~}c}
\includegraphics[width=0.75\textwidth]{images/line_type_supp.png} \\
(a) line type\\
\includegraphics[width=0.75\textwidth]{images/shape_position_supp.png}\\
(b) shape position\\
\includegraphics[width=0.75\textwidth]{images/shape_number_supp.png} \\
(c) shape number\\
\includegraphics[width=0.75\textwidth]{images/shape_type_supp.png} \\
(d) shape type\\
\includegraphics[width=0.75\textwidth]{images/shape_color_supp.png} \\
(e) shape color\\
\includegraphics[width=0.75\textwidth]{images/shape_size_supp.png} \\
(f) shape size\\
\includegraphics[width=0.75\textwidth]{images/line_color_supp.png} \\
(g) line color\\
\end{tabular}
\caption{Generation results for selected rules in PGM, each with 10 problems. The top row is the ground truth answer. The bottom is the generated.
(a) line type. (a) shape position. (c) shape number. (d) shape type.(e) shape color. (f) shape size. (g) line color.}
\label{fig:generation_supp}
\end{figure}
\clearpage
\section{Architecture details}
We detail each sub-module used in our method in Tab.~\ref{table:arch_RM}-\ref{table:arch_G_E_D}.
Since some modules re-use the same blocks, Tab.~\ref{table:arch_ResBlocks} details a set of general modules.
\begin{table}[h]
\caption{ResBlocks, with variable number of channels $c$.}\label{table:arch_ResBlocks}
\centering
\begin{tabular}{llccc}
\toprule
Module & layers & parameters & input & output\\
\midrule
\multirow{7}{*}{ResBlock(c)}
& Conv2D & CcK3S1P1 & $x$ \\
& BatchNorm \\
& ReLU \\
& Conv2D & CcK3S1P1 \\
& BatchNorm & & & $x'$ \\
& Residual & & $(x,x')$ & $x''=x+x'$ \\
& ReLU \\
\midrule
\multirow{10}{*}{DResBlock(c)}
& Conv2D & CcK3S1P1 & $x$ \\
& BatchNorm \\
& ReLU \\
& Conv2D & CcK3S1P1 \\
& BatchNorm & & & $x'$ \\
\cmidrule(lr){2-5}
& Conv2D & CcK1S2P0 & $x$ \\
& BatchNorm & & & $x_d$ \\
\cmidrule(lr){2-5}
& Residual & & $(x_d,x')$ & $x''=x_d+x'$ \\
& ReLU \\
\midrule
\multirow{7}{*}{ResBlock1x1(c)}
& Conv2D & CcK1S1P1 & $x$ \\
& BatchNorm \\
& ReLU \\
& Conv2D & CcK1S1P1 \\
& BatchNorm & & & $x'$ \\
& Residual & & $(x,x')$ & $x''=x+x'$ \\
& ReLU \\
\bottomrule
\end{tabular}
\end{table}
\clearpage
\begin{table}[h]
\caption{$E^C$ and $RM$ modules}\label{table:arch_RM}
\centering
\begin{tabular}{lccc}
\toprule
Module & layers & input & output\\
\midrule
\multirow{6}{*}{$E^C_h$} & Conv2d(1, 32, kernel size=7, stride=2, padding=3, bias=False) & $I_{i}$ & - \\
& BatchNorm2d(32) & - & - \\
& ReLU & - & -\\
& Conv2d(32, 64, kernel size=3, stride=2, padding=1, bias=False) & - &\\
& BatchNorm2d(64) & - & - \\
& ReLU & - & $e^i_h$ \\
\midrule
\multirow{6}{*}{$E^C_m$} & Conv2d(64, 64, kernel size=3, stride=2, padding=1, bias=False) & $e^i_h$ & - \\
& BatchNorm2d(64) & - & - \\
& ReLU & - & - \\
& Conv2d(64, 128, kernel size=3, stride=2, padding=1, bias=False) & - & - \\
& BatchNorm2d(128) & - & - \\
& ReLU & - & $e^i_m$ \\
\midrule
\multirow{6}{*}{$E^C_l$} & nn.Conv2d(128, 128, kernel size=3, stride=2, padding=1, bias=False) & $e^i_m$ & - \\
& BatchNorm2d(128) & - & - \\
& ReLU & - & - \\
& Conv2d(128, 256, kernel size=3, stride=2, padding=0, bias=False) & - & - \\
& BatchNorm2d(256) & - & - \\
& ReLU & - & $e^i_l$ \\
\midrule
\multirow{6}{*}{$RM_h$} & Reshape to (3 * 64, 20, 20) & - & - \\
& Conv2d(3*64, 64, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(64, 64) & - & - \\
& ResBlock(64, 64) & - & - \\
& Conv2d(64, 64, ker size=3, st=1, pad=1, bias=False) & - & - \\
& BatchNorm2d(64) & - & $r^1_h/r^2_h/c^1_h/c^2_h$ \\
\midrule
\multirow{6}{*}{$RM_m$} & Reshape to (3 * 128, 5, 5) & - & - \\
& Conv2d(3 * 128, 128, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(128, 128) & - & - \\
& ResBlock(128, 128) & - & - \\
& Conv2d(128, 128, ker size=3, st=1, pad=1, bias=False) & - & - \\
& BatchNorm2d(128) & - & $r^1_m/r^2_m/c^1_m/c^2_m$ \\
\bottomrule
\multirow{6}{*}{$RM_l$} & Reshape to (3 * 256, 1, 1) & - & - \\
& Conv2d(3*256, 256, ker size=1, st=1, pad=1, bias=False) & - & - \\
& ResBlock1x1(256, 256) & - & - \\
& ResBlock1x1(256, 256) & - & - \\
& Conv2d(256, 256, ker size=3, st=1, pad=1, bias=False) & - & - \\
& BatchNorm2d(256) & - & $r^1_l/r^2_l/c^1_l/c^2_l$ \\
\midrule
\end{tabular}
\end{table}
\begin{table}[h]
\caption{ $P$ and $C_O$ modules}\label{table:arch_general_p}
\centering
\begin{tabular}{lccc}
\toprule
Module & layers & input & output\\
\midrule
\multirow{3}{*}{$P_h$} & Reshape to (3 * 64, 20, 20) & $e^7_h$, $e^8_h$, $q_h$ /$e^3_h$, $e^6_h$, $q_h$ & - \\
& Conv2d(3*64, 64, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(64, 64) & - & $x_h$ \\
\midrule
\multirow{3}{*}{$P_m$} & Reshape to (3 * 128, 5, 5) & $e^7_m$, $e^8_m$, $q_m$ /$e^3_m$, $e^6_m$, $q_m$ & - \\
& Conv2d(3*128, 128, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(128, 128) & - & $x_m$ \\
\midrule
\multirow{3}{*}{$P_l$} & Reshape to (3 * 256, 1, 1) & $e^7_l$, $e^8_l$, $q_l$ /$e^3_l$, $e^6_l$, $q_l$ & - \\
& Conv2d(3*256, 256, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(256, 256) & - & $x_l$ \\
\midrule
\multirow{6}{*}{$C^h_O$} & Reshape to (2 * 64, 20, 20) & cat($x_h$, $e^{a_i}_h$) & - \\
& Conv2d(2*64, 64, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(64, 64) & - & - \\
& DResBlock(64,2 *64, stride=2) & - & - \\
& DResBlock(2 *64,128, stride=2) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_h'$ \\
\midrule
\multirow{6}{*}{$C^m_O$} & Reshape to (2 * 128, 5, 5) & cat($x_m$, $e^{a_i}_m$) & - \\
& Conv2d(2*128, 128, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(128, 128) & - & - \\
& DResBlock(128,2*128, stride=2) & - & - \\
& DResBlock(2*128,128, stride=2) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_m'$ \\
\midrule
\multirow{8}{*}{$C^l_O$} & Reshape to (3 * 256, 1, 1) & cat($x_l$, $e^{a_i}_l$) & - \\
& Conv2d(2*256, 256, ker size=3, st=1, pad=1, bias=False) & - & - \\
& ResBlock(256, 256) & - & - \\
& Conv2d(256, 128, ker size=1, st=1, bias=False) & - & - \\
& BatchNorm2d(128) & - & - \\
& ReLU & - & - \\
& ResBlock1x1(128, 128) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_l'$ \\
\midrule
\multirow{7}{*}{final $C_O$} & Linear(128*3, 256, bias=False) & cat($x_h'$, $x_m'$, $x_l'$) & - \\
& BatchNorm1d(256) & - & - \\
& ReLU & - & - \\
& Linear(256, 128, bias=False) & - & - \\
& BatchNorm1d(128) & - & - \\
& ReLU & - & - \\
& Linear(128, 1, bias=True)) & - & $y_i$ \\
\midrule
\end{tabular}
\end{table}
\begin{table}[h]
\caption{$C_M$ modules}\label{table:arch_general_cm}
\centering
\begin{tabular}{lccc}
\toprule
Module & layers & input & output\\
\midrule
\multirow{3}{*}{$C^h_M$}
& DResBlock(64,2*64, stride=2) & $x_h$ & - \\
& DResBlock(2*64,128, stride=2) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_h'$ \\
\midrule
\multirow{3}{*}{$C^m_M$}
& DResBlock(128,2*128, stride=2) & $x_m$ & - \\
& DResBlock(2*128,128, stride=2) & - & - \\
& AdaptiveAvgPool2d((1, 1)) & - & $x_m'$ \\
\midrule
\multirow{6}{*}{$C^l_M$}
& ResBlock(256, 256) & $x_l$ & - \\
& Conv2d(256, 128, ker size=1, st=1, bias=False) & - & - \\
& BatchNorm2d(128) & - & - \\
& ReLU & - & - \\
& ResBlock1x1(128, 128) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_l'$ \\
\midrule
\multirow{7}{*}{$C_M$} & Linear(128*3, 256, bias=False) & cat($x_h'$, $x_m'$, $x_l'$) & - \\
& BatchNorm1d(256) & - & - \\
& ReLU & - & - \\
& Linear(256, 128, bias=False) & - & - \\
& BatchNorm1d(128) & - & - \\
& ReLU & - & - \\
& Linear(128, 12, bias=True)) & - & $\psi$ \\
\midrule
\end{tabular}
\end{table}
\begin{table}[h]
\caption{ $R$ modules}\label{table:arch_general_R}
\centering
\begin{tabular}{lccc}
\toprule
Module & layers & input & output\\
\midrule
\multirow{3}{*}{$R_h$}
& DResBlock(64,2*64, stride=2) & $x_h$ & - \\
& DResBlock(2*64,128, stride=2) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_h'$ \\
\midrule
\multirow{3}{*}{$R_m$}
& DResBlock(128,2*128, stride=2) & $x_m$ & - \\
& DResBlock(2*128,128, stride=2) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_m'$ \\
\midrule
\multirow{5}{*}{$R_l$}
& Conv2d(256, 128, ker size=1, st=1, bias=False) & $x_l$ & - \\
& BatchNorm2d(128) & - & - \\
& ReLU & - & - \\
& ResBlock1x1(128, 128) & - & - \\
&AdaptiveAvgPool2d((1, 1)) & - & $x_l'$ \\
\midrule
\multirow{5}{*}{T} & Linear(128*3, 128, bias=False) & x = cat($x_h'$, $x_m'$, $x_l'$) & - \\
& ReLU & - & - \\
& Linear(128, 128, bias=False) & - & - \\
& ReLU & - & - \\
& Linear(128, 128, bias=False)) & - & $mu_g$,$sigma_g$ \\
\midrule
\end{tabular}
\end{table}
\begin{table}[h]
\caption{G, E and D}\label{table:arch_G_E_D}
\centering
\begin{tabular}{lccc}
\toprule
Module & layers & input & output\\
\midrule
\multirow{9}{*}{G}
&ConvTranspose2d(64, 64*8,
ker size=5, st=1, pad=0, bias=False) & $h_g = \mu_g + \sigma_g \cdot z'$ & - \\
&BatchNorm2d(64*8)& - & - \\
&ConvTranspose2d(64*8, 64*4,
ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*4)& - & - \\
&ConvTranspose2d(64*4, 64*2,
ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*2)& - & - \\
&ConvTranspose2d(64*2, 64*1,
ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*1)& - & - \\
&ConvTranspose2d(64*1, 1,
ker size=4, st=2, pad=1, bias=False) & - & $I_g$ \\
\midrule
\multirow{13}{*}{E}
&Conv2d(1, 32, kernel size=3, stride=2) & $I_{a_i}$ & - \\
&BatchNorm2d(32) & - & - \\
&ReLU & - & - \\
&Conv2d(32, 32, kernel size=3, stride=2) & - & - \\
&BatchNorm2d(32) & - & - \\
&ReLU & - & - \\
&Conv2d(32, 32, kernel size=3, stride=2 & - & - \\
&BatchNorm2d(32)& - & - \\
&ReLU & - & - \\
&Conv2d(32, 32, kernel size=3, stride=2 & - & - \\
&BatchNorm2d(32) & - & - \\
&ReLU & - & - \\
&Linear(32*4*4, 64*2) & - & $\mu^{a_i}_v$, $\sigma^{a_i}_v$ \\
\midrule
\multirow{15}{*}{D}
&Conv2d(1, 64, ker size=4, st=2, pad=1, bias=False) & $I_{a^{*}}/I_g$ & - \\
&leaky relu & - & - \\
&Conv2d(64,64*2, ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*2) & - & - \\
&leaky relu & - & - \\
&Conv2d(64*2,64*4, ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*4) & - & - \\
&leaky relu & - & - \\
&Conv2d(64*4,64*8, ker size=4, st=2, pad=1, bias=False) & - & - \\
&BatchNorm2d(64*8) & - & - \\
&leaky relu & - & - \\
&Conv2d(64*8, 1, ker size=4, st=1, pad=0, bias=False) & - & - \\
&leaky relu & - & - \\
&Conv2d(1, 1, ker size=2, st=1, pad=0, bias=False) & - & - \\
&sigmoid & - & D out \\
\midrule
\end{tabular}
\end{table} |
2,869,038,154,832 | arxiv | \section{Introduction}
The focus of the present paper is to better detect significant variables in a linear model, with the possibility that the number of explanatory variables is greater than the number of observations and when the error distribution is asymmetrical or heavy-tailed. For this type of law, the using of the least squares (LS) estimation method is not appropriate for the estimator accuracy. One possibility would be to use the quantile method, but it has the disadvantage that the loss function is not derivable, which complicates the theoretical study but also computational methods. A very interesting possibility is to consider the expectile method, introduced by \cite{Newey-Powell.87}, under assumption that the first moments of $\varepsilon$ exist. This method has the advantage that the loss function is differentiable, which simplifies the theoretical study and facilitates the numerical calculation. \\
In application fields (genetics, chemistry, biology, industry, finance), with the development in recent years of storage and/or measurement tools, we are confronted to study the influence of a very large number of variables on a studied process. That is why, let us consider in the present work the following linear model:
\begin{equation}
\label{eq1}
Y_i=\mathbf{X}_i^t \eb+\varepsilon_i, \qquad i=1, \cdots , n,
\end{equation}
with the vector parameter $\eb \in \mathbb{R}^p$ and $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$ its true value (unknown). The size $p$ of $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$ can depend on $n$ but the components $\eb^0_j$ don't depend on $n$, for any $j=1, \cdots , p$. The vector $\mathbf{X}_i=(X_{i1}, \cdots, X_{ip})$ contains the values of the $p$ explanatory deterministic variables and $Y_i$ the values of response variable for observation $i$. The values $(Y_i, \mathbf{X}_i)$ are known for any $i=1, \cdots , n $. Throughout the paper, all vectors are column. If $p$ is very large, in order to find the explanatory variables that significantly influence the response variable $Y$, an automatic selection should be made without performing hypothesis tests. When $p$ is very large, the use of hypothesis tests is not appropriate because they provide results with great variability and the significant explanatory variable final choice is not optimal (see \cite{Breiman-96}). \\
For model (\ref{eq1}), let then the index set of the non-null true parameters,
\[
{\cal A} \equiv \big\{ j \in \{ 1, \cdots, p\}; \, \beta^0_j \neq 0\big\}.
\]
Since $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$ is unknown then, the set ${\cal A}$ is also unknown. We assume, without reducing generality, that ${\cal A}=\{1, \cdots, p_0 \}$ and its complementary set is ${\cal A}^c=\{p_0+1, \cdots , p \}$, with $p_0 \leq p$. Hence the first $p_0$ explanatory variables have a significant influence on the response variable and the last $p-p_0$ variables are irrelevant. Thus, the true parameter vector can be written as $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}=\big(\eb^0_{\cal A}, \eb^0_{{\cal A}^c}\big)=\big(\eb^0_{\cal A},\textbf{0}_{p-p_0}\big)$, with $\textbf{0}_{p-p_0}$ a $(p-p_0)$-vector with all components zero. The number $p_0$ of the nonzero coefficients can depends on $n$. \\
For a vector $\eb$ we use the notational convention $\eb_{\cal A}$ for its subvector containing the corresponding components of ${\cal A}$. For $i=1, \cdots, n$, we denote by $ \mathbf{X}_{i,{\cal A}} $ the $p_0$-vector with the components $X_{ij}$, $j=1, \cdots, p_0$. We also use the notation $|{\cal A}|$ or $Card({\cal A})$ for the cardinality of ${\cal A}$. \\
In order to find the elements of ${\cal A}$, one of the most used techniques is the adaptive LASSO method, introduced by \cite{Zou.06} for $p$ fixed by penalizing the squares sum with an weighted $L_1$ penalty. This type of parameter estimator is interesting if it satisfies the oracle properties, i.e. the two following properties occur:
\begin{itemize}
\item \textit{sparsity of estimation}: the non-zero parameters are estimated as non-zero and the null parameters are shrunk directly as 0, with a probability converging to 1 when $n \rightarrow \infty$;
\item \textit{asymptotic normality} of non-zero parameter estimators.
\end{itemize}
In order to distinguish between different types of adaptive LASSO estimators, we will use the term "adaptive LASSO LS-estimator" to refer to the minimizer of the LS sum penalized with adaptive LASSO. \\
Give some papers from very rich literature that consider the adaptive LASSO LS-estimator when $p$ depends on $n$, with the possibility that $p>n$ are: \cite{Huang-Ma-Zhang.08}, \cite{Wang-Kulasekera.12}, \cite{Yang-Wu.16}. If the moments of the errors do not exist or the distribution of $\varepsilon$ presents outliers, then the LS framework is not appropriate. One possibility is to consider the quantile model with the adaptive LASSO penalty. The recent literature is also very rich: \cite{FFB14}, \cite{Kaul-Koul.15}, \cite{Tang-Song-Wang-Zhu.13}, \cite{Ciuperca-18}, \cite{Zheng-Gallagher-Kulasekera-13}, \cite{Zheng-Peng-He.15}, to give just a few examples. As stated before, the loss function for quantile method not differentiable complicates the theoretical study and its computational implementation, which is a very important aspect in high-dimensionality.
Let us mention another work of \cite{Fan-Li-Wang.17} which is also devoted to the high dimensional regression in absence of symmetry of the errors and which proposes a penalized Huber loss. Only that the Huber loss function is not differentiable.
From where the idea of considering the expectile loss function for an high-dimensional model. In order to introduce the expectile method, for a fixed $\tau \in (0,1)$, let us consider the function $\rho_\tau(.)$ of the form
\[\rho_\tau(x)=|\tau - 1\!\!1_{x <0}|x^2, \qquad \textrm{with} \quad x \in \mathbb{R}.
\]
\hspace*{0.5cm} For the error and the design of model (\ref{eq1}) we make the following basic assumptions. \\
The errors $\varepsilon_i$ satisfy the following assumption:\\
\textbf{(A1)} $(\varepsilon_i)_{1 \leqslant i \leqslant n}$ are i.i.d. such that $\mathbb{E}[\varepsilon^4_i]< \infty$ and $
\mathbb{E}[\varepsilon\big(\tau 1\!\!1_{\varepsilon >0}+ (1-\tau)1\!\!1_{\varepsilon<0} \big)]=0$, that is its $\tau$-th expectile is zero: $\mathbb{E}[\rho'_\tau(\varepsilon)]=0$.\\
While, the design $(\mathbf{X}_i)_{1 \leqslant i \leqslant n}$ satisfies the following assumption:\\
\textbf{(A2)} there exists two positive constants $m_0, M_0$ such that, $0 < m_0 \leq \mu_{min}\big(n^{-1} \sum^n_{i=1} \mathbf{X}_i \mathbf{X}_i^t\big)$ $\leq \mu_{max}\big(n^{-1} \sum^n_{i=1} \mathbf{X}_i \mathbf{X}_i^t\big) \leq M_0 < \infty$.\\
For a positive definite matrix, we denote by $\mu_{min}(.)$ and $\mu_{max}(.)$ its largest and smallest eigenvalues, respectively. Let us consider $\varepsilon$ the generic variable for the sequence $(\varepsilon_i)_{1 \leqslant i \leqslant n}$.
Assumption (A1) is commonly required for the expectile models, see \cite{Zhao-Chen-Zhang.18}, \cite{Gu-Zou.16}, \cite{Liao-Park-Choi.18}, \cite{Newey-Powell.87}, while assumption (A2) is standard in linear model for the parameter identifiability (considered also by \cite{Gao-Huang.10}, \cite{Wang-Wang.14}, \cite{Fan-Li-Wang.17}, \cite{Zou-Zhang-09}).
Other assumptions will be stated about design in the following two sections, depending of the size $p$ which varies in turn with $n$.\\
Quite in general, it is wise to use the expectile method when the moments of $\varepsilon$ exist but its distribution is asymmetric or heavy-tailed. For $\tau=0.5$, we get the classical method of least squares. \\
For model (\ref{eq1}), consider the expectile process
\[
Q_n(\eb)\equiv \sum^n_{i=1} \rho_\tau (Y_i - \mathbf{X}_i^t \eb)
\]
and the one with LASSO adaptive penalty:
\begin{equation}
\label{Rnb}
R_n(\eb)\equiv \sum^n_{i=1} \rho_\tau (Y_i - \mathbf{X}_i^t \eb)+n \lambda_n \sum^p_{j=1} \widehat \omega_{n,j} |\beta_j|.
\end{equation}
The adaptive weights $ \widehat \omega_{n,j}$ will be defined later depending on whether $p$ is smaller or larger than $n$. The tuning parameter $\lambda_n$ is a positive deterministic sequence which together $\widehat \omega_{n,j}$ controls the overall model complexity. Hence, we should choose $\lambda_n$ and $\widehat \omega_{n,j}$ such that $n \lambda_n \widehat \omega_{n,j} \overset{\mathbb{P}} {\underset{n \rightarrow \infty}{\longrightarrow}} 0$ for non-null parameters and $n \lambda_n \widehat \omega_{n,j} \overset{\mathbb{P}} {\underset{n \rightarrow \infty}{\longrightarrow}} \infty $ for null coefficients.
In order to automatically detect the null and non-zero components of $\eb$, we proceed in a similar way as for the adaptive LASSO LS estimation introduced by \cite{Zou.06}, and we consider the adaptive LASSO expectile estimator of $\eb$:
\begin{equation}
\label{hbetan}
\widehat \eb_n \equiv \argmin_{\eb \in \mathbb{R}} R_n(\eb).
\end{equation}
The components of $\widehat \eb_n $ are $\widehat \eb_n =\big( \widehat{\beta}_{n,1}, \cdots , \widehat{\beta}_{n,p} \big)$. Similarly to ${\cal A}$, let's define the index set:
\[
\widehat{{\cal A}}_n \equiv \{ j \in \{1, \cdots , p \}; \; \widehat{\beta}_{n,j} \neq 0 \},
\]
with the non-zero components of the adaptive LASSO expectile estimator.\\
The estimator $\widehat \eb_n$ will satisfy the oracle properties if:
\begin{itemize}
\item \textit{sparsity}: $\lim_{n \rightarrow \infty} \mathbb{P}\big[{\cal A} = \widehat{{\cal A}}_n \big]=1$.
\item \textit{asymptotic normality}: for any vector $\mathbf{{u}} \in \mathbb{R}^{p_0}$ with bounded norm, we have that:\\ $\sqrt{n} (\mathbf{{u}}^t \eU^{-1}_{n,{\cal A}} \mathbf{{u}})^{-1/2} \mathbf{{u}}^t ( \widehat{\eb}_n - \eb^0)_{\cal{A}} $ converges in distribution to a zero-mean Gaussian law, with the $p_0$-squared matrix: $\eU_{n,{\cal A}} \equiv n^{-1} \sum^n_{i=1} \mathbf{X}_{i,{\cal A}} \mathbf{X}_{i,{\cal A}}^t$.
\end{itemize}
The asymptotic distribution of the parameter estimators can be used to construct asymptotic hypothesis tests and confidence intervals for the non-null parameters.
For $p$ fixed, the properties of the estimator $\widehat \eb_n$ have been studied by \cite{Liao-Park-Choi.18} where it is shown that the convergence rate of $\widehat{\eb}_n$ towards $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$ is of order $n^{-1/2}$ and that $\widehat \eb_n$ satisfies the oracle properties. The case $p$ fixed was also studied by \cite{Zhao-Zhang.18} which consider a penalized linear expectile regression with SCAD penalty function and obtain a $n^{-1/2}$-consistent estimator with oracle properties. In the present paper we consider $p$ depends on $n$, more precisely, of the form $p=O(n^c)$, with the constant $c >0$. The size $p_0$ and the set ${\cal A}$ can depend on $n$.
The case when $p$ depends on $n$ was also considered in \cite{Zhao-Chen-Zhang.18} by considering the SCAD penalty for the expectile process. They propose an algorithm that converges, with probability converging to one as $n \rightarrow \infty$, to the oracle estimator after several iterations.
Always for $p$ depending on $n$, especially when $p>n$ and for $(\varepsilon_i)$ sous-Gaussian errors, \cite{Gu-Zou.16} penalize the expectile process with LASSO or nonconvex penalties. They find the convergence rate of the penalized estimator, propose an algorithm for finding this estimator and implement the algorithm in the R language in package \textit{SALES}.\\
The paper of \cite{Spiegel-Sobotka-Kneib.17} introduces several approaches depending on selection criteria and shrinkage methods to perform model selection in semiparametric expectile regression. \\
Give some general notations. For a vector $\textbf{v}$, we denote its transpose by $\textbf{v}^t$, by $\|\textbf{v}\|_1$, $\|\textbf{v}\|_2$ and $\|\textbf{v}\|_\infty$ the $L_1$, $L_2$, $L_\infty$ norms, respectively. The number $p$ of the explanatory variables and $p_0$ of the sugnificant variables can depend on $n$, but for convenience, we do not write the subscript $n$. Throughout the paper, $C$ denotes a positive generic constant not dependent on $n$, which may take a different value in different formula. \\
In order to study the properties of the adapted LASSO expectile estimator $\widehat \eb_n$, we introduce the following function, using the same notations in \cite{Liao-Park-Choi.18}:
\begin{align*}
g_\tau(x) &\equiv \rho'_\tau(x-t) |_{t=0}=2 \tau x 1\!\!1_{x \geq 0}+2(1-\tau)x 1\!\!1_{x<0}, \\
h_\tau(x)& \equiv \rho''_\tau(x-t)_{t=0}=2 \tau 1\!\!1_{x \geq 0}+2(1-\tau) 1\!\!1_{x<0}.
\end{align*}
\hspace*{0.5cm} The paper is organized as follow. In Section \ref{casec<1} we study the asymptotic behavior of the adaptive LASSO expectile estimator when $p=O(n^c)$, with $0 \leq c <1$. We obtain the convergence rate of the $\widehat{\eb}_n$ and the oracle properties. A similar study is realized in Section \ref{casec>1}, when $p \geq n$. In Section \ref{Simus}, a simulation study and an application to real data are presented. All proof are relegated in Section \ref{Proofs}.
\section{Case $c <1$, parameter number less than the sample size}
\label{casec<1}
In this section we study the asymptotic behavior of the adaptive LASSO expectile estimator when the number $p$ of model parameter is $p=O(n^c)$, with $0 \leq c <1$. If $c=0$, that is $p$ fixed, then we get the particular case studied by \cite{Liao-Park-Choi.18}. \\
\hspace*{0.5cm} An additional assumption to (A2) on the design is requested:\\
\textbf{(A3)} $p^{1/2} n^{-1/2} \max_{1 \leqslant i \leqslant n}\|\mathbf{X}_i\|_2 {\underset{n \rightarrow \infty}{\longrightarrow}} 0$.\\
Assumption (A3) is common in works that consider $p$ dependent on $n$ when $p<n$, see for example \cite{Ciuperca-18}. Always for a linear model with the number of parameters to order $n^c$, with $0 \leq c <1$, \cite{Zou-Zhang-09} consider a stronger assumption for design: $\lim_{n \rightarrow \infty} n^{-1} \max_{1 \leqslant i \leqslant n} \| \mathbf{X}_i\|^2_2=0$. \\
Because $p< n$, the regression parameters are identifiable and we can calculated the expectile estimator:
\[
\widetilde \eb_n \equiv \argmin_{\eb \in \mathbb{R}} Q_n(\eb),
\]
the components of $\widetilde \eb_n$ being $\widetilde \eb_n=\big(\widetilde{\beta}_{n,1}, \cdots , \widetilde{\beta}_{n,p} \big)$. This estimator will intervene in the adaptive weight of the penalty, $\widehat \omega_{n,j} =|\widetilde \beta_{n,j}|^{-\gamma}$, for $j =1, \cdots , p$, conditions on the constant $\gamma >0$ will be specified in Theorem \ref{Theorem 2SPL}.
By the following theorem we obtain the convergence rate of the expectile and adaptive LASSO expectile estimators. We obtain that the convergence rate depends on the size $p$ of the vector $\eb$.
\begin{theorem}
\label{th_vconv}
Under assumptions (A1)-(A3) we have:\\
(i) $\| \widetilde{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}} \|_2=O_{\mathbb{P}}\pth{\sqrt{\frac{p}{n}}}$.\\
(ii) If the tuning parameter sequence $(\lambda_n)_{n \in \mathbb{N}}$ satisfies $ p_0^{1/2} n^{(1-c)/2} \lambda_n \rightarrow 0$, as $n \rightarrow \infty$, then, $\| \widehat{\eb}_n- \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\|_2=O_{\mathbb{P}}\pth{\sqrt{\frac{p}{n}}}$.
\end{theorem}
Theorem \ref{th_vconv} provides that the expectile and adaptive LASSO expectile estimators have the same convergence rate. Concerning the adaptive LASSO expectile estimator, the same convergence rate has been obtained for other adaptive LASSO estimators: by the likelihood method for a generalized linear model when $p<n$ in \cite{Wang-Wang.14}, by the least squares approximation method in \cite{Leng-Li.10}.\\
By the following theorem, considering a supplementary condition on $\lambda_n$, $c$ and $\gamma$, in addition to that considered for the convergence rate in Theorem \ref{th_vconv}, we show that the adaptive LASSO expectile estimator $\widehat{\eb}_n$ satisfies the oracle properties. If $\tau=0.5$, that is, for the adaptive LASSO LS-estimator, the variance of the normal limit law is the variance of $\varepsilon$. In fact, we obtained for $\tau=0.5$, the same asymptotic normality as in \cite{Zou-Zhang-09}.
\begin{theorem}
\label{Theorem 2SPL} Suppose that assumptions (A1)-(A3) hold and that the tuning parameter satisfies $\lambda_n n^{(1-c)(1+\gamma)/2} \rightarrow \infty$, $ p_0^{1/2} n^{(1-c)/2} \lambda_n \rightarrow 0$, as $n \rightarrow \infty$. Then:\\
(i) $\mathbb{P} \big[\widehat{\cal A}_n={\cal A}\big]\rightarrow 1$, for $n \rightarrow \infty $.\\
(ii) For any vector $\mathbf{{u}}$ of size $p_0$ such that $\| \mathbf{{u}}\|_2=1$, we have: $n^{1/2} (\mathbf{{u}}^t \eU^{-1}_{n,{\cal A}} \mathbf{{u}})^{-1/2} \mathbf{{u}}^t ( \widehat{\eb}_n - \eb^0)_{\cal{A}} \overset{\cal L} {\underset{n \rightarrow \infty}{\longrightarrow}} {\cal N}\big(0, \frac{\mathbb{V}\mbox{ar}\,[g(\varepsilon)]}{\mathbb{E}^2[h(\varepsilon)]} \big)$.
\end{theorem}
The convergence rate of $ \widehat{\eb}_n$ does not depend on the power $\gamma$, but otherwise, for holding the oracle properties, the choice of $\gamma$ is very important.
Concerning the suppositions and results stated in Theorem \ref{Theorem 2SPL}, let's make some remarks on the regularization parameter $\lambda_n$, the constants $c$, $\gamma$ and the sizes $p_0$, $p$.
\begin{remark}
1) If $p_0=O(p)$, then for that $\lambda_n n^{(1-c)(1+\gamma)/2} \rightarrow \infty$ and $\lambda_n p_0^{1/2} n^{(1-c)/2} \rightarrow 0$ occur, we must choose the constant $\gamma$ and sequence $(\lambda_n)$ such that: $\gamma > c/(1-c)$ and $n^{-1/2}\lambda_n \rightarrow 0$, as $n \rightarrow \infty$.\\
2) If $p_0=O(1)$, we must choose the constant $\gamma >0$ and the tuning parameter such that $n^{(1-c)/2}\lambda_n \rightarrow 0$, as $n \rightarrow \infty$.\\
3) For that $\lambda_n p_0^{1/2} n^{(1-c)/2} \rightarrow 0$ holds, it is necessary that $\lambda_n \rightarrow 0$, as $n \rightarrow \infty$.\\
4) If $c=0$ then the conditions on $\lambda_n$ become: $n^{1/2}\lambda_n \rightarrow 0$ and $ n^{(1+\gamma)/2} \lambda_n \rightarrow \infty$, as $n \rightarrow \infty$, conditions considered by \cite{Liao-Park-Choi.18} for a linear model with $p$ fixed. We also find the same variance of Gaussian distribution in \cite{Liao-Park-Choi.18}.
\end{remark}
\section{Case $c \geq 1$, parameter number greater than the sample size}
\label{casec>1}
In this section, after we propose an adaptive weight, we study the asymptotic behavior of the estimator $\widehat{\eb}_n$ when the number of regressors exceeds the number of observations. \\
\hspace*{0.5cm} Since we consider now that $p \geq n$, instead of assumption (A3), we consider: \\
\textbf{(A4)} There exists a constant $M>0$ such that $\max_{1 \leqslant i \leqslant n}\| \mathbf{X}_i \|_{\infty} < M$.\\
\noindent The same assumption (A4) was considered for a generalized linear model when $p >n$ in \cite{Wang-Wang.14} where the adaptive LASSO likelihood method is proposed.\\
The asymptotic properties of the adaptive LASSO LS-estimators in a linear model, when the number of the explanatory variables is greater than $n$, have been studied by \cite{Huang-Ma-Zhang.08}. They show that if a reasonable initial estimator is available, estimator that enters in the adaptive weight of the penalty, then their adaptive LASSO LS-estimator satisfies the oracle properties.\\
Recall that the expectile estimator is not consistent when $p>n$ and then it can't be used in the weights $\widehat{\omega}_{n,j}$ (see for example \cite{Huang-Ma-Zhang.08}). Then, when $p>n$, we propose in this section to take as adaptive weight $\widehat{\omega}_{n,j}=\min(|\overset{\vee}{\beta}_{n,j}|^{-\gamma} , n^{1/2})$, with $\overset{\vee}{\beta}_{n,j}$ an estimator of $\eb^0_j$ consistent with $a_n \rightarrow 0$ the convergence rate: $\| \overset{\vee}{\eb}_n -\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}} \|_2=O_{\mathbb{P}}(a_n)$. If the estimator $\overset{\vee}{\beta}_{n,j}$ take the value 0, then we consider $n^{1/2}$ as adaptive weight.
An example of such estimator is the LASSO expectile estimator, proposed by \cite{Gu-Zou.16}, defined as:
\[ \argmin_{\eb \in \mathbb{R}^p}\big(n^{-1} \sum^n_{i=1} \rho_\tau(Y_i - \mathbf{X}_i^t \eb)+\nu_n \| \eb\|_1 \big),
\]
with the deterministic sequence $\nu_n \in (0, \infty)$, $\nu_n \rightarrow 0$ as $n \rightarrow \infty$. If $\varepsilon_i$ is sub-Gaussian and $\mathbb{E}[g(\varepsilon)]=0$, under our assumptions (A2), (A4), if $\kappa =\inf_{\textbf{d} \in {\cal C}} \frac{\|\mathbb{X} \textbf{d} \|^2_2}{\|n\textbf{d}\|^2_2} \in (0, \infty)$, with $\mathbb{X}$ the matrix $n \times p $ of design and the set ${\cal C} \equiv \{\textbf{d} \in \mathbb{R}^p ; \|\textbf{d}_{{\cal A}^c}\|_1 \leq 3 \|\textbf{d}_{{\cal A}}\|_1 \neq 0\}$, then $\|\overset{\vee}{{\eb}}_n -\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}} \|_2=O_{\mathbb{P}}\big(p_0^{1/2} \nu_n \big)$. Thus, the sequence $(a_n)$ is in this case $a_n=p_0^{1/2} \nu_n $ (see Theorem 1 of \cite{Gu-Zou.16}).\\
The form of the random process $R_n(\eb)$ of (\ref{Rnb}) and the adaptive LASSO expectile estimator $\widehat{\eb}_n$ of (\ref{hbetan}) remain the same, only the adaptive weight changes. It would be desirable for $\widehat{\eb}_n$ to satisfy the oracle proprties.
For the sparsity property of $\widehat{\eb}_n$ its convergence in $L_1$ norm is required. In the following theorem, $(b_n)_{n \in \mathbb{N}}$ is a deterministic sequence converging to 0 as $n \rightarrow \infty$.
\begin{theorem}
\label{th_vconvn}
Under assumptions (A1), (A2), (A4), the tuning parameter $(\lambda_n)_{n \in \mathbb{N}}$ and sequence $(b_n)$ satisfying $\lambda_n p_0^{1/2} b^{-1}_n \rightarrow 0$, as $n \rightarrow \infty$, we have, $\| \widehat{\eb}_n- \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\|_1=O_{\mathbb{P}}\pth{b_n}$.
\end{theorem}
The result of Theorem \ref{th_vconvn} indicates that the convergence rate of the adaptive LASSO expectile estimator $\widehat{\eb}_n$ depends on the chosen sequence $(\lambda_n)_{n \in \mathbb{N}}$. On the other hand, the convergence rate of $\widehat{\eb}_n$ don't depend on the convergence rate $(a_n)$ of the estimator $\overset{\vee}{{\eb}}_n$. The only thing that matters (see relation (\ref{PPn}) of the proof in Section \ref{Proofs}) is that $\overset{\vee}{{\eb}}_n$ converges in probability to $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$.\\
The result of Theorem \ref{th_vconvn} now allows us to state oracle properties. Still in the case $p>n$, but for a quantile model, \cite{Zheng-Gallagher-Kulasekera-13} obtains that the convergence rate, in $L_2$ norm, of the adaptive LASSO quantile estimator is $(p_0 /n)^{-1/2}$ and that it also satisfies the oracle properties. The sparsity of the adaptive LASSO quantile estimator will be shown in our Section \ref{Simus} by simulations, where we obtain that compared to the adaptive LASSO expectile estimator, it would be necessary to have a larger number $n$ of the observations when the model errors have an asymmetric distribution. From where, a supplementary interest in considering the expectile method instead of the quantile.
\begin{theorem}
\label{Theorem 2SPLn} Suppose that assumptions (A1), (A2), (A4) hold, the tuning parameter $(\lambda_n)$ and sequence $(b_n)$ satisfy $\lambda_n p_0^{1/2} b^{-1}_n \rightarrow 0$, $\lambda_n b_n^{-1}\min\big(n^{1/2} , a_n^{-\gamma}\big) \rightarrow \infty$, as $n \rightarrow \infty$. Then:\\
(i) $\mathbb{P} \big[\widehat{\cal A}_n = {\cal A}\big]\rightarrow 1$, for $n \rightarrow \infty $.\\
(ii) For any vector $\mathbf{{u}}$ of size $p_0$ such that $\| \mathbf{{u}}\|_1=1$, we have $n^{1/2} (\mathbf{{u}}^t \eU^{-1}_{n,{\cal A}} \mathbf{{u}})^{-1/2} \mathbf{{u}}^t ( \widehat{\eb}_n - \eb^0)_{\cal{A}} \overset{\cal L} {\underset{n \rightarrow \infty}{\longrightarrow}} {\cal N}\big(0, \frac{\mathbb{V}\mbox{ar}\,[g(\varepsilon)]}{\mathbb{E}^2[h(\varepsilon)]} \big)$.
\end{theorem}
Hence, even if the parameter number of the model is larger than the observation number, the variance of the normal limit distribution is the same as that obtained when $p<n$. As for the case $p < n$, studied in Section \ref{casec<1}, the convergence rate of the adaptive LASSO espectile estimator don't depend on the power $\gamma$ in the adaptive weight. However, $\gamma$ intervenes in the imposed conditions so that the oracle properties are satisfied. If $\tau=0.5$, that is for the adaptive LASSO LS-estimator, we obtained the same asymptotic normal distribution as that given by \cite{Huang-Ma-Zhang.08}) for their adaptive LASSO LS-estimator.\\
Regarding the tuning parameter sequence we make the following remark, useful for simulations and applications on real data.
\begin{remark}
The supposition $\lambda_n p_0^{1/2} b^{-1}_n \rightarrow 0$ made in the Theorems \ref{th_vconvn} and \ref{Theorem 2SPLn}, implies that the tunning sequence $\lambda_n \rightarrow 0$, as $n \rightarrow \infty$.
\end{remark}
\section{Numerical study}
\label{Simus}
In this section we first perform a numerical simulation study to illustrate our theoretical results on the adaptive LASSO expectile estimation and compare it with estimation obtained by the adaptive LASSO quantile method. Afterwards, an application on real data is presented.\\
We use the following R language packages: package \textit{SALES} (for $p>n$) with function \textit{ernet} for the expectile regression and for quantile regression, the package \textit{quantreg} with function \textit{rq}.\\
Given assumption (A1), the expectile index $\tau$ is:
\begin{equation}
\label{te}
\tau=\frac{\mathbb{E}\big[\varepsilon 1\!\!1_{\varepsilon<0} \big]}{\mathbb{E}\big[\varepsilon(1\!\!1_{\varepsilon<0}- 1\!\!1_{\varepsilon>0}) \big]}.
\end{equation}
In the simulation study, the index $\tau$ is fixed, function of the law of $\varepsilon$, such that assumption (A1) be satisfied. On basis relation (\ref{te}), we will give details in subsection \ref{appli} how an estimation for $\tau$ can be calculated in pratical applications.\\
Taking into account the suppositions imposed on the tuning parameter in Theorems \ref{Theorem 2SPL} and \ref{Theorem 2SPLn}, we consider $\lambda_n=n^{-2/5}$ for the expectile framework. For the power $\gamma$ in the adaptive weights of the penalty, several values will be considered and a variation on a grid of values will be realized to choose the values of $\gamma$ which give the best results in terms of the significant variable identification.
For the quantile method, the tuning parameter is $n^{2/5}$ and the weight in the penalty have the power $1.225$ (see \cite{Ciuperca-16}).
\subsection{\textbf{Simulation study: fixed $p_0$ case}}
In this subsection, we will study the numerical behavior of the adaptive LASSO expectile method and we will compare it with the simulation results obtained by the adaptive LASSO quantile method.
\begin{table}
\caption{\footnotesize Sparsity study for expectile method with adaptive LASSO penalty, $\varepsilon \sim {\cal N}(0,1)$, two values for $\gamma$. }
\begin{center}
{\tiny
\begin{tabular}{|c|c|cc|cc|}\hline
$n$ & $p$ & \multicolumn{2}{c|}{$Card({\cal A} \cap \widehat{{\cal A}}_n)$} & \multicolumn{2}{c|}{$Card(\widehat{{\cal A}}_n \setminus {\cal A})$} \\ \cline{3-6}
& & $\gamma=1/8$ &$\gamma=2$ & $\gamma=1/8$ & $\gamma=2$ \\ \hline
50 & 100 & 5.99&5.58 & 3.8&0.004 \\
& 400 & 5.87&5.12 & 8.7&0.04 \\
& 600 & 5.71&4.98 & 10&0.07 \\ \hline
100 & 100 & 6&5.99 & 2.48&0 \\
& 400 & 6&5.96 & 6.79&0 \\
& 600 & 6&5.97 & 8.8&0 \\ \hline
\end{tabular}
}
\end{center}
\label{Tabl0}
\end{table}
For model (\ref{eq1}), we consider $p_0=6$ and ${\cal A}=\{1, \cdots , 6 \}$. In the all simulations of this subsection we take, $\beta^0_1=1$, $\beta^0_2=4$, $\beta^0_3=-3$, $\beta^0_4=5$, $\beta^0_5=6$, $\beta^0_6=-1$, while $n$ and $p$ are varied. The values of $p$ they can be less than $ n $ but also higher. \\
For the errors $\varepsilon_i$, three distributions are considered: ${\cal N}(0,1)$ which is symmetrical, ${\cal E}xp(-2.5)$ and ${\cal N}(0,4 \cdot 10^{-2})+\chi^2(1)$, the last two being asymmetrical. The exponential law ${\cal E}xp(-2.5)$ has the density function $\exp(-(x+2.5))1\!\!1_{x >-2.5}$. For each value of $n$, $p$ and distribution of $\varepsilon$, 1000 Monte Carlo replications are realized for two possible values for $\gamma$. In Tables \ref{Tabl0} and \ref{Tabl1} we give the average of the 1000 Monte Carlo replications for the cardinalities (number of the true non-zeros estimated as non-zero) $Card({\cal A} \cap \widehat{{\cal A}}_n)$ and $Card(\widehat{{\cal A}}_n \setminus {\cal A})$ (number of the false non-zero) by the expectile (ES) and quantile (Q) penalized methods, each with LASSO adaptive penalty. For a perfect method, we should have: $Card({\cal A} \cap \widehat{{\cal A}}_n)=p_0=6$ and $Card(\widehat{{\cal A}}_n \setminus {\cal A})=0$. In Table \ref{Tabl0}, for standard Gaussian errors, the values considered for $\gamma$ are $1/8$ and $2$. Since for $\gamma=1/8$ the number of false non-zeros, which in addition increases with $n$, is much larger than for $\gamma=2$ and for $\gamma=2$ the number of the true non-zeros decreases with $n$, these values will be dropped, other two values will be considered in Table \ref{Tabl1}. In Table \ref{Tabl1}, taking $\gamma \in \{5/8, 1\}$, for the adaptive LASSO expectile method, all significant variables are detected when $ n $ is large enough, the number $ p $ of variables not coming into play.
\begin{table}
\caption{\footnotesize Sparsity study for expectile (ES) and quantile (Q) methods with adaptive LASSO penalties. }
\begin{center}
{\tiny
\begin{tabular}{|c|c|c|ccc|ccc|}\cline{1-9}
$\varepsilon$ & $n$ & $p$ & \multicolumn{3}{c|}{$Card({\cal A} \cap \widehat{{\cal A}}_n)$} & \multicolumn{3}{c|}{$Card(\widehat{{\cal A}}_n \setminus {\cal A})$} \\ \cline{4-9}
& & & \multicolumn{2}{c}{ ES} & Q &\multicolumn{2}{c}{ ES} & Q \\ \cline{4-5} \cline{7-8}
& & & $\gamma=5/8$ &$\gamma=1$ & & $\gamma=5/8$ & $\gamma=1$ & \\ \cline{1-9}
$ {\cal N} (0,4 \cdot 10^{-2})+\chi^2(1)$ & 50 & 100 & 5.25 & 5&5.11& 0.06 &0.03 & 0.16\\
& & 400 & 4.98 & 4.67 & 4.45 & 0.34 & 0.12 & 0.59 \\
& & 600 & 4.80 & 4.57 & 4.32 & 0.54 & 0.19& 0.67 \\ \cline{2-9}
& 100 & 100 & 5.96 & 5.85 & 5.98& 0 &0 & 0 \\
& & 400 & 5.94 & 5.85 & 5.98 & 0.003 & 0& 0.002 \\
& & 600 & 5.94 & 5.84& 5.96 & 0.006 &0 & 0.001 \\ \cline{2-9}
& 200 & 100 & 6&6 &6& 0 &0 & 0 \\
& & 400 & 6 &6 &6 & 0& 0& 0 \\
& & 600 & 6 &6 &6& 0& 0 & 0 \\ \cline{1-9}
$ {\cal E}xp(-2.5)$ & 50 & 100 & 5.8 &5.66 & 3.18 & 0.33&0.06 & 0.57 \\
& & 400 & 5.58&5.41 & 2.9 & 0.93& 0.27 & 1.1 \\
& & 600 & 5.37 & 5.11 &2.7 & 1.3&0.46 & 1.25 \\ \cline{2-9}
& 100 & 100 & 6 & 5.99&4.9 & 0.06 &0.003 & 0.09 \\
& & 400 & 5.99& 5.99 & 4.67 & 0.12&0.008 & 0.12 \\
& & 600 & 6& 5.99 & 4.68 & 0.14&0.02 & 0.10 \\ \cline{2-9}
& 200 & 100 & 6& 6 & 5.77 & 0.007&0.002 & 0.04 \\
& & 400 & 6& 6 & 5.79 & 0.01& 0 & 0.02 \\
& & 600 & 6& 6 & 5.79 & 0.019& 0 & 0.020 \\ \cline{1-9}
$ {\cal N}(0,1)$ & 50 & 100 & 5.95&5.88 & 5.89 & 0.21&0.02 & 0.44 \\
& & 400 & 5.74 & 5.58 &5.51 & 0.72& 0.28 & 0.72 \\
& & 600 & 5.61&5.36 & 5.33 & 1.2& 0.40 & 0.92 \\ \cline{2-9}
& 100 & 100 & 6 & 6 &6 & 0.01& 0& 0.14 \\
& & 400 & 6& 6 & 6 & 0.04&0.006 & 0.17 \\
& & 600 & 5.99& 6 & 5.99 & 0.04& 0 & 0.15 \\ \cline{2-9}
& 200 & 100 & 6& 6 & 6 & 0.001 & 0 & 0.03 \\
& & 400 & 6& 6 & 6 & 0.001 & 0 &0.06 \\
& & 600 & 6& 6 & 6 & 0 & 0 &0.05 \\ \cline{1-9}
\end{tabular}
}
\end{center}
\label{Tabl1}
\end{table}
We observe that the penalized expectile method better detects non-zero parameters compared to the penalized quantile method, especially for exponential errors. For small values of $ n $, the expectile method makes less false detections of non-significant variables as significant variables. Concerning the two values considered for $\gamma$, when $\gamma = 5/8$, there are a little more true non-zeros detected, while, when $\gamma=1$, there are fewer false non-zeros. This trend will be also confirmed by the following numerical studies.
\subsection{\textbf{Simulation study: case when $p_0$ varies with $n$}}
In this subsection, we always compare expectile and quantile penalized methods, but when the values considered for $ p $ are larger than $ n $. Moreover, the number of non-zero parameters can increase as $n$ increases.
\begin{table}
\caption{\footnotesize Study of the sparsity evolution and of the estimation accuracy for the expectile (ES) and quantile framework, with $p$ and $p_0$ depending on $n$: $p=4n$, $p_0=2[n^{1/2}]$. }
\begin{center}
{\tiny
\begin{tabular}{|c|c|ccc|ccc|ccc|ccc|}\cline{1-14}
$\varepsilon$ & $n$ & \multicolumn{3}{c|}{$100 p_0^{-1} Card({\cal A} \cap \widehat{{\cal A}}_n)$} & \multicolumn{3}{c|}{$100 (n-p_0)^{-1} Card(\widehat{{\cal A}}_n \setminus {\cal A})$} & \multicolumn{3}{c|}{$mean(|\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}|)$} & \multicolumn{3}{c|}{$mean(|(\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\big)_{\cal A}|)$}\\ \cline{3-14}
& & \multicolumn{2}{c}{ ES} & Q &\multicolumn{2}{c}{ ES} & Q & \multicolumn{2}{c}{ ES} & Q &\multicolumn{2}{c}{ ES} & Q\\ \cline{3-4} \cline{6-7} \cline{9-10} \cline{12-13}
& & $\gamma=5/8$ &$\gamma=1$ & & $\gamma=5/8$ & $\gamma=1$ & & $\gamma=5/8$ &$\gamma=1$ & & $\gamma=5/8$ & $\gamma=1$ & \\ \hline \hline
$ {\cal N}(0,1)$ & 75 & 98 &97.4 &98 & 3.3 & 2.7&3.6 & 0.01 &0.01 &0.01 & 0.22& 0.23& 0.22\\
& 100 & 99.4 & 99.2 & 99.3 & 1.1 & 0.46 &1.5 & 0.007 & 0.006 &0.007 & 0.13 &0.13 &0.13 \\
& 200 & 100 &100 &100 & 0.02 & 0.008 &0.07 & 0.002 & 0.002&0.002 & 0.07 & 0.07&0.07\\
&400 & 100 &100 &100 & 0.01 &0.008 &0.002 & 0.001&0.001 &0.001 & 0.05 &0.04 &0.05 \\ \hline
$\varepsilon \sim {\cal E}xp(-2.5)$ & 75 & 97 &97 &43 & 4.2 &2.21 &10.9 & 0.01 & 0.01&0.34 &0.26 &0.23 &5.2\\
& 100 & 99 &98.6 &49 & 1.6 & 0.78&11.6 & 0.008 &0.008 &0.35 & 0.16 &0.15 &5.7 \\
& 200 & 100 & 100&77 & 0.05& 0.01&5 & 0.002 & 0.002&0.13 & 0.08 & 0.08&3.1\\
&400 & 100 & 100 &98 & 0.04 & 0.02&0.37 & 0.001&0.001 &0.007 & 0.05 & 0.05&0.28\\ \cline{1-14}
${\cal N} (0,4 \cdot 10^{-2})+\chi^2(1)$ & 75 & 94 &93.3 &70 & 3.7 &2.55 &12.4 & 0.02 &0.02 &0.18 &0.40& 0.37& 2.58\\
& 100 & 97& 96 &80 & 1.5 & 0.81&9.8 & 0.01 & 0.01&0.13 & 0.26 & 0.22&2.07 \\% \cline{2-10}
& 200 & 99.8 & 99.6&99 & 0.02&0.01 &0.26 & 0.003 & 0.002&0.003 & 0.1 & 0.08&0.1\\
&400 & 100 &100 &100 & 0.02 &0.002& 0 & 0.001& 0.001&0.001 & 0.05& 0.04& 0.02\\ \cline{1-14}
\end{tabular}
}
\end{center}
\label{Tabl2}
\end{table}
In Table \ref{Tabl2} we take $p=4n$, $p_0=2[n^{1/2}]$, with $[x]$ the entire part of $x$, the power $\gamma \in \{5/8, 1\}$ and $\varepsilon \sim {\cal E}xp(-2.5)$. The true value of the non-null parameter vector is $\eb^0_{\cal A}=(1, \cdots , p_0)$. We assess model selection by calculating the percentage ($100 p_0^{-1} Card({\cal A} \cap \widehat{{\cal A}}_n)$) of the non-zero parameters with a non-zero estimation and the percentage of false significant variables ($100 (n-p_0)^{-1} Card(\widehat{{\cal A}}_n \setminus {\cal A})$), by the two estimation methods. We also give the accuracy of the complete estimation vectors ($mean(|\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}|)$) and of the estimations of non-zero parameters ($mean(|(\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\big)_{\cal A}|)$) (average absolute estimation error) obtained on 1000 Monte Carlo replications. More precisely, if $M$ is the Monte Carlo replication number and $\widehat{\eb}^{(m)}_{n,j}$ is the estimation of $\beta^0_j$ obtained for the Monte Carlo replication with the number $m$, then, $mean(|\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}|)=(Mp)^{-1} \sum^{M}_{m=1} \sum^p_{j=1}| \widehat{\eb}^{(m)}_{n,j} - \beta^0_j | $. Similarly we calculate $mean(|(\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})_{\cal A}|)=(Mp)^{-1} \sum^{M}_{m=1} \sum^{p_0}_{j=1} | \widehat{\eb}^{(m)}_{n,j} - \beta^0_j | $. For the Gaussian standardized errors, the results are similar by the two estimation methods. The results remain very good when the errors have an exponential law or a mixing between a Gaussian with a $\chi^2(1)$ law for the proposed adaptive LASSO expectile method, even for small values for $n$, the results being more accurate for $\gamma=1$ than for $\gamma =5/8$. Furthermore, the adaptive LASSO quantile method, provides less accurate estimations, even when it correctly detects significant variables (for values of $ n $ large), which can pose problems of application in practice.
\begin{table}
\caption{\footnotesize Study of the sparsity evolution and of the estimation accuracy for the expectile (ES) and quantile framework, with $p$ and $p_0$ depending an $n$: $p=[n\log(n)]$, $p_0=2[n^{1/4}]$, $\varepsilon \sim {\cal E}xp(-2.5)$. }
\begin{center}
{\tiny
\begin{tabular}{|c|c|ccc|ccc|ccc|ccc|}\hline
$\eb^0_{\cal A}$ & $n$ & \multicolumn{3}{c|}{$100 p_0^{-1} Card({\cal A} \cap \widehat{{\cal A}}_n)$} & \multicolumn{3}{c|}{$100 (n-p_0)^{-1} Card(\widehat{{\cal A}}_n \setminus {\cal A})$} & \multicolumn{3}{c|}{$mean(|\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}|)$} & \multicolumn{3}{c|}{$mean(|(\widehat{\eb}_n-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\big)_{\cal A}|)$}\\ \cline{3-14}
& & \multicolumn{2}{c}{ ES} & Q &\multicolumn{2}{c}{ ES} & Q & \multicolumn{2}{c}{ ES} & Q &\multicolumn{2}{c}{ ES} & Q\\ \cline{3-4} \cline{6-7} \cline{9-10} \cline{12-13}
& & $\gamma=5/8$ &$\gamma=1$ & & $\gamma=5/8$ & $\gamma=1$ & & $\gamma=5/8$ &$\gamma=1$ & & $\gamma=5/8$ & $\gamma=1$ & \\ \hline \hline
$(1, 2, \cdots, p_0)$ & 75 & 99.8 & 99&76 & 0.27 &0.02 &0.22 & 0.002 & 0.002&0.008 & 0.19& 0.19&0.67 \\
& 100 & 100 &99.9& 87 & 0.12 &0.008 &0.11 & 0.002& 0.002 & 0.007 & 0.15&0.14 & 0.53 \\
& 200 & 100 & 100&98 & 0&0 & 0 & 0.0006& 0.0005& 0.001 & 0.10& 0.09& 0.24 \\
& 400 & 100&100 & 100 & 0& 0 & 0 & 0.0002& 0.0005 & 0.0004 & 0.06&0.09 & 0.14 \\ \hline
$(1, \cdots, 1)$ & 75 & 98.4 &97.9&30 & 0.32 &0.03 &0.25 & 0.004& 0.004& 0.01 & 0.31& 0.35&0.83 \\
& 100 & 99.8&99.4 & 36 & 0.26& 0.03& 0.17 & 0.003& 0.003 & 0.01 & 0.25& 0.28& 0.81 \\
& 200 & 100& 100& 82 & 0.02&0 & 0.005 & 0.0009& 0.001 & 0.003 & 0.16& 0.18& 0.51 \\
& 400 & 100&100 & 100 & 0 & 0& 0 & 0.0003 & 0.0003& 0.007 & 0.11&0.11 & 0.23 \\ \cline{1-14}
\end{tabular}
}
\end{center}
\label{Tabl3}
\end{table}
In Table \ref{Tabl3}, taking $p=[n \log(n)]$, the value of $p$ is increased compared to that considered in Table \ref{Tabl2}. Furthermore, the sparsity of the model is more accentuated by considering $p_0=2[n^{1/4}]$. Two values for $\eb^0_{\cal A}$ are considered: $(1, 2, \cdots, p_0)$ and $\textbf{1}_{p_0}=(1,\cdots , 1)$ while for the model errors, only the exponential distribution $\varepsilon \sim {\cal E}xp(-2.5)$ is made. For both values of $\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}$, the expectile method with adaptive LASSO penalty gives very good results for identifying of null and non-null parameters, while the quantile method identifies all significant variables only when $ n $ is large (greater than 200). \\
Comparing Tables \ref{Tabl2} and \ref{Tabl3}, for $n$ fixed and exponential law, we deduce that the penalized expectile estimation quality of the model does not vary for two different $ p $. Furthermore, the quality is better if $ p_0 $ decreases and when $\gamma =1$.
\subsection{\textbf{Sparsity study function of $\gamma$}}
\begin{figure}[h!]
\includegraphics[width=16.5cm,height=6cm,angle=0]{gamma_n75_p100_p200_exp.png}
\caption{ \it Percentage of true (to the left) and false non-zero, for $\varepsilon \sim {\cal E}xp(-2.5)$, $n=75$. Two value for the number of parameters: $p=100$ (doted line) and $p=200$ (solid line).}
\label{Figure n75_exp}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=16.5cm,height=6cm,angle=0]{gamma_n100_p100_p400_exp.png}
\caption{ \it Percentage of true and false non zero, for $\varepsilon \sim {\cal E}xp(-2.5)$, $n=100$. Two value for the number of parameters: $p=100$ (doted line) and $p=400$ (solid line).}
\label{Figure n100_exp}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=16.5cm,height=6cm,angle=0]{gamma_n75_p100_p200_mixt.png}
\caption{ \it Percentage of true and false non zero, for $\varepsilon \sim {\cal N} (0,4 \cdot 10^{-2})+\chi^2(1)$, $n=75$. Two value for the number of parameters: $p=100$ (doted line) and $p=200$ (solid line).}
\label{Figure n75_mixt}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=16.5cm,height=6cm,angle=0]{gamma_n100_p100_p400_mixt.png}
\caption{ \it Percentage of true and false non zero, for $\varepsilon \sim {\cal N} (0,4 \cdot 10^{-2})+\chi^2(1)$, $n=100$. Two value for the number of parameters: $p=100$ (doted line) and $p=400$ (solid line).}
\label{Figure n100_mixt}
\end{figure}
In Figure \ref{Figure n75_exp} we present for a model with sample size $n=75$, errors $\varepsilon \sim {\cal E}xp(-2.5)$, the percentage of the true non-zero parameters estimated by adaptive LASSO expectile method (sub-figure to the left) and the percentage of the true zeros estimated as non-zero (to right), for two values of $p$: $p=100$ (dotes line) and $p=200$ (solid line). We observe that for $\gamma \in ( 0, 1.2]$ the estimation rate of the non-zero parameters by a non-zero estimation exceeds $99 \%$, for $\gamma \geq 0.5$ the rate of false non-zero is less than $1\%$ and this last rate decreases when $\gamma $ increases. A similar study is presented in Figure \ref{Figure n100_exp} for $n=100$, $p \in \{100, 400\}$ and we deduce the same conclusions as in Figure \ref{Figure n75_exp}. \\
In view of all this, we may deduce that for a power $\gamma \in [0.5, 1.25]$, the error rates (of false zeros and false non-zeros) are less than $1 \%$.\\
For errors $\varepsilon \sim {\cal N} (0,4 \cdot 10^{-2})+\chi^2(1)$ (see Figures \ref{Figure n75_mixt} and \ref{Figure n100_mixt}), the detection rate of the true non-zero parameters decreases faster when $\gamma$ increases, the rate of false non-zeros remaining the same as for exponential errors (Figures \ref{Figure n75_exp} and \ref{Figure n100_exp}).
\subsection{\textbf{Application to real data}}
\label{appli}
We use the data \textit{eyedata} of R package \textit{flare} which contains $n=120$ observations (rats) for the response variable of gene \textit{TRIM32} and 200 explanatory variables, other genes probes, from the microarray experiments of mammalian-eye tissue samples in \cite{Scheetz.06}. The objective is to find genes that are correlated with the TRIM32 gene, known to
cause Bardet–Biedl syndrome, a genetically disease of multiple organ systems including the retina.\\
In order to calculate in applications the expectile index $ \tau $, we standardize the values of the explained variable $\widetilde{y}_i=(y_i-\bar y_n)/(\widehat{\sigma}_y)$. Afterwards, based on relation (\ref{te}), we calculate the empirical estimation of $\tau$:
\[
\widehat \tau=\frac{n^{-1}\sum^n_{i=1} \widetilde{y}_i 1\!\!1_{\widetilde{y}_i <0}}{n^{-1} \big( \sum^n_{i=1} \widetilde{y}_i 1\!\!1_{\widetilde{y}_i <0}-\sum^n_{i=1} \widetilde{y}_i 1\!\!1_{\widetilde{y}_i >0}\big)}.
\]
Then, in model (\ref{eq1}), the response variable is $\widetilde{Y}_i=(Y_i-\overline{Y}_n)/\widehat{\sigma}_Y$, with $\widehat{\sigma}_Y$ the empirical standard deviation and $\overline{Y}_n$ empirical mean of $Y$. For this application, we get $ \widehat \tau=0.533$ and $\gamma=5/8$.\\
We obtain, for $\gamma = 5/8$, $\lambda_n=n^{2/5}$, by the method adaptive LASSO expectile that the genes whose expressions influence gene TRIM32 are 87, 153, 180, 185 with the labels: "21092", "25141", "28680" and "28967". The obtained estimations for the coefficients of these four explanatory variables are respectively: \textit{-0.65, 2.24, 0.28} and \textit{-0.35}. In Figure \ref{Figure 1} we illustrate the histogram and the boxplot for response variable TRIM32. We observe that it don't have a symmetrical law. \\
If a classical LS regression of the TRIM32 variable in respect to the four selected covariates is performed, we obtain a model with an adjusted $ R ^ 2 = 0.74 $ and with residual standard error = 0.50. The all four variables are significant and the residuals have Gaussian distribution (the p-value by Shapiro test equal to 0.45). In the sub-figure of the right-hand side of Figure \ref{Figure 1}, we also present, the forecasts beside of the true values of TRIM32. We observe that the scater graph is on the first bisectrix.\\
By the adaptive LASSO quantile method, no variable is selected among the 200 explanatory ones. \\
In literature works that model the same data, variable number 153, tagged "25141", has been selected as the sole regressor by the bayesian shrinkage in \cite{Song-Liang.17} and by a globally adaptive quantile method in \cite{Zheng-Peng-He.15} for quantile index between 0.45 and 0.55. In this last paper, there are other covariates that appear to be significant for other quantile index values. These variables are: "11711", "24565", "25141", "25367", "21092", "29045", "25439", "22140", "15863" and "6222". If we make a classic regression for these ten regressors, we obtain, with a risk of $ 0.05 $ that only the variables "25141", "21092", "29045", "15863", "6222" are significant, in a model of lower quality (adjusted $R^2=0.72$, residual standard error=0.52) than the one with the four explanatory variables found in the present paper.
\begin{figure}[h!]
\includegraphics[width=16.0cm,height=6cm,angle=0]{fig.png}
\caption{ \it Histogram and boxplot of TRIM32. Scater graph between forecast and the true value of TRIM32.}
\label{Figure 1}
\end{figure}
\section{Proofs}
\label{Proofs}
In this section we give the proofs of the results presented in Sections \ref{casec<1} and \ref{casec>1}.
\subsection{\textbf{Result proofs in Section \ref{casec<1}}}
\noindent {\bf Proof of Theorem \ref{th_vconv}}.\\
\textit{(i)} In order to show the convergence rate of the expectile estimator, we show that for all $\epsilon >0$, there exists $B_\epsilon >0$, large enough when $ n $ is large, such that:
\begin{equation}
\label{eq2}
\mathbb{P} \big[ \inf_{\mathbf{{u}} \in \mathbb{R}^p, \; \| \mathbf{{u}} \|_2 =1} Q_n\big(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B_\epsilon \sqrt{\frac{p}{n}} \mathbf{{u}}\big)>Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})\big] \geq 1-\epsilon .
\end{equation}
This part of proof is similar to that of Lemma 1.2 in \cite{Zhao-Chen-Zhang.18}, for the convergence rate of the oracle estimator.
Let $B>0$ be a constant to be determined later and $\mathbf{{u}}$ a vector in $\mathbb{R}^p$ with the norm $\| \mathbf{{u}} \|_2=1$. Let's study the difference:
\begin{align*}
&Q_n\big(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B \sqrt{\frac{p}{n}} \mathbf{{u}} \big) -Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) = \sum^n_{i=1}\big[\rho_\tau\big(\varepsilon_i -B \sqrt{\frac{p}{n}} \mathbf{X}^t_i \mathbf{{u}} \big)- \rho_\tau(\varepsilon_i ) \big]
\end{align*}
\begin{align*}
& \quad = \sum^n_{i=1}\big[\rho_\tau\big(\varepsilon_i -B \sqrt{\frac{p}{n}} \mathbf{X}^t_i \mathbf{{u}} \big)- \rho_\tau(\varepsilon_i ) - \mathbb{E}\big[ \rho_\tau\big(\varepsilon_i -B \sqrt{\frac{p}{n}} \mathbf{X}^t_i \mathbf{{u}} \big)- \rho_\tau(\varepsilon_i ) \big]\big] \\
& \qquad \qquad +\sum^n_{i=1} \mathbb{E}\big[ \rho_\tau\big(\varepsilon_i -B \sqrt{\frac{p}{n}} \mathbf{X}^t_i \mathbf{{u}} \big)- \rho_\tau(\varepsilon_i ) \big] \equiv \Delta_1+\Delta_2.
\end{align*}
We first study the term $\Delta_2$. By Taylor expansion, we have: $\mathbb{E}[\rho_\tau(\varepsilon-t)-\rho_\tau(\varepsilon)]=\mathbb{E} \big[ - g_\tau(\varepsilon)t +2^{-1} h_\tau(\varepsilon) t^2 \big]+o(t^2)=2^{-1}\mathbb{E} \big[ h_\tau(\varepsilon) \big]t^2 +o(t^2)$. By the Cauchy-Schwarz inequality, we have that $| \mathbf{X}^t_i \mathbf{{u}} |^2 \leq \| \mathbf{X}_i\|^2_{2} \| \mathbf{{u}} \|_2^2$ and then, using assumption (A3), we obtain that $(pn^{-1})^{1/2} \max_{1 \leqslant i \leqslant n}| \mathbf{X}^t_i \mathbf{{u}} | {\underset{n \rightarrow \infty}{\longrightarrow}} 0$. Thus,
\[
\Delta_2 =\frac{1}{2} \sum^n_{i=1}\bigg[ B^2\frac{p}{n} \big(\mathbf{X}_i^t \mathbf{{u}} \big)^2\mathbb{E} \big[ h_\tau(\varepsilon) \big]+o\bigg( B^2\frac{p}{n} \big(\mathbf{X}_i^t \mathbf{{u}} \big)^2\mathbb{E} \big[ h_\tau(\varepsilon) \big] \bigg) \bigg].
\]
On the other hand,
\[
2\min(\tau, 1-\tau) \leq \mathbb{E} \big[ h_\tau(\varepsilon) \big]=2 \tau \mathbb{E}[1\!\!1_{\varepsilon \geq 0}]+2(1-\tau)\mathbb{E}[1\!\!1_{\varepsilon<0}] \leq 2\max(\tau, 1-\tau).
\]
Then
\begin{equation}
\label{DD2}
0< \Delta_2 =B^2 \frac{p}{n} \mathbb{E} \big[ h_\tau(\varepsilon) \big] \sum^n_{i=1} \big( \mathbf{X}_i^t \mathbf{{u}} \big)^2\big(1+o(1) \big)=O(B^2p).
\end{equation}
We are now studying the term $\Delta_1$. Let us consider the random variable:
\begin{equation}
\label{Ri}
D_i \equiv \rho_\tau \big(\varepsilon_i - B \sqrt{\frac{p}{n}} \mathbf{X}_i^t \mathbf{{u}} \big) - \rho_\tau(\varepsilon_i) +B \sqrt{\frac{p}{n}} g_\tau(\varepsilon_i) \mathbf{X}_i^t \mathbf{{u}}.
\end{equation}
Then, we can write $\Delta_1$ as:
\[
\Delta_1 = \sum^n_{i=1} \big[ - B \sqrt{\frac{p}{n}} g_\tau(\varepsilon_i) \mathbf{X}^t_i \mathbf{{u}} +D_i -\mathbb{E}[D_i]\big].
\]
Using assumption (A1) we have, $ \mathbb{E}\big[ (pn^{-1})^{1/2} g_\tau(\varepsilon_i) \mathbf{X}^t_i \mathbf{{u}} \big]=0$ and $ \mathbb{V}\mbox{ar}\,\big[ ( pn^{-1})^{1/2} g_\tau(\varepsilon_i) \mathbf{X}^t_i \mathbf{{u}} \big]=p n^{-1} \mathbf{{u}}^t \sum^n_{i=1} \mathbf{X}_i \mathbf{X}_i^t \mathbf{{u}} \mathbb{V}\mbox{ar}\, [g_\tau(\varepsilon)] =O(p)$. Then, we have:
\begin{equation}
\label{D1}
B \sqrt{\frac{p}{n}} g_\tau(\varepsilon_i) \mathbf{X}^t_i \mathbf{{u}} =O_{\mathbb{P}}\big(p^{1/2} B \big).
\end{equation}
By Taylor expansion of $\rho_\tau \big(\varepsilon_i - B (pn^{-1})^{1/2} \mathbf{X}_i^t \mathbf{{u}} \big)$ around of $\varepsilon_i$, we can write $D_i$ also in the form: $D_i=2^{-1}B^2 p n^{-1} |\mathbf{X}_i^t \mathbf{{u}}|^2 h_\tau(\widetilde \varepsilon_i)$, with $\widetilde \varepsilon_i$ a random variable between $\varepsilon_i$ and $\varepsilon_i+B (pn^{-1})^{1/2} \mathbf{X}_i^t \mathbf{{u}}$. On the other hand, given the definition of the function $h$, we have that:
\begin{equation}
\label{RR}
\mathbb{P} \big[2 \min(\tau, 1-\tau) \leq h_\tau(\widetilde \varepsilon_i) \leq 2 \max(\tau, 1-\tau)\big]=1,
\end{equation}
and also $\mathbb{V}\mbox{ar}\, [D_i] \leq \mathbb{E}[D_i^2]=4^{-1}B^4 |\mathbf{X}^t_i \mathbf{{u}}|^4 \mathbb{E}\big[h^2_\tau(\widetilde \varepsilon_i) \big]$. But $h^2_\tau(\widetilde \varepsilon_i)= 4\tau^21\!\!1_{\widetilde \varepsilon_i>0}+4(1-\tau)^21\!\!1_{\widetilde \varepsilon_i<0}$, then, $\mathbb{E}\big[h^2_\tau(\widetilde \varepsilon_i) \big] \leq 4 \max \big(\tau^2, (1-\tau)^2 \big) \leq 4$. Thus, $\mathbb{V}\mbox{ar}\, [D_i] \leq B^4 p^2 n^{-2} |\mathbf{X}^t_i \mathbf{{u}}|^4 $. On the other hand, the random variables $D_i$ defined by (\ref{Ri}), are independent. Then,
\begin{equation}
\label{D2}
\sum^n_{i=1} \big[D_i- \mathbb{E}[D_i] \big]=O_{\mathbb{P}}\bigg(\sqrt{\sum^n_{i=1} \mathbb{V}\mbox{ar}\, [D_i]} \bigg) \leq O_{\mathbb{P}} \bigg(\sqrt{\sum^n_{i=1} \mathbb{E}[D_i^2]} \bigg) =O_{\mathbb{P}}\bigg(B^2 \frac{p}{n^{1/2}} \bigg).
\end{equation}
Relations (\ref{D1}) and (\ref{D2}), imply, since $p n^{-1} \rightarrow 0$, when $n \rightarrow \infty$, that:
\[
\Delta_1= O_{\mathbb{P}}(B p^{1/2})+O_{\mathbb{P}} (B^2 p n^{-1/2})= O_{\mathbb{P}}(B p^{1/2}).
\]
Then, this last relation together relation (\ref{DD2}) imply $\Delta_2 > | \Delta_1|$, with probability converging to one, for $B$ large enough. Relation (\ref{eq2}) follows, which implies the convergence rate of the expectile estimator.\\
\hspace*{0.5cm} \textit{(ii)} For $p$-vector $\mathbf{{u}}=(u_1, \cdots , u_p)$, with $\| \mathbf{{u}}\|_2=1$ and $B>0$ a constant, let us consider the difference
\begin{equation}
\label{RQ}
R_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B \sqrt{\frac{p}{n}} \mathbf{{u}}\big)- R_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})=Q_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B \sqrt{\frac{p}{n}} \mathbf{{u}}\big)- Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})+n \lambda_n \sum^p_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+B\sqrt{\frac{p}{n}} u_j\big|- |\eb^0_j| \big].
\end{equation}
The first term of the right-hand side of (\ref{RQ}) becomes by the above proof for \textit{(i)},
\begin{equation}
\label{QQ}
Q_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B \sqrt{\frac{p}{n}} \mathbf{{u}}\big)- Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})=B^2 \frac{p}{n} \mathbb{E} \big[ h_\tau(\varepsilon) \big] \sum^n_{i=1} (\mathbf{X}_i^t \mathbf{{u}})^2 \big(1+o_{\mathbb{P}}(1)\big)=O_{\mathbb{P}}(B^2 p).
\end{equation}
Furthermore, for the penalty of (\ref{RQ}) we have:
\[
n \lambda_n \sum^p_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+B\sqrt{\frac{p}{n}} u_j\big|- |\eb^0_j| \big] \geq n \lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+B\sqrt{\frac{p}{n}} u_j\big|- |\eb^0_j| \big] \geq - n \lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j} B \sqrt{\frac{p}{n}} |u_j| .
\]
By the Cauchy-Schwarz inequality and afterwards by \textit{(i)}, we have
\begin{equation}
\label{PP}
\geq - n \lambda_n B \sqrt{\frac{p}{n}} \bigg( \sum^{p_0}_{j=1} \widehat{\omega}_{n,j}^2 \bigg)^{1/2} \| \mathbf{{u}} \|_2= - B \sqrt{\frac{p}{n}} n p_0^{1/2} \lambda_n= - B p_0^{1/2} n^{(1+c)/2} \lambda_n.
\end{equation}
Since $ p_0^{1/2} n^{(1-c)/2} \lambda_n \rightarrow 0$, as $n \rightarrow \infty$, we obtain that relation (\ref{QQ}) dominates (\ref{PP}) and the assertion regarding the convergence rate of $\widehat{\eb}_n$ results.
\hspace*{\fill}$\blacksquare$ \\
\noindent {\bf Proof of Theorem \ref{Theorem 2SPL}}.\\
\textit{(i)} Let us consider the parameter set: ${\cal V}_p(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) \equiv \big\{ \eb \in \mathbb{R}^p; \| \eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\|_2 \leq B \sqrt{\frac{p}{n}}\big\}$, with $B>0$ large enough and ${\cal W}_n \equiv \acc{\eb \in {\cal V}_p(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) ; \|\eb_{{\cal A}^c}\|_2 >0}$. According to Theorem \ref{th_vconv}, the estimator $\widehat{\eb}_n$ belongs to the set ${\cal V}_p(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})$ with a probability converging to 1 as $n \rightarrow \infty$. In order to show the sparsity property of claim \textit{(i)}, we will show that, $\lim_{n \rightarrow \infty} \mathbb{P}\big[\widehat{\eb}_n \in {\cal W}_n \big]=0$. Note that if $\eb \in {\cal W}_n$, then $p >p_0$. \\
Let us consider two parameter vectors: $\eb=(\eb_{\cal A}, \eb_{{\cal A}^c}) \in {\cal W}_n$ and $ {\eb}^{(1)}=({\eb}_{\cal A}^{(1)}, {\eb}_{{\cal A}^c}^{(1)}) \in {\cal V}_p(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})$, such that ${\eb}^{(1)}_{\cal A}=\eb_{\cal A}$ and $ {\eb}_{{\cal A}^c}^{(1)}=\textbf{0}_{p-p_0}$. For this parameters, we will study the following difference:
\begin{equation}
\label{DD}
n^{-1} \big[R_n(\eb) - R_n({\eb}^{(1)}) \big] = n^{-1}\big[Q_n(\eb) - Q_n({\eb}^{(1)}) \big] +\lambda_n \sum^p_{j=p_0+1} \widehat{\omega}_{n,j} | \beta_j|.
\end{equation}
For the first term of the right-hand side of relation (\ref{DD}), we have
\begin{align*}
&n^{-1} \sum^n_{i=1} \big[\rho_\tau(Y_i-\mathbf{X}_i^t \eb^{(1)})- \rho_\tau(Y_i-\mathbf{X}_i^t \eb)\big]
\\
& \quad = n^{-1} \sum^n_{i=1} \big[\rho_\tau\big(\varepsilon_i- \mathbf{X}_{i, {\cal A}}^t (\eb_{\cal A}- \eb^0_{\cal A}) \big) -\rho_\tau \big( \varepsilon_i- \mathbf{X}_{i, {\cal A}}^t (\eb_{\cal A}- \eb^0_{\cal A})-\mathbf{X}^t_{i, {\cal A}^c} \eb_{{\cal A}^c} \big) \big]\\
&\quad = n^{-1} \sum^n_{i=1} \big[ g(\varepsilon_i) \mathbf{X}^t_{i,{\cal A} }\big(\eb_{\cal A}- \eb^0_{\cal A} \big)+ \frac{h(\varepsilon_i)}{2}\big(\mathbf{X}^t_{i,{\cal A} }(\eb_{\cal A}- \eb^0_{\cal A} )\big)^2 +o_{\mathbb{P}}\big(\mathbf{X}^t_{i,{\cal A} }(\eb_{\cal A}- \eb^0_{\cal A} )\big)^2 \big] \\
& \quad -n^{-1} \sum^n_{i=1} \big[ g(\varepsilon_i) \mathbf{X}^t_i\big(\eb_{\cal A}- \eb^0_{\cal A},\eb_{{\cal A}^c}\big)+ \frac{h(\varepsilon_i)}{2} \big(\mathbf{X}^t_i (\eb_{\cal A}- \eb^0_{\cal A}, \eb_{{\cal A}^c}) \big)^2+o_{\mathbb{P}}\big(\mathbf{X}^t_i (\eb_{\cal A}- \eb^0_{\cal A}, \eb_{{\cal A}^c}) \big)^2\big] .
\end{align*}
By similar arguments used in the proof of Theorem \ref{th_vconv}\textit{(i)} we have
\begin{align*}
&n^{-1} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}^t_{i,{\cal A} }\big(\eb_{\cal A}- \eb^0_{\cal A} \big) =\mathbb{E}\big[ n^{-1} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}^t_{i,{\cal A} }\big(\eb_{\cal A}- \eb^0_{\cal A} \big)\big] \\
& \hspace{6cm} +O_{\mathbb{P}} \bigg( \mathbb{V}\mbox{ar}\, \big[ n^{-1} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}^t_{i,{\cal A} }\big(\eb_{\cal A}- \eb^0_{\cal A} \big)\big]\bigg)^{1/2}\\
&=O_{\mathbb{P}} \bigg( \mathbb{V}\mbox{ar}\, \big[ n^{-1} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}^t_{i,{\cal A} }\big(\eb_{\cal A}- \eb^0_{\cal A} \big)\big]\bigg)^{1/2} \leq O_{\mathbb{P}} \bigg(\mathbb{E}[g^2(\varepsilon_i)] \frac{1}{n^2} \sum^n_{i=1} \|\mathbf{X}_{i,{\cal A} }\|^2_2 \|\eb_{\cal A}- \eb^0_{\cal A}\|^2_2 \bigg)^{1/2} \\
& =O_{\mathbb{P}} \bigg( \frac{1}{n} \frac{p}{n}\bigg)^{1/2}=O_{\mathbb{P}}\bigg(\frac{p^{1/2}}{n} \bigg).
\end{align*}
Proceeding similarly as above, we get:
\[
n^{-1} \sum^n_{i=1} g(\varepsilon_i)\mathbf{X}^t_i\big(\eb_{\cal A}- \eb^0_{\cal A},\eb_{{\cal A}^c}\big)=O_{\mathbb{P}}\bigg(\frac{p^{1/2}}{n} \bigg).
\]
Taking into account relation (\ref{RR}), we deduce that:
$
0< n^{-1} \sum^n_{i=1} h(\varepsilon_i) \big(\mathbf{X}^t_i (\eb_{\cal A}- \eb^0_{\cal A}, \eb_{{\cal A}^c}) \big)^2=O_{\mathbb{P}}(\| \eb_{\cal A}- \eb^0_{\cal A}\|^2_2 )=O_{\mathbb{P}}( p n^{-1} )$
and also that,
\[
0< n^{-1} \sum^n_{i=1} h(\varepsilon_{i}) \big(\mathbf{X}^t_{i,{\cal A}} (\eb_{\cal A}- \eb^0_{\cal A}) \big)^2=O_{\mathbb{P}}\bigg(\frac{p}{n} \bigg).
\]
By these relations, we obtain that the first term of the right-hand side of relation (\ref{DD}) is of order $p n^{-1}$.
For the penalty of the right-hand side of relation (\ref{DD}), taking into account Theorem \ref{th_vconv}\textit{(i)} and since $\eb \in {\cal W}_n$ we obtain:
\[
\lambda_n \sum^p_{j=p_0+1} \widehat{\omega}_{n,j} | \beta_j| \geq C\lambda_n \bigg( \frac{p}{n}\bigg)^{(1-\gamma)/2}.
\]
Using the supposition $\lambda_n (pn^{-1})^{-(1+\gamma)/2} {\underset{n \rightarrow \infty}{\longrightarrow}} \infty$, that is $\lambda_n n^{(1-c)(1+\gamma)/2} {\underset{n \rightarrow \infty}{\longrightarrow}} \infty$, we have that in the right-hand side of relation (\ref{DD}), it's the penalty that dominates. Then, since $n^{-1}\big[Q_n(\eb) - Q_n({\eb}^{(1)}) \big] =O_\mathbb{P}(p n^{-1})$, we have,
\begin{equation}
\label{Rb}
n^{-1} \big[R_n(\eb) - R_n({\eb}^{(1)}) \big] \geq C\lambda_n \bigg( \frac{p}{n}\bigg)^{(1-\gamma)/2} .
\end{equation}
But, on the other hand, since $\eb^{(1)}_{{\cal A}^c}=\textbf{0}_{p-p_0}$, by similar arguments as above, we have, $n^{-1} \big[R_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) - R_n({\eb}^{(1)}) \big]=O_{\mathbb{P}}(pn^{-1})$. From the last relation together relation (\ref{Rb}), since $\lambda_n (p n^{-1})^{-(1+\gamma)/2} {\underset{n \rightarrow \infty}{\longrightarrow}} \infty$, we deduce, $\lim_{n \rightarrow \infty}\mathbb{P} [\widehat \eb_n \in {\cal W}_n] = 0$.\\
\hspace*{0.5cm} \textit{(ii)} Given the previous result we consider the parameter vector $\eb$ of the form: $\eb=\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+(p n^{-1})^{1/2} \ed$, with $\ed=(\ed_{\cal A}, \ed_{{\cal A}^c})$, $\ed_{{\cal A}^c}=\textbf{0}_{p-p_0}$, $\|\ed_{\cal A}\|_2 \leq C $. We study then the following difference:
\begin{equation}
\label{eq40}
\frac{1}{n}R_n\big(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+\sqrt{\frac{p}{n}} \ed \big)- \frac{1}{n}R_n\big(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}} \big)=\frac{1}{n} \sum^n_{i=1} \big[\rho_\tau\big(Y_i-\mathbf{X}^t_i (\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+\sqrt{\frac{p}{n}} \ed ) \big)-\rho_\tau(\varepsilon_i) \big]+{\cal P}.
\end{equation}
For the penalty ${\cal P}=\lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j}\big(|\beta_j| -|\beta^0_j|\big) $ of the right-hand side of relation (\ref{eq40}) we have, by \ref{th_vconv}\textit{(i)}, $\widehat{\omega}_{n,j}=|\widetilde{\beta}_{n,j}|^{-\gamma}=O_{\mathbb{P}}(1)$ and by the triangular inequality $\big||\beta_j| -|\beta^0_j| \big| \leq |\beta_j -\beta^0_j|$. Then, as in the proof of Theorem \ref{th_vconv}, by relation (\ref{PP}), we obtain:
\begin{equation}
\label{gp}
{\cal P}=O_{\mathbb{P}}\bigg( \lambda_n p_0^{1/2} \bigg( \frac{p}{n}\bigg)^{1/2}\bigg)=O_{\mathbb{P}}\bigg( \lambda_n p_0^{1/2} n^{(c-1)/2 } \bigg).
\end{equation}
For the first term of the right-hand side of relation (\ref{eq40}) we have:
\begin{align*}
&\frac{1}{n} \sum^n_{i=1} \big[\rho_\tau\big(Y_i-\mathbf{X}^t_{i,{\cal A}} (\eb^0_{\cal A}+\sqrt{\frac{p}{n}} \ed _{\cal A}) \big)-\rho_\tau(\varepsilon_i) \big] \nonumber
\\
& \qquad =-\frac{1}{n} \sum^n_{i=1} g(\varepsilon_i) \big(\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big)\sqrt{\frac{p}{n}}+\frac{1}{2n} \sum^n_{i=1} \big[\frac{p}{n}\|\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A}\|^2_2 h(\varepsilon_i) +o_{\mathbb{P}} \big( \|\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A}\|^2\big) \big]
\end{align*}
\begin{align}
& \qquad=\bigg(-\frac{1}{n}\sqrt{\frac{p}{n}} \sum^n_{i=1} g(\varepsilon_i) \big(\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big)+\frac{1}{2n} \frac{p}{n} \sum^n_{i=1} \big(\ed _{\cal A}^t \mathbf{X}_{i,{\cal A}}\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} h(\varepsilon_i) \big)\bigg) \big( 1+o_{\mathbb{P}}(1)\big) \nonumber\\
&\qquad =\bigg(-\frac{1}{n}\sqrt{\frac{p}{n}} \sum^n_{i=1} g(\varepsilon_i) \big(\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big) \nonumber \\
&\qquad \qquad \qquad +\frac{1}{2n} \frac{p}{n} \sum^n_{i=1} \big(\ed _{\cal A}^t \mathbf{X}_{i,{\cal A}}\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big(\mathbb{E}[h(\varepsilon_i)]+h(\varepsilon_i)-\mathbb{E}[h(\varepsilon_i)] \big)\big)\bigg) \big( 1+o_{\mathbb{P}}(1)\big)\nonumber\\
& \qquad=\bigg(-\frac{1}{n}\sqrt{\frac{p}{n}} \sum^n_{i=1} g(\varepsilon_i) \big(\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big)+\frac{1}{2n} \frac{p}{n} \sum^n_{i=1} \big(\ed _{\cal A}^t \mathbf{X}_{i,{\cal A}}\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \mathbb{E}[h(\varepsilon_i)]\big)\bigg) \big( 1+o_{\mathbb{P}}(1)\big),
\label{gg}
\end{align}
which has as minimizer the solution of
\[
-\frac{1}{n}\sqrt{\frac{p}{n}} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}_{i,{\cal A}}+\eU_{n,{\cal A}}\sqrt{\frac{p}{n}} \ed _{\cal A} \mathbb{E}[h(\varepsilon)]=\textbf{0}_{p_0},
\]
from where, we get,
\[
\sqrt{\frac{p}{n}} \ed _{\cal A} =\frac{\eU^{-1}_{n,{\cal A}}}{\mathbb{E}[h(\varepsilon)]}\frac{1}{n} \sum^n_{i=1} g(\varepsilon_i) \mathbf{X}_{i,{\cal A}}.
\]
We deduce that, the minimum value of (\ref{gg}) is of order $O_{\mathbb{P}}\big(p n^{-1}\|\ed _{\cal A}\|_2 \big)=O_{\mathbb{P}}\big(p n^{-1}\big)=O_{\mathbb{P}}\big(n^{c-1}\big)$. Taking into account the supposition $\lambda_n p_0^{1/2} n^{(1-c)/2} {\underset{n \rightarrow \infty}{\longrightarrow}} 0$ and relation (\ref{gp}), we have that, ${\cal P}=o_{\mathbb{P}}(p n^{-1}) $. Then, in the right-hand side of relation (\ref{eq40}), the first term is the dominant one. \\
Let us now consider the following random variable sequence:
\[
W_i \equiv g(\varepsilon_i) \mathbf{{u}}^t \frac{\eU^{-1}_{n,{\cal A}}}{\mathbb{E}[h(\varepsilon)]} \mathbf{X}_{i,{\cal A}},
\]
with $\mathbf{{u}} \in \mathbb{R}^{p_0}$, $\| \mathbf{{u}} \|_2=1$. For the random variable $W_i$, we have that $\mathbb{E}[W_i]=0$ and $\mathbb{V}\mbox{ar}\,[W_i]=\mathbb{E}^{-2}[h(\varepsilon)]\eU^{-1}_{n,{\cal A}} \mathbf{{u}}^t \mathbf{X}_{i,{\cal A}} \mathbf{X}^t_{i,{\cal A}} \mathbf{{u}} \mathbb{V}\mbox{ar}\,[g(\varepsilon_i) ]$. Thus, taking into account assumption (A1), we get:
\[
\sum^n_{i=1}\mathbb{V}\mbox{ar}\,[W_i]=n \frac{\mathbf{{u}}^t \eU^{-1}_{n,{\cal A}} \mathbf{{u}}}{\mathbb{E}^2[h(\varepsilon)]}\mathbb{V}\mbox{ar}\,[g(\varepsilon) ],
\]
which implies
\[
\sqrt{n} \frac{\mathbb{E}[h(\varepsilon)]}{\sqrt{\mathbb{V}\mbox{ar}\,[g(\varepsilon) ]}}\frac{\mathbf{{u}}^t \big( \widehat \eb_n -\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}\big)_{\cal A}}{\big(\mathbf{{u}}^t \eU^{-1}_{n,{\cal A}} \mathbf{{u}} \big)^{1/2} } \overset{\cal L} {\underset{n \rightarrow \infty}{\longrightarrow}} {\cal N}(0,1).
\]
The proof of claim \textit{(ii)} is finished.
\hspace*{\fill}$\blacksquare$ \\
\subsection{\textbf{Result proofs in Section \ref{casec>1}}}
\noindent {\bf Proof of Theorem \ref{th_vconvn}}.\\
The proof is similar to that of Theorem \ref{th_vconv}. Consequently,, we give only the main results. Otherwise, instead of the Cauchy-Schwarz inequality we use Holder's inequality: $|\mathbf{X}_i^t \mathbf{{u}}| \leq \|\mathbf{X}_i\|_{\infty} \|\mathbf{{u}}\|_1$ and then we obtain: $0 <\Delta_2=O(B^2 b^2_n n \|\mathbf{{u}}\|^2_1)$. \\
For a $p$-vector $\mathbf{{u}}=(u_1, \cdots , u_p)$, with $\| \mathbf{{u}}\|_1=1$ and a constant $B>0$, let be the difference
\begin{equation}
\label{RQn}
R_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B b_n \mathbf{{u}}\big)- R_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})=Q_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+Bb_n \mathbf{{u}}\big)- Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})+n \lambda_n \sum^p_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+B b_n u_j\big|- |\eb^0_j| \big].
\end{equation}
By a similar approach made for the terms $\Delta_1$ and $\Delta_2$ of the proof of Theorem \ref{th_vconv}, we obtain:
\begin{equation}
\label{QQn}
Q_n\big( \textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+B b_n \mathbf{{u}}\big)- Q_n(\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})=O_{\mathbb{P}}(B^2 b_n^2 n\|\mathbf{{u}}\|^2_1).
\end{equation}
For the penalty of the right-hand side of relation (\ref{RQn}) we have:
\[
n \lambda_n \sum^p_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+Bb_n u_j\big|- |\eb^0_j| \big] \geq n \lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j} \big[\big|\eb^0_j+Bb_n u_j\big|- |\eb^0_j| \big] \geq - n \lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j} B b_n |u_j| ,
\]
by the Cauchy-Schwarz inequality and afterwards by the estimator consistency of $\overset{\vee}{{\eb}}_n$, we have
\begin{equation}
\label{PPn}
\geq - n \lambda_n B b_n \bigg( \sum^{p_0}_{j=1} \widehat{\omega}_{n,j}^2 \bigg)^{1/2} \| \mathbf{{u}} \|_2\geq - B C b_n n p_0^{1/2} \lambda_n\|\mathbf{{u}}\|^2_1= - B n \lambda_np_0^{1/2} b_n.
\end{equation}
Since $ \lambda_n p_0^{1/2} b^{-1}_n \rightarrow 0$, as $n \rightarrow \infty$, then relation (\ref{QQn}) dominates (\ref{PPn}) and the theorem follows.
\hspace*{\fill}$\blacksquare$ \\
\noindent {\bf Proof of Theorem \ref{Theorem 2SPLn}}.\\
\textit{(i)} Let $j \in {\cal A}^c$ be, then $j >p_0$. Thus, the derivative of the random process $R_n(\eb)$ in respect to $\beta_j$ is:
\begin{equation}
\frac{\partial R_n(\eb)}{\partial \beta_j} = \sum^n_{i=1} g_\tau\big(Y_i- \mathbf{X}^t_i\eb \big) X_{ij}+n \lambda_n \widehat{\omega}_{n,j} \textrm{sgn}(\beta_j).
\label{tt}
\end{equation}
For the first term of the right-hand side of relation (\ref{tt}), we have, $ g_\tau\big(Y_i- \mathbf{X}^t_i\eb \big)=g_\tau\big(\varepsilon_i -\mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) \big)$. We denote $\eta_i=\mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) $ and then $g_\tau(\varepsilon_i-\eta_i)=g_\tau(\varepsilon_i)-\eta_ih_\tau(\tilde \eta_i)$, with $\widetilde{\eta}_i$ a random variable variable between $\varepsilon_i$ and $\varepsilon_i-\eta_i$. Then
\[
\sum^n_{i=1} g_\tau\big(Y_i- \mathbf{X}^t_i\eb \big) X_{ij}=\sum^n_{i=1} g_\tau(\varepsilon_i) X_{ij}- \sum^n_{i=1} \mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) h_\tau(\widetilde \eta_i) X_{ij}.
\]
By the Central Limit Theorem, tacking into account assumption (A4), we have that: $\sum^n_{i=1} g_\tau(\varepsilon_i) X_{ij}=O_{\mathbb{P}}(n^{1/2})$. On the other hand, $0 < h_\tau(\widetilde \eta_i) <2$ with probability 1. Using the Holder's inequality, we have, with probability one,
\[
\bigg| \sum^n_{i=1} \mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) h_\tau(\widetilde \eta_i) X_{ij} \bigg| \leq \sum^n_{i=1} \bigg| \mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) h_\tau(\widetilde \eta_i) X_{ij}\bigg| \leq \sum^n_{i=1}\| \mathbf{X}_i\|_{\infty} \big\|h_\tau(\widetilde \eta_i) X_{ij}(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})\big\|_1 ,
\]
from where, tacking into account assumption (A4), we have: $\sum^n_{i=1} \mathbf{X}_i^t(\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}) h_\tau(\widetilde \eta_i) X_{ij}=O_{\mathbb{P}}(nb_n)$. Thus,
\begin{equation}
\label{t1}
\sum^n_{i=1} g_\tau\big(Y_i- \mathbf{X}^t_i\eb \big) X_{ij}=O_{\mathbb{P}}(n b_n).
\end{equation}
For the penalty of relation (\ref{tt}) we have: $n \lambda_n \widehat{\omega}_{n,j} =O_{\mathbb{P}} \big(n \lambda_n \min \big(n^{1/2} , a_n^{-\gamma}\big)\big)$. Since, $\lambda_n b_n^{-1}\min \big(n^{1/2}, a_n^{-\gamma}\big) \rightarrow \infty$, as $n \rightarrow \infty$, tacking also into account relation (\ref{t1}) we have that:
\[
\frac{\partial R_n(\eb)}{\partial \beta_j} \left\{
\begin{array}{lll}
>0,& &\textrm{if } \beta_j>0,\\
& & \\
<0, & & \textrm{if } \beta_j<0.
\end{array}
\right.
\]
The function $R_n(\eb)$ is continuous in $\eb$. Then, the solution of (\ref{tt}) must be equal to 0. From where $\widehat{\eb}_{n,{\cal A}^c}=\textbf{0}_{p-p_0}$, with probability converging to 1. This relation implies $\widehat{{\cal A}}_n \subseteq {\cal A}$ with probability converging to 1 when $n \rightarrow \infty$.\\
On the basis of this result, from now on we consider the parameters $\eb$ of the form $\eb=\big(\eb_{\cal A}, \textbf{0}_{p-p_0} \big)$. We must show now that ${\cal A} \subseteq \widehat{{\cal A}}_n $. By Theorem \ref{th_vconvn} we have $\| \widehat{\eb}_{\cal A} -\eb^0_{\cal A} \|_1=O_{\mathbb{P}}\big(b_n \big)$, from where for any $j=1, \cdots , p_0$, we obtain, $\widehat{\beta}_{n,j} \overset{\mathbb{P}} {\underset{n \rightarrow \infty}{\longrightarrow}} \beta^0_j \neq 0$. Thus, since $b_n {\underset{n \rightarrow \infty}{\longrightarrow}} 0$, we have that $\widehat{\beta}_{n,j} \neq 0$ with probability converging to 1, from where ${\cal A} \subseteq \widehat{{\cal A}}_n $.\\
\hspace*{0.5cm} \textit{(ii)} Given the previous result \textit{(i)} and Theorem \ref{th_vconvn}, we consider the parameters $\eb$ of the form: $\eb=\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}}+b_n \ed$, with $\ed=(\ed_{\cal A}, \ed_{{\cal A}^c})$, $\ed_{{\cal A}^c}=\textbf{0}_{p-p_0}$, $\|\ed_{\cal A}\|_1 \leq C $.
For the penalty ${\cal P}$ of the right-hand side of relation (\ref{eq40}) we have: $\big|{\cal P} \big|=\lambda_n \big|\sum^{p_0}_{j=1} \widehat{\omega}_{n,j}|\beta_j| -|\beta^0_j|] \big| \leq \lambda_n \sum^{p_0}_{j=1} \widehat{\omega}_{n,j} \big| \beta_j -\beta^0_j \big| \leq \lambda_n \big( \sum^{p_0}_{j=1} \widehat{\omega}_{n,j}^2\big)^{1/2} \| (\eb-\textrm{\mathversion{bold}$\mathbf{\beta^0}$\mathversion{normal}})_{\cal A}\|_2=O_{\mathbb{P}} \big(\lambda_n b_n p^{1/2}_0 \big)$.
For the main part, we have:
\[
\frac{1}{n} \sum^n_{i=1} \big[\rho_\tau\big(Y_i-\mathbf{X}^t_{i,{\cal A}} (\eb^0_{\cal A}+b_n \ed _{\cal A}) \big)-\rho_\tau(\varepsilon_i) \big]
\]
\[
=\bigg(-\frac{1}{n}b_n \sum^n_{i=1} g(\varepsilon_i) \big(\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \big)+\frac{1}{2n} b^2_n \sum^n_{i=1} \big(\ed _{\cal A}^t \mathbf{X}_{i,{\cal A}}\mathbf{X}^t_{i,{\cal A}}\ed _{\cal A} \mathbb{E}[h(\varepsilon_i)]\big)\bigg) \big( 1+o_{\mathbb{P}}(1)\big).
\]
The end of the proof is similar to that of Theorem \ref{th_vconvn}\textit{(ii)}.
\hspace*{\fill}$\blacksquare$ \\
\vspace{0.5cm}
\textbf{References}
|
2,869,038,154,833 | arxiv | \section{Introduction}
It is well known (see e.g. \cite{HS, DS}) that the classical minimal surfaces
do not stick at the boundary. Namely, if~$\Omega$ is a convex
domain and~$E$ is a set that minimizes the perimeter among
its competitors in~$\Omega$, then~$\partial E$ is transverse to~$\partial\Omega$
at their intersection points.
In this paper we show that the situation for the nonlocal
minimal surfaces is completely different.
Indeed, we prove that nonlocal interactions can favor stickiness at
the boundary for minimizers of a fractional perimeter.
The mathematical framework in which we work
was introduced in~\cite{CRS} and is the following. Given~$s\in(0,1/2)$
and an open set~$\Omega\subseteq\mathbb R^n$,
we define the $s$-perimeter of a set~$E\subseteq\mathbb R^n$ in~$\Omega$ as
$$ \,{\rm Per}_s(E,\Omega) := L(E\cap\Omega, E^c)+L(\Omega\setminus E,
E\setminus\Omega),$$
where~$E^c:=\mathbb R^n\setminus E$ and, for any
disjoint sets~$F$ and~$G$, we use the notation
$$ L(F,G):=\iint_{F\times G} \frac{dx\,dy}{|x-y|^{n+2s}}.$$
We say that~$E$ is $s$-minimal in~$\Omega$ if~$\,{\rm Per}_s(E,\Omega)<+\infty$
and~$\,{\rm Per}_s(E,\Omega)\le \,{\rm Per}_s(F,\Omega)$ among all the sets~$F$
which coincide with~$E$ outside~$\Omega$.
With a slight abuse of language, when~$\Omega$
is unbounded, we say that~$E$ is $s$-minimal
in~$\Omega$ if it is $s$-minimal
in any bounded open subsets of~$\Omega$ (for a more
precise distinction between $s$-minimal sets
and locally $s$-minimal sets see e.g.~\cite{luca}).
\medskip
Problems related to the $s$-perimeter naturally arise
in several fields, such as the motion by nonlocal mean curvature
and the nonlocal Allen-Cahn equation, see e.g.~\cite{SOU, GAMMA}.
Also, the $s$-perimeter can be seen as a fractional interpolation
between the classical perimeter (corresponding to the case~$s\to1/2$)
and the Lebesgue measure (corresponding to the case~$s\to0$),
see e.g.~\cite{MAZYA, BREZIS, CV, MARTIN, DFPV}.
\medskip
The field of nonlocal minimal surfaces
is rich of open problems and surprising examples (see e.g.~\cite{LAWSON})
and the interior regularity theory of the nonlocal
minimal surfaces has been established in the plane and when the
fractional parameter is close enough to~$1/2$
(see~\cite{CV-REG, SV-REG}), but, as far as we know,
the boundary behavior of the nonlocal minimal surfaces
has not been studied till now.
\medskip
We show in this paper that the boundary datum
is not, in general, attained continuously. Indeed,
nonlocal minimal surfaces may stick at the boundary
and then detach from the boundary in a~$C^{1,\frac{1}{2}+s}$-fashion.
We will give concrete examples of this stickiness phenomenon
with explicit (and somehow optimal) estimates.
In particular, we will present stickiness phenomena to half-balls,
when the domain is a ball and the datum is a small half-ring,
and to the sides of a two-dimensional box, when the datum is
small on one side and large on the other side.
\medskip
Moreover, we study how small perturbations with compact support may affect
the boundary behavior of a given nonlocal minimal surface.
Quite surprisingly, these perturbations may produce stickiness effects
even in the case of flat objects and in low dimension.
For instance, adding a small perturbation to a half-space
in the plane produces a sticking effect,
with the size of the sticked portion proportional to a power
of the size of the perturbation.
We now present and discuss these results in further detail.
\subsection*{Stickiness to half-balls}
For any~$\delta>0$, we let
\begin{equation}\label{Kdelta}
K_\delta := \big( B_{1+\delta}\setminus B_1\big)\cap \{x_n<0\}.\end{equation}
We define~$E_\delta$ to be the set minimizing~$\,{\rm Per}_s(E,B_1)$
among all the sets~$E$ such that~$E\setminus B_1=K_\delta$.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{half-ball3.pdf}
\caption{The stickiness property in Theorem~\ref{89=THM}.}
\label{NESS}
\end{figure}
Notice that, in the local setting, the minimizer of the perimeter functional
that takes~$K_\delta$ as boundary value at~$\partial B_1$
is the flat set~$B_1\cap \{x_n<0\}$ (independently of~$\delta$).
The picture changes dramatically in the nonlocal framework,
since in this case the nonlocal minimizers stick at~$\partial B_1$
if~$\delta$ is suitably small, see Figure~\ref{NESS}. The formal statement of this
feature is the following:
\begin{theorem}\label{89=THM}
There exists~$\delta_o>0$,
depending on $s$ and $n$, such that
for any~$\delta\in(0,\delta_o]$ we have that
$$ E_\delta=K_\delta.$$
\end{theorem}
\subsection*{Stickiness to the sides of a box}
Given a large~$M>1$ we
consider the $s$-minimal set~$E_M$ in~$(-1,1)\times\mathbb R$
with datum outside~$(-1,1)\times\mathbb R$ given by
the jump
\begin{equation}\label{jump}
\begin{split}
& J_M:=J^-_M \cup J^+_M,\\
{\mbox{where }}\quad
& J^-_M:= (-\infty,-1]\times (-\infty,-M)
\\{\mbox{and }}\quad&J^+_M:= [1,+\infty)\times (-\infty,M).
\end{split}\end{equation}
We prove that, if~$M$ is large enough,
the minimal set~$E_M$ sticks at the boundary
(see Figure~\ref{REG:2}).
Moreover, the stickiness region gets close to the origin,
up to a power of~$M$.
The precise result is the following:
\begin{figure}
\centering
\includegraphics[width=9cm]{cylinder_new.pdf}
\caption{The stickiness property in Theorem~\ref{CYL-THEOR},
with~$\beta:={\frac{1+2s}{2+2s}}$.}
\label{REG:2}
\end{figure}
\begin{theorem}\label{CYL-THEOR}
There exist~$M_o>0$ and~$C_o\ge C_o'>0$, depending on~$s$, such that
if~$M\ge M_o$ then
\begin{eqnarray*}
&& [-1,1)\times [C_o M^{\frac{1+2s}{2+2s}},\,M]\subseteq E_M^c
\\{\mbox{and }}&& (-1,1]\times [-M,\,-C_o M^{\frac{1+2s}{2+2s}}]\subseteq E_M.
\end{eqnarray*}
Also, the exponent~${\frac{1+2s}{2+2s}}$ above is optimal.
For instance, if
either~$[-1,1)\times [b M^{\frac{1+2s}{2+2s}},\,M]\subseteq E_M^c$
or~$(-1,1]\times [-M,\,-C_o M^{\frac{1+2s}{2+2s}}]\subseteq E_M$
for some~$b\ge0$, then~$b\ge C_o'$.
\end{theorem}
\subsection*{Stickiness as~$s\to0^+$}
The stickiness properties of nonlocal minimal surfaces
are a purely nonlocal phenomenon and they become more evident
for small values of~$s$. To provide a confirming example,
we consider the boundary value given by
a sector in~$\mathbb R^2$ outside~$B_1$, i.e. we define
\begin{equation}\label{SECTOR-3456fgg}
\Sigma := \{ (x,y)\in\mathbb R^2\setminus B_1 {\mbox{ s.t. }}
x>0 {\mbox{ and }} y>0\}.\end{equation}
We show that as~$s\to0^+$ the $s$-minimal set in~$B_1$
with datum~$\Sigma$ sticks to~$\Sigma$,
and, more precisely, this stickiness already occurs
for a small~$s_o>0$ (see Figure~\ref{SIMP--1}).
\begin{figure}
\centering
\includegraphics[height=5.9cm]{quadrant.pdf}
\includegraphics[height=5.9cm]{quadrant_new.pdf}
\caption{The stickiness property in Theorem~\ref{SECTOR}.}
\label{SIMP--1}
\end{figure}
\begin{theorem}\label{SECTOR}
Let~$E_s$ be
the $s$-minimizer of~$\,{\rm Per}_s(E,B_1)$
among all the sets~$E$ such that~$E\setminus B_1=\Sigma$.
Then, there exists~$s_o>0$ such that for any~$s\in(0,s_o]$
we have that~$E_s=\Sigma$.
\end{theorem}
\subsection*{Instability of the flat fractional minimal surfaces}
Rather surprisingly, one of our results states that
the flat lines are ``unstable'' fractional minimal surfaces,
in the sense that an arbitrarily small and compactly supported
perturbation can cause a boundary stickiness phenomenon.
We are also able to give a quantification
of the size of the stickiness in terms of the size
of the perturbation: namely the size of the stickiness
is bounded from below by the size of the perturbation
to the power~$\frac{2+\epsilon_0}{1-2s}$,
for any fixed~$\epsilon_0$ arbitrarily small (see Figure~\ref{CAMB}).
We observe that this power tends to~$+\infty$
as~$s\to1/2$, which is consistent with the fact that
classical minimal surfaces do not stick.
The precise result that we obtain is the following:
\begin{figure}
\centering
\includegraphics[height=5.9cm]{FARATH.pdf}
\caption{The stickiness/instability property in Theorem~\ref{UNS},
with~$\beta:=\frac{2+\epsilon_0}{1-2s}$.}
\label{CAMB}
\end{figure}
\begin{theorem}\label{UNS}
Fix~$\epsilon_0>0$ arbitrarily small.
Then, there exists~$\delta_0>0$, possibly depending on~$\epsilon_0$,
such that for any~$\delta\in(0,\delta_0]$ the following statement holds true.
Assume that~$F\supset H\cup F_-\cup F_+$, where~$H:=\mathbb R\times(-\infty,0)$,
$F_-:=
(-3,-2)\times [0,\delta)$ and
$F_+:= (2,3)\times [0,\delta)$. Let~$E$ be the
$s$-minimal set in~$(-1,1)\times\mathbb R$ among all the sets that coincide
with~$F$ outside~$(-1,1)\times\mathbb R$. Then
$$ E\supseteq (-1,1)\times (-\infty, \delta^{\frac{2+\epsilon_0}{1-2s}} ].$$
\end{theorem}
The proof of Theorem~\ref{UNS} is rather delicate
and it is based on the construction of suitable auxiliary barriers,
which we believe are interesting in themselves.
These barriers are used to detach a portion of the set
in a neighborhood of the origin and their construction
relies on some compensations of nonlocal integral terms.
As a matter of fact, the compactly supported barriers are
obtained by glueing other auxiliary barriers with polynomial growth
(the latter barriers are somehow ``self-sustaining solutions''
and can be seen as the geometric counterparts of
the $s$-harmonic function~$x_+^s$).
\medskip
Though quite surprising at a first glance, the sticking
effects that we present in this paper
have some (at least vague)
heuristic explanations.
Indeed, first of all,
the contribution to the fractional mean curvature
which comes from far may bend a nonlocal minimal surface
towards the boundary of the domain: then, the points in
the vicinity of the domain may end up receiving
a contribution which is incompatible with the vanishing
of the fractional mean curvature, due to some transverse
intersection between the datum and the domain itself,
thus forcing these points to stick at the boundary.
Another heuristic explanation of the stickiness phenomenon
comes from the different fractional scalings
that the problem exhibits at different scales.
On the one hand, vanishing of
the fractional mean curvature corresponds to a $s$-harmonicity
property
(i.e. a harmonicity with respect to the fractional operator~$(-\Delta)^s$)
for the characteristic function of the $s$-minimal set,
with~$s\in(0,1/2)$.
If the boundary of the set is the graph
of a smooth function~$u$, this gives an equation for~$u$
whose linearization corresponds to~$(-\Delta)^{\frac{1}{2}+s}$,
which would correspond, roughly speaking, to a regularity theory
of order~$C^{\frac{1}{2}+s}$ at the boundary. On the other hand,
nonlocal minimal surfaces detach from free boundaries in
a~$C^{1,\frac{1}{2}+s}$-fashion (see~\cite{PRO}), which suggests that the linearized
equation of the graph is not a good approximation for the boundary
behavior.
\medskip
The rest of the paper is organized as follows.
In Section~\ref{S:HALF-BALL}, we discuss the case of the stickiness to
a half-ball and we prove Theorem~\ref{89=THM}.
Then, Section~\ref{SEC:BOX} considers the case
of a two-dimensional box with high oscillating datum,
providing the proof of Theorem~\ref{CYL-THEOR}.
The asymptotics as~$s\to0$ is presented in Section~\ref{SEC:to0}.
The second part of the paper is devoted to the proof of
Theorem~\ref{UNS}. In particular, Sections~\ref{SEC:BAR:PTWL}, \ref{sec:GROW:R}
and~\ref{SEC:CMP:SUPP} are devoted to the construction
of the auxiliary barriers. More precisely,
in Section~\ref{SEC:BAR:PTWL} we construct barriers with a linear
growth, by superposing straight lines with slowly varying slopes;
then, in Section~\ref{sec:GROW:R}, we
glue the barrier with linear growth with a power-like function
(this is needed to obtain sharper estimates on the size of the glueing)
and in Section~\ref{SEC:CMP:SUPP} we adapt this construction
to build barriers that are compactly supported.
This will allow us to prove Theorem~\ref{UNS}
in Section~\ref{INST:SEC}. The paper ends with an appendix
that contains a simple, but general, symmetry property,
and an alternative proof of an integral identity.
\section{Stickiness to half-balls}\label{S:HALF-BALL}
This section is devoted to the analysis of the stickiness
phenomena to the half-ball, caused by a small half-ring
as external datum. The main goal of this part is to prove
Theorem~\ref{89=THM}. For this, we
take~$K_\delta$ as in~\eqref{Kdelta}, i.e.
$$ K_\delta := \big( B_{1+\delta}\setminus B_1\big)\cap \{x_n<0\}$$
and~$E_\delta$ to be the set minimizing~$\,{\rm Per}_s(E,B_1)$
among all the sets~$E$ such that~$E\setminus B_1=K_\delta$.
We make some auxiliary observations.
First of all, we check that the~$s$-perimeter of~$K_\delta$
(and then of the minimizer)
must be small if so is~$\delta$:
\begin{lemma}\label{89=IO}
For any~$\varepsilon>0$ there exists~$\delta_\varepsilon>0$ such that
for any~$\delta\in(0,\delta_\varepsilon]$ we have that
$$ \,{\rm Per}_s ( K_\delta,B_1)\le \varepsilon.$$
\end{lemma}
\begin{proof} We have
$$ \,{\rm Per}_s ( K_\delta,B_1) = L(B_1,K_\delta)\le
\iint_{B_1\times \big( B_{1+\delta}\setminus B_1\big)}
\frac{dx\,dy}{|x-y|^{n+2s}}.$$
Now we observe that
\begin{equation}\label{CVFI}
(0,+\infty)\ni \iint_{B_1\times \big( B_{2}\setminus B_1\big)}
\frac{dx\,dy}{|x-y|^{n+2s}}
= \lim_{\delta\to0^+}
\iint_{B_1\times \big( B_{2}\setminus B_{1+\delta}\big)}
\frac{dx\,dy}{|x-y|^{n+2s}}.\end{equation}
Indeed, the first integral in~\eqref{CVFI} is finite,
see for instance Lemma~11 in~\cite{CV} (applied here with~$\varepsilon:=1$,
$\Omega:=B_2$ and~$F:=B_1$).
As a consequence of~\eqref{CVFI}, for any~$\varepsilon>0$
there exists~$\delta_\varepsilon>0$ such that
for any~$\delta\in(0,\delta_\varepsilon]$ we have
$$ \left| \iint_{B_1\times \big( B_{2}\setminus B_{1}\big)}
\frac{dx\,dy}{|x-y|^{n+2s}}-
\iint_{B_1\times \big( B_{2}\setminus B_{1+\delta}\big)}
\frac{dx\,dy}{|x-y|^{n+2s}}\right|\le\varepsilon,$$
which gives the desired result.
\end{proof}
Next result proves that the boundary of the minimal set~$E_\delta$
can only lie in the neighborhood of~$\partial B_1$, if~$\delta$
is small enough. More precisely:
\begin{lemma}\label{89=QW}
For any~$\varepsilon\in(0,1)$ there exists~$\delta_\varepsilon>0$ such that
for any~$\delta\in(0,\delta_\varepsilon]$ we have that
$$ (\partial E_\delta)\cap B_{1-\varepsilon} =\varnothing. $$
\end{lemma}
\begin{proof}
We observe that it is enough to prove the desired claim for small~$\varepsilon$
(since this would imply the claim for bigger~$\varepsilon$).
The proof is by contradiction.
Suppose that there exists~$p\in (\partial E_\delta)\cap B_{1-\varepsilon}$.
Then~$B_{\varepsilon/2}(p)\subset B_1$ and so, by the Clean Ball Condition
(see Corollary 4.3 in~\cite{CRS}), there exist~$p_1$, $p_2\in B_1$ such that
$$ B_{c\varepsilon}(p_1)\subset E\cap B_{\varepsilon/2}(p)\qquad
{\mbox{ and }}\qquad
B_{c\varepsilon}(p_2)\subset E^c\cap B_{\varepsilon/2}(p),$$
for a suitable constant~$c>0$.
In particular, both~$B_{c\varepsilon}(p_1)$ and~$B_{c\varepsilon}(p_2)$ lie inside~$B_1$,
and if~$x\in B_{c\varepsilon}(p_1)$ and~$y\in B_{c\varepsilon}(p_2)$ then~$|x-y|\le\varepsilon$.
As a consequence
$$ \,{\rm Per}_s (E_\delta,B_1)\ge L\big( B_{c\varepsilon}(p_1),B_{c\varepsilon}(p_2)\big)
\ge \frac{|B_{c\varepsilon}(p_1)|\,|B_{c\varepsilon}(p_2)|}{\varepsilon^{n+2s}} = c_o \varepsilon^{n-2s},$$
for some~$c_o>0$.
On the other hand, by Lemma~\ref{89=IO} (used here with~$\varepsilon^n$
in the place of~$\varepsilon$),
we have that~$\,{\rm Per}_s (E_\delta,B_1)\le
\,{\rm Per}_s ( K_\delta,B_1)\le\varepsilon^n$ provided that~$\delta$
is suitably small with respect to~$\varepsilon$. As a consequence,
we obtain that~$\varepsilon^n \ge c_o \varepsilon^{n-2s}$,
which is a contradiction if~$\varepsilon$ is small enough.
\end{proof}
The statement of Lemma~\ref{89=QW} can be better specified,
as follows:
\begin{corollary}\label{89=QW-2}
For any~$\varepsilon\in(0,1)$ there exists~$\delta_\varepsilon>0$ such that
for any~$\delta\in(0,\delta_\varepsilon]$ we have that
$$ E_\delta\cap B_{1-\varepsilon} =\varnothing. $$
\end{corollary}
\begin{proof} Without loss of generality, we
may suppose that~$\varepsilon\in(0,1/2)$.
The proof is by contradiction.
Suppose that~$ E_\delta\cap B_{1-\varepsilon} \ne\varnothing$.
Then, by Lemma~\ref{89=QW}, we have that~$ B_{1-\varepsilon}\subseteq
E_\delta$. Moreover, if we set
$$ H := \big( B_{2}\setminus B_1\big)\cap \{x_n>0\},$$
we have that~$H\subseteq E_\delta^c$.
As a consequence,
$$ \,{\rm Per}_s(E_\delta,B_1)\ge L(B_{1-\varepsilon}, H)\ge L(B_{1/2},H)\ge c,$$
for some~$c>0$. This is in contradiction with Lemma~\ref{89=IO}
and so it proves the desired result.
\end{proof}
With this, we are in the position of completing the proof
of Theorem~\ref{89=THM}:
\begin{proof}[Proof of Theorem~\ref{89=THM}]
We need to show that~$E_\delta\cap B_1=\varnothing$.
By contradiction, suppose not. Then there exists
\begin{equation}\label{TY67}
p\in E_\delta\cap B_1.\end{equation}
By Corollary~\ref{89=QW-2},
we know that
\begin{equation}\label{IOP89}
{\mbox{$B_r\subset E_\delta^c$ if $r\in(0,1-\varepsilon)$.}}\end{equation}
We enlarge~$r$ till $B_r$ hits~$\partial E_\delta$. That is, by~\eqref{TY67},
there exists~$\rho \in[1-\varepsilon,1)$ such that~$B_{\rho}\subset E_\delta^c$
and there exists~$q\in (\partial B_{\rho})\cap (\partial E_\delta)$
(see Figure~\ref{HB1}).
\begin{figure}
\centering
\includegraphics[height=5.9cm]{half-ball1.pdf}
\includegraphics[height=5.9cm]{half-ball2.pdf}
\caption{Touching the set~$E_\delta$ coming from the origin.}
\label{HB1}
\end{figure}
Therefore, using the Euler-Lagrange equation in the viscosity sense
(see Theorem~5.1 in~\cite{CRS}), we conclude that
\begin{equation}\label{89=001}
\int_{\mathbb R^n} \frac{\chi_{E^c_\delta}(y)- \chi_{E_\delta}(y)}{|q-y|^{n+2s}}\,dy\le0.\end{equation}
By \eqref{IOP89}, we know that
$$ E_\delta\subseteq (B_1\setminus B_{\rho})\cup K_\delta
\subseteq B_{1+\delta}\setminus B_{\rho}$$
and so
\begin{equation}\label{89=002}
\int_{\mathbb R^n} \frac{\chi_{E_\delta}(y)-\chi_{E^c_\delta}(y)}{|q-y|^{n+2s}}
\le \int_{B_{1+\delta}\setminus B_{\rho}} \frac{dy}{|q-y|^{n+2s}}
-\int_{B_{\rho}} \frac{dy}{|q-y|^{n+2s}}.\end{equation}
In addition, if~$y\in B_{1/2}$, then~$|q-y|\le |q|+|y|<2$ and so
\begin{equation} \label{09ip}
\int_{B_{1/2}} \frac{dy}{|q-y|^{n+2s}} \ge \tilde c,\end{equation}
for some~$\tilde c>0$.
Now we define~$\lambda:=(\varepsilon+\delta)^\frac{1}{2(n+2s)}$. We
notice that~$\lambda$ is small if so are~$\varepsilon$ and~$\delta$,
and so~$B_\lambda(q)\subset B_{1/2}^c$. Then, formula~\eqref{09ip}
gives that
$$ \int_{B_{\rho}} \frac{dy}{|q-y|^{n+2s}}\ge
\tilde c+ \int_{B_\lambda(q)\cap B_{\rho}} \frac{dy}{|q-y|^{n+2s}}.$$
This, \eqref{89=001} and~\eqref{89=002}
give that
\begin{equation}\label{89=003}
\int_{B_{1+\delta}\setminus B_{\rho}} \frac{dy}{|q-y|^{n+2s}}-
\int_{B_{\lambda}(q)\cap B_{\rho}} \frac{dy}{|q-y|^{n+2s}}
\ge \tilde c.
\end{equation}
Now we define
$$ A_1:=\big( B_{1+\delta}\setminus B_{\rho} \big)\cap B_{\lambda}(q)
\quad{\mbox{ and }} \quad A_2:=
\big( B_{1+\delta}\setminus B_{\rho} \big)\setminus B_{\lambda}(q).$$
We notice that
$$ \int_{A_2} \frac{dy}{|q-y|^{n+2s}}\le\frac{|A_2|}{\lambda^{n+2s}}
\le \frac{|B_{1+\delta}\setminus B_{\rho}|}{\lambda^{n+2s}}
\le \frac{C\,(\varepsilon+\delta)}{\lambda^{n+2s}} = C\,\sqrt{\varepsilon+\delta},$$
for some~$C>0$. Hence, \eqref{89=003}
becomes
\begin{equation}\label{89=004}
\int_{A_1} \frac{dy}{|q-y|^{n+2s}}-
\int_{B_{\lambda}(q)\cap B_{\rho}} \frac{dy}{|q-y|^{n+2s}}
\ge \frac{\tilde c}{2}.\end{equation}
Now we set
$$ A_{1,1}:=A_1\cap B_{\rho}(2q)
\quad{\mbox{ and }} \quad A_{1,2}:= A_1\setminus B_{\rho}(2q),$$
see again Figure~\ref{HB1}.
We remark that~$B_{\rho}(2q)$ is tangent to~$B_{\rho}$ at the point~$q$,
and~$A_{1,1}\subseteq B_{\lambda}(q)\cap B_{\rho}(2q)$.
Therefore, by symmetry
\begin{equation}\label{0oiGH}
\int_{A_{1,1}} \frac{dy}{|q-y|^{n+2s}}\le
\int_{B_{\lambda}(q)\cap B_{\rho}(2q)} \frac{dy}{|q-y|^{n+2s}}
=
\int_{B_{\lambda}(q)\cap B_{\rho}} \frac{dy}{|q-y|^{n+2s}}.\end{equation}
Now we observe that~$A_{1,2}$ is trapped between~$B_{\rho}$
and~$B_{\rho}(2q)$, and it lies in~$B_\lambda(q)$
therefore (see e.g. Lemma~3.1 in~\cite{nostro})
$$ \int_{A_{1,2}} \frac{dy}{|q-y|^{n+2s}}\le
C\rho^{-2s} \lambda^{1-2s}\le C\lambda^{1-2s} =C
\,(\varepsilon+\delta)^{\frac{1-2s}{2(n+2s)}},$$
up to renaming constants.
The latter estimate and~\eqref{0oiGH} give
$$ \int_{A_1} \frac{dy}{|q-y|^{n+2s}}\le
\int_{B_{\lambda}(q)\cap B_{\rho}} \frac{dy}{|q-y|^{n+2s}}+
C\,(\varepsilon+\delta)^{\frac{1-2s}{2(n+2s)}}.$$
By inserting this information into~\eqref{89=004},
we obtain~$2C\,(\varepsilon+\delta)^{\frac{1-2s}{2(n+2s)}}\ge\tilde c$,
which leads to a contradiction by choosing~$\varepsilon$ small enough
(and thus~$\delta\le\delta_\varepsilon$ small).
\end{proof}
\section{Stickiness to the sides of a box}\label{SEC:BOX}
In this section, we discuss the stickiness properties
to the sides of a box with high oscillatory external data
and we prove Theorem~\ref{CYL-THEOR}.
To this
goal,
we recall that the set~$J_M$ has been defined
in~\eqref{jump} and~$E_M$ is the $s$-minimal set
in~$(-1,1)\times\mathbb R$
with datum outside~$(-1,1)\times\mathbb R$ equal to~$J_M$.
We first establish an easier version of
Theorem~\ref{CYL-THEOR}, in which the sticking size is proved
to be at least of the order of the oscillation
(then, a refined estimate will lead to the proof of
Theorem~\ref{CYL-THEOR}).
\begin{proposition}\label{CYL}
There exist~$M_o>0$, $c_o\in(0,1)$, depending on~$s$, such that
if~$M\ge M_o$ then
\begin{eqnarray}
&& [-1,1)\times [c_o M,M]\subseteq E_M^c \label{SL-011}
\\{\mbox{and }}&& (-1,1]\times [-M,-c_o M]\subseteq E_M.
\label{SL-012}
\end{eqnarray}
\end{proposition}
\begin{proof} We denote coordinates in~$\mathbb R^2$ by~$x=(x_1,x_2)$.
We take~$\varepsilon_o>0$, to be chosen conveniently small
in the sequel. Let~$t\in[0,\varepsilon_o^2]$.
We considers balls of radius~$\varepsilon_o M$ with center lying on the
straight line~$\{ x_2=(1-t)M\}$. The idea of the proof
is to slide a ball of this type from left to right till
we touch~$\partial E_M$. We will show that the touching point
can only occur along the boundary~$\{x_1=1\}$.
Hence, by varying~$t\in[0,\varepsilon_o^2]$, we obtain that~$[-1,1)\times
[(1-\varepsilon_o^2) M,M]$ is contained in~$E_M^c$. This would complete
the proof of~\eqref{SL-011}
(and the proof of~\eqref{SL-012} is similar).
The details of the proof of \eqref{SL-011} are the following.
We fix~$t\in[0,\varepsilon_o^2]$.
If~$x_1<-M-2$, then
the ball~$B_{\varepsilon_o M} ( x_1, (1-t)M)$ lies in~$(-\infty,-2)\times\mathbb R$,
and so its closure is contained in~$E_M^c$.
Hence, we consider~$\ell\ge -M-2$ such that~$
\overline{B_{\varepsilon_o M} ( \ell, (1-t)M)}\subseteq E_M^c$
for any~$x_1<\ell$ and there exists~$q=(q_{1},q_{2})\in
(\partial E_M)\cap (\partial B_{\varepsilon_o M} ( \ell, (1-t)M))$.
The proof of~\eqref{SL-011} is complete if we show that
\begin{equation}\label{SL-013}
q_{1} \ge 1.
\end{equation}
To prove this, we argue by contradiction.
If not, then~$q_1\in [-1,1)$, therefore,
by the Euler-Lagrange inequality (see
Theorem~5.1 in~\cite{CRS}),
\begin{equation}\label{89=001=BIS}
\int_{\mathbb R^2} \frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
\le0.\end{equation}
Now we denote by~$z:=(\ell, (1-t)M))$ the center
of the touching ball.
We also consider the extremal point of the touching
ball on the right, that we
denote by~$p:= z+(\varepsilon_o M,0)$.
We claim that
\begin{equation}\label{Y7}
|q_2-p_2|\le 8\sqrt{\varepsilon_o M}.\end{equation}
To prove this, we observe that,
by construction,
both~$q$ and~$p$ lie in~$[-1,1]\times\mathbb R$, hence~$|q_1|$, $|p_1|\le1$,
consequently
\begin{equation}\label{q1p12}
|q_1-p_1|\le2.\end{equation}
Also, both~$q$ and~$p$ lie on the boundary
of the touching ball, namely~$|q-z|=\varepsilon_o M=|p-z|$, therefore
\begin{eqnarray*}
&& 0=|q-z|^2 - |p-z|^2
= |q|^2 -2q\cdot z -|p|^2 +2p\cdot z
= (q-p)\cdot (q+p-2z)
\\ &&\qquad =
(q_1-p_1)(q_1+p_1-2z_1)+
(q_2-p_2)(q_2+p_2-2z_2)
\\ &&\qquad=
(q_1-p_1)(q_1-p_1+2\varepsilon_o M)+
(q_2-p_2)(q_2-p_2)
\\ &&\qquad\ge
-4(1+\varepsilon_o M)+|q_2-p_2|^2.
\end{eqnarray*}
This establishes~\eqref{Y7}, provided that~$M$
is large enough (possibly in dependence of~$\varepsilon_o$).
Now we consider the symmetric ball to the touching ball,
with respect to the touching point~$q$. That is, we define~$\bar z:=
z+2(q-z)$ and consider the ball~$B_{\varepsilon_o M}(\bar z)$.
We remark that
\begin{equation}
{\mbox{$B_{\varepsilon_o M}(z)$ and~$B_{\varepsilon_o M}(\bar z)$
are tangent to each other at~$q$.}}\end{equation}
We also claim that
\begin{equation}\label{Y8}
B_{\varepsilon_o M}(\bar z)\cap
\left\{ x_2> \bar z_2+2\varepsilon_o^2 M\right\}
\subseteq \{ x_2>M\}.
\end{equation}
To prove this, we observe that
$$ -\varepsilon_o^2 M -16\sqrt{\varepsilon_o M}+2\varepsilon_o^2 M
= \varepsilon_o^2 M \left( 1 -\frac{16}{\varepsilon_o^{3/2}\sqrt{M}}\right)>0,$$
if~$M$ is large enough.
Hence, recalling~\eqref{Y7},
\begin{eqnarray*}
&& \bar z_2+ 2\varepsilon_o^2 M =
z_2+2(q_2-z_2) + 2\varepsilon_o^2 M
= (1-t)M+2(q_2-p_2) + 2\varepsilon_o^2 M\\
&&\qquad\ge (1-\varepsilon_o^2)M -16\sqrt{\varepsilon_o M}+ 2\varepsilon_o^2 M> M.
\end{eqnarray*}
This proves~\eqref{Y8}.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{regions.pdf}
\caption{The partition of the plane needed for the proof
of Proposition~\ref{CYL}.}
\label{REG}
\end{figure}
Now we decompose~$\mathbb R^2$ into five nonoverlapping
regions. Namely, we consider
\begin{eqnarray*}
&& R_1:=B_{\varepsilon_o M}(z),\\
&& R_2:=B_{\varepsilon_o M}(\bar z)\cap
\left\{ x_2> \bar z_2+ 2\varepsilon_o^2 M\right\}\\
{\mbox{and }}&&
R_3:=
B_{\varepsilon_o M}(\bar z)\cap
\left\{ x_2\le \bar z_2+ 2\varepsilon_o^2 M\right\}.\end{eqnarray*}
Then we define~$D:=B_{\varepsilon_o M}(z)\cup B_{\varepsilon_o M}(\bar z)$,
$K$ the convex hull
of~$D$ and~$R_4:=K\setminus D$. Finally,
we set~$R_5:=\mathbb R^2 \setminus K$ and consider the partition of~$\mathbb R^2$
given by the regions~$R_1,\dots,R_5$.
We consider the contribution to the integral
in~\eqref{89=001=BIS}
given by these regions. The regions~$R_1$, $R_2$ and~$R_3$
will be considered together: namely, $R_1\subseteq E^c_M$,
and, by~\eqref{Y8}, also~$R_2\subseteq E^c_M$. Therefore,
by symmetry
\begin{equation}\label{Y9-K}
\int_{R_1\cup R_2\cup R_3}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\ge
\int_{R_1\cup R_2}
\frac{dy}{|q-y|^{2+2s}}
-
\int_{R_3}
\frac{dy}{|q-y|^{2+2s}} =
\int_{R_2}
\frac{2\,dy}{|q-y|^{2+2s}}
.\end{equation}
Now, for~$y\in R_2$, we consider the change of variable~$\tilde y=T(y)
:=(y-q)/(\varepsilon_o M)$. We have that
\begin{equation}\label{INC0}\begin{split}
T(R_2) \,&=\,
B_{1}\left( \frac{q-z}{\varepsilon_o M}\right)
\cap
\left\{ \tilde y_2> \frac{q_2-z_2}{\varepsilon_o M}+2\varepsilon_o
\right\} \\
&\supseteq\,
B_{1}\left( \frac{q-z}{\varepsilon_o M}\right)
\cap
\left\{ \tilde y_2> 3\varepsilon_o\right\},
\end{split}\end{equation}
where we used again~\eqref{Y7}
in the last inclusion (provided that~$\varepsilon_o$ is sufficiently
small and~$M$ is sufficiently large, possibly in dependence
of~$\varepsilon_o$).
Now we claim that
\begin{equation}\label{INC-1}
B_{\varepsilon_o}(5\varepsilon_o,5\varepsilon_o)
\subseteq B_{1}\left( \frac{q-z}{\varepsilon_o M}\right)
\cap
\left\{ \tilde y_2> 3\varepsilon_o \right\}
.\end{equation}
To prove this, it is enough to take~$\eta\in B_{\varepsilon_o}$
and show that
\begin{equation} \label{INC-2}
(5\varepsilon_o,5\varepsilon_o)+\eta\in
B_1\left( \frac{q-z}{\varepsilon_o M}\right).\end{equation}
For this, we use~\eqref{Y7} to observe that
\begin{equation}\label{PHG}
|q_1-z_1|^2=|q-z|^2 -|q_2-z_2|^2 \ge
(\varepsilon_o M)^2-64\varepsilon_o M.\end{equation}
Moreover, by~\eqref{q1p12},
$$ q_1-z_1=q_1-p_1+p_1-z_1=q_1-p_1+\varepsilon_o M\ge \varepsilon_o M-2>0.$$
Hence, \eqref{PHG}
gives that
$$ \frac{q_1-z_1}{\varepsilon_o M} =\frac{|q_1-z_1|}{\varepsilon_o M}
\ge \sqrt{1 -\frac{64}{\varepsilon_o M}}
\ge 1 -\frac{128}{\varepsilon_o M},$$
if~$M$ is large enough.
In particular
$$ \frac{q_1-z_1}{\varepsilon_o M} - 5\varepsilon_o -\eta_1
\ge 1 -\frac{128}{\varepsilon_o M} -6\varepsilon_o
\ge\frac12 -\frac{128}{\varepsilon_o M}>0,$$
provided that~$\varepsilon_o$ is small enough and~$M$ large enough
(possibly depending on~$\varepsilon_o$). Therefore
$$ \left|
\frac{q_1-z_1}{\varepsilon_o M} - 5\varepsilon_o -\eta_1\right|
=
\frac{q_1-z_1}{\varepsilon_o M} - 5\varepsilon_o -\eta_1 \le
\frac{|q_1-z_1|}{\varepsilon_o M} - 4\varepsilon_o \le 1-4\varepsilon_o.$$
In addition, by~\eqref{Y7},
$$
\left|
\frac{q_2-z_2}{\varepsilon_o M} - 5\varepsilon_o -\eta_2\right|\le
\frac{|q_2-z_2|}{\varepsilon_o M} +6\varepsilon_o\le 7\varepsilon_o.$$
Therefore
\begin{eqnarray*}
&& \left|
\frac{q-z}{\varepsilon_o M} - (5\varepsilon_o,5\varepsilon_o) -\eta\right|^2 \le \left(
1-4\varepsilon_o\right)^2
+ \left( 7\varepsilon_o\right)^2 \\
&&\qquad= 1 -8\varepsilon_o + 16\varepsilon_o^2+49\varepsilon_o^2
< 1
\end{eqnarray*}
if~$\varepsilon_o$ is small enough.
This establishes~\eqref{INC-2} and therefore~\eqref{INC-1}.
{F}rom~\eqref{INC0} and~\eqref{INC-1}, we see that
$$ T(R_2) \supseteq B_{\varepsilon_o}(5\varepsilon_o,5\varepsilon_o)$$
and then
\begin{equation}\label{INC-12} \int_{R_2} \frac{dy}{|q-y|^{2+2s}}=
\frac{1}{(\varepsilon_o M)^{2s}}\int_{T(R_2)}
\frac{d\tilde y}{|\tilde y|^{2+2s}}\ge
\frac{1}{(\varepsilon_o M)^{2s}}\int_{
B_{\varepsilon_o}(5\varepsilon_o,5\varepsilon_o) }
\frac{d\tilde y}{|\tilde y|^{2+2s}}.\end{equation}
Now, if~$\tilde y\in B_{\varepsilon_o}(5\varepsilon_o,5\varepsilon_o)$
then~$|\tilde y|\le \varepsilon_o+|(5\varepsilon_o,5\varepsilon_o)|\le 10\varepsilon_o$,
and then~\eqref{INC-12} gives that
$$ \int_{R_2} \frac{dy}{|q-y|^{2+2s}}\ge
\frac{\tilde c}{\varepsilon_o^{4s} M^{2s}},$$
for some~$\tilde c>0$.
By inserting this into~\eqref{Y9-K}
we conclude that
\begin{equation}\label{Y10-K}
\int_{R_1\cup R_2\cup R_3}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\ge
\frac{\tilde c}{\varepsilon_o^{4s} M^{2s}}.\end{equation}
Moreover (see e.g. Lemma~3.1 in~\cite{nostro}
with~$R:=\varepsilon_o M$ and~$\lambda:=1$),
we see that
\begin{equation}\label{Y10-K-2}
\left|\int_{R_4}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\right|\le
\int_{R_4}
\frac{dy}{|q-y|^{2+2s}}\le
\frac{C}{\varepsilon_o^{2s} M^{2s}},\end{equation}
for some~$C>0$.
Furthermore, the distance from~$q$ to any point of~$R_5$
is at least~$\varepsilon_o M$, therefore~$R_5\subseteq \mathbb R^2\setminus B_{\varepsilon_o M}(q)$,
and
$$ \left|\int_{R_5}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\right|\le
\int_{\mathbb R^2\setminus B_{\varepsilon_o M}(q)}
\frac{dy}{|q-y|^{2+2s}}=
\frac{\tilde C}{\varepsilon_o^{2s} M^{2s}},$$
for some~$\tilde C>0$.
By combining the latter estimate with~\eqref{Y10-K} and~\eqref{Y10-K-2},
we obtain that
$$ \int_{\mathbb R^2}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\ge
\frac{1}{\varepsilon_o^{2s} M^{2s}}\left( \frac{\tilde c}{\varepsilon_o^{2s}}-
C-\tilde C\right)>0,$$
provided that~$\varepsilon_o$ is suitably small.
This estimate is in contradiction with~\eqref{89=001=BIS}
and therefore the proof of~\eqref{SL-013} is complete.
\end{proof}
The result in Proposition~\ref{CYL} can be refined.
Namely, not only the optimal set~$E_M$ in Proposition~\ref{CYL}
sticks for an amount of order~$M$ is a box of side~$M$,
but it sticks up to an order of~$M^{\frac{1+2s}{2+2s}}$
from the origin, as the following
Proposition~\ref{CYL-RAFF} points out.
As a matter of fact, the exponent~${\frac{1+2s}{2+2s}}$
is sharp, as we will prove in the subsequent
Proposition~\ref{PR-3.2}.
\begin{proposition}\label{CYL-RAFF}
There exist~$M_o$, $C_o>0$, depending on~$s$, such that
if~$M\ge M_o$ then
\begin{eqnarray}
&& [-1,1)\times [C_o M^{\frac{1+2s}{2+2s}},\,M]\subseteq E_M^c \label{XSL-011}
\\{\mbox{and }}&& (-1,1]\times [-M,\,-C_o M^{\frac{1+2s}{2+2s}}]\subseteq E_M.
\label{XSL-012}
\end{eqnarray}
\end{proposition}
\begin{proof}
We let~$\beta:={\frac{1+2s}{2+2s}}$.
We focus on the proof of~\eqref{XSL-011}
(the proof of~\eqref{XSL-012} is similar).
The proof is based on a sliding method: we will
consider a suitable surface and we slide it from left to
right in order to ``clean'' the portion
of space~$[-1,1)\times [C_o M^{\beta},M]$.
As a matter of fact, by Proposition~\ref{CYL},
it is enough to take care of~$[-1,1)\times [C_o M^{\beta},\,c_o M]$,
with~$c_o\in(0,1)$.
For this we fix any
\begin{equation}\label{q2whee-pre}
t\in [C_o M^\beta,\,c_o M]\end{equation}
and, for any~$\mu\in\mathbb R$,
we define
$$ S_\mu := B_{M^{2\beta}} (\mu-M^{2\beta},t)
\cap \{|x_2-t|<4M^\beta\}.$$
Notice that if~$\mu<-1$ then
$$S_\mu\subseteq (-\infty,-1)\times \{|x_2-t|<4M^\beta\}
\subseteq E_M^c.$$
Therefore we increase~$\mu$ till~$S_\mu$ touches~$\partial E_M$.
This value of $\mu$ will be fixed from now on.
We observe that Proposition~\ref{CYL-RAFF} is proved if we show that
$\mu=1$. So we assume by contradiction that~$\mu\in[-1,1)$.
By construction, we have that
\begin{equation}\label{9uIIOk}
S_\mu\subseteq E_M^c
\end{equation}
and there exists~$q\in (\partial S_\mu)\cap(\partial E_M)$,
with~$q_1\in [-1,1)$.
We claim that
\begin{equation}\label{q2whee}
|q_2-t|\le 2 M^\beta.
\end{equation}
To prove this, we observe that~$|q_1-\mu+M^{2\beta}|\ge
M^{2\beta} -|q_1|-|\mu|\ge M^{2\beta}-2$. Moreover,
$q\in \partial S_\mu\subseteq\overline{
B_{M^{2\beta}} (\mu-M^{2\beta},t)}$, therefore
\begin{eqnarray*}
M^{4\beta} \ge \big|q-(\mu-M^{2\beta},t)\big|^2
\ge (M^{2\beta}-2)^2 + |q_2-t|^2
\ge M^{4\beta} -4M^{2\beta}+ |q_2-t|^2,
\end{eqnarray*}
from which we obtain~\eqref{q2whee}.
Now, using the Euler-Lagrange equation in the viscosity sense
(see Theorem~5.1 in~\cite{CRS}), we see that
\begin{equation}\label{89=001=E:L}
\int_{\mathbb R^n} \frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\le0.
\end{equation}
We first estimate the contribution to the integral
above coming from~$B_{M^\beta}(q)$.
For this, we consider the symmetric point
of~$z:=(\mu-M^{2\beta},t)$ with respect to~$q$,
namely we set~$z':= z+2(q-z)$. We also consider the
ball~$B':=B_{M^{2\beta}}(z')$.
Notice that~$B_{M^{2\beta}}(z)$ and~$B'$ are tangent one to the other
at~$q$. We define~$A_1:= B_{M^{2\beta}}(z)\cap B_{M^\beta}(q)$,
$A_2:= B'\cap B_{M^\beta}(q)$ and~$A_3:= B_{M^\beta}(q)\setminus
(A_1\cup A_2)$.
Hence (see e.g.
Lemma~3.1 in~\cite{nostro},
used here with~$R:=M^{2\beta}$ and~$\lambda:=M^{-\beta}$),
we obtain that
\begin{equation}\label{89=001=E:L-1}
\left|\int_{A_3} \frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
\right|\le
\int_{A_3} \frac{dy}{|q-y|^{2+2s}}\le C M^{-\beta(1+2s)}.
\end{equation}
Now we observe that
\begin{equation}\label{A1wh}
A_1\subseteq E_M^c.
\end{equation}
For this, let~$y\in A_1$.
Then~$|y-q|< M^\beta$. Therefore, recalling~\eqref{q2whee},
$$ |y_2-t|\le |y_2-q_2|+|q_2-t| < M^\beta + 2 M^\beta < 4M^\beta.$$
Since also~$y\in B_{M^{2\beta}}(z)$, we obtain that~$y\in S_\mu$.
Then we use~\eqref{9uIIOk}
and we finish the proof of~\eqref{A1wh}.
Then, we use~\eqref{A1wh} and a symmetry argument to see that
$$ \int_{A_1\cup A_2}\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
=
\int_{A_1} \frac{dy}{|q-y|^{2+2s}}+
\int_{A_2} \frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
\ge 0.$$
This and~\eqref{89=001=E:L-1}
give that
$$ \int_{B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\ge- C M^{-\beta(1+2s)}.$$
Consequently,
by~\eqref{89=001=E:L},
\begin{equation}\label{89=001=E:L-2}
\int_{\mathbb R^n\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\le
-\int_{B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\le
C M^{-\beta(1+2s)}.
\end{equation}
Now we observe that
\begin{equation}\label{POL}
\{|x_1-q_1|\le 16\}\setminus B_{M^\beta}(q)
\subseteq \{|x_1-q_1|\le 16\}\times
\left\{|x_2-q_2|\ge \frac{M^\beta}{2}\right\}
.\end{equation}
To prove this, let~$y\in \{|x_1-q_1|\le 16\}\setminus B_{M^\beta}(q)$
and suppose, by contradiction, that~$|y_2-q_2|<M^\beta/2$.
Then
$$ |y-q|^2 \le 16^2+ \frac{M^{2\beta}}{4}< M^{2\beta}.$$
This would say that~$y\in B_{M^\beta}(q)$,
which is a contradiction, and so~\eqref{POL} is proved.
By~\eqref{POL}, we obtain that
\begin{eqnarray*}
&& \left|
\int_{\{|x_1-q_1|\le 16\}\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\right|
\le \int_{ \{|x_1-q_1|\le 16\}\times
\left\{|x_2-q_2|\ge \frac{M^\beta}{2}\right\} }
\frac{dy}{ |q-y|^{2+2s} } \\
&&\qquad\le \int_{q_1-16}^{q_1+16}
\left( \int_{ \{ |q_2-y_2|\ge M^\beta/2\}}
\frac{dy_2}{ |q_2-y_2|^{2+2s} }
\right)
\,dy_1 = CM^{-\beta(1+2s)},
\end{eqnarray*}
for some~$C>0$.
{F}rom this and~\eqref{89=001=E:L-2}, we obtain that
\begin{equation}\label{89=001=E:L-3}
\int_{ \{|x_1-q_1|> 16\} \setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy\le
C M^{-\beta(1+2s)},
\end{equation}
up to renaming~$C>0$.
Now we define~$H_1:=\{ x_1-q_1<-16\}$
and~$H_2:=\{x_1-q_1>16\}$.
Notice that~$H_1\subseteq \{ x_1< -15\}$ and~$H_2\subseteq \{x_1>15\}$.
Therefore~$H_1\cap \{x_2>-M\}\subseteq E^c_M$,
$H_1\cap \{x_2<-M\}\subseteq E_M$,
$H_2\cap \{x_2>M\}\subseteq E^c_M$
and~$H_2\cap \{x_2<M\}\subseteq E_M$.
Then, we define, for any~$i\in\{1,2\}$,
\begin{eqnarray*}
&& H_{i,1}:=H_i\cap \{ x_2>2q_2+M\},\\
&& H_{i,2}:=H_i\cap \{ x_2\in (M,2q_2+M]\},\\
&& H_{i,3}:=H_i\cap \{ x_2\in [-M,M]\},\\
&& H_{i,4}:=H_i\cap \{ x_2<-M\},\end{eqnarray*}
see Figure~\ref{HBQA-HSET}.
\begin{figure}
\centering
\includegraphics[height=7.9cm]{Hset.pdf}
\caption{The geometry involved in the proof of Proposition~\ref{CYL-RAFF}.}
\label{HBQA-HSET}
\end{figure}
By construction, $H_{i,1}\subseteq E^c_M$
and~$H_{i,4}\subseteq E_M$,
therefore, by up/down symmetry,
\begin{equation}\label{89=001=E:L-X1}
\int_{ (H_{1,1}\cup H_{1,4})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy =
0=
\int_{ (H_{2,1}\cup H_{2,4})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
.\end{equation}
Moreover, $H_{1,3}\subseteq E^c_M$
and~$H_{2,3}\subseteq E_M$,
therefore, by left/right symmetry,
\begin{equation}\label{89=001=E:L-X2}
\int_{ (H_{1,3}\cup H_{2,3})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy =0.\end{equation}
Finally, we point out that~$H_{1,2}\cup H_{2,2}\subseteq E_M^c$
and (recalling~\eqref{q2whee} and~\eqref{q2whee-pre}) that
$$ B_{M^\beta}(q)\subseteq\{ x_2<q_2+M^\beta\}
\subseteq\{ x_2< t+3M^\beta\} \subseteq\{ x_2<M\}.$$
Therefore
\begin{equation}
\label{POljU}\begin{split}
& \int_{ (H_{1,2}\cup H_{2,2})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
= \int_{ H_{1,2}\cup H_{2,2}} \frac{dy}{|q-y|^{2+2s}} \\
&\qquad\ge
\int_{ \{y_1-q_1\in (16,16+M),\; y_2\in (M,2q_2+M)} \frac{dy}{|q-y|^{2+2s}}
.\end{split}\end{equation}
Now we observe that if~$y_1-q_1\in (16,16+M)$ and~$y_2\in (M,2q_2+M)$,
then~$|q-y|\le C M$, for some~$C>0$. Then~\eqref{POljU} implies that
$$ \int_{ (H_{1,2}\cup H_{2,2})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
\ge c q_2 M^{-1-2s} ,$$
for some~$c>0$.
As a consequence of~\eqref{q2whee}
and~\eqref{q2whee-pre}, we also know that~$q_2\ge t- 2 M^\beta
\ge (C_o-2)M^\beta\ge C_o M^\beta/2$, if~$C_o$ is taken suitably large.
Hence we obtain
\begin{equation}\label{po09uJ}
\int_{ (H_{1,2}\cup H_{2,2})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy \ge c C_o
M^{\beta-1-2s},\end{equation}
up to renaming~$c>0$.
Now we observe that
$$ \beta-1-2s = \frac{(1+2s) (1-2-2s)}{2+2s} = -\beta(1+2s),$$
so we can write~\eqref{po09uJ} as
\begin{equation*}
\int_{ (H_{1,2}\cup H_{2,2})\setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy \ge c C_o
M^{-\beta(1+2s)}.\end{equation*}
This, together with~\eqref{89=001=E:L-X1}
and~\eqref{89=001=E:L-X2}, gives that
$$ \int_{ \{|x_1-q_1|> 16\} \setminus B_{M^\beta}(q)}
\frac{\chi_{E^c_M}(y)- \chi_{E_M}(y)}{|q-y|^{2+2s}}\,dy
\ge c C_o M^{-\beta(1+2s)}.$$
By comparing this inequality with~\eqref{89=001=E:L-3}, we obtain that
$$ c C_o M^{-\beta(1+2s)}
\le C M^{-\beta(1+2s)},$$
which is a contradiction if~$C_o$ is large enough.
This completes the proof of Proposition~\ref{CYL-RAFF}.
\end{proof}
As a counterpart of Proposition~\ref{CYL-RAFF}, we show that
the stickiness to the boundary of the domain does not
get too close to the origin, as next result points out:
\begin{proposition}\label{PR-3.2}
In the setting of Proposition~\ref{CYL-RAFF}, suppose that
\begin{equation}\label{INCL-1}
[-1,1)\times [b M^{\frac{1+2s}{2+2s}},\,M]\subseteq E_M^c,\end{equation}
with~$p=(1,b M^{\frac{1+2s}{2+2s}})\in\partial E_M$,
for some~$b\ge0$.
Then~$b\ge C_o$,
for some~$C_o>0$, only depending on~$s$,
provided that~$M$ is large enough.
\end{proposition}
\begin{proof} For short, we set~$\beta:={\frac{1+2s}{2+2s}}$.
We remark that
\begin{equation}\label{8u-BETA}
1-\frac{\beta}{1+2s} = 1-\frac{1}{2+2s}=\beta.
\end{equation}
We argue by contradiction, supposing that
\begin{equation}\label{78UI-CON}
b\le C_o
\end{equation}
for some~$C_o\in(0,1)$ that we can take conveniently small in the sequel.
By Lemma~\ref{SYMM-LEMMA} (used here with~$T(x):=-x$),
we have that~$E_M$ is odd with respect to the origin.
This and~\eqref{INCL-1} give that
\begin{equation}\label{INCL-2}
(-1,1]\times [-M,\,-b M^{\beta}]\subseteq E_M.\end{equation}
Now we let~$L:= M-b M^{\beta}$ and
we consider the cube~$Q$ of side~$2L$ that has the point~$p$
on its left side, namely
$$ Q:= (1,1+2L)\times (M-2L,M).$$
Notice that
\begin{equation}\label{PO09GH}
Q\subseteq E_M, \end{equation}
by the boundary datum of the problem.
We also take the symmetric reflection of~$Q$ with
respect to~$\{ x_1=1\}$, that is we set
$$ Q':= (1-2L,1)\times (M-2L,M).$$
We also set
$$ G:= (-1,1)\times (-3b M^{\beta}-2,\,-b M^{\beta}-1).$$
We claim that
\begin{equation}\label{INCL-3}
G\subseteq Q'.\end{equation}
Indeed, if~$x_1\in (-1,1)$
and~$x_2\in (-3b M^{\beta}-2,\,-b M^{\beta}-1)$,
then
$$ 1-2L =1-2M+2b M^{\beta} \le 1-2M+2M^{\beta}< -1<x_1,$$
since~$M$ is large. Also
$$ M-2L = -M+2b M^{\beta}
< -3b M^{\beta}-2 < x_2,$$
using again that~$M$ is large.
Accordingly, $x_1\in (1-2L,1)$ and~$x_2\in(M-2L,M)$,
which proves~\eqref{INCL-3}.
Now we claim that
\begin{equation}\label{INCL-4}
G\subseteq (-1,1]\times [-M,\,-b M^{\beta}].\end{equation}
Indeed, if~$x_2\in
(-3b M^{\beta}-2,\,-b M^{\beta}-1)$, then
$$ -M < -3 M^{\beta}-2
\le -3b M^{\beta}-2<x_2,$$
for large~$M$, and so~$x_2\in[-M,\,-b M^{\beta}]$,
which proves~\eqref{INCL-4}.
{F}rom~\eqref{INCL-2}, \eqref{INCL-3} and~\eqref{INCL-4}, we obtain that
\begin{equation*}
G\subseteq Q'\cap E_M.\end{equation*}
Using this and~\eqref{PO09GH}, by a symmetry argument we conclude that
\begin{equation}\label{INCL-7}
\int_{Q\cup Q'} \frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy
\ge \int_G \frac{dy}{|p-y|^{2+2s}}.
\end{equation}
Now we recall that~$p=(p_1,p_2)=(1,b M^{\beta})$ and we
observe that if~$y\in G$ then
\begin{eqnarray*}
&& |p_2-y_2| = |b M^{\beta}-y_2|\ge
|y_2|-b M^{\beta}
\\&&\qquad
\ge b M^{\beta}+1-b M^{\beta}=1\ge
\frac{|p_1|+|y_1|}{2} \ge \frac{|p_1-y_1|}{2}.
\end{eqnarray*}
Hence, $|p-y|\le C |p_2-y_2|$, for some~$C>0$ and thus~\eqref{INCL-7}
and the substitution~$t:=p_2-y_2$ give
\begin{equation}\label{INCL-77}
\begin{split}
& \int_{Q\cup Q'} \frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy
\ge C \int_G \frac{dy}{|p_2-y_2|^{2+2s}}
\\ &\qquad= C\int_{-3b M^{\beta}-2}^{-b M^{\beta}-1}
\frac{dy_2}{|p_2-y_2|^{2+2s}}
= C \int_{2b M^{\beta}+1}^{4b M^{\beta}+2}
\frac{dt}{t^{2+2s}} =\frac{C}{(2b M^{\beta}+1)^{1+2s}},
\end{split}
\end{equation}
up to renaming~$C$.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{quadr.pdf}
\caption{The geometry involved in the proof of Proposition~\ref{PR-3.2}.}
\label{HBQA}
\end{figure}
Now we define
$$ H:=(-\infty,-1)\times (-M,M-2L),$$
see Figure~\ref{HBQA}. By construction, $H\subseteq E_M^c$.
We notice that the portion on the right of~$Q$ all belongs to~$E_M$,
while the portion on the left of~$Q'$ all belongs to~$E_M^c$,
that is
\begin{eqnarray*}
&&(-\infty,1-2L)\times(M-2L,M)\subseteq E_M^c\\
{\mbox{and }}&&(1+2L,+\infty)\times(M-2L,M)\subseteq E_M.
\end{eqnarray*}
Therefore, by symmetry, these contributions cancel and we have
\begin{equation}\label{INCL-889}
\int_{\mathbb R^2\setminus(Q\cup Q')}
\frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy
=
\int_{\{x_2>M\}\cup\{x_2<M-2L\}}
\frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy.
\end{equation}
Now we observe that~$\{x_2>M\}\subseteq E_M^c$
and~$\{x_2<M-2L\}\setminus H\subseteq E_M$, therefore, by symmetry,
\begin{equation}\label{INCL-889-B}
\int_{\{x_2>M\}\cup\{x_2<M-2L\}}
\frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy
=- 2 \int_{H}
\frac{dy}{|p-y|^{2+2s}}.\end{equation}
Now we observe that if~$y\in H$ then~$|y_2|\ge 2L-M$
and so
$$|y_2-p_2|\ge 2L-M-bM^\beta=M-3bM^\beta\ge M-3M^\beta\ge\frac{M}{2}$$
if~$M$ is large enough. Therefore
\begin{eqnarray*}
&& \int_{H} \frac{dy}{|p-y|^{2+2s}}\le C\int_{-\infty}^{1}
\left(\int_{-M}^{M-2L} \frac{dy_2}{\big( |p_1-y_1|^2 + M^2\big)^{\frac{2+2s}{2}}}
\right)\,dy_1 \\
&&\qquad=
C\,(M-L)\,\int_{-\infty}^{1}
\frac{dy_1}{\big( |1-y_1|^2 + M^2\big)^{\frac{2+2s}{2}}}
\\ &&\qquad\le
C\,(M-L)\,\left(
\int_{-\infty}^{-M}
\frac{dy_1}{|1-y_1|^{2+2s} } +
\int_{-M}^{1}
\frac{dy_1}{M^{2+2s}}
\right) \\
&&\qquad\le
C\,(M-L)\,M^{-1-2s} = C b M^{\beta-1-2s} \le CM^{\beta-1-2s}
,\end{eqnarray*}
for some~$C>0$ (possibly varying from line to line).
Using this, \eqref{INCL-889} and~\eqref{INCL-889-B},
we obtain that
\begin{equation}\label{INCL-8}
\int_{\mathbb R^2\setminus(Q\cup Q')}
\frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy
=
- 2 \int_{H}
\frac{dy}{|p-y|^{2+2s}}\ge -CM^{\beta-1-2s},\end{equation}
up to renaming~$C$.
Now we use the
Euler-Lagrange equation in the viscosity sense at~$p$
and we obtain that
$$ \int_{\mathbb R^2}
\frac{\chi_{E_M}(y)- \chi_{E_M^c}(y)}{|p-y|^{2+2s}}\,dy\le0.$$
Combining this with~\eqref{INCL-77} and~\eqref{INCL-8}, we obtain
$$ 0\ge \frac{C}{(2b M^{\beta}+1)^{1+2s}}
-CM^{\beta-1-2s}.$$
That is, up to renaming constants,
$$ (2b M^{\beta}+1)^{1+2s}\ge c_* M^{1+2s-\beta},$$
for some~$c_*>0$. Using this
and~\eqref{8u-BETA}, we conclude that
$$ 2b M^{\beta}+1 \ge c_*^{\frac{1}{1+2s}} M^{1-\frac{\beta}{1+2s}}
=c_o M^{\beta}.$$
Now we multiply by~$M^{-\beta}$
and we take~$M$ large enough, such that~$M^{-\beta}\le c_o/2$,
so we obtain
$$ 2b \ge -M^{-\beta} + c_o \ge \frac{c_o}{2}.$$
This is in contradiction with~\eqref{78UI-CON},
if we choose~$C_o$ small enough.
\end{proof}
As a combination of Propositions~\ref{CYL-RAFF}
and~\ref{PR-3.2}, we have the optimal statement
in Theorem~\ref{CYL-THEOR}.
\section{Stickiness as~$s\to0^+$}\label{SEC:to0}
This section contains the asymptotic properties as~$s\to0$
and the proof of Theorem \ref{SECTOR}.
For this, we recall that~$\Sigma$ has been defined in~\eqref{SECTOR-3456fgg}
as
$$ \Sigma := \{ (x,y)\in\mathbb R^2\setminus B_1 {\mbox{ s.t. }}
x>0 {\mbox{ and }} y>0\}$$
and~$E_s$
is the $s$-minimizer in~$B_1$
with datum~$\Sigma$ outside~$B_1$.
\begin{proof}[Proof of Theorem \ref{SECTOR}] First, we show that
\begin{equation}\label{UI}
E_s \subseteq \{ x+y=1\}.
\end{equation}
To prove it, we slide the half-plane~$h_t:=\{ x+y\le t\}$.
If~$t\le -3$, we have that~$h_t$ lies below~$\Sigma\cup B_1$
and so~$h_t\subseteq E_s^c$. Then we increase~$t$
until~$h_{t^*}$ intersects~$E_s$, with~$t_*\in [-3,1]$.
Notice that~\eqref{UI} is proved if we show that
\begin{equation}\label{UI2}
t_*=1.
\end{equation}
We prove this arguing by contradiction.
If not, there exists~$p\in B_1\cap (\partial E_s)\cap \{ x+y=t_*\}$.
Hence, using the
Euler-Lagrange equation in the viscosity sense
(see Theorem~5.1 in~\cite{CRS})
and the fact that~$h_{t_*}\subseteq E_s^c$, we obtain
$$ 0\ge
\int_{\mathbb R^2} \frac{\chi_{E^c_s}(y)- \chi_{E_s}(y)}{|p-y|^{2+2s}}\,dy
\ge
\int_{\mathbb R^2} \frac{\chi_{h_{t_*}}(y)- \chi_{h_{t_*}^c}(y)}{|p-y|^{2+2s}}\,dy
=0.$$
This shows that~$h_{t_*}$ must coincide with~$E_s^c$.
This is impossible, since~$E_s$ is not a half-plane outside~$B_1$.
Hence, we have proved~\eqref{UI2} and so~\eqref{UI}.
By~\eqref{UI}, we get that~$B_{\sqrt{2}/2}\subseteq E_s^c$.
So we can enlarge~$r\in [\sqrt{2}/2,1]$ till~$B_r$ touches~$E_s$.
We remark that Theorem~\ref{SECTOR} is proved
if we show that this touching property only occurs at~$r=1$.
Thus, we argue by contradiction and we suppose that there exists
\begin{equation}\label{r-below}
r\in[\sqrt{2}/2,1)\end{equation}
such that~$B_r\subseteq E_s^c$
and there exists~$q\in (\partial B_r)\cap(\partial E_s)$.
Then, by the Euler-Lagrange equation, we have that
\begin{equation}\label{ES-SEC-1}
\int_{\mathbb R^2} \frac{\chi_{E^c_s}(y)- \chi_{E_s}(y)}{|q-y|^{2+2s}}\,dy
\le0.\end{equation}
By construction,
\begin{equation}\label{ES-SEC-2}
E_s \subseteq
\{ (x,y)\in\mathbb R^2\setminus B_r {\mbox{ s.t. }}
x>0 {\mbox{ and }} y>0\}.\end{equation}
Also, $0<q_1,q_2<1$.
Then we consider the translation by~$q$: namely we define~$F_s:=E_s-q$.
It follows from~\eqref{ES-SEC-2} that
\begin{equation}\label{ES-SEC-3}
F_s \subseteq
\{ (x,y)\in\mathbb R^2\setminus B_r(-q) {\mbox{ s.t. }}
x>-1 {\mbox{ and }} y>-1\}.\end{equation}
Also, by~\eqref{ES-SEC-1},
\begin{equation}\label{ES-SEC-4}
\int_{\mathbb R^2} \frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
\le0.\end{equation}
Now we define~$D_r:=B_r(q)\cup B_r(-q)$ and we let
$K_r$ be the convex hull of~$D_r$.
Notice that
\begin{equation}\label{BALL_K}
B_r\subseteq K_r.\end{equation} We also define~$P_r:= K_r\setminus D_r$.
Since~$B_r(-q)\subseteq F_s^c$, by symmetry we obtain that
\begin{equation}\label{ES-SEC-6}
\int_{D_r} \frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
\ge0.\end{equation}
Moreover
(see Lemma~3.1 in~\cite{nostro},
used here with~$\lambda:=1$) and~\eqref{r-below},
\begin{equation*}
\left|
\int_{P_r} \frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy\right|
\le
\frac{C_1 r^{-2s} }{1-2s}\le \frac{C_2}{1-2s},
\end{equation*}
for suitable positive constants~$C_1$ and~$C_2$ that do not depend on~$s$.
Using this, \eqref{ES-SEC-4} and~\eqref{ES-SEC-6}
we obtain that
\begin{equation}\label{ES-SEC-10}
\begin{split}
0\,&\ge \int_{\mathbb R^2\setminus K_r}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
+\int_{D_r}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
+
\int_{P_r}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
\\ &\ge
\int_{\mathbb R^2\setminus K_r}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
-\frac{C_2}{1-2s}.\end{split}\end{equation}
Moreover, recalling~\eqref{BALL_K}
(and using again~\eqref{r-below}), we have that
$$ \left|\int_{B_{2r}\setminus K_r}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy\right|\le
\int_{B_{2r}\setminus B_r}
\frac{dy}{|y|^{2+2s}}\le C_3,$$
for some~$C_3>0$ that does not depend on~$s$. Hence~\eqref{ES-SEC-10}
gives
\begin{equation}\label{ES-SEC-11}
0\ge
\int_{\mathbb R^2\setminus B_{2r}}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
-C_3-\frac{C_2}{1-2s}.\end{equation}
Now we observe that~$B_r(-q)\subseteq B_{2r}$, since~$|q|=r$.
Consequently, recalling~\eqref{ES-SEC-3},
\begin{equation*}
F_s \setminus B_{2r} \subseteq
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }}
x>-1 {\mbox{ and }} y>-1\}.\end{equation*}
That is,~$F_s \setminus B_{2r} \subseteq A_1\cup A_2\cup A_3$, where
\begin{eqnarray*}
&& A_1 :=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x>0 {\mbox{ and }} y\in(-1,1)\},\\
&& A_2 :=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x\in(-1,1) {\mbox{ and }} y>0\}\\
{\mbox{and }}&& A_3:=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x\ge1 {\mbox{ and }} y\ge1\}.
\end{eqnarray*}
\begin{figure}
\centering
\includegraphics[height=5.9cm]{compe.pdf}
\caption{
The partition of the plane needed for the proof
of Theorem~\ref{SECTOR}.}
\label{SIMP}
\end{figure}
On the other hand,~$F_s^c \setminus B_{2r} \supseteq A_1'\cup
A_2'\cup A_3'\cup A_4'$, where
\begin{eqnarray*}
&& A_1' :=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x<0 {\mbox{ and }} y\in(-1,1)\},\\
&& A_2' :=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x\in(-1,1) {\mbox{ and }} y<0\},\\
&& A_3' :=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x\le-1 {\mbox{ and }} y\le-1\},\\
{\mbox{and }}&& A_4':=
\{ (x,y)\in\mathbb R^2 \setminus B_{2r}
{\mbox{ s.t. }} x\ge1 {\mbox{ and }} y\le-1\},
\end{eqnarray*}
see Figure~\ref{SIMP}.
After simplifying~$A_1$ with~$A_1'$, $A_2$ with~$A_2'$
and~$A_3$ with~$A_3'$, we obtain
\begin{equation}\label{78:90P}
\int_{\mathbb R^2\setminus B_{2r}}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
\ge \int_{A'_4} \frac{dy}{|y|^{2+2s}}.\end{equation}
Notice now that~$A'_4$ contains a cone with positive
constant opening with vertex at the origin,
therefore
$$ \int_{A'_4} \frac{dy}{|y|^{2+2s}}\ge
c_1 \int_{2r}^{+\infty} \frac{d\rho}{\rho^{1+2s}}
= \frac{c_2}{s\,r^{2s}} \ge \frac{c_3}{s},$$
where we have used again~\eqref{r-below}, and the positive
constants~$c_1$, $c_2$ and~$c_3$ do not depend on~$s$.
The latter estimate and~\eqref{78:90P} give that
$$ \int_{\mathbb R^2\setminus B_{2r}}
\frac{\chi_{F^c_s}(y)- \chi_{F_s}(y)}{|y|^{2+2s}}\,dy
\ge \frac{c_3}{s}.$$
Therefore, recalling~\eqref{ES-SEC-11},
$$ 0\ge \frac{c_3}{s}
-C_3-\frac{C_2}{1-2s}.$$
This is a contradiction if~$s\in(0,s_o)$ and~$s_o$
is small enough. Hence, we have completed the proof of
Theorem~\ref{SECTOR}.
\end{proof}
\section{Construction of barriers that are piecewise linear}\label{SEC:BAR:PTWL}
This part of the paper is devoted to the proof of
Theorem~\ref{UNS}.
The argument will rely on the construction of a series
of barriers, and the proof of Theorem~\ref{UNS}
will be completed in Section~\ref{INST:SEC}.
In this section, we construct barriers in the plane,
which are subsolutions of the fractional curvature equation when~$\{x_1>0\}$,
which possess a ``vertical'' portion along~$\{x_1=0\}$ and
which are built by joining linear functions
whose slope becomes arbitrarily close to being horizontal
(a precise statement will be given in Proposition~\ref{IT:BARR}).
For this scope,
we start with a simple
auxiliary observation to bound explicitly from below the
fractional curvature of an angle:
\begin{lemma}\label{ANGLE}
Let~$\ell\ge0$,
\begin{eqnarray*}
&&E_1:= (-\infty,0]\times (-\infty,0)\\
&& E_2:= \{ \ell x_2-x_1<0 ,\quad x_1>0\}\\
{\mbox{and }}&& E:=E_1\cup E_2.
\end{eqnarray*}
Then, for any~$p=(p_1,p_2)\in\partial E$ with~$p_2>0$,
\begin{equation}\label{SCD:k}
\int_{\mathbb R^2} \frac{\chi_E(y)-\chi_{E^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{c(\ell)}{|p|^{2s}}, \end{equation}
for a suitable nonincreasing function~$c:[0,+\infty)\to(0,1)$.
More precisely, for large~$\ell$, one has that~$c(\ell)\sim \bar c\ell^{-1}$,
for some~$\bar c>0$.
\end{lemma}
\begin{proof} Let~$\delta:=\arctan (1/\ell)\in \left(0,\frac{\pi}{2}\right]$.
By scaling, it is enough to prove~\eqref{SCD:k} when
\begin{equation}\label{SCD:k2}
|p|=4.\end{equation}
\begin{figure}
\centering
\includegraphics[height=5.9cm]{FsA.pdf}
\caption{The proof of Lemma~\ref{ANGLE}.}
\label{FsA}
\end{figure}
Now, for any~$t>0$, let~$S_t$ be the slab with boundary orthogonal
to the straight line~$\{ \ell x_2-x_1=0\}$
of width~$2t$, having~$p$ on its symmetry axis (see Figure~\ref{FsA}).
For small~$t$, the slab~$S_t$ does not contain the origin,
thus, the ``upper'' half of the slab is contained in~$E^c$,
while the ``lower'' half of the slab is contained in~$E$,
namely
\begin{equation*}
\int_{S_t} \frac{\chi_E(y)-\chi_{E^c}(y)}{|y-p|^{2+2s}}\,dy=0
.\end{equation*}
Enlarging~$t$,
the ``lower'' half of the slab is always contained in~$E$.
As for the ``upper'' half, we have that the triangle~$T$
with vertices~$(0,0)$, $(-\cos\delta,-\sin\delta)$, $(-1,0)$
lies in~$E$. Notice that
$$|T|=\frac{\sin\delta}{2}.$$
Also, if~$y\in T$ then~$|y|\le 2$ and so,
recalling~\eqref{SCD:k2},
$$ |y-p|\le |p|+2 \le 2|p|.$$
Consequently,
$$ \int_{T} \frac{\chi_E(y)-\chi_{E^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{ |T|}{ 2^{2+2s}\,|p|^{2+2s} }
= \frac{ \sin\delta }{ 2^{7+2s}\,|p|^{2s} },$$
which gives the desired result.
\end{proof}
The next result is the building block needed to
construct a barrier iteratively.
Roughly speaking, next result says that
we can tilt a straight line towards infinity
by estimating precisely the effect of this modification
on the fractional curvature.
\begin{lemma}\label{PRE:BAR}
Let~$\ell\ge q\ge0$ and~$\delta:=\arctan (1/\ell)\in \left(0,\frac{\pi}{2}\right]$.
Let~$e:=(\ell-q,1)$.
Let~$\bar\tau\in C^\infty_0( B_1(e))$ with~$\bar\tau=1$ in~$B_{1/2}(e)$.
Let~$\tau_o\in C^\infty(\mathbb R)$ be such that
\begin{equation}
{\mbox{$\tau_o(t)=1$
if~$t\in \left[ \frac{\delta}{2},\frac{3\delta}{2}\right]$
and~$\tau_o(t)=0$ if~$t\in\mathbb R\setminus
\left[ \frac{\delta}{4},\frac{7\delta}{4}\right]$.}}\end{equation}
For any~$x\in\mathbb R^2$, let also~$\alpha(x)\in [0,2\pi)$ be
the angle between the vector~$x-e$ and the $x_1$-axis.
Let
\begin{equation}\label{PIOP11}
\tau(x):= \big(1-\bar\tau(x)\big)\,\tau_o\big( \alpha(x)\big).\end{equation}
For any~$\theta\in\mathbb R$, let~$R_\theta$ be the clockwise rotation by an angle~$\theta$, i.e.
$$ R_\theta(x)=R_\theta(x_1,x_2):=
\left(
\begin{matrix}
\cos\theta & \sin \theta\\
-\sin\theta & \cos\theta
\end{matrix}
\right) \,
\left(
\begin{matrix}
x_1\\ x_2 \end{matrix}
\right). $$
Let also
$$ \Psi_\theta(x):= R_{\tau(x)\theta} \,x.$$
Let~$E\subset\mathbb R^2$
be an epigraph such that
\begin{equation*}
\begin{split}
& E \cap \{x_1 < 0\} = (-\infty,0)\times(-\infty,0), \\
& E\supseteq \mathbb R\times (-\infty,0),\\
& E\cap \{x_2 >1\} = \{\ell x_2-x_1 -q <0\}\cap\{x_2 >1\}\\
{\mbox{and }}\;& E\cap \{x_1>\ell-q\} = \{\ell x_2-x_1 -q <0\}\cap\{x_1 >\ell-q\}.
\end{split}
\end{equation*}
Assume that, for any~$p\in \partial E\cap\{x_2>0\}$,
\begin{equation}\label{STI:STEp0}
\int_{\mathbb R^2} \frac{\chi_E(y)-\chi_{E^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{c}{|p|^{2s}}
\end{equation}
for some~$c\in(0,1)$.
Then, there exist nonincreasing functions~$\phi:[0,+\infty)\to
(0,1)$ and~$c_o:[0,+\infty)\to (0,c)$
such that for any~$\theta\in [0,\,\phi(\ell)]$ the following claim
holds true. Let~$F:=\Psi_\theta(E)$. Then, for any~$p\in (\partial F)
\cap\{x_2>0\}$,
\begin{equation}\label{VBhJ}
\int_{\mathbb R^2} \frac{\chi_F(y)-\chi_{F^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{c_o (\ell)}{|p|^{2s}}
.\end{equation}
More precisely, for large~$\ell$, one has that~$c_o(\ell)\sim
\bar c\min\{c,\ell^{-1}\}$,
for some~$\bar c>0$.
\end{lemma}
\begin{proof} First we point out that
\begin{equation}\label{DE:alpha}
|\nabla\alpha(x)|\le \frac{C}{|x-e|}
,\end{equation}
for some~$C>0$. Indeed, $\alpha(x)$ is identified by the two
conditions
\begin{equation}\label{cos-IU-1}
|x-e|\cos\alpha(x)=|x_1-\ell+q|\end{equation}
and~$|x-e|\sin\alpha(x)=|x_2-1|$.
Assume also that~$\sin^2\alpha(x)\ge 1/2$ (the case~$\cos^2\alpha(x)\ge 1/2$
is similar). Then we differentiate the relation~\eqref{cos-IU-1}
and we obtain
$$ \frac{x-e}{|x-e|}\cos\alpha(x)
-|x-e|\sin\alpha(x)\,\nabla\alpha(x)
=\frac{x_1-\ell+q}{|x_1-\ell+q|}\,(1,0).$$
Therefore
\begin{eqnarray*}
\frac{\sqrt{2}}2 |x-e|\,|\nabla\alpha(x)|
\le |x-e|\,|\sin\alpha(x)|\,|\nabla\alpha(x)|
= \left|
\frac{x-e}{|x-e|}\cos\alpha(x)-\frac{x_1-\ell+q}{|x_1-\ell+q|}\,(1,0)\right|
\le 2,
\end{eqnarray*}
which proves~\eqref{DE:alpha}.
Similarly, taking one more derivative, one sees that
\begin{equation}\label{DE:alpha:second}
|D^2\alpha(x)|\le \frac{C}{|x-e|^2}
.\end{equation}
Now, by~\eqref{PIOP11} and~\eqref{DE:alpha},
\begin{equation}\label{DE:alpha:2}
|\nabla\tau(x)|\le C\left( \chi_{B_1(e)\setminus B_{1/2}(e)}(x)+
\frac{\chi_{\mathbb R^2\setminus B_{1/2}(e)}(x)}{|x-e|}\right).
\end{equation}
Using~\eqref{DE:alpha:second}, one also obtains that
\begin{equation}\label{DE:alpha:2:second}
|D^2\tau(x)|\le C\left( \chi_{B_1(e)\setminus B_{1/2}(e)}(x)+
\frac{\chi_{\mathbb R^2\setminus B_{1/2}(e)}(x)}{|x-e|}\right).
\end{equation}
Let now
$$ \Phi_\theta(x):=
\Psi_\theta(x)-x=
\left(
\begin{matrix}
\cos(\tau(x)\theta)-1 & \sin (\tau(x)\theta)\\
-\sin(\tau(x)\theta) & \cos(\tau(x)\theta)-1
\end{matrix}
\right) \,
\left(
\begin{matrix}
x_1\\ x_2 \end{matrix}
\right).$$
We claim that
\begin{equation}\label{DIF:1}
|D\Phi_\theta(x)|\le C\,(1+\ell)\,\theta,
\end{equation}
for some~$C>0$. To prove it, we consider the first coordinate of~$\Phi_\theta(x)$,
which is
\begin{equation}\label{first coo}
\big(\cos(\tau(x)\theta)-1\big)\,x_1 +
\sin(\tau(x)\theta)\,x_2,\end{equation}
since the computation with the second coordinate is similar.
We bound the derivative of~\eqref{first coo} by
\begin{equation}\label{first coo2}
\big|\cos(\tau(x)\theta)-1\big|+
\big|\sin(\tau(x)\theta)\big|
+ \theta\,\Big( \big|\sin(\tau(x)\theta)\big|+\big|\cos(\tau(x)\theta)\big|\Big)
|\nabla\tau(x)|\,|x|.\end{equation}
Thus, we bound~$\big|\cos(\tau(x)\theta)-1\big|\le C\theta^2$
and~$\big|\sin(\tau(x)\theta)\big|\le C\theta$ and we make use of~\eqref{DE:alpha:2},
to estimate the quantity in~\eqref{first coo2} by
\begin{equation}\label{first coo3}
C\theta \,\left(1+\frac{
\chi_{\mathbb R^2\setminus B_{1/2}(e)}(x)
\;|x|}{|x-e|} \right).\end{equation}
Now we observe that~$|e|=\sqrt{(\ell-q)^2+1}\le\sqrt{\ell^2+1}$, therefore
$$ |x|\le |x-e|+\sqrt{\ell^2+1}$$
and so, if~$|x-e|\ge1/2$,
$$ \frac{|x|}{|x-e|} \le 1 +2\sqrt{\ell^2+1}.$$
By inserting this information into~\eqref{first coo3}
we bound the first coordinate of~$\Phi_\theta(x)$ by~$C\,(1+\ell)\,\theta$.
This proves~\eqref{DIF:1}.
Similarly, making use of~\eqref{DE:alpha:2:second},
one sees that
\begin{equation}\label{DIF:2}
|D^2\Phi_\theta(x)|\le C\,(1+\ell)\,\theta.
\end{equation}
Notice also that, for any fixed~$x\in\mathbb R^2$,
we have that
\begin{equation}\label{NORM:1}
|\Psi_\theta(x)|= |R_{\tau(x)\theta} \,x|=|x|, \end{equation}
therefore
$$ \lim_{|x|\to+\infty} |\Psi_\theta(x)|=+\infty.$$
{F}rom this,~\eqref{DIF:1}, and the Global Inverse Function Theorem
(see e.g. Corollary 4.3 in~\cite{palais-1959-natural}),
we obtain that~$\Psi_\theta$ is a global diffeomorphism of~$\mathbb R^2$,
see Figure~\ref{FsB}.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{FsB.pdf}
\caption{The diffeomorphism of~$\mathbb R^2$ in Lemma~\ref{PRE:BAR}.}
\label{FsB}
\end{figure}
As a consequence, using~\eqref{DIF:1}, \eqref{DIF:2}
and the curvature estimates for diffeomorphisms (see Theorem~1.1
in~\cite{cozzi}), we conclude that
\begin{equation}\label{ST:MAR}
\int_{\mathbb R^2} \frac{\chi_F(y)-\chi_{F^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \int_{\mathbb R^2} \frac{\chi_E(y)-\chi_{E^c}(y)}{|y-q|^{2+2s}}\,dy-
C\,(1+\ell)\,\theta,
\end{equation}
with~$q:=\Psi_\theta^{-1}(p)$, for any~$p\in(\partial F)\cap \{x_2>0\}$.
Now we claim that
\begin{equation}\label{p02:02}
{\mbox{if~$p\in\{x_2>0\}$ then~$\Psi_\theta^{-1}(p)\in\{x_2>0\}$.}}
\end{equation}
Suppose, by contradiction, that~$\Psi_\theta^{-1}(p)\in\{x_2\le0\}$.
Notice that~$\tau$ vanishes in~$\{x_2\le0\}$, therefore
$ \Psi_\theta$ is the identity in~$\{x_2\le0\}$.
As a consequence~$p=\Psi_\theta(\Psi_\theta^{-1}(p))=\Psi_\theta^{-1}(p)
\in\{x_2\le0\}$. This is a contradiction with our assumptions
and so it proves~\eqref{p02:02}.
Using~\eqref{STI:STEp0}, \eqref{NORM:1}, \eqref{ST:MAR}
and~\eqref{p02:02}, we have that
\begin{equation}\label{Go:09-01p1}
\int_{\mathbb R^2} \frac{\chi_F(y)-\chi_{F^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c}{|q|^{2s}}
-C\,(1+\ell)\,\theta = \frac{c}{|p|^{2s}}
-C\,(1+\ell)\,\theta \ge \frac{c}{2\,|p|^{2s}},
\end{equation}
with~$q:=\Psi_\theta^{-1}(p)$, for any~$p\in(\partial F)\cap \{x_2>0\}\cap
B_{r_\theta}$, where~$r_\theta:={\left(\frac{c}{2C\,(1+\ell)\,\theta}\right)^{\frac{1}{2s}}}$
(we stress that~$r_\theta$ is large, for small~$\theta$,
according to the statement of Lemma~\ref{PRE:BAR}).
Now we take~$p\in\big((\partial F)\cap \{x_2>0\}\big)\setminus B_{r_\theta}$
and we observe that~$((\partial F)\setminus B_{r_\theta})\cap \{x_2>0\}$
coincides with a straight line of the form~$\lambda:=\{\ell_\theta x_2-x_1-q_\theta=0\}$,
with~$\ell_\theta\ge \ell$, $|\ell_\theta-\ell|$ as close
to zero as we wish for small~$\theta$, and~$q_\theta:=\ell_\theta-\ell+q$.
The intersections of the straight line~$\lambda$
with~$\{x_2=8\}$ and~$\{x_2=0\}$ occur at points~$x_1=8\ell_\theta-q_\theta$
and~$x_1=-q_\theta$, respectively.
Hence, we consider the triangle~$T$ with vertices~$(8\ell_\theta-q_\theta,0)$, $(8\ell_\theta-q_\theta,8)$
and~$(-q_\theta, 0)$. We observe that~$|T|=32\ell_\theta\le 32(1+\ell)$,
for small~$\theta$. Moreover, if~$y\in T$, then~$|y|\le
C(1+\ell_\theta+q_\theta)\le C(1+\ell)$, up to renaming constants.
Therefore, if~$p\in B_{r_\theta}^c$ and~$y\in T$,
$$ |y-p|\ge |p|- C(1+\ell)\ge \frac{|p|}{2},$$
if~$\theta$ is small. Consequently,
\begin{equation}\label{13UI:0}
\int_{T} \frac{dy}{|y-p|^{2+2s}}\le \frac{C(1+\ell)}{|p|^{2+2s}}
\le \frac{C(1+\ell)}{r_\theta^2\,|p|^{2s}}.
\end{equation}
Now we define~$\tilde F:= F\cup T$. By Lemma~\ref{ANGLE},
$$ \int_{\mathbb R^2} \frac{\chi_{\tilde F}(y)-\chi_{\tilde F^c}(y)}{
|y-p|^{2+2s}}\,dy\ge
\frac{c(\ell_\theta)}{|p-(-q_\theta,0)|^{2s}}.$$
Using that~$\ell_\theta\le \frac{3\ell}{2}$ and that~$c(\cdot)$ is
nonincreasing, we see that~$c(\ell_\theta)\ge c\left(\frac{3\ell}{2}\right)$.
Moreover,
$$ |p-(-q_\theta,0)|\le |p|+q_\theta\le |p|+\ell+1\le 2|p|,$$
so we obtain that
$$ \int_{\mathbb R^2} \frac{\chi_{\tilde F}(y)-\chi_{\tilde F^c}(y)}{
|y-p|^{2+2s}}\,dy\ge
\frac{c\left(\frac{3\ell}{2}\right)}{2^{2s} \, |p|^{2s}}.$$
Exploiting this and~\eqref{13UI:0}, we obtain that,
for any~$p\in\big((\partial F)\cap \{x_2>0\}\big)\setminus B_{r_\theta}$,
\begin{equation}\label{Go:09-01p2}
\begin{split}
& \int_{\mathbb R^2} \frac{\chi_F(y)-\chi_{F^c}(y)}{|y-p|^{2+2s}}\,dy
\ge
\int_{\mathbb R^2} \frac{\chi_{\tilde F}(y)-\chi_{\tilde F^c}(y)}{
|y-p|^{2+2s}}\,dy - \int_{T} \frac{dy}{|y-p|^{2+2s}}
\\ &\qquad\ge \frac{c\left(\frac{3\ell}{2}\right)}{2^{2s} \, |p|^{2s}} - \frac{C(1+\ell)}{r_\theta^2\,|p|^{2s}}
\ge \frac{c\left(\frac{3\ell}{2}\right)}{2^{1+2s} \, |p|^{2s}},
\end{split}\end{equation}
for small~$\theta$. Then, \eqref{VBhJ} follows by combining
\eqref{Go:09-01p1}
and~\eqref{Go:09-01p2}.
\end{proof}
By iterating Lemma~\ref{PRE:BAR} we can
construct the following barrier:
\begin{proposition}\label{IT:BARR}
Fix~$K\ge0$. Then there exist~$a_K\in(0,1)$,
$\ell_K\ge K$, $q_K\ge0$, $c_K\in(0,1)$,
a continuous function~$u_K :[0,+\infty)\to [0,+\infty)$
and a set~$E_K\subset\mathbb R^2$ with $(\partial E_K)\cap \{x_2>0\}$
of class~$C^{1,1}$
and such that:
\begin{itemize}
\item $u_K(x_2) =\ell_K\,x_2-q_K $
for any~$x_2\in [1,+\infty)$,
\item we have that
\begin{eqnarray*}
&& E_K \cap \{x_1 < 0\} = (-\infty,0)\times(-\infty,0), \\
&& E_K\supseteq \mathbb R\times (-\infty,0),\\
&& E_K\supseteq (0,+\infty)\times (-\infty,a_K],\\
&& E_K\cap \{x_2 >1\} = \{x_1> u_K(x_2), \quad x_2 >1\},\\
&& E_K\cap \{x_1>\ell_K-q_K\} =
\{x_1> u_K(x_2),\quad x_1 >\ell_K-q_K\}\end{eqnarray*}
and
$$ \int_{\mathbb R^2} \frac{\chi_{E_K}(y)-\chi_{E_K^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c_K}{|p|^{2s}},$$
for any~$p\in (\partial E_K)\cap \{x_2>0\}$.
\end{itemize}
More precisely, for large~$K$, one has that~$c_K\sim \bar c\ell_K^{-1}$,
for some~$\bar c>0$.
Moreover, one can also prescribe that
\begin{equation}\label{more}
q_K\le K^{-1}.\end{equation}
\end{proposition}
\begin{proof} We apply Lemma~\ref{PRE:BAR} iteratively for a large
(but finite) number of times, see Figure~\ref{FsC}.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{FsC.pdf}
\caption{The barrier of Proposition~\ref{IT:BARR}.}
\label{FsC}
\end{figure}
We start with~$u_0:=0$
and~$E_0:=\mathbb R^2\setminus\{ x_1\le0\le x_2\}$.
By Lemma~\ref{ANGLE} (used here with~$\ell:=0$)
we know that
$$ \int_{\mathbb R^2} \frac{\chi_{E_0}(y)-\chi_{E_0^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c}{|p|^{2s}},$$
for some~$c>0$. Then we apply Lemma~\ref{PRE:BAR}
and we construct a set~$E_1$ whose boundary
coincides with~$\{x_2=0\}$ when~$\{x_1<0\}$
and with a straight line~$\{\ell_1 x_2-x_1-q_1=0\}$
when~$\{x_2>4\}$, whose fractional curvature satisfies the desired estimate
(as a matter of fact, we can take the new slope~$\ell_1$
as the one obtained by~$\phi(0)$
in Lemma~\ref{PRE:BAR}, thus~$\ell_1>0$).
Then we scale~$E_1$ by a factor~$\frac12$ and we apply once again
Lemma~\ref{PRE:BAR}, obtaining
a set~$E_2$ whose boundary
coincides with~$\{x_2=0\}$ when~$\{x_1<0\}$
and with a straight line~$\{\ell_2 x_2-x_1-q_2=0\}$
when~$\{x_2>4\}$, whose fractional curvature satisfies the desired estimate.
Notice again that~$\ell_2$ is obtained
in Lemma~\ref{PRE:BAR} by rotating clockwise
the straight line of slope~$\ell_1$ by an angle~$\phi(\ell_1)>0$,
hence~$\ell_2>\ell_1$.
Iterating this procedure, we obtain a sequence of
increasing slopes~$\ell_j$ and sets~$E_j$
satisfying the desired geometric properties.
We stress that, for large~$j$, the slope~$\ell_j$
must become larger than the quantity~$K$ fixed
in the statement of Proposition~\ref{IT:BARR}.
Indeed, if not, say if~$\ell_j\le \ell_\star$ for some~$\ell_\star>0$,
at each step of the iteration we could
rotate the straight line by an angle of size larger than~$\phi(\ell_\star)$,
which is a fixed positive quantity
(recall that~$\phi$ in
Lemma~\ref{PRE:BAR} is nonincreasing): hence
repeating this argument many times we would make the slope become
bigger than~$\ell_\star$, that is a contradiction.
Thus, we can define~$j_o$ to be the first~$j$ for which~$\ell_j\ge K$.
The set~$E_{j_o}$ obtained in this way satisfies the desired properties,
with the possible exception of~\eqref{more}.
So, to obtain~\eqref{more}, we may suppose that~$q_{j_o}>K^{-1}$,
otherwise we are done, and
we scale the picture once again by a factor~$\mu:= K^{-1} q_{j_o}^{-1}\in(0,1)$.
In this way, the geometric properties of the set
and the estimates on the fractional curvature are preserved,
but the line~$\{ \ell_{j_o} x_2-x_1-q_{j_o}=0\}$
is transformed into the line~$\{ \ell_{j_o} x_2-x_1-\tilde q_{j_o}=0\}$,
with~$\tilde q_{j_o}:= \mu q_{j_o}$.
By construction, we have that~$\tilde q_{j_o}=K^{-1}$,
which gives~\eqref{more}.
\end{proof}
\section{Construction of barriers which grow
like~$x_1^{\frac{1}{2}+s+\epsilon_0}$}\label{sec:GROW:R}
In this section, we construct barriers in the plane,
which are subsolutions of the fractional curvature equation when~$\{x_1>0\}$,
which possess a ``vertical'' portion along~$\{x_1=0\}$ and
which grow like~$x_1^{\frac{1}{2}+s+\epsilon_0}$ at infinity
(here, $\epsilon_0>0$ is arbitrarily small).
This is a refinement of the barrier constructed in
Proposition~\ref{IT:BARR}, which grows linearly (with almost
horizontal slope). Roughly speaking,
the difference with Proposition~\ref{IT:BARR} is
that the results obtained there have nice scaling properties
and an elementary geometry (since the barrier constructed
there is basically the junction of a finite number
of straight lines) but do not possess
an optimal growth at infinity. As a matter of fact,
the power obtained
here at infinity is dictated by the growth of
the functions that are harmonic with respect
to the fractional Laplacian~$(-\Delta)^{\gamma_0}$,
where
\begin{equation}
\gamma_0: =\frac{1}{2}+s.\end{equation}
As a matter of fact, this procedure
provides a good approximation
of the fractional mean curvature equation at points with
nearly horizontal tangent.
Namely, we set
$$ \gamma:= \frac{1}{2}+s+\epsilon_0=\gamma_0+\epsilon_0
\in\left(\frac12,1\right).$$
We will use the fact that~$\gamma>\gamma_0$ to construct
a subsolution of the $\gamma_0$-fractional Laplace equation.
More precisely, the main formula we need in this framework
is the following:
\begin{lemma}\label{TOR}
Let~$\epsilon_0\in(0,1-\gamma_0)$. We have that
$$ \frac{1}{2} \int_{\mathbb R}
\frac{ (1+t)^\gamma_+ + (1-t)^\gamma_+ -2 }{|t|^{2+2s}}\,dt
\ge c_\star \epsilon_0,$$
for some~$c_\star >0$.
\end{lemma}
\begin{proof} Let~$r\ge0$. By a Taylor expansion at~$r=1$, we have that
$$ r^{\frac{\gamma}{\gamma_0}} = 1 +{\frac{\gamma\,(r-1)}{\gamma_0}}
+{\frac{\gamma\,(\gamma-\gamma_0)\,
\xi^{ {\frac{\gamma}{\gamma_0}} -2}\,(r-1)^2 }{\gamma_0^2}},$$
for some~$\xi$ on the segment joining~$r$ to~$1$.
In particular, $\xi\le 1+r$.
Using this with~$r:=(1\pm t)^{\gamma_0}_+$, we obtain
$$ (1\pm t)^\gamma_+
= 1 +{\frac{\gamma\,\big((1\pm t)^{\gamma_0}_+-1\big)}{\gamma_0}}
+{\frac{\gamma\,(\gamma-\gamma_0)\,
\xi^{ {\frac{\gamma}{\gamma_0}} -2}\,
\big((1\pm t)^{\gamma_0}_+-1\big)^2 }{\gamma_0^2}},$$
for some~$\xi\in [0, 2+|t|]$.
Consequently, since
$$ {\frac{\gamma}{\gamma_0}} -2 ={\frac{\epsilon_0}{\gamma_0}} -1<0$$
we obtain
that
$$ \xi^{ {\frac{\gamma}{\gamma_0}} -2} \ge
(2+|t|)^{ {\frac{\gamma}{\gamma_0}} -2}\ge (2+|t|)^{-2}.$$
Accordingly,
$$ (1\pm t)^\gamma_+
\ge 1 +{\frac{\gamma\,\big((1\pm t)^{\gamma_0}_+-1\big)}{\gamma_0}}
+{\frac{\gamma\,(\gamma-\gamma_0)\,
\big((1\pm t)^{\gamma_0}_+-1\big)^2 }{\gamma_0^2
\,( 2+|t| )^{ 2 }
}},$$
and so
$$ (1+ t)^\gamma_+ + (1-t)^\gamma_+ -2
\ge {\frac{\gamma\,\big((1+ t)^{\gamma_0}_+
+(1-t)^{\gamma_0}_+
-2\big)}{\gamma_0}}
+{\frac{\gamma\,(\gamma-\gamma_0)\,\big[
\big((1+t)^{\gamma_0}_+-1\big)^2
+\big((1-t)^{\gamma_0}_+-1\big)^2 \big]}{\gamma_0^2
\,( 2+|t| )^{ 2 }
}}.$$
Hence, we set
$$ \phi(t):=
{\frac{
\big((1+t)^{\gamma_0}_+-1\big)^2
+\big((1-t)^{\gamma_0}_+-1\big)^2 }{ |t|^{2s}
\,( 2+|t| )^{ 2 }
}},$$
we use that~$\gamma=\gamma_0+\epsilon_0>\gamma_0$ and we conclude that
\begin{equation}\label{GH:SP0}
\int_{\mathbb R}
\frac{ (1+t)^\gamma_+ + (1-t)^\gamma_+ -2 }{|t|^{2+2s}}\,dt\ge
\frac{\gamma}{\gamma_0} \int_{\mathbb R}
\frac{ (1+t)^{\gamma_0}_+ + (1-t)^{\gamma_0}_+ -2 }{|t|^{2+2s}}\,dt
+\frac{\epsilon_0}{\gamma_0} \int_\mathbb R \phi(t)\,dt.
\end{equation}
Also, we know (see e.g.~\cite{getoor}) that~$(-\Delta)^{\gamma_0} t_+^{\gamma_0}=0$
for any~$t>0$, therefore, using this formula at~$t=1$
and noticing that~$1+2\gamma_0=2s$, we see that
$$ \int_{\mathbb R}
\frac{ (1+t)^{\gamma_0}_+ + (1-t)^{\gamma_0}_+ -2 }{|t|^{2+2s}}\,dt =0.$$
Using this and~\eqref{GH:SP0}, we obtain
$$ \int_{\mathbb R}
\frac{ (1+t)^\gamma_+ + (1-t)^\gamma_+ -2 }{|t|^{2+2s}}\,dt\ge
\frac{\epsilon_0}{\gamma_0} \int_\mathbb R \phi(t)\,dt,$$
which
implies the desired result.
\end{proof}
Throughout this section, we will consider~$m$ and~$\epsilon_0$
(to be taken appropriately small in the sequel, namely~$\epsilon_0>0$
can be fixed as small as one wishes, and then~$m>0$ is taken to
be small possibly in dependence of~$\epsilon_0$) and~$c_m\in\mathbb R$,
and let
\begin{equation}\label{DEF:v1}
v(x_1):=\frac{m\,(x_1+c_m)^\gamma_+ }{\gamma}.\end{equation}
The parameter~$c_m$ will be conveniently chosen in the sequel,
see in particular the following formula~\eqref{J56GFJJ},
but for the moment it is free.
Also, given~$p:=(p_1,p_2)$ with~$p_1\ge1-c_m$ and~$p_2=v(p_1)$,
we consider the tangent line at~$v$ through~$p$, namely
\begin{equation}\label{DEF:v2}
\Lambda(x_1):=v'(p_1) (x_1-p_1)+ v(p_1)
= m\,(p_1+c_m)^{\gamma-1} (x_1-p_1)+
\frac{m\,(p_1+c_m)^\gamma }{\gamma}.\end{equation}
We observe that the tangent line above
meets the $x_1$-axis at the point~$q=(q_1,0)$, with
\begin{equation}\label{q1q1}
q_1:= p_1-\frac{v(p_1)}{v'(p_1)} = p_1- \frac{p_1+c_m}{\gamma}.\end{equation}
We also consider the region~$A$ which lies above
the graph of~$v$ and below the graph of~$\Lambda$
and the region~$B$ which lies above
the graph of~$\Lambda$ and below the $x_1$-axis, see Figure~\ref{AB:fig}.
More explicitly, we have
\begin{equation}\label{A:Bdef}\begin{split}
& A:= \{ (x_1,x_2) {\mbox{ s.t. }} x_1>q_1 {\mbox{ and }}
v(x_1)<x_2<\Lambda(x_1)\}\\ {\mbox{and }}\quad&
B:= \{ (x_1,x_2) {\mbox{ s.t. }} x_1<q_1 {\mbox{ and }}
\Lambda(x_1)<x_2<0\}.
\end{split}\end{equation}
\begin{figure}
\centering
\includegraphics[height=5.9cm]{TANG.pdf}
\caption{The sets involved in Section~\ref{sec:GROW:R}.}
\label{AB:fig}
\end{figure}
The first technical result that we need is the following:
\begin{lemma}\label{TECH-LM1}
Let~$\epsilon_0\in(0,1-\gamma_0)$.
There exist~$c$, $c'\in(0,1)$ such that if~$m\in (0,c\epsilon_0]$
then
\begin{equation}\label{T:SPP:1}
\int_{B} \frac{dy}{|y-p|^{2+2s}} -\int_{A} \frac{dy}{|y-p|^{2+2s}} \ge
\frac{c' \,\epsilon_0\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}},
\end{equation}
for any~$p:=(p_1,p_2)$ with~$p_1\ge1-c_m$ and~$p_2=v(p_1)$.
\end{lemma}
\begin{proof} First of all, we observe that~$|y-p|\ge |y_1-p_1|$,
therefore
\begin{equation}\label{FI6}
\int_{A} \frac{dy}{|y-p|^{2+2s}} \le
\int_{A} \frac{dy}{|y_1-p_1|^{2+2s}}
= \int_{q_1}^{+\infty} \frac{\Lambda(y_1)-v(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1
=:H.\end{equation}
Recalling~\eqref{DEF:v1} and~\eqref{DEF:v2}, we have that
\begin{eqnarray*}
H &=&
\int_{q_1}^{+\infty} \frac{
m\,(p_1+c_m)^{\gamma-1} (y_1-p_1)+
\gamma^{-1} m\,(p_1+c_m)^\gamma_+ -\gamma^{-1}
m\,(y_1+c_m)^\gamma_+ }{|y_1-p_1|^{2+2s}}\,dy_1
\\ &=& \frac{ m\,(p_1+c_m)^{\gamma} }{\gamma}
\int_{q_1}^{+\infty} \frac{
\gamma\,(p_1+c_m)^{-1}(y_1-p_1)+1-
(p_1+c_m)^{-\gamma}(y_1+c_m)^\gamma_+ }{|y_1-p_1|^{2+2s}}\,dy_1.
\end{eqnarray*}
Now we recall~\eqref{q1q1}
and use the change of variable from the variable~$y_1$
to the variable~$t$ given by
\begin{equation}\label{SU:id}
y_1+c_m = (p_1+c_m)(t+1).\end{equation}
In this way, we obtain that
\begin{equation*} H
=
\frac{ m}{\gamma \,(p_1+c_m)^{1+2s-\gamma}}
\int_{-\frac{1}{\gamma}}^{+\infty} \frac{
\gamma t+1-
(t+1)^\gamma_+ }{|t|^{2+2s}}\,dt=\frac{C_A\, m}{(p_1+c_m)^{1+2s-\gamma}},\end{equation*}
where
\begin{equation*} C_A :=
\int_{-\frac{1}{\gamma}}^{+\infty} \frac{
\gamma t+1-
(t+1)^\gamma_+ }{|t|^{2+2s}}\,dt .\end{equation*}
Therefore, recalling \eqref{FI6}, we conclude that
\begin{equation}\label{JKUFG}
\int_{A} \frac{dy}{|y-p|^{2+2s}} \le
\frac{C_A\, m}{(p_1+c_m)^{1+2s-\gamma}}.\end{equation}
Now we claim that
\begin{equation}\label{POg33h}
{\mbox{if $y\in B$, then~$|y_2-p_2|\le m |y_1-p_1|$.}}
\end{equation}
To prove this, we take~$y\in B$. Then~$\Lambda(y_1)< y_2<0$, therefore,
since~$p_2\ge0$, we have
$$ |y_2-p_2|=p_2-y_2\le p_2-\Lambda(y_1)=v(p_1)-
\Big( v'(p_1) (y_1-p_1)+ v(p_1) \Big)
\le m(p_1+c_m)^{\gamma-1} |y_1-p_1|.$$
Now we have that~$p_1+c_m\ge 1$, by our assumptions.
Hence, since~$\gamma-1<0$,
we conclude that~$ |y_2-p_2|\le m|y_1-p_1|$,
thus proving~\eqref{POg33h}.
As a consequence of~\eqref{POg33h}, we have that if~$y\in B$
then~$|y-p|\le (1+Cm)|y_1-p_1|$, for some~$C>0$, and therefore
\begin{equation}\label{Y67P}
\int_{B} \frac{dy}{|y-p|^{2+2s}} \ge
(1-Cm) \int_{B} \frac{dy}{|y_1-p_1|^{2+2s}}=(1-Cm)\,I ,\end{equation}
up to renaming~$C>0$, where
$$ I:=\int_{B} \frac{dy}{|y_1-p_1|^{2+2s}}=
\int_{-\infty}^{q_1} \frac{-\Lambda(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1.$$
Recalling the definition of~$H$ in~\eqref{FI6}, we have that
$$ J:= H-I =
\int_{q_1}^{+\infty} \frac{\Lambda(y_1)-v(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1
+
\int_{-\infty}^{q_1} \frac{\Lambda(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1.$$
Accordingly, since~$v(y_1)=0$ if~$y_1\le q_1$, we obtain that
$$ J= \int_{-\infty}^{+\infty}
\frac{\Lambda(y_1)-v(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1
= \int_{-\infty}^{+\infty}
\frac{v(p_1)-v(y_1)}{|y_1-p_1|^{2+2s}}\,dy_1,$$
where we have used~\eqref{DEF:v2} in the last identity and the
integrals are taken in the principal value sense.
Hence, we use \eqref{DEF:v1} and the substitution in~\eqref{SU:id},
and we conclude that
$$ J=
\frac{m}{\gamma}
\int_{-\infty}^{+\infty}
\frac{ (p_1+c_m)^\gamma - (y_1+c_m)^\gamma_+}{|y_1-p_1|^{2+2s}}\,dy_1
=
\frac{m}{\gamma (p_1+c_1)^{1+2s-\gamma}}
\int_{-\infty}^{+\infty}
\frac{ 1- (t+1)^\gamma_+}{|t|^{2+2s}}\,dt =
-\frac{C_B\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}},$$
where
$$ C_B:= \int_{-\infty}^{+\infty}
\frac{ (t+1)^\gamma_+-1}{|t|^{2+2s}}\,dt.$$
{F}rom Lemma~\ref{TOR}, we have that~$C_B \ge c_\star \epsilon_0$,
for some~$c_\star>0$. As a consequence,
$$ I = H-J =
\frac{(C_A+C_B)\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}}
\ge \frac{(C_A+c_\star\epsilon_0)\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}},$$
and so, by~\eqref{Y67P}
\begin{equation*}
\int_{B} \frac{dy}{|y-p|^{2+2s}} \ge
\frac{(1-Cm)(C_A+c_\star\epsilon_0)\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}}
.\end{equation*}
Putting together this and~\eqref{JKUFG}, we obtain that
$$
\int_{B} \frac{dy}{|y-p|^{2+2s}} - \int_{A} \frac{dy}{|y-p|^{2+2s}}\ge
\frac{\big[ (1-Cm)(C_A+c_\star\epsilon_0)-C_A\big]\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}},$$
which implies the desired result.
\end{proof}
\begin{figure}
\centering
\includegraphics[height=5.5cm]{Fig-BAR-inf.pdf}
\caption{The barrier constructed in Proposition~\ref{BAR-gr}.}
\label{SK}
\end{figure}
Now we are in the position of
improving the behavior at infinity of the barrier constructed in
Proposition~\ref{IT:BARR}. The idea is to ``glue'' the barrier
of Proposition~\ref{IT:BARR} with the graph of
the ``right'' power function at infinity. The construction
is sketched in Figure~\ref{SK} and the precise result obtained
is the following:
\begin{proposition}\label{BAR-gr}
Let~$\epsilon_0\in(0,1-\gamma_0)$.
There exists~$c>0$ such that if~$m\in (0,c\epsilon_0]$,
then the following statement holds.
There exist~$a_m>0$, $d_m>1>\alpha_m>0$, $c_m\in\mathbb R$
and a set~$E_m\subset\mathbb R^2$
with $(\partial E_m)\cap \{x_2>0\}$
of class~$C^{1,1}$
and such that:
\begin{eqnarray*}
&& E_m \cap \{x_1 < 0\} = (-\infty,0)\times(-\infty,0), \\
&& E_m\supseteq \mathbb R\times (-\infty,0),\\
&& E_m\supseteq (0,+\infty)\times (-\infty,a_m],\\
&& E_m\cap \{\alpha_m\le x_1 \le d_m\} =
\{ x_2<v'(d_m)(x_1-d_m)+v(d_m),\;\alpha_m\le x_1 \le d_m\}\\
{\mbox{and }}&& E_m\cap \{x_1 >d_m\} = \{x_2<v(x_1), \;x_1 >d_m\}
,\end{eqnarray*}
where~$v$ was introduced in~\eqref{DEF:v1}.
Moreover, there exist~$c'\in(0,1)$ and~$N>1$ such that
\begin{equation}\label{PLFG}
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c'\,\epsilon_0\,m}{|p|^{\frac{1}{2}+s-\epsilon_0}},\end{equation}
for any~$p\in (\partial E_m)\cap \{x_1>\frac{d_m}{N}\}$, and
\begin{equation}\label{PLFGbar}
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c'\,m}{d_m^{1-\gamma}\, |p|^{2s}},\end{equation}
for any~$p\in (\partial E_m)\cap \{x_1\in\left(0,\frac{d_m}{N}\right]\}$.
\end{proposition}
\begin{proof} We use Proposition~\ref{IT:BARR} with a large~$K$.
In this way, we may suppose that~$\ell_K\ge K$ is as large as
we wish, while~$q_K\le K^{-1}$ is as small as we wish.
We fix~$N>0$, to be chosen
appropriately large (independently on~$K$) and we set
\begin{equation}\label{UIj:M}
\begin{split}
& d_m:=N^2\\
{\mbox{and }}\quad& m:=
\ell_K^{-1} \big( \gamma\,(d_m+q_K)\big)^{1-\gamma}
.\end{split}\end{equation}
We stress that~$m>0$ is small when~$K$ is large, since
$$ m\le K^{-1} \big( \gamma\,(N^2+K^{-1})\big)^{1-\gamma},$$
that is small when~$K$ is large (much larger than the fixed~$N$).
Hence Proposition~\ref{IT:BARR} provides a set, say~$F_m$,
whose boundary agrees with a straight line~$\lambda_m$
of the form~$x_2=\ell_K^{-1}(x_1+q_K)$
when~$x_1\ge \alpha_m$, for suitable~$q_K\in[0,K^{-1}]$ and~$\alpha_m>0$.
Now we join such a straight line with the function~$v$ defined
in~\eqref{DEF:v1}, at the point~$(d_m,v(d_m))$, with~$\beta_m:=d_m-\alpha_m$
suitably large. To this goal, we define
\begin{equation}\label{J56GFJJ}
c_m:=(\gamma-1)d_m+\gamma q_K.\end{equation}
Notice that
\begin{equation}\label{HJ:LKJ}
d_m+c_m =\gamma(d_m+q_K).
\end{equation}
This and~\eqref{UIj:M} give that
$$ v(d_m)=\frac{m\,(d_m+c_m)^\gamma_+ }{\gamma}
= \frac{m\,\big(\gamma (d_m+q_K)\big)^\gamma }{\gamma}
=
\ell_K^{-1} \big( \gamma\,(d_m+q_K)\big)^{1-\gamma}
\cdot\frac{\big(\gamma (d_m+q_K)\big)^\gamma }{\gamma}
=\ell_K^{-1}(d_m+q_K),$$
which says that~$v$ meets the straight line~$\lambda_m$
at the point~$(d_m,v(d_m))$.
Also, by~\eqref{UIj:M} and~\eqref{HJ:LKJ}, we see that
$$ v'(d_m)= m(d_m+c_m)^{\gamma-1} =
\ell_K^{-1} \big( \gamma\,(d_m+q_K)\big)^{1-\gamma}\cdot
\big( \gamma\,(d_m+q_K)\big)^{\gamma-1} =\ell_K^{-1},$$
therefore $v$ and~$\lambda_m$ have the same slope at
the meeting point~$(d_m,v(d_m))$.
Therefore, the set~$E_m$ which coincides with~$F_m$
when~$\{x_1\le d_m\}$ and with the subgraph of~$v$ when~$\{x_1>d_m\}$
satisfy the geometric properties listed in the statement of
Proposition~\ref{BAR-gr}, and it only remains to prove~\eqref{PLFG}
and~\eqref{PLFGbar}.
For this scope, we first consider the case in which~$p_1\ge d_m$.
Then, we take~$\Lambda$ as in~\eqref{DEF:v2}
and $A$ and~$B$ as in~\eqref{A:Bdef}. Let also~$T$ be the subgraph
of~$\Lambda$. Then, by symmetry
$$ \int_{\mathbb R^2} \frac{\chi_{T}(y)-\chi_{T^c}(y)}{|y-p|^{2+2s}}\,dy=0.$$
Notice that~$T\setminus E_m\subseteq A$
and~$E_m\setminus T \supseteq B$, therefore
\begin{eqnarray*}
&& \int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy\\
&=&
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)-\chi_{T}(y)+\chi_{T^c}(y)
}{|y-p|^{2+2s}}\,dy\\
&=&2
\int_{\mathbb R^2} \frac{\chi_{E_m\setminus T}(y) - \chi_{T\setminus E_m}(y)
}{|y-p|^{2+2s}}\,dy\\
&\ge& 2
\int_{\mathbb R^2} \frac{\chi_{B}(y) - \chi_{A}(y)
}{|y-p|^{2+2s}}\,dy\\
&=& 2\left(
\int_{B} \frac{dy}{|y-p|^{2+2s}}-\int_{A} \frac{dy}{|y-p|^{2+2s}}
\right).
\end{eqnarray*}
Notice also that
\begin{equation}\label{HJLKK9}
1-c_m = 1 -(\gamma-1)d_m-\gamma q_K\le 1-\gamma d_m +d_m\le d_m\end{equation}
thanks to \eqref{J56GFJJ}
and \eqref{UIj:M}. Hence, in this case, $p_1\ge d_m\ge 1-c_m$,
and so the assumptions of Lemma~\ref{TECH-LM1} are fulfilled.
Therefore, by~\eqref{T:SPP:1},
\begin{equation}\label{JK:PLL}
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge
\frac{c' \,\epsilon_0\, m}{(p_1+c_m)^{\frac{1}{2}+s-\epsilon_0}},
\end{equation}
for some~$c'>0$.
Now we notice that, by~\eqref{J56GFJJ} and \eqref{UIj:M},
$$ p_1+c_m = p_1+ (\gamma-1)d_m+\gamma q_K\le 2 p_1 \le 2|p|.$$
Using this and~\eqref{JK:PLL}, we see that~\eqref{PLFG}
holds true in this case.
Hence, it remains to prove~\eqref{PLFG} and~\eqref{PLFGbar}
when~$p_1\in(0,d_m)$.
In this case, we use that, by
Proposition~\ref{IT:BARR},
$$ \int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{\bar c}{\ell_K\,|p|^{2s}},$$
for some~$\bar c>0$.
Also~$F_m\setminus E_m$ coincides with the portion comprised above
the graph of~$v$ and below the straight line~$\lambda_m$, that is
$$ G:=\{ x_1> d_m,\; v(x_1)< x_2< v'(d_m)(x_1-d_m)+v(d_m)\},$$
while~$E_m\setminus F_m$ is empty. Therefore
\begin{equation}\label{PL:PL1}
\begin{split}
& \frac{\bar c}{\ell_K\,|p|^{2s}}
-
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\le
\int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)-
\chi_{E_m}(y)+\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\\ &\quad=
2\int_{G} \frac{dy}{|y-p|^{2+2s}}
\le 2\int_{G} \frac{dy}{|y_1-p_1|^{2+2s}}
\\ &\quad= 2\int_{d_m}^{+\infty} \frac{v'(d_m)(y_1-d_m)+v(d_m)-v(y_1)
}{ |y_1-p_1|^{2+2s} }\,dy_1.\end{split}\end{equation}
Now, we distinguish the cases~$p_1\in \left(0,\frac{d_m}{N}\right)$
and~$p_1\in\left[\frac{d_m}{N},d_m\right)$.
If~$p_1\in \left(0,\frac{d_m}{N}\right)$, we use~\eqref{PL:PL1}
and observe that~$v(y_1)\ge v(d_m)$ if~$y_1\ge d_m$, to conclude that
\begin{eqnarray*}
&&\frac{\bar c}{\ell_K\,|p|^{2s}}
-
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\le
2 v'(d_m) \int_{d_m}^{+\infty} \frac{y_1-d_m}{ |y_1-p_1|^{2+2s} }\,dy_1
\\ &&\quad\le
2 v'(d_m) \int_{d_m}^{+\infty} \frac{dy_1}{ (y_1-p_1)^{1+2s} }
\le\frac{Cm\, (d_m+c_m)^{\gamma-1} }{(d_m-p_1)^{2s}}
\\ &&\quad\le \frac{Cm\, (d_m+c_m)^{\gamma-1} }{d_m^{2s}},\end{eqnarray*}
up to renaming constants. Therefore, recalling~\eqref{UIj:M}
and~\eqref{HJ:LKJ},
\begin{equation}\label{JK:PJgH}
\begin{split}
& \int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{\bar c\,m}{ \big( \gamma\,(d_m+q_K)\big)^{1-\gamma}\,|p|^{2s}} -
\frac{Cm }{ (d_m+c_m)^{1-\gamma}\,
d_m^{2s}} \\&\qquad= \frac{m}{ (d_m+c_m)^{1-\gamma} }\left(\frac{\bar c}{|p|^{2s}}-
\frac{C}{d_m^{2s}}
\right).\end{split}\end{equation}
Now we observe that, when~$p_1\le \frac{d_m}{N}$, we have
that~$p_2\le 1 +\ell_K^{-1}\left( \frac{d_m}{N}+q_K\right)\le
2+\frac{d_m}{N}\le \frac{d_m}{N^{1/2}}$, and so~$|p|\le \frac{d_m}{N^{1/4}}$.
Therefore
$$ \frac{C}{d_m^{2s}} \le \frac{C}{N^{s/2}\,|p|^{2s}}\le
\frac{\bar c}{2\,|p|^{2s}},$$
if~$N$ is large enough (independently on~$m$ and~$K$).
This and~\eqref{JK:PJgH} imply that
\begin{equation*} \int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{m\,\bar c}{ 2(d_m+c_m)^{1-\gamma} |p|^{2s}}.\end{equation*}
By recalling~\eqref{HJ:LKJ},
we see that the latter estimate implies~\eqref{PLFGbar}
in this case.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{VpTAN.pdf}
\caption{A power-like function tangent at~$p\in \partial E_m$,
with $p_1\in\left[\frac{d_m}{N},d_m\right)$.}
\label{VTAN}
\end{figure}
It remains to prove~\eqref{PLFG} when~$p_1\in\left[\frac{d_m}{N},d_m\right)$.
In this case, we argue like this.
For any $p=(p_1,p_2)\in \partial F_m$
with~$p_1\in\left[\frac{d_m}{N},d_m\right)$, we have~$p_2=
v'(d_m)(p_1-d_m)+v(d_m)$,
and we
define~$v_p$
the power function whose graph passes
through~$p$ and tangent to the line~$\{ x_2 = v'(d_m)(x_1-d_m)+v(d_m)\}$
at~$p$, see Figure~\ref{VTAN}.
Explicitly, we define
\begin{eqnarray*}
&& v_p(x_1):=\frac{m_p \,(x_1+c_{m,p} )^\gamma_+ }{\gamma},\\
{\mbox{with }}&& m_p:=(\gamma p_2)^{1-\gamma} m^\gamma
(d_m+c_m)^{\gamma(\gamma-1)}\\
{\mbox{and }}&& c_{m,p}:=\frac{\gamma p_2}{m\,(d_m+c_m)^{\gamma-1}}-p_1.
\end{eqnarray*}
We remark that
$$ v_p (p_1)=p_2 {\mbox{ and }}
v_p'(p_1) = v'(d_m).$$
Since~$p_2< v(d_m)=m\,\gamma^{-1}(d_m+c_{m} )^\gamma$, we have that
\begin{equation}\label{LP:mp small}
m_p<\big(
m\,(d_m+c_{m} )^\gamma \big)^{1-\gamma} m^\gamma
(d_m+c_m)^{\gamma(\gamma-1)}=m.
\end{equation}
Moreover~$p_2=v'(d_m)(p_1-d_m)+v(d_m)=
m\,(d_m+c_m)^{\gamma-1} (p_1-d_m) + m\,\gamma^{-1} (d_m+c_m)^{\gamma}$,
therefore
\begin{equation}\label{HJLKK9-cmp:PRE}
\begin{split} c_{m,p}\;&=\,\frac{\gamma
m\,(d_m+c_m)^{\gamma-1} (p_1-d_m) + m\,(d_m+c_m)^{\gamma}
}{m\,(d_m+c_m)^{\gamma-1}}-p_1
\\&= \,\gamma(p_1-d_m)+d_m+c_m-p_1= (1-\gamma)(d_m-p_1)+c_m.\end{split}\end{equation}
Hence, since~$p_1<d_m$,
\begin{equation}\label{HJLKK9-cmp}
c_{m,p}> \,c_m
.\end{equation}
Also, from~\eqref{J56GFJJ} and~\eqref{HJLKK9-cmp:PRE},
\begin{equation} \label{KL:PP-2}
c_{m,p} = (1-\gamma)(d_m-p_1)+
(\gamma-1)d_m+\gamma q_K = -(1-\gamma)\,p_1+\gamma q_K.\end{equation}
Therefore, since~$p_1\ge\frac{d_m}{N}=N$,
\begin{equation}\label{HJ:pl}
c_{m,p} \le -(1-\gamma)\,N+\gamma q_K\le -(1-\gamma)\,N+1
< -1 \le -\alpha_m,
\end{equation}
provided that~$N$ is large enough.
Furthermore, using again~\eqref{KL:PP-2},
\begin{equation}\label{PKFDc}
p_1+c_{m,p} = \gamma p_1+\gamma q_K\ge \frac{\gamma d_m}{N}=\gamma N\ge 1.
\end{equation}
In addition,
$$ m_p (p_1+c_{m,p})_+^{\gamma-1}=
v_p'(p_1) = v'(d_m) = m (d_m+c_m)_+^{\gamma-1},$$
therefore, by~\eqref{HJ:LKJ}
and~\eqref{KL:PP-2},
\begin{equation}\label{SRTG-9}
\begin{split}
& \frac{m_p}{m} =\frac{(p_1+c_{m,p})_+^{1-\gamma} }{ (d_m+c_m)_+^{1-\gamma} }
=\frac{(\gamma p_1+\gamma q_K)_+^{1-\gamma} }{ (\gamma(d_m+q_K))_+^{1-\gamma} }
= \frac{(p_1+q_K)_+^{1-\gamma} }{ (d_m+q_K)_+^{1-\gamma} }
\\
&\qquad\ge \frac{\left(\frac{d_m}{N}\right)_+^{1-\gamma} }{
(2d_m)_+^{1-\gamma} }
= \frac{1}{(2N)^{1-\gamma}}.
\end{split}\end{equation}
Now we claim that
\begin{equation}\label{OILJ:889}
{\mbox{if~$x_1\ge d_m$, then $v_p(x_1)\le v(x_1)$.}}\end{equation}
To prove this,
we use~\eqref{HJLKK9}
and~\eqref{HJLKK9-cmp} to see that
$$ x_1+c_{m,p}\ge x_1+c_m\ge d_m+c_m\ge1,$$ therefore
$$ \psi(x_1):=\gamma\,\big(v_p(x_1)-v(x_1)\big)=
{m_p \,(x_1+c_{m,p} )^\gamma }-
{m \,(x_1+c_{m} )^\gamma }.$$
Also, $v_p$ is concave, therefore
\begin{eqnarray*}
&& v_p(d_m) \le v_p (p_1) + v_p'(p_1)(d_m-p_1)
=p_2+v'(d_m)(d_m-p_1)\\
&&\qquad=v'(d_m)(p_1-d_m)+v(d_m)+v'(d_m)(d_m-p_1)=v(d_m).\end{eqnarray*}
As a consequence, $ \psi(d_m)\le 0$. Moreover, for any~$x_1\ge d_m$,
$$ \psi'(x_1)=
{m_p \gamma \,(x_1+c_{m,p} )^{\gamma-1} }-
{m \gamma \,(x_1+c_{m} )^{\gamma-1} }
\le m\gamma\,\big[ {(x_1+c_{m,p} )^{\gamma-1} }-
{(x_1+c_{m} )^{\gamma-1} } \big]\le0,$$
thanks to~\eqref{LP:mp small} and~\eqref{HJLKK9-cmp}. {F}rom
these considerations, we obtain that~$\psi\le0$
in~$[d_m,+\infty)$, which proves~\eqref{OILJ:889}.
Also, by concavity,
\begin{equation}\label{COPP}
\begin{split}
& {\mbox{if $x_1\in[-c_{m,p},d_m]$, then}}\\
&\qquad v_p(x_1)\le
v_p'(p_1)(x_1-p_1)+v_p(p_1)=
v'(d_m)(x_1-p_1)+p_2
=v'(d_m)(x_1-d_m)+v(d_m).\end{split}\end{equation}
Now we claim that
\begin{equation}\label{is co}
{\mbox{the subgraph of~$v_p$ is contained in~$E_m$.}}
\end{equation}
To check this, let~$x=(x_1,x_2)$ be such that~$x_2< v_p(x_1)$.
Then, if~$x_1 < -c_{m,p}$ then~$v_p(x_1)=0$ and so~\eqref{is co}
plainly follows. If~$x_1\in [-c_{m,p},d_m]$,
then~\eqref{is co} is implied by~\eqref{HJ:pl} and \eqref{COPP}.
Finally, if~$x_1>d_m$, then~\eqref{is co}
is a consequence of~\eqref{OILJ:889}.
Hence, we define~$S:= \{x_2<v_p(x_1)\}$,
we use~\eqref{is co}
and Lemma~\ref{TECH-LM1} (which can be exploited in this
framework with the power-like function~$v_p$,
thanks to~\eqref{PKFDc}) and we obtain that
\begin{equation}\label{LGH:LK}
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge
\int_{\mathbb R^2} \frac{\chi_{S}(y)-\chi_{S^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c' \,\epsilon_0\, m_p}{(p_1+c_{m,p})^{\frac{1}{2}+s-\epsilon_0}},\end{equation}
for some~$c'>0$. Now we recall~\eqref{KL:PP-2} and we see that~$
p_1+c_{m,p}\le p_1\le |p|$. Using this and~\eqref{SRTG-9}
(notice that~$N$ has now been fixed),
we obtain~\eqref{PLFG} if~$p_1\in\left[\frac{d_m}{N},d_m\right)$
as a consequence of~\eqref{LGH:LK}.
This completes the proof of \eqref{PLFG} in all cases and finishes the
proof of Proposition~\ref{BAR-gr}.
\end{proof}
\section{Construction of compactly supported barriers}\label{SEC:CMP:SUPP}
In this section, we construct a suitable barrier for the fractional
mean curvature equation in the plane which is flat and horizontal
outside a vertical slab, and whose geometric properties
inside the slab are under control. Roughly speaking,
we will take the barrier constructed in Proposition~\ref{BAR-gr}
and a reflected version of it and join it smoothly in the middle.
The effect of this surgery is negligible at
the points of the barrier that are near the horizontal part,
and give a bounded contribution in the middle.
\begin{figure}
\centering
\includegraphics[height=3.2cm]{F-BAR-Comp.pdf}
\caption{The barrier constructed in Proposition~\ref{LK:PO}.}
\label{LLM}
\end{figure}
This barrier is described in Figure~\ref{LLM}
and the precise result obtained is the following:
\begin{proposition}\label{LK:PO}
Let~$\epsilon_0\in(0,1-\gamma_0)$.
There exists~$m_{\epsilon_0}>0$ such that if~$m\in (0,\,m_{\epsilon_0}]$
then the following statement holds.
There exist~$a_m>0$, $L_m>A_m>d_m>1$, $c_m\in\mathbb R$, $C_\star>0$
and a set~$F_m\subset\mathbb R^2$
with $(\partial F_m)\cap \{x_2>0\}$
of class~$C^{1,1}$ and
such that:
\begin{eqnarray*}
&& F_m \cap \{x_1 < 0\} = (-\infty,0)\times(-\infty,0), \\
&& F_m\supseteq \mathbb R\times (-\infty,0),\\
&& F_m\supseteq (0,L_m+1)\times (-\infty,a_m],\\
&& F_m \subseteq \{ x_2\le C_\star m\, L_m^{\frac{1}{2}+s+\epsilon_0} \}\\
{\mbox{and }}&&
F_m\cap \{d_m<x_1<L_m\} = \{x_2<v(x_1), \;d_m<x_1 <L_m\}
,\end{eqnarray*}
where~$v$ was introduced in~\eqref{DEF:v1}.
In addition, one can suppose that
\begin{equation}\label{L emme}
L_m = 10 A_m \ge 2+m^{-1} + e^{\frac{1}{a_m}}.
\end{equation}
Moreover, the set~$F_m$ is even symmetric with respect to
the vertical axis~$\{x_1=L_m+1\}$,
and
there exists $C'>0$ such that
\begin{equation}\label{PLFG-SYM}
\int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge 0,\end{equation}
for any~$p\in (\partial F_m)\cap \{x_1\in (0,A_m)\}$,
and
\begin{equation}\label{PLFG-SYM:2}
\int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge -\frac{C' m^{2s}}{L_m^{\frac{1}{2}+s-\epsilon_0}},\end{equation}
for any~$p\in (\partial F_m)\cap \{x_1\in\left[A_m, L_m+1\right]\}$.
\end{proposition}
\begin{proof} We let~$E_m$ be the set constructed in
Proposition~\ref{BAR-gr}. Let~$E'_m$
be the even reflection
of~$E_m$ with respect to
the vertical axis~$\{x_1=L_m+1\}$.
We take a smooth function~$w:[L_m,\,L_m+2]\to [v(L_m),\,Cm \,L_m^\gamma]$
that is even with respect to~$\{x_1=L_m+1\}$, with~$w(L_m)=v(L_m)$
and such that
its derivatives agree with the ones of~$v$ at the point~$L_m$.
The set~$F_m$ is then defined as
$$ \big( E_m\cap \{x_1 \le L_m\} \big)
\;\cup\;
\big\{ x_2<w(x_1),\; x_1\in (L_m,L_m+2)\big\}
\;\cup\;
\big( E_m'\cap \{x_1 \ge L_m+2\} \big).$$
For completeness, let us describe the above function~$w$ explicitly.
One takes an odd function~$\tau\in C^\infty(\mathbb R,[-1,1])$ such that~$\tau=-1$
in~$(-\infty,-1]$ and~$\tau=1$ in~$[1,+\infty)$ and defines~$w$ by
$$ w(x_1):=\frac{\big(1-\tau(x_1-L_m-1)\big)\,v(x_1)+
\big(1+\tau(x_1-L_m-1)\big)\,v(2L_m+2-x_1)
}{2}.$$
Then~$w(L_m+1+x_1)=w(L_m+1-x_1)$,
hence~$w$ is even with respect to~$\{x_1=L_m+1\}$.
The set~$F_m$ has the desired geometric properties, so
it remains to prove~\eqref{PLFG-SYM}
and~\eqref{PLFG-SYM:2}. For this, we take~$L_m=10 A_m$
appropriately large. In particular, we suppose that~$L_m\ge c_m+2A_m$,
and therefore, for any~$y_1\in [L_m,+\infty)$ and~$p_1\in(0,A_m)$
we have that~$ y_1+c_m\le 2(y_1-p_1)$, and so, by~\eqref{DEF:v1},
$$ v(y_1)=\frac{m\,(y_1+c_m)^\gamma_+ }{\gamma}\le
\frac{2^\gamma m\,(y_1-p_1)^\gamma }{\gamma}.$$
We also notice that~$E_m\setminus F_m\subseteq\{x_1>L_m,\;0<x_2<v(x_1)\}$.
Therefore, for every~$p\in (\partial F_m)\cap \{x_1\in (0,A_m)\}$,
\begin{equation}\label{CVXKK}
\begin{split} &
\int_{\mathbb R^2} \frac{\chi_{E_m}(y)-\chi_{E_m^c}(y)}{|y-p|^{2+2s}}\,dy
-\int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy\\
&\qquad=
2\int_{\mathbb R^2} \frac{
\chi_{E_m\setminus F_m}(y)-
\chi_{F_m\setminus E_m}(y)
}{|y-p|^{2+2s}}\,dy
\le
2\int_{ E_m\setminus F_m }\frac{dy}{ |y-p|^{2+2s} }
\\ &\qquad\le 2\int_{L_m}^{+\infty} \frac{v(y_1)\,dy_1}{ (y_1-p_1)^{2+2s} }
\le Cm \int_{L_m}^{+\infty} (y_1-p_1)^{\gamma-2-2s}
= \frac{Cm}{(L_m-p_1)^{1+2s-\gamma}}\\
&\qquad\le \frac{Cm}{L_m^{1+2s-\gamma}} =
\frac{Cm}{L_m^{\frac{1}{2}+s-\epsilon_0}},
\end{split}\end{equation}
up to changing the names of the constant~$C>0$ line after line.
Hence, recalling~\eqref{PLFG},
$$ \int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy\ge
\frac{c'\,\epsilon_0\,m}{|p|^{\frac{1}{2}+s-\epsilon_0}}
-\frac{Cm}{L_m^{\frac{1}{2}+s-\epsilon_0}}
\ge \frac{c'\,\epsilon_0\,m}{2\,|p|^{\frac{1}{2}+s-\epsilon_0}}$$
for any~$p\in (\partial F_m)\cap \{x_1\in \left(\frac{d_m}{N},A_m\right)\}$,
as long as~$L_m$ is large enough (possibly in dependence of~$
\sup_{q_1\in(0,A_m)} |q|$). This establishes~\eqref{PLFG-SYM}
if~$p\in (\partial F_m)\cap \{x_1\in \left(\frac{d_m}{N},A_m\right)\}$.
If instead~$p\in (\partial F_m)\cap \{x_1\in \left(0,\frac{d_m}{N}\right]\}$,
we use~\eqref{CVXKK}
and~\eqref{PLFGbar} to obtain that
$$ \int_{\mathbb R^2} \frac{\chi_{F_m}(y)-\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy
\ge \frac{c'\,m}{d_m^{1-\gamma}\, |p|^{2s}} -
\frac{Cm}{L_m^{\frac{1}{2}+s-\epsilon_0}}
\ge
\frac{c'\,N^{2s}\,m}{d_m^{1-\gamma}\, d_m^{2s}} -
\frac{Cm}{L_m^{\frac{1}{2}+s-\epsilon_0}}
=\frac{c'\,N^{2s}\,m}{d_m^{ {\frac{1}{2}+s-\epsilon_0} }} -
\frac{Cm}{L_m^{\frac{1}{2}+s-\epsilon_0}}\ge0
,$$
as long as~$L_m$ is large enough, and this proves~\eqref{PLFG-SYM}
also in this case.
Now we prove~\eqref{PLFG-SYM:2}.
For this, we take~$p\in (\partial F_m)\cap \{x_1\in\left[A_m, L_m+1\right)\}$.
By~\eqref{DEF:v1}, the curvature of~$F_m$ at~$p$
is bounded (in absolute value) by~$C m L_m^{\gamma-2}$.
Hence (see Lemma~3.1 in~\cite{nostro}, applied
here with~$\lambda:=L_m^{\gamma-1}$
and~$R:=m^{-1} L^{2-\gamma}$, so that~$\lambda R= \frac{L_m}{m}$,
and canceling the contribution coming from
the tangent line) one obtains that
\begin{equation}\label{POlfG-1}
\left|\int_{B_{\frac{L_m}{m}} (p)} \frac{\chi_{F_m}(y)-
\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy\right|\le C \big( L_m^{\gamma-1}\big)^{1-2s}
\big( m^{-1} L_m^{2-\gamma}\big)^{-2s}
= C m^{2s} L_m^{\gamma-1-2s}=\frac{C m^{2s}}{ L_m^{\frac12+s-\epsilon_0}}
,\end{equation}
for some~$C>0$, possibly varying from step to step.
Moreover, to compute the contribution coming from
outside~$B_{\frac{L_m}{m}}(p)$, we can compare the set~$F_m$
with the horizontal line passing through~$p$.
Notice indeed that~$F_m\setminus B_{\frac{L_m}{m}}(p)=
\{x_2<0\}\setminus B_{\frac{L_m}{m}}(p)$.
Thus, since~$p_2$ is controlled by~$C m L_m^\gamma$
\begin{equation*}\begin{split}
& \left|\int_{\mathbb R^2\setminus B_{ \frac{L_m}{m} }(p)} \frac{\chi_{F_m}(y)-
\chi_{F_m^c}(y)}{|y-p|^{2+2s}}\,dy\right|
\le 2 \int_{\{0<y_2<C m L_m^\gamma\}\setminus B_{ \frac{L_m}{m} (p)}} \frac{dy}{|y-p|^{2+2s}}
\\ &\qquad\le C m L_m^\gamma \int_{ \{ |y_1-p_1|\ge L_m\} } \frac{dy_1}{|y_1-p_1|^{2+2s}}
= C m L_m^{\gamma-1-2s} =
\frac{C m}{ L_m^{\frac12+s-\epsilon_0}}
.\end{split}\end{equation*}
up to renaming~$C>0$. This and~\eqref{POlfG-1} imply~\eqref{PLFG-SYM:2},
as desired.
\end{proof}
By scaling Proposition~\ref{LK:PO},
one obtains the following result:
\begin{corollary}\label{COR 2.2-BAR}
Fix~$\epsilon_0>0$ arbitrarily small.
There exist an infinitesimal sequence of
positive~$\delta$'s and sets~$H_\delta\subseteq\mathbb R^2$,
with $(\partial H_\delta)\cap \{x_2>0\}$
of class~$C^{1,1}$,
that are even symmetric with respect to the axis~$\{x_1=0\}$ and
satisfy the following properties:
\begin{eqnarray*}
&& H_\delta \cap \{x_1 < -1\} = (-\infty,-1)\times(-\infty,0), \\
&& H_\delta \supseteq \mathbb R\times (-\infty,0),\\
&& H_\delta \supseteq (-1,1)\times (-\infty,\delta^{\frac{2+\epsilon_0}{1-2s}}]\\
{\mbox{and }}&& H_\delta \subseteq \{ x_2\le \delta \}
.\end{eqnarray*}
Moreover,
\begin{equation}\label{PLFG-SYM:cor}
\int_{\mathbb R^2} \frac{\chi_{H_\delta}(y)-\chi_{H_\delta^c}(y)}{|y-p|^{2+2s}}\,dy
\ge 0,\end{equation}
for any~$p\in (\partial H_\delta)\cap \{x_1\in \left(-1,-1+\frac{1}{100}\right)\}$
and
\begin{equation}\label{PLFG-SYM:2:cor}
\int_{\mathbb R^2} \frac{\chi_{H_\delta}(y)-\chi_{H_\delta^c}(y)}{|y-p|^{2+2s}}\,dy
\ge -\delta,\end{equation}
for any~$p\in (\partial H_\delta)\cap \{x_1\in\left[-1+\frac{1}{100},\;0\right]\}$.
\end{corollary}
\begin{proof} We scale the set $F_m$ constructed in
Proposition~\ref{LK:PO} by a factor of order~$\frac{1}{L_m}$
(then we also translate to the left
by a horizontal vector of length~$1$)
and take~$\delta:= \frac{1}{L_m^{\frac12-s-\epsilon_0}}$.
Notice that~$\delta$ is infinitesimal, due to~\eqref{L emme}.
Also, the estimates in~\eqref{PLFG-SYM:cor}
and~\eqref{PLFG-SYM:2:cor} follow from the
ones in~\eqref{PLFG-SYM}
and~\eqref{PLFG-SYM:2}, since the fractional curvature scales
by a factor proportional to~$L_m^{2s}$.
We also remark that the vertical stickiness
of~$F_m$ in Proposition~\ref{LK:PO} was bounded from below by~$a_m$,
and~$L_m \ge e^{\frac{1}{a_m}}$, by~\eqref{L emme}.
As a consequence, by scaling, the vertical stickiness of~$H_\delta$
here is bounded by
an order of~$\frac{a_m}{L_m}\ge \frac{1}{L_m\log L_m}$.
This quantity is in turn bounded by an order of~$
\frac{\delta^{\frac{2}{1-2s-2\epsilon_0}} }{|\log\delta|}$,
which we can bound by~$\delta^{\frac{2+\epsilon_0}{1-2s}}$,
up to renaming~$\epsilon_0$.
\end{proof}
We observe that while in~\eqref{PLFG-SYM:cor}
we obtained that the fractional mean curvature of the set is
nonnegative near~$\{x_1=\pm1\}$, from~\eqref{PLFG-SYM:2:cor}
we can only say that the fractional mean curvature of the set
near~$\{x_1=0\}$ is controlled by a small negative quantity
(and this cannot be improved, since at the points in which
the set reaches its highest level the fractional mean
curvature must be negative). By adding an additional small
contribution to the set in~$\{|x_1|\in (2,3)\}$, we can
obtain a complete subsolution, i.e. a set whose
fractional mean curvature is nonnegative.
Such subsolution has the important geometric feature
that the points along~$\{x_1=0\}$ detach from~$\{x_2=0\}$,
see Figure~\ref{FARA}.
The precise statement goes as follows:
\begin{figure}
\centering
\includegraphics[height=3.6cm]{F-BAR-con-AGGIUNTA.pdf}
\caption{The barrier constructed in Proposition~\ref{LK:FARA}.}
\label{FARA}
\end{figure}
\begin{proposition}\label{LK:FARA}
Fix~$\epsilon_0>0$ arbitrarily small.
There exist~$C>0$,
an infinitesimal sequence of
positive~$\delta$'s and sets~$E_\delta\subseteq\mathbb R^2$,
with $(\partial E_\delta)\cap \big((-\frac32,\frac32)\times(0,+\infty)\big)$
of class~$C^{1,1}$,
that are even symmetric with respect to the axis~$\{x_1=0\}$ and
satisfy the following properties:
\begin{eqnarray*}
&& E_\delta \cap \{x_1 \in(-\infty,-3)\cup(-2,-1)\} =
\big( (-\infty,-3)\cup(-2,-1)\big)\times(-\infty,0), \\
&& E_\delta \cap \{x_1 \in[-3,-2]\} = [-3,-2]\times (-\infty, C\delta),\\
&& E_\delta \supseteq \mathbb R\times (-\infty,0),\\
&& E_\delta \supseteq (-1,1)\times (-\infty,\delta^{\frac{2+\epsilon_0}{1-2s}}]\\
{\mbox{and }}&& E_\delta \cap \{ |x_1|\le 1\}\subseteq \{ x_2\le \delta \}
.\end{eqnarray*}
Moreover, for any~$p\in(\partial E_\delta)\cap \{|x_1|<1\}$,
\begin{equation}\label{PLFG-SYM:FIN}
\int_{\mathbb R^2} \frac{\chi_{E_\delta}(y)-\chi_{E_\delta^c}(y)}{|y-p|^{2+2s}}\,dy
\ge 0.\end{equation}
\end{proposition}
\begin{proof} Let~$H_\delta$ be as in
Corollary~\ref{COR 2.2-BAR}.
We define~$E_\delta:= H_\delta\cup F_-\cup F_+$, where
$F_-:= (-3,-2)\times [0,C\delta)$ and
$F_+:= (2,3)\times [0,C\delta)$.
Then~$E_\delta$ satisfies all the desired geometric properties,
and~$E_\delta\supset H_\delta$. Therefore,
when~$p\in (\partial E_\delta)
\cap \{|x_1|\in \left(1-\frac{1}{100},\,1\right)\}$,
we have that~\eqref{PLFG-SYM:FIN}
follows from~\eqref{PLFG-SYM:cor}.
Moreover,
when~$p\in (\partial E_\delta)
\cap \{|x_1|\le1-\frac{1}{100}\}$,
we have that~\eqref{PLFG-SYM:FIN} follows from~\eqref{PLFG-SYM:2:cor}
and the fact that~$|F_+|=|F_-|=C\delta$
(and one can choose $C>0$ conveniently large).
\end{proof}
\begin{remark}{\rm Concerning the statement of Proposition~\ref{LK:FARA},
by~\eqref{PLFG-SYM:FIN} (see in addition
Lemma~3.3 in~\cite{nostro}),
we also obtain that
\begin{equation}\label{PLFG-SYM:FIN:c}
\int_{\mathbb R^2} \frac{\chi_{E_\delta}(y)-\chi_{E_\delta^c}(y)}{|y-p|^{2+2s}}\,dy
\ge 0\end{equation}
for any~$p\in\overline{(\partial E_\delta)\cap \{|x_1|<1\}}$.
}\end{remark}
\section{Instability of the flat fractional minimal surfaces}\label{INST:SEC}
With the barrier constructed in Proposition~\ref{LK:FARA}
we are now in the position of proving Theorem~\ref{UNS}.
For this, we will take~$E$ and~$F$ as in the statement of
Theorem~\ref{UNS}.
\begin{proof}[Proof of Theorem~\ref{UNS}] Let~$E_\delta$ be as in
Proposition~\ref{LK:FARA}. The idea is to slide~$E_\delta$ (or, more precisely,
$E_{\frac{\delta}{C}}$)
from below. Namely, for any~$t\ge0$
we consider the set~$E(t):=E_{\frac{\delta}{C}} - te_2$.
For large~$t$, we have that~$E(t)\subseteq E$.
So we take the smallest~$t\ge0$ for which
such inclusion holds. We observe that Theorem~\ref{UNS}
would be proved if we show that such~$t$ equals to~$0$.
Then suppose, by contradiction, that
\begin{equation}\label{KJrf78}
t>0.\end{equation}
By construction,
\begin{equation}\label{JG5rt}
E(t)\subseteq E\end{equation}
and
there exists a contact point between the two sets.
{F}rom the data outside~$[-1,1]\times\mathbb R$,
we have that all the contact points must lie
in~$[-1,1]\times\mathbb R$.
Furthermore,
\begin{equation}\label{no co}
{\mbox{no contact point can occur in~$(-1,1)\times\mathbb R$.}}\end{equation}
To check this, suppose that there exists~$p=(p_1,p_2)\in (\partial E(t))\cap
(\partial E)$ with~$|p_1|<1$. Then, using the
Euler-Lagrange equation in the viscosity sense for~$E$
(see Theorem~5.1 in~\cite{CRS})
and~\eqref{PLFG-SYM:FIN} we have that
$$ \int_{\mathbb R^2} \frac{\chi_{E}(y)-\chi_{E^c}(y)}{|y-p|^{2+2s}}\,dy
\le0\le
\int_{\mathbb R^2} \frac{\chi_{E(t)}(y)-\chi_{E^c(t)}(y)}{|y-p|^{2+2s}}\,dy.$$
Also, the opposite inequality holds, thanks to~\eqref{JG5rt},
and therefore~$E(t)$ and~$E$ must coincide. This would give that~$t=0$,
against our assumption. This proves~\eqref{no co}.
As a consequence, we have that all the contact points lie
on~$\{\pm1\}\times\mathbb R$. Since both~$\partial E(t)$
and~$\partial E$ are closed set, we can take the contact point
with lower vertical coordinate along~$\{x_1=\pm1\}$, and we denote it
by~$x_{o}^\pm =(\pm1,x_{o,2}^\pm)$.
Now, for any~$k\in\mathbb N$ (to be taken as large as we wish)
and any~$h\in [0,\,1/k]$
we consider the ball of small radius~$r>0$ (smaller than
the radius of curvature of~$E(t)$)
centered on the line~$\{x_2=x_{o,2}^\pm+h\}$
and we slide such ball to the left (towards~$\{x_1=-1\}$)
or to the right (towards~$\{x_1=1\}$) till it touches
either~$\partial E\cap \{ |x_1|<1\}$ or~$\{x_1=\pm 1\}$,
see Figure~\ref{ARCHI}.
\begin{figure}
\centering
\includegraphics[height=5.9cm]{PROV.pdf}
\caption{Sliding the balls from the barriers towards~$\partial E\cap \{ |x_1|<1\}$.}
\label{ARCHI}
\end{figure}
We claim that there exists a sequence~$k\to+\infty$ for which
there exists~$h_k\in
[0,\,1/k]$ such that the sliding of this ball (either to the right or to
the left)
touches~$\partial E\cap \{ |x_1|<1\}$. Indeed, if not,
we have that~$\partial E$, near~$\{x_1=\pm1\}$, stays
above~$\{x_2=x_{o,2}^\pm+\alpha\}$,
for some~$\alpha>0$. But this would imply that we can keep
sliding~$E(t)$ a little more upwards, in contradiction
with the minimality of~$t$.
Therefore, we can assume that, for a suitable sequence~$k\to+\infty$,
we have that there exist points~$x_k=(x_{k,1},x_{k,2})
\in (\partial E)\cap\{|x_1|<1\}$
with~$x_{k,2}=x_{o,2}^\pm + h_k$ and~$h_k\in[0,\,1/k]$. By construction,
the points~$x_k$ must lie outside~$E(t)$, hence,
if~$r$ is small enough,
we have that~$|x_{k,1}|\to1$
as~$k\to+\infty$.
Hence, we assume that~$x_k\in (\partial E)\cap\{|x_1|<1\}$
and~$x_k\to x_o:=x_o^-$ as~$k\to+\infty$ (the case in which~$x_k\to x_o^+$
is completely analogous).
Then, by the Euler-Lagrange equation at the points~$x_k$ (see
Lemma~3.4 in~\cite{nostro}),
we obtain that
\begin{equation}\label{PLFG-SYM:FIN:c:NINI}
\int_{\mathbb R^n} \frac{\chi_{E}(y)- \chi_{E^c}(y)}{|
x_o-y|^{n+2s}}\,dy\le0.\end{equation}
On the other hand, by~\eqref{PLFG-SYM:FIN:c},
\begin{equation}\label{PLFG-SYM:FIN:c:NI}
\int_{\mathbb R^2} \frac{\chi_{E(t)}(y)-\chi_{E^c(t)}(y)}{|x_o-y|^{2+2s}}\,dy
\ge 0.\end{equation}
Combining~\eqref{JG5rt}, \eqref{PLFG-SYM:FIN:c:NINI}
and~\eqref{PLFG-SYM:FIN:c:NI}, it follows that~$E(t)=E$.
Thus, from the values of~$E_\delta$ and~$E$ outside~$\{|x_1|\le 1\}$,
we conclude that~$t=0$.
This is in contradiction with~\eqref{KJrf78}
and so the desired result is proved.
\end{proof}
\begin{appendix}
\section{Symmetry properties and a variation on the proof
of Lemma \ref{TOR}}
Here we prove that the minimizers inherit the symmetry
properties of the boundary data:
\begin{lemma}\label{SYMM-LEMMA}
Let~$T:\mathbb R^n\to\mathbb R^n$ be an isometry,
with~$T(\Omega)=\Omega$.
Assume that there exists~$N\in\mathbb N$ such that~$T^N(x)=x$ for every~$x\in
\Omega$.
Let~$E\subseteq\mathbb R^n$ be such that~$T(E)=E$.
Let~$E_*$ be the $s$-minimal set in a domain~$\Omega$
among all the sets~$F$ such that~$F\setminus\Omega=E\setminus\Omega$.
Then~$T(E_*)=E_*$.
\end{lemma}
\begin{proof} We let
$$ {\mathcal{F}}(u):=\frac12\iint_{\mathbb R^{2n}\setminus (\Omega^c)^2}
\frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy.$$
We observe that
$$ {\mathcal{F}}(\chi_E)=\,{\rm Per}_s(E,\Omega).$$
Moreover, by Lemma~3 on page~685
in~\cite{PSV-AnnMat}, we have that
$$ {\mathcal{F}}(\min\{u, v\} ) + {\mathcal{F}}(\max\{u,v\}) \le
{\mathcal{F}} (u ) + {\mathcal{F}} (v),$$
and the equality holds if and only if
either~$u(x) \le v(x)$ or $v(x) \le u(x)$ for any~$ x\in\Omega$.
We use the observations above with~$u:=\chi_{E_*}$ and~$v:=\chi_{T(E_*)}$.
Notice that, in this case, $\min\{u, v\}=\chi_{E_*\cap T(E_*)}$
and~$\max\{u, v\}=\chi_{E_*\cup T(E_*)}$. Hence, we obtain
\begin{equation}\label{89-987}
\,{\rm Per}_s\big( E_*\cap T(E_*),\Omega\big)+
\,{\rm Per}_s\big( E_*\cup T(E_*),\Omega\big)\le
\,{\rm Per}_s( E_*,\Omega)+\,{\rm Per}_s( T(E_*),\Omega),
\end{equation}
and the equality holds if and only if
either~$\chi_{E_*}(x) \le \chi_{T(E_*)}(x)$ or $\chi_{T(E_*)}(x) \le \chi_{E_*}(x)$
for any~$ x\in\Omega$, that is,
if and only if
\begin{equation}\label{89j67-0}
{\mbox{either~$E_*\cap\Omega\subseteq T(E_*)\cap\Omega$
or~$T(E_*)\cap\Omega\subseteq E_*\cap\Omega$.}}\end{equation}
Now we observe that
\begin{eqnarray*}
\,{\rm Per}_s\big(T(E_*),\Omega\big) &=& L\big(
T(E_*)\cap\Omega, \mathbb R^n\setminus T(E_*)\big)+
L\big(\Omega\setminus T(E_*),
T(E_*)\setminus\Omega\big) \\
&=& L\big(T(E_*)\cap T(\Omega), \mathbb R^n\setminus T(E_*)\big)+
L\big(T(\Omega)\setminus T(E_*),
T(E_*)\setminus T(\Omega)\big)\\
&=&
L\big(T(E_*\cap\Omega), T(\mathbb R^n\setminus E_*)\big)+
L\big(T(\Omega\setminus E_*),
T(E_*\setminus \Omega)\big) \\
&=&
L(E_*\cap\Omega,\mathbb R^n\setminus E_*)+
L(\Omega\setminus E_*, E_*\setminus \Omega)
\\ &=& \,{\rm Per}_s(E_*,\Omega).
\end{eqnarray*}
Substituting this in~\eqref{89-987}, we obtain that
\begin{equation}\label{89-987-2}
\,{\rm Per}_s\big( E_*\cap T(E_*),\Omega\big)+
\,{\rm Per}_s\big( E_*\cup T(E_*),\Omega\big)\le
2\,{\rm Per}_s( E_*,\Omega).\end{equation}
On the other hand,
\begin{equation}\label{0989uo}
T(E_*)\setminus\Omega=
T(E_*)\setminus T(\Omega)= T(E_*\setminus\Omega)
=T(E\setminus\Omega)=T(E)\setminus\Omega=E\setminus \Omega.\end{equation}
This says that~$E_*\cap T(E_*)$ and~$E_*\cup T(E_*)$
are admissible competitors for~$E_*$ and therefore
$$ \,{\rm Per}_s( E_*,\Omega)\le\,{\rm Per}_s\big( E_*\cap T(E_*),\Omega\big)
\;{\mbox{ and }}\;
\,{\rm Per}_s( E_*,\Omega)\le\,{\rm Per}_s\big( E_*\cup T(E_*),\Omega\big).$$
This implies that the equality holds in~\eqref{89-987-2},
and so in~\eqref{89-987}.
Therefore, \eqref{89j67-0} holds true.
So we suppose that~$E_*\cap\Omega\subseteq T(E_*)\cap\Omega$
(the case in which~$T(E_*)\cap\Omega\subseteq E_*\cap\Omega$
can be dealt with in a similar way).
Then we have that~$E_*\cap\Omega\subseteq T(E_*\cap\Omega)$.
By applying~$T$, we obtain~$T(E_*\cap\Omega)\subseteq T^2(E_*\cap\Omega)$,
and so, iterating the procedure
$$ E_*\cap\Omega\subseteq T(E_*\cap\Omega)\subseteq\dots
\subseteq T^{N-1}(E_*\cap\Omega)
\subseteq T^N(E_*\cap\Omega)=E_*\cap\Omega.$$
This shows that~$E_*\cap\Omega=T(E_*\cap\Omega)$, that is~$E_*\cap\Omega=
T(E_*)\cap\Omega$.
Also, by~\eqref{0989uo}, $E_*\setminus\Omega=
T(E_*)\setminus\Omega$. Therefore~$E_*=T(E_*)$, as desired.
\end{proof}
Now we give a different (and more general)
proof of Lemma~\ref{TOR}, according to the following result:
\begin{lemma}\label{lemma:trasformata}
Let $\sigma$, $\sigma_0\in (0,1)$, with $\sigma<2\sigma_0$. Then, for any $t>0$, we have
\begin{equation}\label{prima112}
(-\Delta)^{\sigma_0} t_+^{\sigma} =-4\,\Gamma(1+\sigma)\,\Gamma(2\sigma_0-\sigma)\,
\sin\big(\pi(\sigma-\sigma_0)\big)t^{\sigma-2\sigma_0},\end{equation}
where $\Gamma$ is the gamma function.
In particular,
\begin{itemize}
\item if $\sigma=\sigma_0$, then, for any $t>0$,
$$ (-\Delta)^{\sigma} t_+^{\sigma}=0,$$
\item if $\sigma>\sigma_0$, then for any $t>0$,
$$ (-\Delta)^{\sigma_0} t_+^{\sigma}<0, $$
\item if $\sigma<\sigma_0$, then for any $t>0$,
$$ (-\Delta)^{\sigma_0} t_+^{\sigma}>0. $$
\end{itemize}
\end{lemma}
\begin{proof} The proof is a modification of an argument
given in~\cite{claudia}.
In order to prove Lemma \ref{lemma:trasformata},
we will use the Fourier transform of~$|t|^q$
in the sense of distribution, where~$q\in\mathbb C\setminus\mathbb Z$.
Namely (see e.g.
Lemma~2.23 on page~38 of~\cite{KOLDO})
\begin{equation}\label{KSp}
{\mathcal{F}} (|t|^q) = C_q\,|\xi|^{-1-q},
\end{equation}
with
\begin{equation}\label{J5K:1} C_q:=-2\Gamma(1+q)\,\sin\frac{\pi q}{2}.
\end{equation}
Notice that the map~$\mathbb R\ni t\mapsto|t|^q$
is even, and so we can rewrite~\eqref{KSp} as
\begin{equation}\label{KSp2}
{\mathcal{F}}^{-1} (|\xi|^q) =(2\pi)^{-1} C_q\,|t|^{-1-q}.
\end{equation}
Moreover,
$$ |t|^\sigma +\frac{1}{\sigma+1} \partial_t |t|^{\sigma +1} = 2t_+^\sigma.$$
Therefore, taking the Fourier transform and using~\eqref{KSp} with~$q:=\sigma$
and~$q:=\sigma+1$, we obtain that
\begin{eqnarray*}
2{\mathcal{F}}(t_+^\sigma) &=&
{\mathcal{F}}(|t|^\sigma) +\frac{1}{\sigma +1} {\mathcal{F}}\big(\partial_t |t|^{\sigma +1}\big)\\
&=&
{\mathcal{F}}(|t|^\sigma) +\frac{2 i\xi}{\sigma +1} {\mathcal{F}}(|t|^{\sigma +1})
\\ &=& C_\sigma \,|\xi|^{-1-\sigma}+\frac{2 i\xi}{\sigma +1} C_{\sigma +1}\,|\xi|^{-2-\sigma}.
\end{eqnarray*}
So, multiplying the equality above by $|\xi|^{2\sigma_0}$, we obtain that
$$ 2 |\xi|^{2\sigma_0} {\mathcal{F}}(t_+^\sigma)
= C_\sigma\,|\xi|^{2\sigma_0 -\sigma -1}+\frac{2 i\xi}{\sigma +1}
C_{\sigma +1}\,|\xi|^{2\sigma_0-\sigma-2},$$
and so
\begin{equation}\label{senzapi}
2 {\mathcal{F}}^{-1}\Big( |\xi|^{2\sigma_0} {\mathcal{F}}(t_+^\sigma)\Big)
=
C_\sigma \,{\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-1})
+\frac{2 C_{\sigma+1}\,i}{\sigma+1}
{\mathcal{F}}^{-1}(\xi) * {\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-2})
\end{equation}
Now we claim that, for any test function~$g$,
\begin{equation}\label{spero1534}
\left({\mathcal{F}}^{-1}(\xi) * g\right)(t) = -i \partial_t g(t).
\end{equation}
Indeed,
\begin{eqnarray*}
&& \left({\mathcal{F}}^{-1}(\xi) * g\right)(t) =
{\mathcal{F}}^{-1}\left(\xi {\mathcal{F}}g(\xi)\right)(t)\\
&&\qquad =
\frac{1}{2\pi}\int_{\mathbb R}dy\int_{\mathbb R}d\xi\,e^{iy\cdot(t-\xi)}\,y\,g(\xi)
= -
\frac{1}{2\pi i}\int_{\mathbb R}dy\int_{\mathbb R}d\xi\,\partial_\xi e^{iy\cdot(t-\xi)} \,g(\xi)\\&&\qquad =
\frac{1}{2\pi i}\int_{\mathbb R}dy\int_{\mathbb R}d\xi\, e^{iy\cdot(t-\xi)}\,\partial_\xi g(\xi)
=
\frac{1}{2\pi i}\int_{\mathbb R}dy\, e^{iy\cdot t}\,{\mathcal{F}}\big(\partial_\xi g\big)(y)\\ &&\qquad
= \frac{1}{i}{\mathcal{F}}^{-1}\left({\mathcal{F}}(\partial_\xi g)\right)(t)
= -i\, \partial_\xi g(t),
\end{eqnarray*}
which shows~\eqref{spero1534}.
Using~\eqref{spero1534} into~\eqref{senzapi}, we obtain that
\begin{eqnarray*}
2 {\mathcal{F}}^{-1}\Big( |\xi|^{2\sigma_0} {\mathcal{F}}(t_+^\sigma)\Big)
&=&C_\sigma\,{\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-1})
-\frac{C_{\sigma+1}\,i}{\sigma+1}\cdot i\, \partial_t
{\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-2})
\\ &=&C_\sigma\,{\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-1})
+\frac{C_{\sigma+1}}{\sigma+1}\partial_t {\mathcal{F}}^{-1}(|\xi|^{2\sigma_0-\sigma-2}).
\end{eqnarray*}
As a consequence, exploiting~\eqref{KSp2} with~$q:=2\sigma_0-\sigma-1$ and~$q:=2\sigma_0-\sigma-2$,
we have that
\begin{eqnarray*}
2 {\mathcal{F}}^{-1}\Big( |\xi|^{2\sigma_0} {\mathcal{F}}(t_+^\sigma)\Big)
&=& C_\sigma \,C_{2\sigma_0-\sigma-1}\,|t|^{\sigma-2\sigma_0}
+\frac{C_{\sigma+1}\,C_{2\sigma_0-\sigma-2}}{\sigma+1}
\partial_t |t|^{\sigma-2\sigma_0+1}\\
&=&C_\sigma\,C_{2\sigma_0-\sigma-1}\,|t|^{\sigma-2\sigma_0}
+\frac{\sigma-2\sigma_0+1}{\sigma+1}\cdot C_{\sigma+1}\,C_{2\sigma_0-\sigma-2}
\,t\,|t|^{\sigma-2\sigma_0-1}.
\end{eqnarray*}
This gives that, for $t>0$,
$$ 2 {\mathcal{F}}^{-1}\Big( |\xi|^{2\sigma_0} {\mathcal{F}}(t_+^\sigma)\Big)
= \left( C_\sigma\,C_{2\sigma_0-\sigma-1} +
\frac{\sigma-2\sigma_0+1}{\sigma+1}\cdot C_{\sigma+1}\,C_{2\sigma_0-\sigma-2}\right)
t^{\sigma-2\sigma_0}.$$
So we obtain that, up to a dimensional constant, for any $t>0$,
\begin{equation}\label{plug}
(-\Delta)^{\sigma_0} (t_+^\sigma) = \left( C_\sigma\,C_{2\sigma_0-\sigma-1} +
\frac{\sigma-2\sigma_0+1}{\sigma+1}\cdot C_{\sigma+1}\,C_{2\sigma_0-\sigma-2}\right)
t^{\sigma-2\sigma_0}.\end{equation}
Now, we observe that
\begin{equation}\label{C sigma}
C_\sigma\,C_{2\sigma_0-\sigma-1}= 4 \,\Gamma(1+\sigma)\,\Gamma(2\sigma_0-\sigma)\,
\sin\left(\frac{\pi}{2}\sigma\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right).
\end{equation}
Moreover,
$$ \Gamma(2+\sigma)=(1+\sigma) \Gamma(1+\sigma)\;{\mbox{ and }}\;
\Gamma(2\sigma_0-\sigma)=(2\sigma_0-\sigma-1) \Gamma(2\sigma_0-\sigma-1).$$
As a consequence, recalling \eqref{J5K:1} and \eqref{C sigma},
\begin{eqnarray*}
&&\frac{\sigma-2\sigma_0+1}{\sigma+1}\cdot C_{\sigma+1}\,C_{2\sigma_0-\sigma-2}
\\& =& \frac{\sigma-2\sigma_0+1}{\sigma+1}\, 4\, \Gamma(2+\sigma)\, \Gamma(2\sigma_0-\sigma-1)
\,\sin\left( \frac{\pi}{2}(\sigma+1)\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right) \\
&=& -4\, \Gamma(1+\sigma)\, \Gamma(2\sigma_0-\sigma)\,
\sin\left( \frac{\pi}{2}(\sigma+1)\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right)\\
&=& - C_\sigma\,C_{2\sigma_0-\sigma-1}\cdot
\frac{\sin\left( \frac{\pi}{2}(\sigma+1)\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right)}{
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}.\end{eqnarray*}
Plugging this into \eqref{plug}, we get
\begin{equation*}
(-\Delta)^{\sigma_0} (t_+^\sigma) = C_\sigma\,C_{2\sigma_0-\sigma-1}\left(1 -
\frac{\sin\left( \frac{\pi}{2}(\sigma+1)\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right)}{
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}
\right)
t^{\sigma-2\sigma_0}.\end{equation*}
Now, by elementary trigonometry, we see that
$$ \sin\left( \frac{\pi}{2}(\sigma+1)\right) = \cos\left(\frac{\pi}{2}\sigma\right)
\;{\mbox{ and }}\;
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right) = -
\cos\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right).
$$
Therefore,
\begin{eqnarray*}
&& 1 -
\frac{\sin\left( \frac{\pi}{2}(\sigma+1)\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-2)\right)}{
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\\
&=& 1 +
\frac{\cos\left( \frac{\pi}{2}\sigma\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\cos\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}{
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\\
&=& \frac{\cos\left(\frac{\pi}{2}\sigma\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}
\left[ \frac{\sin\left(\frac{\pi}{2}\sigma\right)}{\cos\left(\frac{\pi}{2}\sigma\right)}
+ \frac{\cos\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}{
\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\right]\\
&=& \frac{\cos\left(\frac{\pi}{2}\sigma\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\sin\left(\frac{\pi}{2}\sigma\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)
+\cos\left(\frac{\pi}{2}\sigma\right)\,\cos\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}{
\cos\left(\frac{\pi}{2}\sigma\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\\
&=& \frac{\cos\left(\frac{\pi}{2}\sigma\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\cos\left( \pi(\sigma-\sigma_0) +\frac{\pi}{2} \right)}{
\cos\left(\frac{\pi}{2}\sigma\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\\
&=& - \frac{\cos\left(\frac{\pi}{2}\sigma\right)}{\sin\left(\frac{\pi}{2}\sigma\right)}\cdot
\frac{\sin\left( \pi(\sigma-\sigma_0)\right)}{
\cos\left(\frac{\pi}{2}\sigma\right)\,\sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}\\
&=& - \frac{\sin\left( \pi(\sigma-\sigma_0)\right)}{
\sin\left(\frac{\pi}{2}\sigma\right)\, \sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}.
\end{eqnarray*}
Accordingly, up to a dimensional constant,
$$ (-\Delta)^{\sigma_0} (t_+^\sigma) = -C_\sigma\,C_{2\sigma_0-\sigma-1}
\frac{\sin\left( \pi(\sigma-\sigma_0)\right)}{
\sin\left(\frac{\pi}{2}\sigma\right)\, \sin\left(\frac{\pi}{2}(2\sigma_0-\sigma-1)\right)}
t^{\sigma-2\sigma_0}.$$
So, recalling \eqref{C sigma}, we obtain that, for any $t>0$,
$$ (-\Delta)^{\sigma_0} (t_+^\sigma) =-4\,\Gamma(1+\sigma)\,\Gamma(2\sigma_0-\sigma)\,
\sin\left(\pi(\sigma-\sigma_0)\right),$$
which shows \eqref{prima112}.
We finish the proof of Lemma \ref{lemma:trasformata} by noticing that
\begin{itemize}
\item if $\sigma=\sigma_0$, then $\sin\left(\pi(\sigma-\sigma_0)\right)=0$,
\item if $\sigma>\sigma_0$, then $\sin\left(\pi(\sigma-\sigma_0)\right)>0$,
\item if $\sigma<\sigma_0$, then $\sin\left(\pi(\sigma-\sigma_0)\right)<0$.
\end{itemize}
This implies the desired result.
\end{proof}
\end{appendix}
|
2,869,038,154,834 | arxiv | \section*{Acknowledgment}
\title{Reliable NLU}
\begin{abstract}
Humans understand language by extracting information (meaning) from sentences, combining it with existing commonsense knowledge, and then performing reasoning to draw conclusions. While large language models (LLMs) such as GPT-3 and ChatGPT are able to leverage patterns in the text to solve a variety of NLP tasks, they fall short in problems that require reasoning. They also cannot reliably explain the answers generated for a given question. In order to emulate humans better, we propose STAR, a framework that combines LLMs with Answer Set Programming (ASP). We show how LLMs can be used to effectively extract knowledge---represented as predicates---from language. Goal-directed ASP is then employed to reliably reason over this knowledge.
We apply the STAR framework to three different NLU tasks requiring reasoning: qualitative reasoning, mathematical reasoning, and goal-directed conversation. Our experiments reveal that STAR is able to bridge the gap of reasoning in NLU tasks, leading to significant performance improvements, especially for smaller LLMs, i.e., LLMs with a smaller number of parameters. NLU applications developed using the STAR framework are also explainable: along with the predicates generated, a justification in the form of a proof tree can be produced for a given output.
\end{abstract}
\begin{keywords}
Commonsense Reasoning, Question Answering, Conversational Agent, Large Language Models, Explainable AI
\end{keywords}
\begin{quote}
\textit{Words are not in themselves carriers of meaning, but merely pointers to shared understanding. }
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- David Waltz~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{quote}
\vspace{-0.2in}
\section{Introduction}
The long-term goal of natural language understanding (NLU) research is to build systems that are as good as humans in understanding language. This is a challenging task since there are multiple skills that humans employ to understand a typical sentence. First, a person needs to be proficient in the language to be able to interpret the sentence and understand its surface-level meaning. Second, they need to be able to interpret the meaning of the sentence in the current context, using the commonsense knowledge they already possess. This helps resolve ambiguities in the sentence and assess if any information is missing. Third, if required, they should be able to pose a question that would seek to fill in any information that is missing. Finally, once they attain a complete understanding of the sentence, they should be able to explain what they understood. We believe that all of these skills are also important for an NLU system that seeks to reliably answer questions or hold a conversation with a human.
In recent years, Large Language Models (LLMs) have been trained on massive amounts of text extracted from the internet. They have shown language proficiency to the extent that they are able to perform reading comprehension, translate languages and generate text to complete stories, poems, or even code (\cite{gpt3, codex}). However, they can fall short when applied to problems that require complex reasoning. When tested on commonsense reasoning or mathematics word problems, LLMs such as GPT-3 have been shown to make simple reasoning errors (\cite{gpt3-scope}). Though such errors may be mitigated with strategies such as chain-of-thought prompting (\cite{chain}), they continue to make mistakes that originate from calculation errors or missing reasoning steps in the solution, making it difficult to rely completely on such systems. While it is possible to prime LLMs to generate explanations for their answers, they sometimes generate the right explanation along with a wrong answer and vice versa (\cite{chain}). This brings into question the dependability of such explanations. The lack of a clear separation of the reasoning process also makes it difficult to assess the models' state of knowledge and identify commonsense knowledge that needs to be integrated as necessary. These shortcomings point to the need for better NLU systems that also employ reasoning.
\begin{figure}[t]
\includegraphics[scale = 0.45]{images/STAR.png}
\caption{STAR framework Design}
\label{fig:star}
\vspace{-0.15in}
\end{figure}
With this motivation, we propose the STAR (\underline{S}emantic-parsing \underline{T}ransformer and \underline{A}SP \underline{R}easoner") framework that closely aligns with the way human-beings understand language. STAR maps a sentence to the semantics it represents, augments it with commonsense knowledge related to the concepts involved---just as humans do---and then uses the combined knowledge to perform the required reasoning and draws conclusions (see Figure \ref{fig:star}). The STAR framework relies on LLMs to perform semantic parsing (converting sentences to predicates that capture its semantics) and shifts the burden of reasoning to an answer set programming (ASP) system (\cite{gelfondkahl,cacm-asp}). For our experiments, we use variants of GPT-3 (\cite{gpt3}) to generate predicates from the text. LLMs can be taught to do this either using fine-tuning or in-context learning using a small number of text-predicate pairs, resulting in a 'Specialized LLM'. Commonsense knowledge related to these predicates is coded in advance using ASP. Depending on the problem, a query is either pre-defined or can also be similarly generated from the problem using LLMs. The query is executed on the s(CASP) (\cite{scasp}) goal-directed ASP system against the LLM-generated predicates and ASP-coded commonsense knowledge to generate a response.
In this paper, we use the STAR framework for three different NLU applications: (i) a system for solving qualitative reasoning problems, (ii) a system for solving math word problems, and (iii) a system representing a hotel concierge that holds a conversation with a human user who is looking for a restaurant recommendation. All three tasks require different types of reasoning. Qualitative reasoning and mathematical reasoning tasks require the system to perform a few steps of reasoning involving qualitative relationships and arithmetic operations, respectively. On the other hand, the conversation bot task requires the system to interact with the user to seek missing information, ``understand" it, and reason over it.
Our experiments involve two main variants of GPT-3; Davinci ($\sim$ 175B parameters) and Curie ($\sim$ 6.7B parameters). To measure the performance with STAR, we perform direct answer prediction using both models and compare them to the corresponding answers produced using our framework. The results show that STAR shows an increase in answer prediction accuracy and the difference is especially large for the smaller LLM, which might be weaker at reasoning. In both question-answering tasks, we are able to produce proof trees for the generated response, making them \textit{explainable}. The knowledge predicates also help us understand the shortcomings and potential design improvements, which is not possible when the models are run using the LLMs alone. In the conversation bot task that requires in-depth reasoning, we observe that STAR provides much better control wrt seeking information from the user and giving restaurant suggestions faithfully based on the database of restaurants it has. However, when used directly for the purpose of restaurant recommendation, Davinci alters restaurant information based on user interaction. Hence, our method is more \textit{reliable}. Since reasoning is performed using s(CASP) in our approach, we can handle a database of restaurants that can be arbitrarily large. This is not possible when LLMs are used end-to-end for this conversation bot, as there is a limit on the maximum prompt size. Thus, our approach is also \textit{scalable}, unlike a purely LLM-based approach.
\vspace{-0.2in}
\section{Background}
\label{sec:background}
\subsection{Large Language Models}
Until recently, transformer-based deep learning models have been applied to NLP tasks by training and fine-tuning them on task-specific datasets (\cite{pre_trained_transformers}). With the advent of Large Language Models, the paradigm changed to teaching a language model any arbitrary task using just a few demonstrations, called \textit{in-context learning}. \cite{gpt3} introduced an LLM called GPT-3 containing approximately 175 billion parameters that has been trained on a massive corpus of filtered online text, on which the well-known ChatGPT is based (\cite{chatgpt})).
The model was able to perform competitively on several tasks such as question-answering, semantic parsing (\cite{LLM_sem_parse}) and machine translation. However, such LLMs tend to make simple mistakes in tasks such as semantic (commonsense) and mathematical reasoning (\cite{gpt3-scope, chain}).
In our work, we use GPT-3 for semantic parsing and leave the reasoning part to ASP. We theorize that given the vast pre-training they go through, LLMs can be used to automatically extract knowledge inherent in the text, just like humans do. In fact, just as humans can, the same set of predicates should be extracted even if the text is expressed in different ways, including ungrammatical forms. For example, given the sentence "Alice loves Bob", "Alice is in love with Bob" or "loves Alice Bob", we are be able to learn the predicate {\tt loves(alice,bob)} from the sentence. We would also infer the implicit predicates; {\tt person(alice)}, {\tt person(bob)}, {\tt female(alice)} and {\tt male(bob)}. Our experiments confirm that Davinci and Curie are able to extract such knowledge as predicates from sentences---with high accuracy---after learning from a few example demonstrations.
Thus, our experiments show that LLMs are able to extract, what linguists call, the \textit{deep structure} of a sentence, given a sentence's \textit{surface structure}.
\subsection{Answer Set Programming and the s(CASP) system}
The s(CASP) system (developed by \cite{scasp}) is an answer set programming (\cite{cacm-asp}) system that supports predicates, constraints over non-ground variables, uninterpreted functions, and, most importantly, a top-down, query-driven execution strategy.
These features make it possible to return answers with non-ground variables (possibly including constraints among them) and compute partial models by returning only the fragment of a stable model that is necessary to support the answer to a given query.
The s(CASP) system supports constructive negation based on a disequality constraint solver, and unlike Prolog's negation as failure and ASP's
default negation, %
{\tt not p(X)} can return bindings for \texttt{X} on success, i.e., bindings for which the call \texttt{p(X)} would have failed. Additionally, s(CASP) system's interface with a constraint solver (over reals) allows for sound non-monotonic reasoning with constraints (useful for solving algebra problems in one of the NLU applications we discuss later).
Complex commonsense knowledge can be represented in ASP and the s(CASP) query-driven predicate ASP system can be used for querying it (\cite{gupta-csr,murder-trial,gelfondkahl}). Commonsense knowledge can be emulated using (i) default rules, (ii) integrity constraints, and (iii) multiple possible worlds~(\cite{gelfondkahl,gupta-csr}). Default rules are used for jumping to a conclusion in the absence of exceptions, e.g., a bird normally flies, unless it's a penguin. Thus, if we are told that Tweety is a bird, we jump to the conclusion that Tweety flies. Later, if we are told that Tweety is a penguin, we withdraw the conclusion that Tweety can fly. Default rules with exceptions represent an elaboration-tolerant way of representing knowledge~(\cite{gelfondkahl}).
{\tt
flies(X) :- bird(X), not abnormal\_bird(X).
abnormal\_bird(X) :- penguin(X).}
\smallskip
\noindent Integrity constraints allow us to express impossible situations and invariants. For example, a person cannot sit and stand at the same time.
{\tt false :- person(X), sit(X), stand(X).}
\smallskip
\noindent Finally, multiple possible worlds allow us to construct alternative universes that may have some of the parts common but other parts inconsistent. For example, the cartoon world of children's books has a lot in common with the real world (e.g., birds can fly in both worlds), yet in the former birds can talk like humans but in the latter they cannot.
Default rules are used to model a bulk of our commonsense knowledge. Integrity constraints help in checking the consistency of the information extracted. Multiple possible worlds allow us to perform assumption-based reasoning (for example, knowing that ``Alice loves Bob", we could assume that either Bob also loves Alice or he does not).
A large number of commonsense reasoning applications have already been developed using ASP and the s(CASP) system (\cite{blawx,logical-english,chef,murder-trial}). In the three applications reported in this paper, we have kept the commonsense reasoning component simple, as our main goal is to illustrate our framework for combining LLMs and ASP to develop NLU applications that are explainable and reliable. Because of use of ASP, it is also possible to detect inconsistencies or biases in the text by reasoning over the predicates extracted. Justification for each response can also be given, as the s(CASP) system can generate justifications as proof trees (\cite{scasp-justification}).
\vspace{-0.15in}
\section{Qualitative Reasoning}
\label{sec:qualitative}
Qualitative reasoning tests a model's ability to reason about properties of objects and events in the World. \cite{quarel} introduced the QuaRel dataset in order to test question answering about qualitative relationships of a set of physical properties, which forms a perfect test-bed for our approach.
Our experimental results show that our effort based on the STAR framework advances the state-of-the-art
for the Quarel dataset (\cite{quarel}).
We show that the STAR framework also results in significant performance improvement compared to the case where the LLMs are applied directly to question answering.
\vspace{-0.15in}
\subsection{The QuaRel Dataset}
The QuaRel dataset consists of 2771 questions designed around 19 different properties such as `friction', `heat', `speed', `time', etc. In order to answer these questions, one must account for the correlation between these properties. Each question has a certain observation made about the two worlds where a property has a higher (or lower) value in one world compared to the other. Based on this observation, a (commonsense) inference needs to be drawn about other related properties described in the two worlds. This inference helps pick one of the two choices as the answer for the given question \cite{quarel}.
A question from the dataset is given in example \ref{verb:quarel-example}. In this example, the two worlds are 'Carpet' and 'Floor'. The observation made is that the \textit{distance} traveled by a toy car is more in world1 (floor). From this, the model needs to infer that the resistance or \textit{friction} would be higher in world2 (carpet), which should lead to picking option A as the answer.
\medskip
\noindent
\textbf{Example 3.1:}
{\small
\begin{verbatim}
Question: Alan noticed that his toy car rolls further on a wood
floor than on a thick carpet. This suggests that:
(world1: wood floor, world2: thick carpet)
(A) The carpet has more resistance (Solution)
(B) The floor has more resistance
\end{verbatim}\label{verb:quarel-example}
}
\noindent Along with each question, \cite{quarel} provides a logical form that captures the semantics of the question and we use it to extract the predicates needed for our method. For the above question (example \ref{verb:quarel-example}), the logical form given is as follows:
\vspace{-0.15in}
{\small
\begin{equation}\label{eqn:quarel-example}
qrel(distance, higher, world1) \rightarrow qrel(friction, higher, world2)\ ;\ qrel(friction, higher, world1)
\end{equation}
}
\noindent The predicate $qrel(distance,higher,world1)$ refers to the observation that the \textit{distance} is higher in world1, while $qrel(friction,higher,world2)$ and $qrel(friction,higher,world1)$ refer to the conclusions drawn in the two answer options, respectively.
\vspace{-0.15in}
\subsection{Predicate Generation Step}
We use GPT-3 to convert the Quarel dataset's natural language question (including the two answers) into appropriate predicates. Since we have a training dataset available, we \textit{fine-tune} the two GPT-3 model variants, namely, Davinci and Curie, (\cite{gpt3}) for the QuaRel dataset, instead of just using \textit{in-context learning}\footnote{Fine-tuning an LLM involves using additional training data to refine the LLM for the task at hand; in-context learning refers to giving some examples from the training data, along with the question posed, to the LLM as a part of its input.}.
Our input prompt consists of the question (including answer options), followed by the world descriptions. The world descriptions are included to enable the model to link the two worlds to the ones in the predicates (\textit{obs} and \textit{conc}) that are generated in the output. The prompt and completion formats for fine-tuning are given below:
\noindent \textbf{Prompt format:}
{\small {\small\tt <Question-Answers>$\backslash$n\ world1:<world1>$\backslash$n world2:<world2>$\backslash$n$\backslash$n\#\#$\backslash$n$\backslash$n}}
\noindent \textbf{Completion format:}
\centerline{{\small{\small\tt obs(<p>, <h/l>, <w1/w2>)} $\rightarrow${\tt \ conc(<p>, <h/l>, <w1/w2>)\ ;}}}
\centerline{\small{{\small\tt conc(<p>, <h/l>, <w1/w2>)\ <EOS>}}}
\noindent where p is a property, h/l is either higher or lower and w1/w2 is either world1 or world2. After fine-tuning on the training set using the prompt and completion pairs, we use the prompt to generate the completion during testing. The {\tt <EOS>} token helps cut off the generation when apt, avoiding completions that are either too long or too short. The extracted \textit{obs} and \textit{conc} predicates are then used by the logic program to determine the correct answer.
\subsection{Commonsense Reasoning Step}
The commonsense knowledge required to answer the questions is encoded in ASP as facts and rules. First, we ground the 19 properties using facts such as,
{\small\begin{verbatim}
property(friction). property(heat). property(speed).
\end{verbatim}}
\noindent Next, we define the relationships between the properties, noting their positive and negative correlations and also the symmetry,
{\small\begin{verbatim}
qplus(friction, heat). qminus(friction, speed).
qplus(speed, distance). qminus(distance, loudness).
positive(X, Y) :- qplus(X, Y). negative(X, Y) :- qminus(X, Y).
positive(X, Y) :- qplus(Y, X). negative(X, Y) :- qminus(Y, X).
\end{verbatim}}
In the QuaRel dataset, we are only dealing with two worlds. Hence, if a property P is higher in world1, it must be lower in world2 and vice versa. We capture this logic using the \textit{opposite} predicates and the rules below:
{\small\begin{verbatim}
opposite_w(world1,world2). opposite_v(higher,lower).
opposite_w(world2,world1). opposite_v(lower,higher).
conc(P, V, W) :- obs(P, Vr, Wr), property(P),
opposite_w(W,Wr), opposite_v(V,Vr).
\end{verbatim}}
\noindent In order to capture the relationship between each pair of properties, we need to account for 4 different cases that may arise. If properties P and Pr are positively correlated, then (i) if P is higher in world W, Pr must also be higher in W, and (ii) if P is higher in world W, Pr must be lower in the other world Wr. Similarly, if P and Pr are negatively correlated, then (i) if P is higher in world W, Pr must be lower in W, and (ii) if P is higher in world W, Pr must be higher in the other world Wr. Note that the higher/lower relations may be swapped in all cases above.
These 4 possible scenarios can be encoded in logic using the following rules:
{\small\begin{verbatim}
conc(P,V,W) :- obs(Pr,V,W), property(P), property(Pr),
positive(P,Pr).
conc(P,V,W) :- obs(Pr,Vr,Wr), property(P), property(Pr),
opposite_w(W,Wr), opposite_v(V,Vr), positive(P,Pr).
conc(P,V,W) :- obs(Pr,Vr,W), property(P), property(Pr),
opposite_v(V,Vr), negative(P,Pr).
conc(P,V,W) :- obs(Pr,V,Wr), property(P), property(Pr),
opposite_w(W,Wr), negative(P,Pr).
\end{verbatim}}
\noindent Using this knowledge base, asserting a fact as an observation (\textit{obs}) allows us to check for the correct conclusion (\textit{conc}) that is entailed. For the example question in example ~\ref{verb:quarel-example}, we can arrive at the answer by checking for entailment of the two possible conclusions as shown:
$assert(obs(distance,\ higher,\ world1)),\ conc(friction,\ higher,\ world2). \rightarrow\ True$
\noindent and
$assert(obs(distance,\ higher,\ world1)),\ conc(friction,\ higher,\ world1). \rightarrow\ False$
\vspace{-0.12in}
\subsection{Results and Evaluation}
We compare the results of our models to those reported by \cite{quarel} in Table \ref{table:quarel-results}. Accuracy for four QuaRel datasets is considered ($QuaRel^F$ refers to the subset of the dataset which only focuses on friction-related questions). The first 8 rows show the accuracy of baseline models proposed by \cite{quarel}. Curie-Direct and Davinci-Direct rows report the performance of Curie and Davinci models which directly predict the answer after fine-tuning on the QuaRel's training set. The Curie-STAR and Davinci-STAR rows show the performance for our approach, i.e. first generating the predicates and then reasoning using ASP and commonsense knowledge. The values in bold represent the highest accuracy values obtained for each dataset.
\begin{table}[h!]
\centering
\caption{Comparison of Models on the QuaRel Dataset (Qualitative Reasoning)}\label{table:quarel-results}
{\tablefont\begin{tabular}{@{\extracolsep{\fill}}llcccc}
\topline
No.~~~&Model&
Acc. QuaRel Dev&
Acc. QuaRel Test&
Acc. $QuaRel^F$ Dev&
Acc. $QuaRel^F$ Test\midline
1.&Random & 50.0 & 50.0 & 50.0 & 50.0 \\
2.&Human & 96.4 & - & 95.0 & - \\
3.&IR & 50.7 & 48.6 & 50.7 & 48.9 \\
4.&PMI & 49.3 & 50.5 & 50.7 & 52.5 \\
5.&Rule-Based & - & - & 55.0 & 57.7 \\
6.&BiLSTM & 55.8 & 53.1 & 59.3 & 54.3 \\
7.&QUASP & 62.1 & 56.1 & 69.2 & 61.7 \\
8.&QUASP+ & 68.9 & 68.7 & 79.6 & 74.5 \\
9.&Curie-Direct & 67.6 & 63.5 & 45.7 & 52.7 \\
10.&Curie-STAR (ours) & 86.2 & 85.2 & 87.9 & 85.9 \\
11.&Davinci-Direct & \textbf{93.1} & \textbf{90.5} & 90.0 & 91.3 \\
12.&Davinci-STAR (ours) & 90.6 & \textbf{90.5} & \textbf{90.6} & \textbf{93.5}
\botline
\end{tabular}}
\end{table}
The results show that Davinci-STAR achieves better than state-of-the-art performance on QuaRel and beats Davinci-Direct on 3 out of the 4 datasets considered.
Table 1 also show a large improvement in accuracy for the Curie model on all the datasets. We infer from this that while Davinci has some ability to reason, Curie lacks the reasoning skill required for the task. Hence, our framework helps bridge the gap by reasoning externally. However, interestingly, we see that Davinci-Direct outperforms Davinci-STAR on the QuaRel-Dev dataset. Since our framework is explainable, we were able to analyze the cases where our approach makes a mistake. We found that the LLM is generating properties that are not in the domain in some predicates (such as 'smoke' instead of 'heat' since the question mentions smoke). Such errors can be fixed by adding relevant examples to the training set (for example one that links the appearance of smoke to heat property). Also, since a majority of the questions in QuaRel are on friction, the training data might contain a larger variation of questions on this property. This, we believe led to more accurate predicate generation on $QuaRel^F$ datasets (which focus on friction-based questions), leading to the higher accuracy obtained by our models.
\vspace{-0.15in}
\section{Solving Word Problems in Algebra}
\label{sec:math}
Solving word problems in algebra requires extracting information from the question (interpreting its language) and performing mathematical reasoning to come up with an answer. Hence, it forms a good experiment to test our framework. We choose a specific type of addition and subtraction problems from the dataset used by \cite{MWP}.
We define the predicates {\tt has/4}, {\tt transfer/5} and {\tt total/4} as shown below to encode the knowledge in the problems:
{\small\tt has(entity, quantity, time\_stamp, k/q).}
{\small\tt transfer(entity1, entity2, quantity, time\_stamp, k/q).}
{\small\tt total(entity, quantity, time\_stamp, k/q).}
\noindent The predicate {\tt has/4} defines that an {\tt entity} has a certain {\tt quantity} of some objects, at a particular {\tt time\_stamp} and it either constitutes knowledge facts (denoted {\tt k}) or a question (denoted {\tt q}).
The {\tt transfer/5} predicate defines that an {\tt entity1} has transferred a certain {\tt quantity} of objects, to {\tt entity2} at a particular {\tt time\_stamp} and that this information is part of the knowledge facts {\tt k} or is the query {\tt q}.
Finally, the {\tt total/4} predicate defines that an {\tt entity} has a total amount of some objects equal to the {\tt quantity}, at a particular {\tt time\_stamp} and that this information is part of the knowledge facts {\tt k} or is the query {\tt q}. We design these based on what information a human might glean from the problem in order to solve it.
The computation of the answer is done by simple s(CASP) rules. The rules are not shown due to lack of space and can be thought of as commonsense knowledge required to solve simple Algebra word problems given the {\tt has/4}, {\tt transfer/5}, and {\tt total/4} predicates.
\iffalse
The rules written in s(CASP) are as follows:
\begin{verbatim}
total(E3,X,T2,q) :- has(E1,A,T0,k), has(E2,B,T1,k),
X .=. A + B., T1 .=. T0+1, T2 .=. T0+2.
has(E1,X,T2,q):- has(E1,A,T0,k), transfer(E2,E1,B,T1,k),
X .=. A + B, T1 .=. T0+1, T2 .=. T0+2.
has(E1,X,T2,q):- has(E1,A,T0,k), transfer(E1,E2,B,T1,k),
A .>. B, X .=. A - B, T1 .=. T0+1, T2 .=. T0+2.
transfer(E1,E2,X,T1,q) :- has(E1,A,T0,k), has(E1,B,T2,k),
A .>=. B, X .=. A - B, T1 .=. T0+1, T2 .=. T0+2.
transfer(E1,E2,X,T1,q):- has(E2,A,T0,k), has(E2,B,T2,k), B .>. A,
X .=. B - A, T1 .=. T0+1, T2 .=. T0+2.
\end{verbatim}
\fi
\iffalse
\noindent The first rule states that, the total quantity that {\tt E3} has at time stamp {\tt 2} is {\tt X} if, {\tt E1} had quantity {\tt A} at time stamp {\tt 0}, and {\tt E2} had quantity {\tt B} at time stamp {\tt 1}, and {\tt X} can be calculated as {\tt A + B}.
The second rule states that, {\tt E1} has quantity {\tt X} at time stamp {\tt 2} if, {\tt E1} had quantity {\tt A} at time stamp {\tt 0}, and {\tt E2} transfered quantity {\tt B} to {\tt E1} at time stamp {\tt 1}, and {\tt X} can be calculated as {\tt A + B}.
The third rule is similar to the second rule but instead of {\tt E2} transferring the quantity to {\tt E1} it captures the situation where {\tt E1} transferred some quantity to {\tt E2}. Hence {\tt X} is finally calculated as {\tt A - B}.
The fourth rule states that {\tt E1} transferred quantity {\tt X} to {\tt E2} at time stamp {\tt 1} if, {\tt E1} had quantity {\tt A} at time stamp {\tt 0}, and {\tt E1} had quantity {\tt B} at time stamp {\tt 2}, and A is greater than or equal to B, and {\tt X} is calculated as {\tt A - B}.
The fifth rule is similar to the fourth rule and it states that if {\tt E2} had a greater quantity at time step {\tt 2} than at time step {\tt 0}, and {\tt X} can be calculated as {\tt B - A}.
\fi
An example problem and corresponding predicates generated to represent the knowledge are shown below.
\medskip\noindent\textbf{Ex 1:}
\textit{Joan found 70 seashells on the beach. Joan gave Sam some of her seashells . Joan has 27 seashells left. How many seashells did Joan give to Sam?}
{\tt has(joan,70,0,k). transfer(joan,sam,X,1,q). has(joan,27,2,k). }
\medskip \noindent
Following the STAR approach, we convert the knowledge in the chosen algebraic problem to the predicates defined above using an LLM. The predicates thus obtained (including the query) along with the rules then constitute the logic program. The query predicate is then executed against the program to solve the word problem.
\vspace{-0.15in}
\subsection{Experiments and Results}
Our dataset contains $91$ problems drawn from a collection of word problems provided by \cite{MWP}. We select these problems because they have a similar structure which could be represented by a few pre-defined predicates. We use text-davinci-003 which is the most capable GPT-3 model for in-context learning. We did not use the text-curie-001 model as done in the qualitative reasoning experiment because the model requires fine-tuning on a larger set of questions to be effective. We provide a context of a few problems with their corresponding predicates to the GPT-3 models and then use each problem as a prompt along with the context for the model to generate the facts and the query predicate(s) corresponding to the new problem. We then use the commonsense rules we defined along with the generated predicates (facts) as the logic program and query the program using the query predicate. We then compare the answer generated by the logic program with the actual, human-computed answer for each problem. As a baseline we use the GPT-3 model for direct answer prediction. Here, with the
algebra problems as the context, we provide the correct answer as the expected completion.
For our experiments, we initially started with a smaller context of $12$ problems and examined the mistakes the LLM was making in generating the predicates. Since our approach is explainable (unlike the direct answer prediction approach), we were able to analyze the mistakes and added more problems to the context that might fix them. Repeating this process a few times, we end up with $24$ problems as the final context for the GPT-3 model. Results of our experiments are shown in table \ref{table:math_results}.
\begin{table}[h!]
\centering
\caption{Performance comparison between the baseline model and our approach}\label{table:math_results}
{\tablefont\begin{tabular}{@{\extracolsep{1.0in}}lr}
\topline
Model&
Accuracy\midline
text-davinci-003-Direct & $\textbf{1.00}$ \\
text-davinci-003-STAR & $\textbf{1.00}$
\botline
\end{tabular}}
\end{table}
Both text-davinci-003-Direct and text-davinci-003-STAR result in $100\%$ accuracy on the test set of $67$ problems. Our experiments show that algebraic word problems that require simple reasoning can be solved easily by large LLMs. While our STAR approach achieves the same accuracy, it can also generate justification, making it explainable. Below we show a justification tree generated by s(CASP) (\cite{scasp-justification}) for the problem shown in Example 1 above, where the computed answer is 43.
{\small
\begin{verbatim}
JUSTIFICATION_TREE:
transfer(joan,sam,43,1,q) :-
has(joan,70,0,k),
has(sam,27,2,k),
43 #= 70-27.
global_constraint.
\end{verbatim}
}
A major weakness of LLMs is that they can perform poorly on mathematical tasks. Since the computation in our approach is done by the reasoning process externally (as seen above), it circumvents any potential calculation mistakes the LLM might make, making it reliable.
\vspace{-0.15in}
\section{Concierge Bot: A Goal-Directed Conversational Bot}
\label{sec:conversation}
In this section, we describe the conversational bot that we have developed using our STAR framework. We first leverage LLMs to extract appropriate predicates from user's dialogue to reveal the implicit as well as the explicit knowledge predicates, and then reason over the extracted predicates with s(CASP), to satisfy the user's request.
To illustrate this application of our method to conversational AI, we narrow the domain of the bot to a very specific problem: restaurant recommendations. This ensures that commonsense knowledge that needs to be modeled is within a limited domain that can be expanded later as necessary. We call this bot the concierge bot.
Since the conversational bot is an interactive application, the commonsense reasoning involved is more complex. Here, the reasoner also has to keep track of the state of the conversation. In order to make a restaurant recommendation, the bot needs to ask the user for some information (preferred cuisine, price range, etc.) and a recommendation will be presented to the user once all information is procured. At any stage, the bot can explain why it's asking for a given piece of information as well as its recommendation.
The benefits of the STAR framework are more apparent in the interactive chatbot application. The bot can ask the user questions to acquire more information when necessary or to verify if any errors are made in predicate generation. We show an example transcript of an interaction session below of the concierge bot with a human user. The English responses are also generated using GPT-3 after learning from predicate-text demonstrations (essentially, using reverse of the
text-to-predicate generation process).
{\small
\begin{verbatim}
Bot: Hi, what can I assist you with?
Human: Can you help me find a place for food with curry?
I don't want a pricey one.
Bot: Are you looking for a place with a specific customer rating?
Human: No, I don't mind the rating.
Bot: What kind of restaurant would you like to visit?
Human: A normal restaurant.
Bot: Would you like somewhere family-friendly for tonight?
Human: No. Just for myself.
Bot: Unfortunately, we cannot provide the results to your request.
Human: How about one with a high price?
But it should be then at least above average quality.
Bot: Are you looking for a Thai restaurant?
The Rice Boat may be suitable but it can be pricey and
customer feedback has been mixed. Additionally, it may
not have the best atmosphere for families.
Human: Yes, that's what I need! Tell me where it is.
Bot: The Rice Boat, located on 901 Bangkok Road, has an average
customer rating and offers Thai cuisine at a premium cost.
Unfortunately, it is not suitable for children.
Human: Great! Thank you for the service!
Bot: It's no problem, I'm happy to assist.
\end{verbatim}
}
\vspace{-0.15in}
\subsection{Predicate Generation using LLMs}
\label{sec:convbot_test}
\cite{novikova2017e2e} provide a dataset (called E2E) containing restaurant information (in a form similar to predicates) and corresponding natural language descriptions. We employ in-context learning where we give the LLM examples of pairs containing a natural language sentence and the corresponding predicates.
The dataset contains the following eight predicates: \textbf{name}, \textbf{eatType} (as establishment type), \textbf{food} (as cuisine), \textbf{priceRange}, \textbf{customerRating}, \textbf{familyFriendly}, \textbf{area}, \textbf{near}.
The \textbf{area} and \textbf{near} predicates are not included in the output during in-context learning, which is a cue to GPT-3 to ignore that information in the input sentence (later we add an address predicate to tell the user where the restaurant is located).
We use the STAR framework with in-context learning where we provide the GPT-3 model with 11 selected examples from the dataset, which covers all the predicates along with the possible arguments for each predicate. This ensures that the LLM is aware of every possible predicate as well as every possible argument value these predicates can take. We tested the model using the first 500 examples in the E2E training set, and obtained an accuracy of 89.33\%. The accuracy metric we use is designed to account for the proportion of predicates produced with their correct arguments. The high accuracy of predicate generation supports the feasibility of using our STAR framework for the restaurant recommendation bot. Our framework can be applied, in general, to build robust domain-specific conversational bots such as a front desk office receptionist or an airline reservation assistant.
\vspace{-0.15in}
\subsection{Concierge Bot System Construction}
To build the conversation bot, we modify the predicates used in E2E. To make GPT-3 better understand the meaning of each predicate, we change the predicate names as follows: \textbf{name}, \textbf{typeToEat}, \textbf{cuisine}, \textbf{priceRange}, \textbf{customerRating}, \textbf{familyFriendly}. We also add two predicates \textbf{address} and \textbf{phoneNumber} to record the location and contact information for the user's query. An external predicate \textbf{prefer} is also added to capture the user's preference (such as curry, spicy, etc.) The information asked by the user is expressed by the value ``query". We specialized GPT-3 with about a dozen example sentences along with the corresponding predicate(s). Below we show some examples of the sentences and the predicates generated after this specialization.
{\small
\begin{verbatim}
Sentence: Fitzbillies coffee shop provides a kid-friendly venue for
Chinese food at an average price point in the riverside area.
It is highly rated by customers.
Predicates: name(Fitzbillies), typeToEat(coffee shop), cuisine(Chinese),
priceRange(moderate), customerRating(high), familyFriendly(yes)
Sentence: Can you find a place for food at a low price? Both English
and French cuisine is fine for me.
Predicates: name(query), cuisine([Engish, French]), priceRange(cheap)
\end{verbatim}
}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{images/Chatbot_flowchart2.pdf}
\caption{The framework of the reasoning system in Concierge Bot. The green boxes indicate the steps done by LLMs and the orange ones indicate the steps done by s(CASP).}
\label{fig:chatbot_flow}
\end{figure}
Commonsense knowledge involved in making a restaurant recommendation is coded using s(CASP). The interactive bot will take in the user's response, then convert it to predicates using GPT-3. The predicates become part of the state. At this stage, we check for user preference. For example, if the user wants curry, Indian and Thai cuisine would be automatically added to the state through appropriate rules. The bot then examines the state to assess if all the information needed is present so that it can make a recommendation, and if not, it will generate a question to ask the user for that information. This logic, shown in Figure \ref{fig:chatbot_flow}, can be thought of as a state machine and has been referred to as a conversational knowledge template (CKT) by \cite{ckt}. The concierge bot determines which predicates are missing in its state to make a recommendation. One of the missing predicates is then selected and a question is drafted around it. Note that we deploy GPT-3 again to generate natural-sounding text from the predicate(s) that correspond to the response that our bot computes. The users can also change their preferences during the conversations and our bot can handle that.
Take the conversation mentioned above as an example. When the user asks ``Can you help me find a place for food with curry? I don't want a pricey one.", following predicates are generated by the GPT-3 text-davinci-003 model: \textit{``name(query), prefer(curry), priceRange([cheap,moderate])"}. The predicates are then added to the memory of the bot, where the log of the user requirements of the current conversation is stored. Note that the predicates \textit{``prefer(curry)"} and \textit{ ``cuisine([indian,thai])"} are also added to the state using the specific commonsense rule we wrote for this purpose (not shown due to lack of space).
\iffalse
{\small
\begin{verbatim}
cuisine_style(['Indian', 'Thai', 'Japanese', 'English'], 'curry').
set_prefer([], Final, Final).
set_prefer([prefer([])|Rest], Next, Updated)
:- set_prefer(Rest, Next, Updated).
set_prefer([prefer(Values)|Rest], Current, Updated)
:- select(Value, Values, Rest_Values),
append([prefer(Rest_Values)], Rest, R1),
cuisine_style(Cuisine_List, Value),
merge_cuisine(Current, [cuisine(Cuisine_List)], Next),
set_prefer(R1, Next, Updated).
set_prefer([Attr|Rest], Next, Updated)
:- set_prefer(Rest, Next, Updated).
\end{verbatim}
}
\fi
After the state gets updated, the bot checks to see whether any other key information is missing using another commonsense rule (not shown also due to lack of space).
\iffalse
{\small
\begin{verbatim}
key_info([typeToEat, cuisine, priceRange, customerRating,
familyFriendly]).
next_info(Attr, State) :- key_info(L), member(Attr, L),
member(require(Attr, ['query']), State).
next_info(Attr, State) :- key_info(L), member(Attr, L),
forall(A, not member(require(Attr, A), State)).
\end{verbatim}
}
\fi
If no information is missing, the bot comes to the recommendation step where it searches for the appropriate restaurant in its database, consistent with the user's input, and outputs it as a recommendation.
Since we use GPT-3 to generate predicates from user utterances, the user has complete freedom in terms of what they say. GPT-3 is able to extract the appropriate predicates from the corresponding deep form of the sentence.
GPT-3 is also used to generate the natural language response of the bot. As a result, the interaction appears quite natural, as the example above shows.
\vspace{-0.15in}
\subsection{Experiments}
To explore GPT-3's ability to generate goal-directed conversation without the reasoning system that STAR contains, we tested the conversation directly on GPT-3 with the sample conversations as input. We showed GPT-3 three full paragraphs of conversations and asked GPT-3 to generate a similar conversation according to the user input. The database of restaurants used by our concierge bot is also included in the input to the GPT-3 model.
An example interactive conversation directly with GPT-3 is as follows:
{\small
\begin{verbatim}
Bot: Hello, what can I do for you today?
Human: I'd like to have a family meal with my children.
Bot: What type of cuisine would you like?
Human: Mexican food is suitable for me.
Bot: What is your desired price range for the place?
Human: At a higher price, please.
Bot: I recommend The Wrestlers, a coffee shop serving Mexican
cuisine at a moderate price range. It is family-friendly
and has an average customer rating. The address is 470 Main Rd.
\end{verbatim}
}
The responses given by GPT-3 in the above conversation are correct except for the price range. In the given database, the restaurant recommended only serves cheap food. Hence, GPT-3 modified the information to align with the user's request. GPT-3 also follows the given examples and asks about the cuisine and price, but does not request for other information like our STAR framework does. This is because these questions are not motivated by missing information, unlike in our approach. This example shows that although GPT-3 used on its own as a conversational bot is able to generate natural-sounding sentences fluently given the required information, it is also unreliable and does not understand the knowledge given. Bots developed using our framework do not face such problems because they employ commonsense reasoning.
\vspace{-0.15in}
\section{Related Work}
\label{sec:related}
This work follows our earlier research where we advocate a combination of machine learning and commonsense reasoning to carry out intelligent tasks in a human-like manner (\cite{aqua,square,auto-discern}). In the SQuARE question-answering system (\cite{square}), knowledge was extracted using the Stanford CoreNLP Parser (\cite{corenlp}) and then mapped to templates from VerbNet. This method was only usable for simple sentences such as for the bAbI dataset. Along similar lines, \cite{quarel} report models that convert problems in English to logical forms, which are then processed using a custom interpreter. The semantic parsers discussed in their paper are variations of LSTMs that
generate CFG-like grammar rules which create the logic form. The main bottleneck of these two approaches was in the parsers used. In our work, we use LLMs that can extract predicates from arbitrary sentences which makes our approach applicable to more complex problems and translates to better performance. \cite{LLM_sem_parse} also demonstrate that LLMs are effective as semantic parsers, but don't show how the generated parse might be used in a downstream task.
\cite{restKB} provide an approach to reason about the use-case of dining in a restaurant by extracting knowledge from sentences (using CoreNLP) and using commonsense knowledge defined in an ASP-based action language. However, it reasons only about actions in a restaurant story that is already given.
A recent line of research on improving the reasoning capabilities of LLMs focus on prompt engineering. \cite{chain} show that generating a chain of thought before the answer leads to a significant improvement in performance in a variety of reasoning tasks. However, in some cases a wrong reasoning chain can lead to the right answer or vice versa. \cite{star} extend this by generating rationales using a self-taught approach. Instead of depending on LLMs end-to-end, our approach instead relies on s(CASP) to perform reasoning explicitly. This explicit reasoning is not only more reliable, but it is also explainable.
\vspace{-0.2in}
\section{Conclusions and Future Work}
\label{sec:conc}
In this paper we described the STAR framework that combines LLMs and ASP for NLU tasks. We show that our system is reliable, explainable, and scalable using three different reasoning tasks. For the qualitative reasoning task, STAR beats purely LLM-based approaches and advances the state-of-the-art wrt performance on the QuaRel dataset. The performance difference is more significant for Curie, indicating that it helps bridge the reasoning gap that might be present in smaller LLMs. In all three tasks, STAR can explain its reasoning process by producing a justification tree. In the LLM-only approach for developing a concierge bot, we noticed that the LLM mixes up information collected during the conversation and leads to incorrect suggestions, while our STAR-based approach stays faithful to the information given in the restaurant database. Our approach allows holding long, interactive, and meaningful conversations where the bot actually ``understands" what the human user is saying.
The potential applications of our STAR framework are immense. It can help in any NLU application that requires reasoning about knowledge in text or utterances. In the future, we aim to use our STAR framework for more applications such as automatically extracting formal software requirements from textual specifications and building conversational agents for other domains. We believe that the STAR framework can also provide a more reliable basis for machine translations. Instead of mapping a sentence from language A to language B directly, we can extract the predicates from the sentence using meaning assigned by language A. These predicates are then transformed into predicates consistent with language B, taking language-appropriate idioms and conventions into account. The language B predicates can then be used to obtain a faithful translation to an appropriate sentence of language B. Problems such as the \textit{Winograd Schema Challenge} can also be tackled with the STAR framework. For tasks that contain more complex reasoning, we believe, the performance improvement using STAR will be more pronounced. We also plan to develop a general commonsense knowledge-base that applications developed using the framework can employ.
\smallskip
\noindent{\bf Acknowledgment:}
Authors acknowledge support from NSF grants IIS 1910131, IIP 1916206, and US DoD.
We are grateful to Farhad Shakerin, Kinjal Basu, Joaquin Arias, Elmer Salazar, and members of the UT Dallas ALPS Lab for discussions.
\vspace{-0.2in}
\bibliographystyle{tlplike}
|
2,869,038,154,835 | arxiv | \section{Introduction}
\label{section:introduction}
Co-representations of compact quantum groups correspond to certain involutive monoidal categories (\cite{Wo87}, \cite{Wo88}, \cite{Wo98}). Banica and Speicher showed how to construct examples of such categories by taking rows of points as objects and partitions of two such rows as morphisms (\cite{BaSp09}). By additionally painting the points different colors, Freslon, Tarrago and the second author (\cite{FrWe14}, \cite{TaWe15a}, \cite{TaWe15b}) extended this construction to produce even more categories. For two (mutually inverse) colors, one obtains quantum subgroups of the free unitary quantum group $U_n^+$ of Wang's (\cite{Wa95}). For the precise definitions of two-col\-ored partitions and their categories the reader is referred to \cite[Sec\-tions~2 and~3]{MWNHO1}. See also \cite{TaWe15a} for more details and examples.
\par
In \cite{TaWe15a} Tarrago and the second author initiated a program to classify all categories of two-col\-ored partitions. Different subclasses have since been indexed (\cite{TaWe15a}, \cite{Gr18}, \cite{MaWe18a}, \cite{MaWe18b}) by various contributors. The present article is the second part of a series aiming to determine and describe all so-called non-hy\-per\-oc\-ta\-he\-dral categories, i.e., all categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ with $\PartSinglesWBTensor\in \mathcal C$ or $\PartFourWBWB\notin\mathcal C$.
\par
In this regard the first article \cite{MWNHO1} and the present one pursue complementary approaches for detecting whether a given set of partitions is a non-hy\-per\-oc\-ta\-he\-dral category: Part~I gave sufficient conditions for being a non-hy\-per\-octa\-he\-dral category, Part~II now provides necessary ones.
\par
\vspace{0.5em}
Let us take a closer look at the findings of Part~I, \cite{MWNHO1}.
Every two-col\-ored partition can be equipped with two natural structures on its set of points: a measure-like one, the \emph{color sum}, and a metric-like one, the \emph{color distance}. Both \cite{MWNHO1} and the present article study tuples of six properties of any given partition:
\begin{enumerate}[label=(\arabic*)]
\item the set of block sizes,
\item the set of block color sums,
\item the color sum of the set of all points,
\item the set of color distances between subsequent legs of the same block with identical (normalized) colors,
\item the set of color distances between subsequent legs of the same block with different (normalized) colors and
\item the set of color distances between legs belonging to crossing blocks.
\end{enumerate}
By forming unions, one can aggregate these data over a given set of partitions. This information extracted from a set $\mathcal S\subseteq \mc{P}^{\circ\bullet}$ of partitions was called $Z(\mathcal S)$ in \cite{MWNHO1}.
\par
There it was shown that one can give constraints on the above six properties which are preserved under category operations: A partially ordered set $(\mathsf Q,\leq)$ of parameters was introduced to prove that the sets of the form
\begin{IEEEeqnarray*}{rCl}
\mathcal R_Q\coloneqq\{p\in \mc{P}^{\circ\bullet}\mid Z(\{p\})\leq Q\}\quad\text{for }Q\in\mathsf Q
\end{IEEEeqnarray*}
form non-hy\-per\-octa\-he\-dral categories.
\par
The current article now shows that these constraints encoded in $Z$ and $(\mathsf Q,\leq)$ are natural in the following sense. (See also Sec\-tion~\ref{section:reminder} for the definitions.)
\begin{umain}{\normalfont [{Theorem~\ref{theorem:main}}]}
Given any non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ of two-col\-ored partitions, we have $Z(\mathcal C)\in\mathsf Q$.
\end{umain}
The importance of this result comes from its role in the overall program of the article series. On the one hand, it will be crucial to proving the main assertions of the ensuing articles. On the other hand, once those have been established, it will combine with them to show the final result of the entire series, roughly:
\begin{umainseries}[Excerpt]
$Z$ restricts to a one-to-one correspondence between the set $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$ of non-hy\-per\-oc\-ta\-he\-dral categories of two-col\-ored partitions and the parameter set $\mathsf Q$.
\end{umainseries}
The proof will go as follows: By Part~I of the series, $\mathcal R_Q\subseteq \mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$ for every $Q\in \mathsf Q$. Conversely, by the above Main Theorem of Part~II, $Z(\mathcal C)\in \mathsf Q$ for any $\mathcal C\in\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$. In the subsequent articles we will define a set $\mathcal G_{Z(\mathcal C)}\subseteq \mc{P}^{\circ\bullet}$ and show
\begin{IEEEeqnarray*}{rCl}
\mathcal G_{Z(\mathcal C)}\subseteq \mathcal C\subseteq \langle \mathcal G_{Z(\mathcal C)}\rangle \quad\text{and} \quad \mathcal G_{Z(\mathcal R_{Z(\mathcal C)})}\subseteq \mathcal R_{Z(\mathcal C)}\subseteq \langle \mathcal G_{Z(\mathcal R_{Z(\mathcal C)})}\rangle.
\end{IEEEeqnarray*}
Proving $Z(\mathcal R_{Z(\mathcal C)})=Z(\mathcal C)$ will then let us conclude $\mathcal C=\langle \mathcal G_{Z(\mathcal C)}\rangle=\langle \mathcal G_{Z(\mathcal R_{Z(\mathcal C)})}\rangle=\mathcal R_{Z(\mathcal C)}$.
\section{Reminder on Definitions from Part~I}
\label{section:reminder}
For the convenience of the reader we briefly repeat those definitions from \cite[Sec\-tions~3--5]{MWNHO1} which are relevant to the current article. For definitions of partitions and categories of partitions see \cite[Sec\-tions~3.1 and~4.2]{MWNHO1}. Throughout this article we will use the notations and definitions from \cite[Sec\-tions~3--5]{MWNHO1}.
\begin{notation}
For every set $S$ denote its power set by $\mathfrak P(S)$.
\end{notation}
\begin{definition}{{\normalfont\cite[Definition~5.2]{MWNHO1}}}
The \emph{parameter domain} $\mathsf L$ is the sixfold Cartesian product of $\mathfrak{P}(\mathbb{Z})$.
\end{definition}
\begin{definition}{{\normalfont\cite[Definition~5.3]{MWNHO1}}}
\label{definition:Z}
Using the notation from \cite[Sec\-tions~3--5]{MWNHO1}, we define the \emph{analyzer} $Z: \, \mathfrak{P}(\mc{P}^{\circ\bullet})\to \mathsf L$ by
\begin{align*}
Z\coloneqq (\, F,\, V,\, \Sigma,\, L,\, K,\, X\,)
\end{align*}
where, for all $\mathcal S\subseteq \mc{P}^{\circ\bullet}$,
\begin{enumerate}[label=(\alph*)]
\item \(F(\mathcal S)\coloneqq \{\, |B| \mid p\in \mathcal S,\, B\text{ block of } p\}\) is the set of block sizes,
\item \(V(\mathcal S)\coloneqq \{\,\sigma_p(B)\mid p\in \mathcal S,\, B\text{ block of }p\}\) is the set of block color sums,
\item \(\Sigma(\mathcal S)\coloneqq \{\,\Sigma(p)\mid p\in \mathcal S\}\) is the set of total color sums,
\item $\begin{aligned}[t]
L(\mathcal S)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2)\mid&\, p\in \mathcal S, \, B\text{ block of }p,\, \alpha_1,\alpha_2\in B,\, \alpha_1\neq \alpha_2,\\
&\, ]\alpha_1,\alpha_2[_p\cap B=\emptyset,\, \sigma_p(\{\alpha_1,\alpha_2\})\neq 0\}
\end{aligned}$ \\
is the set of color distances between any two subsequent legs of the \emph{same} block having the \emph{same} normalized color,
\item $\begin{aligned}[t]
K(\mathcal S)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2)\mid\,& p\in \mathcal S, \, B\text{ block of }p,\, \alpha_1,\alpha_2\in B,\, \alpha_1\neq \alpha_2,\\
&\, ]\alpha_1,\alpha_2[_p\cap B=\emptyset,\, \sigma_p(\{\alpha_1,\alpha_2\})= 0\}
\end{aligned}$\\
is the set of color distances between any two subsequent legs of the \emph{same} block having \emph{different} normalized colors and
\item $\begin{aligned}[t]
X(\mathcal S)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2) \mid \, & p\in \mathcal S,\, B_1,B_2\text{ blocks of }p, \, B_1\text{ crosses } B_2,\\
&\, \alpha_1\in B_1,\,\alpha_2\in B_2\}
\end{aligned}$ \\
is the set of color distances between any two legs belonging to two \emph{crossing} blocks.
\end{enumerate}
\end{definition}
\begin{notation}
\label{integer-notations}
\begin{enumerate}[label=(\alph*)]
\item\label{integer-notations-2} For all $x,y\in \mathbb{Z}$ and $A,B\subseteq \mathbb{Z}$ write
\begin{align*}
xA+yB\coloneqq \{xa+yb\mid a\in A, \,b\in B\}.
\end{align*}
Moreover, put $xA-yB\coloneqq xA+(-y)B$. Per $A=\{1\}$ expressions like $x+yB$ are defined as well, and per $x=1$ so are such like $A+yB$.
\item\label{integer-notations-3} Let $\pm S\coloneqq S\cup(-S)$ for all sets $S\subseteq \mathbb{Z}$.
\item\label{integer-notations-4} For all $m\in \mathbb{Z}$ and $D\subseteq \mathbb{Z}$ define
\begin{align*}
D_m\coloneqq (D\cup(m\!-\! D))+m\mathbb{Z} \quad\text{and}\quad D_m'\coloneqq (D\cup(m\!-\!D)\cup \{0\})+m\mathbb{Z}.
\end{align*}
\item\label{integer-notations-5} Use the abbreviations $\dwi{0}\coloneqq\emptyset$ and $\dwi{k}\coloneqq \{1,\ldots,k\}$ for all $k\in \mathbb{N}$.
\end{enumerate}
\end{notation}
\begin{definition}[{\cite[Definition~5.7]{MWNHO1}}]
\label{definition:Q}
Define the \emph{parameter range} $\mathsf Q$ as the subset of $\mathsf L$ comprising all tuples $(f,v,s,l,k,x)$
listed below, where $u\in\{0\}\cup \mathbb{N}$, where $m\in \mathbb{N}$, where $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$, where $E\subseteq \{0\}\cup \mathbb{N}$ and where $N$ is a sub\-se\-mi\-group of $(\mathbb{N},+)$:
\begin{align*}
\begin{matrix}
f&v&s&l&k& x \\ \hline \\[-0.85em]
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\\
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z} \\
\{2\} & \pm \{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash m\mathbb{Z} \\
\{2\} & \{0\} & \{0\} & \emptyset & m\mathbb{Z} & \mathbb{Z}\\
\{2\} & \pm\{0, 2\} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash N_0 \\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash N_0 \\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} &\mathbb{Z}\backslash N_0' \\
\{1,2\}&\pm\{0, 1, 2\} & um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{1,2\}&\pm\{0, 1, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{1,2\}&\pm \{0, 1\} & um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m \\
\{1,2\}&\pm\{0, 1, 2\} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{1,2\}&\pm\{0, 1\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0\\
\mathbb{N} & \mathbb{Z} & um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\mathbb{N} & \mathbb{Z} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0\\
\end{matrix}
\end{align*}
\end{definition}
The goal of this article, as sketched in the introduction, is to prove that $Z$ restricts to a map $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}\to \mathsf Q$ (see Theorem~\ref{theorem:main}). Evidently, $\mathsf Q$ is not a Cartesian product; the six entries of the tuples cannot vary independently. Rather, only very special tuples of sets are allowed. Hence, if the claim $Z:\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}\to \mathsf Q$ is to be true, then it is not enough to study the components of $Z$ individually. We must also investigate the relations between them. In consequence, the argument follows a winding path, taking components into and out of consideration underway as required or convenient.
\section{Tools: Equivalence and Projection}
\label{section:equivalence}
We introduce an equivalence relation on pairs of partitions and consecutive sets therein by which to compare partitions locally (cf.\ \cite[Definition~6.2]{MaWe18a}).
\begin{definition}
For all $i\in \{1,2\}$, let $P_{p_i}$ denote the set of all points of $p_i\in \mc{P}^{\circ\bullet}$ and let $S_i\subseteq P_{p_i}$ be consecutive. We call $(p_1,S_1)$ and $(p_2,S_2)$ \emph{equivalent} if $S_1=S_2=\emptyset$ or if the following is true:
There exist $n\in \mathbb N$ and for each $i\in \{1,2\}$ pairwise distinct points $\gamma_{i,1},\ldots,\gamma_{i,n}$ in $p_i$ such that $(\gamma_{i,1},\ldots, \gamma_{i,n})$ is ordered in $p_i$ and $S_i=\{\gamma_{i,1},\ldots,\gamma_{i,n}\}$ and such that for all $j,j'\in \{1,\ldots,n\}$ (possibly $j=j'$) the following are true:
\begin{enumerate}[label=(\arabic*)]
\item The normalized colors of $\gamma_{1,j}$ in $p_1$ and $\gamma_{2,j}$ in $p_2$ agree.
\item The points $\gamma_{1,j}$ and $\gamma_{1,j'}$ both belong to a block $B_1$ of $p_1$ with $B_1\subseteq S_1$ if and only if $\gamma_{2,j}$ and $\gamma_{2,j'}$ both belong to a block $B_2$ of $p_2$ with $B_2\subseteq S_2$.
\item The points $\gamma_{1,j}$ and $\gamma_{1,j'}$ both belong to a block $B_1$ of $p_1$ with $B_1\not\subseteq S_1$ if and only if $\gamma_{2,j}$ and $\gamma_{2,j'}$ both belong to a block $B_2$ of $p_2$ with $B_2\not\subseteq S_2$.
\end{enumerate}
\begin{mycenter}[1em]
\begin{tikzpicture}[baseline=2.5cm*0.666]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{5*\scp*1cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({2*\dx+\scp*0.5cm},{2*\dy});
%
\node [whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l4) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l5) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l6) at ({0+5*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l7) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
%
\node [blp] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u2) at ({0+1*\scp*1cm},{0+1*2*\dy}) {};
\node [blp] (u3) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u4) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u5) at ({0+4*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u6) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\node [lk, yshift={2*\scp*1cm}] at (l1) {};
\node [lk, yshift={2*\scp*1cm}] at (l5) {};
\node [lk, yshift={2*\scp*1cm}] at (l6) {};
\node [lk, yshift={-1*\scp*1cm}] at (u1) {};
\node [lk, yshift={-1*\scp*1cm}] at (u3) {};
\node [lk, yshift={-2*\scp*1cm}] at (u2) {};
\node [lk, yshift={-2*\scp*1cm}] at (u4) {};
\node [lk, yshift={-2*\scp*1cm}] at (u5) {};
\node [lk, yshift={3*\scp*1cm}] at (l7) {};
%
\draw [->] (l4) --++ (0,{\scp*1cm});
\draw [->] (u6) --++ (0,{-\scp*1cm});
\draw (l2) -- (u2);
\draw (l3) -- (u3);
\draw (l1) --++(0,{2*\scp*1cm}) -| (l5);
\draw (l5) --++(0,{2*\scp*1cm}) -| (l6);
\draw (u1) --++(0,{-1*\scp*1cm}) -| (u3);
\draw (u2) --++(0,{-2*\scp*1cm}) -| (u4);
\draw (u4) --++(0,{-2*\scp*1cm}) -| (u5);
\draw (u5) --++(0,{-2*\scp*1cm}) -| (l7);
%
\draw [densely dashed] ({-\scp*0.5cm},{\scp*0.3cm}) -- ($(l4)+({\scp*0.3cm},{\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -- ({-\scp*0.5cm},{-\scp*0.3cm});
\draw [densely dashed] ({-\scp*0.5cm},{\scp*0.3cm+2*\dy}) -- ($(u3)+({\scp*0.3cm},{\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -- ({-\scp*0.5cm},{-\scp*0.3cm+2*\dy});
%
\node at ({2*\dx+\scp*1cm},{0.5*2*\dy}) {$p$};
\node (lab1) at ({-2*\scp*1cm},{0.5*2*\dy}) {$S$};
\draw [densely dotted, ->, shorten >=5pt] (lab1.east) -- ({-\scp*0.5cm},0);
\draw [densely dotted, ->, shorten >=5pt] (lab1.east) -- ({-\scp*0.5cm},{2*\dy});
\end{tikzpicture}\quad\quad
\begin{tikzpicture}[baseline=2cm*0.666]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{4*\scp*1cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({2*\dx+\scp*0.5cm},{2*\dy});
%
\node [whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l4) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l5) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l6) at ({0+5*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l7) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
%
\node [blp] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {};
\node [blp] (u2) at ({0+1*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u3) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
\node [blp] (u4) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
%
\node [lk, yshift={-1*\scp*1cm}] at (u3) {};
\node [lk, yshift={3*\scp*1cm}] at (l6) {};
\node [lk, yshift={2*\scp*1cm}] at (l5) {};
\node [lk, yshift={2*\scp*1cm}] at (l7) {};
\node [lk, yshift={2*\scp*1cm}] at (l2) {};
%
\draw [->] (u1) --++ (0,{-\scp*1cm});
\draw (l4) -- (u4);
\draw (l3) --++(0,{3*\scp*1cm});
\draw (l1) --++(0,{1*\scp*1cm}) -| (l2);
\draw (l5) --++(0,{2*\scp*1cm}) -| (l7);
\draw (l5) --++(0,{2*\scp*1cm}) -| (u2);
\draw (l6) --++(0,{3*\scp*1cm}) -| (u3);
%
\draw [densely dashed] ({2*\dx+\scp*0.5cm},{\scp*0.3cm}) -- ($(l5)+({-\scp*0.3cm},{\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -- ({2*\dx+\scp*0.5cm},{-\scp*0.3cm});
\draw [densely dashed] ({\txu+\scp*0.5cm},{\scp*0.3cm+2*\dy}) -- ($(u1)+({-\scp*0.3cm},{\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -- ({2*\dx+\scp*0.5cm},{-\scp*0.3cm+2*\dy});
%
\node at ({-\scp*1cm},{0.5*2*\dy}) {$p'$};
\node (lab1) at ({2*\dx+2*\scp*1cm},{0.5*2*\dy}) {$S'$};
\draw [densely dotted, ->, shorten >=5pt] (lab1.west) -- ({2*\dx+\scp*0.5cm},0);
\draw [densely dotted, ->, shorten >=5pt] (lab1.west) -- ({2*\dx+\scp*0.5cm},{2*\dy});
\end{tikzpicture}
\\[1em]
\begin{tikzpicture}
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{4*\scp*1cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
%
\node [whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l4) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l5) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l6) at ({0+5*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l7) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
%
\node [lk, yshift={2*\scp*1cm}] at (l1) {};
\node [lk, yshift={2*\scp*1cm}] at (l3) {};
\node [lk, yshift={2*\scp*1cm}] at (l6) {};
\node [lk, yshift={1*\scp*1cm}] at (l2) {};
\node [lk, yshift={1*\scp*1cm}] at (l5) {};
%
\draw [->] (l7) --++ (0,{\scp*1cm});
\draw (l2) --++(0,{3*\scp*1cm});
\draw (l4) --++(0,{3*\scp*1cm});
\draw (l5) --++(0,{3*\scp*1cm});
\draw (l1) --++(0,{2*\scp*1cm}) -| (l3);
\draw (l3) --++(0,{2*\scp*1cm}) -| (l6);
\draw (l2) --++(0,{1*\scp*1cm}) -| (l5);
%
\node at ({0.5*2*\dx},{-\scp*1cm}) {class of $(p,S)\cong (p',S')$};
\end{tikzpicture}
\end{mycenter}
\end{definition}
If $(p_1,S_1)$ and $(p_2,S_2)$ are equivalent, then $S_1$ and $S_2$ agree in size and normalized coloring up to a rotation $\varrho$ and the induced partitions $\{B_1\cap S_1\,\vert\, B_1\text{ block of }p_1\}$ of $S_1$ and $\{B_2\cap S_2\,\vert\, B_2\text{ block of }p_2\}$ of $S_2$ concur up to $\varrho$. However, this is only a necessary condition. Equivalence further requires that a block $B_1\cap S_1$ of the restriction of $p_1$ stems from a block $B_1$ of $p_1$ which has legs outside $S_1$ if and only if the corresponding statement $B_2\not\subseteq S_2$ is true for the block $B_2$ of $p_2$ which $B_1$ is mapped to under $\varrho$.
\par
We define and construct special representatives of the classes of this equivalence relation. Recall that a partition $p\in \mc{P}^{\circ\bullet}$ is called \emph{projective} if $p$ is self-adjoint, i.e., $p=p^\ast$, and idempotent, i.e., the pair $(p,p)$ is composable and $pp=p$.
\begin{definition}
For every consecutive set $S$ in $p\in \mc{P}^{\circ\bullet}$ we call the unique projective partition $q$ with lower row $M$ such that $(q,M)$ and $(p,S)$ are equivalent the \emph{projection} $P(p,S)$ of $(p,S)$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=1.5cm*0.666]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{4*\scp*1cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
%
\node [whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l4) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l5) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l6) at ({0+5*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l7) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
%
\node [lk, yshift={2*\scp*1cm}] at (l1) {};
\node [lk, yshift={2*\scp*1cm}] at (l3) {};
\node [lk, yshift={2*\scp*1cm}] at (l6) {};
\node [lk, yshift={1*\scp*1cm}] at (l2) {};
\node [lk, yshift={1*\scp*1cm}] at (l5) {};
%
\draw [->] (l7) --++ (0,{\scp*1cm});
\draw (l2) --++(0,{3*\scp*1cm});
\draw (l4) --++(0,{3*\scp*1cm});
\draw (l5) --++(0,{3*\scp*1cm});
\draw (l1) --++(0,{2*\scp*1cm}) -| (l3);
\draw (l3) --++(0,{2*\scp*1cm}) -| (l6);
\draw (l2) --++(0,{1*\scp*1cm}) -| (l5);
%
\node at ({0.5*2*\dx},{-2*\scp*1cm}) {class of $(p,S)\cong (q,M)$};
\end{tikzpicture}\quad\quad
\begin{tikzpicture}[baseline=2.5cm*0.666]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{5*\scp*1cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({2*\dx+\scp*0.5cm},{2*\dy});
%
\node [whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l4) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node [blp] (l5) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l6) at ({0+5*\scp*1cm},{0+0*2*\dy}) {};
\node [whp] (l7) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
%
\node [whp] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {};
\node [blp] (u2) at ({0+1*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u3) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u4) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
\node [blp] (u5) at ({0+4*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u6) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
\node [whp] (u7) at ({0+6*\scp*1cm},{0+1*2*\dy}) {};
%
\node [lk, yshift={2*\scp*1cm}] at (l1) {};
\node [lk, yshift={2*\scp*1cm}] at (l3) {};
\node [lk, yshift={2*\scp*1cm}] at (l6) {};
\node [lk, yshift={1*\scp*1cm}] at (l2) {};
\node [lk, yshift={1*\scp*1cm}] at (l5) {};
\node [lk, yshift={-2*\scp*1cm}] at (u1) {};
\node [lk, yshift={-2*\scp*1cm}] at (u3) {};
\node [lk, yshift={-2*\scp*1cm}] at (u6) {};
\node [lk, yshift={-1*\scp*1cm}] at (u2) {};
\node [lk, yshift={-1*\scp*1cm}] at (u5) {};
%
\draw [->] (l7) --++ (0,{\scp*1cm});
\draw [->] (u7) --++ (0,{-\scp*1cm});
\draw (l2) -- (u2);
\draw (l4) -- (u4);
\draw (l5) -- (u5);
\draw (l1) --++(0,{2*\scp*1cm}) -| (l3);
\draw (l3) --++(0,{2*\scp*1cm}) -| (l6);
\draw (l2) --++(0,{1*\scp*1cm}) -| (l5);
\draw (u1) --++(0,{-2*\scp*1cm}) -| (u3);
\draw (u3) --++(0,{-2*\scp*1cm}) -| (u6);
\draw (u2) --++(0,{-1*\scp*1cm}) -| (u5);
%
\node at ({0.5*2*\dx},{-\scp*1cm}) {$P(p,S)=q$};
\node at ({2*\dx+\scp*1cm},0) {$M$};
\end{tikzpicture}
\end{mycenter}
\end{definition}
In truth, of course, for any consecutive set $S$ in $p\in \mc{P}^{\circ\bullet}$ the projection $P(p,S)$ depends only on the equivalence class of $(p,S)$. The following lemma constitutes a generalization of \cite[Lem\-ma~6.4]{MaWe18a}.
\begin{lemma}
\label{lemma:projection}
$P(p,S)\in \langle p\rangle$ for any consecutive set $S$ in any $p\in \mc{P}^{\circ\bullet}$.
\end{lemma}
\begin{proof}
As $S=\emptyset$ implies $P(p,S)=\emptyset\in \langle p\rangle$, let $S\neq \emptyset$. By rotation we can assume that $S$ is the lower row of $p$. Then $S$ has the same size and coloring in $p$ as in $q\coloneqq pp^*$. We show $q=P(p,S)$. By the nature of composition the blocks of $p$ which are contained in $S$ are blocks of $q$ as well. We only need to care about the other blocks of $q$. If we identify the upper row of $p$ and the lower row of $p^\ast$, the same partition $s$ is induced there by $p$ and $p^\ast$. Consequently, the meet of the two induced partitions is identical with $s$ as well. That means that every block $D$ of $s$ intersects exactly one block $B$ of $p$ and exactly one block of $p^*$, namely the mirror image of $B$. The block of $q$ resulting from $D$ therefore contains exactly the restriction of $B$ to the lower row and the mirror image of that set on the upper row. That means $q=P(p,S)$, which proves the claim.
\end{proof}
\section{Step~1: Component $F$ in Isolation}
\label{section:block-sizes}
We now take our first step towards proving the main result that the analyzer $Z$ from Definition~\ref{definition:Z} restricts to a map $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}\to \mathsf Q$ (see Theorem~\ref{theorem:main}). Namely, we verify (see Proposition~\ref{proposition:result-F}) that, for every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the set
\begin{align*}
F(\mathcal C)\coloneqq \{|B| \mid p\in\mathcal C,\, B\text{ block of }p\}
\end{align*}
of block sizes appearing in $\mathcal C$ can only be one of the three sets of integers admissible as a first component for tuples in $\mathsf Q$ by Definition~\ref{definition:Q}.
\begin{lemma}{\normalfont \cite[Lem\-mata~1.3 (b), 2.1 (a)]{TaWe15a}}
\label{lemma:singletons}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a category.
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:singletons-1}
\(\langle \PartSinglesWBTensor\rangle =\langle \PartSinglesBWTensor\rangle=\langle \PartSinglesProjW\rangle=\langle \PartSinglesProjB\rangle\).
\item\label{lemma:singletons-2} The following statements are equivalent:
\begin{enumerate}[label=(\arabic*)]
\item There exists in $\mathcal C$ a partition with a singleton block.
\item $\PartSinglesWBTensor\in \mathcal C$.
\end{enumerate}
\item\label{lemma:singletons-3} If $\PartSinglesWBTensor\in \mathcal C$, then $\mathcal C$ is closed under disconnecting points from their blocks.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[wide,label=(\alph*)]
\item All transformations can be achieved by basic and cyclic rotations.
\item Projecting to a singleton block produces $\PartSinglesProjW$ or $\PartSinglesProjB$. Hence, Part~\ref{lemma:singletons-1} and Lem\-ma~\ref{lemma:projection} prove the claim.
\item Rotate a given partition such that the leg to disconnect from its block is the only lower point. Composing from below with $\PartSinglesProjW$ or $\PartSinglesProjB$, depending on the color of the leg, and reversing the rotation achieves what is claimed. Hence, Part~\ref{lemma:singletons-1} concludes the proof. \qedhere
\end{enumerate}
\end{proof}
\begin{lemma}{\normalfont \cite[Lem\-mata~1.3 (d), 2.1 (b)]{TaWe15a}}
\label{lemma:companies}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a category.
\begin{enumerate}[label=(\alph*)]
\item \label{lemma:companies-1}
\(\langle \PartFourWBWB\rangle =\langle \PartFourBWBW\rangle=\langle \PartFourProjWB\rangle=\langle \PartFourProjBW\rangle\).
\item \label{lemma:companies-2} The following statements are equivalent:
\begin{enumerate}[label=(\arabic*)]
\item There exists in $\mathcal C$ a partition with a block with at least three legs.
\item $\PartFourWBWB\in \mathcal C$.
\end{enumerate}
\item \label{lemma:companies-3} If $\PartFourWBWB\in \mathcal C$, then $\mathcal C$ is closed under connecting the two points in any turn.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[label=(\alph*),wide]
\item Once again, by basic and cyclic rotations we can transform the partitions into each other.
\item Suppose $B$ is a block in $p\in \mathcal C$ with at least three legs, $\alpha,\beta\in B$, $\alpha\neq \beta$ and $]\alpha,\beta[_p\cap B=\emptyset$. Let $T$ be the set of the first lower and the first upper point of $P(p,[\alpha,\beta]_p)$. The partition $P(P(p,[\alpha,\beta]_p),T)$ is either $\PartFourProjWB$ or $\PartFourProjBW$. Thus follows the claim by Part~\ref{lemma:companies-1} and Lem\-ma~\ref{lemma:projection}.
\item Let $T$ be the turn in $p\in \mathcal C$ whose points we want to connect. By rotation we can assume that $T$ is the upper row of $p$. By composing $p$ from above with $\PartFourProjWB$ or $\PartFourProjBW$, depending on the sequence of colors in $T$, and reversing the initial rotation we achieve exactly what is claimed. So, Part~\ref{lemma:companies-1} implies the assertion.\qedhere
\end{enumerate}
\end{proof}
Recall the cases $\mathcal O$, $\mathcal B$, $\mathcal S$ from \cite[Definition~4.1]{MWNHO1}.
\begin{proposition}
\label{proposition:result-F}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a non-hy\-per\-oc\-ta\-hed\-ral category.
\begin{enumerate}[label=(\alph*)]
\item \label{proposition:result-F-0} The set $F(\mathcal C)$ is given by $\{2\}$, $\{1,2\}$ or $\mathbb{N}$.
\item\label{proposition:result-F-1} If $\mathcal C$ is case~$\mathcal O$, then $F(\mathcal C)=\{2\}$.
\item\label{proposition:result-F-2} If $\mathcal C$ is case~$\mathcal B$, then $F(\mathcal C)=\{1,2\}$.
\item\label{proposition:result-F-3} If $\mathcal C$ is case~$\mathcal S$, then $F(\mathcal C)=\mathbb{N}$.
\end{enumerate}
\end{proposition}
\begin{proof}
By definition of a category, $\PartIdenLoWB\in \mathcal C$ and thus $\{2\}\subseteq F(\mathcal C)$.
\begin{enumerate}[label=(\alph*),wide]
\item The first claim follows from the other three.
\item Because $\PartSinglesWBTensor\notin \mathcal C$ and $\PartFourWBWB\notin \mathcal C$, Lem\-mata~\hyperref[lemma:singletons-2]{\ref*{lemma:singletons}~\ref*{lemma:singletons-2}} and~\hyperref[lemma:companies-2]{\ref*{lemma:companies}~\ref*{lemma:companies-2}} show that every block in every partition of $\mathcal C$ has exactly two legs, i.e., $F(\mathcal C)=\{2\}$.
\item The assumption $\PartFourWBWB\notin \mathcal C$ implies by Lem\-ma~\hyperref[lemma:companies-2]{\ref*{lemma:companies}~\ref*{lemma:companies-2}} that no partition of $\mathcal C$ has blocks with more than two legs: $F(\mathcal C)\subseteq \{1,2\}$. Because $\PartSinglesWBTensor\in \mathcal C$, it is clear that $\{1\}\subseteq F(\mathcal C)$. Thus, $F(\mathcal C)=\{1,2\}$ has been proven.
\item It suffices to show $\mathbb{N}\subseteq F(\mathcal C)$. Let $n\in \mathbb{N}$ be arbitrary. Then, \[p\coloneqq (\PartSinglesWBTensor)^{\otimes \left\lceil \frac{n}{2}\right\rceil}\in \mathcal C.\] Thanks to $\PartFourWBWB\in \mathcal C$ we can, by Lem\-ma~\hyperref[lemma:companies-3]{\ref*{lemma:companies}~\ref*{lemma:companies-3}}, connect the first $n$ points in $p$ to produce a partition in $\mathcal C$ containing a block with $n$ points, proving $\{n\}\subseteq F(\mathcal C)$. \qedhere
\end{enumerate}
\end{proof}
\section{Step~2: Component $V$ and its Relation to $F$ and $L$}
\label{section:block-color-sums}
The next objective is to narrow down the range of the component $V$ of $Z$ over $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$. Given a non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, we show that the set
\begin{align*}
V(\mathcal C)\coloneqq \{\sigma_p(B)\mid p\in \mathcal C,\, B\text{ block of }p\}
\end{align*}
of block color sums occurring in $\mathcal C$
can only be one of the five sets allowed as second components for tuples of $\mathsf Q$ by Definition~\ref{definition:Q}. Beyond that, we can use Proposition~\ref{proposition:result-F} to show a result about the three parameters $V(\mathcal C)$, $F(\mathcal C)$ and
\begin{align*}
L(\mathcal C)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2)\mid&\, p\in \mathcal C, \, B\text{ block of }p,\, \alpha_1,\alpha_2\in B,\, \alpha_1\neq \alpha_2,\\
&\, ]\alpha_1,\alpha_2[_p\cap B=\emptyset,\, \sigma_p(\{\alpha_1,\alpha_2\})\neq 0\},
\end{align*}
the set of color distances between legs of the same block with identical normalized colors appearing in $\mathcal C$: Viewed together as $(F,V,L)(\mathcal C)$, they satisfy the conditions necessary for $Z(\mathcal C)$ to be element of $\mathsf Q$ by Definition~\ref{definition:Q}.
\begin{proposition}
\label{proposition:result-V}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a non-hy\-per\-oc\-ta\-hed\-ral category.
\begin{enumerate}[label=(\alph*)]
\item\label{proposition:result-V-0} The set $V(\mathcal C)$ is given by $\{0\}$, $\pm\{0,2\}$, $\pm \{0,1\}$, $\pm \{0,1,2\}$ or $\mathbb{Z}$.
\item\label{proposition:result-V-1} If $\mathcal C$ is case $\mathcal O$, then
\begin{align*}
V(\mathcal C)=
\begin{cases}
\pm\{0,2\}&\text{if }L(\mathcal C)\neq \emptyset,\\
\phantom{\pm}\{0\}&\text{otherwise.}
\end{cases}
\end{align*}
\item\label{proposition:result-V-2} If $\mathcal C$ is case $\mathcal B$, then
\begin{align*}
V(\mathcal C)=
\begin{cases}
\pm\{0,1,2\}&\text{if }L(\mathcal C)\neq \emptyset,\\
\pm \{0,1\}&\text{otherwise.}
\end{cases}
\end{align*}
\item\label{proposition:result-V-3} If $\mathcal C$ is case $\mathcal S$, then $L(\mathcal C)\neq \emptyset$ and $V(\mathcal C)=\mathbb{Z}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Two general facts about $V(\mathcal C)$ in advance: In any case, $0\in V(\mathcal C)$ since $V(\{\PartIdenLoWB\})=\{0\}$. And \cite[Lem\-ma~6.4]{MWNHO1}, using the fact that $p\in \mathcal C$ implies $\tilde p\in \mathcal C$, showed $V(\mathcal C)=-V(\mathcal C)$.
\par
\begin{enumerate}[label=(\alph*),wide]
\item Claim~\ref{proposition:result-V-0} follows from the other three.
\item A pair block $B$ in $p\in \mathcal C$ satisfies $\sigma_p(B)=0$ if and only if that block has no two (necessarily subsequent) legs of the same normalized colors. Otherwise it has color sum $-2$ or $2$.
\item And a singleton block always has color sums $-1$ or $1$. The rest follows from the proof of Part~\ref{proposition:result-V-1}.
\item If $\mathcal C$ is case~$\mathcal S$, then $\PartFourWBWB\in \mathcal C$ and $\PartSinglesWBTensor\in \mathcal C$. Hence, we can use $\PartSinglesWBTensor$ to disconnect the left black point in $\PartFourWBWB$ by Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}} to obtain $p\coloneqq\PartSpecThreeCoSingleWBWB\in \mathcal C$ with $V(\{p\})=\{-1, 1\}$. Given any $n\in \mathbb{N}$, we use $\PartFourWBWB$ to connect in $p^{\otimes n}\in \mathcal C$ all the $n$ many three-leg blocks together (leaving the disconnected singletons alone) in accordance with Lem\-ma~\hyperref[lemma:companies-3]{\ref*{lemma:companies}~\ref*{lemma:companies-3}}. That procedure results in the partition $q\in \mathcal C$ with $V(\{q\})=\{-1,n\}$. By $V(\mathcal C)=-V(\mathcal C)$ it then follows $V(\mathcal C)=\mathbb{Z}$ as claimed.\qedhere
\end{enumerate}
\end{proof}
\section{\texorpdfstring{Step~3: Component $\Sigma$ in Isolation}{Step~3: Parameter Sigma in Isolation}}
\label{section:total-color-sums}
Easily, we can confirm that for all non-hy\-per\-octa\-he\-dral categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ the set
\begin{align*}
\Sigma(\mathcal C)\coloneqq \{\Sigma(p)\mid p\in \mathcal C\}
\end{align*}
of all total color sums appearing in $\mathcal C$
is within the range of allowed third entries of tuples in $\mathsf Q$ by Definition~\ref{definition:Q}.
The following proposition contains a generalization of \cite[Lem\-ma~2.6]{TaWe15a} and \cite[Proposition~2.7]{TaWe15a}.
\begin{proposition}
\label{proposition:result-S}
For every category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ the set $\Sigma(\mathcal C)$ is a subgroup of $\mathbb{Z}$.
\end{proposition}
\begin{proof}
\cite[Lem\-ma~6.5~(c)]{MWNHO1} implies $\Sigma(\mathcal C)+\Sigma(\mathcal C)\subseteq \Sigma(\mathcal C)$. And $-\Sigma(\mathcal C)\subseteq \Sigma(\mathcal C)$ was shown in \cite[Lem\-ma~6.4]{MWNHO1}. As also $\Sigma(\PartIdenLoWB)=0$ and $\PartIdenLoWB\in \mathcal C$ by definition, the set $\Sigma(\mathcal C)$ is indeed a subgroup of $\mathbb{Z}$.
\end{proof}
\section{\texorpdfstring{Step~4: General Relations between $\Sigma$, $L$, $K$ and $X$}{Step~4: General Relations between Sigma, L, K and X}}
\label{section:color-lengths}
The goal remains proving that $Z$ (see Definition~\ref{definition:Z}) maps the set $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$ of non-hy\-per\-octa\-he\-dral categories to $\mathsf Q$ (see Definition~\ref{definition:Q}). So far, we have tackled this problem, more or less, one component of $Z$ at a time. In that way, what we have managed to show is, mostly, that the values over $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$ of each of the three maps $F$, $V$ and $\Sigma$, viewed individually, are confined to the range of parameters allowed by $\mathsf Q$ as corresponding entries of its elements. To complete this picture, we would also like to see that for any non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ the three sets $L(\mathcal C)$,
\begin{align*}
K(\mathcal C)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2)\mid\,& p\in \mathcal C, \, B\text{ block of }p,\, \alpha_1,\alpha_2\in B,\, \alpha_1\neq \alpha_2,\\
&\, ]\alpha_1,\alpha_2[_p\cap B=\emptyset,\, \sigma_p(\{\alpha_1,\alpha_2\})= 0\}\\
\text{and}\quad X(\mathcal C)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2) \mid \, & p\in \mathcal C,\, B_1,B_2\text{ blocks of }p, \, B_1\text{ crosses } B_2,\\
&\, \alpha_1\in B_1,\,\alpha_2\in B_2\},
\end{align*}
too, can only be of the kinds allowed as fourth, fifth and sixth components of tuples in $\mathsf Q$, respectively, by Definition~\ref{definition:Q}. However, due to the strong interdependences between these three components of $Z$, it is not even possible to prove this basic claim about the ranges of the individual maps by studying them one at a time. Instead, now, the reasonable thing to do is to consider the tuple $(\Sigma, L, K,X)$ and make inferences about its range over $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$. That will give us (see Proposition~\ref{lemma:result-s-l-k-x}) the claim about the individual ranges of $L$, $K$ and $X$ but also many more of the relations between them (and $\Sigma$), which we need to verify the main result.
\subsection{Abstract Arithmetic Lemma}
As a first step, it is best to study the relationship between the $\Sigma$-, $L$-, $K$- and $X$-components of $Z$ in an abstract context, merely talking about arbitrary subsets of $\mathbb{Z}$ subject to certain axioms. Our goal for this sub\-sec\-tion is to prove the \hyperref[lemma:arithmetic]{Arithmetic Lem\-ma (\ref*{lemma:arithmetic})}: Assuming certain axioms (\ref{axioms:arithmetic}), we may deduce a certain parameter range. We will show in Sub\-sec\-tion~\ref{section:verifying-the-axioms} that for non-hy\-per\-oc\-ta\-he\-dral categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ our sets $\Sigma(\mathcal C)$, $L(\mathcal C)$, $K(\mathcal C)$ and $X(\mathcal C)$ satisfy these axioms. Recall $\overline\bullet\coloneqq\circ$ and $\overline\circ\coloneqq\bullet$.
\begin{axioms}
\label{axioms:arithmetic}
Let $\sigma$ as well as $\kappa_{c_1,c_2}$ and $\xi_{c_1,c_2}$ for all $c_1,c_2\in\{\circ,\bullet\}$ be subsets of $\mathbb{Z}$.
Throughout this sub\-sec\-tion, make the following assumptions:
\begin{enumerate}[label=(\roman*)]
\item\label{lemma:gradedsets-condition-0} $\sigma$ is a subgroup of $\mathbb{Z}$.
\end{enumerate}
For all $(\omega_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}}\in \{(\kappa_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}},(\xi_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}}\}$ and for all $c_1,c_2\in\{\circ,\bullet\}$:
\begin{enumerate}[label=(\roman*), start=2]
\item\label{lemma:gradedsets-condition-1} $\omega_{c_1,c_2}+\sigma\subseteq \omega_{c_1,c_2}$.
\item\label{lemma:gradedsets-condition-3} $\omega_{c_1,c_2}\subseteq - \omega_{\overline{c_2},\overline{c_1}}$.
\item\label{lemma:gradedsets-condition-2} $\omega_{c_1,c_2}\subseteq -\omega_{c_2,c_1}+\sigma$.
\end{enumerate}
For all $c_1,c_2,c_3\in \{\circ,\bullet\}$:
\begin{enumerate}[label=(\roman*), start=5]
\item\label{lemma:gradedsets-condition-6} $\xi_{c_1,c_2}\subseteq \xi_{c_1,\overline{c_2}}\cup \left(-\xi_{c_2,\overline{c_1}}+\sigma\right)$.
\item\label{lemma:gradedsets-condition-4} $0\in \kappa_{\circ\bullet}\cap \kappa_{\bullet\circ}$.
\item\label{lemma:gradedsets-condition-5} $\kappa_{c_1,c_2}+\kappa_{\overline{c_2},c_3}\subseteq \kappa_{c_1,c_3}$.
\item\label{lemma:gradedsets-condition-7} $\kappa_{c_1,c_2}+\xi_{\overline{c_2},c_3}\subseteq \xi_{c_1,c_3}$.
\end{enumerate}
\end{axioms}
Let us first study how much $\kappa_{c_1,c_2}$ and $\xi_{c_1,c_2}$ depend on $c_1,c_2\in\{\circ,\bullet\}$.
\begin{lemma}
\label{lemma:omegas}
For any $(\omega_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}}\in \{(\kappa_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}},(\xi_{c_1,c_2})_{c_1,c_2\in \{\circ,\bullet\}}\}$:
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:omegas-1} $\omega_{\circ\circ}=\omega_{\bullet\bullet}$ and $\omega_{\circ\circ}=-\omega_{\circ\circ}=\omega_{\circ\circ}+\sigma$.
\item\label{lemma:omegas-2} $\omega_{\circ\bullet}=\omega_{\bullet\circ}$ and $\omega_{\circ\bullet}=-\omega_{\circ\bullet}=\omega_{\circ\bullet}+\sigma$.
\end{enumerate}
\end{lemma}
\begin{proof}
Because $0\in \sigma$ by Assumption~\ref{lemma:gradedsets-condition-0}, the Assumption~\ref{lemma:gradedsets-condition-1} actually means
\begin{align}
\label{eq:gradedsets-1-alternative}
\omega_{c_1,c_2}=\omega_{c_1,c_2}+\sigma\tag{ii'}
\end{align}
for all $c_1,c_2\in \{\circ,\bullet\}$. And with this new identity we can, for all $c_1,c_2\in \{\circ,\bullet\}$, refine Assumption~\ref{lemma:gradedsets-condition-2} to
\begin{align}
\label{eq:gradedsets-2-alternative}
\omega_{c_1,c_2}\subseteq -\omega_{c_2,c_1}\tag{iv'}
\end{align}
as $-\omega_{c_2,c_1}+\sigma=-(\omega_{c_2,c_1}-\sigma)=-(\omega_{c_2,c_1}+ \sigma)=-\omega_{c_2,c_1}$ due to $\sigma=-\sigma$.
\begin{enumerate}[wide, label=(\alph*)]
\item Version~\eqref{eq:gradedsets-1-alternative} of Assumption~\ref{lemma:gradedsets-condition-1} yields $\omega_{\circ\circ}=\omega_{\circ\circ}+\sigma$ as claimed.
And Assumption~\ref{lemma:gradedsets-condition-2} in the form of \eqref{eq:gradedsets-2-alternative} proves
\begin{align*}
\omega_{\circ\circ}\overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }-\omega_{\circ\circ} \overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }\omega_{\circ\circ} \quad \text{and}\quad \omega_{\bullet\bullet}\overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }-\omega_{\bullet\bullet} \overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }\omega_{\bullet\bullet},
\end{align*}
thus verifying $\omega_{\circ\circ}=-\omega_{\circ\circ}$ and $\omega_{\bullet\bullet}=-\omega_{\bullet\bullet}$.
Now, if we apply Assumption~\ref{lemma:gradedsets-condition-3} to conclude
\begin{align*}
\omega_{\circ\circ}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq }-\omega_{\bullet\bullet}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq } \omega_{\circ\circ},
\end{align*}
we can infer $\omega_{\circ\circ}=\omega_{\bullet\bullet}$.
That proves the remainder of the claims about $\omega_{\circ\circ}$ and $\omega_{\bullet\bullet}$.
\item Here also, Version~\eqref{eq:gradedsets-1-alternative} of Assumption~\ref{lemma:gradedsets-condition-1} implies $\omega_{\circ\bullet}=\omega_{\circ\bullet}+\sigma$.
Now, though, for $\omega_{\circ\bullet}$ and $\omega_{\bullet\circ}$ the roles of Assumptions~\ref{lemma:gradedsets-condition-3} and~\ref{lemma:gradedsets-condition-2} reverse. First, we apply the former to conclude
\begin{align*}
\omega_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq }-\omega_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq } \omega_{\circ\bullet}\quad\text{and}\quad \omega_{\bullet\circ}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq }-\omega_{\bullet\circ}\overset{\ref{lemma:gradedsets-condition-3}}{\subseteq } \omega_{\bullet\circ},
\end{align*}
which shows the claims $\omega_{\circ\bullet}=-\omega_{\circ\bullet}$ and $\omega_{\bullet\circ}=-\omega_{\bullet\circ}$. Then, it is the refined version \eqref{eq:gradedsets-2-alternative} of As\-sump\-tion~\ref{lemma:gradedsets-condition-2} that yields
\begin{align*}
\omega_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }-\omega_{\bullet\circ}\overset{\ref{lemma:gradedsets-condition-2}}{\subseteq }\omega_{\circ\bullet},
\end{align*}
implying $\omega_{\circ\bullet}=\omega_{\bullet\circ}$ and thus completing the proof.\qedhere
\end{enumerate}
\end{proof}
In the case of $(\omega_{c_1,c_2})_{c_1,c_2\in\{\circ,\bullet\}}=(\xi_{c_1,c_2})_{c_1,c_2\in\{\circ,\bullet\}}$ of Lem\-ma~\ref{lemma:omegas} we can go even further and combine the objects of Parts~\ref{lemma:omegas-1} and~\ref{lemma:omegas-2}.
\begin{lemma}
\label{lemma:xi-well-defined}
$\xi_{\circ\circ}=\xi_{\circ\bullet}$.
\end{lemma}
\begin{proof}
Since $\xi_{c_2,\overline{c_1}}=\xi_{c_2,\overline{c_1}}+\sigma$ for all $c_1,c_2\in \{\circ,\bullet\}$ by Version~\eqref{eq:gradedsets-1-alternative} of Axiom~\ref{lemma:gradedsets-condition-1}, our Assumption~\ref{lemma:gradedsets-condition-6} actually spells
\begin{align}
\label{eq:gradedsets-6-alternative}
\xi_{c_1,c_2}\subseteq \xi_{c_1,\overline{c_2}}\cup (-\xi_{c_2,\overline{c_1}})\tag{v'}
\end{align}
for all $c_1,c_2\in \{\circ,\bullet\}$ as $\sigma=- \sigma$. Using this version of the assumption twice, we conclude
\begin{align*}
\xi_{\circ\circ}\overset{\ref{lemma:gradedsets-condition-6}}\subseteq \xi_{\circ\bullet}\cup (-\xi_{\circ\bullet})=\xi_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-6}}{\subseteq}\xi_{\circ\circ}\cup(-\xi_{\bullet\bullet})=\xi_{\circ\circ},
\end{align*}
where we have used the results $\xi_{\circ\bullet}=-\xi_{\circ\bullet}$ and $\xi_{\circ\circ}=-\xi_{\bullet\bullet}$ of Lem\-ma~\ref{lemma:omegas}. It follows that indeed $\xi_{\circ\circ}=\xi_{\circ\bullet}$.
\end{proof}
\begin{definition}
\label{definition:xi}
\label{definition:lambda-kappa}
Write $\lambda\coloneqq \kappa_{\circ\circ}=\kappa_{\bullet\bullet}$ and $\kappa\coloneqq \kappa_{\circ\bullet}=\kappa_{\bullet\circ}$ and
$\xi\coloneqq \xi_{\circ\circ}=\xi_{\bullet\bullet}=\xi_{\circ\bullet}=\xi_{\bullet\circ}$.
\end{definition}
Our next step is to show that the pair $(\lambda,\kappa)$ is of a very simple form (Lem\-ma~\ref{lemma:lambda-kappa-summary}).
\begin{definition}
\label{definition:d-l}
Define the non-negative integers
\begin{align*}
d\coloneqq
\begin{cases}
\min\left(\kappa\cap \mathbb{N}\right) &\text{if }\kappa\cap \mathbb{N}\neq \emptyset,\\
0&\text{otherwise},
\end{cases}
\quad\text{and}\quad
l\coloneqq
\begin{cases}
\min\left(\lambda\cap \mathbb{N}\right) &\text{if }\lambda\cap \mathbb{N}\neq \emptyset,\\
0&\text{otherwise}.
\end{cases}
\end{align*}
\end{definition}
\begin{lemma}
\label{lemma:lambda-kappa}
\phantomsection
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:lambda-kappa-1} $\kappa=d\mathbb{Z}$.
\item\label{lemma:lambda-kappa-2} If $\lambda\neq \emptyset$, then $l\in\lambda$ and $\lambda-l\supseteq \kappa$.
\item\label{lemma:lambda-kappa-3} $\lambda-l\subseteq \kappa$.
\item\label{lemma:lambda-kappa-4} If $\lambda\neq \emptyset$ and $d\neq 0$, then $l\leq d$.
\item\label{lemma:lambda-kappa-5} If $\lambda\neq \emptyset$ and $d\neq 0$, then $l\neq 0$.
\item\label{lemma:lambda-kappa-6} If $\lambda\neq \emptyset$, then $2l\mathbb{Z}\subseteq d\mathbb{Z}$.
\item\label{lemma:lambda-kappa-7} If $\lambda\neq \emptyset$, then $d=l$ or $d=2l$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[label=(\alph*),wide]
\item Of course, $0\in \kappa$ by Assumption~\ref{lemma:gradedsets-condition-4}. And $-\kappa= \kappa$ was established in Lem\-ma~\hyperref[lemma:omegas-2]{\ref*{lemma:omegas}~\ref*{lemma:omegas-2}}. And with the choices $c_1=\circ$, $c_2=c_3=\bullet$, Assumption~\ref{lemma:gradedsets-condition-5} implies that
\begin{align*}
\kappa+\kappa=\kappa_{\circ\bullet}+\kappa_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-5}}{\subseteq} \kappa_{\circ\bullet}=\kappa.
\end{align*}
Hence, $\kappa$ is indeed a subgroup of $\mathbb{Z}$. The definition of $d$ makes $d$ a generator of $\kappa$, implying $\kappa=d\mathbb{Z}$.
\item
As $\lambda=-\lambda$ by Lem\-ma~\hyperref[lemma:omegas-1]{\ref*{lemma:omegas}~\ref*{lemma:omegas-1}}, assuming $\lambda\neq \emptyset$ ensures $\lambda\cap (\{0\}\cup\mathbb{N})\neq \emptyset$. Hence, under this assumption, $l\in \lambda$ by definition of $l$.
If we choose $c_1=c_3=\circ$ and $c_2=\bullet$ in As\-sump\-tion~\ref{lemma:gradedsets-condition-5}, it follows that
\begin{align*}
\kappa+\lambda=\kappa_{\circ\bullet}+\kappa_{\circ\circ}\overset{\ref{lemma:gradedsets-condition-5}}{\subseteq} \kappa_{\circ\circ}=\lambda.
\end{align*}
Since $l\in \lambda$, we can specialize the $\lambda$ on the left hand side of that inclusion to $l$ and then subtract $l$ on both sides. We obtain $\kappa\subseteq \lambda-l$.
\item If $\lambda= \emptyset$, there is nothing to prove. Hence, let $\lambda\neq \emptyset$, implying $l\in \lambda$ by Part~\ref{lemma:lambda-kappa-2}. Using Assumption~\ref{lemma:gradedsets-condition-5} once more, this time with the choices $c_1=c_2=\circ$ and $c_3=\bullet$, yields
\begin{align*}
\lambda-\lambda=\lambda+\lambda=\kappa_{\circ\circ}+\kappa_{\bullet\bullet}\overset{\ref{lemma:gradedsets-condition-5}}{\subseteq} \kappa_{\circ\bullet}=\kappa,
\end{align*}
where we have used $\lambda=-\lambda$ (Lem\-ma~\hyperref[lemma:omegas-1]{\ref*{lemma:omegas}~\ref*{lemma:omegas-1}}) in the first step. Specializing on the left hand side the second instance of $\lambda$ to $l$ yields $\lambda-l\subseteq \kappa$.
\item Actually, we show the contraposition. Hence, suppose $\lambda\neq \emptyset$ and $l>d$. Since $\lambda=l+d\mathbb{Z}$ by Parts~\ref{lemma:lambda-kappa-1}--\ref{lemma:lambda-kappa-3}, it then follows that $l-d\in \lambda\cap \mathbb{N}$. The definition of $l$ consequently requires $l\leq l-d$, i.e.\ $d\leq 0$. As $d\geq 0$ by definition, $d=0$ is the only possibility.
\item We prove the contraposition indirectly. As $\lambda=l+d\mathbb{Z}$ by Parts~\ref{lemma:lambda-kappa-1}--\ref{lemma:lambda-kappa-3}, supposing $l=0$ entails $\lambda=d\mathbb{Z}$. Thus, if $d\neq 0$ were true, then $\emptyset\neq d\mathbb{Z}\cap \mathbb{N} = \lambda\cap \mathbb{N}$ would yield the contradiction $0<\min(\lambda\cap \mathbb{N})=l=0$ by definition of $l$.
\item In the proof of Part~\ref{lemma:lambda-kappa-3} we saw $\lambda+\lambda\subseteq \kappa$. Specializing therein both instances of $\lambda$ on the left hand side to $l$ (which we can do due to $\lambda\neq \emptyset$ by Part~\ref{lemma:lambda-kappa-2}) yields $2l\in \kappa=d\mathbb{Z}$. It follows $2l\mathbb{Z}\subseteq d\mathbb{Z}$ as asserted.
\item
From $2l\mathbb{Z}\subseteq d\mathbb{Z}$, as shown in Part~\ref{lemma:lambda-kappa-6}, it is immediate that, if $d=0$, then $l=0=d$ as claimed. If $d\neq 0$, we know, firstly, $l\leq d$ by Part~\ref{lemma:lambda-kappa-4}, secondly, $l\neq 0$ by Part~\ref{lemma:lambda-kappa-5} and, thirdly, $2l\mathbb{Z}\subseteq d\mathbb{Z}$ by Part~\ref{lemma:lambda-kappa-6}. That is only possible if $d=l$ or $d=2l$: Indeed, if $c\in \mathbb{Z}$ is such that $2l=cd$, then $l> 0$ and $d\geq 0$ ensure $c> 0$. Moreover, $l\leq d$ implies $2l\leq 2d$, i.e., $cd\leq 2d$. We infer $c\leq 2$ by $d> 0$. Hence, $c\in \{1,2\}$ by $c>0$.\qedhere
\end{enumerate}
\end{proof}
\begin{lemma}
\label{lemma:lambda-kappa-summary}
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:lambda-kappa-summary-1} If $\lambda=\emptyset$, then $(\lambda,\kappa)=(\emptyset,d\mathbb{Z})$.
\item\label{lemma:lambda-kappa-summary-2} If $\lambda\neq \emptyset$, then $(\lambda,\kappa)$ is equal to $(l\!+\!2l\mathbb{Z},2l\mathbb{Z})$ or $(l\mathbb{Z},l\mathbb{Z})$.
\end{enumerate}
\end{lemma}
\begin{proof}
In Lem\-ma~\ref{lemma:lambda-kappa} we established that $\kappa=d\mathbb{Z}$ (Part~\ref{lemma:lambda-kappa-1}) and that $\lambda=\emptyset$ or $\lambda=l+d\mathbb{Z}$ (Parts~\ref{lemma:lambda-kappa-2} and~\ref{lemma:lambda-kappa-3}), where $d=l$ or $d=2l$ (Part~\ref{lemma:lambda-kappa-7}). In other words, we have proven that $(\lambda,\kappa)$ is of the asserted form.
\end{proof}
We can immediately relate $\sigma$ to $\kappa$.
\begin{definition}
\label{definition:k}
Define
\begin{align*}
k\coloneqq \begin{cases}
\min\left(\sigma\cap \mathbb{N}\right) &\text{if }\sigma\cap \mathbb{N}\neq \emptyset,\\
0&\text{otherwise}.
\end{cases}
\end{align*}
\end{definition}
\begin{lemma}
\label{lemma:sigma}
$\sigma=k\mathbb{Z}\subseteq d\mathbb{Z}=\kappa$.
\end{lemma}
\begin{proof}
Because $\sigma$ is a subgroup of $\mathbb{Z}$, the definition of $k$ implies $\sigma=k\mathbb{Z}$. Moreover, we know $\kappa=\kappa+\sigma$ by Lem\-ma~\hyperref[lemma:omegas-2]{\ref*{lemma:omegas}~\ref*{lemma:omegas-2}}. Hence Assumption~\ref{lemma:gradedsets-condition-4}, namely $0\in \kappa$, implies $k\mathbb{Z}=\sigma\subseteq \kappa+\sigma\subseteq \kappa=d\mathbb{Z}$.
\end{proof}
Let us now turn to the description of $\xi$.
\begin{lemma}
\label{lemma:xi}
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:xi-1} $\xi=\xi+d\mathbb{Z}$.
\item\label{lemma:xi-2} If $\lambda\neq \emptyset$, then $\xi=\xi+l\mathbb{Z}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[label=(\alph*), wide]
\item Picking $c_1=\circ$, $c_2=c_3=\bullet$, As\-sump\-tion~\ref{lemma:gradedsets-condition-7} implies the inclusion
\begin{align*}
\kappa+\xi=\kappa_{\circ\bullet}+\xi_{\circ\bullet}\overset{\ref{lemma:gradedsets-condition-7}}{\subseteq}\xi_{\circ\bullet}=\xi.
\end{align*}
As the reverse inclusion is trivially true by $0\in \kappa$ (Assumption~\ref{lemma:gradedsets-condition-4}), we have thus verified our claim $\xi=\xi+d\mathbb{Z}$ by Lem\-ma~\hyperref[lemma:lambda-kappa-1]{\ref*{lemma:lambda-kappa}~\ref*{lemma:lambda-kappa-1}}.
\item
Assumption~\ref{lemma:gradedsets-condition-7}, applied a second time, now with $c_1=c_2=c_3=\circ$, allows us to conclude
\begin{align*}
\lambda+\xi=\kappa_{\circ\circ}+\xi_{\bullet\circ}\overset{\ref{lemma:gradedsets-condition-7}}{\subseteq} \xi_{\circ\circ}=\xi.
\end{align*}
If $\lambda\neq \emptyset$, then $l\in \lambda$ by Lem\-ma~\hyperref[lemma:lambda-kappa-2]{\ref*{lemma:lambda-kappa}~\ref*{lemma:lambda-kappa-2}}. Hence, the above inclusion shows in particular $\xi+l\subseteq \xi$. Using this, induction proves $\xi+l\mathbb{N}\subseteq \xi$. Lem\-ma~\hyperref[lemma:lambda-kappa-7]{\ref*{lemma:lambda-kappa}~\ref*{lemma:lambda-kappa-7}} established that $d=l$ or $d=2l$. Either way, $\xi=\xi+d\mathbb{Z}$, as seen in Part~\ref{lemma:xi-1}, then ensures $\xi-2l\subseteq \xi$. Combining this conclusion with $\xi+l\subseteq \xi$ lets us infer $\xi-l=(\xi+l)-2l\subseteq \xi$. Again, it follows $\xi-l\mathbb{N}\subseteq \xi$ by induction. Hence, altogether we have shown $\xi+l\mathbb{Z}=(\xi-l\mathbb{N})\cup \xi\cup (\xi+l\mathbb{N})\subseteq \xi$. Of course, the converse inclusion is true as well because $0\in \mathbb{Z}$, proving $\xi=\xi+l\mathbb{Z}$ as claimed. \qedhere
\end{enumerate}
\end{proof}
In order to obtain a refined understanding of $\xi$ we need the following preparatory lemma.
\begin{lemma}
\label{lemma:chi}
Let $\chi\subseteq \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\chi=-\chi=\chi+m\mathbb{Z}$.
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:chi-1} $\chi=(\chi\cap (\{0\}\cup \dwi{m\!-\!1}))_m$.
\item\label{lemma:chi-2} $\chi\cap \dwi{m\!-\! 1}=m-(\chi\cap \dwi{m\!-\!1})$.
\item\label{lemma:chi-3} $\chi=(\chi\cap (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor}))_m$.
\item\label{lemma:chi-4} $\chi=\mathbb{Z}\backslash D_m$ for $D=(\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor})\backslash \chi$.
\end{enumerate}
\end{lemma}
\begin{proof}
The mapping $S\mapsto S_m\coloneqq (S\cup (m-S))+m\mathbb{Z}$ of subsets $S\subseteq \mathbb{Z}$ is a closure operator with respect to $\subseteq$, i.e., for all $S,T\subseteq \mathbb{Z}$ with $S\subseteq T$ we have $S\subseteq S_m$ and $S_m\subseteq T_m$ and $(S_m)_m=S_m$. In particular $S=S_m$ if and only if $S=-S=S+m\mathbb{Z}$.
\begin{enumerate}[label=(\alph*), wide]
\item The assumption $\chi=-\chi=\chi+m\mathbb{Z}$ implies $\chi=\chi_m$. Hence, $\chi=\chi_m\supseteq (\chi\cap (\{0\}\cup \dwi{m\!-\!1}))_m$ is clear by monotonicity of $S\mapsto S_m$. We show the converse: If $x\in \chi$, we find $x'\in \{0\}\cup\dwi{m\!-\!1}$ such that $x'-x\in m\mathbb{Z}$. Consequently, $x'\in x+m\mathbb{Z}\subseteq \chi+m\mathbb{Z}\subseteq \chi$ by assumption. We conclude $x\in x'+m\mathbb{Z}\subseteq (\chi\cap (\{0\}\cup \dwi{m\!-\!1})) +m\mathbb{Z}\subseteq (\chi\cap (\{0\}\cup \dwi{m\!-\!1}))_m$, which is what we needed to show.
\item We further deduce from $\chi=-\chi=\chi+m\mathbb{Z}$ that $m-\chi\subseteq \chi$. Naturally, $m-\left(\chi\cap \dwi{m\!-\!1}\right)\subseteq m-\dwi{m\!-\!1} =\dwi{m\!-\!1}$. Combining this with $m-\left(\chi\cap \dwi{m\!-\!1}\right)\subseteq m-\chi\subseteq \chi$ yields $m-\left(\chi\cap\dwi{m\!-\!1}\right)\subseteq \chi\cap \dwi{m\!-\!1}$. We conclude $\chi\cap\dwi{m\!-\!1}=m-\left(m-\left(\chi\cap\dwi{m\!-\!1}\right)\right)\subseteq m-\left(\chi\cap\dwi{m\!-\!1}\right)$, which proves one inclusion.
\par
Now, the converse. From $\chi=-\chi=\chi+m\mathbb{Z}$ we can infer $m-\chi=-(m-\chi)=(m-\chi)+m\mathbb{Z}$. In consequence we can apply the inclusion we just proved to the set $m-\chi$ in the role of $\chi$. Since $m-\dwi{m\!-\!1}=\dwi{m\!-\!1}$, the resulting inclusion $(m-\chi)\cap \dwi{m\!-\!1}\subseteq m-\left((m-\chi)\cap\dwi{m\!-\!1}\right)$ actually spells $m-(\chi\cap \dwi{m\!-\!1})\subseteq \chi\cap \dwi{m\!-\!1}$. That is just what we had to show.
\item Due to the monotonicity and idempotency of the mapping $S\mapsto S_m$, it suffices by Part~\ref{lemma:chi-1} to prove $\chi\cap (\{0\}\cup \dwi{m\!-\!1})\subseteq (\chi\cap (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor}))_m$. Let $x\in \chi\cap (\{0\}\cup \dwi{m\!-\!1})$ be arbitrary. If $x\leq \lfloor\frac{m}{2}\rfloor$, then, naturally, $x \in \chi\cup (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor})\subseteq (\chi\cap (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor}))_m$. Hence, we can assume $x>\lfloor\frac{m}{2}\rfloor$. By Part~\ref{lemma:chi-2} we know $m-x\in \chi$. By assumption, $m-x<m-\lfloor\frac{m}{2}\rfloor$. If $m$ is even, then this inequality says $m-x<m-\frac{m}{2}=\frac{m}{2}=\lfloor\frac{m}{2}\rfloor$. Should $m$ be odd instead, it means $m-x<m-\frac{m-1}{2}=\frac{m+1}{2}$, which implies $m-x\leq \frac{m+1}{2}-1=\frac{m-1}{2}=\lfloor\frac{m}{2}\rfloor$. Thus, $m-x\leq \lfloor\frac{m}{2}\rfloor$ in all cases. Hence we have shown $m-x\in \chi\cap (\{0\}\cup \dwi{\lfloor \frac{m}{2}\rfloor})$. It follows $x=m-(m-x)\in m-(\chi\cap (\{0\}\cup \dwi{\lfloor \frac{m}{2}\rfloor}))\subseteq (\chi(\{0\}\cup \dwi{\lfloor \frac{m}{2}\rfloor}))_m$. That is what we needed to see.
\item The assumption $\chi=-\chi=\chi+m\mathbb{Z}$ implies $\mathbb{Z}\backslash \chi=-(\mathbb{Z}\backslash \chi )=(\mathbb{Z}\backslash \chi)+m\mathbb{Z}$. Hence, we can apply Part~\ref{lemma:chi-3} to the set $\mathbb{Z}\backslash \chi$ in the role of $\chi$ and obtain $\mathbb{Z}\backslash \chi=((\mathbb{Z}\backslash \chi)\cap (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor}))_m$. Since $(\mathbb{Z}\backslash \chi)\cap (\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor})=(\{0\}\cup \dwi{\lfloor\frac{m}{2}\rfloor})\backslash \chi=D$ we have shown $\mathbb{Z}\backslash \chi=D_m$. It follows $\chi=\mathbb{Z}\backslash D_m$ as claimed. \qedhere
\end{enumerate}
\end{proof}
\begin{lemma}
\label{lemma:xi-summary}
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:xi-summary-1} If $d=0$, then $\xi=\mathbb{Z}\backslash E_0$ for $E=(\{0\}\cup\mathbb{N})\backslash \xi$.
\item\label{lemma:xi-summary-2} If $d\geq 1$ and $\lambda\neq \emptyset$, then $\xi=\mathbb{Z}\backslash D_l$ for $D=(\{0\}\cup \dwi{\lfloor\frac{l}{2}\rfloor})\backslash \xi$.
\item\label{lemma:xi-summary-3} If $d\geq 1$ and $\lambda= \emptyset$, then $\xi=\mathbb{Z}\backslash D_d$ for $D=(\{0\}\cup \dwi{\lfloor\frac{d}{2}\rfloor})\backslash \xi$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[label=(\alph*),wide]
\item
The defining equations $E=(\{0\}\cup \mathbb{N})\backslash \xi$ and $E_0=E\cup (-E)$ imply $E_0=((\{0\}\cup \mathbb{N}) \backslash \xi)\cup ((-(\{0\}\cup \mathbb{N}))\backslash (-\xi))$. Hence, $\xi=-\xi$ (by Lem\-ma~\ref{lemma:omegas}) shows $E_0=\mathbb{Z}\backslash \xi$ and thus the claim $\xi=\mathbb{Z}\backslash E_0$.
\item Because $\lambda \neq \emptyset$, Lem\-ma~\hyperref[lemma:lambda-kappa-7]{\ref*{lemma:lambda-kappa}~\ref*{lemma:lambda-kappa-7}} guarantees $d=l$ or $d=2l$. Hence, the assumption $d\geq 1$ implies $l\geq 1$. Moreover, Lem\-ma~\hyperref[lemma:xi-2]{\ref*{lemma:xi}~\ref*{lemma:xi-2}} assures us that $\xi=\xi+l\mathbb{Z}$. And, we already know $\xi=-\xi$ by Lem\-ma~\ref{lemma:omegas}. Hence, Lem\-ma~\hyperref[lemma:chi-4]{\ref*{lemma:chi}~\ref*{lemma:chi-4}} yields the claim.
\item Still, $\xi=-\xi$, of course. And $\xi=\xi+d\mathbb{Z}$ by Lem\-ma~\hyperref[lemma:xi-1]{\ref*{lemma:xi}~\ref*{lemma:xi-1}} as $d\geq 1$. Thus, once more, Lem\-ma~\hyperref[lemma:chi-4]{\ref*{lemma:chi}~\ref*{lemma:chi-4}} proves the claim. \qedhere
\end{enumerate}
\end{proof}
In conclusion we have shown the following auxiliary result.
\begin{lemma}[Arithmetic Lem\-ma]
\label{lemma:arithmetic}
If the nine sets of integers $\sigma$ and $\kappa_{c_1,c_2}$, $\xi_{c_1,c_2}$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axioms~\ref{axioms:arithmetic}, then
\begin{align*}
\kappa_{\circ\circ}=\kappa_{\bullet\bullet}=\colon \lambda,\quad \kappa_{\circ\bullet}=\kappa_{\bullet\circ}=\colon \kappa \quad\text{and}\quad \xi_{\circ\circ}=\xi_{\bullet\bullet}=\xi_{\circ\bullet}=\xi_{\bullet\circ}=\colon \xi
\end{align*}
and there exist $u\in\{0\}\cup \mathbb{N}$, $m\in \mathbb{N}$, $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$ and $E\subseteq \{0\}\cup \mathbb{N}$ such that the tuple $(\sigma,\lambda,\kappa,\xi)$ is given by one of the following:
\begin{align*}
\begin{matrix}
\sigma& \lambda&\kappa& \xi \\ \hline \\[-0.85em]
um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
2um\mathbb{Z} & m\!+\!2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0
\end{matrix}
\end{align*}
\end{lemma}
\begin{proof}
That $\lambda$, $\kappa$ and $\xi$ are well-defined was shown in Lem\-mata~\ref{lemma:omegas} and~\ref{lemma:xi-well-defined}. Hence, we can let $k$, $d$ and $l$ be as in Definitions~\ref{definition:k} and \ref{definition:d-l}. We distinguish five cases in total.
\par
\textbf{Case~1:} First, suppose that $\lambda=\emptyset$. Then, $\kappa=d\mathbb{Z}$. By Lem\-ma~\hyperref[lemma:lambda-kappa-summary-1]{\ref*{lemma:lambda-kappa-summary}~\ref*{lemma:lambda-kappa-summary-1}}. There are now two possibilities depending on the value of $d\in\{0\}\cup\mathbb{N}$.
\par
\textbf{Case~1.1:} If $d=0$, which is to say $\kappa=\{0\}$, then Lem\-ma~\hyperref[lemma:xi-summary-1]{\ref*{lemma:xi-summary}~\ref*{lemma:xi-summary-1}} yields $\xi=\mathbb{Z}\backslash E_0$ for $E\coloneqq(\{0\}\cup \mathbb{N})\backslash \xi$. And Lem\-ma~\ref{lemma:sigma} proves $\sigma=k\mathbb{Z}\subseteq d\mathbb{Z}=\{0\}$, implying $k=0$ and thus $\sigma=\{0\}$. As, naturally, $E\subseteq \{0\}\cup\mathbb{N}$, the tuple $(\sigma,\lambda,\kappa,\xi)$ is indeed as claimed in the fifth row of the table.
\par
\textbf{Case~1.2:} Should $d\geq 1$ on the other hand, then by Lem\-ma~\hyperref[lemma:xi-summary-3]{\ref*{lemma:xi-summary}~\ref*{lemma:xi-summary-3}} we infer $\xi=\mathbb{Z}\backslash D_d$ for $D\coloneqq(\{0\}\cup \dwi{\lfloor\frac{d}{2}\rfloor})\backslash \xi$. Since $\sigma=k\mathbb{Z}\subseteq d\mathbb{Z}$ by Lem\-ma~\ref{lemma:sigma}, if we put $u\coloneqq\frac{k}{d}$, then $\sigma=ud\mathbb{Z}$. Recognizing $D\subseteq \{0\}\cup \dwi{\lfloor\frac{d}{2}\rfloor}$ and defining $m\coloneqq d$ thus proves that $(\sigma,\lambda,\kappa,\xi)$ is as asserted by the third row of the table.
\par
\textbf{Case~2:} Now, let $\lambda\neq \emptyset$ instead. Then, $(\lambda,\kappa)=(l\!+\!2l\mathbb{Z},2l\mathbb{Z})$ or $(\lambda,\kappa)=(l\mathbb{Z},l\mathbb{Z})$ by Lem\-ma~\hyperref[lemma:lambda-kappa-summary-1]{\ref*{lemma:lambda-kappa-summary}~\ref*{lemma:lambda-kappa-summary-1}}. Respectively, $d=2l$ or $d=l$. We now distinguish two cases based on the value of $l\in\{0\}\cup \mathbb{N}$.
\par
\textbf{Case~2.1:} Assuming $l=0$ lets us conclude $l\mathbb{Z}=2l\mathbb{Z}=l\!+\!2l\mathbb{Z}=\{0\}$, which implies $(\lambda,\kappa)=(\{0\},\{0\})$. Lem\-ma~\ref{lemma:sigma} gives $\sigma=k\mathbb{Z}\subseteq \kappa=\{0\}$ and thus $k=0$ and $\sigma=\{0\}$. Because $d=l=2l=0$ we can infer
$\xi=\mathbb{Z}\backslash E_0$ for $E\coloneqq(\{0\}\cup \mathbb{N})\backslash \xi$ by Lem\-ma~\hyperref[lemma:xi-summary-1]{\ref*{lemma:xi-summary}~\ref*{lemma:xi-summary-1}}. As $E\subseteq \{0\}\cup\mathbb{N}$, the tuple $(\sigma,\lambda,\kappa,\xi)$ is hence given by the fourth row of the table.
\par
\textbf{Case~2.2:} Finally, let $l\geq 0$. Then, also $d\geq 0$, no matter whether $d=l$ or $d=2l$. In conclusion, $\xi=\mathbb{Z}\backslash D_l$ for $D\coloneqq(\{0\}\cup \dwi{\lfloor\frac{l}{2}\rfloor})\backslash \xi$ by Lem\-ma~\hyperref[lemma:xi-summary-3]{\ref*{lemma:xi-summary}~\ref*{lemma:xi-summary-3}}.
\par
\textbf{Case~2.2.1:} If $(\lambda,\kappa)=(l\!+\!2l\mathbb{Z},2l\mathbb{Z})$, i.e., $d=2l$, then the implication $\sigma=k\mathbb{Z}\subseteq d\mathbb{Z}=2l\mathbb{Z}$ of Lem\-ma~\ref{lemma:sigma} lets us define $u\in\{0\}\cup \mathbb{N}$ by $u\coloneqq \frac{k}{2l}$ and obtain $\sigma=2ul\mathbb{Z}$. Hence, choosing $m\coloneqq l$ proves that $(\sigma,\lambda,\kappa,\xi)$ fits the second row of the table.
\par
\textbf{Case~2.2.2:} If instead, $(\lambda,\kappa)=(l\mathbb{Z},l\mathbb{Z})$, i.e., $d=l$, then Lem\-ma~\ref{lemma:sigma} yields $\sigma=k\mathbb{Z}\subseteq d\mathbb{Z}=l\mathbb{Z}$, thus permitting us to define $u\in\{0\}\cup \mathbb{N}$ by $u\coloneqq \frac{k}{l}$ and obtain $\sigma=ul\mathbb{Z}$. The choice $m\coloneqq l$ hence shows $(\sigma,\lambda,\kappa,\xi)$ to be given by the first row.
\end{proof}
As mentioned before, our goal will be to show (Sec\-tion~\ref{section:verifying-the-axioms}) that for every non-hy\-per\-oc\-ta\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ the tuple $(\Sigma,L,K,X)(\mathcal C)$ is of the form given in the table of the \hyperref[lemma:arithmetic]{Arithmetic Lem\-ma}.
\subsection{Reduction to Singleton and Pair Blocks} Let us return to categories of partitions. To elucidate the ranges of $K$, $L$ and $X$ over $\mathsf{PCat}^{\circ\bullet}_{\mathrm{NHO}}$ and central relations between $\Sigma(\mathcal C)$, $K(\mathcal C)$, $L(\mathcal C)$ and $X(\mathcal C)$ for non-hy\-per\-octa\-he\-dral categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, we must consider certain decompositions of $K$, $L$ and $X$ according to leg colors.
\begin{definition}
Let $\mathcal S\subseteq \mc{P}^{\circ\bullet}$ and $c_1,c_2\in\{\circ,\bullet\}$ be abitrary. Then, define
\begin{align}
K_{c_1,c_2}(\mathcal S)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2)\mid &\;p\in \mathcal S, \, B\text{ block of }p,\, \alpha_1,\alpha_2\in B,\, \alpha_1\neq \alpha_2,\tag{a}\\
&\; ]\alpha_1,\alpha_2[_p\cap B=\emptyset,\, \forall i=1,2: \alpha_i\text{ of normalized color }c_i\},\notag\\
X_{c_1,c_2}(\mathcal S)\coloneqq \{\,\delta_p(\alpha_1,\alpha_2) \mid &\; p\in \mathcal S,\, B_1,B_2\text{ blocks of }p,\, B_1 \text{ and }B_2 \text{ cross}, \tag{b}\\
&\; \alpha_1\in B_1,\,\alpha_2\in B_2,\, \forall i=1,2: \alpha_i\text{ of normalized color }c_i\}.\notag
\end{align}
\end{definition}
$L$, $K$ and $X$ can then be written as, where the union occurs pointwise,
\begin{IEEEeqnarray*}{rCl}
L=\bigcup_{\substack{c_1,c_2\in \{\circ,\bullet\}\\c_1= c_2}} K_{c_1,c_2},\quad K=\bigcup_{\substack{c_1,c_2\in \{\circ,\bullet\}\\c_1\neq c_2}} K_{c_1,c_2},\quad\text{and}\quad X=\bigcup_{c_1,c_2\in \{\circ,\bullet\}}X_{c_1,c_2}.
\end{IEEEeqnarray*}
\par
Recall that $\mc{P}^{\circ\bullet}_{\leq 2}$ denotes the set of all partitions with block sizes one or two and that it is a category (see \cite[Lem\-ma~4.4~(a)]{MWNHO1}). By the next lemma we may always restrict to partitions in $\mc{P}^{\circ\bullet}_{\leq 2}$ when studying $K_{c_1,c_2}$ and $X_{c_1,c_2}$. This is trivial in cases~$\mathcal O$ and~$\mathcal B$, while for case~$\mathcal S$ this basically follows from Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}}.
\begin{lemma}
\label{lemma:simplification-k-x}
For all non-hy\-per\-octa\-he\-dral categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ and $c_1,c_2\in \{\circ,\bullet\}$:
\begin{enumerate}[label=(\alph*)]
\item $K_{c_1,c_2}(\mathcal C)=K_{c_1,c_2}(\mathcal C\cap\mc{P}^{\circ\bullet}_{\leq 2})$.
\item $X_{c_1,c_2}(\mathcal C)=X_{c_1,c_2}(\mathcal C\cap\mc{P}^{\circ\bullet}_{\leq 2})$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[label=(\alph*),wide]
\item If $\mathcal C$ is case~$\mathcal O$ or case~$\mathcal B$, i.e., if $\mathcal C\subseteq \mc{P}^{\circ\bullet}_{\leq 2}$ by Proposition~\ref{proposition:result-F}, there is nothing to show. Hence, suppose that $\mathcal C$ is case~$\mathcal S$ and let $c_1,c_2\in \{\circ,\bullet\}$. We only need to prove $K_{c_1,c_2}(\mathcal C)\subseteq K_{c_1,c_2}(\mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2})$. Let $\alpha_1$ and $\alpha_2$ with $\alpha_1\neq \alpha_2$ be points in $p\in \mathcal C$ such that $\alpha_i$ is of normalized color $c_i$ for every $i\in \{1,2\}$ and such that $\alpha_1,\alpha_2\in B$ and $]\alpha_1,\alpha_2[_p\cap B=\emptyset$ for some block $B$ in $p$. Because $\mathcal C$ is case~$\mathcal S$, by Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}} we do not violate the assumption $p\in \mathcal C$ by assuming that every block other than $B$ is a singleton. In the same way we can assume that $\alpha_1$ and $\alpha_2$ are the only legs of $B$. None of these assumptions affect $\delta_p(\alpha_1,\alpha_2)$ or the normalized colors of $\alpha_1$ or $\alpha_2$. As they ensure $p\in \mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2}$ though, we have shown $\delta_p(\alpha_1,\alpha_2)\in K_{c_1,c_2}(\mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2})$, which is what we needed to see.
\item Again, all that we need to prove is that $X_{c_1,c_2}(\mathcal C)\subseteq X_{c_1,c_2}(\mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2})$ if $\mathcal C$ is case~$\mathcal S$ and if $c_1,c_2\in \{\circ,\bullet\}$. Let the points $\alpha_1$ of normalized color $c_1$ and $\alpha_2$ of normalized color $c_2$ in $p\in \mathcal C$ belong to the blocks $B_1$ and $B_2$, respectively, and suppose that $B_1$ and $B_2$ cross. Because $\mathcal C$ is case~$\mathcal S$ we can, by Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}}, assume that all other blocks of $p$ besides $B_1$ and $B_2$ are singletons. Now the only thing standing in the way of $p\in \mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2}$ is the possibility of at least one of $B_1$ and $B_2$ having more than two legs. We would like to assume that $B_1$ and $B_2$ have only two legs each and still maintain all the other assumptions including $\alpha_1\in B_1$ and $\alpha_2\in B_2$ and, of course, not alter $\delta_p(\alpha_1,\alpha_2)$.
By Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}}, we can always remove surplus legs of $B_1$ and $B_2$. But it is not immediately clear that we can remove legs without affecting the other assumptions. A priori, the crossing between $B_1$ and $B_2$ only implies that we can find points $\beta_1,\gamma_1\in B_1$ and $\beta_2,\gamma_2\in B_2$ such that $(\beta_1,\beta_2,\gamma_1,\gamma_2)$ is ordered in $p$. If now $\alpha_1\in \{\beta_1,\gamma_1\}$ and $\alpha_2\in \{\beta_2,\gamma_2\}$, then we can certainly remove all legs except $\{\beta_i,\gamma_i\}$ from $B_i$ for all $i\in \{1,2\}$ and still maintain the other assumptions. In fact, we can do so in general as well:
\par
Let us only consider the \enquote{worst case} that $\alpha_1\notin \{\beta_1,\gamma_1\}$ and $\alpha_2\notin\{\beta_2,\gamma_2\}$. There are $20$ possible arrangements of the points $\{\alpha_1,\beta_1,\gamma_1,\alpha_2,\beta_2,\gamma_2\}$ relative to each other with respect to the cyclic order respecting that $(\beta_1,\beta_2,\gamma_1,\gamma_2)$ is ordered.
\begin{align*}
\begin{array}{ c | c | c | c }
\overset{\downarrow}{\alpha_1}\beta_1\beta_2\gamma_1\gamma_2 & \beta_1\overset{\downarrow}{\alpha_1}\beta_2\gamma_1\gamma_2 &
\beta_1\beta_2\overset{\downarrow}{\alpha_1}\gamma_1\gamma_2 &
\beta_1\beta_2\gamma_1\overset{\downarrow}{\alpha_1}\gamma_2\\ [0.1em]\hline\Tstrut
\underline{\alpha_2\alpha_1}\beta_1\underline{\beta_2\gamma_1}\gamma_2 & \underline{\alpha_2}\beta_1\underline{\alpha_1\beta_2\gamma_1}\gamma_2 &
\underline{\alpha_2\beta_1\beta_2\alpha_1}\gamma_1\gamma_2 &
\underline{\alpha_2\beta_1\beta_2}\gamma_1\underline{\alpha_1}\gamma_2\\
\underline{\alpha_1\alpha_2\beta_1\beta_2}\gamma_1\gamma_2 &
\beta_1\underline{\alpha_2\alpha_1\beta_2\gamma_1}\gamma_2 &
\underline{\beta_1\alpha_2}\beta_2\underline{\alpha_1}\gamma_1\underline{\gamma_2} &
\underline{\beta_1\alpha_2}\beta_2\gamma_1\underline{\alpha_1\gamma_2} \\[0.1em]
\underline{\alpha_1}\beta_1\underline{\alpha_2}\beta_2\underline{\gamma_1\gamma_2} &
\beta_1\underline{\alpha_1\alpha_2}\beta_2\underline{\gamma_1\gamma_2} &
\underline{\beta_1}\beta_2\underline{\alpha_2\alpha_1}\gamma_1\underline{\gamma_2} &
\underline{\beta_1}\beta_2\underline{\alpha_2}\gamma_1\underline{\alpha_1\gamma_2} \\[0.1em]
\underline{\alpha_1}\beta_1\beta_2\underline{\alpha_2\gamma_1\gamma_2} &
\beta_1\underline{\alpha_1}\beta_2\underline{\alpha_2\gamma_1\gamma_2} &
\beta_1\beta_2\underline{\alpha_1\alpha_2\gamma_1\gamma_2} &
\underline{\beta_1}\beta_2\gamma_1\underline{\alpha_2\alpha_1\gamma_2} \\[0.1em]
\underline{\alpha_1}\beta_1\underline{\beta_2\gamma_1\alpha_2}\gamma_2 &
\beta_1\underline{\alpha_1\beta_2\gamma_1\alpha_2}\gamma_2 &
\underline{\beta_1\beta_2\alpha_1}\gamma_1\underline{\alpha_2}\gamma_2 &
\underline{\beta_1\beta_2}\gamma_1\underline{\alpha_1\alpha_2}\gamma_2
\end{array}
\end{align*}
We remove all legs of $B_1$ and $B_2$ except for the underlined ones. Then the above table shows that we can always turn $B_1$ and $B_2$ into crossing pair blocks containing $\alpha_1$ and $\alpha_2$, respectively. That concludes the proof.\qedhere
\end{enumerate}
\end{proof}
\subsection{Verifying the Axioms}
\label{section:verifying-the-axioms}
We want to apply the \hyperref[lemma:arithmetic]{Arithmetic Lem\-ma~\ref*{lemma:arithmetic}} to the sets $\sigma\coloneqq \Sigma(\mathcal C)$, $\kappa_{c_1,c_2}\coloneqq K_{c_1,c_2}(\mathcal C)$ and $\xi_{c_1,c_2}\coloneqq X_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ and non-hy\-per\-octa\-he\-dral categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$. In order to be able to do so, we, of course, need to show that these sets actually satisfy the prerequisite Axioms~\ref{axioms:arithmetic}. Proving that will crucially utilize the reduction to singleton and pair blocks from Lem\-ma~\ref{lemma:simplification-k-x}.
\begin{lemma}
\label{lemma:verifying-axioms-1}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the set $\sigma\coloneqq \Sigma(\mathcal C)$ satisfies Axiom~\ref{lemma:gradedsets-condition-0} of \ref{axioms:arithmetic}: $\sigma$ is a subgroup of $\mathbb{Z}$.
\end{lemma}
\begin{proof}
That was shown in Proposition~\ref{proposition:result-S}.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-2}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\kappa_{c_1,c_2}\coloneqq K_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axioms~\ref{lemma:gradedsets-condition-1}--\ref{lemma:gradedsets-condition-2} of \ref{axioms:arithmetic}:
\begin{IEEEeqnarray*}{rCCrCCrC}
\text{(ii)}\hspace{0.5em}& \kappa_{c_1,c_2}+\sigma\subseteq \kappa_{c_1,c_2}, &\hspace{1.5em} &\text{(iii)}\hspace{0.5em}& \kappa_{c_1,c_2}\subseteq - \kappa_{\overline{c_2},\overline{c_1}}, &\hspace{1.5em} &\text{(iv)}\hspace{0.5em}& \kappa_{c_1,c_2}\subseteq - \kappa_{c_2,c_1}+\sigma
\end{IEEEeqnarray*}
for all $c_1,c_2\in\{\circ,\bullet\}$.
\end{lemma}
\begin{proof}
Let $c_1,c_2\in \{\circ,\bullet\}$ be arbitrary and let $\alpha_1$ and $\alpha_2$ be distinct points of the same block $B$ in $p\in \mathcal C$ such that $]\alpha_1,\alpha_2[_p\cap B=\emptyset$ and such that $\alpha_i$ has normalized color $c_i$ for every $i\in \{1,2\}$. In other words, let $\delta_{p}(\alpha_1,\alpha_2)$ be a generic element of $K_{c_1,c_2}(\mathcal C)=\kappa_{c_1,c_2}$.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-1}:} Let $q\in \mathcal C$ be arbitrary. None of the assumptions about $p$, $\alpha_1$, $\alpha_2$ and $\delta_p(\alpha_2,\alpha_2)$ are impacted by assuming that $p$ is rotated in such a way that $\alpha_1$ is the rightmost lower point of $p$. Then, $B$ is a block of $p\otimes q\in \mathcal C$ as well and $]\alpha_1,\alpha_2[_{p\otimes q}\cap B=\emptyset$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{3*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
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%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_2}$};
%
\draw[sstr] (l1) --++(0,{0.5*2*\dy}) -| (u1);
\draw[dashed] ($(u1)+(0,{-0.5*2*\dy})$) -- ++ ({-1.25*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [above ={\scp*0.3cm} of u1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) -- ++ ({-2*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0+0*\scp*1cm-\scp*0.4cm}) -- ++ ({-0*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{7*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
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\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
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\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
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%
%
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%
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%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({3*\scp*1cm+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm}) -- cycle;
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
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%
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\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0+0*\scp*1cm-\scp*0.4cm}) -- ++ ({-3*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0+0*\scp*1cm+\scp*0.4cm});
%
\node at ({-1.5*\scp*1cm},{0.5*2*\dy}) {$p\otimes q$};
%
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\end{tikzpicture}
\end{mycenter}
\noindent
Now, because all points stemming from $q$ lie within $]\alpha_1,\alpha_2[_{p\otimes q}$, \[\delta_{p\otimes q}(\alpha_1,\alpha_2)=\delta_p(\alpha_1,\alpha_2)+\Sigma(q).\]
That proves $\delta_{p}(\alpha_1,\alpha_2)+\Sigma(q)\in K_{c_1,c_2}(\mathcal C)=\kappa_{c_1,c_2}$, which is what we needed to see.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-3}:} The verticolor reflection $\tilde p$ of $p$ belongs to $\mathcal C$. The set $]\alpha_1,\alpha_2]_p$ in $p$ is mapped by the reflection $\rho$ to the set $[\rho(\alpha_2),\rho(\alpha_1)[_{\tilde p}$ in $\tilde p$. As the operation of ver\-ti\-col\-or re\-flec\-tion inverts normalized colors, $\sigma_{p}(S)=-\sigma_{\tilde p}(\rho(S))$ for any set $S$ of points in $p$.
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%
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%
%
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%
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\draw [draw=gray, pattern = north east lines, pattern color = lightgray]
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
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%
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({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) -- ++ ({-1*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) -- ++ ({4*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
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%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$c_1$};
%
\draw[sstr] (u1) --++(0,{-0.5*2*\dy}) -| (l1);
\draw[dashed] ($(u1)+(0,{-0.5*2*\dy})$) -- ++ ({-1.25*\scp*1cm},0);
%
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray]
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-3*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\rho(\alpha_1)$};
\node [below ={\scp*0.3cm} of l1] {$\rho(\alpha_2)$};
%
\draw [draw=darkgray, densely dashed]
({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) -- ++ ({-2*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) -- ++ ({-4*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$\tilde p$};
\end{tikzpicture}
\end{mycenter}
Using the case distinction free formula for $\delta_p(\alpha_1,\alpha_2)$ given in the proof of \cite[Lem\-ma~3.1~(b)]{MWNHO1}, we thus compute
\begin{align*}
\delta_{p}(\alpha_1,\alpha_2)&=\sigma_{p}(]\alpha_1,\alpha_2]_{ p})+{\textstyle\frac{1}{2}}(\sigma_{ p}(\alpha_1)-\sigma_{p}(\alpha_2)) \\
&=-\sigma_{\tilde p}([\rho(\alpha_2),\rho(\alpha_1)[_{\tilde p})-{\textstyle\frac{1}{2}}(\sigma_{\tilde p}(\rho(\alpha_1))-\sigma_{\tilde p}(\rho(\alpha_2))) \\
&=-\sigma_{\tilde p}(]\rho(\alpha_2),\rho(\alpha_1)]_{\tilde p})-\sigma_{\tilde p}(\rho(\alpha_2))+\sigma_{\tilde p}(\rho (\alpha_1)) \\
&\phantom{{}={}}-{\textstyle\frac{1}{2}}(\sigma_{\tilde p}(\rho(\alpha_1))-\sigma_{\tilde p}(\rho(\alpha_2))) \\
&=-\sigma_{\tilde p}(]\rho(\alpha_2),\rho(\alpha_1)]_{\tilde p})-{\textstyle\frac{1}{2}}(\sigma_{\tilde p}(\rho (\alpha_2))-\sigma_{\tilde p}(\rho(\alpha_1))) \\
&=-\delta_{\tilde p}(\rho(\alpha_2),\rho(\alpha_1)).
\end{align*}
Because, for every $i\in \{1,2\}$, the point $\rho(\alpha_i)$ has normalized color $\overline{c_i}$ in $\tilde p$ and because $\rho(B)$ is a block of $\tilde p$ with $]\rho(\alpha_2),\rho(\alpha_1)[_{\tilde p}\cap \rho(B)=\emptyset$, we conclude $\delta_p(\alpha_1,\alpha_2)\in -K_{\overline{c_2},\overline{c_1}}(\mathcal C)=-\kappa_{\overline{c_2},\overline{c_1}}$. And that is what we had to show.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-2}:} So far, we have not made use of Lem\-ma~\ref{lemma:simplification-k-x}. Now, though, we employ it to additionally assume $p\in \mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2}$. In particular, then, $B= \{\alpha_1,\alpha_2\}$ is a pair block. Consequently, not only $]\alpha_1,\alpha_2[_p\cap B=\emptyset$ but also $]\alpha_2,\alpha_1[_p\cap B=\emptyset$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+4*\scp*1cm},{0+1*2*\dy}) {$\overline {c_1}$};
%
\draw[sstr] (u1) --++(0,{-0.5*2*\dy}) -| (l1);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray]
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) -- ++ ({2*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) -- ++ ({-1*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) -- ++ ({4*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+4*\scp*1cm},{0+1*2*\dy}) {$\overline {c_1}$};
%
\draw[sstr] (u1) --++(0,{-0.5*2*\dy}) -| (l1);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray]
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed]
({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) -- ++ ({-2*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) -- ++ ({-1*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\end{mycenter}
\noindent
By \cite[Lem\-ma~2.1~(b)]{MWNHO1} we infer \[\delta_p(\alpha_1,\alpha_2)= -\delta_{p}(\alpha_2,\alpha_1)\mod \Sigma(p).\] As $\Sigma(p)\in \Sigma(\mathcal C)$, it follows $\delta_p(\alpha_1,\alpha_2)\in -K_{c_2,c_1}(\mathcal C)+\Sigma(\mathcal C)=-\kappa_{c_2,c_1}+\sigma$, which is what we wanted to see.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-3}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\xi_{c_1,c_2}\coloneqq X_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axioms~\ref{lemma:gradedsets-condition-1}--\ref{lemma:gradedsets-condition-2} of \ref{axioms:arithmetic}:
\begin{IEEEeqnarray*}{rCCrCCrC}
\text{(ii)}\hspace{0.5em}& \xi_{c_1,c_2}+\sigma\subseteq \xi_{c_1,c_2}, &\hspace{1.5em} &\text{(iii)}\hspace{0.5em}& \xi_{c_1,c_2}\subseteq - \xi_{\overline{c_2},\overline{c_1}}, &\hspace{1.5em} &\text{(iv)}\hspace{0.5em}& \xi_{c_1,c_2}\subseteq - \xi_{c_2,c_1}+\sigma
\end{IEEEeqnarray*}
for all $c_1,c_2\in\{\circ,\bullet\}$.
\end{lemma}
\begin{proof}
The proof is similar to that of Lem\-ma~\ref{lemma:verifying-axioms-2}.
Let $c_1,c_2\in \{\circ,\bullet\}$, let $B_1$ and $B_2$ be crossing blocks of $p\in \mathcal C$ and let $\alpha_1\in B_1$ and $\alpha_2\in B_2$ have normalized colors $c_1$ and $c_2$, respectively. That makes $\delta_{p}(\alpha_1,\alpha_2)$ a generic element of $X_{c_1,c_2}(\mathcal C)=\xi_{c_1,c_2}$.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-1}:} Just like in the proof of Lem\-ma~\ref{lemma:verifying-axioms-2}, we can assume that $\alpha_1$ is the rightmost lower point. Given arbitrary $q\in \mathcal C$, the sets $B_1$ and $B_2$ are crossing blocks of $p\otimes q$ as well,
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+6*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (l2) at ({0+2*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l1);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l2);
\draw[dashed] ($(l2)+ (0,{2*\dy / 3})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(l1)+ (0,{2*\dy *2/3})$) --++ ({0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({5*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({6*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l2] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({2*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-0*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({-\gx},{2*\dy-\scp*0.4cm}) -- ({\txu+\gx},{2*\dy-\scp*0.4cm}) ({\txu+\gx},{2*\dy+\scp*0.4cm}) -- ({-\gx},{2*\dy+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\quad$\to$\quad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{9*\scp*1cm}
\def2*\dx{9*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+6*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (l2) at ({0+2*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l1);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l2);
\draw[dashed] ($(l2)+ (0,{2*\dy / 3})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(l1)+ (0,{2*\dy *2/3})$) --++ ({0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({5*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({6*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l2] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({2*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({-\gx},{2*\dy-\scp*0.4cm}) -- ({\txu+\gx},{2*\dy-\scp*0.4cm}) ({\txu+\gx},{2*\dy+\scp*0.4cm}) -- ({-\gx},{2*\dy+\scp*0.4cm});
%
\node at ({-1.5*\scp*1cm},{0.5*2*\dy}) {$p\otimes q$};
%
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\end{tikzpicture}
\end{mycenter}
\noindent
which proves \[\delta_{p}(\alpha_1,\alpha_2)+\Sigma(q)=\delta_{p\otimes q}(\alpha_1,\alpha_2)\in X_{c_1,c_2}(\mathcal C)=\xi_{c_1,c_2}.\]
Thus, $\xi_{c_1,c_2}+\sigma\subseteq \xi_{c_1,c_2}$ as claimed.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-3}:} Likewise, the sets $\rho(B_1)$ and $\rho(B_2)$ are still crossing blocks in $\tilde p\in \mathcal C$. There, $\alpha_i$ has normalized color $\overline{c_i}$ for every $i\in \{1,2\}$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{5*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+1*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (l2) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node[vp] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {$\overline {c_1}$};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l2);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l1);
\draw[dashed] ($(l1)+ (0,{2*\dy / 3})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(l2)+ (0,{2*\dy *2/3})$) --++ ({0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({3*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({2*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{5*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[cc] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node[vp] (u1) at ({0+5*\scp*1cm},{0+1*2*\dy}) {$c_1$};
\node[cc] (u2) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l2);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l1);
\draw[dashed] ($(l2)+ (0,{2*\dy *2 / 3})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(l1)+ (0,{2*\dy /3})$) --++ ({0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({2*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({3*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\rho(\alpha_1)$};
\node [below ={\scp*0.3cm} of l1] {$\rho(\alpha_2)$};
%
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-1*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-2*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$\tilde p$};
\end{tikzpicture}
\end{mycenter}
\noindent
By the same calculation as in the proof of Lem\-ma~\ref{lemma:verifying-axioms-2}, we obtain $\delta_p(\alpha_1,\alpha_2)=-\delta_{\tilde p}(\rho(\alpha_2),\rho(\alpha_1))$. Hence, $\xi_{c_1,c_2}=X_{c_1,c_2}(\mathcal C)\subseteq -X_{\overline{c_2},\overline{c_1}}(\mathcal C)=-\xi_{\overline{c_2},\overline{c_1}}$ as claimed.
\par
\emph{Axiom~\ref{lemma:gradedsets-condition-2}:} Lastly, as crossing each other is a symmetric $2$-relation on blocks,
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+1*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (l2) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node[vp] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {$\overline {c_1}$};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l2);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l1);
\draw[dashed] ($(l2)+ (0,{2*\dy *2 / 3})$) --++ ({0.75*\scp*1cm},0);
\draw[dashed] ($(l1)+ (0,{2*\dy /3})$) --++ ({-0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({3*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({2*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+1*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (l2) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
\node[vp] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {$\overline {c_1}$};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (u1) -- ++ (0,{-1/3*2*\dy}) -| (l2);
\draw[dashed] ($(u1)+ (0,{-1/3*2*\dy})$) --++ ({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+ (0,{-2/3*2*\dy})$) --++ ({0.75*\scp*1cm},0);
\draw[sstr] (u2) -- ++ (0,{-2/3*2*\dy}) -| (l1);
\draw[dashed] ($(l2)+ (0,{2*\dy *2 / 3})$) --++ ({0.75*\scp*1cm},0);
\draw[dashed] ($(l1)+ (0,{2*\dy /3})$) --++ ({-0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({3*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l1] {$\alpha_2$};
%
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-5*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-5*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\end{mycenter}
\noindent
and because, \(\delta_p(\alpha_1,\alpha_2)\equiv -\delta_{p}(\alpha_2,\alpha_1)\mod \Sigma(p)\) (by \cite[Lem\-ma~2.1~(b)]{MWNHO1}), we can immediately conclude $\delta_p(\alpha_1,\alpha_2)\in -X_{c_2,c_1}(\mathcal C)+\Sigma(\mathcal C)$. Thus, $\xi_{c_1,c_2}\subseteq -\xi_{c_2,c_1}+\sigma$. Differently from
Lem\-ma~\ref{lemma:verifying-axioms-2}, we did not need Lem\-ma~\ref{lemma:simplification-k-x} to see this.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-4}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\xi_{c_1,c_2}\coloneqq X_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axiom~\ref{lemma:gradedsets-condition-6} of~\ref{axioms:arithmetic}: For all $c_1,c_2\in\{\circ,\bullet\}$,
\begin{IEEEeqnarray*}{rCl}
\xi_{c_1,c_2}\subseteq \xi_{c_1,\overline{c_2}}\cup \left(-\xi_{c_2,\overline{c_1}}+\sigma\right).
\end{IEEEeqnarray*}
\end{lemma}
\begin{proof}
Let $B_1$ and $B_2$ be crossing blocks in $p\in \mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2}$ and let $\alpha_1\in B_1$ and $\alpha_2\in B_2$ have normalized colors $c_1\in \{\circ,\bullet\}$ and $c_2\in \{\circ,\bullet\}$, respectively. According to Lem\-ma~\ref{lemma:simplification-k-x} then, every element of $\xi_{c_1,c_2}=X_{c_1,c_2}(\mathcal C)=X_{c_1,c_2}(\mathcal C\cap \mc{P}^{\circ\bullet}_{\leq 2})$ is of the form $\delta_p(\alpha_1,\alpha_2)$. Because $p\in \mc{P}^{\circ\bullet}_{\leq 2}$, the blocks $B_1$ and $B_2$ are pairs. Hence, the crossing between these blocks means that we find points $\beta_1\in B_1$ and $\beta_2\in B_2$ with $\alpha_1\neq \beta_1$ and $\alpha_2\neq \beta_2$ such that either $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ or $(\alpha_2,\alpha_1,\beta_2,\beta_1)$ is ordered.
\par
\emph{Case~1:} First, we suppose that $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ is ordered and show $\delta_p(\alpha_1,\alpha_2)\in X_{c_1,\overline{c_2}}(\mathcal C)$. We can assume that $\alpha_1$ is the leftmost and $\beta_1$ the rightmost lower point.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{8*\scp*1cm}
\def2*\dx{7*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (l2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (l3) at ({0+7*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (u1) at ({0+4*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (l1) --++(0,{2*\dy /3}) -| (l3);
\draw[sstr] (l2) --++(0,{2*\dy *2/3}) -| (u1);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({1*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({4*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({3*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({8*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l2] {$\alpha_2$};
\node [below ={\scp*0.3cm} of l3] {$\beta_1$};
\node [above ={\scp*0.3cm} of u1] {$\beta_2$};
%
\draw [draw=darkgray, densely dashed] ({0*\scp*1cm-\gx},{-\scp*0.4cm}) rectangle ({3*\scp*1cm+\gx},{\scp*0.4cm});
%
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\end{mycenter}
By Lem\-ma~\ref{lemma:projection}, the partition $p'\coloneqq P(p,[\alpha_1,\beta_1]_p)$ belongs to $\mathcal C$. The definition of the projection operation has the following consequences: The three lower points $\alpha_1,\alpha_2$ and $\beta_1$ of $p$, also points of $p'$, all retain their normalized colors in $p'$; the set $B_1=\{\alpha_1,\beta_1\}$ is still a block of $p'$; the point $\alpha_2$ is now connected to its counterpart $\beta_2'$ on the upper row of $p'$, implying in particular that the blocks of $\alpha_1$ and $\alpha_2$ still cross in $p'$; and it holds \[\delta_{p'}(\alpha_1,\alpha_2)=\delta_p(\alpha_1,\alpha_2).\]
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{7*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (l2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[cc] (l3) at ({0+7*\scp*1cm},{0+0*2*\dy}) {};
\node[vp] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {$c_1$};
\node[vp] (u2) at ({0+3*\scp*1cm},{0+1*2*\dy}) {$c_2$};
\node[cc] (u3) at ({0+7*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (l1) --++(0,{\scp*1cm}) -| (l3);
\draw[sstr] (l2) to (u2);
\draw[sstr] (u1) --++(0,{-\scp*1cm}) -| (u3);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({1*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({4*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({1*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({4*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [below ={\scp*0.3cm} of l2] {$\alpha_2$};
\node [below ={\scp*0.3cm} of l3] {$\beta_1$};
\node [above ={\scp*0.3cm} of u2] {$\beta_2'$};
%
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({2*\dx+\scp*1cm},{0.5*2*\dy}) {$p'$};
\node (lab1) at ({-2.5*\scp*1cm},{0.5*2*\dy}) {$[\beta_2',\alpha]_{p'}$};
\draw[dotted] (lab1) -- ({-\scp*0.5cm-\scp*0.3cm},0);
\draw[dotted] (lab1) -- ({-\scp*0.5cm-\scp*0.3cm},{2*\dy});
\end{tikzpicture}
\end{mycenter}
We apply Lem\-ma~\ref{lemma:projection} a second time to infer $p''\coloneqq P(p',[\beta_2',\alpha_2])\in \mathcal C$. Denote the images of the points $\beta_2'$, $\alpha_1$ and $\alpha_2$ of $p'$ in $p''$ by $\beta_2''$, $\alpha_1''$ and $\alpha_2''$, respectively. Now, $\beta_2''$ is the leftmost lower point and $\alpha_2''$ the rightmost lower point of $p''$ and the two form a block; the point $\alpha_1''\in [\beta_2'',\alpha_2'']_{p''}$ is connected to its counterpart on the upper row; and \[\delta_{p''}(\alpha_1'',\alpha_2'')=\delta_{p'}(\alpha_1,\alpha_2)=\delta_p(\alpha_1,\alpha_2).\]
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{7*\scp*1cm}
\def2*\dx{7*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (l2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$\overline{c_1}$};
\node[vp] (l3) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_1$};
\node[vp] (l4) at ({0+7*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {$\overline{c_2}$};
\node[vp] (u2) at ({0+3*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
\node[vp] (u3) at ({0+4*\scp*1cm},{0+1*2*\dy}) {$c_1$};
\node[vp] (u4) at ({0+7*\scp*1cm},{0+1*2*\dy}) {$c_2$};
%
\draw[sstr] (l1) --++(0,{\scp*1cm}) -| (l4);
\draw[sstr] (l2) to (u2);
\draw[sstr] (l3) to (u3);
\draw[sstr] (u1) --++(0,{-\scp*1cm}) -| (u4);
%
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({1*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({5*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north west lines, pattern color = darkgray] ({1*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\beta_2''$};
\node [below ={\scp*0.3cm} of l3] {$\alpha_1''$};
\node [below ={\scp*0.3cm} of l4] {$\alpha_2''$};
\node [above ={\scp*0.3cm} of u4] {$\gamma''$};
%
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-0*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({-\scp*1cm},{0.5*2*\dy}) {$p''$};
\end{tikzpicture}
\end{mycenter}
There are two crucial observations to make about the successor $\gamma''$ of $\alpha_2''$ in $p''$, the rightmost upper point of $p''$. Firstly, $\gamma''$ has the inverse normalized color $\overline{c_2}$ of $\alpha_2''$, which in particular implies that $\delta_{p''}(\alpha_2'',\gamma'')=0$. Secondly, $\gamma''$ forms a block of $p''$ together with the leftmost upper point of $p''$, which entails that its block crosses the block of $\alpha_1''$ in $p''$. Hence, $\delta_{p''}(\alpha_1'',\gamma'')\in X_{c_1,\overline{c_2}}(\mathcal C)$ and
\begin{align*}
\delta_{p''}(\alpha_1'',\gamma'')=\delta_{p''}(\alpha_1'',\alpha_2'')+\delta_{p''}(\alpha_2'',\gamma'')=\delta_p(\alpha_1,\alpha_2)
\end{align*}
together show $\delta_p(\alpha_1,\alpha_2)\in X_{c_1,\overline{c_2}}(\mathcal C)= \xi_{c_1,\overline{c_2}}$, which is what we set out to prove.
\par
\emph{Case~2:} Now, let $(\alpha_2,\alpha_1,\beta_2,\beta_1)$ be ordered instead. By Case~1 then, $\delta_p(\alpha_2,\alpha_1)\in X_{c_2,\overline{c_1}}(\mathcal C)$. \cite[Lem\-ma~2.1~(b)]{MWNHO1} shows $\delta_p(\alpha_2,\alpha_1)\equiv -\delta_p(\alpha_1,\alpha_2)\mod \Sigma(p)$. That implies $\delta_p(\alpha_1,\alpha_2)\in -X_{c_2,\overline{c_1}}(\mathcal C)+\Sigma(\mathcal C)=-\xi_{c_2,\overline{c_1}}+ \sigma$, which is what we needed to see.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-5}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\kappa_{c_1,c_2}\coloneqq K_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axiom~\ref{lemma:gradedsets-condition-4} of~\ref{axioms:arithmetic}: $0\in\kappa_{\circ\bullet}\cap \kappa_{\bullet\circ}$.
\end{lemma}
\begin{proof}
Since $\PartIdenLoWB\in \mathcal C$ and $K_{\circ\bullet}(\{\PartIdenLoWB\})=K_{\bullet\circ}(\{\PartIdenLoWB\})=\{0\}$, this is clear.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-6}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\kappa_{c_1,c_2}\coloneqq K_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axiom~\ref{lemma:gradedsets-condition-5} of~\ref{axioms:arithmetic}:
\begin{IEEEeqnarray*}{rCl}
\kappa_{c_1,c_2}+\kappa_{\overline{c_2},c_3}\subseteq \kappa_{c_1,c_3}
\end{IEEEeqnarray*}
for all $c_1,c_2,c_3\in\{\circ,\bullet\}$.
\end{lemma}
\begin{proof}
Let $c_1,c_2,c_3\in \{\circ,\bullet\}$ be arbitrary and let $\eta_1$ and $\eta_2$ be distinct points of the same block $B$ of $p\in \mathcal C$ such that $]\eta_1,\eta_2[_p\cap B=\emptyset$ and such that $\eta_i$ has normalized color $c_i$ in $p$ for every $i\in \{1,2\}$. Furthermore, let $\theta_1$ and $\theta_2$ be distinct points of the same block $C$ of $q\in \mathcal C$ with $]\theta_1,\theta_2[_q\cap C=\emptyset$ such that $\theta_1$ has normalized color $\overline{c_2}$ in $q$ and $\theta_2$ normalized color $c_3$. None of these assumptions are impacted and neither $\delta_p(\eta_1,\eta_2)$ nor $\delta_q(\theta_1,\theta_2)$ altered by assuming that $\eta_2$ is the rightmost lower point of $p$ and $\theta_1$ the leftmost lower point of $q$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{4*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
%
\draw[sstr] (l1) --++(0,{\scp*1cm})-| (u1);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\eta_2$};
\node [above ={\scp*0.3cm} of u1] {$\eta_1$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({4*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({-\scp*1cm},{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\otimes$\qquad
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{3*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {$\overline{c_3}$};
%
\draw[sstr] (l1) --++(0,{\scp*1cm})-| (u1);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
%
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({0*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({1*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\theta_1$};
\node [above ={\scp*0.3cm} of u1] {$\theta_2$};
%
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$q$};
\end{tikzpicture}
\end{mycenter}
Denote the images of the points $\theta_1$ and $\theta_2$ of $q$ in $p\otimes q\in \mathcal C$ by $\theta_1'$ and $\theta_2'$, respectively.
The assumptions about the normalized colors of $\eta_2$ and $\theta_1$ imply that $T\coloneqq \{\eta_2,\theta_1'\}$ is a turn in $p\otimes q$, meaning in particular $\delta_{p\otimes q}(\eta_2,\theta_1')=0$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{10*\scp*1cm}
\def2*\dx{8*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
\node[vp] (l2) at ({0+5*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (u2) at ({0+7*\scp*1cm},{0+1*2*\dy}) {$\overline{c_3}$};
%
\draw[sstr] (l1) --++(0,{\scp*1cm})-| (u1);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
\draw[sstr] (l2) --++(0,{\scp*1cm})-| (u2);
\draw[dashed] ($(l2) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\eta_2$};
\node [above ={\scp*0.3cm} of u1] {$\eta_1$};
\node [below = {\scp*0.3cm} of l2] {$\theta_1'$};
\node [above ={\scp*0.3cm} of u2] {$\theta_2'$};
%
\draw [densely dotted, draw=gray] ({4*\scp*1cm-\gx},{-\scp*0.4cm}) rectangle ({5*\scp*1cm+\gx},{\scp*0.4cm});
\node at ({4.5*\scp*1cm},{0.35*2*\dy}) {$T$};
%
\node at ({-1.5*\scp*1cm},{0.5*2*\dy}) {$p\otimes q$};
\end{tikzpicture}
\end{mycenter}
\noindent
Moreover, $\delta_{p\otimes q}(\eta_1,\eta_2)=\delta_p(\eta_1,\eta_2)$ and $\delta_{p\otimes q}(\theta_1',\theta_2')=\delta_{q}(\theta_1,\theta_2)$ by nature of the tensor product.
\par
Let $\theta_2''$ denote the image of $\theta_2'$ in $r\coloneqq E(p\otimes q,T)\in \mathcal C$, the partition obtained from $p\otimes q$ by erasing the turn $T$ (see \cite[Sec\-tion~4.3]{MWNHO1}). By definition of the erasing operation, $\eta_1$ and $\theta_2''$ belong to the same block $D$ in $r$ with $]\eta_1,\theta_2''[_{r}\cap D=\emptyset$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{10*\scp*1cm}
\def2*\dx{6*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
\node[vp] (u2) at ({0+7*\scp*1cm},{0+1*2*\dy}) {$\overline{c_3}$};
%
\coordinate (l1) at ({0+4*\scp*1cm},{0+0*2*\dy});
\coordinate (l2) at ({0+5*\scp*1cm},{0+0*2*\dy});
%
\draw[sstr] (u1) --++(0,{-\scp*1cm})-| (u2);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
\draw[dashed] ($(l2) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-2*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above= {\scp*0.3cm} of u1] {$\eta_1$};
\node [above ={\scp*0.3cm} of u2] {$\theta_2''$};
%
\draw [draw=darkgray, densely dashed] ({-\gx},{-\scp*0.4cm}) -- ({2*\dx+\gx},{-\scp*0.4cm}) ({2*\dx+\gx},{\scp*0.4cm}) -- ({-\gx},{\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
\node at ({-\scp*1cm},{0.5*2*\dy}) {$r$};
\end{tikzpicture}
\end{mycenter}
\noindent
Hence, from $\delta_{r}(\eta_1,\theta_2'')\in K_{c_1,c_3}(\mathcal C)=\kappa_{c_1,c_3}$ and from
\begin{align*}
\delta_{r}(\eta_1,\theta_2'')&=\delta_{p\otimes q}(\eta_1,\theta_2)-\sigma_{p\otimes q}(T)\\
&=\delta_{p\otimes q}(\eta_1,\theta_2)\\
&=\delta_{p\otimes q}(\eta_1,\eta_2)+\delta_{p\otimes q}(\eta_2,\theta_1')+\delta_{p\otimes q}(\theta_1',\theta_2')\\
&=\delta_{p\otimes q}(\eta_1,\eta_2)+\delta_{p\otimes q}(\theta_1',\theta_2')\\
&=\delta_p(\eta_1,\eta_2)+\delta_q(\theta_1,\theta_2)
\end{align*}
it follows $\delta_p(\eta_1,\eta_2)+\delta_q(\theta_1,\theta_2)\in\kappa_{c_1,c_3}$. And that is what we needed to show.
\end{proof}
\begin{lemma}
\label{lemma:verifying-axioms-7}
For every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, the sets $\sigma\coloneqq \Sigma(\mathcal C)$ and $\kappa_{c_1,c_2}\coloneqq K_{c_1,c_2}(\mathcal C)$ and $\xi_{c_1,c_2}\coloneqq X_{c_1,c_2}(\mathcal C)$ for $c_1,c_2\in \{\circ,\bullet\}$ satisfy Axiom~\ref{lemma:gradedsets-condition-7} of~\ref{axioms:arithmetic}: For all $c_1,c_2,c_3\in\{\circ,\bullet\}$,
\begin{IEEEeqnarray*}{rCl}
\kappa_{c_1,c_2}+\xi_{\overline{c_2},c_3}\subseteq \xi_{c_1,c_3}.
\end{IEEEeqnarray*}
\end{lemma}
\begin{proof}
We adapt the proof of Lem\-ma~\ref{lemma:verifying-axioms-6}.
Let $c_1,c_2\in \{\circ,\bullet\}$ be arbitrary. Let $p,q\in \mathcal C$, let $B$ be a block in $p$, and let $C$ and $D$ be two crossing blocks in $q$. Let $\gamma_1$ and $\gamma_2$ be two distinct points of $B$ of normalized colors $c_1$ respectively $c_2$ in $p$ with $]\gamma_1,\gamma_2[_p\cap B=\emptyset$. In $q$, let $\eta_1\in C$ have normalized color $\overline{c_2}$ and $\theta_1\in D$ normalized color $c_3$. Then, $\delta_p(\gamma_1,\gamma_2)$ is a generic element of $K_{c_1,c_2}(\mathcal C)=\kappa_{c_1,c_2}$ and $\delta_q(\eta_1,\theta_1)$ one of $X_{\overline{c_2},c_3}(\mathcal C)=\xi_{\overline{c_2},c_3}$. No generality is lost assuming that $\gamma_2$ is the rightmost lower point of $p$ and $\eta_1$ the leftmost lower point of $q$. We find $\eta_2\in C$ and $\theta_2\in D$ such that $\eta_1\neq \eta_2$ and $\theta_1\neq \theta_2$ and such that $(\eta_1,\theta_1,\eta_2,\theta_2)$ or $(\eta_1,\theta_2,\eta_2,\theta_1)$ is ordered in $q$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{4*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
%
\draw[sstr] (l1) --++(0,{\scp*1cm}) -| (u1);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\gamma_2$};
\node [above ={\scp*0.3cm} of u1] {$\gamma_1$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({4*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({-\scp*1cm},{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\qquad$\otimes$\qquad
\begin{tikzpicture}[baseline=0.666*1.5cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{6*\scp*1cm}
\def2*\dx{5*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (l2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$c_3$};
\node[cc] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (l2) --++(0,{2*\scp*1cm}) -| (u1);
\draw[sstr] (l1) --++(0,{1*\scp*1cm}) -| (u2);
\draw[dashed] ($(l1) +(0,{\scp*1cm})$) --++(0,{0.75*\scp*1cm});
\draw[dashed] ($(u2) -(0,{2*\scp*1cm})$) --++({0.75*\scp*1cm},0);
\draw[dashed] ($(l2) +(0,{2*\scp*1cm})$) --++({0.75*\scp*1cm},0);
\draw[dashed] ($(u1) -(0,{1*\scp*1cm})$) --++({-0.75*\scp*1cm},0);
%
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({1*\scp*1cm-\scp*0.2cm},{0*2*\dy-\scp*0.2cm}) rectangle ({2*\scp*1cm+\scp*0.2cm},{0*2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({0*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({1*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\eta_1$};
\node [below = {\scp*0.3cm} of l2] {$\theta_1$};
\node [above ={\scp*0.3cm} of u1] {$\theta_2$};
\node [above ={\scp*0.3cm} of u2] {$\eta_2$};
%
\draw [draw=darkgray, densely dashed] ({0*\scp*1cm-\gx},{-\scp*0.4cm}) rectangle ({3*\scp*1cm+\gx},{\scp*0.4cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$q$};
\end{tikzpicture}
\end{mycenter}
Let $\eta_1'$, $\eta_2'$, $\theta_1'$ and $\theta_2'$ denote the images of, respectively, $\eta_1$, $\eta_2$, $\theta_1$ and $\theta_2$ in $p\otimes q\in \mathcal C$. By nature of the tensor product, $B$ is a block of $p\otimes q$. Likewise, $\eta_1'$ and $\eta_2'$ belong to the same block in $p\otimes q$ and so do $\theta_1'$ and $\theta_2'$. And each involved point has the same normalized color in $p\otimes q$ as the corresponding preimage in $p$ or $q$. The set $T\coloneqq \{\gamma_2,\eta_1'\}$ is a turn in $p\otimes q$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{11*\scp*1cm}
\def2*\dx{10*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (l1) at ({0+4*\scp*1cm},{0+0*2*\dy}) {$c_2$};
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
\node[vp] (l2) at ({0+5*\scp*1cm},{0+0*2*\dy}) {$\overline{c_2}$};
\node[vp] (l3) at ({0+8*\scp*1cm},{0+0*2*\dy}) {$c_3$};
\node[cc] (u2) at ({0+7*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u3) at ({0+10*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (l3) --++(0,{2*\scp*1cm}) -| (u2);
\draw[sstr] (l2) --++(0,{1*\scp*1cm}) -| (u3);
%
\draw[sstr] (l1) --++(0,{1*\scp*1cm}) -| (u1);
\draw[dashed] ($(l1)+(0,{\scp*1cm})$) --++ (0,{0.75*\scp*1cm});
\draw[dashed] ($(l2)+(0,{1*\scp*1cm})$) --++ (0,{0.75*\scp*1cm});
\draw[dashed] ($(u3)+(0,{-2*\scp*1cm})$) --++ ({0.75*\scp*1cm},0);
\draw[dashed] ($(l3)+(0,{2*\scp*1cm})$) --++ ({0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+(0,{-1*\scp*1cm})$) --++ ({-0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({6*\scp*1cm-\scp*0.2cm},{0*2*\dy-\scp*0.2cm}) rectangle ({7*\scp*1cm+\scp*0.2cm},{0*2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({8*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({9*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\gamma_2$};
\node [above ={\scp*0.3cm} of u1] {$\gamma_1$};
\node [below = {\scp*0.3cm} of l2] {$\eta_1'$};
\node [below = {\scp*0.3cm} of l3] {$\theta_1'$};
\node [above ={\scp*0.3cm} of u2] {$\theta_2'$};
\node [above ={\scp*0.3cm} of u3] {$\eta_2'$};
%
%
\draw [densely dotted, draw=gray] ({4*\scp*1cm-\gx},{-\scp*0.4cm}) rectangle ({5*\scp*1cm+\gx},{\scp*0.4cm});
\node at ({4.5*\scp*1cm},{0.35*2*\dy}) {$T$};
%
\node at ({-1.5*\scp*1cm},{0.5*2*\dy}) {$p\otimes q$};
\end{tikzpicture}
\end{mycenter}
If we denote by $\eta_2''$, $\theta_1''$ and $\theta_2''$ the images of $\eta_1'$, $\theta_1'$ and $\theta_2'$ in $r\coloneqq E(p\otimes q,T)\in \mathcal C$, then $\gamma_1$ and $\eta_2''$ belong to the same block in $r$ and so do $\theta_1'$ and $\theta_2'$. Because $(\gamma_1,\gamma_2,\eta_1',\theta_i',\eta_2',\theta_{\neg i}')$ is ordered in $p\otimes q$ for some $i,\neg i\in \{1,2\}$ with $\{i,\neg i\}=\{1,2\}$, the tuple $(\gamma_1,\theta_i'',\eta_2'',\theta_{\neg i}'')$ is then ordered in $r$. Thus, the blocks of $\gamma_1$ and $\eta_2''$ and of $\theta_1''$ and $\theta_2''$ cross in $r$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{11*\scp*1cm}
\def2*\dx{8*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[vp] (u1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline{c_1}$};
\node[vp] (l3) at ({0+6*\scp*1cm},{0+0*2*\dy}) {$c_3$};
\node[cc] (u2) at ({0+7*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u3) at ({0+10*\scp*1cm},{0+1*2*\dy}) {};
%
\draw[sstr] (l3) --++(0,{2*\scp*1cm}) -| (u2);
\draw[sstr] (u1) --++(0,{-2*\scp*1cm}) -| (u3);
\draw[dashed] ($(u1)+(0,{-2*\scp*1cm})$) --++({-0.75*\scp*1cm},0);
\draw[dashed] ($(u3)+(0,{-2*\scp*1cm})$) --++({0.75*\scp*1cm},0);
\draw[dashed] ($(l3)+(0,{2*\scp*1cm})$) --++({-0.75*\scp*1cm},0);
\draw[dashed] ($(u2)+(0,{-1*\scp*1cm})$) --++({0.75*\scp*1cm},0);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({3*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({4*\scp*1cm-\scp*0.2cm},{0*2*\dy-\scp*0.2cm}) rectangle ({5*\scp*1cm+\scp*0.2cm},{0*2*\dy+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({0*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-0*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({8*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({9*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=gray, fill = lightgray] ({5*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({6*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above ={\scp*0.3cm} of u1] {$\gamma_1$};
\node [below = {\scp*0.3cm} of l3] {$\theta_1''$};
\node [above ={\scp*0.3cm} of u2] {$\theta_2''$};
\node [above ={\scp*0.3cm} of u3] {$\eta_2''$};
%
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({6*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({1*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\node at ({-1*\scp*1cm},{0.5*2*\dy}) {$r$};
\end{tikzpicture}
\end{mycenter}
\noindent
Consequently, from $\delta_{r}(\gamma_1,\theta_1'')\in X_{c_1,c_3}(\mathcal C)$ and from
\begin{align*}
\delta_{r}(\gamma_1,\theta_1'')&=\delta_{p\otimes q}(\gamma_1,\theta_1')-\sigma_{p\otimes q}(T)\\
&=\delta_{p\otimes q}(\gamma_1,\theta_1')\\
&=\delta_{p\otimes q}(\gamma_1,\gamma_2)+\delta_{p\otimes q}(\gamma_2,\eta_1')+\delta_{p\otimes q}(\eta_1',\theta_1')\\
&=\delta_{p\otimes q}(\gamma_1,\gamma_2)+\delta_{p\otimes q}(\eta_1',\theta_1')\\
&=\delta_p(\gamma_1,\gamma_2)+\delta_q(\eta_1,\theta_1)
\end{align*}
it follows $\delta_p(\gamma_1,\gamma_2)+\delta_q(\eta_1,\theta_1)\in X_{c_1,c_3}(\mathcal C)= \xi_{c_1,c_3}$. And that is what we needed to see.
\end{proof}
Finally, we can give the final result of this sec\-tion.
\begin{proposition}
\label{lemma:result-s-l-k-x}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a non-hy\-per\-octa\-he\-dral category. Then,
\begin{gather*}
L(\mathcal C)=K_{\circ\circ}(\mathcal C)=K_{\bullet\bullet}(\mathcal C),\qquad K(\mathcal C)=K_{\circ\bullet}(\mathcal C)=K_{\bullet\circ}(\mathcal C)
\end{gather*}
and
\begin{gather*}
X(\mathcal C)=X_{\circ\circ}(\mathcal C)=X_{\bullet\bullet}(\mathcal C)=X_{\circ\bullet}(\mathcal C)=X_{\bullet\circ}(\mathcal C)
\end{gather*}
and there exist $u\in \{0\}\cup \mathbb{N}$, $m\in \mathbb{N}$, $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$ and $E\subseteq \{0\}\cup\mathbb{N}$ such that the tuple $(\Sigma,L,K,X)(\mathcal C)$ is one of the following:
\begin{align*}
\begin{matrix}
\Sigma(\mathcal C)& L(\mathcal C)&K(\mathcal C)& X(\mathcal C) \\ \hline \\[-0.85em]
um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
2um\mathbb{Z} & m\!+\!2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0
\end{matrix}
\end{align*}
\end{proposition}
\begin{proof}
Follows from Lem\-mata~\ref{lemma:verifying-axioms-1} -- \ref{lemma:verifying-axioms-7} and the \hyperref[lemma:arithmetic]{Arithmetic Lem\-ma~\ref*{lemma:arithmetic}}.
\end{proof}
\section{\texorpdfstring{Step~5: Special Relations between $\Sigma$, $L$, $K$ and $X$\\depending on $F$ and $V$}{Step~5: Special Relations between Sigma, L, K and X depending on F and V}}
\label{section:special-restrictions}
Our objective remains proving $Z(\mathcal C)\in\mathsf Q$ for any non-hy\-per\-oc\-ta\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$. After studying components $F$ (Sec\-tion~\ref{section:block-sizes}) and $\Sigma$ (Sec\-tion~\ref{section:total-color-sums}) in isolation and after in\-ves\-ti\-ga\-ting the images of the mappings $(F,V,L)$ (Sec\-tion~\ref{section:block-color-sums}) and $(\Sigma,L,K,X)$ (Sec\-tion~\ref{section:color-lengths}), we have arrived at the point where we must take all six components of $Z=(F,V,\Sigma,L,K,X)$ into account simultaneously. Fortunately, we can capitalize on the results of Sec\-tions~\ref{section:block-sizes}--\ref{section:color-lengths} in this endeavor. In consequence, it largely suffices to understand better the behavior of $(\Sigma,L,K,X)$ as dependent on $(F,V)$ or, roughly, on $F$.
\par
Recall from \cite[Definition~4.1]{MWNHO1} that a category is non-hy\-per\-oc\-ta\-he\-dral if and only if it is case~$\mathcal O$, $\mathcal B$ or~$\mathcal S$ and that these cases are mutually exclusive.
\subsection{Special Relations in Case~$\mathcal S$}
\label{section:special-restrictions-s}
For case~$\mathcal S$ categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, i.e., by Proposition~\ref{proposition:result-F} assuming $F(\mathcal C)=\mathbb{N}$, there is just a single fact about $(\Sigma,L,K, X)(\mathcal C)$ we have to note, one about $L(\mathcal C)$.
\begin{proposition}
\label{lemma:restriction-case-s}
$0\in L(\mathcal C)$ for every case~$\mathcal S$ category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$.
\end{proposition}
\begin{proof}
As $\PartSinglesWBTensor\in \mathcal C$, we can, by Lem\-ma~\hyperref[lemma:singletons-3]{\ref*{lemma:singletons}~\ref*{lemma:singletons-3}}, disconnect the black points in $\PartFourWBWB\in \mathcal C$ and obtain $\PartCoSingleWBWB\in \mathcal C$. It follows $\{0\}=L(\{\PartCoSingleWBWB\})\subseteq L(\mathcal C)$.
\end{proof}
\subsection{Special Relations in Case~$\mathcal O$}
\label{section:special-restrictions-o}
For case~$\mathcal O$ categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, i.e., assuming $F(\mathcal C)=\{2\}$, more than what Proposition~\ref{lemma:result-s-l-k-x} is able to discern can be said about $\Sigma(\mathcal C)$ and $X(\mathcal C)$.
\subsubsection{\texorpdfstring{Relation of $\Sigma$ to $L$ and $K$ in Case~$\mathcal O$}{Relation of Sigma to L and K in Case~O}} First, we treat the total color sums of case~$\mathcal O$ categories.
\begin{proposition}
\label{lemma:k-v-l}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a case~$\mathcal O$ category and let $m\in \mathbb{N}$.
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:k-v-l-1} If $(L,K)(\mathcal C)=(\emptyset,m\mathbb{Z})$, then $\Sigma(\mathcal C)=\{0\}$.
\item\label{lemma:k-v-l-2} If $(L,K)(\mathcal C)=(m\mathbb{Z},m\mathbb{Z})$ or $(L,K)(\mathcal C)=(m\!+\!2m\mathbb{Z},2m\mathbb{Z})$, then \[\Sigma(\mathcal C)=2um\mathbb{Z}\] for some $u\in \{0\}\cup\mathbb{N}$.
\end{enumerate}
\end{proposition}
\begin{proof}
\begin{enumerate}[wide,label=(\alph*)]
\item By Proposition~\ref{lemma:result-s-l-k-x} there exists $\tilde u\in \{0\}\cup \mathbb{N}$ such that $\Sigma(\mathcal C)=\tilde um\mathbb{Z}$. We suppose $\tilde u\neq 0$ and derive a contradiction. As $\mathcal C$ is closed under erasing turns and as erasing turns does not affect total color sum, we find $p\in\mathcal C$ with no turns such that $\Sigma(p)=\tilde um$. Because $\tilde um>0$, the partition $p$ has at least one block. As all blocks of $p$ are pairs by Proposition~\ref{proposition:result-F}, there is a block $B$ of $p$ with (necessarily subsequent) legs $\alpha,\beta\in B$ and $\alpha\neq \beta$. Since $p$ has no turns, all points of $p$ have normalized color $\circ$. In particular, $\alpha$ and $\beta$ do. That proves $L(\mathcal C)\neq \emptyset$, contradicting the assumption.
\item Proposition~\ref{lemma:result-s-l-k-x} guarantees that $\Sigma(\mathcal C)=\tilde um\mathbb{Z}$ for some $\tilde u\in \{0\}\cup\mathbb{N}$ and that $\tilde u$ is even if $(L,K)(\mathcal C)=(m\!+\!2m\mathbb{Z},2m\mathbb{Z})$. We want to show that $\tilde u$ is even also if $(L,K)(\mathcal C)=(m\mathbb{Z},m\mathbb{Z})$. If $\tilde u=0$, this claim is true. Hence, suppose $\tilde u>0$. As in Part~\ref{lemma:k-v-l-1}, we utilize $p\in\mathcal C$ with no turns such that $\Sigma(p)=\tilde um>0$ and, this time, also with no upper points.
\par
For every $i\in \mathbb{N}$ with $i\leq m$ consider the set
\begin{align*}
S_i=\{\lop{j}\mid j\in (i+m{\mathbb{N}_0}),\, j\leq \tilde um\}
\end{align*}
comprising the $i$-th lower point and all its $m$-th neighbors to the right. Then, $\bigcup_{i=1}^m S_i$ comprises all points of $p$ and $|S_i|=\tilde u$ for every $i\in \mathbb{N}$ with $i\leq m$.
\par
The sets $S_1,\ldots,S_{m}$ must all be subpartitions of $p$: Otherwise, we find $j,j'\in \mathbb{N}$ with $j<j'\leq \tilde u m$ and $j'-j\notin m\mathbb{Z}$ such that $\lop{j}$ and $\lop{j'}$ belong to the same block. As all of $]\lop{j},\lop{j'}]_p$ has normalized color $\circ$, \[\delta_p(\lop{j},\lop{j'})=\sigma_p(]\lop{j},\lop{j'}]_p)=|]\lop{j},\lop{j'}]_p|=j'-j\notin m\mathbb{Z}.\] That contradicts the assumption $L(\mathcal C)= m\mathbb{Z}$.
\par
Because all blocks of $p$ are pairs by Proposition~\ref{proposition:result-F}, subpartitions of $p$ have even cardinality. We conclude $\tilde u=|S_1|\in 2\mathbb{Z}$, which then proves the claim.
\qedhere
\end{enumerate}
\end{proof}
\subsubsection{Relation of $X$ to $L$ and $K$ in Case~$\mathcal O$}
When studying $X(\mathcal C)$ further for case~$\mathcal O$ categories $\mathcal C\subseteq \mc{P}^{\circ\bullet}$, it is best to distinguish whether $(L\cup K)(\mathcal C)$ contains non-zero elements or not.
\begin{proposition}
\label{lemma:result-x-case-o-positive-w}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a case~$\mathcal O$ category and let $m\in \mathbb{N}$.
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:result-x-case-o-positive-w-1} If $(L,K)(\mathcal C)=(m\!+\!2m\mathbb{Z},2m\mathbb{Z})$, then $X(\mathcal C)=\mathbb{Z}$ or $X(\mathcal C)=\mathbb{Z}\backslash m\mathbb{Z}$.
\item\label{lemma:result-x-case-o-positive-w-2} If $(L,K)(\mathcal C)=(m\mathbb{Z},m\mathbb{Z})$ or $(L,K)(\mathcal C)=(\emptyset,m\mathbb{Z})$, then $X(\mathcal C)=\mathbb{Z}$.
\end{enumerate}
\end{proposition}
\begin{proof} No matter which of the three values $(L,K)(\mathcal C)$ takes, by Proposition~\ref{lemma:result-s-l-k-x} the set $X(\mathcal C)$ is $m$-periodic. Therefore, showing $\dwi{m}\subseteq X(\mathcal C)$ already implies $X(\mathcal C)=\mathbb{Z}$. Likewise, provided $m\geq 2$, establishing $\dwi{m\!-\!1}\subseteq X(\mathcal C)$ forces the conclusion that $X(\mathcal C)=\mathbb{Z}\backslash m\mathbb{Z}$ or $X(\mathcal C)=\mathbb{Z}$.
\begin{enumerate}[wide, label=(\alph*)]
\item First, let $(L,K)(\mathcal C)=(m\!+\!2m\mathbb{Z},2m\mathbb{Z})$. If $m=1$, the $1$-periodicity of $X(\mathcal C)$ immediately implies $X(\mathcal C)=\emptyset$ or $X(\mathcal C)=\mathbb{Z}$. Hence, we can suppose $m\geq 2$ and only need to prove $\dwi{m\!-\!1}\subseteq X(\mathcal C)$ by the initial remark.
\par
Proposition~\ref{lemma:result-s-l-k-x} lets us infer $K_{\circ\circ}(\mathcal C)=m\!+\!2m\mathbb{Z}$. Hence, we find a partition $p\in \mathcal C\subseteq \mc{P}^{\circ\bullet}_{2}$, therein a block $\{\alpha,\beta\}$ with $\alpha$ and $\beta$ both of normalized color $\circ$, with $\alpha\neq \beta$ and with $\delta_p(\alpha,\beta)=m$. Without infringing on any of these assumptions we can additionally suppose that there are no turns $T$ in $p$ such that $T\subseteq ]\alpha,\beta[_p$ (otherwise we erase them). Then, all of $]\alpha,\beta[_p$ has the same normalized color $c\in \{\circ,\bullet\}$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
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\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
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\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
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\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
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%
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\node[blp] (u1) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
%
\draw (l1) -- ++(0,{\scp*1cm}) -| (u1);
%
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({2*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node (y1) at ({0*\scp*1cm},{2*\dy}) {$\overline c$};
\node (y2) at ({2*\scp*1cm},{2*\dy}) {$\overline c$};
\path (y1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (y2);
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\node (x2) at ({1*\scp*1cm},{0}) {$c$};
\path (x1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots$}} (x2);
%
\node [below ={\scp*0.3cm} of l1] {$\beta$};
\node [above ={\scp*0.3cm} of u1] {$\alpha$};
%
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$p$};
\node at (-1*\scp*1cm,{0.5*2*\dy}) {$\delta_p(\alpha,\beta)=m$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*0.5cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
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\def\scp*1cm{0.666*1cm}
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%
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\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
%
\node[whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l3) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l4) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
%
\draw (l1) -- ++ (0,{\scp*1cm}) -| (l4);
\draw (l2) -- ++ (0, {1.5*\scp*1cm});
\draw (l3) -- ++ (0, {1.5*\scp*1cm});
%
%
\path (l2) -- node[pos=0.5, yshift=0.35cm] {$\ldots$} (l3);
%
\node [below ={\scp*0.3cm} of l1] {$\lop{1}$};
\node [xshift=0.5em, below ={\scp*0.3cm} of l4] {$\lop{(m\!+\!1)}$};
%
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$P(p,[\alpha,\beta]_p)$};
\draw[densely dotted] ($(l2)+({-\scp*0.3cm},{-\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -| node[pos=0.25, below] {$m-1$} ($(l3)+({\scp*0.3cm},{-\scp*0.3cm})$);
\end{tikzpicture}
\end{mycenter}
Because $\alpha$ and $\beta$ also identically have normalized color $\circ$,
\begin{align*}
m=\delta_p(\alpha,\beta)=\sigma_p(]\alpha,\beta]_p)=
\begin{cases}
|]\alpha,\beta]_p| & \text{if }c=\circ,\\
-|]\alpha,\beta]_p| &\text{otherwise}.
\end{cases}
\end{align*}
As $m>0$, the only option is $c=\circ$. That means $[\alpha,\beta]_p$ consists of $m+1$ points of normalized color $\circ$.
\par
By definition of the projection operation and by Lem\-ma~\ref{lemma:projection}, it is possible to further add the premise $p=P(p,[\alpha,\beta]_p)$ without impacting any of the previous assumptions. Now, $p$ is also projective and $[\alpha,\beta]_p=[\lop{1},\lop{(m+1)}]_p$ is its lower row.
\par
For every $j\in \mathbb{N}$ with $1<j<m+1$ the point $\lop{j}$ belongs to a through block: Assuming otherwise, forces us to accept the existence of $j,j'\in \mathbb{N}$ with $1<j<j'<m+1$ such that $\lop{j}$ and $\lop{j'}$ belong to the same block. But then, the uniform color $\circ$ of $[\alpha,\beta]_p$ implies \[1\leq \delta_p(\lop{j},\lop{j'})=j'-j\leq m-2\leq m-1\] and thus $L(\mathcal C)\cap \{1,\ldots,m-1\}\neq \emptyset$, contradicting $L(\mathcal C)\subseteq m\mathbb{Z}$.
\par
Thus we have shown that $\alpha=\lop{1}$ and $\lop{j}$ belong to crossing blocks for every $j\in \mathbb{N}$ with $1<j<m+1$. Because $\delta_p(\alpha,\lop{j})=j-1$ for every such $j$, this proves $\dwi{m\!-\!1}\subseteq X(\mathcal C)$. And that is what we needed to show.
\item Let $(L,K)(\mathcal C)$ be given by $(m\mathbb{Z},m\mathbb{Z})$ or $(\emptyset,m\mathbb{Z})$. We adapt the proof of Part~\ref{lemma:result-x-case-o-positive-w-1}. However, this time, we do \emph{not} yet impose any restriction on $m$.
\par
Proposition~\ref{lemma:result-s-l-k-x} assures us that $K_{\circ\bullet}(\mathcal C)=K(\mathcal C)=m\mathbb{Z}$. Hence, we again find $p\in \mathcal C$, a block $B$ of $p$ and legs $\alpha,\beta\in B$ with $\alpha\neq \beta$, with $]\alpha,\beta[_p\cap B=\emptyset$ and with $\delta_p(\alpha,\beta)=m$, but this time, such that $\alpha$ is of normalized color $\circ$ and $\beta$ of normalized color $\bullet$. By the same argument as before we can assume that all points of $]\alpha,\beta[_p$ share the same normalized color. Then, the deviating assumption on the colors of $\alpha$ and $\beta$ implies $m=\delta_p(\alpha,\beta)=\sigma_p(]\alpha,\beta[_p)=|]\alpha,\beta[_p|$, which forces $[\alpha,\beta]_p$ to consist of exactly $m+2$ points (rather than $m+1$ as in Part~\ref{lemma:result-x-case-o-positive-w-1}), the first $m+1$ of which have normalized color $\circ$. Once more, we can assume $p=P(p,[\alpha,\beta]_p)$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
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\def\scp*0.4cm{0.666*0.4cm}
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%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[blp] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node[blp] (u1) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
%
\draw (l1) -- ++(0,{1*\scp*1cm}) -| (u1);
%
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({2*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node (y1) at ({0*\scp*1cm},{2*\dy}) {$\overline c$};
\node (y2) at ({2*\scp*1cm},{2*\dy}) {$\overline c$};
\path (y1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (y2);
\node (x1) at ({0*\scp*1cm},{0}) {$c$};
\node (x2) at ({1*\scp*1cm},{0}) {$c$};
\path (x1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots$}} (x2);
%
\node [below ={\scp*0.3cm} of l1] {$\beta$};
\node [above ={\scp*0.3cm} of u1] {$\alpha$};
%
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$p$};
\node at (-1*\scp*1cm,{0.5*2*\dy}) {$\delta_p(\alpha,\beta)=m$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*0.5cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
%
\node[whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l3) at ({0+3*\scp*1cm},{0+0*2*\dy}) {};
\node[blp] (l4) at ({0+4*\scp*1cm},{0+0*2*\dy}) {};
%
\draw (l1) -- ++ (0,{\scp*1cm}) -| (l4);
\draw (l2) -- ++ (0, {1.5*\scp*1cm});
\draw (l3) -- ++ (0, {1.5*\scp*1cm});
%
%
\path (l2) -- node[pos=0.5, yshift=0.35cm] {$\ldots$} (l3);
%
\node [below ={\scp*0.3cm} of l1] {$\lop{1}$};
\node [xshift=0.65em, below ={\scp*0.3cm} of l4] {$\lop{(m\!+\!2)}$};
%
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$P(p,[\alpha,\beta]_p)$};
\draw[densely dotted] ($(l2)+({-\scp*0.3cm},{-\scp*0.3cm})$) --++ (0,{-2*\scp*0.3cm}) -| node[pos=0.25, below] {$m$} ($(l3)+({\scp*0.3cm},{-\scp*0.3cm})$);
\end{tikzpicture}
\end{mycenter}
\par
If $m=1$, then $F(\{p\})=\{2\}$ requires the unique point $\lop{2}\in ]\lop{1},\lop{3}[_p$ to belong to a through block, proving $1\in X(\mathcal C)$ and thus $X(\mathcal C)=\mathbb{Z}$ as claimed. Hence, suppose $m\geq 2$ in the following.
\par
We prove that only through blocks intersect $]\lop{1},\lop{(m\!+\!2)}[_p$: Supposing that $\lop{j}$ and $\lop{j'}$, where $j,j'\in \mathbb{N}$ and $1<j<j'<m+2$, belong to the same block requires us to believe, as both $\lop{j}$ and $\lop{j'}$ are $\circ$-colored, that \[1\leq \delta_p(\lop j,\lop j')=j'-j\leq (m+1)-2=m-1\] and thus $L(\mathcal C)\cap \{1,\ldots,m-1\}\neq \emptyset$. As this would contradict the assumption $L(\mathcal C)\subseteq m\mathbb{Z}$, this cannot be the case.
\par
Now, the conclusion that the blocks of $\lop{1}$ and of $\lop{j}$ cross for every $j\in \mathbb{N}$ with $1<j<m+2$ and the fact $\delta_p(\lop{1},\lop{j})=j-1$ let us deduce $\dwi{m}\subseteq X(\mathcal C)$, which is what needed to see.\qedhere
\end{enumerate}
\end{proof}
\begin{proposition}
\label{lemma:result-x-subsemigroup}
Let $\mathcal C\subseteq \mc{P}^{\circ\bullet}$ be a case~$\mathcal O$ category.
\begin{enumerate}[label=(\alph*)]
\item\label{lemma:result-x-subsemigroup-1} If $(L,K)(\mathcal C)=(\{0\},\{0\})$, then $X(\mathcal C)=\mathbb{Z}\backslash N_0$ for a sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$.
\item\label{lemma:result-x-subsemigroup-2} If $(L,K)(\mathcal C)=(\emptyset,\{0\})$, then there exists a sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$ such that $X(\mathcal C)=\mathbb{Z}\backslash N_0$ or $X(\mathcal C)=\mathbb{Z}\backslash N_0'$.
\end{enumerate}
\end{proposition}
\begin{proof} Let $(L,K)(\mathcal C)$ be given by $(\{0\},\{0\})$ or $(\emptyset,\{0\})$. We show the two claims jointly in two steps:
\par
\textbf{Step~1:} First, we prove that there exists a sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$ such that $X(\mathcal C)=\mathbb{Z}\backslash N_0$ or $X(\mathcal C)=\mathbb{Z}\backslash N_0'$. That in itself requires two steps as well.
\par
\textbf{Step~1.1:} Recall from \cite[Definition~4.1]{MaWe18a} that by $\mathcal S_0$ we denote the set of all $p\in \mc{P}^{\circ\bullet}_{2}$ with $\sigma_p(B)=0$ and $\delta_p(\alpha,\beta)=0$ for all blocks $B$ of $p$ and all $\alpha,\beta\in B$. We justify that it suffices to prove
\begin{align}
\label{eq:result-x-subsemigroup}
\{|z|\mid z\in X(\mathcal C)\}\backslash\{0\}\overset{!}{\subseteq}\{|z|\mid z\in X(\mathcal C\cap \mathcal S_0)\}\tag{$\ast$}
\end{align}
in order to verify the assertion of Step~1.
\par
Indeed, in \cite[Theorem~8.3, Lem\-mata~8.1~(b) and~7.16~(c)]{MaWe18b} it was shown that for every category $\mathcal I\subseteq \mathcal S_0$ there exists a sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$ such that
\begin{align*}
\{|z|\mid z\in X(\mathcal I)\}\backslash \{0\}=\mathbb{N}\backslash N.
\end{align*}
The set $\mathcal S_0$ is a category by \cite[Proposition~5.3]{MaWe18a}, which means that so is $\mathcal C\cap \mathcal S_0$. Thus, we find a corresponding sub\-se\-mi\-group $N$ for the special case $\mathcal I= \mathcal C\cap \mathcal S_0$. If we now suppose \eqref{eq:result-x-subsemigroup}, which can immediately be sharpened to
\begin{align*}
\{|z|\mid z\in X(\mathcal C)\}\backslash\{0\}=\{|z|\mid z\in X(\mathcal C\cap \mathcal S_0)\}\backslash \{0\},
\end{align*}
that implies
\begin{align*}
\{|z|\mid z\in X(\mathcal C)\}\backslash \{0\}=\mathbb{N}\backslash N.
\end{align*}
As we know $X(\mathcal C)=-X(\mathcal C)$ by Proposition~\ref{lemma:result-s-l-k-x}, this is equivalent to
\begin{align*}
X(\mathcal C)\backslash \{0\}=\mathbb{Z}\backslash N_0'
\end{align*}
and thus the claim of Step~1. Hence, it is indeed sufficient to show \eqref{eq:result-x-subsemigroup}.
\par
\textbf{Step~1.2:} We prove \eqref{eq:result-x-subsemigroup}.
As $\mathcal C\subseteq \mc{P}^{\circ\bullet}_{2}$ by Proposition~\ref{proposition:result-F}, we are assured by Lem\-ma~\ref{lemma:simplification-k-x} and Proposition~\ref{lemma:result-s-l-k-x} that $X(\mathcal C)=X_{c_1,c_2}(\mathcal C\cap \mc{P}^{\circ\bullet}_{2})$ for all $c_1,c_2\in \{\circ,\bullet\}$. Now, let $z\in X(\mathcal C)\backslash \{0\}$ be arbitrary. By definition we find $p\in \mathcal C\cap \mc{P}^{\circ\bullet}_{2}$ and therein crossing blocks $B_1$ and $B_2$ as well as points $\alpha_1\in B_1$ and $\alpha_2\in B_2$ such that $\delta_p(\alpha_1,\alpha_2)=z$. Then, there exist points $\beta_1\in B_1$ and $\beta_2\in B_2$ such that $\alpha_1\neq \beta_1$ and $\alpha_2\neq \beta_2$ and such that either $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ or $(\alpha_2,\alpha_1,\beta_2,\beta_1)$ is ordered in $p$. As $\Sigma(\mathcal C)= \{0\}$ by Proposition~\ref{lemma:result-s-l-k-x} and thus $\Sigma(p)=0$, we know $\delta_p(\alpha_2,\alpha_1)=-\delta_p(\alpha_1,\alpha_2)$ by \cite[Lem\-ma~2.1]{MWNHO1}. Hence, by renaming $B_1$ and $B_2$ if necessary we can, at the cost of weakening $\delta_p(\alpha_1,\alpha_2)=z$ to $|\delta_p(\alpha_1,\alpha_2)|=|z|$, assume that $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ is ordered.
\par As $\mathcal C\cap \mc{P}^{\circ\bullet}_{2}$ is closed under erasing turns and as $(B_1\cup B_2)\cap ]\alpha_1,\alpha_2[_p=\emptyset$ we can further suppose that no turns $T$ exist in $p$ with $T\subseteq ]\alpha_1,\alpha_2[_p$. In other words, there is $c\in \{\circ,\bullet\}$ such that every point in $]\alpha_1,\alpha_2[_p$ has normalized color $c$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{8*\scp*1cm}
\def2*\dx{9*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{3*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\draw [draw=white, fill=white] ({7*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({9*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=white, fill=white] ({6*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({8*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node[cc] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l2) at ({0+6*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (u1) at ({0+2*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+5*\scp*1cm},{0+1*2*\dy}) {};
%
\node[vp] (d1) at ({0+7*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d2) at ({0+9*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (e1) at ({0+6*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
\node[vp] (e2) at ({0+8*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
%
\draw[sstr] (l2) -- ++ (0,{2*\scp*1cm}) -| (u1);
\draw[sstr] (l1) -- ++ (0,{1*\scp*1cm}) -| (u2);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({5*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({3*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({4*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [above = {\scp*0.3cm} of u1] {$\beta_1$};
\node [above = {\scp*0.3cm} of u2] {$\alpha_2$};
\node [below ={\scp*0.3cm} of l1] {$\beta_2$};
\node [below ={\scp*0.3cm} of l2] {$\alpha_1$};
%
\draw [draw=darkgray, densely dashed] ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({2*\dx+0*\scp*1cm+\gx},{0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dashed] ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({-3*\scp*1cm-2*\gx},0) |- ({\txu+0*\scp*1cm+\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
%
\path (d1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d2);
\path (e1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e2);
\node at (-\scp*1cm,{0.5*2*\dy}) {$p$};
\node at ({2*\dx+2*\scp*1cm},{0.5*2*\dy}) {$|\delta_p(\alpha_1,\alpha_2)|=|z|$};
\end{tikzpicture}
\end{mycenter}
Even further, by Lem\-ma~\ref{lemma:projection} none of the previous assumptions are violated by assuming that $p=P(p,[\alpha_1,\beta_1]_p)$. Then, $\beta_2$ is the counterpart of $\alpha_2$ on the upper row, $\alpha_1\in [\beta_2,\alpha_2]_p$ and $\beta_1\notin [\beta_2,\alpha_2]_p$. If we let $\epsilon$ be the predecessor of $\alpha_1$, i.e., if $\epsilon$ is the leftmost upper point of $p$, then $(\beta_2,\epsilon,\alpha_1,\alpha_2,\beta_1)$ is ordered.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{11*\scp*1cm}
\def2*\dx{11*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{5*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\draw [draw=white, fill=white] ({1*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({7*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=white, fill=white] ({1*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({7*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node[cc] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l2) at ({0+8*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l3) at ({0+11*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+8*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u3) at ({0+11*\scp*1cm},{0+1*2*\dy}) {};
%
\node[vp] (d1) at ({0+1*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d3) at ({0+5*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d4) at ({0+7*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (e1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e2) at ({0+3*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e3) at ({0+5*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e4) at ({0+7*\scp*1cm},{0+1*2*\dy}) {$c$};
%
\draw (l1) --++(0,{2*\scp*1cm}) -| (l3);
\draw (u1) --++(0,{-2*\scp*1cm}) -| (u3);
\draw[gray] (d2) -- (e2);
\draw[gray, shorten >= {\scp*0.2cm}] (d3) --++(0,{\scp*1cm}) -| ({9.5*\scp*1cm},0);
\draw[gray, shorten >= {\scp*0.2cm}] (e3) --++(0,{-\scp*1cm}) -| ({9.5*\scp*1cm},{2*\dy});
\draw (l2) to (u2);
%
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({9*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({10*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({9*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({10*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node [below = {\scp*0.3cm} of l1] {$\alpha_1$};
\node [below = {\scp*0.3cm} of l2] {$\alpha_2$};
\node [below ={\scp*0.3cm} of l3] {$\beta_1$};
\node [above ={\scp*0.3cm} of u2] {$\beta_2$};
\node [above ={\scp*0.3cm} of u1] {$\epsilon$};
%
%
\path (d1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d2)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d3)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d4);
\path (e1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e2) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e3) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e4);
\node at ({-\scp*1cm},{0.5*2*\dy}) {$p$};
\end{tikzpicture}
\end{mycenter}
Recall that there are no turns $T$ in $p$ with $T\subseteq ]\alpha_1,\alpha_2[_p$. As $p=p^\ast$, there are none with $T\subseteq ]\beta_2,\epsilon[_p$ either. That means every point in $]\alpha_1,\alpha_2[_p$ has normalized color $c$ and every point in $]\beta_2,\epsilon[_p$ normalized color $\overline{c}$. We can also say a lot about the blocks of $p$ which intersect $[\beta_2,\alpha_2]_p$: If a point $\theta_1\in ]\alpha_1,\alpha_2[_p$ belongs to a through block it must be connected to its counterpart on the upper row because $p\in\mc{P}^{\circ\bullet}_{2}$ is projective. If $\theta_1$ belongs to a non-through block instead, then the partner $\theta_2$ of $\theta_1$ must lie outside $[\beta_2,\alpha_2]_p$: Supposing otherwise, i.e., $\theta_2\in ]\alpha_1,\alpha_2[_p$, produces a contradiction: If $(\alpha_1,\theta_i,\theta_{\neg i},\beta_2)$ with $i,\neg i\in \{1,2\}$ and $\{i,\neg i\}=\{1,2\}$ is ordered, then, as all points in $[\theta_i,\theta_{\neg i}]_p$ are $c$-colored, the consequence $|\delta_p(\theta_i,\theta_{\neg i})|=|]\theta_i,\theta_{\neg i}]_p|>0$ violates $L(\mathcal C)\subseteq \{0\}$, which follows from $K(\mathcal C)=\{0\}$ by Proposition~\ref{lemma:result-s-l-k-x}.
\par
Define $p'\coloneqq P(p,[\beta_2,\alpha_2]_p)\in \mathcal C\cap \mc{P}^{\circ\bullet}_{2}$ and denote by $\beta_2'$, $\epsilon'$, $\alpha_1'$ and $\alpha_2'$ the images in $p'$ of $\beta_2$, $\epsilon$, $\alpha_1$ and $\alpha_2$, respectively. In $p'$ the leftmost lower point $\beta_2'$ and the rightmost lower point $\alpha_2'$ form a block. The points $\epsilon',\alpha_1'\in [\beta_2',\alpha_2']$ are each paired with their respective counterpart on the upper row. In particular the blocks of $\alpha_1'$ and $\alpha_2'$ cross in $p'$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{17*\scp*1cm}
\def2*\dx{17*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{5*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{cc/.style={cross out, minimum size={1.5*\scp*0.25cm-\pgflinewidth}, inner sep=0pt, outer sep=0pt, draw=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 3pt, shorten >= 3pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\draw [draw=white, fill=white] ({1*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({7*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=white, fill=white] ({1*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({7*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
\draw [draw=white, fill=white] ({10*\scp*1cm-\scp*0.2cm},{-\scp*0.2cm}) rectangle ({16*\scp*1cm+\scp*0.2cm},{\scp*0.2cm});
\draw [draw=white, fill=white] ({10*\scp*1cm-\scp*0.2cm},{2*\dy-\scp*0.2cm}) rectangle ({16*\scp*1cm+\scp*0.2cm},{2*\dy+\scp*0.2cm});
%
\node[cc] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l2) at ({0+8*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l3) at ({0+9*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (l4) at ({0+17*\scp*1cm},{0+0*2*\dy}) {};
\node[cc] (u1) at ({0+0*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u2) at ({0+8*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u3) at ({0+9*\scp*1cm},{0+1*2*\dy}) {};
\node[cc] (u4) at ({0+17*\scp*1cm},{0+1*2*\dy}) {};
%
\node[vp] (d1) at ({0+1*\scp*1cm},{0+0*2*\dy}) {$\overline c$};
\node[vp] (d2) at ({0+3*\scp*1cm},{0+0*2*\dy}) {$\overline c$};
\node[vp] (d3) at ({0+5*\scp*1cm},{0+0*2*\dy}) {$\overline c$};
\node[vp] (d4) at ({0+7*\scp*1cm},{0+0*2*\dy}) {$\overline c$};
\node[vp] (e1) at ({0+1*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
\node[vp] (e2) at ({0+3*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
\node[vp] (e3) at ({0+5*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
\node[vp] (e4) at ({0+7*\scp*1cm},{0+1*2*\dy}) {$\overline c$};
%
\node[vp] (d5) at ({0+10*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d6) at ({0+12*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d7) at ({0+14*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (d8) at ({0+16*\scp*1cm},{0+0*2*\dy}) {$c$};
\node[vp] (e5) at ({0+10*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e6) at ({0+12*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e7) at ({0+14*\scp*1cm},{0+1*2*\dy}) {$c$};
\node[vp] (e8) at ({0+16*\scp*1cm},{0+1*2*\dy}) {$c$};
%
\draw (l1) --++(0,{2*\scp*1cm}) -| (l4);
\draw (u1) --++(0,{-2*\scp*1cm}) -| (u4);
\draw (l2) to (u2);
\draw (l3) to (u3);
\draw[gray] (d2) -- (e2);
\draw[gray] (d7) -- (e7);
\draw[gray] (d3) --++(0,{\scp*1cm}) -| (d6);
\draw[gray] (e3) --++(0,{-\scp*1cm}) -| (e6);
%
%
\node [below = {\scp*0.3cm} of l1] {$\beta_2'$};
\node [below = {\scp*0.3cm} of l2] {$\epsilon'$};
\node [below ={\scp*0.3cm} of l3] {$\alpha_1'$};
\node [below ={\scp*0.3cm} of l4] {$\alpha_2'$};
%
%
\path (d1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d2)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d3)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d4);
\path (e1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e2) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e3) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e4);
\path (d5) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d6)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d7)-- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (d8);
\path (e5) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e6) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e7) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (e8);
\node at ({-1*\scp*1cm},{0.5*2*\dy}) {$p'$};
\end{tikzpicture}
\end{mycenter}
Our knowledge about the blocks of $p$ intersecting $[\beta_2,\alpha_2]_p$ lets us draw the following conclusions about the blocks of $p'$: A point in $]\alpha_1',\alpha_2'[_p$ is either partnered with its reflection at the center $[\epsilon',\alpha_1']_{p'}$ of the lower row of $p'$ in $]\beta_2',\epsilon'[_{p'}$ or, as $p'$ is projective, it is partnered with its counterpart on the opposite row. As $]\beta_2',\epsilon'[_{p'}$ is uniformly $\overline c$-colored and $]\alpha_1',\alpha_2'[_{p'}$ uniformly $c$-colored, that means that all blocks emanating from $]\beta_2',\epsilon'[_{p'}\cup ]\alpha_1',\alpha_2'[_p$ are neutral. But then, all blocks of $p'$ are neutral. Due to $L(\mathcal C)\subseteq \{0\}$ and $K(\mathcal C)=\{0\}$, this is already enough to know $p'\in \mathcal S_0$. Because $\delta_{p'}(\alpha_1',\alpha_2')=\delta_p(\alpha_1,\alpha_2)$, that proves $|z|=|\delta_p(\alpha_1,\alpha_2)|=|\delta_{p'}(\alpha_1',\alpha_2)|\in \{|z|\mid z\in X(\mathcal C\cap \mathcal S_0)\}$. As $z$ was arbitrary, \eqref{eq:result-x-subsemigroup} holds true and Part~\ref{lemma:result-x-subsemigroup-2} has been proven.
\par
\textbf{Step~2:} In order to prove Part~\ref{lemma:result-x-subsemigroup-1} it remains to show $0\in X(\mathcal C)$ provided $L(\mathcal C)=\{0\}$. Under this latter assumption, by Proposition~\ref{lemma:result-s-l-k-x} we infer $K_{\circ\circ}(\mathcal C)=\{0\}$. Hence, we find $p\in \mathcal C$, therein a block $B$ and legs $\alpha,\beta\in B$ of normalized color $\circ$ with $\alpha\neq \beta$, with $]\alpha,\beta[_p\cap B=\emptyset$ and with $\delta_p(\alpha,\beta)=0$. As in the proof of Proposition~\ref{lemma:result-x-case-o-positive-w} we can assume that there are no turns $T$ in $p$ such that $T\subseteq]\alpha,\beta[_p$, i.e.\ that all points in $]\alpha,\beta[_p$ have the same normalized color $c\in \{\circ,\bullet\}$. From
\begin{align*}
0=\delta_p(\alpha,\beta)=\sigma_p(]\alpha,\beta]_p)=\sigma_p(]\alpha,\beta[_p)+\sigma_p(\{\beta\})=
\begin{cases}
|]\alpha,\beta[_p|+1&\text{if }c=\circ,\\
-|]\alpha,\beta[_p|+1&\text{otherwise}
\end{cases}
\end{align*}
and from $|]\alpha,\beta[_p|\geq 0$ it follows that $c=\bullet$ and that $]\alpha,\beta[_p$ is a singleton set. Emulating the proof of Proposition~\ref{lemma:result-x-case-o-positive-w} further, we can assume $p=P(p,[\alpha,\beta]_p)$.
\begin{mycenter}[0.5em]
\begin{tikzpicture}[baseline=0.666*1cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{5*\scp*1cm}
\def2*\dx{4*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
\draw[dotted] ({0-\scp*0.5cm},{2*\dy}) -- ({\txu+\scp*0.5cm},{2*\dy});
%
\node[whp] (l1) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
\node[blp] (u1) at ({0+3*\scp*1cm},{0+1*2*\dy}) {};
%
\draw (l1) --++(0,{1*\scp*1cm}) -| (u1);
%
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm-\scp*0.2cm}) -- ++ ({1*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{0+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
\draw [draw=white, fill=white] ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({2*\scp*1cm+2*\scp*0.2cm},0) |- ({0+0*\scp*1cm-\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
\draw [draw=gray, pattern = north east lines, pattern color = lightgray] ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({\txu+0*\scp*1cm+\scp*0.2cm},{2*\dy+0*\scp*1cm+\scp*0.2cm});
%
\node (y1) at ({0*\scp*1cm},{2*\dy}) {$\overline c$};
\node (y2) at ({2*\scp*1cm},{2*\dy}) {$\overline c$};
\path (y1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots\,\hdots$}} (y2);
\node (x1) at ({0*\scp*1cm},{0}) {$c$};
\node (x2) at ({1*\scp*1cm},{0}) {$c$};
\path (x1) -- node[pos=0.5] {\raisebox{-0.25cm}{$\hdots$}} (x2);
%
\node [below ={\scp*0.3cm} of l1] {$\beta$};
\node [above ={\scp*0.3cm} of u1] {$\alpha$};
%
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{0+0*\scp*1cm+\scp*0.4cm});
\draw [draw=darkgray, densely dotted] ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm-\scp*0.4cm}) --++ ({3*\scp*1cm+2*\gx},0) |- ({0+0*\scp*1cm-\gx},{2*\dy+0*\scp*1cm+\scp*0.4cm});
({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm-\scp*0.2cm}) -- ++ ({-1*\scp*1cm-2*\scp*0.2cm},0) |- ({2*\dx+0*\scp*1cm+\scp*0.2cm},{0*\scp*1cm+\scp*0.2cm});
%
\node at ({\txu+\scp*1cm},{0.5*2*\dy}) {$p$};
\node at (-1*\scp*1cm,{0.5*2*\dy}) {$\delta_p(\alpha,\beta)=0$};
\end{tikzpicture}
\qquad$\to$\qquad
\begin{tikzpicture}[baseline=0.666*0.5cm-0.25em]
\def0.666{0.666}
\def\scp*0.075cm{0.666*0.075cm}
\def\scp*0.25cm{0.666*0.25cm}
%
\def\scp*0.5cm{0.666*0.5cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def\txu{2*\scp*1cm}
\def2*\dx{2*\scp*1cm}
\def\scp*1cm{0.666*1cm}
\def\scp*0.3cm{0.666*0.3cm}
\def2*\dy{2*\scp*1cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.2cm{0.666*0.2cm}
\def\scp*0.4cm{0.666*0.4cm}
\def\gx{0.666*0.4cm}
%
\tikzset{whp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=white}}
\tikzset{blp/.style={circle, inner sep=0pt, text width={\scp*0.25cm}, draw=black, fill=black}}
\tikzset{lk/.style={regular polygon, regular polygon sides=4, inner sep=0pt, text width={\scp*0.075cm}, draw=black, fill=black}}
\tikzset{vp/.style={circle, inner sep=0pt, text width={1.5*\scp*0.25cm}, fill=white}}
\tikzset{sstr/.style={shorten <= 5pt, shorten >= 5pt}}
%
%
\draw[dotted] ({0-\scp*0.5cm},{0}) -- ({2*\dx+\scp*0.5cm},{0});
%
\node[whp] (l1) at ({0+0*\scp*1cm},{0+0*2*\dy}) {};
\node[blp] (l2) at ({0+1*\scp*1cm},{0+0*2*\dy}) {};
\node[whp] (l3) at ({0+2*\scp*1cm},{0+0*2*\dy}) {};
%
\draw (l1) -- ++ (0,{\scp*1cm}) -| (l3);
\draw (l2) -- ++ (0, {1.5*\scp*1cm});
%
%
%
\node [below ={\scp*0.3cm} of l1] {$\lop{1}$};
\node [below ={\scp*0.3cm} of l3] {$\lop{3}$};
%
%
\node at ({\txu+2*\scp*1cm},{0.5*2*\dy}) {$P(p,[\alpha,\beta]_p)$};
\end{tikzpicture}
\end{mycenter}
\noindent
Then, the lower row $[\alpha,\beta]_p=[\lop{1},\lop{3}]_p$ of $p$ has coloration $\circ\bullet\circ$. As $p\in \mc{P}^{\circ\bullet}_{2}$ and as $p$ is projective, the block of $\lop{2}$ is the pair $\{\lop 2, \upp2\}$. That means the blocks of $\alpha=\lop{1}$ and $\lop 2$ cross, implying $0=\delta_p(\lop{1},\lop 2)\in X(\mathcal C)$. That concludes the proof.
\end{proof}
\section{Step~6: Synthesis}
Combining the results from Sec\-tions~\ref{section:block-sizes}--\ref{section:special-restrictions}, we are able to show the main theorem.
\begin{theorem}
\label{theorem:main}
$Z(\mathcal C)\in \mathsf Q$ for every non-hy\-per\-octa\-he\-dral category $\mathcal C\subseteq \mc{P}^{\circ\bullet}$.
\end{theorem}
\begin{proof}
By Lem\-ma~\ref{lemma:result-s-l-k-x} there exist $u\in \{0\}\cup \mathbb{N}$, $m\in \mathbb{N}$, $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$ and $E\subseteq \{0\}\cup\mathbb{N}$ such that the tuple $(\Sigma,L,K,X)(\mathcal C)$ is given by one of the following:
\begin{align}
\label{eq:main-table-1}
\begin{matrix}
\Sigma& L&K& X \\ \hline \\[-0.85em]
um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
2um\mathbb{Z} & m\!+\!2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0
\end{matrix}\tag{$\ast$}
\end{align}
We treat the three cases $\mathcal O$, $\mathcal B$ and $\mathcal S$ individually. The formulaic presentation will mirror that of Definition~\ref{definition:Q} exactly, to facilitate cross-checking.
\par
\textbf{Case~$\mathcal B$:} First, let $\mathcal C$ be case~$\mathcal B$. Proposition~\hyperref[proposition:result-F-2]{\ref*{proposition:result-F}~\ref*{proposition:result-F-2}} implies $F(\mathcal C)=\{1,2\}$. So, we can immediately add the column for $F(\mathcal C)$ to table \eqref{eq:main-table-1}. Further, Proposition~\hyperref[proposition:result-V-2]{\ref*{proposition:result-V}~\ref*{proposition:result-V-2}} shows $V(\mathcal C)=\pm\{0,1,2\}$ if and only if $L(\mathcal C)\neq \emptyset$ and $V(\mathcal C)=\pm\{0,1\}$ otherwise. That allows us to fill in the column for $V(\mathcal C)$ as well. The result is that $Z(\mathcal C)$ concurs with a row of the table
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{1,2\}&\pm\{0, 1, 2\} & um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{1,2\}&\pm\{0, 1, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{1,2\}&\pm \{0, 1\} & um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m \\
\{1,2\}&\pm\{0, 1, 2\} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{1,2\}&\pm\{0, 1\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0\\
\end{matrix}
\end{align*}
for some $u\in \{0\}\cup \mathbb{N}$, $m\in \mathbb{N}$, $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$ and $E\subseteq \{0\}\cup\mathbb{N}$.
Hence, by Definition~\ref{definition:Q}, we have shown $Z(\mathcal C)\in \mathsf Q$ if $\mathcal C$ is case~$\mathcal B$.
\par
\textbf{Case~$\mathcal S$:} Next, let $\mathcal C$ be case~$\mathcal S$. Propositions~\hyperref[proposition:result-F-3]{\ref*{proposition:result-F}~\ref*{proposition:result-F-3}} and~\hyperref[proposition:result-V-3]{\ref*{proposition:result-V}~\ref*{proposition:result-V-3}} guarantee $F(\mathcal C)=\mathbb{N}$ and $V(\mathcal C)=\mathbb{Z}$. Hence, we can fill in the columns for $F$ and $V$ in \eqref{eq:main-table-1} once more. Moreover, $0\in L(\mathcal C)$ by Proposition~\ref{lemma:restriction-case-s}. Thus, we can exclude that $(\Sigma,L,K,X)(\mathcal C)$ is given by the second, third or fifth rows of \eqref{eq:main-table-1}.
In other words, there are $u\in \{0\}\cup \mathbb{N}$, $m\in \mathbb{N}$, $D\subseteq \{0\}\cup\dwi{\lfloor\frac{m}{2}\rfloor}$ and $E\subseteq \{0\}\cup\mathbb{N}$ such that $Z(\mathcal C)$ is given by one of the rows of the following table:
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\mathbb{N} & \mathbb{Z} & um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\mathbb{N} & \mathbb{Z} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0\\
\end{matrix}
\end{align*}
And, by Definition~\ref{definition:Q}, this means $Z(\mathcal C)\in\mathsf Q$ for $\mathcal C$ in case~$\mathcal S$.
\par
\textbf{Case~$\mathcal O$:} Lastly, let $\mathcal C$ be case~$\mathcal O$. Once more, Propositions~\hyperref[proposition:result-F-1]{\ref*{proposition:result-F}~\ref*{proposition:result-F-1}} and~\hyperref[proposition:result-V-1]{\ref*{proposition:result-V}~\ref*{proposition:result-V-1}} give, on the one hand, $F(\mathcal C)=\{2\}$ and, on the other hand, $V(\mathcal C)=\pm\{0,2\}$ if $L(\mathcal C)\neq \emptyset$ and $V(\mathcal C)=\{0\}$ otherwise. That enables us to fill in the columns for $F(\mathcal C)$ and $V(\mathcal C)$ in \eqref{eq:main-table-1}:
\begin{align}
\label{eq:main-table-2}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \pm\{0, 2\} & um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash D_m \\
\{2\} & \{0\} & um\mathbb{Z} & \emptyset & m\mathbb{Z} & \mathbb{Z}\backslash D_m\\
\{2\} & \pm\{0, 2\} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash E_0 \\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash E_0 \\
\end{matrix}\tag{$\ast\ast$}
\end{align}
This is not yet what we claim as this range is not contained in $\mathsf Q$. We need to exclude certain values for $u$, $D$ and $E$ by taking into account the results of Sec\-tion~\ref{section:special-restrictions-o}. This we shall do on a row-by-row basis.
\par
\textbf{Case~$\mathcal O$.1:} First, suppose $(L,K)(\mathcal C)=(m\mathbb{Z},m\mathbb{Z})$ for some $m\in \mathbb{N}$, as in the first row of Table~\eqref{eq:main-table-2}. Then $\Sigma(\mathcal C)\subseteq 2m\mathbb{Z}$ (corresponding to parameters $u\in 2\mathbb{Z}$) according to Pro\-po\-si\-tion~\hyperref[lemma:k-v-l-2]{\ref*{lemma:k-v-l}~\ref*{lemma:k-v-l-2}}. Moreover, $X(\mathcal C)=\mathbb{Z}$ (corresponding to $D=\emptyset$) as seen in
Pro\-po\-si\-tion~\hyperref[lemma:result-x-case-o-positive-w-2]{\ref*{lemma:result-x-case-o-positive-w}~\ref*{lemma:result-x-case-o-positive-w-2}}. Hence, we can replace the first row of Table~\eqref{eq:main-table-2} by
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}
\end{matrix}
\end{align*}
still for parameters $u\in \{0\}\cup \mathbb{N}$ and $m\in \mathbb{N}$ exactly as before.
\par
\textbf{Case~$\mathcal O$.2:} Now, proceeding to the second row of Table~\eqref{eq:main-table-2}, let $(L,K)(\mathcal C)=(m\!+\!2m\mathbb{Z},2m\mathbb{Z})$ for some $m\in \mathbb{N}$. By Proposition~\hyperref[lemma:result-x-case-o-positive-w-1]{\ref*{lemma:result-x-case-o-positive-w}~\ref*{lemma:result-x-case-o-positive-w-1}} the only two values $X(\mathcal C)$ can possibly take are $\mathbb{Z}$ and $\mathbb{Z}\backslash m\mathbb{Z}$ (corresponding to $D=\emptyset$ and $D=\{0\}$, respectively). Thus, we can delete the second row of Table~\eqref{eq:main-table-2} and insert the two new rows
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z} \\
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash m\mathbb{Z} \\
\end{matrix}
\end{align*}
in its stead, still for parameters $m\in \mathbb{N}$ and $u\in \{0\}\cup \mathbb{N}$.
\par
\textbf{Case~$\mathcal O$.3:} Next, assume $(L,K)(\mathcal C)=(\emptyset,m\mathbb{Z})$ for some $m\in\mathbb{N}$ as in row three of Table~\eqref{eq:main-table-2}. Then, in fact, $\Sigma(\mathcal C)=\{0\}$ as seen in Proposition~\hyperref[lemma:k-v-l-1]{\ref*{lemma:k-v-l}~\ref*{lemma:k-v-l-1}}. Furthermore, $X(\mathcal C)=\mathbb{Z}$ by Proposition~\hyperref[lemma:result-x-case-o-positive-w-2]{\ref*{lemma:result-x-case-o-positive-w}~\ref*{lemma:result-x-case-o-positive-w-2}}. Hence, we rewrite the third row of \eqref{eq:main-table-2} as
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \{0\} & \{0\} & \emptyset & m\mathbb{Z} & \mathbb{Z}
\end{matrix}
\end{align*}
depending only on the parameter $m\in \mathbb{N}$.
\par
\textbf{Case~$\mathcal O$.4:} Let $(L,K)(\mathcal C)=(\{0\},\{0\})$, i.e., consider the fourth row of Table~\eqref{eq:main-table-2}. Then, $X(\mathcal C)=\mathbb{Z}\backslash N_0$ for some sub\-se\-mi\-group of $(\mathbb{N},+)$ by Proposition~\hyperref[lemma:result-x-subsemigroup-1]{\ref*{lemma:result-x-subsemigroup}~\ref*{lemma:result-x-subsemigroup-1}} (corresponding to $E=N$ being a sub\-se\-mi\-group). Accordingly, we can replace the fourth row of Table~\eqref{eq:main-table-2} by
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \pm\{0,2\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash N_0
\end{matrix}
\end{align*}
for a new table parameter $N$, running through all sub\-se\-mi\-groups of $(\mathbb{N},+)$.
\par
\textbf{Case~$\mathcal O$.5:} Lastly, suppose $(L,K)(\mathcal C)=(\emptyset,\{0\})$ as in the fifth row of Table~\eqref{eq:main-table-2}. In Proposition~\hyperref[lemma:result-x-subsemigroup-1]{\ref*{lemma:result-x-subsemigroup}~\ref*{lemma:result-x-subsemigroup-1}} we showed $X(\mathcal C)$ is of the form $\mathbb{Z}\backslash N_0$ or $\mathbb{Z}\backslash N_0'$ for some sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$ (corresponding to $E=N$ and $E=\{0\}\cup N$, respectively). Thus, strike the last row of Table~\eqref{eq:main-table-2} and append the two rows
\begin{align*}
\begin{matrix}
F&V&\Sigma&L&K& X \\ \hline \\[-0.85em]
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash N_0\\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash N_0'
\end{matrix}
\end{align*}
to the table, with $N$ being a sub\-se\-mi\-group of $(\mathbb{N},+)$.
\par
\textbf{Synthesis in case~$\mathcal O$:} If we combine the results of Cases~1--5, then we can say that there exist $m\in \mathbb{N}$, $u\in \{0\}\cup \mathbb{N}$ and a sub\-se\-mi\-group $N$ of $(\mathbb{N},+)$ such that $Z(\mathcal C)$ is given by one of the rows of the following table:
\begin{align*}
\begin{matrix}
F&V&S&L&K& X \\ \hline \\[-0.85em]
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\mathbb{Z} & m\mathbb{Z} & \mathbb{Z}\\
\{2\} & \pm\{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z} \\
\{2\} & \pm \{0, 2\} & 2um\mathbb{Z} & m\hspace{-2.5pt}+\hspace{-2.5pt}2m\mathbb{Z} & 2m\mathbb{Z} & \mathbb{Z}\backslash m\mathbb{Z} \\
\{2\} & \{0\} & \{0\} & \emptyset & m\mathbb{Z} & \mathbb{Z}\\
\{2\} & \pm\{0, 2\} & \{0\} & \{0\} & \{0\} & \mathbb{Z}\backslash N_0 \\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} & \mathbb{Z}\backslash N_0 \\
\{2\} & \{0\} & \{0\} & \emptyset & \{0\} &\mathbb{Z}\backslash N_0' \\
\end{matrix}
\end{align*}
Definition~\ref{definition:Q} thus yields $Z(\mathcal C)\in \mathsf Q$ if $\mathcal C$ is case~$\mathcal O$. Hence, the overall claim is true.
\end{proof}
\printbibliograph
\end{document}
|
2,869,038,154,836 | arxiv |
\section{Introduction}
\label{sec:introduction}
\noindent Susceptible-Infected-Removed (SIR)
is a classical
differential equation
model of infectious diseases \cite{anderson_may_1992}. It divides the total population into three compartments and models their evolution by the system of equations
$$\frac{dS}{dt}=\, -\beta \, I\, S$$
$$\frac{dI}{dt}=\, \beta \, I\, S-\gamma\, I$$
$$\frac{dR}{dt}=\, \gamma \, I$$
where $\beta$ and $\gamma$ are two positive parameters. SIR is a simple and efficient model of temporal data for a given region, see also \cite{hethcote_2000} for related compartment models with social structures.
\medskip
Yet the infectious disease data are often spatio-temporal as in
the case of COVID-19, see \cite{Italy_data}.
A natural question is how to extend SIR to a space time model of suitable complexity so that it can be quickly trained from the available public data sets and applied in real-time forecasts. See
\cite{roosa_2020} for temporal model real-time forecasts on cumulative cases of China in Feb 2020.
In this paper, we explore spatial infectious disease information to model the latent effect due to the in-flow of the infected people from the geographical neighbors. The in-flow data is not observed. To this end, machine learning tools such as regression and neural network models are more convenient.
Auto-regressive model (AR) and its variants are linear statistical models to forecast time-series data.
The Long Short Term Memory (LSTM) neural networks, originally designed for natural language processing \cite{hochreiter1997long}, have more representation power and can be applied to disease time-series data as well. With spatial structures added,
the graph-structured LSTM models can achieve state-of-the-art performance on spatiotemporal influenza data \cite{li_2019}, crime and traffic data \cite{wang,wang2018graph}.
However, they require a large enough supply of training data.
For COVID-19, we only have limited daily data since the outbreaks began in early 2020. Applying space-time LSTM models \cite{li_2019,wang2018graph} directly to COVID-19 turns out to produce poor results.
In view of the limited COVID-19 data, we shall propose a hybrid SIR-LSTM model.
\section{Related Work}
\noindent In \cite{yang2015accurate}, the authors designed a variant of AR, the AutoRegression with Google search data (ARGO), that utilizes external feature of google search data to forecast influenza data from Centers for Disease Control and Prevention (CDC). Based on google search trend data that correlated to influenza as external feature, ARGO is a linear model that processes historical observations and external features. The prediction of influenza activity level at time $t$, defined as $\hat{y}_t$, is given by:
$$\hat{y}_t=u_t+\sum_{j=1}^{52}\alpha_jy_{t-j}+\sum_{i=1}^{100}\beta_iX_{i,t}.$$
ARGO is optimized as:
$$\min_{\mu_y,\vec{\alpha},\vec{\beta}}\Big(y_t-u_t-\sum_{j=1}^{52}\alpha_jy_{t-j}-\sum_{i=1}^{100}\beta_iX_{i,t}\Big)^2$$
$$+\lambda_a||\vec{\alpha}||_1+\eta_a||\vec{\beta}||_1+\lambda_b||\vec{\alpha}||_2^2+\eta_b||\vec{\beta}||_2^2$$
where $\vec{\alpha}=(\alpha_1,\cdots,\alpha_{52})$ and $\vec{\beta}=(\beta_1,\cdots,\beta_{100})$. The $y_{t-j}$'s, $ 1\leq j\leq 52$, are historical observations of previous 52 weeks and $X_{i,t}$ are the google search trend measures of top 100 terms that are most correlated to influenza at time $t$.
Essentially, ARGO is a linear regression with regularization terms. In \cite{yang2015accurate}, ARGO is shown to outperform standard machine learning models such as LSTM, AR, and ARIMA.
\medskip
In \cite{li_2019}, graph structured recurrent neural network (GSRNN) further improved ARGO in the forecasting accuracy of CDC influenza activity level. The CDC partitions the US into 10 Health and Human
Services (HHS) regions for reporting. GSRNN treats the 10 regions as a graph with nodes $v_1,\cdots,v_{10}$, and $E$ be the collection of edges (i.e $E=\{(v_i,v_j)|v_i, v_j \,\, \text{are adjacent}\}$). Based on the average history of activity levels, the 10 HHS regions are divided into two groups, relatively active group, $\mathcal{H}$, and relatively inactive group, $\mathcal{L}$. There are 3 types of edges, $\mathcal{L}-\mathcal{L}$, $\mathcal{H}-\mathcal{L}$, and $\mathcal{H}-\mathcal{H}$, and each edge type has a corresponding RNN to train the edge features. There are also two node RNNs for each group to output the final prediction. Given a node (region) $v$, suppose $v\in \mathcal{H}$. GSRNN generates the edge features of $v$ at time $t$, $e_{v,\mathcal{H}}^t$ and $e_{v,\mathcal{L}}^t$, by averaging the history of neighbors of $v$ in the corresponding groups. Next, the edge features are fed into the corresponding edge RNNs:
$$f_v^t=\text{edgeRNN}_{\mathcal{H}-\mathcal{L}}(e_{v,\mathcal{L}}^t), \ \ h_v^t=\text{edgeRNN}_{\mathcal{H}-\mathcal{H}}(e_{v,\mathcal{L}}^t)$$
Then, the outputs of edgeRNNs are fed into the nodeRNN of group $\mathcal{H}$ together with the node feature of $v$ at time $t$, denoted as $v^t$, to output the prediction of the activity level of node $v$ at time $t+1$, or $y_{v}^{t+1}$:
$$y_{v}^{t+1}=\text{nodeRNN}_{\mathcal{H}}(v^t,f_v^t,h_v^t).$$
\section{Our Contribution: IeRNN model}
We propose a novel spatiotemporal model integrating LSTM \cite{hochreiter1997long} with a discrete time I-equation derived from SIR differential equations. The LSTM is utilized to model latent spatial information. The I-equation models the observed temporal information. Our model, named IeRNN, differs from \cite{li_2019,wang,wang2018graph} in that a difference equation
with 3 parameters (the I-equation)
fits the limited temporal data, which is far more compact than LSTM.
\subsection{Derivation of Discrete-Time I-Equation}
Based on SIR model, we add an additional feature $I_e$ that represents the external infection influence from the neighbors of a region. Then the SIR nonlinear system with $I_e$ as external forcing becomes
\begin{equation}
\frac{dS}{dt}=\, -\beta_1\, S\, I-\, \beta_2\, S\, I_e
\label{eq:I1}
\end{equation}
\begin{equation}
\frac{dI}{dt}=\, \beta_1\, S\, I+\, \beta_2\, S\, I_e-\, \gamma \, I
\label{eq:I2}
\end{equation}
\begin{equation}
\frac{dR}{dt}=\gamma \, I
\label{eq:I3}
\end{equation}
which conserves the total mass (normalized to 1): $ S+I + R=1$.
It follows from (\ref{eq:I3}) that
$$R(t)=R(t_0)+\int_{t_0}^{t}\gamma \, I \, d\tau$$
Hence,
$$S(t)=1-I(t)-R(t_0)-\, \gamma\, \int_{t_0}^t \, I \, d\tau$$
Substituting $S(t)$ into (\ref{eq:I2}) we have:
$$\frac{dI}{dt}=(\beta_1\, I+\beta_2\, I_e)\left (1-I(t)-R(t_0)-\gamma \int_{t_0}^tI(\tau) d\tau\right ) - \gamma I$$
Combining forward Euler method and Riemann sum approximation of the integral, we have a discrete approximation:
$$I(t+1)=(1-\gamma)I(t)+
\big(\beta_1I(t) +\beta_2I_e(t)\big)$$ $$\cdot \Big(1-I(t)-R(t_0)-\gamma\frac{t-t_0}{p+1}\sum_{j=0}^{p}I(t-j)\Big)$$
As we model $I(t)$ from the beginning of the infection, we have $t_0=0$ and $R(t_0)=0$. We arrive at the following discrete time I-equation:
\[ I(t+1)=(1-\gamma)I(t)+\big(\beta_1I(t)+\beta_2I_e(t)\big)\]
\begin{equation}
\cdot \Big(1-I(t)-\gamma\frac{t}{p+1}\sum_{j=0}^{p}I(t-j)\Big)
\label{eq:sire}
\end{equation}
Note that if we let $I_e(t)=0$, then we have an approximation of $I(t)$ for the original SIR model, which is a solely temporal model (named I-model):
\[
I(t+1)= (1-\gamma-\beta)I(t)-\beta\, I^2(t)\]
\begin{equation}
-\beta \, \gamma\, \frac{t}{p+1}\, I(t)\,
\sum_{j=0}^{p}I(t-j) \label{Ieq}
\end{equation}
In reality, it is hard to know how a population of a region interacts with populations of neighboring regions. As a result, $I_e(t)$ is a latent information that is difficult to model by a mathematical formula or equation. In order to retrieve latent spatial information, we
employ recurrent neural networks made of LSTM cells \cite{hochreiter1997long}, see Fig. \ref{fig:lstm}.
\subsection{Generating Edge Feature and Computing Latent $I_e$}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\includegraphics[scale=0.35]{italy_mapD.png}
\caption{Italian Region Map}
\label{fig:map}
\end{center}
\end{figure}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\scalebox{0.85}{%
\begin{tikzpicture}[
font=\sf \scriptsize,
>=LaTeX,
cell/.style=
rectangle,
rounded corners=5mm,
draw,
very thick,
},
operator/.style=
circle,
draw,
inner sep=-0.5pt,
minimum height =.2cm,
},
function/.style=
ellipse,
draw,
inner sep=1pt
},
ct/.style=
circle,
draw,
line width = .75pt,
minimum width=1cm,
inner sep=1pt,
},
gt/.style=
rectangle,
draw,
minimum width=4mm,
minimum height=3mm,
inner sep=1pt
},
mylabel/.style=
font=\scriptsize\sffamily
},
ArrowC1/.style=
rounded corners=.25cm,
thick,
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ArrowC2/.style=
rounded corners=.5cm,
thick,
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\node [cell, minimum height =4cm, minimum width=6cm] at (0,0){} ;
\node [gt] (ibox1) at (-2,-0.75) {$\sigma$};
\node [gt] (ibox2) at (-1.5,-0.75) {$\sigma$};
\node [gt, minimum width=1cm] (ibox3) at (-0.5,-0.75) {Tanh};
\node [gt] (ibox4) at (0.5,-0.75) {$\sigma$};
\node [operator] (mux1) at (-2,1.5) {$\times$};
\node [operator] (add1) at (-0.5,1.5) {+};
\node [operator] (mux2) at (-0.5,0) {$\times$};
\node [operator] (mux3) at (1.5,0) {$\times$};
\node [function] (func1) at (1.5,0.75) {Tanh};
\node[ct, label={[mylabel]}] (c) at (-4,1.5) {\empt{c}{t-1}};
\node[ct, label={[mylabel]}] (h) at (-4,-1.5) {\empt{h}{t-1}};
\node[ct, label={[mylabel]left:Input}] (x) at (-2.5,-3) {\empt{x}{t}};
\node[ct, label={[mylabel]}] (c2) at (4,1.5) {\empt{c}{t}};
\node[ct, label={[mylabel]}] (h2) at (4,-1.5) {\empt{h}{t}};
\node[ct, label={[mylabel]left:Output}] (x2) at (2.5,3) {\empt{h}{t}};
connecting all.
\draw [ArrowC1] (c) -- (mux1) -- (add1) -- (c2);
\draw [ArrowC2] (h) -| (ibox4);
\draw [ArrowC1] (h -| ibox1)++(-0.5,0) -| (ibox1);
\draw [ArrowC1] (h -| ibox2)++(-0.5,0) -| (ibox2);
\draw [ArrowC1] (h -| ibox3)++(-0.5,0) -| (ibox3);
\draw [ArrowC1] (x) -- (x |- h)-| (ibox3);
\draw [->, ArrowC2] (ibox1) -- (mux1);
\draw [->, ArrowC2] (ibox2) |- (mux2);
\draw [->, ArrowC2] (ibox3) -- (mux2);
\draw [->, ArrowC2] (ibox4) |- (mux3);
\draw [->, ArrowC2] (mux2) -- (add1);
\draw [->, ArrowC1] (add1 -| func1)++(-0.5,0) -| (func1);
\draw [->, ArrowC2] (func1) -- (mux3);
\draw [-, ArrowC2] (mux3) |- (h2);
\draw (c2 -| x2) ++(0,-0.1) coordinate (i1);
\draw [-, ArrowC2] (h2 -| x2)++(-0.5,0) -| (i1);
\draw [-, ArrowC2] (i1)++(0,0.2) -- (x2);
\end{tikzpicture}
}
\caption{LSTM cell: input $x_t$, output: $h_t$; $\sigma$ is a sigmoid function.}
\label{fig:lstm}
\end{center}
\end{figure}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\includegraphics[scale=0.6]{stacked_LSTM_1.png}
\caption{Edge RNN consisting of 3 stacked LSTM cells.}
\label{fig:ernn}
\end{center}
\end{figure}
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1.7]{SIR_LSTM2.pdf}
\caption{Computing $I_e$ of Lazio region. Edge RNN is as shown in Fig. \ref{fig:ernn}. Dense layer is fully connected (see Fig. \ref{fig:fcNN}).}
\label{fig:edge}
\end{figure*}
We utilize the spatial information based on the Italy region map, Fig. \ref{fig:map}. In order to learn the latent information $I_e$ of a region $v$, we first generate the edge feature of $v$. Let $C$ be the collection of neighbors of $v$. Then, the edge feature of $v$ at time $t$ is formulated as:
$$f_e^t=\frac{1}{|C|}\; \sum_{i: \, v_i\in C}\; \big[I_i(t-1),\cdots,I_i(t-p)\big]$$
where $I_i(t)$ is the infection population percentage of region $v_i$ at time $t$.
Then, we feed $f_e^t$ into an Edge RNN, an RNN with 3 stacked LSTM cells (see Fig. \ref{fig:ernn}), followed by a dense layer for computing $I_e$. The activation function of the dense layer is
hyperbolic tangent function. Figure \ref{fig:edge} illustrates the procedure of computing $I_e$ for Lazio as an illustration. We hence {\it call our model IeRNN due to its integrated design of I-equation and edge RNN.}
\section{Experiment}
To calibrate our model IeRNN, we use the Italy COVID-19 data \cite{Italy_data} for training and testing. Although the US has the most infected cases, the recovered cases are largely missing. On the other hand, the Italian COVID-19 data is more accurately reported and better maintained, reflecting a nearly complete duration of the rise and fall of infection. We collect the data of daily new (current) cases from 2020-02-24 to 2020-06-18 of 20 Italian regions. We set $p=3$ in \eqref{eq:sire} based on experimental performance. As a result, we have the current data for 113 days, with 81 days to train our model and 32 days to test our model (or 70\%/30\% training/testing data split). Our training loss function is the mean squared error of the model output and training data:
$$\hat{y}(t)=(1-\gamma)y(t-1)
+\big(\beta_1y(t-1)+\beta_2I_e(t-1)\big)$$
$$\cdot \Big(1-y(t-1)-\gamma\frac{t}{p+1}\sum_{j=1}^{p+1}y(t-j)\Big)$$
$$Loss=\frac{1}{T-p-1}\sum_{t=p+1}^T\big(y(t)-\hat{y}(t)\big)^2$$
\medskip
{\it Since the training is minimization of the above loss fucntion over parameters in both I-equation and RNN, the two components of IeRNN are coupled while learning from data}.
We use Adam gradient descent to learn the weights of LSTM and the dense layer, as well as I-equation parameters
$\beta_1$, $\beta_2$, and $\gamma$.
\medskip
To evaluate the performance of our model, we compare IeRNN, I-model (\ref{Ieq}), a fully-connected neural network (fcNN, Fig. \ref{fig:fcNN}) with hyperbolic tangent activation function, and auto-regression model (ARIMA).
As the standard setting of ARIMA is 1-day ahead prediction, we shall only compare with it in such a very short-term case. Since infectious disease evolution is intrinsically nonlinear,
we shall compare nonlinear models for 3-day and 1-week
ahead forecasting.
Based on experimental performance, we set the number of hidden units to be 100, 150, and 100 for the three layers of fcNN respectively.
\tikzset{%
every neuron/.style={
circle,
draw,
minimum size=1cm
},
neuron missing/.style={
draw=none,
scale=4,
text height=0.333cm,
execute at begin node=\color{black}$\vdots$
},
}
\begin{figure}[ht!]
\caption{Schematic of fcNN for modeling time series.}
\begin{center}
\scalebox{0.6}{%
\begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth]
\foreach \m/\l [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2-\y) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden1-\m) at (1.5,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden2-\m) at (3,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden3-\m) at (4.5,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1}
\node [every neuron/.try, neuron \m/.try ] (output-\m) at (6,0) {};
\foreach \l [count=\i] in {1,p}
\draw [<-] (input-\i) -- ++(-1,0)
node [above, midway] {$\pmb{y_{t-\l}}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden1-\i.north) {$\pmb{w^{(1)}_\l}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden2-\i.north) {$\pmb{w^{(2)}_\l}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden3-\i.north) {$\pmb{w^{(3)}_\l}$};
\foreach \l [count=\i] in {1}
\node [above] at (output-\i.north) {$\pmb{tanh}$};
\foreach \l [count=\i] in {1}
\draw [->] (output-\i) -- ++(1,0)
node [above, midway] {$\pmb{y_t}$};
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (input-\i) -- (hidden1-\j);
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (hidden1-\i) -- (hidden2-\j);
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (hidden2-\i) -- (hidden3-\j);
\foreach \i in {1,2}
\foreach \j in {1}
\draw [->] (hidden3-\i) -- (output-\j);
\foreach \l [count=\x from 0] in {\textbf{Input}, \textbf{Hidden}, \textbf{Output}}
\node [align=center, above] at (\x*3,2.2) {\l \\ \textbf{layer}};
\end{tikzpicture}
}
\end{center}
\label{fig:fcNN}
\end{figure}
\newpage
\begin{figure*}[ht!]
\caption{Training and 1-day ahead forecast of 4 models (IeRNN, fcNN, I-model, and ARIMA) in 4 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio.png}}
\quad \quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc.png}}
\quad \quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR.png}}
\quad \quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_AR.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_AR.png}}
\quad \quad
\vspace{0.5 in}
\label{fig:results}
\end{figure*}
\newpage
\begin{figure*}[ht!]
\caption{Training and
1-day ahead forecast
of 4 models with reduced (40\%) training data. The 4 rows are IeRNN, fcNN, I-model, and ARIMA respectively.}
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_AR1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_AR1.png}}
\label{fig:results2}
\end{figure*}
\newpage
\begin{figure}[]
\caption{IeRNN training and 1-day ahead forecast on four additional regions.}
\centering
\subfloat[Piemonte]{\includegraphics[scale=0.45]{Piemonte.png}}
\qquad
\subfloat[Campania]{\includegraphics[scale=0.45]{Campania.png}}
\qquad
\subfloat[Molise]{\includegraphics[scale=0.45]{Molise.png}}
\qquad
\subfloat[Umbria]{\includegraphics[scale=0.45]{Umbria.png}}
\qquad
\label{fig:additional}
\end{figure}
\begin{figure}[]
\caption{Visualization of the latent information $I_e$ in Fig. \ref{fig:additional} learned by IeRNN.}
\centering
\subfloat[Piemonte]{\includegraphics[scale=0.45]{Piemonte_Ie.png}}
\qquad
\subfloat[Campania]{\includegraphics[scale=0.45]{Campania_Ie.png}}
\qquad
\subfloat[Molise]{\includegraphics[scale=0.45]{Molise_Ie.png}}
\qquad
\subfloat[Umbria]{\includegraphics[scale=0.45]{Umbria_Ie.png}}
\label{fig:Ie}
\end{figure}
\newpage
\begin{figure*}[ht!]
\caption{Training and 7-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio7.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc7.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR7.png}}
\quad
\label{fig:result 7-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 7-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) with reduced (40\%) training data in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio71.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc71.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR71.png}}
\quad
\label{fig:result2 7-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 3-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio2.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc2.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR2.png}}
\quad
\label{fig:result 3-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 3-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) with reduced (40 \%) training data in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio22.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc22.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR22.png}}
\quad
\label{fig:result 3-day1}
\end{figure*}
\subsection{One-Day Ahead Forecast}
As we see in Fig. \ref{fig:results}, fcNN can perform poorly. This is not a surprise, as both \cite{li_2019} and \cite{yang2015accurate} relied on hundreds of historical observations to train their models. The I-model based on only sequential data in time of one region merely follows the trend of the true data but cannot provide accurate predictions. Our IeRNN model, with the help of additional spatial information, is able to make accurate predictions and outperform other models. We also test the IeRNN with training data reduced to 40\% (46 days). IeRNN is still able to track the general trend of the infected population percentage.
\medskip
We measure the test accuracy with the Root Mean Square Error (RMSE)
averaged over a few trials in training. In Tables \ref{tab:error1a} and \ref{tab:error1b} on 1-day ahead forecast, IeRNN achieves the smallest RMSE errors, and I-model has the largest errors. The compact I-model with 2 parameters cannot do 1-day ahead prediction as accurately. ARIMA outperforms I-model and does better on Emilia-Romagna and Lazio regions than fcNN. ARIMA, a linear model, has simpler structure than fcNN whose nonlinearity does not play out in such a short time task. Fig. \ref{fig:additional} shows
1-day ahead forecast of IeRNN model on other regions with the learned latent external forcing $I_e$ in Fig. \ref{fig:Ie}.
\medskip
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 1-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 1.027e-04 & 6.333e-05 & 3.251e-05\\
\hline
I-model & 1.175e-03 & 3.284e-04 & 2.439e-04\\
\hline
fcNN& 1.580e-04&4.614e-04&2.294e-04\\
\hline
ARIMA&9.789e-04&3.627e-04&4.365e-05\\
\hline
\end{tabular}
\label{tab:error1a}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 1-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 9.850e-05 & 1.778e-04 & 3.617e-05\\
\hline
I-model & 1.871e-03 &1.252 e-03 & 5.443e-04\\
\hline
fcNN& 3.364e-04&6.204e-04&8.030e-04\\
\hline
ARIMA&1.277e-03&1.082e-03&4.018e-05\\
\hline
\label{tab:error1b}
\end{tabular}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 7-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 3.513e-04 & 4.423e-04 & 1.161e-04\\
\hline
I-model & 2.004e-03 & 6.627e-04 & 5.586e-04\\
\hline
fcNN& 6.608e-04&4.804e-04&4.508e-04\\
\hline
\end{tabular}
\label{tab:error2}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 7-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 3.061e-04 & 4.324e-04 & 7.754e-05\\
\hline
I-model & 2.196e-03 & 1.167e-03 & 6.011e-04\\
\hline
fcNN& 2.224e-03&6.889e-04&1.851e-04\\
\hline
\end{tabular}
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 3-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 2.479e-04 & 3.668e-04 & 5.979e-05\\
\hline
I-model & 5.609e-04 & 1.724e-04 & 1.383e-04\\
\hline
fcNN& 8.165e-04&6.757e-04&1.689e-04\\
\hline
\end{tabular}
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 3-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 1.987e-04 & 3.256e-04 & 5.297e-05\\
\hline
I-model & 1.114e-03 & 7.337e-04 & 3.507e-04\\
\hline
fcNN& 8.611e-04&1.374e-03&5.290e-04\\
\hline
\end{tabular}
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{Average model training (tr) and inference (inf) times in seconds on Macbook Pro with Intel i5 CPU. The first two columns are for 70 \% training (tr70) data and the last two columns are for 40 \% training (tr40) data. }
\begin{tabular}{c |c| c| c| c}
\hline \hline
Model& tr70 & inf70 & tr40 & inf40 \\
\hline
IeRNN & 0.58s &0.018s & 0.51s &0.02s\\
\hline
I-model & 0.14s &0.004s & 0.11s &0.004s\\
\hline
fcNN& 0.09s&0.003s &0.09s&0.003s\\
\hline
ARIMA&0.23s&0.014s &0.19s&0.015s\\
\hline
\end{tabular}
\label{tab:time}
\end{table}
\subsection{Multi-Day Ahead Forecast}
In model training for multi-day ahead forecast,
the training loss function is modified so that the model input comes from multiple days in the past.
In 7-day ahead forecast, IeRNN leads the other two nonlinear models especially in the 40\% training data case, by as much as a factor of 7 in Lombardy.
In the 3-day ahead forecast, IeRNN leads fcNN by a factor of 4 in the 40\% training data case, as much as a factor of 10 in Lazio.
Figs. 10-13 show model comparison in training and forecast phases for Lombardy and Lazio.
\subsection{Model Size and Computing Time}
IeRNN (fcNN) has about 16400 (1800)
parameters.
The optimized $(\beta_1,\beta_2, \gamma)=
(0.685,0.158,0.044)$ in Lombardy, similarly in other regions.
Table \ref{tab:time}
lists average model training and inference times.
\section{Conclusions and Future Work}
We developed a novel spatiotemporal infectious disease model consisting of a discrete epidemic equation for the region of interest and RNNs for interactions with nearest geographic regions. Our model can be trained under 1 second. Its inference takes a fraction of a second, suitable for real-time applications. Our model out-performs temporal models in one-day and multi-day ahead forecasts in limited training data regime. In future work, we shall consider social and control mechanisms \cite{Pareschi_2020,Levin_2020} to strengthen the I-equation, as well as traffic data to expand interaction beyond nearest neighbors.
\newpage
\bibliographystyle{apalike}
{\small
\section{Introduction}
\label{sec:introduction}
\noindent Susceptible-Infected-Removed (SIR)
is a classical
differential equation
model of infectious diseases \cite{anderson_may_1992}. It divides the total population into three compartments and models their evolution by the system of equations
$$\frac{dS}{dt}=\, -\beta \, I\, S$$
$$\frac{dI}{dt}=\, \beta \, I\, S-\gamma\, I$$
$$\frac{dR}{dt}=\, \gamma \, I$$
where $\beta$ and $\gamma$ are two positive parameters. SIR is a simple and efficient model of temporal data for a given region, see also \cite{hethcote_2000} for related compartment models with social structures.
\medskip
Yet the infectious disease data are often spatio-temporal as in
the case of COVID-19, see \cite{Italy_data}.
A natural question is how to extend SIR to a space time model of suitable complexity so that it can be quickly trained from the available public data sets and applied in real-time forecasts. See
\cite{roosa_2020} for temporal model real-time forecasts on cumulative cases of China in Feb 2020.
\medskip
In this paper, we explore spatial infectious disease information to model the latent effect due to the in-flow of the infected people from the geographical neighbors. The in-flow data is not observed. To this end, machine learning tools such as regression and neural network models are more convenient.
Auto-regressive model (AR) and its variants are linear statistical models to forecast time-series data.
The Long Short Term Memory (LSTM) neural networks, originally designed for natural language processing \cite{hochreiter1997long}, have more representation power and can be applied to disease time-series data as well. With spatial structures added,
the graph-structured LSTM models can achieve state-of-the-art performance on spatiotemporal influenza data \cite{li_2019}, crime and traffic data \cite{wang,wang2018graph}.
However, they require a large enough supply of training data.
For COVID-19, we only have limited daily data since the outbreaks began in early 2020. Applying space-time LSTM models \cite{li_2019,wang2018graph} directly to COVID-19 turns out to produce poor results.
In view of the limited COVID-19 data, we shall propose a hybrid SIR-LSTM model.
\section{Related Work}
\noindent In \cite{yang2015accurate}, the authors designed a variant of AR, the AutoRegression with Google search data (ARGO), that utilizes external feature of google search data to forecast influenza data from Centers for Disease Control and Prevention (CDC). Based on google search trend data that correlated to influenza as external feature, ARGO is a linear model that processes historical observations and external features. The prediction of influenza activity level at time $t$, defined as $\hat{y}_t$, is given by:
$$\hat{y}_t=u_t+\sum_{j=1}^{52}\alpha_jy_{t-j}+\sum_{i=1}^{100}\beta_iX_{i,t}.$$
ARGO is optimized as:
$$\min_{\mu_y,\vec{\alpha},\vec{\beta}}\Big(y_t-u_t-\sum_{j=1}^{52}\alpha_jy_{t-j}-\sum_{i=1}^{100}\beta_iX_{i,t}\Big)^2$$
$$+\lambda_a||\vec{\alpha}||_1+\eta_a||\vec{\beta}||_1+\lambda_b||\vec{\alpha}||_2^2+\eta_b||\vec{\beta}||_2^2$$
where $\vec{\alpha}=(\alpha_1,\cdots,\alpha_{52})$ and $\vec{\beta}=(\beta_1,\cdots,\beta_{100})$. The $y_{t-j}$'s, $ 1\leq j\leq 52$, are historical observations of previous 52 weeks and $X_{i,t}$ are the google search trend measures of top 100 terms that are most correlated to influenza at time $t$.
Essentially, ARGO is a linear regression with regularization terms. In \cite{yang2015accurate}, ARGO is shown to outperform standard machine learning models such as LSTM, AR, and ARIMA.
\medskip
In \cite{li_2019}, graph structured recurrent neural network (GSRNN) further improved ARGO in the forecasting accuracy of CDC influenza activity level. The CDC partitions the US into 10 Health and Human
Services (HHS) regions for reporting. GSRNN treats the 10 regions as a graph with nodes $v_1,\cdots,v_{10}$, and $E$ be the collection of edges (i.e $E=\{(v_i,v_j)|v_i, v_j \,\, \text{are adjacent}\}$). Based on the average history of activity levels, the 10 HHS regions are divided into two groups, relatively active group, $\mathcal{H}$, and relatively inactive group, $\mathcal{L}$. There are 3 types of edges, $\mathcal{L}-\mathcal{L}$, $\mathcal{H}-\mathcal{L}$, and $\mathcal{H}-\mathcal{H}$, and each edge type has a corresponding RNN to train the edge features. There are also two node RNNs for each group to output the final prediction. Given a node (region) $v$, suppose $v\in \mathcal{H}$. GSRNN generates the edge features of $v$ at time $t$, $e_{v,\mathcal{H}}^t$ and $e_{v,\mathcal{L}}^t$, by averaging the history of neighbors of $v$ in the corresponding groups. Next, the edge features are fed into the corresponding edge RNNs:
$$f_v^t=\text{edgeRNN}_{\mathcal{H}-\mathcal{L}}(e_{v,\mathcal{L}}^t), \ \ h_v^t=\text{edgeRNN}_{\mathcal{H}-\mathcal{H}}(e_{v,\mathcal{L}}^t)$$
Then, the outputs of edgeRNNs are fed into the nodeRNN of group $\mathcal{H}$ together with the node feature of $v$ at time $t$, denoted as $v^t$, to output the prediction of the activity level of node $v$ at time $t+1$, or $y_{v}^{t+1}$:
$$y_{v}^{t+1}=\text{nodeRNN}_{\mathcal{H}}(v^t,f_v^t,h_v^t).$$
\section{Our Contribution: IeRNN model}
We propose a novel spatiotemporal model integrating LSTM \cite{hochreiter1997long} with a discrete time I-equation derived from SIR differential equations. The LSTM is utilized to model latent spatial information. The I-equation models the observed temporal information. Our model, named IeRNN, differs from \cite{li_2019,wang,wang2018graph} in that a difference equation
with 3 parameters (the I-equation)
fits the limited temporal data, which is far more compact than LSTM.
\subsection{Derivation of Discrete-Time I-Equation}
Based on SIR model, we add an additional feature $I_e$ that represents the external infection influence from the neighbors of a region. Then the SIR nonlinear system with $I_e$ as external forcing becomes
\begin{equation}
\frac{dS}{dt}=\, -\beta_1\, S\, I-\, \beta_2\, S\, I_e
\label{eq:I1}
\end{equation}
\begin{equation}
\frac{dI}{dt}=\, \beta_1\, S\, I+\, \beta_2\, S\, I_e-\, \gamma \, I
\label{eq:I2}
\end{equation}
\begin{equation}
\frac{dR}{dt}=\gamma \, I
\label{eq:I3}
\end{equation}
which conserves the total mass (normalized to 1): $ S+I + R=1$.
It follows from (\ref{eq:I3}) that
$$R(t)=R(t_0)+\int_{t_0}^{t}\gamma \, I \, d\tau$$
Hence,
$$S(t)=1-I(t)-R(t_0)-\, \gamma\, \int_{t_0}^t \, I \, d\tau$$
Substituting $S(t)$ into (\ref{eq:I2}) we have:
$$\frac{dI}{dt}=(\beta_1\, I+\beta_2\, I_e)\left (1-I(t)-R(t_0)-\gamma \int_{t_0}^tI(\tau) d\tau\right ) - \gamma I$$
Combining forward Euler method and Riemann sum approximation of the integral, we have a discrete approximation:
$$I(t+1)=(1-\gamma)I(t)+
\big(\beta_1I(t) +\beta_2I_e(t)\big)$$ $$\cdot \Big(1-I(t)-R(t_0)-\gamma\frac{t-t_0}{p+1}\sum_{j=0}^{p}I(t-j)\Big)$$
As we model $I(t)$ from the beginning of the infection, we have $t_0=0$ and $R(t_0)=0$. We arrive at the following discrete time I-equation:
\begin{equation}
I(t+1)=(1-\gamma)I(t)+\big(\beta_1I(t)+\beta_2I_e(t)\big)
\cdot \Big(1-I(t)-\gamma\frac{t}{p+1}\sum_{j=0}^{p}I(t-j)\Big)
\label{eq:sire}
\end{equation}
\medskip
Note that if we let $I_e(t)=0$, then we have an approximation of $I(t)$ for the original SIR model, which is a solely temporal model (named I model):
\begin{equation}
I(t+1)= (1-\gamma-\beta)I(t)-\beta\, I^2(t)
-\beta \, \gamma\, \frac{t}{p+1}\, I(t)\,
\sum_{j=0}^{p}I(t-j) \label{Ieq}
\end{equation}
In reality, it is hard to know how a population of a region interacts with populations of neighboring regions. As a result, $I_e(t)$ is a latent information that is difficult to model by a mathematical formula or equation. In order to retrieve latent spatial information, we
employ recurrent neural networks made of LSTM cells \cite{hochreiter1997long}, see Fig. \ref{fig:lstm}.
\subsection{Generating Edge Feature and Computing Latent $I_e$}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\includegraphics[scale=0.35]{italy_mapD.png}
\caption{Italian Region Map}
\label{fig:map}
\end{center}
\end{figure}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\includegraphics[scale=0.8]{LSTM_cell.pdf}
\caption{LSTM cell: input $x_t$, output: $h_t$; $\sigma$ is a sigmoid function.}
\label{fig:lstm}
\end{center}
\end{figure}
\begin{figure}[ht!]
\hspace*{-2.5cm}
\begin{center}
\includegraphics[scale=0.6]{stacked_LSTM_1.png}
\caption{Edge RNN consisting of 3 stacked LSTM cells.}
\label{fig:ernn}
\end{center}
\end{figure}
\begin{figure*}[ht!]
\centering
\includegraphics[scale=1.7]{SIR_LSTM2.pdf}
\caption{Computing $I_e$ of Lazio region. Edge RNN is as shown in Fig. \ref{fig:ernn}. Dense layer is fully connected (see Fig. \ref{fig:fcNN}).}
\label{fig:edge}
\end{figure*}
We utilize the spatial information based on the Italy region map, Fig. \ref{fig:map}. In order to learn the latent information $I_e$ of a region $v$, we first generate the edge feature of $v$. Let $C$ be the collection of neighbors of $v$. Then, the edge feature of $v$ at time $t$ is formulated as:
$$f_e^t=\frac{1}{|C|}\; \sum_{i: \, v_i\in C}\; \big[I_i(t-1),\cdots,I_i(t-p)\big]$$
where $I_i(t)$ is the infection population percentage of region $v_i$ at time $t$.
Then, we feed $f_e^t$ into an Edge RNN, an RNN with 3 stacked LSTM cells (see Fig. \ref{fig:ernn}), followed by a dense layer for computing $I_e$. The activation function of the dense layer is
hyperbolic tangent function. Figure \ref{fig:edge} illustrates the procedure of computing $I_e$ for Lazio as an illustration. We hence {\it call our model IeRNN due to its integrated design of I-equation and edge RNN.}
\section{Experiment}
To calibrate our model IeRNN, we use the Italy COVID-19 data \cite{Italy_data} for training and testing. Although the US has the most infected cases, the recovered cases are largely missing. On the other hand, the Italian COVID-19 data is more accurately reported and better maintained, reflecting a nearly complete duration of the rise and fall of infection. We collect the data of daily new (current) cases from 2020-02-24 to 2020-06-18 of 20 Italian regions. We set $p=3$ in \eqref{eq:sire} based on experimental performance. As a result, we have the current data for 113 days, with 81 days to train our model and 32 days to test our model (or 70\%/30\% training/testing data split). Our training loss function is the mean squared error of the model output and training data:
$$\hat{y}(t)=(1-\gamma)y(t-1)
+\big(\beta_1y(t-1)+\beta_2I_e(t-1)\big)$$
$$\cdot \Big(1-y(t-1)-\gamma\frac{t}{p+1}\sum_{j=1}^{p+1}y(t-j)\Big)$$
$$Loss=\frac{1}{T-p-1}\sum_{t=p+1}^T\big(y(t)-\hat{y}(t)\big)^2$$
\medskip
{\it Since the training is minimization of the above loss fucntion over parameters in both I-equation and RNN, the two components of IeRNN are coupled while learning from data}.
We use Adam gradient descent to learn the weights of LSTM and the dense layer, as well as I-equation parameters
$\beta_1$, $\beta_2$, and $\gamma$.
\medskip
To evaluate the performance of our model, we compare IeRNN, I-model (\ref{Ieq}), a fully-connected neural network (fcNN, Fig. \ref{fig:fcNN}) with hyperbolic tangent activation function, and auto-regression model (ARIMA).
As the standard setting of ARIMA is 1-day ahead prediction, we shall only compare with it in such a very short-term case. Since infectious disease evolution is intrinsically nonlinear,
we shall compare nonlinear models for 3-day and 1-week
ahead forecasting.
Based on experimental performance, we set the number of hidden units to be 100, 150, and 100 for the three layers of fcNN respectively.
\tikzset{%
every neuron/.style={
circle,
draw,
minimum size=1cm
},
neuron missing/.style={
draw=none,
scale=4,
text height=0.333cm,
execute at begin node=\color{black}$\vdots$
},
}
\begin{figure}[ht!]
\caption{Schematic of fcNN for modeling time series.}
\begin{center}
\scalebox{0.6}{%
\begin{tikzpicture}[x=1.5cm, y=1.5cm, >=stealth]
\foreach \m/\l [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try] (input-\m) at (0,2-\y) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden1-\m) at (1.5,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden2-\m) at (3,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1,missing,2}
\node [every neuron/.try, neuron \m/.try ] (hidden3-\m) at (4.5,2.5-\y*1.25) {};
\foreach \m [count=\y] in {1}
\node [every neuron/.try, neuron \m/.try ] (output-\m) at (6,0) {};
\foreach \l [count=\i] in {1,p}
\draw [<-] (input-\i) -- ++(-1,0)
node [above, midway] {$\pmb{y_{t-\l}}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden1-\i.north) {$\pmb{w^{(1)}_\l}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden2-\i.north) {$\pmb{w^{(2)}_\l}$};
\foreach \l [count=\i] in {1,n}
\node [above] at (hidden3-\i.north) {$\pmb{w^{(3)}_\l}$};
\foreach \l [count=\i] in {1}
\node [above] at (output-\i.north) {$\pmb{tanh}$};
\foreach \l [count=\i] in {1}
\draw [->] (output-\i) -- ++(1,0)
node [above, midway] {$\pmb{y_t}$};
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (input-\i) -- (hidden1-\j);
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (hidden1-\i) -- (hidden2-\j);
\foreach \i in {1,2}
\foreach \j in {1,2}
\draw [->] (hidden2-\i) -- (hidden3-\j);
\foreach \i in {1,2}
\foreach \j in {1}
\draw [->] (hidden3-\i) -- (output-\j);
\foreach \l [count=\x from 0] in {\textbf{Input}, \textbf{Hidden}, \textbf{Output}}
\node [align=center, above] at (\x*3,2.2) {\l \\ \textbf{layer}};
\end{tikzpicture}
}
\end{center}
\label{fig:fcNN}
\end{figure}
\newpage
\begin{figure*}[ht!]
\caption{Training and 1-day ahead forecast of 4 models (IeRNN, fcNN, I-model, and ARIMA) in 4 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_AR.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_AR.png}}
\label{fig:results}
\end{figure*}
\newpage
\begin{figure*}[ht!]
\caption{Training and
1-day ahead forecast
of 4 models with reduced (40\%) training data. The 4 rows are IeRNN, fcNN, I-model, and ARIMA respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR1.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_AR1.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_AR1.png}}
\label{fig:results2}
\end{figure*}
\newpage
\begin{figure}[]
\caption{IeRNN training and 1-day ahead forecast on four additional regions.}
\centering
\subfloat[Piemonte]{\includegraphics[scale=0.5]{Piemonte.png}}
\qquad
\subfloat[Campania]{\includegraphics[scale=0.5]{Campania.png}}
\qquad
\subfloat[Molise]{\includegraphics[scale=0.5]{Molise.png}}
\qquad
\subfloat[Umbria]{\includegraphics[scale=0.5]{Umbria.png}}
\qquad
\label{fig:additional}
\end{figure}
\begin{figure}[]
\caption{Visulization of the latent information $I_e$ in Fig. \ref{fig:additional} learned by IeRNN.}
\centering
\subfloat[Piemonte]{\includegraphics[scale=0.5]{Piemonte_Ie.png}}
\qquad
\subfloat[Campania]{\includegraphics[scale=0.5]{Campania_Ie.png}}
\qquad
\subfloat[Molise]{\includegraphics[scale=0.5]{Molise_Ie.png}}
\qquad
\subfloat[Umbria]{\includegraphics[scale=0.5]{Umbria_Ie.png}}
\label{fig:Ie}
\end{figure}
\newpage
\begin{figure*}[ht!]
\caption{Training and 7-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio7.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc7.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR7.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR7.png}}
\quad
\label{fig:result 7-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 7-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) with reduced (40\%) training data in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio71.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc71.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR71.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR71.png}}
\quad
\label{fig:result2 7-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 3-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio2.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc2.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR2.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR2.png}}
\quad
\label{fig:result 3-day}
\end{figure*}
\begin{figure*}[ht!]
\caption{Training and 3-day ahead forecast of 3 models (IeRNN, fcNN, and I-model) with reduced (40 \%) training data in 3 rows respectively.}
\centering
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio22.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_fc22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_fc22.png}}
\quad
\subfloat[Lombardy]{\includegraphics[scale=0.45]{Lombardia_SIR22.png}}
\subfloat[Lazio]{\includegraphics[scale=0.45]{Lazio_SIR22.png}}
\quad
\label{fig:result 3-day1}
\end{figure*}
\subsection{One-Day Ahead Forecast}
As we see in Fig. \ref{fig:results}, fcNN can perform poorly. This is not a surprise, as both \cite{li_2019} and \cite{yang2015accurate} relied on hundreds of historical observations to train their models. The I-model based on only sequential data in time of one region merely follows the trend of the true data but cannot provide accurate predictions. Our IeRNN model, with the help of additional spatial information, is able to make accurate predictions and outperform other models. We also test the IeRNN with training data reduced to 40\% (46 days). IeRNN is still able to track the general trend of the infected population percentage.
\medskip
We measure the test accuracy with the Root Mean Square Error (RMSE)
averaged over a few trials in training. In Tables \ref{tab:error1a} and \ref{tab:error1b} on 1-day ahead forecast, IeRNN achieves the smallest RMSE errors, and I-model has the largest errors. The compact I-model with 2 parameters cannot do 1-day ahead prediction as accurately. ARIMA outperforms I-model and does better on Emilia-Romagna and Lazio regions than fcNN. ARIMA, a linear model, has simpler structure than fcNN whose nonlinearity does not play out in such a short time task. Fig. \ref{fig:additional} shows
1-day ahead forecast of IeRNN model on other regions with the learned latent external forcing $I_e$ in Fig. \ref{fig:Ie}.
\medskip
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 1-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 1.027e-04 & 6.333e-05 & 3.251e-05\\
\hline
I-model & 1.175e-03 & 3.284e-04 & 2.439e-04\\
\hline
fcNN& 1.580e-04&4.614e-04&2.294e-04\\
\hline
ARIMA&9.789e-04&3.627e-04&4.365e-05\\
\hline
\end{tabular}
\label{tab:error1a}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 1-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 9.850e-05 & 1.778e-04 & 3.617e-05\\
\hline
I-model & 1.871e-03 &1.252 e-03 & 5.443e-04\\
\hline
fcNN& 3.364e-04&6.204e-04&8.030e-04\\
\hline
ARIMA&1.277e-03&1.082e-03&4.018e-05\\
\hline
\label{tab:error1b}
\end{tabular}
\medskip
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 7-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 3.513e-04 & 4.423e-04 & 1.161e-04\\
\hline
I-model & 2.004e-03 & 6.627e-04 & 5.586e-04\\
\hline
fcNN& 6.608e-04&4.804e-04&4.508e-04\\
\hline
\end{tabular}
\medskip
\label{tab:error2}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 7-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna. }
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 3.061e-04 & 4.324e-04 & 7.754e-05\\
\hline
I-model & 2.196e-03 & 1.167e-03 & 6.011e-04\\
\hline
fcNN& 2.224e-03&6.889e-04&1.851e-04\\
\hline
\end{tabular}
\medskip
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 3-day ahead forecast trained with 70 \% of data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 2.479e-04 & 3.668e-04 & 5.979e-05\\
\hline
I-model & 5.609e-04 & 1.724e-04 & 1.383e-04\\
\hline
fcNN& 8.165e-04&6.757e-04&1.689e-04\\
\hline
\end{tabular}
\medskip
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{RMSE test errors in 3-day ahead forecast trained with reduced (40 \% of) data. E-R= Emilia-Romagna.}
\begin{tabular}{c c c c}
\hline \hline
Model& Lombardy & E-R & Lazio \\
\hline
IeRNN & 1.987e-04 & 3.256e-04 & 5.297e-05\\
\hline
I-model & 1.114e-03 & 7.337e-04 & 3.507e-04\\
\hline
fcNN& 8.611e-04&1.374e-03&5.290e-04\\
\hline
\end{tabular}
\medskip
\label{tab:error3}
\end{table}
\begin{table}[ht!]
\centering
\caption{Average model training (tr) and inference (inf) times in seconds on Macbook Pro with Intel i5 CPU. The first two columns are for 70 \% training (tr70) data and the last two columns are for 40 \% training (tr40) data. }
\begin{tabular}{c |c| c| c| c}
\hline \hline
Model& tr70 & inf70 & tr40 & inf40 \\
\hline
IeRNN & 0.58s &0.018s & 0.51s &0.02s\\
\hline
I-model & 0.14s &0.004s & 0.11s &0.004s\\
\hline
fcNN& 0.09s&0.003s &0.09s&0.003s\\
\hline
ARIMA&0.23s&0.014s &0.19s&0.015s\\
\hline
\end{tabular}
\medskip
\label{tab:time}
\end{table}
\subsection{Multi-Day Ahead Forecast}
In model training for multi-day ahead forecast,
the training loss function is modified so that the model input comes from multiple days in the past.
In 7-day ahead forecast, IeRNN leads the other two nonlinear models especially in the 40\% training data case, by as much as a factor of 7 in Lombardy.
In the 3-day ahead forecast, IeRNN leads fcNN by a factor of 4 in the 40\% training data case, as much as a factor of 10 in Lazio.
Figs. 10-13 show model comparison in training and forecast phases for Lombardy and Lazio.
\subsection{Model Size and Computing Time}
IeRNN (fcNN) has about 16400 (1800) parameters.
The optimized $(\beta_1,\beta_2, \gamma)=
(0.685,0.158,0.044)$ in Lombardy, similarly in other regions.
Table \ref{tab:time}
lists average model training and inference times.
\section{Conclusions and Future Work}
We developed a novel spatiotemporal infectious disease model consisting of a discrete epidemic equation for the region of interest and RNNs for interactions with nearest geographic regions. Our model can be trained under 1 second. Its inference takes a fraction of a second, suitable for real-time applications. Our model out-performs temporal models in one-day and multi-day ahead forecasts in limited training data regime. In future work, we shall consider social and control mechanisms \cite{Pareschi_2020,Levin_2020} to strengthen the I-equation, as well as traffic data to expand interaction beyond nearest neighbors.
\section{Acknowledgements}
\noindent The work was partially supported by NSF grants IIS-1632935, and DMS-1924548. JX would like to thank Prof. Fred Wan for helpful communications on disease modeling.
\newpage
\bibliographystyle{plain}
|
2,869,038,154,837 | arxiv | \section{Introduction}
Computing the topological entropy of a one-dimensional dynamical system is in general a very difficult task. Motivated by this problem, in \cite{MR}, the entropy was computed following backward trajectories in a way that at each step every preimage can be chosen with equal probability introducing a new concept of entropy, the fair entropy. Fair entropy gives a lower bound for topological entropy and is simple to compute.
As in \cite{MR} let us denote by $c(x)$ the cardinality of the set $f^{-1}(x)$. We start with a point $x_0$ and proceed by induction. Given $x_n$, we choose $x_{n+1}$ from the set $f^{-1}(x_n)$ randomly, that is, the probability of choosing any of these points is $1/c(x_n)$. Then we go to the limit with the geometric averages of $c(x_0), c(x_1), \ldots, c(x_n)$ as $n$ goes to infinity.
Also in \cite{MR}, convergence of the geometric averages of $c(x_0), c(x_1), \ldots, c(x_n)$ as $n$ goes to infinity for a random choice of the backward trajectory as well as convergence of the measures equidistributed along longer and longer pieces of a random backward trajectory were investigated. Indeed, these questions were answered for some special classes of maps, namely, transitive subshifts of finite type and piecewise monotone (with a finite number of pieces) topologically mixing interval maps.
In this paper we define fair measures in a broad setting, flexible enough to handle noncompact spaces and discontinuous maps.
Let us state our main definition. Let $(X,\mathcal{F})$ be a measurable space and $f:X\to X$ a surjection. Assume that $X$ admits a countable measurable partition $\mathcal{X}=\{X_i\}_{i=1}^{\infty}$ so that each $f(X_i)$ is also measurable and each restriction $f|_{X_i} : X_i \to f(X_i)$ is a measurable isomorphism, i.e. images and preimages of measurable sets are again measurable. Let $\mathcal{M}(X,f)$ be the space of all $f$-invariant probability measures on $X$. To avoid pathologies, we impose the mild topological assumption that $X$ is a Polish space and $\mathcal{F}$ is the Borel $\sigma$-algebra. Then for each $\mu\in\mathcal{M}(X,f)$, the measure-theoretic completion of $(X,\mathcal{F},\mu)$ is a standard probability space (Lebesgue space).
We define another countable measurable partition $\mathcal{A}$ as the common refinement of the partitions $\left\{f(X_i),X\setminus f(X_i)\right\}$, $i=1,2,\ldots$. Thus, if a set $B$ is a subset of an element of $\mathcal{A}$ (in symbols, $B\prec\mathcal{A}$), then we know which branches of $f^{-1}$ are defined on $B$. We write $p(B)=\{i; B\subset f(X_i)\}$ to identify those branches and $c(B)=\# p(B)$ to count how many there are. A singleton $\{x\}$ is always contained in an element of $\mathcal{A}$ and we write simply $c(x)$ for the number of preimages of $x$. This number is always positive since $f$ is surjective, but may be infinite.
Now we define a fair measure as a special kind of invariant measure for which the measure of a set is divided equally among the pieces of its preimage.
\begin{definition}\label{defnFair} An invariant measure $\mu\in\mathcal{M}(X,f)$ is called \emph{fair} if each measurable set $B\prec \mathcal{A}$ satisfies
\begin{equation}\label{fair}
\mu(X_i\cap f^{-1}(B)) = \frac{\mu(B)}{c(B)} \text{, for all } i\in p(B).
\end{equation}
\end{definition}
By a simple common refinement argument, the definition of a fair measure does not depend on the choice of the partition $\mathcal{X}$.
With our definition we are able to study the behaviour of a random backward trajectory for transitive countable state Markov shift maps. We then extend the results from shift maps to maps on the interval. To do this, we introduce the notion of {\it isomorphism modulo countable invariant sets} and we then study Markov and mixing interval maps.
This paper is organized as follows. In Section 2 we provide some more definitions and describe the general setting. In Section 3 we discuss random backward trajectories for transitive countable state Markov shift maps and in Section 5 we introduce the main tool that will allow us to extend our results to countably Markov and mixing interval maps in Section 6. We finally introduce Lebesgue fair models in Section 7 and in Section 8 we look for fair measures on graph maps.
\section{General Case}
In this section we study some properties of fair measures. Immediately from our Definition \ref{defnFair} we get the following properties of a fair measure.
\begin{lemma}\label{lem:basic}
If $\mu$ is a fair measure, then
(a) For a measurable set $B\prec\mathcal{A}$ with $c(B)=\infty$ we have $\mu(B)=0$.
(b) If $B$ is measurable and $\mu(B)=0$, then $\mu(f(B))=0$.
(c) $\mu(\{x\in X ; \#f^{-n}(x)=\infty \text{ for some }n\in\mathbb{N}\})=0$.
\end{lemma}
\begin{proof}
For $B\prec\mathcal{A}$ we use invariance of the measure to write $\mu(B)=\sum \mu(X_i\cap f^{-1}(B))$ where the sum extends over all $i\in p(B)$. By~\eqref{fair} each summand is zero. This proves (a).
By cutting up a given measurable set $B$ into pieces, we may assume that $B\prec\mathcal{X}$ and $f(B)\prec\mathcal{A}$. If $B\subset X_i$, then $B = X_i\cap f^{-1}(f(B))$. Then by~\eqref{fair}, $\mu(f(B))=c(f(B))\cdot\mu(B)$, which is zero if $c(f(B))$ is finite. But if $c(f(B))$ is infinite, then $\mu(f(B))=0$ by (a). This completes the proof of (b).
To prove (c) note that this set can be written as the union of the sets $f^n(A)$ where $A\in\mathcal{A}$, $c(A)=\infty$, and $n\geq0$. By (a) if $c(A)=\infty$, then $\mu(A)=0$. Now the result follows from (b).
\end{proof}
Thus for maps like the Gauss map, where every point has countably many preimages, we have no hope of finding a fair measure. In the search for fair measures, we must focus our attention on the part of the space where the cardinalities of preimage sets are finite.
In~\cite{MR} if a system $(X,f)$ has a unique fair measure, then the measure-theoretic entropy of that measure is referred to as the \emph{fair entropy} of the system. For systems with more than one fair measure, we generalize that definition as follows:
\begin{definition}\label{def:fairentropy}
The \emph{fair entropy} of a system $(X,f)$ is the supremum of measure-theoretic entropies of its fair measures, $h_{\textnormal{fair}}(f)=\sup\left\{h_{\mu}(f)~|~\mu\in\mathcal{M}(X,f)\text{ is fair}\right\}$.
\end{definition}
It may also happen that a system has no fair measures, as in Examples~\ref{ex:null} and~\ref{ex:transient} below. In this case, we take the supremum over the empty set in Definition~\ref{def:fairentropy} to be zero.
Next we record two properties of fair measures which were proved in a more restrictive setting in~\cite{MR}. But the proofs need no modification; the arguments involved are purely measure-theoretic and do not require compactness of $X$, continuity of $f$, or finiteness of $\mathcal{X}$.
\begin{lemma}[\cite{MR}]\label{lem:Jac} A measure $\mu\in\mathcal{M}(X,f)$ is fair if and only if its measure-theoretic Jacobian is $x\mapsto c(f(x))$.
\end{lemma}
\begin{lemma}[\cite{MR}]\label{lem:erg-dec}If $\mu$ is a fair measure, then so is almost every component of its ergodic decomposition.
\end{lemma}
The \emph{Jacobian} of $\mu\in\mathcal{M}(X,f)$ referred to in Lemma~\ref{lem:Jac} is the measurable function $J:X\to[0,\infty)$, unique up to changes on sets of $\mu$-measure zero, such that for every Borel set $B\subset X$,
\begin{equation}\label{Jac}
\text{If } f|_B \text{ is injective, then } \mu(f(B))=\int_B J \, d\mu.
\end{equation}
In particular, for the Jacobian to exist, the measure $\mu$ must be \emph{non-singular}, i.e. every measure-zero set must have a measure zero image. Fair measures always fulfill this condition -- Lemma~\ref{lem:basic}~(b) -- and their Jacobians always exist, see~\cite[Proposition 9.7.2]{VO}.
\section{Random Backward Trajectories}
The motivation for studying fair measures is to understand what happens along a random backward trajectory $y_0 \mapsfrom y_1 \mapsfrom y_2 \mapsfrom \cdots$ of a given point $y_0\in X$, where the backward trajectory is chosen as follows: given $y_i$, we choose $y_{i+1}$ from $f^{-1}(y_i)$ by ``rolling a dice'' with $c(y_i)$ sides. Thus, at each stage of the process, the choice of the next preimage is ``fair.'' We hope that by distributing point masses along longer and longer pieces of this backward trajectory and passing to a weak-* limit
we can generate a fair measure. And we hope that we can calculate the entropy of this measure by taking geometric averages of the function $c$ along longer and longer pieces of the orbit.
One way to formalize what we mean by a random backward trajectory of a point $y_0$ is as follows.
\begin{definition}\label{def:rbt} Let $y_0\in X$ satisfy $\# f^{-n}(y_0)<\infty$ for all $n\geq0$. Consider the Markov sequence of random variables $Y_0, Y_1, \ldots$ with
\begin{itemize}
\item initial distribution $P[Y_0=y_0]=1$, and
\item transition probabilities $P[Y_{i+1}=y_{i+1} \, | \, Y_i=y_i] = 1/c(y_i)$ for each $y_{i+1}\in f^{-1}(y_i)$.
\end{itemize}
A property is said to hold for a \emph{random choice of the backward trajectory} of $y_0$ if the property holds for almost every outcome $(y_i)_{i=0}^\infty$ of this Markov chain.
\end{definition}
Note that we cannot discuss random backward trajectories of points $y_0$ for which $\# f^{-n}(y_0)=\infty$ for some $n\geq0$. When forced to choose among infinitely many preimages, there is no fair way to do it. In light of Lemma~\ref{lem:basic}~(c) this does not bother us too much.
Next, we state clearly the meaning of weak-* convergence, remembering that our space $X$ need not be compact. Let $\mu_n, \mu$ be Borel probability measures on $X$. We say that $\mu$ is the \emph{weak-* limit} of the measures $\mu_n$ and write $\mu_n \xrightarrow{weak-*} \mu$ if one (all) of the following equivalent conditions is satisfied (see \cite[Proposition 2.7]{DGS}):
\vspace{0.1in}
(a) $\int_X \phi \, d\mu_n \to \int_X \phi \, d\mu$ for all bounded (!) continuous functions $\phi:X\to\mathbb{R}$.
(b) $\limsup \mu_n(C) \leq \mu(C)$ for every closed subset $C\subset X$.
(c) $\liminf \mu_n(U) \geq \mu(U)$ for every open subset $U\subset X$.
(d) $\lim \mu_n(B) = \mu(B)$ for every subset $B\subset X$ whose boundary has measure $\mu(\partial B)=0$.
\vspace{0.1in}
Finally, we say that a sequence of points $y_n\in X$ \emph{equidistributes} for the measure $\mu$ if $\mu$ is the weak-* limit of the measures $\frac{1}{N}\sum_{n=0}^{N-1} \delta_{y_n}$.
Just as ergodic invariant measures can be used to understand the behavior of typical forward trajectories of a system, so also ergodic fair measures give us information about typical backward trajectories. This is the meaning of the following theorem -- it is a straightforward adaptation of~\cite[Theorem 3.4]{MR}.
\begin{theorem}[\cite{MR}]\label{th:back}
Let $\mu$ be an ergodic fair measure. Then:
(a) For each integrable function $\phi:X\to\mathbb{R}$, for $\mu$-almost every $y_0\in X$, for a random choice $(y_n)$ of the backward trajectory of $y_0$, $\frac{1}{N}\sum_{n=0}^{N-1} \phi(y_n) \to \int_X \phi\, d\mu$.
(b) If $X$ is compact, then for $\mu$-almost every $y_0\in X$, a random choice $(y_n)$ of the backward trajectory of $y_0$ equidistributes for the measure $\mu$.
(c) For $\mu$-almost every $y_0\in X$, for a random choice $(y_n)$ of the backward trajectory of $y_0$, the geometric averages $\sqrt[n]{c(y_0) \cdot c(y_1) \cdot \ldots \cdot c(y_n)}$ converge as $n\to\infty$ to the (possibly infinite) number $\exp\left(\int\log c\, d\mu\right)$.
(d) If additionally $f$ has a one-sided generator of finite entropy, then the limit in~(c) is the exponential of the entropy of $\mu$.
\end{theorem}
\begin{proof}[Sketch of proof]
(a) Form the natural extension $(\tilde{X}, \tilde{f}, \tilde{\mu})$ of $(X,f,\mu)$. Then apply Birkhoff's ergodic theorem using $\tilde{f}^{-1}$. For more details, see~\cite[Theorem 3.4]{MR}.
(b) Let $C(X)$ be the space of continuous real-valued functions on $X$ with the topology of uniform convergence. By compactness of $X$, $C(X)$ contains a countable dense subset $\{\phi_i\}_{i=1}^\infty$, see~\cite{PU}. Applying (a) to each $\phi_i$, we find a full-measure set of points $y_0$ such that $\frac1N \sum_{n=0}^{N-1} \phi_i(y_n) \to \int_X \phi_i \, d\mu$ for a random backward trajectory of $y_0$. If we choose $y_0$ from the intersection of these countably many full-measure sets, then a random backward trajectory will equidistribute for the measure $\mu$.
(c) If $\phi(x)=\log(c(x))$ is integrable, then apply~(a) directly. Otherwise, approximate $\phi$ from below by truncations $\phi_m(x)=\min(\phi(x),m)$ and apply~(a) anyway.
(d) This is Rohlin's entropy formula $h_\mu(f)=\int \log J \, d\mu$ for systems with a one-sided generator \cite[Theorem 2.9.7]{PU}, together with the observation that $\int \log c\circ f\, d\mu = \int \log c \, d\mu$ by the invariance of $\mu$.
\end{proof}
Theorem~\ref{th:back} gives us only partial information about random backward trajectories. One problem is that we do not know if there are any fair measures to apply it to. Another problem is that the results hold only almost everywhere, which may mean almost nowhere with respect to some other natural measure. The situation is much better, however, for countable state Markov shifts.
\section{Countable State Markov Shifts}\label{sec:cms}
Let $\mathcal{I}$ be a countable set of indices (states) and $M=(m_{ij})_{i,j\in \mathcal{I}}$ a 0-1 matrix. Consider the corresponding one-sided Markov shift $(\Sigma_M,\sigma)$. We assume transitivity of $\sigma$, which is the same as irreducibility of $M$. The number of preimages $c(x)$ of a point $x\in\Sigma_M$ depends only on the cylinder of length 1 to which it belongs, and if it is the $j$th cylinder, then it is equal to
\begin{equation}\label{cj}
c_j=\sum_{i} m_{ij}.
\end{equation}
If one of the column sums $c_j$ is infinite, then by Lemma~\ref{lem:basic} (c) every fair measure assigns the value zero to the set $\bigcup_{n=0}^\infty \sigma^n([j])$. But by transitivity of our Markov shift, this union is the whole space $\Sigma_M$. We may conclude that there are no fair measures.
Assume from now on that all the column sums $c_j$ are finite. One natural way to look for a fair measure is to use stochastic Markov chains. Form a matrix $Q$ with entries
\begin{equation}\label{Q}
q_{ji}=\frac{m_{ij}}{c_j}.
\end{equation}
It is a nonnegative matrix with rows summing to 1 and nonzero entries in the same positions as the transpose matrix $M^T$. So we can ask whether the time-reversed Markov shift $\Sigma_{M^T}$ supports a shift-invariant Markov measure with transition probabilities $Q$. What is needed is a vector $\pi$ satisfying
\begin{equation}\label{pi}
\pi Q=\pi,\qquad \text{with each }\pi_i\geq0 \text{ and }\sum_{i\in\mathcal{I}} \pi_i=1,
\end{equation}
to serve as the initial distribution. The theory of stochastic Markov chains tells us that there is at most one such vector $\pi$, its entries are necessarily strictly positive, and it exists if and only if $Q$ is positive recurrent~\cite{Fel}. In this case, we can use $\pi$ to construct a stochastic matrix $P$ with entries given by
\begin{equation}\label{P}
\pi_i p_{ij} = \pi_j q_{ji}.
\end{equation}
Summing~\eqref{P} over $j$ we get that $\pi P = \pi$, so that the measure $\mu=\text{Markov}(\pi, P)$ is shift-invariant. $\mu$ defines a measure on $\Sigma_M$ because $P, M$ have their nonzero entries in the same positions. To show that $\mu$ is fair, it suffices to check equation~\eqref{fair} on cylinder sets. Let $B=[j j_1 j_2 \cdots j_n]$ be a cylinder set. We are using $\mathcal{X}=\{[i]; i\in \mathcal{I}\}$. Then $p(B)=\{i; m_{ij}=1\}$ and $c(B)=c_j$. For each $i\in p(B)$ we have $[i]\cap \sigma^{-1}(B) = [i j j_1 \cdots j_n]$. The measure of this set is
$\pi_i p_{ij} p_{jj_1} \cdots p_{j_{n-1}j_n}$, which by~\eqref{P} equals $q_{ji}\pi_j p_{jj_1} \cdots p_{j_{n-1}j_n}$, which by~\eqref{Q} equals $\frac{1}{c(B)}\mu(B)$. This shows that $\mu$ is fair.
\begin{theorem}\label{th:cms}
Let $(\Sigma_M,\sigma)$ be a transitive countable-state Markov shift with all $c_j$ finite. Given any point $y_0\in\Sigma_M$ the behavior of a random backward trajectory $(y_n)$ is as follows:
(a) If $Q$ is positive recurrent, then $(y_n)$ equidistributes for the fair measure $\mu=\textnormal{Markov}(\pi, P)$.
(b) If $Q$ is null recurrent, then $(y_n)$ is dense in $\Sigma_M$, but visits each cylinder set $[i]$ with limiting frequency zero.
(c) If $Q$ is transient, then $(y_n)$ visits each cylinder set $[i]$ only finitely often.
\end{theorem}
\begin{proof}
Write $y_0=\omega_0\omega_1\omega_2\cdots$. Let $\delta_{\omega_0}$ be the probability vector on the state space $\mathcal{I}$ with a $1$ in position $\omega_0$ and $0$'s elsewhere. Consider the measure $\nu_0=\text{Markov}(\delta_{\omega_0},Q)$ on the space $\Sigma_{M^T}$. In the positive recurrent case there is also a stationary probability vector $\pi$ for $Q$ and the corresponding measure $\nu=\text{Markov}(\pi,Q)$ on $\Sigma_{M^T}$.
We want to choose a backward trajectory for $y_0$ in a fair way. Any point $\omega'_0 \omega'_1 \cdots \in \Sigma_{M^T}$ with $\omega'_0=\omega_0$ can be used as a possible history. One way to think of this is that we extend our one-sided sequence $y_0$ to a two-sided sequence
\begin{equation*}
\lefteqn{\underbrace{\phantom{\cdots\, \omega'_3\, \omega'_2\, \omega'_1\, \omega'_0\,}}_{\text{taken from } \Sigma_{M_T}}} \cdots \omega'_3\, \omega'_2\, \omega'_1\, \!\overbrace{\omega'_0\, \omega_1\, \omega_2\, \omega_3\, \cdots}^{\text{initial point }y_0}.
\end{equation*}
Then the backward trajectory consists of the points $y_n = \omega'_n \omega'_{n-1} \cdots \omega'_0 \omega_1 \omega_2 \cdots$. To make this choice in a fair way, we use the Markov chain $(\Sigma_{M_T}, \nu_0)$. We define a sequence of random variables $Y_0, Y_1, \ldots$ on the probability space $(\Sigma_{M^T},\nu_0)$ by
\begin{equation}\label{recover}
Y_n(\omega'_0\omega'_1\omega'_2\cdots) = \omega'_n\omega'_{n-1}\cdots\omega'_1\omega'_0\omega_1\omega_2\cdots.
\end{equation}
It follows immediately that $P[Y_0=y_0]=1$ and
\begin{multline*}
P[Y_{n+1} = y_{n+1} ~|~ Y_n = y_n] = q_{ji}
= \frac{m_{ij}}{c_{j}}
= \begin{cases} 1/c(y_n), & \text{ if }y_{n+1}\in\sigma^{-1}(y_n) \\ 0, & \text{ otherwise} \end{cases}, \\
\text{ where } y_{n+1}\in[i],\, y_n\in[j].
\end{multline*}
In this way we recover the stochastic process $Y_0, Y_1,\ldots$ from Definition~\ref{def:rbt}. All we've done is construct one realization of an underlying probability space for this process.
Let us first prove (a). We work with cylinder sets, i.e.\ sets of the form $C=[i_0\cdots i_{m}] \subset \Sigma_M$ with $m\geq0$. The length of this cylinder is $m+1$. The \emph{reverse} of this cylinder is $\overline{C}=[i_m \cdots i_0] \subset \Sigma_{M^T}$ and is nonempty if and only if $C$ is nonempty in $\Sigma_M$. By applying~\eqref{P} several times we get $\nu(\overline{C})=\mu(C)$. We define the collection $\mathcal{E}_m$ to contain all unions of length $m+1$ cylinder sets, and the reverse of a set $B=\cup_\alpha C_\alpha \in \mathcal{E}_m$ is simply $\overline{B}=\cup_\alpha \overline{C_\alpha}$. Again, we get $\nu(\overline{B})=\mu(B)$. Moreover, the characteristic function $\mathds{1}_B$ is clearly integrable over $(\Sigma_{M^T},\nu)$. Applying Birkhoff's ergodic theorem to all these countably many characteristic functions at once, we get that
\begin{equation*}
W=\left\{\omega'=\omega'_0\omega'_1\omega'_2\cdots \in \Sigma_{M^T} ; \lim_{N\to\infty} \frac{1}{N} \sum_{n=0}^{N-1} \mathds{1}_{\overline{B}}(\sigma^n\omega')=\nu(\overline{B}) \text{ for all }B\in\mathcal{E}_m, \, m\geq1 \right\}
\end{equation*}
has full measure $\nu(W)=1$. Let $W_0$ be the intersection of $W$ with the cylinder set $[\omega_0]\subset \Sigma_{M^T}$. Since $\nu_0$ is just the (normalized) restriction of $\nu$ to $[\omega_0]$, we get also $\nu_0(W_0)=1$. Thus we can choose our random backward trajectory by choosing $\omega'\in W_0$ and setting $y_n=Y_n(\omega')$, $n=1,2,\ldots$. We need to show that this backward trajectory equidistributes for $\mu$.
Let $E$ be any open subset of $\Sigma_M$. For each $m\geq0$ let $E_m$ be the maximal element of $\mathcal{E}_m$ contained in $E$. Clearly $E_0 \subset E_1 \subset \cdots$ and since cylinder sets form a basis for the topology we have $E=\cup_m E_m$. Therefore $\mu(E)=\lim \mu(E_m)$. For $n\geq m$ we see by~\eqref{recover} that $y_n\in E_m$ if and only if $\sigma^{n-m}(\omega')\in \overline{E_m}$. Then we can calculate
\begin{multline*}
\liminf_{N\to\infty} \left(\frac{1}{N} \sum_{n=0}^{N-1} \delta_{y_n}\right)(E) \geq
\lim_{N\to\infty} \frac{1}{N} \sum_{n=m}^{N-1} \mathds{1}_{E_m}(y_n) =\\
= \lim_{N\to\infty} \frac{1}{N}\sum_{n=0}^{N-m-1}\mathds{1}_{\overline{E_m}}(\sigma^n\omega')=\nu(\overline{E_m})=\mu(E_m).
\end{multline*}
Since this holds for arbitrary $m$, we get that the limes inferior is at least $\mu(E)$. Since $E$ was an arbitrary open set, this shows that $\frac{1}{N}\sum_{n=0}^{N-1}\delta_{y_n} \xrightarrow{weak-*} \mu$.
We now prove (b). Null recurrence of $Q$ means that our Markov chain $(\Sigma_{M^T},\nu_0)$ has the following property: with probability 1, the orbit under $\sigma$ of a randomly chosen point $\omega'=\omega'_0\omega'_1\cdots$ visits each cylinder $[i]$ infinitely often but with limiting frequency zero. By the Markov property, if there are infinitely many visits to $[i]$, then with probability $1$ each subcylinder $\overline{C}\subset [i]$ is also visited infinitely often. By the definition of $\nu_0$ we also get $\omega'_0=\omega_0$ with probability 1. Choose $\omega'$ with all of these properties and let $(y_n)=(Y_n(\omega'))$ be the corresponding backward trajectory of $y_0$. For each nonempty cylinder $C=[i_0\cdots i_m]\subset \Sigma_M$ there is $n\geq m$ with $\sigma^{n-m}(\omega')\in\overline{C}$, which gives $y_n\in C$. This shows the density in $\Sigma_M$ of our backward trajectory. Moreover, the visits of $y_n$ to each $[i]$ occur with the same limiting frequency as the visits of $\sigma^n(\omega')$ to $[i]$, and this frequency is zero.
Finally, we prove (c). Transience of $Q$ means that our Markov chain $(\Sigma_{M^T},\nu_0)$ has the following property: with probability 1, the orbit under $\sigma$ of a randomly chosen point $\omega'=\omega'_0\omega'_1\cdots$ visits each cylinder set $[i]$ only finitely many times. Proceeding as before, we see that the randomly chosen backward trajectory $(y_n)$ visits each $[i]$ only finitely many times.
\end{proof}
\begin{corollary}\label{cor:cms}
In the positive recurrent case, $\mu$ is ergodic and is the only fair measure on $\Sigma_M$. In the null recurrent and transient cases, there are no fair measures on $\Sigma_M$.
\end{corollary}
\begin{proof}
Suppose first that $Q$ is positive recurrent so that the fair measure $$\mu=\text{Markov}(\pi,P)$$ exists. We wish to show that $\mu$ is the only fair measure. By Lemma~\ref{lem:erg-dec}, it suffices to show that each ergodic fair measure $\mu'$ is equal to $\mu$. Let $C=[i_0\cdots i_m]\subset \Sigma_{M}$ be a cylinder set. By Theorem~\ref{th:back} (a) applied to $\mathds{1}_C$ we can find a point $y_0$ from the $\mu'$-full measure set whose almost every backward trajectory visits $C$ with limiting frequency $\mu'(C)$. But by Theorem~\ref{th:cms} (a), the random backward trajectory $(y_n)$ of $y_0$ equidistributes for $\mu$. Since the cylinder set $C$ is both closed and open, we get $\mu(\partial C)=\mu(\emptyset)=0$. So equidistribution tells us that $(y_n)$ visits $C$ with limiting frequency $\mu(C)$. In this way we get equality $\mu(C)=\mu'(C)$ for all cylinder sets, from which it follows that $\mu'=\mu$.
Next we show the ergodicity of $\mu$. This follows immediately from Lemma~\ref{lem:erg-dec} and the fact that $\mu$ is the only fair measure. Alternatively, ergodicity follows because $\mu$ is a Markov measure whose transition matrix $P$ is positive recurrent.
Next, we consider what happens when $Q$ is null recurrent or transient. We wish to show that there are no fair measures. By Lemma~\ref{lem:erg-dec}, it suffices to show that there are no ergodic fair measures. Suppose to the contrary that $\mu'$ is an ergodic fair measure. There must be some cylinder set $[i]$ with $\mu'([i])>0$. By Theorem~\ref{th:back} (a) applied to $\mathds{1}_{[i]}$ we can find a point $y_0$ from the $\mu'$-full measure set whose almost every backward trajectory visits $[i]$ with limiting frequency $\mu'([i])$. But this contradicts Theorem~\ref{th:cms}.
\end{proof}
\begin{corollary}\label{cor:cms2}
In the positive recurrent case, for each $y_0\in\Sigma_M$, for a random choice $(y_n)$ of the backward trajectory,
\begin{equation}\label{limJac}
\lim_{n\to\infty} \sqrt[n]{c(y_0)\cdots c(y_{n-1})} = \exp \int \log(c)\,d\mu = \exp \sum_{i\in\mathcal{I}} \pi_i \log c_i,
\end{equation}
where all three expressions may be infinite. If additionally $- \sum \pi_i \log(\pi_i) < \infty$, then
\begin{equation}\label{Jacfair}
\int \log(c)\,d\mu = h_\mu(\sigma) = -\sum_{i,j\in\mathcal{I}} \pi_i p_{ij} \log(p_{ij}) = h_{\textnormal{fair}}(\sigma) < \infty.
\end{equation}
\end{corollary}
\begin{proof}
Theorem~\ref{th:back}~(c) applied to the measure $\mu$ gives~\eqref{limJac} for $\mu$-almost every $y_0$. In particular, in each cylinder set $[i]$ we get the result for at least one point $y'_0$. Now let $y_0$ be any other point in the same cylinder $[i]$. There is a natural way to identify backward trajectories of $y_0$ and $y'_0$. In the language of the probability model $(\Sigma_{M^T},\nu_0)$ developed earlier, we identify $(y_n)$ with $(y'_n)$ if they both arise from the same point $\omega'\in W_0$. But then $c(y_n)=c(y'_n)$ for all $n$. This shows that~\eqref{limJac} holds not just almost everywhere, but in fact for every $y_0\in\Sigma_M$.
The partition by length 1 cylinder sets $\mathcal{X}=\{[i] ; i\in \mathcal{I} \}$ is a one-sided generating partition and the condition $- \sum \pi_i \log(\pi_i) < \infty$ just says that the (Shannon) entropy $H_\mu(\eta)$ of this partition is finite. Then by Theorem~\ref{th:back}~(d) we get the first equality in~\eqref{Jacfair}. The sum in~\eqref{Jacfair} is just the well-known formula for the entropy of a Markov measure. This is also the fair entropy, since $\mu$ is the unique fair measure. Finally, since $\eta$ is a finite-entropy generating partition, the (Kolmogorov-Sinai) entropy $h_\mu(\sigma)$ is less than or equal to $H_\mu(\mathcal{X})$ and is therefore finite, see~\cite[Equation (9.1.15) and Corollary 9.2.5]{VO}.
\end{proof}
We now show some examples on known shift spaces. Later we will associate them with interval maps too.
\vspace{0.1in}
\begin{example}\label{ex:null} (Null recurrent case)
We consider a classic unbiased random walk on $\mathbb Z $, where we can go 1 step forward or 1 step backward from each state. Here is the transition diagram:
\begin{center}
\begin{tikzcd
{\cdots} &[-30]
{\circ} \arrow[r, bend left=25]
& \circ
\arrow[r, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[r, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[l, bend left=25]
&[-30pt]\cdots
\end{tikzcd}
\end{center}
\vspace{.5em}
We calculate the transition matrix $M$ in the standard way and by (\ref{Q}) we get the corresponding stochastic matrix $Q$.
\begin{equation*}
M =
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& 0 & 1 & 0 & 0 &0 & \cdots \\
\cdots&1 & 0 & 1 & 0 &0 & \cdots \\
\cdots &0& 1 & 0 & 1 & 0 & \cdots \\
\cdots &0& 0& 1 & 0 & 1 & \cdots \\
\cdots &0& 0& 0& 1 & 0 & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}
,\qquad
\renewcommand{\arraystretch}{1.2}
Q=
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& 0 & \frac12 & 0 & 0 &0 & \cdots \\
\cdots& \frac12 & 0 & \frac12 & 0 &0 & \cdots \\
\cdots &0& \frac12 & 0 & \frac12 & 0 & \cdots \\
\cdots &0& 0& \frac12 & 0 & \frac12 & \cdots \\
\cdots &0& 0& 0& \frac12 & 0 & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}.
\end{equation*}
We already know from probability theory (see \cite{Fel}) that
$Q$ is null recurrent, so by Corollary \ref{cor:cms} there is no fair measure in this case. Nevertheless $ \sqrt[n]{c(y_0)\cdots c(y_{n-1})} \rightarrow 2$ as $n \rightarrow \infty.$
\end{example}
\begin{example}\label{ex:transient} (Transient case)
A biased random walk on $\mathbb Z $ can be defined as the option to go 2 steps forward or 1 step backward from any state. Here is the transition diagram:
\begin{center}
\begin{tikzcd
{\cdots} \arrow[rr, dashed, bend left=25]
&[-30]
{\circ} \arrow[rr, bend left=25]
& \circ
\arrow[rr, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[rr, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[rr, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[rr, dashed, bend left=25]
\arrow[l, bend left=25]
& \circ
\arrow[l, bend left=25]
&[-30pt]\cdots
\end{tikzcd}
\end{center}
\vspace{.5em}
The matrices $M$ and $Q$ can be found the same way as before.
\begin{equation*}
M =
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& 0& 0 & 1 & 0 & 0 &0 & \cdots \\
\cdots&1& 0 & 0 & 1 & 0 &0 & \cdots \\
\cdots &0& 1 & 0& 0 & 1 & 0 & \cdots \\
\cdots &0& 0& 1 & 0& 0 & 1 & \cdots \\
\cdots &0& 0& 0& 1 & 0& 0 & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}
,\qquad
\renewcommand{\arraystretch}{1.2}
Q=
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& 0 & \frac12 & 0 & 0 & 0 &0 & \cdots \\
\cdots& 0 & 0 & \frac12 & 0 & 0 &0 & \cdots \\
\cdots & \frac12 & 0 & 0 & \frac12 & 0 & 0 & \cdots \\
\cdots &0 & \frac12& 0 & 0 & \frac12 & 0 & \cdots \\
\cdots &0& 0& \frac12& 0 & 0 & \frac12 & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots& \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}.
\end{equation*}
Then $\left(Q^{3n}\right)_{00}= \dfrac{(3n)!}{(2n)!n!}\cdot\dfrac{1}{2^{3n}}$, and so $\sum_{n\in \mathbb N} \left(Q^{n}\right)_{00} < \infty$ which by \cite[pg. 389]{Fel} shows us that the stochastic matrix is transient and therefore by Corollary \ref{cor:cms} there is no fair measure.
Another classic shift example is the unbiased random walk on $\mathbb Z^3$. We will get similar results. The stochastic matrix $Q$ is transient and so again by Corollary \ref{cor:cms} there is no fair measure in this case.
\end{example}
\begin{example} (Positive recurrent case)
For our last example on shift spaces we choose a process which can be defined as an option to go anywhere from the origin or 1 step backward from any other state, as is shown in the following transition diagram and matrices:
\begin{center}
\begin{tikzcd
\circ
\ar[loop,out=150,in=210,distance=30]
\arrow[r, bend left=20]
\arrow[rr, bend left=20]
\arrow[rrr, bend left=20]
\arrow[rrrr, bend left=20]
& \circ
\arrow[l, bend left=20]
& \circ
\arrow[l, bend left=20]
& \circ
\arrow[l, bend left=20]
& \circ
\arrow[l, bend left=18]
&[-30pt]\cdots
\end{tikzcd}
\end{center}
\begin{equation*}
M =
\begin{bmatrix}
& 1& 1 & 1 & 1 & 1 &1 & \cdots \\
&1& 0 & 0 & 0 & 0 &0 & \cdots \\
&0& 1 & 0& 0 & 0 & 0 & \cdots \\
&0& 0& 1 & 0& 0 & 0 & \cdots \\
&0& 0& 0& 1 & 0& 0 & \cdots \\
& \vdots & \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}
,\qquad
\renewcommand{\arraystretch}{1.2}
Q=
\begin{bmatrix}
& \frac12 & \frac12 & 0 & 0 & 0 &0 & \cdots \\
& \frac12 & 0 & \frac12 & 0 & 0 &0 & \cdots \\
& \frac12 & 0 & 0 & \frac12 & 0 &0 & \cdots \\
& \frac12 & 0 & 0 & 0 & \frac12 &0 & \cdots \\
& \frac12 & 0 & 0 & 0 & 0 &\frac12 & \cdots \\
& \vdots & \vdots& \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}.
\end{equation*}
The reader can easily check that
$\pi= \begin{bmatrix} \frac{1}{2} \, \frac{1}{4} \cdots \frac{1}{2^i} \cdots \end{bmatrix}$
satisfies~\eqref{pi}. Therefore $Q$ is positive recurrent and there is a unique fair measure. We can also calculate $P$ using~\eqref{P}:
\begin{equation*}
P =
\begin{bmatrix}
& \frac12& \frac14 & \frac18 & \frac{1}{16} & \cdots & \frac{1}{2^i} & \cdots \\
&1& 0 & 0 & 0 & \cdots &0 & \cdots \\
&0& 1 & 0& 0 & \cdots & 0 & \cdots \\
&0& 0& 1 & 0&\cdots & 0 & \cdots \\
&0& 0& 0& 1 & \cdots & 0 & \cdots \\
& \vdots & \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}
\end{equation*}
And so the measure $\textnormal{Markov}(\pi,P)$ is the fair measure.
Finally, we may calculate the fair entropy as the entropy of the measure $\textnormal{Markov}(\pi,P)$ on the shift space, using the well-known formula for the entropy of a Markov measure,
\begin{equation*}
h_{\textnormal{fair}}(f)=-\sum_{ij} \pi_i p_{ij} \log p_{ij} = \log 2.
\end{equation*}
This is the same as the Gurevich entropy of $(\Sigma_M, \sigma)$, and so the fair measure is the maximal measure.
\end{example}
\section{Isomorphisms for Fair Measures}\label{sec:iso}
We would like to extend our results from shift spaces to various maps on the interval, dendrites, etc. through the use of Markov partitions. But before we can proceed, we need to develop a notion of isomorphism. Our notion is inspired by the isomorphisms modulo small sets in~\cite{Hof}. For a system $(X,f)$, a set $B\subset X$ is called \emph{totally invariant} if $f^{-1}(B)=B$.
\begin{definition}
Two systems $(X_1,f_1)$, $(X_2, f_2)$ are called \emph{isomorphic modulo countable invariant sets} if there exist totally invariant countable sets $N_1\subset X_1$, $N_2\subset X_2$ and a bijection $\psi:X_1\setminus N_1 \to X_2\setminus N_2$, bimeasurable with respect to the Borel $\sigma$-algebras, such that $f_2\circ\psi=\psi\circ f_1$.
\end{definition}
From the point of view of fair measures, deleting countable totally invariant sets is rather harmless. For the points that remain, the tree of preimages is unchanged, so that the meaning of a random backward orbit is the same as before. And countable sets have measure zero for every non-atomic measure. Letting $\mathcal{M}_{\text{n.a.}}(\cdot,\cdot)$ denote the set of non-atomic invariant Borel probability measures for a system, we get bijections
\begin{equation*}
\mathcal{M}_{\text{n.a.}}(X_i,f_i) \to \mathcal{M}_{\text{n.a.}}(X_i\setminus N_i,f_i), \quad i=1,2,
\end{equation*}
given by restriction $\mu \mapsto \mu|_{X_i\setminus N_i}$, as well as a bijection
\begin{equation*}
\mathcal{M}_{\text{n.a.}}(X_1\setminus N_1,f_1) \to \mathcal{M}_{\text{n.a.}}(X_2\setminus N_2, f_2)
\end{equation*}
given by composition $\mu \mapsto \mu \circ \psi^{-1}$.
If $\mu_1\in\mathcal{M}_{\text{n.a.}}(X_1,f_1)$ and $\mu_2\in\mathcal{M}_{\text{n.a.}}(X_2,f_2)$ correspond to each other under these bijections, then $\psi:(X_1,f_1,\mu_1) \to (X_2,f_2,\mu_2)$ is a conjugacy in the sense of measure theory (where null sets are negligible). Consequently,
\begin{itemize}
\item the entropies are equal $h_{\mu_1}=h_{\mu_2}$,
\item $\mu_1$ is ergodic if and only if $\mu_2$ is, and
\item The Jacobians are related by $J_{\mu_1} = J_{\mu_2} \circ \psi$.
\end{itemize}
But total invariance of our deleted sets gives $c_1\circ f_1 = c_2 \circ f_2 \circ \psi$ on $X_1\setminus N_1$, i.e. $\mu_1$-almost everywhere. In light of Lemma~\ref{lem:Jac} we can conclude that if $\mu_2$ is fair, then so is $\mu_1$. By the symmetry of the situation, the reverse implication holds also. In particular, we have proved
\begin{theorem}\label{th:isom}
A Borel isomorphism modulo countable invariant sets induces an entropy-preserving bijection of non-atomic fair measures. Moreover, this implies that the two systems have the same fair entropy.
\end{theorem}
We remark that in general, fairness of a measure is not an invariant of measure theoretic conjugacy. This is because adding or deleting a set of measure zero can change the function $c\circ f$ almost everywhere, cf. Lemma~\ref{lem:Jac}. Figure~\ref{fig:1} illustrates how this might happen.
\begin{figure}[htb!!]
\vspace{1em}
\scalebox{.95}{
\includegraphics[width=.3\textwidth]{fig1a.jpg}
\begin{picture}(0,0)
\put(0,90){\scriptsize 3 preimages}
\put(0,30){\scriptsize 1 preimage}
\put(-133,-2){$\underbrace{\hspace{5.2em}}_{\substack{\text{zero}\\\text{measure}}}$}
\put(-72,-2){$\underbrace{\hspace{5.2em}}_{\substack{\text{normalized}\\\text{Lebesgue measure}}}$}
\put(-100,135){\small (a) Not fair}
\end{picture}
\hspace{.2\textwidth}
\includegraphics[width=.3\textwidth]{fig1b.jpg}
\begin{picture}(0,0)
\put(0,60){\scriptsize 2 preimages}
\put(-131,-2){$\underbrace{\hspace{10.4em}}_{\text{Lebesgue measure}}$}
\put(-90,135){\small (b) Fair}
\end{picture}
}
\\[1.5em]
\caption{Deleting the (measure zero) left half of the interval and rescaling gives a measure theoretic conjugacy, but system (a) is not fair while system (b) is fair.}\label{fig:1}
\end{figure}
Fair measures with atoms are not addressed in Theorem~\ref{th:isom}, but are rather simple to understand. If $(X,f)$ has a totally invariant periodic orbit, then we can equidistribute point masses along this orbit and we obtain a purely atomic ergodic fair measure. Conversely, we observe that
\begin{proposition}\label{prop:atoms}
All the atoms of a fair measure belong to totally invariant periodic orbits.
\end{proposition}
\begin{proof}
As a general fact regarding invariant probability measures, atoms can only occur in periodic orbits and with each point of a periodic orbit $P$ having the same mass $m$. If $P$ is not totally invariant, then at least one point $x\in P$ has $c(x)\geq2$. Then the preimage of $x$ in $P$ has mass under a fair measure equal to both $m$ and $\frac{m}{c(x)}$, which implies that $m=0$.
\end{proof}
\section{Countably Markov and mixing Interval Maps}\label{LIM}
Throughout this section our space is the unit interval $X=[0,1]$, our map $f$ is allowed to have countably many pieces of continuity and monotonicity, and our partition $\mathcal{X}$ is Markov. That means that
\begin{itemize}
\item $\mathcal{X}$ consists of open intervals $(a,b)$ and singletons $\{x\}$. We write $$\mathcal{I}=\left\{i=(a,b);~(a,b)\in\mathcal{X}\right\}, \quad C=\left\{x;~\{x\}\in\mathcal{X}\right\}$$
for the sets of \emph{partition intervals} and \emph{partition points}.
\item For each $i\in\mathcal{I}$, the restriction $f|_i$ is continuous and strictly monotone.
\item For each pair $i,j\in\mathcal{I}$, either $f(i)\supset j$ or else $f(i)\cap j=\emptyset$.
\item The partition points form a forward invariant set $f(C)\subset C$.
\end{itemize}
We assume additionally that $f$ is \emph{mixing}, i.e. for each pair $U,V$ of nonempty open sets there is $N\geq0$ such that for all $n\geq N$, $f^n(U)\cap V\neq\emptyset$. If all these assumptions are satisfied, then we call $f$ \emph{countably Markov and mixing}.
A \emph{homterval} for an interval map $f$ is a nonempty open interval $U\subset[0,1]$ such that for all $n\geq0$, $f$ maps $f^n(U)$ homeomorphically onto $f^{n+1}(U)$.
\begin{lemma}
A countably Markov and mixing interval map has no homtervals.
\end{lemma}
\begin{proof}
A mixing interval map has at least one \emph{critical point} $x\in(0,1)$, such that $f$ is not monotone on any neighborhood of $x$ . Let $U$ be any nonempty open interval. By the mixing property applied to $U$, $V=[0,x)$, and $V'=(x,1]$, there is a common value $n$ such that $f^n(U)$ meets both $V$ and $V'$. If $U$ were a homterval, then by the intermediate value theorem $f^n(U)$ would be a neighborhood of $x$, but then $f$ could not map $f^n(U)$ homeomorphically onto $f^{n+1}(U)$.
\end{proof}
To a countably Markov and mixing interval map $f$ we associate a \emph{transition matrix} $M$ with rows and columns indexed by $\mathcal{I}$ and entries
\begin{equation}
\label{mij}
m_{ij}=\begin{cases}
0,&\text{ if }f(i)\cap j=\emptyset\\
1,&\text{ if }f(i)\supset j
\end{cases}.
\end{equation}
Using the mixing property of $f$ it is easy to see that $M$ is irreducible. Thus we may form the shift space $\Sigma_M$ and look for fair measures as in Section~\ref{sec:cms}.
All of our results about Markov interval maps flow out of the following isomorphism theorem.
\begin{theorem}\label{th:cmm}
Let $f$ be countably Markov and mixing with transition matrix $M$. Then $(\Sigma_M,\sigma)$ and $([0,1],f)$ are isomorphic modulo countable invariant sets.
\end{theorem}
Before beginning the proof we record one more or less standard lemma.
\begin{lemma}\label{lem:symbolic}
Let $f$ be countably Markov and mixing with transition matrix $M$.
\begin{enumerate}[label=(\alph*)]
\item\label{lem:symbolic_a}
For each nonempty cylinder set $[i_0\cdots i_n]\subset \Sigma_M$, the set $$U=i_0 \cap f^{-1}(i_1) \cap \cdots \cap f^{-n}(i_n)$$ is a nonempty open interval mapped homeomorphically by $f^n$ onto $i_n$.
\item\label{lem:symbolic_b}
If $[i'_0 \cdots i'_n]\neq[i_0 \cdots i_n]$ is another nonempty cylinder set, then the corresponding set $U'$ is disjoint from $U$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{lem:symbolic_a}: By induction in the length of the cylinder. The base case $n=0$ is clear. Given $[i_0\cdots i_n]$, the induction hypothesis applied to $[i_1\cdots i_n]$ gives us a nonempty open interval $V=i_1 \cap f^{-1}(i_2) \cap \cdots f^{-n+1}(i_n)$ contained in $i_1$ and mapped homeomorphically by $f^{n-1}$ onto $i_n$. But $f(i_0) \supset i_1$ and $f|_{i_0}$ is continuous and strictly monotone. We get that $U=i_0\cap f^{-1}(V)=f|_{i_0}^{-1}(V)$ is a nonempty open interval mapped by $f$ homeomorphically onto $V$, and the result follows.
\ref{lem:symbolic_b}: If $i_j\neq i'_j$, then the sets $f^j(U)\subset i_j$ and $f^j(U')\subset i'_j$ are disjoint. Therefore $U, U'$ are disjoint.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:cmm}]
Elements of $\Sigma_M$ are called \emph{itineraries}. Given an itinerary $\omega=i_0 i_1 \cdots$ we put $U_n(\omega)=i_0\cap f^{-1}(i_1)\cap \cdots \cap f^{-n}(i_n)$ for each $n\geq0$. We have a nested sequence of open intervals $U_0(\omega) \supset U_1(\omega) \supset \cdots$ as well as a nested sequence of closed intervals $\overline{U_0(\omega)} \supset \overline{U_1(\omega)} \supset \cdots$. Taking the intersection, we obtain a non-empty closed interval $\bigcap_{n=0}^\infty \overline{U_n(\omega)}$. This interval must be degenerate (a singleton), for otherwise its interior is a homterval for $f$. In this way we get a map
\begin{equation}\label{psi}
\psi:\Sigma_M\to[0,1], \quad \psi(\omega)=x\text{, where }\{x\}=\bigcap_{n=0}^\infty \overline{U_n(\omega)}.
\end{equation}
We say that $\omega$ is an itinerary of the point $x=\psi(\omega)$. If in~\eqref{psi} the closures are not needed so that $x\in\bigcap_{n=0}^\infty U_n(\omega)$, then we call $\omega$ a \emph{true itinerary}; otherwise we call $\omega$ a \emph{false itinerary}.
True itineraries behave well. If $\omega=i_0i_1\cdots$ is a true itinerary of the point $x$, then $f^n(x)\in i_n$ for all $n\geq0$. In particular, $x$ does not belong to the set of \emph{pre-critical points} $D=\bigcup_{n=0}^\infty f^{-n}(C)$. Conversely, each point $x\in[0,1]\setminus D$ has a true itinerary given by
\begin{equation}\label{phi}
\phi:[0,1]\setminus D \to \Sigma_M, \quad \phi(x)=i_0i_1i_2\cdots\text{, where }f^n(x)\in i_n\in\mathcal{I} \text{ for all }n\geq0.
\end{equation}
Even better, if $\omega$ is a true itinerary for $x$, we see immediately that $\sigma(\omega)$ is a true itinerary for $f(x)$. Thus, we get the conjugacy relation $\psi(\sigma(\omega))=f(\psi(\omega))$ for all true itineraries $\omega$.
Conversely, suppose $\omega=i_0i_1\cdots$ is an itinerary whose shift $\sigma(\omega)$ is true. Write $y=\psi(\sigma(\omega))$. Since $i_0\cap f^{-1}(i_1)$ is mapped homeomorphically by $f$ onto $i_1$, we find a point $x\in i_0\cap f^{-1}(y)$. Since $y$ was not a pre-partition point, neither is $x$, and we see immediately from~\eqref{phi} that $\omega$ is a true itinerary for $x$. We have shown that an itinerary $\omega$ is true if and only if $\sigma(\omega)$ is true.
False itineraries behave much worse. If $\omega=i_0 i_1 \cdots$ is a false itinerary of the point $x$, then there is a minimal number $n\geq0$ such that $x\notin U_n(\omega)$, called the \emph{order} of the false itinerary. Recall that $U_n(\omega)$ is an open interval contained in $i_0$ and mapped homeomorphically by $f^n$ onto $i_n$. Since $x\in\overline{U_n(\omega)}$ and $f|_{i_0}$ is continuous and $x\in i_0$ (assuming $n\geq1$), we get $f^n(x)\in\overline{i_n}$. Therefore $f^n(x)$ is an endpoint of $i_n$ (this holds also in the case $n=0$). In particular, $f^n(x)\in C$, so $x\in D$. Thus, false itineraries are only associated with pre-partition points. The conjugacy relation may not hold: if $\omega$ is a false itinerary for $x$ of order $0$, then $\sigma(\omega)$ is still a false itinerary but perhaps not for $f(x)$. This is because we have no control over the value $f(x)$ when $x\in C$ is not a continuity point for $f$.
A pre-partition point $x\in D$ may have zero, one, or two false itineraries, but not more. For let $n\geq0$ be minimal such that $f^n(x)\in C$. We have already seen that each false itinerary $\omega=i_0i_1\cdots$ for $x$ has order $n$. Thus $x\in U_{n-1}(\omega)$ and $x$ is an endpoint of $U_n(\omega)$. Because of the nesting $\overline{U_n(\omega)} \supset \overline{U_{n+1}(\omega)} \supset \cdots$, we see that $x$ is an endpoint of $U_j(\omega)$ for all $j\geq n$. By Lemma~\ref{lem:symbolic}~\ref{lem:symbolic_b}, the condition $x\in U_{n-1}(\omega)$ uniquely determines the symbols $i_0, \cdots, i_{n-1}$. There are at most two choices for $i_n$, namely, the at most two partition intervals with common endpoint $f^n(x)$. Now let $\omega'=i'_0i'_1\cdots$ be another false itinerary. We claim that if $i_0\cdots i_n=i'_0\cdots i'_n$, then $\omega=\omega'$. This follows inductively, for if $i_0\cdots i_j=i'_0\cdots i'_j$ with $j\geq n$ but $i_{j+1}\neq i'_{j+1}$, then $U_{j+1}(\omega)$ and $U_{j+1}(\omega')$ are disjoint subintervals of $U_j(\omega)=U_j(\omega')$, and all three of these intervals have $x$ as an endpoint, which is impossible.
Let $N\subset \Sigma_M$ denote the set of false itineraries. We have shown so far that $N=\psi^{-1}(D)$ and that $\phi,\psi,f,\sigma$ are related by the following commutative diagram:
\begin{equation}\label{cd}
\begin{tikzcd}[row sep=1cm, column sep=1.5cm]
\Sigma_M\setminus N \arrow[r, "\sigma"] \arrow[d, shift left=-1, "\psi" swap] & \Sigma_M\setminus N \arrow[d, shift left=-1, "\psi" swap] \\
\left[0,1\right]\setminus D \arrow[r, "f"] \arrow[u, shift left=-1, "\phi" swap] & \left[0,1\right]\setminus D \arrow[u, shift left=-1, "\phi" swap]
\end{tikzcd}
\end{equation}
We still need to show that
\begin{enumerate}[label=(\roman*)]
\item\label{it:cd1} $N$ and $D$ are countable and totally invariant, and
\item\label{it:cd2} $\psi:\Sigma_M\setminus N \to [0,1]\setminus D$ and its inverse $\phi$ are both measurable with respect to the Borel $\sigma$-algebras.
\end{enumerate}
We start with~\ref{it:cd1}. Since $f$ is at most countable-to-one and $C$ is countable, it follows that $D=\cup_{n=0}^\infty f^{-n}(C)$ is countable. Backward invariance $f^{-1}(D)\subset D$ is clear from construction, and forward invariance $f(D)\subset D$ is inherited from $C$. Therefore $D$ is totally invariant. Since $N=\psi^{-1}(D)$ and $\psi$ is at most two-to-one, it follows that $N$ is countable. We already showed that an itinerary $\omega$ is true if and only if $\sigma(\omega)$ is true. Thus, the set of false itineraries $N$ is also totally invariant.
To prove~\ref{it:cd2}, it suffices to show continuity of the maps $\psi:\Sigma_M\to[0,1]$ and $\phi:[0,1]\setminus D \to \Sigma_M$. To see that $\psi$ is continuous at a given point $\omega=i_0i_1\cdots$, let $\epsilon>0$ be given. Since the nested intersection in~\eqref{psi} contains only the one point $x=\psi(\omega)$, it follows that there is $n\geq0$ such that $\overline{U_n(\omega)}$ is contained in the $\epsilon$-neighborhood of $x$. But the cylinder set $[i_0\cdots i_n]$ is an open neighborhood of $\omega$ mapped by $\psi$ into $\overline{U_n(\omega)}$.
To see that $\phi$ is continuous consider any nonempty cylinder set $[i_0\cdots i_n]\subset\Sigma_M$ and define $U$ as in Lemma~\ref{lem:symbolic}~\ref{lem:symbolic_a}. Then $U$ is open in $[0,1]$, so $U\setminus D$ is an open subset of $[0,1]\setminus D$ in the subspace topology. But $U\setminus D=\phi^{-1}([i_0\cdots i_n])$. We have shown that each cylinder set has an open preimage under $\phi$, and since cylinder sets form a basis for the topology on $\Sigma_M$, we are done.
\end{proof}
Now we explore the consequences of our isomorphism theorem for our interval map.
\begin{theorem}\label{th:cmmfair}
Let $f$ be countably Markov and mixing with transition matrix $M$. Form $Q$ using~\eqref{cj} and~\eqref{Q}. Let $\Acc(C)$ denote the set of accumulation points of $C$.
\begin{enumerate}[label=(\alph*)]
\item\label{th:cmmfair_a} If $Q$ is positive recurrent, then
$f$ has exactly one non-atomic fair measure $\mu$. It is ergodic and has full support.
For any non-partition point $y_0\in[0,1]\setminus C$, a randomly chosen backward trajectory $(y_n)$ equidistributes for $\mu$, and the geometric averages $\sqrt[n]{c(y_0)\cdots c(y_{n-1})}$ converge to $\exp h_\mu(f)$.
\item\label{th:cmmfair_b} If $Q$ is null recurrent, then
$f$ does not have any non-atomic fair measure.
For any non-partition point $y_0\in[0,1]\setminus C$, a randomly chosen backward trajectory $(y_n)$ is dense in $[0,1]$, but visits each set $E$ bounded away from $\Acc(C)$ with limiting frequency zero.
\item\label{th:cmmfair_c} If $Q$ is transient, then
$f$ does not have any non-atomic fair measure.
For any non-partition point $y_0\in[0,1]\setminus C$, a randomly chosen backward trajectory $(y_n)$ converges to $\Acc(C)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Most of these results follow immediately from Theorem~\ref{th:cms} and its corollaries together with our isomorphism result Theorem~\ref{th:cmm}. It is also critical to note that the maps $\phi,\psi$ in~\eqref{cd} are continuous. Therefore if $U\subset[0,1]$ is open, then there is an open set $V\subset\Sigma_M$ whose symmetric difference with $\psi^{-1}(U)$ is contained in the countable invariant set $N$. This gives us the full support result in~\ref{th:cmmfair_a}, because the fair measure $\text{Markov}(\pi,P)$ on $\Sigma_M$ has full support. It also gives us the density result in~\ref{th:cmmfair_b} because $y_n\in U$ if and only if $\phi(y_n)\in V$.
Since $C$ is closed and countable, so is its set of accumulation points $\Acc(C)$. Now if $E\subset[0,1]$ is bounded away from this set, i.e. $\overline{E}\cap\Acc(C)=\emptyset$, then $E$ is contained in a finite union of partition intervals $i_1\cup \cdots \cup i_n$. Then $y_n$ cannot visit $E$ unless $\phi(y_n)$ is in the corresponding union of cylinders $[i_1]\cup\cdots\cup[i_n]$. This gives us the frequency of visits to $E$ in part~\ref{th:cmmfair_b}. It also gives us the convergence of $y_n$ to $\Acc(C)$ in part~\ref{th:cmmfair_c}, where convergence means that the distance between $y_n$ and the nearest point of $\Acc C$ goes to zero.
\end{proof}
\section{Lebesgue fair models.}
\begin{definition}
An interval map is called \emph{Lebesgue fair} if Lebesgue measure is fair for it. A \emph{Lebesgue fair model} for an interval map $f$ is a Lebesgue fair map $g$ conjugate to $f$ by a monotone increasing homeomorphism $\phi:[0,1]\to[0,1]$, $g\circ\phi=\phi\circ f$.
\end{definition}
The main result of this section is a construction of the Lebesgue fair models for countably Markov and mixing interval maps. In a sense, it allows us to visualize the fair measures which we found. We start with two easy lemmas.
\begin{lemma}\label{lem:fmfm}
The Lebesgue fair models for $f$ are in bijective correspondence with the non-atomic fair measures of full support.
\end{lemma}
\begin{proof}
Let $\mu$ be such a measure and define $\phi_\mu:[0,1]\to[0,1]$ by $x\mapsto\mu([0,x])$. This is a monotone increasing homeomorphism with $\mu\circ\phi_\mu^{-1}$ equal to the Lebesgue measure $\lambda$. Thus $\phi_\mu : ([0,1],f,\mu) \to ([0,1],g,\lambda)$ is a measure theoretic isomorphism everywhere, i.e., without the removal of measure zero sets. Therefore $\lambda$ is fair for $g$ (see Section~\ref{sec:iso}).
Conversely, let $g$ be a Lebesgue fair model for $f$ with conjugating homeomorphism $\phi$. Then $\mu_\phi=\lambda\circ\phi$ is a Borel probability measure and $\phi:([0,1],f,\mu_\phi)\to([0,1],g,\lambda)$ is a measure theoretic isomorphism everywhere. It follows that $\mu_\phi$ is a non-atomic fully supported fair measure for $f$.
Since the operations $\mu\mapsto\phi_\mu$ and $\phi\mapsto\mu_\phi$ are clearly inverse to each other, we've found the desired bijective correspondence.
\end{proof}
\begin{lemma}\label{lem:determined}
A homeomorphism $g:(a,b)\to(c,d)$ is uniquely determined by its orientation (increasing or decreasing) and the Jacobian for Lebesgue measure, provided that Lebesgue measure is non-singular for $g$.
\end{lemma}
\begin{proof}
Assume $g$ is an increasing homeomorphism. Given $x\in(a,b)$, put $B=(a,x)$ so that $g(B)=(c,g(x))$. By~\eqref{Jac} we get $g(x)=c+\int_B J\,d\lambda$, where $J$ is the Jacobian and $\lambda$ is the Lebesgue measure.
\end{proof}
Now let $f$ be countably Markov and mixing with transition matrix $M$. If $f$ falls into case~\ref{th:cmmfair_b} or~\ref{th:cmmfair_c} of Theorem~\ref{th:cmmfair}, then there are no Lebesgue fair models. But in case~\ref{th:cmmfair_a}, we see that there is a unique Lebesgue fair model $g$, and we would like to know what it looks like.
Here is a construction for the graph of $g$. On the horizontal axis we draw the sets $\phi(i\cap f^{-1}(j))$, $i,j\in\mathcal{X}$, $i\cap f^{-1}(j)\neq\emptyset$, where $\phi=\phi_\mu$ is the conjugacy given by Lemma~\ref{lem:fmfm} using the unique non-atomic fair measure $\mu$ for $f$. Thankfully, there is no need to calculate $\phi$ explicitly; it suffices to calculate $\pi$ and $P$ from~\eqref{pi} and~\eqref{P}. We know that $\phi(i\cap f^{-1}(j))$ is an open interval if $i,j\in\mathcal{I}$ and a singleton otherwise. We also know that these sets form a partition. And we know exactly where to draw these sets in $[0,1]$ because we know their ordering ($\phi$ preserves ordering) and their lengths (denoted $\len$) $$\len(\phi(i\cap f^{-1}(j)))=\mu(i\cap f^{-1}(j)) = \begin{cases}\pi_i p_{ij},&\text{if }i,j\in\mathcal{I}\\0,&\text{otherwise}\end{cases}.$$
On the vertical axis we draw the partition sets $\phi(j)$, $j\in\mathcal{X}$. We get open intervals when $j\in\mathcal{I}$ and singletons otherwise. Again, we know exactly where to draw these sets because we know their ordering and their lengths $$\len(\phi(j))=\mu(j)=\begin{cases}\pi_j,&\text{if }j\in\mathcal{I}\\0,&\text{otherwise}\end{cases}.$$
Now we fill in the graph of $g$. If $\phi(i\cap f^{-1}(j))$ is a singleton, then $g$ maps this point to the single element of $\phi(j)$. If $\phi(i\cap f^{-1}(j))$ is an interval, then $g$ carries this interval homeomorphically onto $\phi(j)$. The Jacobian here for the fair (non-singular) Lebesgue measure is the constant $c_j=\#f^{-1}(x)$, $x\in j$. So by Lemma~\ref{lem:determined}, this piece of $g$ is affine with slope $\pm c_j$, with the plus or minus determined by the orientation of $f|_i$. Notice that this slope agrees with the lengths of these intervals, because $$\frac{\pi_j}{\pi_i p_{ij}} = \frac{\pi_j}{\pi_j q_{ji}} = \frac{c_j}{m_{ij}}=c_j.$$
We may summarize our construction as a theorem.
\begin{theorem}\label{th:fairmodel}
Let $f$ be a countably Markov and mixing interval map whose associated Markov shift $\Sigma_M$ has a fair measure $\textnormal{Markov}(\pi,P)$. Then the unique Lebesgue fair model for $f$ is the piecewise affine map $g$ defined as follows: For each pair $i,j\in\mathcal{X}$ with $i\cap f^{-1}(j)\neq\emptyset$,
if $i,j\in\mathcal{I}$, then $g$ maps $(x,x')$ affinely onto $(y,y')$ with the same orientation as $f|_i$, and
if $j=\{c\}$ is a singleton, then $g$ maps $x$ to $y$, where
\begin{gather*}
x=\sum_{\mathclap{(k,l)\in W_{ij}}} \pi_k p_{kl}, \qquad x'=x+\pi_i p_{ij}, \qquad y=\sum_{l\in W_j}\pi_l, \qquad y'=y+\pi_j,\\
W_{ij}=\left\{ (k,l)\in\mathcal{I}\times\mathcal{I} ;~ \emptyset \neq k\cap f^{-1}(l) \text{ lies to the left of }i\cap f^{-1}(j)\right\}\\
W_j=\left\{ l\in\mathcal{I} ;~ l \text{ lies to the left of }j\right\}.
\end{gather*}
\end{theorem}
In this way, we have constructed the Lebesgue fair model $g$ for $f$ using only combinatorial information.
Incidentally, this shows that there is quite a lot of flexibility in the definition of the function $f$ -- as long as we keep the right Markov structure and the mixing property, we automatically get a topological conjugacy to the same Lebesgue fair model $g$.
\vspace{0.2in}
\begin{example}
Applying the results of section \ref{LIM}, especially Theorem \ref{th:cmmfair}, we can associate interval maps with shift spaces. The next example is just one of a pile of transitive mappings which are a lot like a random walk. We choose one which has all partition intervals the same length. One partition interval will cover itself and another 4 partition intervals when the map there is increasing or another 2 intervals when it decreases. Here are an exact formula for $f(x)$, a picture, and also the corresponding matrices $M$ and $Q$ with the main diagonals shown in bold:
\begin{minipage}[c][0.55cm][c]{.55\textwidth}
\begin{equation*}\label{53}
f(x) = \begin{cases}
5x-8n-2, & {\rm if} ~ x \in I_{2n},\\\\
-3x+8n+6, & {\rm if} ~ x \in I_{2n+1} ,\\
\end{cases}
\end{equation*}\\[1em]
\centering
where $n\in \mathbb Z$ and $I_k= \langle k, k +1\rangle$.
\end{minipage}%
\begin{minipage}[c][6cm][c]{.45\textwidth}
\includegraphics[width=.7\textwidth]{PMM.jpg}
\end{minipage}
\vspace{-1em}
\begin{equation*}
{\small
\setcounter{MaxMatrixCols}{20}
M =
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& \mathbf{1} & 1 & 1 & 0 & {0} & 0 & 0 & \cdots \\
\cdots& 1 & \mathbf{1} & 1 & 0 & {0} & 0 & 0 & \cdots \\
\cdots& 1 & 1 & \mathbf{1} & 1 & {1} & 0 & 0 & \cdots \\
\cdots& 0 & 0 & 1 & \mathbf{1} & {1} & 0 & 0 & \cdots \\
\cdots & 0 & 0 & 1 & 1 & \mathbf{1} & 1 & 1 & {\cdots} \\
\cdots & 0 & 0& 0 & 0 & {1} & \mathbf{1} & 1 & \cdots \\
\cdots & 0 & 0& 0 & 0 & {1} & 1 &\mathbf{1} & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots & \vdots & \vdots& \vdots& \vdots & \vdots &\ddots
\end{bmatrix}
,\quad
\renewcommand{\arraystretch}{1.2}
Q=
\begin{bmatrix}
\ddots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \reflectbox{$\ddots$} \\
\cdots& \frac15& \frac15 & \mathbf{\frac15} & \frac15 & {\frac15} & 0 & 0 & 0 & \cdots \\
\cdots & 0 & 0 & \frac13 &\mathbf{\frac13} & {\frac13} & 0 & 0 & 0& \cdots \\
\cdots & 0 & 0 & \frac15 & \frac15 & \mathbf{\frac15} & \frac15 & \frac15 & 0 & \cdots \\
\cdots & 0 & 0 & 0 & 0 & {\frac13} & \mathbf{\frac13} & \frac13 & 0 & \cdots \\
\cdots & 0 & 0 & 0 & 0 & {\frac15} & \frac15 & \mathbf{\frac15} & \frac15 &\cdots \\
\cdots & 0& 0& 0 & 0 & {0} & 0 & \frac13 & \mathbf{\frac13} & \cdots \\
\reflectbox{$\ddots$} & \vdots & \vdots & \vdots& \vdots & \vdots & \vdots & \vdots& \vdots &\ddots
\end{bmatrix}.
}
\end{equation*}
The map $f$ has no totally invariant periodic orbits, so if there are any fair measures, they must be non-atomic (see Proposiiton~\ref{prop:atoms}).
It is not so hard to check that $\pi = (\cdots 3 \, 5 \, 3\, 5\, 3 \cdots)$, where $\pi_{2i} = 5, \pi_{2i+1} = 3$, satisfies the formula $\pi Q=\pi$. But even if we rescale $\pi,$ $ \sum_{i\in \mathbb Z} \pi_i = \infty$. Therefore, our $\pi$ cannot satisfy formula (\ref{pi}) and by \cite[Section XV.11]{Fel} there is no summable solution, and so the stochastic matrix is not positive recurrent and there is no fair measure for $f$.
\end{example}
\begin{example}\label{ex:BruinTodd}
Bruin and Todd studied the thermodynamic formalism for a countably piecewise linear interval map $f_{\lambda}:[0,1] \rightarrow [0,1]$, as defined in~\eqref{BT} below. To be complete, at the partition points $C=\{0,1,\lambda,\lambda^2,\ldots\}$ we make $f_\lambda$ continuous from the left, and we put $f_\lambda(0)=0$. All choices of the parameter $\lambda\in(0,1)$ give a map from the same topological conjugacy class. There is no measure of maximal entropy, and the topological entropy (supremum of entropies of invariant probability measures) is equal to $\log 4$, see~\cite{BT}. \\[-1em]
\begin{minipage}[c][6cm][c]{.6\textwidth}
\begin{equation}\label{BT}
f_{\lambda}(x) = \begin{cases}
\dfrac{x- \lambda}{1- \lambda}, & {\rm if} ~ x \in i_1,\\[1.5em]
\dfrac{x- \lambda^n}{\lambda(1- \lambda)}, & {\rm if} ~ x \in i_n, n \geq 2,\\
\end{cases}
\end{equation}\\[.5em]
\centering
where $i_n = ( \lambda^n, \lambda^{n-1})$
\end{minipage}%
\begin{minipage}[c][6cm][c]{.4\textwidth}
\scalebox{.9}{
\includegraphics[width=.8\textwidth]{bt.jpg}
\begin{picture}(0,0)
\put(-30,-10){$W_1$}
\put(-60,-10){$W_2$}
\put(-90,-10){$\cdots$}
\end{picture}
}
\end{minipage}
We want to calculate the non-atomic fair measures for this map. Therefore we write down the transition matrix $M$ and the corresponding stochastic matrix $Q$,
\begin{equation*}
M =
\begin{bmatrix}
1 & 1 & 1 & 1 & \cdots \\
1 & 1 & 1 & 1 & \cdots \\
0 & 1 & 1 & 1 & \cdots \\
0 & 0 & 1 & 1 & \cdots \\
0 & 0 & 0 & 1 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{bmatrix}
,\qquad
\renewcommand{\arraystretch}{1.2}
Q=
\begin{bmatrix}
\tfrac12 & \tfrac12 & 0 & 0 & 0 & \cdots \\
\tfrac13 & \tfrac13 & \tfrac13 & 0 & 0 & \cdots \\
\tfrac14 & \tfrac14 & \tfrac14 & \tfrac14 & 0 & \cdots \\
\tfrac15 & \tfrac15 & \tfrac15 & \tfrac15 & \tfrac15 & \cdots \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{bmatrix}
.
\end{equation*}
The reader can easily check that
$\pi=\frac1e \begin{bmatrix} \frac{1}{0!} \frac{1}{1!} \frac{1}{2!} \frac{1}{3!} \cdots \end{bmatrix}$
satisfies~\eqref{pi}. Therefore $Q$ is positive recurrent and our interval map $f_\lambda$ has a unique non-atomic fair measure. We may calculate $P$ using~\eqref{P} and the Lebesgue fair model $g$ using Theorem~\ref{th:fairmodel}.
\begin{minipage}[c][6cm][c]{.3\textwidth}
\begin{equation*}
\renewcommand{\arraystretch}{1.2}
P=
\begin{bmatrix}
\tfrac12 & \tfrac13 & \tfrac18 & \tfrac{1}{30} & \cdots \\
\tfrac12 & \tfrac13 & \tfrac18 & \tfrac{1}{30} & \cdots \\
0 & \tfrac23 & \tfrac28 & \tfrac{2}{30} & \cdots \\
0 & 0 & \tfrac68 & \tfrac{6}{30} & \cdots \\
0 & 0 & 0 & \tfrac{24}{30} & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{bmatrix}
\end{equation*}
\end{minipage}\hspace{.1\textwidth}%
\begin{minipage}[c][6cm][c]{.26\textwidth}
\includegraphics[width=\textwidth]{btfair.jpg}
\end{minipage}\hspace{.04\textwidth}%
\begin{minipage}[c][6cm][c]{.26\textwidth}
\centering
\small
The Lebesgue fair model. Notice that\\
$|g'(x)|=\#g^{-1}(g(x))$
Lebesgue almost everywhere.
\end{minipage}
Finally, we may use Corollary~\ref{cor:cms2} to calculate the fair entropy as the entropy of the measure $\textnormal{Markov}(\pi,P)$ on the shift space $\Sigma_M$,
\begin{equation*}
h_{\textnormal{fair}}(f)=-\sum_{ij} \pi_i p_{ij} \log p_{ij} \approx \log(2.85053).
\end{equation*}
\end{example}
\section{Maps on Tame Graphs}
In this section we move beyond the interval and look for fair measures for graph maps. Our main theorem is for tame graphs, a generalization of finite graphs introduced in~\cite{BBPRV}. We start by recalling the definition.
A \emph{continuum} is a nonempty, compact, connected metric space. By $E(G)$ we denote the \emph{endpoints} of the continuum $G$, i.e. the points $x\in G$ having arbitrarily small neighborhoods $V$ with one-point boundaries $\#\partial V=1$. Similarly, by $B(G)$ we denote the \emph{branching points}, i.e. the points $x\in G$ such that any sufficiently small neighborhood $V$ of $x$ has at least three points in its boundary $\#\partial V\geq 3$. A continuum $G$ is called a \emph{tame graph} if the set $E(G)\cup B(G)$ has a countable closure.
An arc $\alpha$ in a continuum $G$ is called a \emph{free arc} if the set $\alpha^\circ = \alpha \setminus E(\alpha)$ is open in $G$. A \emph{tame partition} for $G$ is a countable family $\P$ of free arcs with pairwise disjoint interiors covering $G$ up to a countable set of points. In~\cite{BBPRV} it was shown that a continuum $G$ is a tame graph if and only if it admits a tame partition, which happens if and only if all but countably many points of $G$ have a neighborhood in $G$ which is a finite graph. It was also shown that every tame graph is locally connected (and thus a Peano continuum).
Let $g:G\to G$ be a continuous map on a tame graph. Suppose that there is a tame partition $\P$ such that
\begin{itemize}
\item For every $i\in\P$ the restriction $g|_i : i \to g(i)$ is a homeomorphism, and
\item For every pair $i,j\in\P$ if $g(i)\cap j^\circ\neq\emptyset$, then $g(i)\supset j$.
\end{itemize}
Then we will call $\P$ a \emph{countable Markov partition} for $g$. As a reminder, $g$ is called \emph{mixing} if for each pair of nonempty open sets $U,V\subset G$ there is $N\geq0$ such that for all $n\geq N$, $g^n(U)\cap V\neq\emptyset$.
Our main idea for studying tame graph maps is to cut up the graph into arcs and glue those arcs back together to get an interval. The resulting interval map will be discontinuous, but for our purposes it does not matter much.
\begin{definition}\label{def:cutandpaste}
Let $g:G\to G$ be a continuous mixing map of a tame graph with countable Markov partition $\P$. Let $\psi:\bigcup_{i\in\P}i^\circ \to [0,1]$ and $f:[0,1]\to[0,1]$ be maps such that
\begin{itemize}
\item $\psi\left(\bigcup_{i\in\P}i^\circ\right)$ has a countable complement in $[0,1]$,
\item For each $i\in\P$, $\psi$ maps $i^\circ$ homeomorphically onto its image,
\item For distinct $i,j\in\P$, $\psi(i^\circ)\cap\psi(j^\circ)=\emptyset$ (thus $\psi$ is injective), and
\item $f(x)=(\psi\circ g\circ\psi^{-1})(x)$ if this composition of maps is defined at $x$, and otherwise $f(x)\not\in\psi\left(\bigcup_{i\in\P}i^\circ\right)$.
\end{itemize}
Then the system $([0,1],f)$ will be called a \emph{cut-and-paste model} for $(G,g)$.
\end{definition}
\begin{lemma}\label{lem:exists-cap}
If $G$ is a tame graph and $g:G\to G$ is a continuous mixing map with a countable Markov partition, then there exists a cut-and-paste model for $(G,g)$.
\end{lemma}
\begin{proof}
Let $\P$ denote the countable Markov partition. It may be finite or countably infinite; we give the proof for the countably infinite case. Choose an enumeration $i_1, i_2, i_3, \ldots$ of the partition arcs. Define $\psi$ on $i_n^\circ$ to be any homeomorphism of $i_n^\circ$ onto $(2^{-n}, 2^{-n+1})$. Finally, define $f$ by
\begin{equation*}
f(y)=\begin{cases}
0, &\text{if }y=2^{-n} \text{ for some } n\geq0,\\
0, &\text{if }g(\psi^{-1}(y)) \text{ is not in the interior of any partition arc},\\
\psi(g(\psi^{-1}(y))), & \text{otherwise}.
\end{cases}
\end{equation*}
Then $([0,1],f)$ is a cut-and-paste model for $(G,g)$.
\end{proof}
\begin{theorem}\label{th:iso-cap}
If $([0,1],f)$ is a cut-and-paste model for $(G,g)$, then they are isomorphic modulo countable invariant sets.
\end{theorem}
\begin{proof}
Let $\P$, $\psi$ be as in Definition~\ref{def:cutandpaste}. Let $U_1$ denote the open set $\bigcup_{i\in\P}i^\circ$ and $C_1$ its complement in $G$. Let $N_1=\bigcup_{n=0}^\infty g^{-n}(C_1)$. Now $C_1$ is countable by the definition of a tame partition and is forward-invariant under $g$ by~\cite[Lemma 2.5(v)]{BBPRV}. Since $g$ is an at most countable-to-one map, $N_1$ is countable and totally invariant. Similarly, let $U_2$ denote the open set $\bigcup_{i\in\P} \psi(i^\circ)=\psi(U_1)$ and let $C_2$ denote its complement in $[0,1]$. Let $N_2=\bigcup_{n=0}^\infty f^{-n}(C_2)$. Applying Definition~\ref{def:cutandpaste}, we see that $f$ is also at most countable-to-one, and that $N_2$ is also countable and totally invariant.
Consider the restricted map $\psi: G\setminus N_1 \to [0,1]\setminus N_2$. We wish to show that $\psi$ is well-defined, bijective, and bimeasurable. Then the identity $\psi\circ g = f\circ\psi$ clearly follows, so that $\psi$ is the desired isomorphism.
\emph{(Well-defined):} Let $x\in G\setminus N_1$. We must show that $\psi(x)\not\in N_2$. We have $g^n(x)\in U_1$ for all $n\geq0$. It follows inductively that $f^n(\psi(x))=\psi(g^n(x))\in U_2$ for all $n\geq0$.
\emph{(Bijective):} Injectivity is free from Definition~\ref{def:cutandpaste}. To prove surjectivity, choose $y\in[0,1]\setminus N_2$. Then $f^n(y)\in U_2$ for each $n\geq0$, so that $\psi^{-1}$ is defined at each point along the forward orbit of $y$. It follows inductively that $f^n(y)=\psi\circ g^n\circ\psi^{-1}(y)$ for all $n\geq0$. Therefore $g^n(\psi^{-1}(y))\in U_1$ for all $n\geq0$, that is, $\psi^{-1}(y)\in G\setminus N_1$.
\emph{(Bimeasurable):} In fact, we show that both $\psi, \psi^{-1}$ are continuous at every point where they are defined. For if $y=\psi(x)$, then there is $i\in\P$ with $x\in i^\circ$. Then $i^\circ$ is an open neighbourhood of $x$ in $G$, $\psi(i^\circ)$ is an open neighbourhood of $y$ in $[0,1]$, and $\psi$ gives a homeomorphism between these two neighbourhoods.
\end{proof}
Given a tame graph $G$ and a continuous, mixing map $g:G\to G$ with a countable Markov partition $\P$, we may define the \emph{transition matrix} $M=M(g,\P)=(m_{ij})_{i,j\in\P}$ by the rule
\begin{equation*}
m_{ij}=\begin{cases}
1, &\text{if }g(i)\supset j,\\
0, &\text{if }g(i)\cap j^\circ=\emptyset.
\end{cases}
\end{equation*}
\begin{theorem}
Let $G$ be a tame graph, $g:G\to G$ a continuous mixing map with countable Markov partition $\P$, and let $M$ be the associated transition matrix. Then $(G,g)$ is isomorphic modulo countable invariant sets to the shift space $(\Sigma_M,\sigma)$.
\end{theorem}
\begin{proof}
Consider the cut-and-paste model $([0,1], f)$ and the map $\psi$ constructed in Lemma \ref{lem:exists-cap}. Then $f$ is a countably Markov and mixing interval map (in the sense of Section~\ref{LIM}) with respect to the partition intervals $\mathcal{I}=\{\psi(i^\circ \cap g^{-1}(j^\circ));~i,j\in\P\}$. The transition matrix for $f$ is the same as the transition matrix for $g$ with respect to the refined tame partition $\P\vee g^{-1}(\P) = \{i\cap g^{-1}(j);~i,j\in\P\}$, which is again a countable Markov partition for $g$~\cite[Lemma 2.5]{BBPRV}. Combining Theorems~\ref{th:cmm} and \ref{th:iso-cap} we see that $(G,g)$ is isomorphic modulo countable invariant sets with $(\Sigma_{M(g,\P\vee g^{-1}(\P))}, \sigma)$. But this system is in turn topologically conjugate to $(\Sigma_M,\sigma)$, being nothing more than its higher block presentation with blocks of length 2 (see \cite[Section 1.4]{K}).
\end{proof}
\begin{example} Figure~\ref{fig:dendrite} illustrates a countably Markov and mixing map $g$ on a dendrite $G$ sometimes called the star or the locally connected fan. $G$ is the union in $\mathbb{R}^2$ of countably many line segments $A_i, i=1,2, \ldots$ called blades. Each blade has one endpoint at the origin; the other endpoint is called the tip of the blade. No blade contains any other and the lengths of the blades converge to zero. Each blade is subdivided into arcs at countably many points converging to the tip of the blade -- this defines the Markov partition $\P$. The map $g$ fixes the origin. If we denote the sequence of points subdividing $A_i$ as $(x^i_n)_{n=0}^\infty$, ordering them along $A_i$ from the origin to the tip, then $g$ maps $x^i_{2n}$ to the origin and $x^i_{2n+1}$ to the tip of blade $A_{\max(1,i-1)+n}$ for all $n$, and is piecewise affine between these partition points. In the left part of the figure, the label next to a pair of partition arcs indicates which blade those arcs will be mapped onto; in the right part of the figure, the labels name the blades. Continuity of $g$ is clear. The topological mixing property is not clear, but can be ensured by an appropriate choice of the lengths of the blades and of the subarcs into which they are partitioned -- we omit the calculations. It is also fairly easy to show that $g$ has no totally invariant periodic orbits, and thus no atomic fair measures.
\begin{figure}[htb!!]
\includegraphics[width=.8\textwidth]{dendrite.jpg}
\begin{picture}(0,0)
\put(-276,23){\tiny $A_1$}
\put(-239,23){\tiny $A_2$}
\put(-217,23){\tiny $\cdots$}
\put(-287,48){\tiny $A_1$}
\put(-278,76){\tiny $A_2$}
\put(-271,90){\rotatebox{73}{\tiny $\cdots$}}
\put(-307,52){\rotatebox{-45}{\tiny $A_2$}}
\put(-321,70){\rotatebox{-45}{\tiny $A_3$}}
\put(-330,82){\rotatebox{-45}{\tiny $\cdot\!\cdot\!\cdot$}}
\put(-315,38){\rotatebox{-5}{\tiny $A_3$}}
\put(-333,40){\rotatebox{-5}{\tiny $A_4$}}
\put(-344,41){\rotatebox{-5}{\tiny $\cdot\!\cdot\!\cdot$}}
\put(-4,29){\small $A_1$}
\put(-84,113){\small $A_2$}
\put(-157,87){\small $A_3$}
\put(-176,36){\small $A_4$}
\put(-156,4){\small $A_5$}
\put(-132,-6){\small $A_6$}
\put(-115,-4){\small $A_7$}
\put(-185,125){\small $g$}
\end{picture}
\caption{A dendrite map on the star dendrite $G$.}
\label{fig:dendrite}
\end{figure}
\begin{figure}[htb!!]
\includegraphics[width=.3\textwidth]{cutandpaste.jpg}
\begin{picture}(0,0)
\put(-35,-10){\small $A_1$}
\put(-85,-10){\small $A_2$}
\put(-116,-10){\small $A_3$}
\put(-130,-10){\small $\cdot\!\cdot\!\cdot$}
\put(-148,100){\small $A_1$}
\put(-148,55){\small $A_2$}
\put(-148,22){\small $A_3$}
\put(-145,4){\rotatebox{90}{\small $\cdot\!\cdot\!\cdot$}}
\end{picture}
\caption{A cut-and-paste model for $(G,g)$.}
\label{fig:cutandpaste}
\end{figure}
A cut-and-paste model $([0,1],f)$ for $(G,g)$ is shown in Figure~\ref{fig:cutandpaste} -- the labels show how the blades of $G$ have been placed within the interval $[0,1]$. Notice that the dendrite map and the interval map have the same symbolic dynamics. Moreover, the interval map $f$ is Lebesgue fair -- it is the same as the Lebesgue fair model from Example~\ref{ex:BruinTodd}, but with each affine piece of the graph replaced by two pieces with twice the slope. We conclude that $(G,g)$ has a unique fair measure. Moreover, we may calculate the fair entropy via the Rohlin formula as $\int_0^1 \log|f'(x)| dx \approx \log(2.85053)+\log(2)$ -- this just adds $\log(2)$ to the fair entropy from Example~\ref{ex:BruinTodd}, which makes sense heuristically, since each point has twice as many preimages.
\end{example}
|
2,869,038,154,838 | arxiv | \section{Motivation and history of the problem}
\label{intro}
The task for a general and useful classification of probability distributions with respect to the their tails seems to be still open.
Embrechts et al. (1997) \cite{EMK} have made a very useful figure about the relations between different subclasses of heavy-tailed distributions in the sense of the infinite moment generating function for all positive arguments. However, this classification puts too many distributions with very different tail behavior in one and the same class. According to this classification for example Pareto($\alpha$), Fr$\acute{e}$chet($\alpha$) and Hill-horror($\alpha$) distributions, with one and the same fixed parameter $\alpha > 0$ belong to one and the same class of distributions with regularly varying tails with parameter $\alpha$. However, the chance to observe "unexpected" value in these three cases is very different, especially for the Hill-Horror distribution. See Figure \ref{fig:peRalphaE}. Comparison of the corresponding hazard rate functions $r_X(x)$, cumulative distribution functions $F_X(x)$ (c.d.fs), and probability density functions $f_X(x)$ (p.d.fs) when $x$ is fixed, and close to the ends of the support of the corresponding distributions is a relatively good approach, but all these characteristics depend on the center of the distribution and the scale parameters. In order to delete these dependencies usually, we normalize the considered random variable(r.v.) with the variance. However, in order to do this, we need existence not only of the first but also of the second moment of the observed distribution. In the most important cases of heavy-tailed distributions, these moments do not exist and this approach is not applicable. Therefore we need some characteristics which describe separately the left and the right tail of the distribution and do not depend on the moments. Index of regular variation, in cases when it is meaningful, is not enough. These lead us to the idea about usage of quantiles, VaR and expectiles, described e.g. in Daouia et al. (2018) \cite{Daouia2018a} or Marinelli et al. (2007) \cite{MarinelliCarlo2007}. They are good characteristics, but also depend on the center of the distribution and the scaling factor. Therefore they are very appropriate for predicting "big looses" within a fixed family of distributional type, but first, you need a right characterization of the type of the "the profit and loss" distribution. A long critical review about the kurtosis can be seen in Balanda and MacGillary (1988) \cite{BalandaMacGillary1988}. Harter (1959) \cite{Harter} uses sample quasi-ranges in estimating population standard deviation. Mosteller (1946) \cite{Monsteller1946} and Sarhan (1954) \cite{SarhanGeneral} propose estimators of the mean and standard deviation which are functions of order statistics. They give us the idea to work with something related to the quantile spread
Due to lack of information outside the range of the data the tails of the distribution should be described via many characteristics.
Through the paper, we show that probabilities for different orders of outside values can be appropriate characteristics for solving this task. Their properties outperform the properties of the kurtosis, tail index and hazard function when speaking about classification with respect to the tail of the observed distribution. The main their advantages are that they do not depend on the center and the scaling factor of the distribution, and do not need the existence of the moments. They are useful for answering the question:
\begin{center}
{\it At what extend we should observe "unexpected" values?}
\end{center}
The idea origins from Tukey's box plots (1977) \cite{Tukey1977} and Balanda and MacGillary's (1990) \cite{BalandaMacGillary} spread-spread plot. However instead of the quantiles here we use probabilities. This allows us to obtain one and the same characteristic of the tail of the observed distribution within all distributional type with respect to increasing affine transformations.
In Section 2 we define and investigate the general properties of probabilities for $p$-outside values. In Section 3 their explicit forms are calculated and plotted for the most popular probability distributions. Section 4 investigates asymptotic properties of empirical left and right $p$-fences, and the estimators of probabilities for left and right $p$-outside values. A result about strong consistency of relative frequency estimator completes that part.
Different estimators of the exponent of regular variation are proposed in Hill (1975) \cite{Hill}, Pickands (1975)\cite{Pickands} and Deckers-Einmahl-de Haan Dekkers (1989) \cite{Dekkers1989}, Einmahl and Guillou (2008) \cite{EinmahlGuillou}, t-Hill Stehlik et al. (2010) \cite{Stehlik2010}, Pancheva and Jordanova (2012) \cite{JordanovaPancheva,JordanovaMilan2012}, Jordanova et al. (2016) \cite{OurExtremes} among others. The mean of order $p$ generalization of the t-Hill and Hill statistics is introduced by Beran et al. (2014) \cite{Beran}, Caeiro et al. (2016) \cite{CaeiroGomes} and Paulauskas and Vaiciulis (2017) \cite{Paulauskas}. Another approach can be seen in Huisman et al. (2001) \cite{Huisman} who recommend to correct small-sample bias of Hill estimators via weighted averages of its values for different thresholds.
In Sections 5 the previous results are applied and a completely new approach for estimating the parameter of the heaviness of the tail of the observed distribution is demonstrated. Four of the examples consider cumulative distribution functions (c.d.fs.) with regularly varying right tail. These are Pareto, Fr$\acute{e}$chet, Log-logistic and Hill-horror cases. It is easy to realize that in order to estimate their indexes of regular variation working with small samples only distribution sensitive estimators can be useful. The main idea of this section is to show that our approach works also in the case when the c.d.f. of the observed r.v. does not have regularly varying right tail. We depict this result via an example about $H_1$ distribution (\ref{H1CDF}) which tail is not regularly varying. The paper finishes with some conclusive remarks. The proofs are sent to the Appendix section.
All plots and computations are made via software R (2018)\cite{R}.
In this work we do not use the second order regular variation introduced in de Haan and Stadtmueller (1996) \cite{deHaanStadtmueller} for distributions with regularly varying tails, because it is applicable only for huge samples. A very comprehensive study of analyzing extreme values under the second order regularly varying condition can be found e.g. in de Haan and Ferreira (2006) \cite{deHaanFerreira}. The corresponding properties of the convolutions and the central limit theorem are obtained by Geluk and de Haan (1997) \cite{GelukdeHaan1997}.
Through the paper we use the following notations: $ {\mathop{=}\limits_{}^{d}}$ is for the equality in distribution, $ {\mathop{\to}\limits_{}^{d}}$ means convergence in distribution, ${\mathop{\to}\limits_{}^{a.s.}}$ denotes almost sure convergence. $\in$ means that the considered r.v. belongs to the corresponding probability type. $\sim$ is asymptotic equivalence. $B(\alpha, \beta) = \int_0^1 x^{\alpha - 1}(1- x)^{\beta-1} dx$ is for the beta function, and $Beta(\alpha, \beta)$ denotes Beta distribution with parameters $\alpha > 0$ and $\beta > 0$.
$\mathbf{X} \in Par(\alpha, \delta)$, means that a r.v. $\mathbf{X}$ has Pareto c.d.f.
\begin{equation}\label{Pareto}
F_{\mathbf{X}}(x) = \left\{
\begin{array}{ccc}
0 & , & x \leq \delta \\
1 - \left(\frac{\delta}{x}\right)^{\alpha} & , & x > \delta
\end{array}
\right..
\end{equation}
More general definitions of Pareto distributions, together with very useful descriptions of the relations between them and the most important other distributions could be seen e.g. in Arnold (2015) \cite{Arnold2015}.
$\mathbf{X} \in Fr(\alpha, c)$, means that a r.v. $\mathbf{X}$ has Fr$\acute{e}$chet c.d.f.
\begin{equation}\label{Frechet}
F_{\mathbf{X}}(x) = \left\{
\begin{array}{ccc}
0 & , & x \leq 0 \\
e^{-(cx)^{-\alpha}} & , & x > 0
\end{array}
\right..
\end{equation}
$\mathbf{X} \in NegWeibull(\alpha, \sigma, \mu)$ is an abbreviation of the fact that the r.v. $\mathbf{X}$ has a Negative Weibull c.d.f.
\begin{equation}\label{Weibull}
F_{\mathbf{X}}(x) =\left\{\begin{array}{ccc}
exp\left\{-\left(-\frac{x-\mu}{\sigma}\right)^{\alpha}\right\} & , & x \leq \mu\\
1 & , & x > \mu
\end{array}
\right..
\end{equation}
We consider only absolutely continuous distributions. For $p \in [0, 1]$ the theoretical quantile function of the c.d.f. $F$ is defined as $$F^\leftarrow(p) = inf\{x \in R: F(x) \geq p\} = sup \{x \in R: F(x) \leq p\}.$$
Let $X_1, X_2, ..., X_n$ be a sample of independent observations on a r.v. $X$ with c.d.f. $F$. Here we denote the corresponding order statistics by $X_{(1, n)} \leq X_{(2, n)} \leq ... \leq X_{(n, n)}$. In Parzen (1979) \cite{Parzen}, Hyndman et al. (1996) \cite{Hyndman}, Langford (2006) \cite{Langford} among others, one can find different definitions of empirical $p$-quantiles, $p \in \left[\frac{1}{n+1}, \frac{n}{n+1}\right]$. We use the following one $F^\leftarrow_n(p):= X_{([(n+1)p],n)}$, where $[a]$ means the integer part of $a$. {\footnote{As it is noticed in Chu (1957) \cite{Chu}, for large samples, these methods are equivalent because we consider only absolutely continuous distributions.}}
\section{Using probabilities for $p$-outside values for characterising the tails of the observed distribution}
\label{sec:1}
The idea about classification of distributions based on quartiles and box plots comes from Tukey (1977) \cite{Tukey1977}, and recently was reminded by Devore (2015) \cite{Devore} and Jordanova and Petkova (2017) \cite{MoniPoli2017}. Here we generalize this concept and introduce one more parameter in the definition of outside values, which allows the researchers to decide at what extend atypical observations would be called "outside value".
Denote by $R_n(F,p) = R_n(X,p)$
$$ = F^\leftarrow_n(1 - p) + \frac{1-p}{p}(F^\leftarrow_n(1 - p) - F^\leftarrow_n(p)) = \frac{1}{p}F^\leftarrow_n(1 - p) - \frac{1-p}{p} F^\leftarrow_n(p)$$
and by $L_n(F,p) = L_n(X,p)$
$$ = F^\leftarrow_n(p) - \frac{1-p}{p}(F^\leftarrow_n(1 - p) - F^\leftarrow_n(p)) = \frac{1}{p}F^\leftarrow_n(p) - \frac{1-p}{p} F^\leftarrow_n(1 - p),$$
correspondingly {\bf empirical p-right-} and {\bf empirical p-left-fence}.
Their sum is equal to $F^\leftarrow_n(1 - p) + F^\leftarrow_n(p)$. The difference $F^\leftarrow_n(1 - p) - F^\leftarrow_n(p)$ between these quantiles is very well known. It is called {\bf empirical quantile spread(quasi range)}, and is considered e.g. in Gumbel(1944)\cite{Gumbel1944}, Monsteller (1946) \cite{Monsteller1946}, and Balanda and MacGillary (1990) \cite{BalandaMacGillary}.
The meaning of these values comes from the expression
$$p[R_n(F,p) - F^\leftarrow_n(p)] = F^\leftarrow_n(1 - p) - F^\leftarrow_n(p) = p[F^\leftarrow_n(1 - p) - L_n(F,p)].$$
It is clear that analogously to Tukey's box-plot (1977) \cite{Tukey1977}, one can use {\bf empirical box plot of order p}.
Its borders are determined via the values
$$L_n(F,p),\,\, F^\leftarrow_n(p),\,\, F^\leftarrow_n(0.5),\,\, F^\leftarrow_n(1 - p),\,\, R_n(F, p).$$
The most frequently $p = 0,25$. This case is partially investigated in the supplementary material of Soza et al. (2019) \cite{JordanovaStehlik2018}.
Sample right or left $p$-outside values are the observations which fall outside the interval $[L_n(F,p), R_n(F,p)]$. Their absolute frequencies, strongly depend on the sample size.
\begin{defi}\label{def1} Assume $p \in (0, 0.5]$. We call on observation $Y$ sample(empirical)
\begin{itemize}
\item {\bf right $p$-outside values} if $Y > R_n(X,p);$
\item {\bf left $p$-outside values} if $Y < L_n(X,p).$
\end{itemize}
\end{defi}
For $p = 0.25$, the definition coincides with the one in Devore (2015) \cite{Devore}. He calls them "extreme right" and "extreme left outliers".
Denote by $n_{R}(p, n)$, and $n_{L}(p, n)$ the numbers of these outside values in a sample of $n$ independent observations. According to the A.Kolmogorov's Zero-one law, never mind how small, but strictly positive is $p$ one can almost sure observe such outside values in a large enough sample of observations on a r.v. with many light tailed distributions, e.g. Gaussian. Therefore the number of outside values is not too informative.
In order to classify distributions with respect to their tail behaviour we propose to compare their theoretical probabilities an observation to be an outside value of the considered type. Denote by
\begin{eqnarray*}
p_{L,p}(X) &=& p_{L,p}(F) = P(X < L(X,p)),\\
p_{R, p}(X) &=& p_{R, p}(F) = P(X > R(X,p))
\end{eqnarray*}
the observed r.v. $X$ to be left or right $p$-outside value.
Here, analogously to $R_n(F,p)$ and $L_n(F,p)$ we have denoted by
\begin{eqnarray*}
R(F,p) &=& R(X,p) = F^\leftarrow(1 - p) + \frac{1-p}{p}[F^\leftarrow(1 - p) - F^\leftarrow(p)]\\
&=& \frac{1}{p}F^\leftarrow(1 - p) - \frac{1-p}{p} F^\leftarrow(p)
\end{eqnarray*}
{\bf theoretical p-right fence} and by
\begin{eqnarray*}
L(F,p) &=& L(X,p) = F^\leftarrow(p) - \frac{1-p}{p}[F^\leftarrow(1 - p) - F^\leftarrow(p)]\\
&=& \frac{1}{p}F^\leftarrow(p) - \frac{1-p}{p} F^\leftarrow(1 - p)
\end{eqnarray*}
{\bf theoretical p-left fence}.
Their properties are analogous to the properties of empirical p-right- and p-left-fences.
Let us note that for any absolutely continuous c.d.f. $F$
$$R(X,0.5) = L(X,0.5) = F^\leftarrow(0.5),$$
is the median and $p_{R, 0.5}(X) = p_{L, 0.5}(X) = 0.5.$
It is not difficult to check that $R(F, p)$ and $L(F, p)$ are monotone. Therefore by monotonicity of probability measures, the characteristics $p_{L,p}(X)$ and $p_{R,p}(X)$ are also monotone.
\begin{thm}\label{thm:monotonicity} For a fixed $p \in (0, 0.5]$ if there exist $f(F^\leftarrow(p)) \in (0, \infty)$, and $f(F^\leftarrow(1 - p)) \in (0, \infty)$, then,
\begin{description}
\item[a)] $L(F, p)$ is increasing in $p$;
\item[b)] $R(F, p)$ is decreasing in $p$;
\item[c)] $p_{L,p}(X)$ and $p_{R,p}(X)$ are non-decreasing in $p$.
\end{description}
\end{thm}
For $p = 0.25$, $p_{L, p}(X)$ and $p_{R, p}(X)$ are correspondingly the probabilities an observation to be left- or right- extreme outlier. Jordanova and Petkova (2018) \cite{MoniPoli2018} and Soza et al.(2019) \cite{JordanovaStehlik2018} denote these probabilities by $p_{eL} = p_{L, 0.25}$ and $p_{eR} = p_{R, 0.25}$. In that case, the authors obtain them for Pareto, Fr$\acute{e}$chet, $H_1$, $H_2$, and Hill-Horror distributions. Further on, we generalize these results.
\begin{thm}\label{thm:thm1} Assume $p \in (0, 0.5]$, and $g(x)$ and $F := F_X$ are strictly monotone, well defined, and continuous function in the considered values. The characteristics $p_{L,p}(X)$, $p_{R,p}(X)$, possess the following properties:
\begin{description}
\item[a)] $p_{L,p}(X) \in [0, p]$ and $p_{R,p}(X) \in [0, p]$.
\item[b)] $p_{L,p}(X) = p_{L,p}(X + c)$, $p_{R,p}(X) = p_{R,p}(X + c)$, $c \in R$.
\item[c)] If $c > 0$, then $p_{L,p}(cX) = p_{L,p}(X)$, $p_{R,p}(cX) = p_{R,p}(X)$.
\item[d)] If $c < 0$, then $p_{L,p}(cX) = p_{R,p}(X)$, $p_{R,p}(cX) = p_{L,p}(X)$.
\item[e)] If $g(x)$ is continuous and strictly increasing
\begin{eqnarray}
\label{eL} p_{L,p}(g(X)) &=& F \left\{g^\leftarrow\left[\frac{1}{p} g[F^\leftarrow(p)] - \frac{1-p}{p}g[F^\leftarrow(1 - p)]\right]\right\}, \\
\label{eR} p_{R,p}(g(X)) &=& 1 - F \left\{g^\leftarrow\left[\frac{1}{p} g[F^\leftarrow(1-p)] - \frac{1-p}{p}g[F^\leftarrow(p)]\right]\right\}
\end{eqnarray}
and
\begin{equation}\label{auxTh1e1}
\frac{1}{p} g[F^\leftarrow(1-p)] - \frac{1-p}{p}g[F^\leftarrow(p)] \geq g[R(X,p)] \iff p_{R,p}(X) \geq p_{R,p}[g(X)],
\end{equation}
\begin{equation}\label{auxTh1e2}
\frac{1}{p} g[F^\leftarrow(p)] - \frac{1-p}{p}g[F^\leftarrow(1 - p)] \geq g[L(X,p)] \iff p_{L,p}(X) \leq p_{L,p}[g(X)].
\end{equation}
\item[f)] If $g(x)$ is continuous and strictly decreasing
\begin{eqnarray}
\label{fL} p_{L, p}(g(X)) &=& 1 - F \left\{g^\leftarrow\left[\frac{1}{p} g[F^\leftarrow(1 - p)] - \frac{1-p}{p}g[F^\leftarrow(p)]\right]\right\},\\
\label{fR} p_{R, p}(g(X)) &=& F \left\{g^\leftarrow\left[ \frac{1}{p} g[F^\leftarrow(p)] - \frac{1-p}{p}g[F^\leftarrow(1-p)]\right]\right\},
\end{eqnarray}
and
\begin{equation}\label{auxTh1e3}
\frac{1}{p} g[F^\leftarrow(p)] - \frac{1-p}{p}g[F^\leftarrow(1-p)] \leq g[L(X,p)] \iff p_{L,p}(X) \leq p_{R,p}[g(X)],
\end{equation}
\begin{equation}\label{auxTh1e4}
\frac{1}{p} g[F^\leftarrow(1 - p)] - \frac{1-p}{p}g[F^\leftarrow(p)] \leq g[R(X,p)] \iff p_{L,p}(g(X)) \leq p_{R,p}(X).
\end{equation}
\item[g)] $p_{R,p}(X_{(n, n)}) = 1 - F_X^n\left[\frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{p})\right]$
$p_{L,p}(X_{(n, n)}) = F_X^n\left[\frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{1 - p})\right]$
\item[h)] $p_{R,p}(X_{(1, n)}) = \left\{1 - F_X\left[\frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{1 - p})\right]\right\}^n$
$p_{L,p}(X_{(1, n)}) = 1 - \left\{1 - F_X \left[\frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{p})\right]\right\}^n$
\item[i)] For all $t > 0$, $$p_{R,p}(X-t|X>t) = p_{R,p}(X|X>t), \quad p_{L,p}(X-t|X>t) = p_{L,p}(X|X>t).$$
\item[k)] Let $l < u$, and $l, u \in R$. Denote correspondingly the right-, left-, and double-truncated r.vs. by $X_{RT} = (X|X<u)$, $X_{LT} = (X|X>l)$, $X_{DT} = (X|l<X<u)$. If $0< F(u)$, $F(l) < 1$ and $F(l) \not= F(u)$, then the relations between their probabilities for $p$-outside values, $F^\leftarrow:=F_X^\leftarrow$, and $F:=F_X$ are the following:
\end{description}
\begin{eqnarray*}
p_{L,p}(X_{RT}) &=& \frac{F\left\{\frac{1}{p}F^\leftarrow[pF(u)]-\frac{1-p}{p}F^\leftarrow[(1-p)F(u)]\right\}}{F(u)} \\
p_{R,p}(X_{RT}) &=& 1-\frac{F\left\{\frac{1}{p}F^\leftarrow[(1-p)F(u)]-\frac{1-p}{p}F^\leftarrow[pF(u)]\right\}}{F(u)}\\
p_{L,p}(X_{LT}) &=& \frac{F\left\{\frac{1}{p}F^\leftarrow[p+(1-p)F(l)]-\frac{1-p}{p}F^\leftarrow[(1-p)+pF(l)]\right\}-F(l)}{1 - F(l)}\\
p_{R,p}(X_{LT}) &=& \frac{1-F\left\{\frac{1}{p}F^\leftarrow[1-p+pF(l)]-\frac{1-p}{p}F^\leftarrow[p+(1-p)F(l)]\right\}}{1 - F(l)}\\
p_{L,p}(X_{DT}) &=& \frac{F\left\{\frac{1}{p}F^\leftarrow[pF(u)+(1-p)F(l)]-\frac{1-p}{p}F^\leftarrow[(1-p)F(u)+pF(l)]\right\}-F(l)}{F(u) - F(l)}\\
p_{R,p}(X_{DT}) &=& \frac{F(u)-F\left\{\frac{1}{p}F^\leftarrow[(1-p)F(u)+pF(l)]-\frac{1-p}{p}F^\leftarrow[pF(u)+(1-p)F(l)]\right\}}{F(u) - F(l)}.
\end{eqnarray*}
\item[l)] If $P(X > 0) = 1$, and
\begin{itemize}
\item $a > 1$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(log_a(X)) &=& F \left\{\sqrt[p]{\frac{F^\leftarrow(p)}{[F^\leftarrow(1 - p)]^{1 - p}}}\right\},\\
p_{R,p}(log_a(X)) &=& 1 - F\left\{\sqrt[p]{\frac{F^\leftarrow(1 - p)}{[F^\leftarrow(p)^{1 - p}]}}\right\}.
\end{eqnarray*}
\begin{itemize}
\item $0 < a < 1$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(log_a(X)) &=& 1 - F \left\{\sqrt[p]{\frac{F^\leftarrow(1 - p)}{[F^\leftarrow(p)]^{1 - p}}}\right\},\\
p_{R,p}(log_a(X)) &=& F\left\{\sqrt[p]{\frac{F^\leftarrow(p)}{[F^\leftarrow(1 - p)]^{1 - p}}}\right\}.
\end{eqnarray*}
\item[m)] For any r.v. $X$, such that $P(X > 0) = 1$, if
\begin{itemize}
\item If $\alpha < 0$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(X^\alpha) &=& 1 - F\left\{\sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(1 - p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(p)]^\alpha }\right\},\\
p_{R,p}(X^\alpha) &=& F\left\{\sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(1 - p)]^\alpha }\right\}.
\end{eqnarray*}
\begin{itemize}
\item If $\alpha > 0$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(X^\alpha)&=& F\left\{\sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(1 - p)]^\alpha }\right\},\\
p_{R,p}(X^\alpha) &=& 1 - F\left\{\sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(1 - p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(p)]^\alpha }\right\}.
\end{eqnarray*}
\item[n)] If
\begin{itemize}
\item $a \in (0, 1)$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(a^X) &=& 1 - F\left\{log_a\left[\frac{1}{p}a^{F^\leftarrow(1 - p)} - \frac{1 - p}{p}a^{F^\leftarrow(p)}\right]\right\},\\
p_{R,p}(a^X) &=& F\left\{log_a\left[\frac{1}{p}a^{F^\leftarrow(p)} - \frac{1 - p}{p}a^{F^\leftarrow(1 - p)}\right]\right\}.
\end{eqnarray*}
\begin{itemize}
\item If $\alpha > 1$, then
\end{itemize}
\begin{eqnarray*}
p_{L,p}(a^X) &=& F\left\{log_a\left[\frac{1}{p}a^{F^\leftarrow(p)} - \frac{1 - p}{p}a^{F^\leftarrow(1 - p)}\right]\right\},\\
p_{R,p}(a^X) &=& 1 - F\left\{log_a\left[\frac{1}{p}a^{F^\leftarrow(1 - p)} - \frac{1 - p}{p}a^{F^\leftarrow(p)}\right]\right\}.
\end{eqnarray*}
\end{thm}
\begin{rem} Note that in Theorem 1, l), the expressions for $p_{R,p}(log_a(X))$ and $p_{L,p}(log_a(X))$ do not depend on the exact value of $a$ but only on the fact if $0 < a < 1$, or $a > 1$. The last means that, according to this classification of absolutely continuous probability distributions with respect to the tails of their c.d.fs., if we decide to change these characteristics and take a logarithm, the exact value of the basis of the logarithm is not important for probabilities for outside values of the transformed distribution. Only the fact that it is bigger or less than $1$ can influence $p_{R,p}(log_a(X))$ and $p_{L,p}(log_a(X))$.
\end{rem}
{\bf Corollary of f):} Let $p \in (0, 0.5]$ be fixed.
\begin{itemize}
\item If $P(X > 0) = 1$, then $p_{L,p}(X) \leq p_{R,p}(\frac{1}{X}).$
\item If $P(X < 0) = 1$, then $p_{R,p}(X) \leq p_{L,p}(\frac{1}{X}).$
\end{itemize}
The next corollary corresponds to the well-known experience that taking a logarithm with basis bigger than one of the data we decrease the chance to observe right $p$-outside values and increase the chance to observe left $p$-outside values. Together with Theorem \ref{thm:thm1aux}, they show once again the appropriateness of these characteristics when speaking about the tail of the observed distribution.
{\bf Corollary of e):}\label{cor:Cor1e} Let $p \in (0, 0.5]$ be fixed. Suppose $P(X > 0) = 1$,
\begin{itemize}
\item if $a > 1$, then
\end{itemize}
\begin{equation}\label{LogarithmsFirstinequalityL}
p_{L,p}(a^X) \leq p_{L,p}(X) \leq p_{L, p}[log_a(X)],
\end{equation}
\begin{equation}\label{LogarithmsFirstinequalityR}
p_{R,p}[log_a(X)] \leq p_{R,p}(X) \leq p_{R, p}(a^X).
\end{equation}
\begin{itemize}
\item If $0 < a < 1$, then
\end{itemize}
\begin{equation}\label{LogarithmsFirstinequalityLalessthan1}
p_{L,p}(a^X) \leq p_{R, p}(X), \quad p_{L, p}\left(\frac{1}{a^X}\right) \leq p_{L,p}(X) \leq p_{R, p}[log_a(X)],
\end{equation}
\begin{equation}\label{Logarithms1}
p_{L,p}[log_a(X)] \leq p_{R, p}(X) \leq p_{R, p}\left(\frac{1}{a^X}\right), \quad p_{L,p}(X) \leq p_{R, p}(a^X).
\end{equation}
According to $p_{R, p}$ characteristics, taking powers bigger than 1 of the data we increase the chance to observe right $p$-outside values, and decrease the chance to observe left $p$-outside values in the observed distribution.
\begin{thm}\label{thm:thm1aux} For $p \in (0, 0.5]$, and $P(X > 0) = 1$,
\begin{description}
\item[a)] $0 < \alpha_1 \leq \alpha_2$, then
\begin{equation}\label{Cor2ofe1}
p_{L,p}(X^{\alpha_2}) \leq p_{L,p}(X^{\alpha_1}), \quad p_{R,p}(X^{\alpha_1}) \leq p_{R,p}(X^{\alpha_2}).
\end{equation}
\item[b)] If $\alpha > 1$, and $\frac{1}{1 - p} \leq \left[\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}\right]^\alpha$, then $p_{L,p}(X^{\alpha}) = 0$ and
\begin{equation}\label{Cor2ofe21}
p_{R,p}(X^{1/\alpha}) \leq p_{R,p}(X) \leq p_{R,p}(X^{\alpha}).
\end{equation}
\item[c)] If $\alpha > 1$, and $\frac{1}{1 - p} \geq \left[\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}\right]^{\alpha}$, then $p_{R,p}(X^{\alpha}) = 0$ and
\begin{equation}\label{Cor2ofe2}
p_{L,p}(X^{\alpha}) \leq p_{L,p}(X) \leq p_{L,p}(X^{1/\alpha}).
\end{equation}
\end{description}
\end{thm}
Application of these probabilities requires knowledge about their values for different distributions. Therefore we have calculated some of their explicit forms in the next section.
\section{The most important particular cases}
\label{ParticularCases}
In order to choose the most appropriate class for modeling the tails of the c.d.f. of the observed r.v. we can first calculate the probabilities for left and right $p$-outside values, for as more as possible distributional types, and then to compare these probabilities with corresponding estimators. This approach is analogous to the comparison of the means in cases when we are interested in the center of the distribution.
Let us now present the exact values of these characteristics in some of the most popular cases of probability distributions used in practice for modeling heavy tails. Till the end of this section, we assume that $p \in (0, 0.5]$.
The dependencies of $p_{R, 0.25}$ on the parameter $\alpha$, which characterises the tail of the corresponding distribution in cases when $F$ is $Gamma(\alpha, \beta)$, or Fr$\acute{e}$chet, Pareto, Stable, Weibull positive, $H_1$, $H_4$, log-Pareto, Hill horror or Burr distributed are depicted on Figure \ref{fig:peRalphaE} and Figure \ref{fig:peRalphaExtremes}, and could be seen also in Jordanova and Petkova (2018) \cite{MoniPoli2018}, and in the supplementary material of Soza et al. (2019) \cite{JordanovaStehlik2018}.
\begin{itemize}
\item Exponential distribution. Let $\lambda > 0$, and $X$ be Exponential with mean $\frac{1}{\lambda}$.
\end{itemize}
It is well known that $\lambda$ is a scale parameter of the exponential distribution, therefore due to Th. 1, c) without lost of generality (w.l.g.) we can assume that $\lambda = 1$ and this will not change the values of $p_{L, p}(X)$ and $p_{R, p}(X)$. In this case,
$$ p_{L,p}(X) = P\left[X < log\frac{p^{\frac{1-p}{p}}}{(1-p)^{\frac{1}{p}}}\right] = \left\{
\begin{array}{ccc}
0 & , & p \in (0, p_0), \\
p\frac{(1-p)^{\frac{1}{p}}}{p^{\frac{1}{p}}} & , & p \in [p_0, \frac{1}{2})
\end{array}
\right.,$$
where $p_0$ is the solution of the equation $(1-p_0)log(p_0)=log(1-p_0)$ and $p_0 \approx 0.4096$.
$$p_{R, p}(X) = P\left\{X > log\left[\left(\frac{1 - p}{p}\right)^{\frac{1}{p}}\frac{1}{1-p}\right]\right\} = (1-p)\left(\frac{p}{1-p}\right)^{\frac{1}{p}}.$$
In particular, for $p_{L, 0.25}(X) = 0$, the empirical right fence is asymptotically unbiased and efficient estimator for the theoretical right fence
$$\lim_{i \to \infty} ER_{4i-1}(X,0.25) = R(X,0.25) = 2log(2) + 3log(3) = log(108)$$
$$\lim_{i \to \infty} DR_{4i-1}(X,0.25) = \lim_{i \to \infty} [16\psi'(i) - \psi'(4i) - 15\psi'(3i)] = 0.$$
where $\psi'(i) = \frac{\partial^2 log \Gamma(z)}{\partial z^2}$ is the Polygamma function (for the last limit see Guo and Feng (2013) \cite{GuoFeng2013}), and $p_{R, 0.25}(X) = \frac{1}{108} = 0.00925(925).$
See Jordanova and Petkova (2017-2018) \cite{MoniPoli2018,MoniPoli2017}.
Further on in this section (due to properties b) and c) Theorem \ref{thm:thm1}, w.l.g. we assume that $\mu = 0$ and $\sigma = 1$.
\begin{itemize}
\item Generalized Pareto distribution (GPD). Consider $\mu \in R$ and $\sigma > 0$.
\end{itemize}
\begin{eqnarray}\label{GPD}
F_X(x) = \left\{\begin{array}{ccc}
1 - (1 + \xi \frac{x - \mu}{\sigma})^{-1/\xi} &, & x > \mu, \xi > 0 \\
1 - (1 + \xi \frac{x - \mu}{\sigma})^{-1/\xi} &, & \mu \leq x \leq \mu - \frac{\sigma}{\xi}, \xi < 0 \\
1- e^{-\frac{x - \mu}{\sigma}} & , & x > \mu, \xi \to 0.
\end{array}
\right.
\end{eqnarray}
We have already considered the case $\xi \to 0$. In that case, it is well known that, the GPD coincides with Exponential distribution. So, here we assume that $\xi \not = 0$. Then the quantile function is
$F_X^\leftarrow(p) = \frac{1}{\xi}[(1-p)^{-\xi}-1].$ We replace it in the formula for $L(X; p)$, then in the definition for $p_{L, p}(X)$, and obtain
$$ p_{L,p}(X) = P\left\{X < \frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi} -1 + p^{-\xi} - p^{-\xi-1}\right]\right\}.$$
In order to replace (\ref{GPD}) in the last probability we need to consider separately the following two cases:
\begin{itemize}
\item[$\cdot$] Case $\xi > 0$. In this case
$\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi} -1 + p^{-\xi} - p^{-\xi-1}\right] < \frac{1}{\xi}$ $\left[\frac{1}{p}-1 +\right.$ $\left. p^{-\xi} - p^{-\xi-1} \right]$. The last expression is equal to $ \frac{1}{\xi} \left(1- \frac{1}{p}\right)\left(\frac{1}{p^\xi}-1\right) < 0 $, therefore $p_{L,p}(X) = 0$.
\item[$\cdot$] Case $\xi < 0$. In this case $\frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi} -1 + p^{-\xi} - p^{-\xi-1}\right] > 0$, therefore
\end{itemize}
\begin{eqnarray*}
p_{L,p}(X) &=& P\left\{X < \frac{1}{\xi}\left[\frac{1}{p}(1-p)^{-\xi} -1 + p^{-\xi} - p^{-\xi-1}\right]\right\}\\
&=& 1 - (\frac{1}{p}(1-p)^{-\xi} + p^{-\xi} - p^{-\xi-1})^{-1/\xi}
\end{eqnarray*}
Analogously we replace the quantile function in the definition for $R(X; p)$, then in $p_{R, p}(X)$, and obtain
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > \frac{1}{\xi}[p^{-\xi}-1] + \frac{1-p}{p}\left[\frac{1}{\xi}(p^{-\xi}-1) - \frac{1}{\xi}[(1-p)^{-\xi}-1]\right]\right\}\\
&=& P\left\{X > \frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right]\right\}
\end{eqnarray*}
In order to calculate this expression we need to determine the sign of $\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right]$. Therefore again we consider two cases.
\begin{itemize}
\item[$\cdot$] Case $\xi > 0$. Because of $p \in (0, 0.5]$, we have $p^{-\xi} > (1-p)^{-\xi}$, $p(1-p)^{-\xi} > p$ and
\end{itemize}
$$(1-p)^{-\xi} + p < p^{-\xi} + p(1-p)^{-\xi}$$
$$1-(1-p)^{-\xi}-p[1-(1-p)^{-\xi}] > 1 - p^{-\xi}$$
$$(1-p)[1-(1-p)^{-\xi}] > 1 - p^{-\xi}$$
$$\frac{1-p}{p}[1-(1-p)^{-\xi}] > \frac{1}{p}\left(1 - p^{-\xi}\right)$$
Therefore
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > \frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right]\right\}\\
&=& \left\{1 + \frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right\}^{-1/\xi}\\
&=& \left\{\frac{p^{-\xi}}{p}+\left(1 - \frac{1}{p}\right)(1 - p)^{-\xi}\right\}^{-1/\xi}
\end{eqnarray*}
In case $p = 0.25$, and $\alpha = \frac{1}{\xi}$ this expression coincide with the one in Jordanova and Petkova (2018)\cite{MoniPoli2018}
\begin{equation}\label{PR025Pareto}
p_{R, 0.25}(X) = \frac{3}{4(4.3^{\frac{1}{\alpha}}-3)^{\alpha}}.
\end{equation}
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.45\linewidth}
\includegraphics[scale=.43]{peRalphaE}\vspace{-0.3cm}
\caption{The dependence of $p_{R, 0.25}(X)$ on $\alpha$}
\label{fig:peRalphaE}
\end{minipage} $ $
\begin{minipage}[t]{0.45\linewidth}
\includegraphics[scale=.43]{peRalphaExtremes}\vspace{-0.3cm}
\caption{The dependence of $p_{R, 0.25}(X)$ on $\alpha$}
\label{fig:peRalphaExtremes}
\end{minipage}
\end{center}
\end{figure}
\begin{itemize}
\item[$\cdot$] Case $\xi < 0$. Because of $p \in (0, 0.5)$ we have that $\frac{1-p}{p} < \frac{1}{p}$, $1-(1-p)^{-\xi} < 1 - p^{-\xi}$ and $\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right] > 0.$
\end{itemize}
In case
$$\frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right] > -\frac{1}{\xi}$$
$$(1-p)[1-(1-p)^{-\xi}] - 1 + p^{-\xi} < -p$$
$$p^{-\xi} < (1 - p)^{1-\xi}$$
$$\xi < \frac{log(1 - p)}{log\left(\frac{1 - p}{p}\right)}$$
we have that $p_{R, p}(X) =0$.
If $0 > \xi > \frac{log(1 - p)}{log\left(\frac{1 - p}{p}\right)}$,
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > \frac{1}{\xi}\left[\frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}\left(1 - p^{-\xi}\right)\right]\right\}\\
&=& \left\{1 + \frac{1-p}{p}[1-(1-p)^{-\xi}]-\frac{1}{p}(1 - p^{-\xi})\right\}^{-\frac{1}{\xi}} \\
&=& \left\{(1-p)^{-\xi}\left(1 - \frac{1}{p}\right) + \frac{1}{p} p^{-\xi}\right\}^{-\frac{1}{\xi}}.
\end{eqnarray*}
When speaking about heavy tails we can not forget about Extreme value distributions with respect to linear transformations. Therefore in the next three points, we will consider them. At the beginning of the last century Fisher and Tippet (1928)\cite{FisherTippet}, Gnedenko (1943) \cite{Gnedenko1943}, and Gumbel (1958)\cite{Gumbel1958} have shown that they appear as limiting distributions of maxima of i.i.d. r.vs. after appropriate affine transformations.
\begin{itemize}
\item Fr$\acute{e}$chet distribution. Let $\alpha > 0$. W.l.g. we assume that in (\ref{Frechet}) $c = 1$.
\end{itemize}
Therefore for $p \in (0, 1)$, $F^\leftarrow(p) = (-log(p))^{-\frac{1}{\alpha}}$.
$$ p_{L,p}(X) = P\left\{X < (-log(p))^{-\frac{1}{\alpha}} - \frac{1-p}{p}[(-log(1 - p))^{-\frac{1}{\alpha}} - (-log(p))^{-\frac{1}{\alpha}}]\right\}$$
and because of $(-log(p))^{-\frac{1}{\alpha}} > \frac{1-p}{p}[(-log(1 - p))^{-\frac{1}{\alpha}} - (-log(p))^{-\frac{1}{\alpha}}]$ $\Longleftrightarrow $
$$-log(p) < (1-p)^{-\alpha}(-log(1-p)) \Longleftrightarrow \alpha_0: = -\frac{log\left(\frac{log(p)}{log(1-p)}\right)}{log(1-p)} < \alpha$$
$$p_{L, p}(X) = exp\left\{-\left[\frac{1}{p}(-log\,p)^{-1/\alpha}+ \left(1-\frac{1}{p}\right)[-log\,(1-p)]^{-1/\alpha}\right]^{-\alpha}\right\},$$ when $\alpha > \alpha_0$, and $p_{L, p}(X) =0$, for $\alpha \in (0, \alpha_0]$.
Analogously
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > (-log(1 - p))^{-\frac{1}{\alpha}} + \frac{1-p}{p}[(-log(1 - p))^{-\frac{1}{\alpha}} - (-log(p))^{-\frac{1}{\alpha}}]\right\}\\
&=& P\left[X > \frac{1}{p}(-log(1 - p))^{-\frac{1}{\alpha}} - \frac{1-p}{p} (-log(p))^{-\frac{1}{\alpha}}\right].
\end{eqnarray*}
Now we need to consider the expression after the inequality. As far as for all $p \in (0, 0.5)$ the following four expressions are equivalent
$$(-log(1 - p))^{-\frac{1}{\alpha}} > (1-p)(-log(p))^{-\frac{1}{\alpha}}$$
$$\frac{-log(1 - p)}{-log(p)} < (1-p)^{-\alpha}$$
$$log\left[\frac{-log(1 - p)}{-log(p)}\right] < -\alpha \,\,log(1-p)$$
$$-\frac{log\left[\frac{-log(1 - p)}{-log(p)}\right]}{log(1-p)} < 0 < \alpha $$
we have
$$p_{R, p}(X) = exp\left\{-\left(\frac{1}{p}(-log(1 - p))^{-\frac{1}{\alpha}} - \frac{1-p}{p} (-log(p))^{-\frac{1}{\alpha}}\right)^{-\alpha}\right\}.$$
Figure \ref{fig:peRalphaE} represents the dependence of $p_{R, p}(X)$ on $\alpha$ in case $p = 0.25$. The explicit formula for this case can be seen also in Jordanova and Petkova (2018) \cite{MoniPoli2018}.
\begin{itemize}
\item Weibull negative distribution. Consider $\alpha > 0$. W.l.g. $X \in NegWeibull(\alpha,$ $1,$ $0)$. See (\ref{Weibull}). Therefore for $p \in (0, 1)$, $F^\leftarrow(p) = -(-log(p))^{\frac{1}{\alpha}}.$
\end{itemize}
Note that its positive version $-X$ coincides in distribution with a standard Exponentially distributed r.v. raised to the power $\frac{1}{\alpha}$. Therefore, for $\alpha > 1$, according to our classification, this distribution has heavier right tail than the exponential one and for $\alpha < 1$ vice versa. This can be seen also on Figure \ref{fig:peRalphaE}, where standard exponential distribution is depicted as $\Gamma(1,1)$. More precisely
\begin{eqnarray*}
p_{L,p}(X) &=& P\left\{X < -[-log(p)]^{\frac{1}{\alpha}} - \frac{1-p}{p}\left[[-log(p)]^{\frac{1}{\alpha}} - [-log(1 - p)]^{\frac{1}{\alpha}}\right]\right\}\\
&=& P\left\{X < \frac{1-p}{p}[-log(1 - p)]^{\frac{1}{\alpha}} - \frac{1}{p}[-log(p)]^{\frac{1}{\alpha}} \right\}
\end{eqnarray*}
and because of for all $p \in (0, 0.5]$ we have $(1-p)[-log(1 - p)]^{\frac{1}{\alpha}} < [-log(p)]^{\frac{1}{\alpha}}$, therefore
\begin{equation}\label{WNalpha}
p_{L, p}(X) = exp\left\{-\left[\frac{1}{p}[-log(p)]^{\frac{1}{\alpha}} - \frac{1-p}{p}[-log(1 - p)]^{\frac{1}{\alpha}}\right]^{ \alpha}\right\}
\end{equation}
Figure \ref{fig:peRalphaE}, depicts the dependence of $p_{L, 0.25}(X) = p_{R, 0.25}(-X)$, (i.e. $-X$ is Weibull positive) on $\alpha$.
Analogously
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > -(-log(1 - p))^{\frac{1}{\alpha}} + \frac{1-p}{p}[(-log(p))^{\frac{1}{\alpha}} -(-log(1 - p))^{\frac{1}{\alpha}} ]\right\}\\
&=& P\left[X > \frac{1-p}{p}(-log(p))^{\frac{1}{\alpha}} - \frac{1}{p} (-log(1 - p))^{\frac{1}{\alpha}}\right]
\end{eqnarray*}
As far as for all $p \in (0, 0.5]$
$$(-log(1 - p))^{\frac{1}{\alpha}} < (1-p)(-log(p))^{\frac{1}{\alpha}}$$
$$\frac{-log(1 - p)}{-log(p)} > (1-p)^{-\alpha}$$
$$log\left[\frac{-log(1 - p)}{-log(p)}\right] > \alpha \,\,log(1-p)$$
$$\alpha_1: = \frac{log\left[\frac{-log(1 - p)}{-log(p)}\right]}{log(1-p)} > \alpha $$
we have
$$p_{R, p}(X) = \left\{\begin{array}{ccc}
0 & , & \alpha \in (0, \alpha_1] \\
1 - exp\left\{-\left(\frac{1}{p}(-log(1 - p))^{\frac{1}{\alpha}} - \frac{1-p}{p} (-log(p))^{\frac{1}{\alpha}}\right)^\alpha\right\}& , & \alpha > \alpha_1 \end{array}
\right..$$
\begin{itemize}
\item Gumbell distribution. Let $\alpha > 0$, $\mu \in R$ and $\sigma > 0$
\end{itemize}
$$F_X(x) = exp\left\{-exp\left[-\frac{x-\mu}{\sigma}\right]\right\},\quad x \in R.$$
W.l.g. $\mu = 0$ and $\sigma = 1$. $F^\leftarrow(p) = -log(-log(p))$,
\begin{eqnarray*}
p_{L,p}(X) &=& P\left\{X < -log(-log(p)) - \frac{1-p}{p}[log(-log(p)) - log(-log(1 - p))]\right\}\\
&=& P\left\{X < \frac{1-p}{p}log(-log(1 - p)) - \frac{1}{p}log(-log(p))\right\} = p^{\left[\frac{log(p)}{log\,(1 - p)}\right]^\frac{1}{p}}
\end{eqnarray*}
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > -log(-log(1 - p)) + \frac{1-p}{p}[log(-log(p))-log(-log(1 - p))]\right\}\\
&=& P\left\{X > \frac{1-p}{p}log(-log(p)) - \frac{1}{p}log(-log(1 - p))\right\} = 1 - p^{\left[\frac{log(1-p)}{log\,(p)}\right]^\frac{1}{p}}
\end{eqnarray*}
Jordanova and Petkova (2018) \cite{MoniPoli2018} have calculated that $p_{L, 0.25}(X) \approx 4.264\times10^{-68}$ and $p_{R, 0.25}(X) \approx 0.002568$.
\begin{itemize}
\item Logistic distribution. Assume $\mu \in R$, $\sigma > 0$ and
\end{itemize}
$$F_X(x) = \frac{1}{1+exp\left(-\frac{x-\mu}{\sigma}\right)},\quad x \in R.$$
W.l.g. $\mu = 0$ and $\sigma = 1$. $F^\leftarrow(p) = log\left(\frac{p}{1 - p}\right)$,
$$p_{R, p}(X) = p_{L,p}(X) = P\left(X < \frac{2-p}{p}log\frac{p}{1 - p}\right) = \left[1 + \left(\frac{1-p}{p}\right)^{\frac{2-p}{p}}\right]^{-1}$$
$\leq \left[1 + 3^5\right]^{-1} = p_{R, 0.25}(X)\approx 0.000457, p \in (0, 0.25).$
\begin{itemize}
\item Log-logistic distribution. Assume $\mu \in R$, $\sigma > 0$ and
\end{itemize}
\begin{equation}\label{LolLogidticCDF}
F_X(x) = \frac{1}{1+\left(\frac{x-\mu}{\sigma}\right)^{-\alpha}},\quad x > \mu.
\end{equation}
W.l.g. $\mu = 0$ and $\sigma = 1$. As far as $F^\leftarrow(p) = \sqrt[\alpha]{\frac{p}{1 - p}}$, and for $p \in (0, 0.5]$
$$R(X,p) = \frac{1}{p}\sqrt[\alpha]{\frac{1-p}{p}}-\frac{1 - p}{p}\sqrt[\alpha]{\frac{p}{1 -p}} > 0$$
therefore by definition
\begin{equation}\label{pRpLogLogistic}
p_{R, p}(X) = \frac{1}{1+\left(\frac{1}{p}\sqrt[\alpha]{\frac{1-p}{p}}-\frac{1 - p}{p}\sqrt[\alpha]{\frac{p}{1 -p}}\right)^{\alpha}}.
\end{equation}
The plot of this function of $\alpha > 0$ for $p = 0.25$ can be seen on Figure \ref{fig:peRalphaLogLogistic}. We observe that for a fixed $\alpha > 0$ its right tail is very similar to the corresponding tails of Pareto and Fr$\acute{e}$chet distribution and heavier than the Stable one.
Analogously
$$L(X,p) = \frac{1}{p}\sqrt[\alpha]{\frac{p}{1-p}}-\frac{1 - p}{p}\sqrt[\alpha]{\frac{1-p}{p}},$$
therefore for $0 < \alpha < 2 [log_{1-p}(p) - 1]$, and $p \in (0, 0.5]$
$$p_{L, p}(X) = 1 - \frac{1}{1+\left(\frac{1}{p}\sqrt[\alpha]{\frac{p}{1 - p}}-\frac{1 - p}{p}\sqrt[\alpha]{\frac{1 - p}{p}}\right)^{\alpha}}.$$
and $p_{L,p}(X) = 0$ otherwise.
In the next two cases we assume that $\tau > 0$ and $\alpha > 0$. W.l.g. $\mu = 0$, because it is a location parameter, and $\sigma = 1$ and $\delta = 1$ because they are scale parameters. See Burr (1942) \cite{Burr} or Einmahl et al. (2008) \cite{EinmahlGuillou}.
\begin{itemize}
\item Burr distribution. Let
\end{itemize}
\begin{equation}\label{f}
F_X(x) =\left\{\begin{array}{ccc}
0 & , & x < \mu \\
1 - \left[\frac{\delta}{\delta + \left(\frac{x-\mu}{\sigma}\right)^{\tau}}\right]^\alpha & , & x \geq \mu
\end{array}
\right..
\end{equation}
For $p \in (0, 1)$, the quantile function is
$$F^\leftarrow(p) = \sqrt[\tau]{(1 - p)^{-1/\alpha}-1} $$
(see also Nair et al. (2013) \cite{Nair2013}). Therefore in our context from (\ref{f}) we have that if
$$\frac{(1-p)^{-\frac{1}{\alpha}}-1}{p^{-\frac{1}{\alpha}}-1} > (1-p)^\tau,$$
then
\begin{eqnarray*}
p_{L,p}(X) &=& P\left\{X < \frac{1}{p} \sqrt[\tau]{(1 - p)^{-1/\alpha}-1} - \frac{1-p}{p}\sqrt[\tau]{p^{-1/\alpha}-1}\right\}\\
&=& 1 - \left\{\frac{p^\tau}{p^\tau + \left[\sqrt[\tau]{(1 - p)^{-1/\alpha}-1} - (1-p)\sqrt[\tau]{p^{-1/\alpha}-1}\right]^{\tau}}\right\}^\alpha
\end{eqnarray*}
and $p_{L,p}(X) = 0$ otherwise.
For $p \in (0, 0.25)$ and $\tau > 0$ the inequality $$ \frac{p^{-\frac{1}{\alpha}}-1}{(1-p)^{-\frac{1}{\alpha}}-1} \geq 1 \geq (1-p)^\tau,$$ the definition of $p_{R, p}(X)$, and (\ref{f}) entail
\begin{eqnarray*}
p_{R, p}(X) &=& P\left[X > \frac{1}{p} \sqrt[\tau]{p^{-1/\alpha}-1} - \frac{1-p}{p}\sqrt[\tau]{(1 - p)^{-1/\alpha}-1}\right]\\
&=& \left[\frac{p^\tau}{p^\tau + \left(\sqrt[\tau]{p^{-1/\alpha}-1} - (1-p)\sqrt[\tau]{(1 - p)^{-1/\alpha}-1}\right)^{\tau}}\right]^\alpha.
\end{eqnarray*}
The dependence of $p_{R,0.25}(X)$ on $\alpha$ and for different values of $\tau$ is depicted on Figure \ref{fig:peRalphaExtremes}. We see that when $\tau$ increases, the chance to observe $p$-outside values in the considered distribution decreases. The last means that for Burr distribution not only $\alpha$ but also $\tau$ influences the tail-behaviour.
\begin{itemize}
\item Reverse Burr distribution.
\end{itemize}
\begin{equation}\label{f1}
F_X(x) =\left\{\begin{array}{ccc}
1 - \left[\frac{\delta}{\delta + \left(\frac{\mu-x}{\sigma}\right)^{-\tau}}\right]^\alpha & , & x \leq \mu\\
1 & , & x > \mu
\end{array}
\right..
\end{equation}
For $p \in (0, 1)$, the corresponding quantile function is
$$F^\leftarrow(p) = -[(1 - p)^{-1/\alpha}-1]^{-1/\tau}.$$
We are interested in the case when $p \in (0, 0.5]$. It guarantees that
$$(1-p)[p^{-1/\alpha}-1]^{-1/\tau} < [(1 - p)^{-1/\alpha}-1]^{-1/\tau}.$$
Therefore from (\ref{f1}) we have
\begin{eqnarray*}
p_{L,p}(X) &=& P\left\{X < \frac{1-p}{p}[p^{-1/\alpha}-1]^{-1/\tau} - \frac{1}{p} [(1 - p)^{-1/\alpha}-1]^{-1/\tau}\right\}\\
&=& 1 - \left\{\frac{1}{1 + p^\tau\left[[(1 - p)^{-1/\alpha}-1]^{-1/\tau}-(1-p)(p^{-1/\alpha}-1)^{-1/\tau}\right]^{-\tau}}\right\}^\alpha.
\end{eqnarray*}
The dependence of $p_{R,0.25}(-X) = p_{L,0.25}(X)$ on $\alpha$ and for different values of $\tau$ is depicted on Figure \ref{fig:peRalphaExtremes3}.
We observe that for $\alpha > 0.5$ the tail behavior of the Reverse-Burr distribution is much more sensitive on $\tau$ than on $\alpha$.
If $(1-p)[(1 - p)^{-1/\alpha}-1]^{-1/\tau} < [p^{-1/\alpha}-1]^{-1/\tau},$
\begin{eqnarray*}
p_{R, p}(X) &=& P\left[X > \frac{1-p}{p}[(1 - p)^{-1/\alpha}-1]^{-1/\tau} - \frac{1}{p} [p^{-1/\alpha}-1]^{-1/\tau} \right]\\
&=& \left[\frac{1}{1 + p^\tau\left((1-p)[(1 - p)^{-1/\alpha}-1]^{-1/\tau} - [p^{-1/\alpha}-1]^{-1/\tau}\right)^{-\tau}}\right]^\alpha.
\end{eqnarray*}
and $p_{R, p}(X) = 0$ otherwise.
\begin{figure}
\begin{center}
\begin{minipage}[t]{0.45\linewidth}
\includegraphics[scale=.43]{peRalphaExtremes3}\vspace{-0.3cm}
\caption{The dependence of $p_{R, 0.25}(-X)$ on parameter $\alpha$ in Reverse-Burr case for $ $ $\tau = 0.5, 1, 1.5, 2$}
\label{fig:peRalphaExtremes3}
\end{minipage} $ $
\begin{minipage}[t]{0.45\linewidth}
\includegraphics[scale=.43]{peRalphaLogLogistic}\vspace{-0.3cm}
\caption{The dependence of $p_{R, 0.25}(X)$ on $\alpha$}
\label{fig:peRalphaLogLogistic}
\end{minipage}
\end{center}
\end{figure}
\begin{itemize}
\item Gompertz distribution. Assume $\mu \in R$, $\sigma > 0$, $\alpha > 0$, and
\end{itemize}
\begin{equation}\label{f11}
F_X(x) =\left\{\begin{array}{ccc}
0 & , & x \leq \mu\\
1 - exp\left\{-\alpha\left[exp\left(\frac{x-\mu}{\sigma}\right)-1\right]\right\} & , & x > \mu
\end{array}
\right..
\end{equation}
For $\mu = 0$, $\sigma = 1$ and $p \in (0, 1)$,
$F^\leftarrow(p) = log\left[1 - \frac{log(1 - p)}{\alpha}\right].$
For $1-\frac{1}{\alpha}log(1 - p) \geq \left[1-\frac{1}{\alpha}log(p)\right]^{1-p}$ we have that $p_{L,p}(X)$ is equal to
\begin{eqnarray*}
& & P\left\{X < \frac{1}{p}log\left[1 - \frac{log(1 - p)}{\alpha}\right] - \frac{1 - p}{p}log\left[1 - \frac{log(p)}{\alpha}\right]\right\}\\
&=& 1 - exp\left\{-\alpha\left[exp\left(\frac{1}{p}log\left[1 - \frac{log(1 - p)}{\alpha}\right] - \frac{1 - p}{p}log\left[1 - \frac{log(p)}{\alpha}\right]\right)-1\right]\right\}\\
&=& 1 - exp\left\{-\alpha\left[\sqrt[p]{\frac{1-\alpha^{-1}log(1 - p)}{[1-\alpha^{-1}log(p)]^{1-p}}}-1\right]\right\}
\end{eqnarray*}
and $p_{L, p}(X) = 0$ otherwise. In particular $p_{L, 0.25}(X) = 0$.
As far as for all $p \in (0, 0.5]$
$\frac{log[1-\frac{1}{\alpha}log(p)]}{log[1-\frac{1}{\alpha}log(1 - p)]} > 1 - p,$ therefore from (\ref{f11}) and the definition of $p_{R, p}(X)$ we have that it is equal to
\begin{eqnarray*}
& & P\left\{X > \frac{1}{p}log\left[1 - \frac{log(p)}{\alpha}\right]-\frac{1-p}{p}log\left[1 - \frac{log(1 - p)}{\alpha}\right]\right\}\\
&=& exp\left\{-\alpha\left[exp\left[\frac{1}{p}log\left[1 - \frac{log(p)}{\alpha}\right]-\frac{1-p}{p}log\left[1 - \frac{log(1 - p)}{\alpha}\right]\right]-1\right]\right\}\\
&=& exp\left\{-\alpha\left[\sqrt[p]{\frac{1-\alpha^{-1}log(p)}{[1-\alpha^{-1}log(1 - p)]^{1-p}}}-1\right]\right\} \approx 0.
\end{eqnarray*}
Note that or all fixed $\alpha > 0$, and within the considered distributions with $p_{R, p}(X) > 0$ in this study, this distribution has smallest value of $p_{R, p}(X)$.
The next two distributions that we consider belong to the class of so-called p-max-stable laws. They can be generalized to a strictly increasing affine transformation and $p_{L,p}$, and $p_{R,p}$ characteristics will not change.
Using power normalizations Pancheva (1985) \cite{Pancheva} obtained them as limiting laws of power transformed maximums. Falk et al. (2004) \cite{Falk2004} describe domains of attraction of these laws under power normalization. We have already seen in Corollary 1 of e), Theorem 1 that these transformations increase the values of $p_{R, p}$. Ravi and Saeb (2012) \cite{Ravi} have obtained their entropies.
\begin{itemize}
\item $H_{1}$ type. (See Pancheva (1985) \cite{Pancheva}). Let $\alpha > 0$, and $X \in H_1(\alpha)$, i.e.
\end{itemize}
\begin{equation}\label{H1CDF}
F_X(x) =\left\{\begin{array}{ccc}
0 & , & x < 1 \\
exp\left\{-(log\,\,x)^{-\alpha}\right\} & , & x \geq 1
\end{array}
\right..
\end{equation}
The quantile function of this distribution is
$F_X^\leftarrow(p) = exp\left\{(-log\,\, p)^{-1/\alpha}\right\}$. \footnote{It can be seen e.g. in the supplementary material of Soza et al. (2019) \cite{JordanovaStehlik2018} who consider the case $p = 0.25$.}
Therefore
$$p_{L,p}(X) = P\left\{X < \frac{1}{p}exp\left\{[-ln\,\,(p)]^{-1/\alpha}\right\} - \frac{1-p}{p}exp\left\{[-ln\,\,(1- p)]^{-1/\alpha}\right\}\right\}.$$
and for $exp\left\{[-ln\,\,(p)]^{-1/\alpha}\right\} - (1-p)exp\left\{[-ln\,\,(1- p)]^{-1/\alpha}\right\} > p$, $p_{L,p}(X)$ is equal to
$$ exp\left\{-\left\{log\left[\frac{1}{p}exp\left\{[-ln\,\,(p)]^{-1/\alpha}\right\} - \frac{1-p}{p}exp\left\{[-ln\,\,(1- p)]^{-1/\alpha}\right\}\right]\right\}^{-\alpha}\right\},$$
otherwise $p_{L,p}(X) = 0$. The last mean that for large $\alpha > 0$ it is possible to observe also left outside values.
Now let us consider the right tail. As far as for all $\alpha > 0$
$$\frac{1}{p}exp\left\{[-log\,\, (1-p)]^{-1/\alpha}\right\} - \frac{1-p}{p}exp\left\{[-log(p)]^{-1/\alpha}\right\} > 1$$
$$exp\left\{[-log\,\, (1-p)]^{-1/\alpha}\right\} - (1 - p)exp\left\{[-log(p)]^{-1/\alpha}\right\} > p$$
$$exp\left\{[-log\,\, (1-p)]^{-1/\alpha}\right\} - exp\left\{[-log(p)]^{-1/\alpha}\right\} > 0$$
and $$0 > p\left\{1 - exp\left\{[-log(p)]^{-1/\alpha}\right\}\right\}$$
\begin{eqnarray}
\nonumber p_{R, p}(X) &=& P\left\{X > \frac{1}{p}exp\left\{[-log\,\, (1-p)]^{-1/\alpha}\right\} - \frac{1-p}{p}exp\left\{[-log(p)]^{-1/\alpha}\right\}\right\} =\\
\nonumber &=& 1 - exp\left\{-\left\{log\left\{\frac{1}{p}exp\left\{[-log\,\, (1-p)]^{-1/\alpha}\right\} \right.\right.\right.\\
&-& \left. \left. \left. \frac{1-p}{p}exp\left\{[-log(p)]^{-1/\alpha}\right\}\right\} \right\} ^{-\alpha}\right\} \label{peRH1}
\end{eqnarray}
Figure \ref{fig:peRalphaExtremes} shows their dependence on $\alpha$ in case $p = 0.25$. This case is considered in the supplementary material of Soza et al. (2019) \cite{JordanovaStehlik2018}.
\begin{itemize}
\item $H_{4}$ type. Let $\alpha > 0$,
$$F_X(x) =\left\{\begin{array}{ccc}
exp\left\{-[ln(-\,x)]^{\alpha}\right\} & , & x < -1\\
1 & , & x \geq -1
\end{array}
\right..$$
\end{itemize}
It is one of the limiting distributions of power-transformed maxima obtained in Pancheva (1984) \cite{Pancheva}. Ravi and Saeb (2012) \cite{Ravi} calculate its Shannon entropy. $H_{4}$ type is known also as log-Weibull law, and it is one of the p-max stable laws.
The quantile function has the form $F_{Y}^\leftarrow(p) = -exp\{[-log(p)]^{1/\alpha}\}$, $p \in (0, 1).$
Therefore
$$p_{L,p}(X) = P\left\{X < \frac{1-p}{p}exp\{[-log(1 - p)]^{1/\alpha}\}-\frac{1}{p}exp\{[-log(p)]^{1/\alpha}\} \right\}.$$
And for $exp\{[-log(p)]^{1/\alpha}\} - (1-p)exp\{[-log(1 - p)]^{1/\alpha}\} > p$ we have that $p_{L,p}(X)$ is equal to
$$exp\left\{-\left\{log\left\{\frac{1}{p}exp\{[-log(p)]^{1/\alpha}\} - \frac{1-p}{p}exp\{[-log(1 - p)]^{1/\alpha}\}\right\}\right\}^{\alpha}\right\},$$
and $p_{L,p}(X) = 0$ otherwise.
When consider the right tail for all $\alpha > 0$ as far as $p < 1 - p$
$$\frac{1-p}{p}exp\{[-log(p)]^{1/\alpha}\} - \frac{1}{p}exp\{[-log(1 - p)]^{1/\alpha}\} < - 1$$
$$(1-p)exp\{[-log(p)]^{1/\alpha}\}-exp\{[-log(1 - p)]^{1/\alpha}\} < -p$$
$$-(1-p)exp\{[-log(p)]^{1/\alpha}\}-\left\{-exp\{[-log(1 - p)]^{1/\alpha}\}\right\} > $$ $$> -(1-p)exp\{[-log(p)]^{1/\alpha}\}-\left\{-exp\{[-log(p)]^{1/\alpha}\}\right\}$$
The last expression is equal to $-(-p)exp\{[-log(p)]^{1/\alpha}\} > p$, therefore $p_{R, p}(X)$ is equal to
\begin{eqnarray*}
& & P\left\{X > \frac{1-p}{p}exp\{[-log(p)]^{1/\alpha}\} - \frac{1}{p}exp\{[-log(1 - p)]^{1/\alpha}\}\right\}\\
&=& 1 - exp\left\{-\left\{log\left\{ \frac{1}{p}exp\{[-log(1 - p)]^{1/\alpha}\} - \frac{1-p}{p}exp\{[-log(p)]^{1/\alpha}\}\right\}\right\}^{\alpha}\right\}
\end{eqnarray*}
As in the previous case for large values of $\alpha > 0$ it is possible to observe both left and right outside values.
The dependance of $p_{R, 0.25}(-X)$ on $\alpha$ is depicted on Figure \ref{fig:peRalphaExtremes}. We call the distribution of $-X$, "$H_{4}$ positive", and we have denoted it by $H_{4}^+$.
Note that the function
$g(\alpha) = exp\{[log(4)]^{1/\alpha}\} - \frac{3}{4}exp\{[log\frac{4}{3}]^{1/\alpha}\} - \frac{1}{4}$ is decreasing in $\alpha$ and $g(2) \approx 1.713627$ therefore for $\alpha \in (0, 2]$
\begin{eqnarray*}
p_{L,0.25}(X) &=& p_{R,0.25}(-X) \\
&=& exp\left\{-\left\{log\left\{4exp\{[log(4)]^{1/\alpha}\} - 3exp\{[log(\frac{4}{3})]^{1/\alpha}\}\right\}\right\}^{\alpha}\right\}.
\end{eqnarray*}
See the supplementary material of Soza et al. \cite{JordanovaStehlik2018}.
\begin{itemize}
\item $log-Par_\alpha$ type. Log-Pareto law with parameter $\alpha$ seems to be introduced in Cormann and Reiss (2009) \cite{CormannAndReiss2009}.
More precisely here we assume tha
\end{itemize}
$$F_X(x) =\left\{\begin{array}{ccc}
0 & , & x < e \\
1 -(log(x))^{-\alpha} & , & x \geq e
\end{array}
\right..$$
This distribution belongs to $\Pi$ class considered e.g. in de Haan and Ferreira (2006) \cite{deHaanFerreira} or Embrehts et al. (1997) \cite{EMK}. Ravi and Saeb (2012) \cite{Ravi} investigate their entropies.
Due to Corollary 1 of Theorem 1 its tail is heavier than the tail of Fr$\acute{e}$chet distribution. The quantile function of this distribution is
$F_X^\leftarrow(p) = exp\left\{(1 - p)^{-1/\alpha}\right\}$. Therefore
for $\frac{1}{p}exp\left\{(1 - p)^{-1/\alpha}\right\} - \frac{1-p}{p}exp\left\{p^{-1/\alpha}\right\} > e$
$$p_{L,p}(X) = 1 -\left\{log\left[\frac{1}{p}exp\left[(1 - p)^{-1/\alpha}\right] - \frac{1-p}{p}exp\left(p^{-1/\alpha}\right)\right]\right\}^{-\alpha},$$
otherwise $p_{L,p}(X) = 0$.
When consider the right tail for all $\alpha > 0$ as far as $p < 1 - p$
$$exp\left(p^{-1/\alpha}\right) + p\, exp\left[(1-p)^{-1/\alpha}\right] \geq exp\left[(1-p)^{-1/\alpha}\right] + ep$$
$$exp\left(p^{-1/\alpha}\right) - (1-p)exp\left[(1 - p)^{-1/\alpha}\right] \geq ep$$
$$\frac{1}{p}exp\left(p^{-1/\alpha}\right) - \frac{1-p}{p}exp\left[(1 - p)^{-1/\alpha}\right] \geq e$$
therefore from the definition of $p_{R, p}(X)$ we obtain
\begin{eqnarray*}
p_{R, p}(X) &=& P\left\{X > \frac{1}{p}exp\left(p^{-1/\alpha}\right) - \frac{1-p}{p}exp\left[(1 - p)^{-1/\alpha}\right]\right\}\\
&=& \left\{log\left\{\frac{1}{p}exp\left(p^{-1/\alpha}\right) - \frac{1-p}{p}exp\left[(1 - p)^{-1/\alpha}\right]\right\}\right\}^{-\alpha}.
\end{eqnarray*}
The dependence of $p_{R, 0.25}(X)$ on $\alpha$ could be seen on Figure \ref{fig:peRalphaExtremes}. We observe that its tail behaviour almost coincide with log-Fr$\acute{e}$chet, i.e. $H_1$, and within the considered distributions, for fixed $\alpha$ according to $p_{R, 0.25}(X)$ the last two distributions have highest probabilities to observe extreme outside values. Having in mind that without transformations the tails of Fr$\acute{e}$chet and Pareto distributions are heavy-tailed Cl. Neves et al. (2008) \cite{ClNeves}, Corman and Reiss (2009) \cite{CormannAndReiss2009} or Falk (2004) \cite{Falk2004} call them "super heavy-tailed".
Further on in this section, we consider distributions which quantile function have no explicit form. Therefore we use R software (2018) \cite{R} in order to obtain obtain $F^\leftarrow(0.25)$ and $F^\leftarrow(0.75)$. Then we come back to the well-known formulas for c.d.f. and obtain $p_{R, 0.25}(X)$ characteristics.
\begin{itemize}
\item Normal distribution. Assume $\mu \in R$, $\sigma^2 > 0$ and $X \in N(\mu, \sigma^2)$.
\end{itemize}
W.l.g. $\mu = 0$ and $\sigma^2 = 1$. Due to the symmetry of this distribution with respect to (w.r.t.) Oy, for all $p \in (0,0.5)$ we have that $p_{L, p}(X) = p_{R, p}(X)$. In particular $p_{L, 0.25}(X) = p_{R, 0.25}(X) \approx 0.000001171$.
\begin{itemize}
\item $t$-distribution. Assume $n \in N$ and $X \in t(n)$.
\end{itemize}
The symmetry of the p.d.fs. of these distributions w.r.t. $Oy$ implies $p_{L, p}(X) = p_{R, p}(X)$, for all $p \in (0,0.5)$. The values of these characteristics for $n = 1, 2, ..., 10$ are presented in Table \ref{tab:t}, and could be seen also in Jordanova and Petkova \cite{MoniPoli2018}, and in the supplementary material of Soza et al. \cite{JordanovaStehlik2018}.
\begin{center}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 \\
\hline
$p_{L, 0.25}(X) = p_{R, 0.25}(X)$ & 0.0452 & 0.0146 & 0.0064 & 0.0033 & 0.0019 \\
\hline
\hline
n & 6 & 7 & 8 & 9 & 10 \\
\hline
$p_{L, 0.25}(X) = p_{R, 0.25}(X)$& 0.0012 & 0.0008 & 0.0006 & 0.0004 & 0.0003 \\
\hline
\end{tabular}
\caption{The dependence of $p_{L, 0.25}(X) = p_{R, 0.25}(X)$ on $n$. $X \in t(n)$.}\label{tab:t}
\end{center}
\end{table}
\end{center}
\begin{itemize}
\item Gamma distribution. Assume $\alpha > 0$, $\beta > 0$ and $X \in Gamma(\alpha, \beta)$ which means that
\end{itemize}
$$F_X(x) =\left\{\begin{array}{ccc}
0 & , & x < 0 \\
\int_0^x \frac{\beta^\alpha}{\Gamma(\alpha)} y^{\alpha - 1} e^{-\beta y} dy & , & x \geq 0
\end{array}
\right..$$
W.l.g. we assume that $\beta = 1$. The plot of the dependence of $p_{R, 0.25}(X)$ on $\alpha$ is depicted on Figure \ref{fig:peRalphaE}.
\begin{itemize}
\item Hill-horror distribution. For $\alpha > 0$ Embrechts et al. (1997) \cite{EMK} define it via its quantile function
\end{itemize}
\begin{equation}\label{HHQuantilefunction}
F_X^\leftarrow(p) = \frac{-log(1 - p)}{\sqrt[\alpha]{1 - p}}, \quad p \in (0, 1).
\end{equation}
Then $p_{L, 0.25}(X) = 0$. For the values of $p_{R, 0.25}(X)$ see Figure \ref{fig:peRalphaE}. We observe that this distribution has one of the heaviest right tails within the considered probability types.
Following this approach, we can find explicit values or plots of $p_{L, p}$ and $p_{R, p}$ characteristics for many other distributions and in this way to compare their tails. For example log-Positive Weibull, log-Gumbel, Invence-Gamma, Log-Gamma, Beta prime, or powers bigger than one, of these and other distributions.
\section{Properties of the estimators}
\label{sec:3}
In the previous section, we have considered some particular cases of probability laws and we have shown which parameter governs the heaviness of the tail of the corresponding distribution according to our classification. For the distributions with regularly varying tails, it coincides with the very well-known index of regular variation. In this section, we obtain different asymptotic properties of the estimators of the corresponding parameters of heaviness of the tails. The general formula for the joint distribution of order statistics is very well known. Together with the formula for their conditional distributions they could be found e.g. in Nevzorov (2001) \cite{Nevzorov} or in Arnold et al. (1992) \cite{Arnold1992}. The following lemma is their immediate corollary. Its first part summarises the same results in the terms of equality in distributions. In vi) we have expressed the bivariate vector of order statistics as a bivariate function of independent r.vs. This allows as to make the same with the fences in the next property. Finally two explicit formulae for the probability mass functions of the numbers of left and right $p$-outside values in a sample of independent observations are presented. Due to their complicated forms further on the section proceeds with asymptotic results.
\begin{lm}\label{lem:lm1} If $\xi \in Beta(i, j)$, $i \in N$, $j \in N$, $F(z) = P(X < z)$ is some c.d.f. of a r.v. $X$ and $c > 0$ is a constant, then
\begin{enumerate}
\item[i)] $-\frac{log\, (\xi)}{c} \stackrel{d}{=} \epsilon_{(j, i+j-1)},$
where $\epsilon_{(j, i+j-1)}$ is the $j$-th order statistics in a sample of $i+j-1$ independent observations on i.i.d. Exponential r.vs. with parameter $c > 0$.
\item[ii)] $\frac{1}{c\xi^\alpha} \stackrel{d}{=} \nu_{(j, i+j-1)}$,
where $\nu_{(j, i+j-1)}$ is the $j$-th order statistics in a sample of $i+j-1$ independent observations on i.i.d. Pareto distributed r.vs. with parameters $\frac{1}{\alpha} > 0$ and $\delta = \frac{1}{c}$.
\item[iii)] $\frac{-1}{c \,\, log(\xi)} \stackrel{d}{=} \phi_{(j, i+j-1)}$, where $\phi_{(j, i+j-1)}$ is the $j$-th order statistics in a sample of $i+j-1$ independent observations on i.i.d. Fr$\acute{e}$chet distributed r.vs. with parameter $\alpha = 1$ and scale parameter $c$.
\item[iv)] $F^\leftarrow(\xi)\stackrel{d}{=} \kappa_{(i, i+j-1)},$ and $(1 - F)^\leftarrow(\xi)\stackrel{d}{=} \kappa_{(j, i+j-1)}$, where $\kappa_{(s, i+j-1)}$ is the $s$-th order statistic of a sample of $i+j-1$ independent observations on a r.v. with absolutely continuous c.d.f. $F$, $s = i, j$.
\item[v)] For $1 \leq i < j$
$$(X_{(j, i+j-1)}|X_{(i, i+j-1)} = x) \stackrel{d}{=} \tilde{\tilde{\theta}}_{(j-i, j-1)} \stackrel{d}{=} G_x^\leftarrow(\xi^*),$$
$$(X_{(i, i+j-1)}|X_{(j, i+j-1)} = x) \stackrel{d}{=} \tilde{\theta}_{(i, j-1)} \stackrel{d}{=} T_x^\leftarrow(\xi^{**}),$$ where $\tilde{\tilde{\theta}}_{(j-i, j-1)}$ is the $j-i$-th order statistics in a sample of $j-1$ independent observations on i.i.d. r.vs. with $X_{LT}$ c.d.f.
$G_x(y) = \frac{F(y) - F(x)}{1-F(x)}$, $y > x$, $\tilde{\theta}_{(i, j-1)}$ is the $i$-th order statistics in a sample of $j-1$ independent observations on i.i.d. r.vs. with $X_{RT}$ c.d.f.
$T_x(y) = \frac{F(y)}{F(x)}$, $y < x$, $\xi^* \in Beta(j - i, i)$, and $\xi^{**} \in Beta(i, j - i)$.
\item[vi)] Assume $\xi$ and $\xi^*$ are independent, and $\xi^* \in Beta(j-i, i)$. Denote by $\xi^{**} = 1 - \xi^*$, and $\xi^{***} = 1 - \xi$.\footnote{Then $1 - \xi^* =: \xi^{**} \in Beta(i, j-i)$ and $1 - \xi =: \xi^{***} \in Beta(j, i)$.} Then for $n = i + j - 1$, $i, j \in N$, $1 \leq i < j$,
\begin{enumerate}
\item $(X_{(j, i+j-1)}, X_{(i, i+j-1)})$
\begin{eqnarray*}
&\stackrel{d}{=}& \{ F^\leftarrow(1 - \xi^{**} \xi^{***}),F^\leftarrow (1 - \xi^{***})\}\\
&\stackrel{d}{=}& \left\{(1-F)^\leftarrow(\xi^{**} \xi^{***}), (1-F)^\leftarrow(\xi^{***})\right\}\\
&\stackrel{d}{=}& \left\{\left(\frac{1}{1-F}\right)^\leftarrow\left(\frac{1}{\xi^{**} \xi^{***}}\right), \left(\frac{1}{1-F}\right)^\leftarrow\left(\frac{1}{\xi^{***}}\right)\right\}
\end{eqnarray*}
Moreover $\xi^{**} \xi^{***} \stackrel{d}{=} \xi$.
\item The empirical $\frac{i}{i+j}$-right fences
\begin{eqnarray} \label{rightfences}
R_n(X,\frac{i}{i+j}) &=& \left(1+\frac{j}{i}\right)X_{j, i + j - 1} - \frac{j}{i} X_{i, i + j - 1} \\
\nonumber &\stackrel{d}{=}& \left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(\xi^{**}\xi^{***}) - \frac{j}{i}(1- F)^\leftarrow (\xi^{***}).
\end{eqnarray}
\item The empirical $\frac{i}{i+j}$-left fences
\begin{eqnarray} \label{leftfences}
L_n(X,\frac{i}{i+j}) &=& \left(1+\frac{j}{i}\right)X_{i, i + j - 1} - \frac{j}{i} X_{j, i + j - 1} \\
\nonumber &\stackrel{d}{=}& \left(1+\frac{j}{i}\right)(1- F)^\leftarrow (\xi^{***}) - \frac{j}{i}(1 - F)^\leftarrow(\xi^{**}\xi^{***}).
\end{eqnarray}
\end{enumerate}
\item[vii)] For $k = 0, 1, ..., i+j$,
$P(n_R(\frac{i}{i+j},i+j-1) = k)$
\begin{eqnarray*}
&=& \frac{(i+j-1)!}{k!(i+j-k-1)!}\\
&.&\int_0^1\int_0^1\left\{\frac{1-F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(xy) - \frac{j}{i}(1 - F)^\leftarrow (x)\right]}{F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(xy) - \frac{j}{i}(1 - F)^\leftarrow (x)\right]}\right\}^k \\
&.&\frac{x^{j-1}(1-x)^{i-1}}{B(i,j)}\frac{y^{i-1}(1-y)^{j-i-1}}{B(j-i,i)}\\
&.&\left\{F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(xy) - \frac{j}{i}(1 - F)^\leftarrow (x)\right]\right\}^{i+j-1}dydx
\end{eqnarray*}
$P(n_L(\frac{i}{i+j},i+j-1) = k)$
\begin{eqnarray*}
&=& \frac{(i+j-1)!}{k!(i+j-k-1)!}\\
&.&\int_0^1\int_0^1\left\{\frac{F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(x) - \frac{j}{i}(1 - F)^\leftarrow(xy)\right]}{1-F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(x) - \frac{j}{i}(1 - F)^\leftarrow (xy)\right]}\right\}^k \\
&.&\frac{x^{j-1}(1-x)^{i-1}}{B(i,j)}\frac{y^{i-1}(1-y)^{j-i-1}}{B(j-i,i)}\\
&.&\left\{1-F\left[\left(1+\frac{j}{i}\right)(1 - F)^\leftarrow(x) - \frac{j}{i}(1 - F)^\leftarrow (xy)\right]\right\}^{i+j-1}dydx
\end{eqnarray*}
\end{enumerate}
\end{lm}
\begin{rem} As an additional result, we can use the above theorem to obtain different univariate and bivariate distributions and new relations between them. This approach is analogous to the one applied by Eugene et al. (2002) \cite{Eugene} or Cordeiro et al. (2012) \cite{Cordeiro} among others, who consider Generalized-Beta generalized distributions.
\end{rem}
\begin{rem} Using the general formula for the moments of order statistics, for $m = 1, 2, ..., n$, and $r \in R$
$$E[X_{(m,n)}^r] = \frac{n!}{(m-1)!(n-m)!}\int_{-\infty}^\infty x^r[F(x)]^{m-1}[1-F(x)]^{n-m}dF(x),$$
which could be seen e.g. in the books of Arnold et al. (1992)\cite{Arnold1992} or Nevzorov (2001) \cite{Nevzorov}, we can easily obtain the general formulae for the mean and the variance of $L_n(X,\frac{i}{i+j})$ and $R_n(X,\frac{i}{i+j})$ in cases when they exist. For example in Pareto (\ref{Pareto}) case, for $i = k$, $j = sk$, $s = 2, 3, ...,$ $L_{k(s+1)-1}(X,\frac{1}{1+s})$ and $R_{k(s+1)-1}(X,\frac{1}{1+s})$, are asymptotically unbiased estimators correspondingly for $L(F, \frac{1}{s + 1})$ and $R(F, \frac{1}{s + 1})$. More precisely
$$\lim_{k\to \infty} EL_{k(s+1)-1}(X,\frac{1}{s+1}) = L(F, \frac{1}{s + 1}), \quad and$$
$$\lim_{k\to \infty} ER_{k(s+1)-1}(X,\frac{1}{s+1}) = R(F, \frac{1}{s + 1}).$$
\end{rem}
The following result is an immediate corollary of the definition of convergence in probability, quantile transform, a.s. convergence of empirical quantiles to the corresponding theoretical one, and Slutsky's theorem about continuous functions. See e.g. Embrechts et al. (1997) \cite{EMK}.
\begin{thm}\label{thm:Lemma2} Given a sample of independent observations, for any fixed $s = 2, 3, ...$
\begin{eqnarray}
L_{(s+1)k - 1}\left(F, \frac{1}{s + 1}\right) &\stackrel{P}{\to }& L(F, \frac{1}{s + 1}), \quad k \to \infty,\\ \label{LFencesConsistency}
R_{(s+1)k - 1}\left(F, \frac{1}{s + 1}\right) &\stackrel{P}{\to }& R(F, \frac{1}{s + 1}), \quad k \to \infty. \label{RFencesConsistency}
\end{eqnarray}
\end{thm}
Cadwell (1953) \cite{Cadwell} finds the distribution of quasi-ranges in samples from a normal population. He gives us the idea about the next result. Rider (1959) \cite{Rider} obtains their exact distribution in case of samples from an exponential population. Sarhan et al. (1963) \cite{SarhanExponential} propose simplified estimates in this case. The asymptotic normality of the appropriately normalized univariate distributions of the central order statistics is investigated in Smirnov (1949) \cite{Smirnov1949}. Note that in the next theorem, because of the special choice of the numbers of order statistics, $p$ and $n$ his conditions $\frac{k}{n} \to p \in(0, 1)$ and $\sqrt{n}(\frac{k}{n} - p) \to \mu \in (-\infty, \infty)$ are satisfied. Moreover, in our case $\mu = 0$. The theorem about the joint distribution of the central order statistics of i.i.d. observations, could be seen e.g. in Nair (2013) \cite{Nair2013}, p.330, or Arnold et al. (1992) \cite{Arnold1992}, p. 226, among others. The multivariate delta method is a very powerful technique for obtaining confidence intervals in such cases. It can be seen e.g. in Sobel (1982) \cite{MultivariateDeltaMethod}. In the next theorem we use these results and obtain the limiting distribution of the fences of central order statistics.
\begin{thm}\label{thm:ThmGeneralAsumptoticNormalityFences} Consider a sample of $n = (s+1)k-1$, $s = 1, 2, 3, ...$ observations on a r.v. $X$ with c.d.f. $F$ and p.d.f. $f = F'$. Suppose that there exists $c_{F,s} : = f\left[F^\leftarrow(\frac{1}{s+1})\right] \in (0, \infty)$ and $d_{F,s} = f\left[F^\leftarrow(\frac{s}{s+1})\right] \in (0, \infty)$. Then
$$\lim_{k \to \infty} E L_{n}\left(F, \frac{1}{s + 1}\right) = L(F, \frac{1}{s + 1})$$
$$\lim_{k \to \infty} E R_{n}\left(F, \frac{1}{s + 1}\right) = R(F, \frac{1}{s + 1})$$
and for $k \to \infty$
\begin{equation}\label{General_CI_L_fence}
\sqrt{(s+1)k-1}\left|L_{n}\left(F, \frac{1}{s + 1}\right) - L(F, \frac{1}{s + 1})\right| \stackrel{d}{\to } N\left(0; V_{L,F,s}\right),
\end{equation}
\begin{equation}\label{General_CI_R_fence}
\sqrt{(s+1)k-1}\left|R_{n}\left(F, \frac{1}{s + 1}\right) - R(F, \frac{1}{s + 1})\right| \stackrel{d}{\to} N\left(0; V_{R,F,s}\right),
\end{equation}
where $V_{L,F,s} = \frac{s}{(s+1)^2} \left[\frac{(s + 1)^2}{c_{F,s}^2} - \frac{2(s+1)}{c_{F,s}d_{F,s}} + \frac{s^2}{d_{F,s}^2}\right]$ and \\$V_{R,F,s} = \frac{s}{(s+1)^2} \left[\frac{s^2}{c_{F,s}^2} - \frac{2(s+1)}{c_{F,s}d_{F,s}} + \frac{(s + 1)^2}{d_{F,s}^2}\right].$
\end{thm}
These allows as to compute the asymptotic confidence intervals of these estimators for $k \to \infty$, $n = (s+1)k - 1$, and $s = 1, 2, 3, ...$, fixed. Denote $L_{F,s,k} = L_{(s+1)k - 1}\left(F, \frac{1}{s + 1}\right)$ and $R_{F,s,k} = R_{(s+1)k - 1}\left(F, \frac{1}{s + 1}\right)$. If the conditions of Theorem \ref{thm:ThmGeneralAsumptoticNormalityFences} are satisfied, then given $\alpha \in (0, 1)$,
\begin{equation}\label{CI_L}
L_{F,s,k} - z_{\alpha} \sqrt{\frac{V_{L,F,s}}{(s+1)k-1}} \leq L(F, \frac{1}{s + 1}) \leq L_{F,s,k} + z_{\alpha} \sqrt{\frac{V_{L,F,s}}{(s+1)k-1}},
\end{equation}
\begin{equation}\label{CI_R}
R_{F,s,k} - z_{\alpha} \sqrt{\frac{V_{R,F,s}}{(s+1)k-1}} \leq R(F, \frac{1}{s + 1}) \leq R_{F,s,k} + z_{\alpha} \sqrt{\frac{V_{R,F,s}}{(s+1)k-1}}.
\end{equation}
The next theorem explains why different probabilities for outside values can be useful for estimating the tail behaviour of the observed distribution.
For a fixed $s = 2, 3, ...$ and $k \to \infty$, we apply the approach of Dembinska (2012) \cite{Dembinska2012NZ}, for the bivariate case, and obtain that
$\frac{n_L\left(\frac{1}{s+1}; (s+1)k-1\right)}{(s + 1)k - 1}$ and $\frac{n_R\left(\frac{1}{s+1};(s+1)k-1\right)}{ (s + 1)k - 1}$
are strongly consistent estimators correspondingly of $p_{L,\frac{1}{s+1}}(X)$ and $p_{R,\frac{1}{s+1}}(X)$. Moreover Dembinska (2017) \cite{Dembinska2017Metrika} shows that this approach works not only for i.i.d., but also also for strictly stationary and ergodic sequences.
\begin{thm}\label{thm:Strongconsistencofpel} Let $s = 2, 3, ...$ be fixed. Assume $f[F^\leftarrow(\frac{1}{s+1})] \in (0, \infty)$ and \\ $f[F^\leftarrow(\frac{s}{s+1})] \in (0, \infty)$.
\begin{enumerate}
\item If $f[F^\leftarrow(\frac{1}{s+1})] \in (0, \infty)$, then $$\frac{n_L\left(\frac{1}{s+1}; (s+1)k-1\right)}{(s + 1)k - 1} {\mathop{\to}\limits_{k \to \infty}^{a.s.}} p_{L,\frac{1}{s+1}}(X).$$
\item If $f[F^\leftarrow(\frac{1}{s+1})] \in (0, \infty)$, then $$\frac{n_R\left(\frac{1}{s+1};(s+1)k-1\right)}{ (s + 1)k - 1} {\mathop{\to}\limits_{k \to \infty}^{a.s.}} p_{R,\frac{1}{s+1}}(X).$$
\end{enumerate}
\end{thm}
\section{Simulation study}
\label{sec:sym}
In this section we assume that $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ are independent realizations of $\mathbf{X}$, with c.d.f. $F$.
Jordanova and Petkova (2017-2018) \cite{MoniPoli2017,MoniPoli2018} assume that $F$ has regularly varying tail. More precisely they consider only the cases when there exists $\alpha > 0$, such that for all $x > 0$,
\begin{equation}\label{RV}
\lim_{t \to \infty} \frac{1 - F(xt)}{1 - F(t)} = x^{-\alpha}.
\end{equation}
The number $-\alpha$ is called "index of regular variation of the tail of c.d.f.", see e.g. de Haan and Ferreira (2006) \cite{deHaanFerreira} or Resnick (1987) \cite{Resnick87}. Using the explicit form of the corresponding probabilities for extreme outliers according the definition in Devore(2015) \cite{Devore},($0.25$-outliers) Jordanova and Petkova (2017-2018) \cite{MoniPoli2017,MoniPoli2018} obtain distribution sensitive estimators of the unknown parameter $\alpha$ which governs the tail of the considered distributional type. The algorithm consists of the following three main steps.
\begin{enumerate}
\item Using the results from the previous two sections the explorer chooses the most appropriate probability type (let us call it $T$) for modeling the tail of the distribution of the observed r.v.
\item Using the formula for $p_{R, 0.25}$ in case $T$ one expresses the unknown parameter $\alpha$.
\item Replace the theoretical characteristics in the previous step with the corresponding estimators and obtain a new estimator for the parameter which governs the tail behavior.
\end{enumerate}
In their work Jordanova and Petkova (2017-2018) \cite{MoniPoli2017,MoniPoli2018} compare the obtained in this way estimators in Pareto, Fr$\acute{e}$chet, and Hill-Horror case with Hill, t-Hill, Pickands, and Deckers-Einmahl-de Haan estimators and depict the results via a simulation study. Here we consider two more cases: Log-Logistic case (\ref{LolLogidticCDF}) and $H_1$ case (\ref{H1CDF}). Although in the last case the right tail of the c.d.f. is not regularly varying the next study shows that the approach still gives very good results.
In any of the following five examples using the functions implemented in R (2018), \cite{R} we have simulated $m = 1000$ samples of $n$ independent observations separately on $X$. Then for any fixed $n = 10, 11, ..., 500$ and for any fixed sample we have computed the estimators $\hat{p}_R(0.25, n)= \frac{n_{R}(0.25, n)}{n},$ $F_n^\leftarrow(0.25)$, $F_n^\leftarrow(0.75)$, and $\hat{\alpha}_{\bullet,n}$. Here $n_{R}(0.25, n)$ is the numbers of right extreme outside values in the considered sample of $n$ independent observations, and $\bullet$ means one of the abbreviations $Par$, $Fr$, $HH$, $H_1$ or $LL$ explained below. Finally we have fixed one of the last estimators and we have averaged the corresponding values of $\hat{\alpha}_{\bullet,n}$ over the considered $n$. The next Figures \ref{fig:SymStudy1}-\ref{fig:SymStudy10} depict the dependence of these values, together with the corresponding asymptotic normal 95\% confidence intervals, on the real type of the observed r.v., and on the sample size, for $\alpha = 0.5$, or $1$. We have chosen only the cases when $\alpha$ is small because our observations show that for a fixed sample size the more the outside values, the heavier of the tail of the c.d.f. is and the better the corresponding estimator is.
Let us depict this approach with some examples. In any of them we suppose that $F_n^\leftarrow\left(\frac{3}{4}\right) > 1$ and $\hat{p}_R(0.25, n) > 0$.
\begin{ex} Assime $\mathbf{X} \in Par(\alpha, \delta)$. See (\ref{Pareto}). Having a sample of $n$ independent observations on $\mathbf{X}$, analogously to the generalized method of moments, and following the above algorithm Jordanova and Petkova (2018) \cite{MoniPoli2018} obtain
\begin{equation}\label{EstParn}
\hat{\alpha}_{Par,n} = -\frac{log\,\,\hat{p}_R(0.25, n)}{log\left\{F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right) - F_n^\leftarrow\left(\frac{1}{4}\right)\right]\right\}}.
\end{equation}
\end{ex}
\begin{ex} If $\mathbf{X} \in Fr(\alpha, c)$, see (\ref{Frechet}), Jordanova and Petkova (2018) \cite{MoniPoli2018} propose
\begin{equation}\label{EstFr}
\hat{\alpha}_{Fr,n} = -\frac{log\{-log[1-\hat{p}_R(0.25, n)]\}}{log\left\{F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right)- F_n^\leftarrow\left(\frac{1}{4}\right)\right]\right\}}.
\end{equation}
\end{ex}
\begin{ex} Let $\mathbf{X}$ be Hill-Horror distributed. This distribution is usually defined via its quantile function (\ref{HHQuantilefunction}). Given $F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right)- F_n^\leftarrow\left(\frac{3}{4}\right)\right] \not= -log\, \hat{p}_R(0.25, n)$ Jordanova and Petkova (2018) \cite{MoniPoli2018} use
\begin{equation}\label{EstHH}
\hat{\alpha}_{HH,n} = \frac{log\, \hat{p}_R(0.25, n)}{log\,\left\{\frac{-log\,\hat{p}_R(0.25, n)}{F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right)- F_n^\leftarrow\left(\frac{1}{4}\right)\right]}\right\}}.
\end{equation}
\end{ex}
The next two estimators seems to be new. They show that this approach can be applied in much wider than the regularly varying case.
\begin{ex} Let $\mathbf{X} \in H_1$, see (\ref{H1CDF}). It is difficult to solve (\ref{peRH1}) with respect to $\alpha$, therefore we solve the equation
$$ p_{R, 0.25}(X) = 1 - exp\{-[log\,R(X, 0.25)]^{-\alpha}\}.$$
When express $\alpha$ and replace the theoretical characteristics with the corresponding empirical one we obtain the estimator
\begin{equation}\label{EstH1}
\hat{\alpha}_{H_1,n} = -\frac{log\{-log[1-\hat{p}_R(0.25, n)]\}}{log\left\{log\left\{F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right) - F_n^\leftarrow\left(\frac{1}{4}\right)\right]\right\}\right\}}.
\end{equation}
\end{ex}
\begin{ex} Suppose $\mathbf{X}$ follow Log-logistic probability law (\ref{LolLogidticCDF}). The equation (\ref{pRpLogLogistic}) have no explicit solution for $\alpha$, therefore we solve the equation
$$ p_{R, 0.25}(X) = \frac{1}{1 + R(X, 0.25)^{\alpha}}.$$
Then we replace the theoretical characteristics with the corresponding empirical one we obtain the estimator
\begin{equation}\label{EstLL}
\hat{\alpha}_{LL,n} = \frac{log\left[\frac{1}{\hat{p}_R(0.25, n)}-1\right]}{log\left\{F_n^\leftarrow\left(\frac{3}{4}\right) + 3\left[F_n^\leftarrow\left(\frac{3}{4}\right) - F_n^\leftarrow\left(\frac{1}{4}\right)\right]\right\}}.
\end{equation}
\end{ex}
Figures \ref{fig:SymStudy1}-\ref{fig:SymStudy10} depict the dependence of these estimators, together with their empirical 95\% confidence intervals on the sample size, probability law of the simulated r.v., and the estimated parameter $\alpha$. The names of the estimators in these figures are abbreviated as follows: $\hat{\alpha}_{Par,n} = aParn$, $\hat{\alpha}_{Fr,n} = aFrn$, $\hat{\alpha}_{HH,n} = aHHn$, $\hat{\alpha}_{LL,n} = aLLn$, and $\hat{\alpha}_{H_1,n} = aH1n$.
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy1}\vspace{-0.3cm}
\caption{Pareto case, $\alpha = 0.5$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy1}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy2}\vspace{-0.3cm}
\caption{Pareto case, $\alpha = 1$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy2}
\end{minipage}
\end{figure}
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy3}\vspace{-0.3cm}
\caption{Fr$\acute{e}$chet case, $\alpha = 0.5$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy3}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy4}\vspace{-0.3cm}
\caption{Fr$\acute{e}$chet case, $\alpha = 1$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy4}
\end{minipage}
\end{figure}
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy5}\vspace{-0.3cm}
\caption{Hill-Horror case, $\alpha = 0.5$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy5}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy6}\vspace{-0.3cm}
\caption{Hill-Horror case, $\alpha = 1$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy6}
\end{minipage}
\end{figure}
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy7}\vspace{-0.3cm}
\caption{$H_1$ case, $\alpha = 0.5$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy7}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy8}\vspace{-0.3cm}
\caption{$H_1$ case, $\alpha = 1$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy8}
\end{minipage}
\end{figure}
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy9}\vspace{-0.3cm}
\caption{Log-Logistic case, $\alpha = 0.5$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy9}
\end{minipage}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.47]{SymStudy10}\vspace{-0.3cm}
\caption{Log-Logistic case, $\alpha = 1$: Dependence of $\hat{\alpha}_{Par,n} $, $\hat{\alpha}_{Fr,n} $, $\hat{\alpha}_{HH,n}$, $\hat{\alpha}_{H_1,n}$, $\hat{\alpha}_{LL,n}$ on the sample size}
\label{fig:SymStudy10}
\end{minipage}
\end{figure}
The above simulation study shows that within the considered set of distributions given a small sample of observations the considered estimators outperform the properties of the well-known estimators proposed by Hill (1975) \cite{Hill}, Pickands (1975)\cite{Pickands} and Deckers-Einmahl-de Haan Dekkers (1989) \cite{Dekkers1989}. Within the right probability type, the rate of convergence of any of them increases when the sample size increases and $\alpha > 0$ decreases. However, according to our investigation, these estimators are too distribution sensitive. The biggest their advantage is that they are applicable for relatively small samples.
\section{Conclusive remarks}
\label{sec:4}
To the best knowledge of the author, a universal numerical characteristic of the tail of the c.d.f., which is invariant within distributional type (with respect to increasing affine transformation) is still not known. Here we show that probabilities of the events an observation to be $p$-outside value can be very useful in this sense. They can be used for making a reasonable classification of the tails of probability distributions. They outperform the role e.g. of the excess in characterizing the tail of the observed distribution because they do not depend on the moments of the observed r.v. and could be applied also in cases when moments do not exist. Their estimators are appropriate for usage in preliminary statistical analysis in presence of corresponding outside values. They can help the practitioners to find the most appropriate classes of probability laws for modeling the tails of the distribution of the observed r.v. Within that family the parameter which influences the tails needs further estimation. According to our simulation study, the proposed algorithm for making estimators gives better results when $\alpha>0$ decreases. The fast rate of convergence allows one to apply these estimators also for relatively small samples. However, the main disadvantage of all these estimators is that they are distribution sensitive. The last means that their good properties may disappear if the distributional type is not correctly determined.
\section{Acknowledgements}
The author would like to thank Prof. Milan Stehlik for bringing her to the question about statistical modeling of extremes given small samples
\section{Apendix}
{\bf Proof of Theorem \ref{thm:monotonicity}}:
a) By definition of $L(X, p)$ and formula for derivative of the inverse function $\frac{\partial F^\leftarrow(p)}{\partial p} = \frac{1}{f(F^\leftarrow(p))}$ we obtain
$$\frac{\partial L(X, p)}{\partial p} = \frac{1}{pf[F^\leftarrow(p)]} + \frac{F^\leftarrow(1 - p) - F^\leftarrow(p)}{p^2} + \frac{1 - p }{pf[F^\leftarrow(1 - p)]}.$$
$f(x)$ is a density function of the r.v. $X$, therefore it is non-negative. The difference $F^\leftarrow(1 - p) - F^\leftarrow(p) \geq 0$ because $p \in (0; 0.5]$. Therefore $\frac{\partial L(X, p)}{\partial p} \geq 0$.
b) By definition of $R(X, p)$ and the same formula for derivative of the inverse function we have that
$$\frac{\partial R(X, p)}{\partial p} = \frac{F^\leftarrow(p) - F^\leftarrow(1 - p)}{p^2} - \frac{1}{pf[F^\leftarrow(p)]} - \frac{1 - p }{pf[F^\leftarrow(1 - p)]}.$$
Now the difference $F^\leftarrow(1 - p) - F^\leftarrow(p) \leq 0$ because $p \in (0; 0.5]$. Therefore $\frac{\partial R(X, p)}{\partial p} \leq 0$.
c) follows by a), b), and monotonicity of probability measures.
\hfill Q.A.D
{\bf Proof of Theorem \ref{thm:thm1}}:
b) For $c \in R$ from the definition of the quantile function we have that $F_{X+c}^\leftarrow(p) = F_X^\leftarrow(p)+c$. Therefore
\begin{eqnarray*}
p_{R,p}(X + c) &=& P(X+c >\frac{1}{p} F_{X+c}^\leftarrow(1-p) - \frac{1-p}{p}F_{X+c}^\leftarrow(p)) \\
&=& P(X + c > \frac{1}{p} F_{X}^\leftarrow(1-p) + \frac{c}{p} - \frac{1-p}{p}F_{X}^\leftarrow(p)- \frac{c}{p}(1-p))\\
&=& p_{R,p}(X).
\end{eqnarray*}
c) For $c > 0$ again from the definition of the quantile function $F_{cX}^\leftarrow(p) = cF_X^\leftarrow(p)$. Therefore
\begin{eqnarray*}
p_{R,p}(cX) &=& P(cX >\frac{1}{p} F_{cX}^\leftarrow(1-p) - \frac{1-p}{p}F_{cX}^\leftarrow(p)) \\
&=& P(cX > \frac{c}{p} F_{X}^\leftarrow(1-p) - c\frac{1-p}{p}F_{X}^\leftarrow(p)) = p_{R,p}(X).
\end{eqnarray*}
d) In this case $c < 0$, therefore $F_{cX}^\leftarrow(p) = cF^\leftarrow(1-p)$ and
\begin{eqnarray*}
p_{R,p}(cX) &=& P(cX >\frac{1}{p} F_{cX}^\leftarrow(1-p) - \frac{1-p}{p}F_{cX}^\leftarrow(p)) \\
&=& P(cX > \frac{c}{p} F_{X}^\leftarrow(p) - c\frac{1-p}{p}F_{X}^\leftarrow(1-p)) \\
&=& P(X < \frac{1}{p} F_{X}^\leftarrow(p) - \frac{1-p}{p}F_{X}^\leftarrow(1-p)) = p_{L,p}(X).
\end{eqnarray*}
e) Because of $g$ is a strictly increasing and continuous function we have that $F_{g(X)}^\leftarrow(p) = g(F^\leftarrow(p))$ and $F_{g(X)}^\leftarrow(1-p) = g(F^\leftarrow(1-p))$. Therefore
\begin{eqnarray*}
p_{R,p}(g(X)) &=& P\left[g(X) > \frac{1}{p} F_{g(X)}^\leftarrow(1-p) - \frac{1-p}{p}F_{g(X)}^\leftarrow(p)\right] \\
&=& P\left\{g(X) > \frac{1}{p} g[F^\leftarrow(1-p)] - \frac{1-p}{p}g[F^\leftarrow(p)]\right\} \\
&\leq& P\left\{g(X) > g[\frac{1}{p}F^\leftarrow(1-p) - \frac{1-p}{p}F^\leftarrow(p)]\right\} \\
&=& P\left[X > \frac{1}{p}F^\leftarrow(1-p) - \frac{1-p}{p}F^\leftarrow(p)\right] = p_{R,p}(X)
\end{eqnarray*}
These, together with the definition of $p_{R,p}(X)$ and monotonicity of probability measures entail (\ref{auxTh1e1}).
(\ref{auxTh1e2}) is a corollary of (\ref{auxTh1e3}), applied for $\tilde{g}(x) = - g(x)$.
f) The equalities $F^\leftarrow_{g(X)}(1-p) = g(F^\leftarrow_X(p))$ and $F^\leftarrow_{g(X)}(p) = g(F^\leftarrow_X(1-p))$ entail
\begin{eqnarray*}
p_{R, p}(g(X)) &=& P\left[g(X) > \frac{1}{p}F_{g(X)}^\leftarrow(1-p)- \frac{1-p}{p}F_{g(X)}^\leftarrow(p)\right] \\
&=& P\left\{g(X) > \frac{1}{p} g[F^\leftarrow(p)] - \frac{1-p}{p}g[F^\leftarrow(1-p)]\right\}\\
&\geq& P\left\{g(X) > g[\frac{1}{p} F^\leftarrow(p) - \frac{1-p}{p}F^\leftarrow(1-p)]\right\} \\
&=& P\left[X < \frac{1}{p}F^\leftarrow(p) - \frac{1-p}{p}F^\leftarrow(1-p)\right] = p_{L,p}(X).
\end{eqnarray*}
Now the definitions of $p_{R,p}(X)$, and $p_{L,p}(X)$, and the monotonicity of probability measures entail (\ref{auxTh1e3}).
(\ref{auxTh1e4}) follows by (\ref{auxTh1e1}), when replace the function $g(x)$ with $\tilde{g}(x) = - g(x)$ and take into account that $p_{R, p}(-g(X)) = p_{L, p}(g(X))$.
g) As far as $F^\leftarrow_{X_{(n, n)}}(p) = F^\leftarrow_{X}(\sqrt[n]{p})$
\begin{eqnarray*}
p_{R, p}(X_{(n, n)}) &=& P\left\{X > \frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{p})\right\}\\
&=& 1 - F_X^n\left[\frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{p})\right]
\end{eqnarray*}
\begin{eqnarray*}
p_{L,p}(X_{(n, n)}) &=& P\left\{X < \frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{1 - p})\right\} \\
&=&F_X^n\left[\frac{1}{p}F^\leftarrow_{X}(\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(\sqrt[n]{1 - p})\right]
\end{eqnarray*}
h) Using $F^\leftarrow_{X_{(1, n)}}(p) = F^\leftarrow_{X}(1-\sqrt[n]{1 - p})$ we obtain
\begin{eqnarray*}
p_{R,p}(X_{(1, n)}) &=& P\left\{X > \frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{1 - p})\right\}\\
&=&\left\{1 - F_X\left[\frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{1 - p})\right]\right\}^n
\end{eqnarray*}
\begin{eqnarray*}
p_{L,p}(X_{(1, n)}) &=& P\left\{X < \frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{p})\right\} \\
&=&1 - \left\{1 - F_X \left[\frac{1}{p}F^\leftarrow_{X}(1-\sqrt[n]{1 - p}) - \frac{1-p}{p} F^\leftarrow_{X}(1-\sqrt[n]{p})\right]\right\}^n.
\end{eqnarray*}
i) Consider $t > 0$. The relation between the quantile function of the exceedances and the c.d.f. of $F$ (see e.g. in Nair et al. (2013) \cite{Nair2013}) entails
\begin{eqnarray*}
& & p_{R,p}(X-t|X>t)\\
&=& P\left\{X - t > \frac{1}{p}F^\leftarrow_{X-t|X>t}(1 - p) - \frac{1-p}{p} F^\leftarrow_{X-t|X>t}(p)|X > t\right\} \\
&=&P\left\{X - t > \frac{1}{p}F^\leftarrow_{X}[1 - p + pF_X(p)] - \frac{t}{p} \right.\\
&-& \frac{1-p}{p} F_X^\leftarrow[p + (1-p)F_X(t)] + \left. \frac{t(1-p)}{p}|X > t\right\} \\
&=&\frac{P\left\{X > \frac{1}{p}F^\leftarrow_{X}[1 - p + pF_X(p)] - \frac{1-p}{p} F^\leftarrow_{X}[p + (1-p)F_X(t)] \right\}}{P(X>t)} \\
&=& \frac{P\left\{X > \frac{1}{p}F^\leftarrow_{X|X>t}(1 - p) - \frac{1-p}{p} F^\leftarrow_{X|X>t}(p) \right\}}{P(X>t)}\\
&=& P\left\{X > \frac{1}{p}F^\leftarrow_{X|X>t}(1 - p) - \frac{1-p}{p} F^\leftarrow_{X|X>t}(p)| X>t \right\} = p_{R,p}(X|X>t).
\end{eqnarray*}
Analogously for $p_{L,p}(X-t|X>t)$. \footnote{This property can be obtained also as a corollary of b).}
k) We obtain this property when replace the relations between the quantile functions of left-, right-, and double-truncated r.vs., $F_X^\leftarrow$ and $F_X$, i.e.
\begin{eqnarray*}
F_{X_{LT}}^\leftarrow(p) &=& F_X^\leftarrow(p+(1-p)F_X(l)) \\
F_{X_{RT}}^\leftarrow(p) &=& F_X^\leftarrow(pF_X(u)) \\
F_{X_{DT}}^\leftarrow(p) &=& F_X^\leftarrow(pF_X(u)+(1-p)F_X(l))
\end{eqnarray*}
in the definitions of $L(X,p)$ and $R(X,p)$. The above equalities is not difficult to calculate, and could be found e.g. in Nair et al. (2013) \cite{Nair2013}.
Finally we use the definitions of $p_{L,p}(X)$ and $p_{R, p}(X)$.
l) Assume $p \in (0, 0.5]$.
- Case $a > 1$. In this case $F_{log_a(X)}^\leftarrow(p) = log_a[F^\leftarrow(p)]$, and the function $a^x$ is increasing in $x$, therefore
\begin{eqnarray*}
p_{L,p} [log_a(X)] &=& P\left[log_a(X) < \frac{1}{p}F_{log_a(X)}^\leftarrow(p) - \frac{1 - p}{p}F_{log_a(X)}^\leftarrow(1 - p)\right] \\
&=& P\left\{log_a(X) < \frac{1}{p}log_a[F^\leftarrow(p)] - \frac{1 - p}{p}log_a[F^\leftarrow(1 - p)]\right\} \\
&=& F \left\{\sqrt[p]{\frac{F^\leftarrow(p)}{[F^\leftarrow(1 - p)]^{1 - p}}}\right\}.
\end{eqnarray*}
\begin{eqnarray*}
p_{R,p} [log_a(X)] &=& P\left[log_a(X) > \frac{1}{p}F_{log_a(X)}^\leftarrow(1 - p) - \frac{1 - p}{p}F_{log_a(X)}^\leftarrow(p)\right] \\
&=& P\left\{log_a(X) > \frac{1}{p}log_a[F^\leftarrow(1 - p)] - \frac{1 - p}{p}log_a[F^\leftarrow(p)]\right\} \\
&=& 1 - F \left\{\sqrt[p]{\frac{F^\leftarrow(1 - p)}{[F^\leftarrow(p)]^{1 - p}}}\right\}.
\end{eqnarray*}
- Case $0 < a < 1$. here we use the fact that $1/a > 1$, therefore our computations in the previous case imply
\begin{eqnarray*}
p_{L,p} [log_a(X)] &=& p_{L,p} [-log_{1/a}(X)] = p_{R,p} [log_{1/a}(X)] \\
&=& 1 - F \left\{\sqrt[p]{\frac{F^\leftarrow(1 - p)}{[F^\leftarrow(p)]^{1 - p}}}\right\}.
\end{eqnarray*}
\begin{eqnarray*}
p_{R,p} [log_a(X)] &=& p_{R,p} [-log_{1/a}(X)] = p_{L,p} [log_{1/a}(X)] \\
&=& F \left\{\sqrt[p]{\frac{F^\leftarrow(p)}{[F^\leftarrow(1 - p)]^{1 - p}}}\right\}.
\end{eqnarray*}
m) Case $\alpha > 0$. In this case $F_{X^\alpha}^\leftarrow(p) = [F^\leftarrow(p)]^\alpha$, and the function $x^\alpha$ is increasing in $x$, therefore
we apply Theorem 1, e) (\ref{eL}) and (\ref{eR}) and obtain the desired result.
Case $\alpha < 0$. Now $F_{X^\alpha}^\leftarrow(p) = [F^\leftarrow(1 - p)]^\alpha$, and the function $x^\alpha$ is decreasing in $x$, therefore
we apply Theorem 1, f) (\ref{fL}) and (\ref{fR}) complete the proof.
n) In case $\alpha \in (0, 1)$ we take into account that $F_{a^X}^\leftarrow(p) = a^[F^\leftarrow(1 - p)]$, and the functions $a^x$ and $log_a (x)$ are decreasing in $x$, then we apply Theorem 1, f) (\ref{fL}) and (\ref{fR}) and finish the proof of this case.
Analogously, if $\alpha > 1$ then $F_{a^X}^\leftarrow(p) = a^[F^\leftarrow(p)]$, and the function $x^\alpha$ is increasing in $x$. Therefore
we apply Theorem 1, e) and after some algebra complete the proof.
\hfill Q.A.D
{\bf Proof of Corollary of f):} Assume $p \in (0, 0.5]$. Consider the case $P(X > 0) = 1$. Theorem 1, f), applied for $g(x) = x^{-1}$ entails
$$p_{R, p}\left(\frac{1}{X}\right) = F\left[\frac{pF^\leftarrow(p)F^\leftarrow(1 - p)}{F^\leftarrow(1 - p) - (1 - p)F^\leftarrow(p)}\right].$$
By the definition for $p_{L, p}(X)$ and monotonicity of probability measures in order to prove that $p_{L, p}(X) \leq p_{R, p}(X^{-1})$ we need to show that
\begin{equation}\label{aux1}
\frac{1}{p}F^\leftarrow(p) - \frac{1-p}{p}F^\leftarrow(1 - p) \leq \frac{pF^\leftarrow(p)F^\leftarrow(1 - p)}{F^\leftarrow(1 - p) - (1 - p)F^\leftarrow(p)}
\end{equation}
The last inequality is equivalent to
$$1 - (1 - p)\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} \leq p^2\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}\left[\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} - 1 + p\right]^{-1}.$$
For $z := \frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} \geq 1$, $z - 1 + p \geq 0$, therefore $(z - 1 + p)[1 - (1 - p)z] \leq p^2z$, which completes the proof of this part.
The assertion for the case $P(X < 0) = 1$ follows by the fact that $$p_{L, p}\left(\frac{1}{X}\right) = p_{R, p}\left(\frac{1}{-X}\right).$$
\hfill Q.A.D
{\bf Proof of Corollary of e):} Assume $p \in (0, 0.5]$.
- Case $a > 1$. Because of $F_{log_a(X)}^\leftarrow(p) = log_a[F^\leftarrow(p)]$, in order to use Theorem \ref{thm:thm1}, e) for $g(x) = log_a(x)$, which is increasing in $x$ we need to prove that
$$\frac{1}{p}log_a[F^\leftarrow(1-p)] - \frac{1-p}{p}log_a[F^\leftarrow(p)] \geq log_a\left[\frac{1}{p}F^\leftarrow(1-p) - \frac{1 - p}{p}F^\leftarrow(p)\right].$$
The function $a^x$ is also increasing in $x$, therefore the above inequality is equivalent to
\begin{equation}\label{smile}
\sqrt[p]{\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}} \geq \frac{1}{p}\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} - \frac{1}{p} + 1.
\end{equation}
The function $\sqrt[p]{z} - \frac{1}{p}z + \frac{1}{p} - 1$ is increasing in $z \geq 1$, and for $p \in (0, 0.5]$, $\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} \geq 1$. Therefore, for $z \geq 1$, $\sqrt[p]{z} - \frac{1}{p}z + \frac{1}{p} - 1 \geq 0$ proves (\ref{smile}) and completes the proof of (\ref{LogarithmsFirstinequalityR}).
By Theorem \ref{thm:thm1}, f), applied for $g(x) = log_a(x)$, (because now $log_a(x)$ is also increasing in $x$) in order to compare $p_{L, p}[log_a(X)]$ and $p_{L, p}(X)$ in (\ref{LogarithmsFirstinequalityL}) we have to show that
\begin{equation}\label{smile1}
\sqrt[p]{\frac{F^\leftarrow(p)}{[F^\leftarrow(1 - p)]^{1 - p}}} \geq \frac{1}{p}F^\leftarrow(p) - \frac{1-p}{p} F^\leftarrow(1 - p).
\end{equation}
Which is the same as
$$\sqrt[p]{\frac{F^\leftarrow(p)}{F^\leftarrow(1 - p)}} \geq \frac{1}{p}\frac{F^\leftarrow(p)}{F^\leftarrow(1 - p)} - \frac{1}{p} + 1.$$
The function $\sqrt[p]{z} - \frac{1}{p}z + \frac{1}{p} - 1$ is decreasing in $z \in (0, 1)$, and because of by assumption $P(X > 0) = 1$, for $p \in (0, 0.5]$, $0 < \frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)} \leq 1$. Therefore, for $z \geq 1$, $\sqrt[p]{z} - \frac{1}{p}z + \frac{1}{p} - 1 \geq 0$ proves (\ref{smile1}) and (\ref{LogarithmsFirstinequalityL}).
- Case $0 < a < 1$. Because of $1/a\geq 1$, by the previous case, and the definitions for $p_{R, p}$ and $p_{L, p}$, entail
$$p_{R, p}(log_a(X)) = p_{R, p}(-log_{1/a}(X)) = p_{L, p}(log_{1/a}(X)) \geq p_{L, p}(X)$$
and complete the proof of (\ref{LogarithmsFirstinequalityLalessthan1}).
The fact that $1/a\geq 1$ and (\ref{LogarithmsFirstinequalityR}) entail
$$p_{L, p}(log_a(X)) = p_{L, p}(-log_{1/a}(X)) = p_{R, p}(log_{1/a}(X)) \leq p_{R, p}(X) \leq p_{R, p}(\frac{1}{a^X}), $$
and this proves inequalities in (\ref{Logarithms1}).
\hfill Q.A.D.
{\bf Proof of Theorem \ref{thm:thm1aux}}: a) Consider $p \in (0, 0.5]$, $0 \leq \alpha_1 \leq \alpha_2$ and $P$ almost sure positive r.v. $X$, i.e. $P(X > 0) = 1$. According to monotonicity of probability measures, the definition of $p_{R,p}$, and the equalities $F_{X^{\alpha_i}}^\leftarrow(p) = [F^\leftarrow(p)]^{\alpha_i}$, $i = 1, 2$ we need to show that
\begin{equation}\label{aux2}
\sqrt[\alpha_1]{\frac{1}{p}[F^\leftarrow(1-p)]^{\alpha_1} - \frac{1 - p}{p}[F^\leftarrow(p)]^{\alpha_1}} \geq \sqrt[\alpha_2]{\frac{1}{p}[F^\leftarrow(1-p)]^{\alpha_2} - \frac{1 - p}{p}[F^\leftarrow(p)]^{\alpha_2}}.
\end{equation}
It is the same as
$$\sqrt[\alpha_1]{\frac{1}{p}\left[\frac{F^\leftarrow(1-p)}{F^\leftarrow(p)}\right]^{\alpha_1} - \frac{1}{p} + 1} \geq \sqrt[\alpha_2]{\frac{1}{p}\left[\frac{F^\leftarrow(1-p)}{F^\leftarrow(p)}\right]^{\alpha_2} - \frac{1}{p} + 1}.$$
Denote by $z:=\frac{F^\leftarrow(1-p)}{F^\leftarrow(p)} \geq 1$. The last inequality is true, because of for any fixed $z \geq 1$ and $p \in (0; 0.5]$ the function
$$h(\alpha) = \sqrt[\alpha]{\frac{1}{p}z^{\alpha} - \frac{1}{p} + 1}$$
is decreasing in $\alpha > 0$. Therefore (\ref{aux2}) is also true, and the proof of (\ref{Cor2ofe1}) is completed.
b) The r.v. $X$ is almost sure positive, so we have the same for $X^\alpha$. Therefore, by the definition of $p_{L, p}(X^\alpha)$ it is enough to show that, in this case $L(X^\alpha, p) \leq 0$. Now we use the equality $F^\leftarrow_{X^\alpha}(p) = [F^\leftarrow_{X}(p)]^\alpha$ in the definition of $L(X^\alpha, p)$ and obtain that given the condition in c), the value of
$$L(X^\alpha, p) = \frac{1}{p}\left\{[F^\leftarrow(p)]^\alpha - (1 - p)[F^\leftarrow(1 - p)]^\alpha\right\} \leq 0.$$
c) It is enough to prove the second inequality in (\ref{Cor2ofe2}). It is equivalent to
$$P\left[X < \frac{1}{p}F^\leftarrow(p) - \frac{1 - p}{p}F^\leftarrow(1 - p)\right] \geq P\left[X^\alpha < \frac{1}{p}F_{X^\alpha}^\leftarrow(p) - \frac{1 - p}{p}F_{X^\alpha}^\leftarrow(1 - p)\right],$$
and using the equality $F_{X^\alpha}^\leftarrow(p) = [F^\leftarrow(p)]^\alpha$ it is the same as
$$P\left[X < \frac{1}{p}F^\leftarrow(p) - \frac{1 - p}{p}F^\leftarrow(1 - p)\right] \geq P\left[X <\sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(1 - p)]^\alpha}\right].$$
The monotonicity of probability measures entails that the above inequality would be true if
$$\frac{1}{p}F^\leftarrow(p) - \frac{1 - p}{p}F^\leftarrow(1 - p) \geq \sqrt[\alpha]{\frac{1}{p}[F^\leftarrow(p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(1 - p)]^\alpha},\,\, i.e.$$
$$\left[\frac{1}{p}F^\leftarrow(p) - \frac{1 - p}{p}F^\leftarrow(1 - p)\right]^\alpha \geq \frac{1}{p}[F^\leftarrow(p)]^\alpha - \frac{1 - p}{p}[F^\leftarrow(1 - p)]^\alpha, \,\, that \,\, is$$
\begin{equation}\label{aux3}
\left[\frac{1}{p} - \frac{1 - p}{p}\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}\right]^\alpha \geq \frac{1}{p}- \frac{1 - p}{p}\left[\frac{F^\leftarrow(1 - p)}{F^\leftarrow(p)}\right]^\alpha.
\end{equation}
Again denote by $z:=\frac{F^\leftarrow(1-p)}{F^\leftarrow(p)} \geq 1$, and consider the function
$$t(z): = \frac{1}{p} - \frac{1 - p}{p}z^\alpha - \left[\frac{1}{p} - \frac{1 - p}{p}z\right]^\alpha.$$
Given $\alpha > 1$, it is decreasing for $z \geq 1$ and $t(1) = 0$. Therefore $t(z) \leq 0$, for $z \geq 1$. This entails (\ref{aux3}) and completes the proof of (\ref{Cor2ofe2}). \hfill Q.A.D.
{\bf Proof of Theorem \ref{thm:ThmGeneralAsumptoticNormalityFences}:} Let us fix $s \in N$. From the theorem about the joint asymptotic normality of the order statistics, because the limit exists we have that for any subsequence $\{n: n = (s+1)k-1, s \in N\}$,
$$\sqrt{(s+1)k-1}\left(\begin{array}{c}
X_{(k, (s+1)k-1)} - F^\leftarrow(\frac{1}{s+1}) \\
X_{(ki, (s+1)k-1)} - F^\leftarrow(\frac{i}{s+1}) \\
\end{array}
\right) \stackrel{d}{\to} N\left[\left(\begin{array}{c}
0\\
0 \\
\end{array}\right); D\right], \quad k \to \infty,$$
where the asymptotic covariance matrix of this bivariate distribution is
$$D = \frac{1}{(s+1)^2}\left(\begin{array}{cc}
\frac{s}{f^2\left[F^\leftarrow(\frac{1}{s+1})\right]} & \frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} & \frac{s}{f^2\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\end{array}\right)$$
and the asymptotic correlation between these two order statistics is $\frac{1}{i}$.
1.) Consider the function $g(x, y) = (s+1)x - sy$. For $x > 0$ and $y > 0$ it is continuously differentiable.
The asymptotic mean is
\begin{eqnarray*}
\lim_{k \to \infty} E L_n\left(X, \frac{1}{s+1}\right) &=& g\left[F^\leftarrow(\frac{1}{s+1}), F^\leftarrow(\frac{s}{s+1})\right]\\
&=& (s+1)F^\leftarrow(\frac{1}{s+1}) - i F^\leftarrow(\frac{s}{s+1}) = L\left(X,\frac{1}{s+1}\right).
\end{eqnarray*}
The Jacobian of the transformation is
$$I_L : = \left[\frac{\partial g(x, y)}{\partial x}, \frac{\partial g(x, y)}{\partial y}\right] = \left(s+1, -s\right).$$
Now we apply the Multivariate Delta method Sobel (1982) \cite{MultivariateDeltaMethod}, and obtain that the asymptotic variance of $L_n\left(X, \frac{1}{s+1}\right)$ is
\begin{eqnarray*}
V_L : &=& I_L \times D \times I_L' = \left(s+1, -s\right)\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&\times& \frac{1}{(s+1)^2}\left(\begin{array}{cc}
\frac{s}{f^2\left[F^\leftarrow(\frac{1}{s+1})\right]} & \frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} & \frac{i}{f^2\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\end{array}\right) \\
&\times& \left(
\begin{array}{c}
s+1 \\
-s \\
\end{array}
\right)\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&=& \frac{1}{(s+1)^2} \\
&\times& \left[\left(s+1, -s\right)\left(\begin{array}{cc}
\frac{i}{f^2(x)} & \frac{1}{f(x)f(y)} \\
\frac{1}{f(x)f(y)} & \frac{i}{f^2(y)} \\
\end{array}\right) \left(
\begin{array}{c}
s+1 \\
-s\\
\end{array}
\right)\right]\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&=& \frac{1}{(s+1)^2} \left[\frac{s(s+1)^2}{f^2(x)} - \frac{2s(s+1)}{f(x)f(y)} + \frac{s^3}{f^2(y)}\right] \left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&=& \frac{s}{(s+1)^2} \left[\frac{(s+1)^2}{c_{F,s}^2} - \frac{2(s+1)}{c_{F,s}d_{F,s}} + \frac{s^2}{d_{F,s}^2}\right].
\end{eqnarray*}
2.) Analogously, because of
\begin{eqnarray*}
R_{n}\left(F, \frac{1}{s + 1}\right) &=& (s + 1)X_{(ks, (s + 1)k-1)} - i X_{(k, (s + 1)k-1)}, \,\, and \\
R\left(X, \frac{1}{s+1}\right) &=& (s + 1)F^\leftarrow\left(\frac{s}{s + 1}\right) + s F^\leftarrow\left(\frac{1}{s + 1}\right),
\end{eqnarray*}
we consider function $g_R(x, y) = -sx + (s + 1)y$.
For $x > 0$ and $y > 0$ it is continuously differentiable.
The asymptotic mean is
\begin{eqnarray*}
\lim_{k \to \infty} E R_{n}\left(F, \frac{1}{s + 1}\right) &=& g_R\left[F^\leftarrow(\frac{1}{s+1}), F^\leftarrow(\frac{s}{s+1})\right]\\
&=& -sF^\leftarrow(\frac{1}{s+1}) + (s + 1)F^\leftarrow(\frac{s}{s+1}) = R\left(X, \frac{1}{s+1}\right).
\end{eqnarray*}
In order to obtain the asymptotic variance of $R_n\left(X, \frac{1}{s+1}\right)$ we calculate the Jacobian of the transformation. It is
$$I_R : = \left[\frac{\partial g_R(x, y)}{\partial x}, \frac{\partial g_R(x, y)}{\partial y}\right] = \left(s+1, -s\right).$$
Now we apply the Multivariate Delta method (see e.g. Sobel (1982) \cite{MultivariateDeltaMethod}), calculate $I_R \times D \times I_R'$ and obtain the asymptotic variance of $R_{n}\left(F, \frac{1}{s + 1}\right)$ is
$$V_R := I_R \times D \times I_R' = \frac{s}{(s+1)^2} \left[\frac{s^2}{c_{F,s}^2} - \frac{2(s+1)}{c_{F,s}d_{F,s}} + \frac{(s + 1)^2}{d_{F,s}^2}\right].$$
\hfill Q.A.D.
|
2,869,038,154,839 | arxiv | \section{Introduction}
Partition statistics are often defined with an eye toward proving a congruence property. An application of this principal can be found in the proofs of the famous congruence results of Ramanujan \textbf{\cite{raman}}, which were eventually established using the rank and crank statistics \textbf{\cite{crank,rankproof}}. On the other hand, a partition statistic may present itself without prior appeal to an anticipated congruence property. The recent index theory of seaweed algebras \textbf{\cite{Coll3, Coll1, Coll2, dk}} provides just such an instance. Indeed, seaweed subalgebras of $\mathfrak{sl}(n)$ -- or simply, \textit{seaweeds} -- which are naturally defined in terms of two compositions of a single integer $n$, provide a sort of partition statistic generator. One begins with a pair $(\lambda, \mu)$ of partitions of $n$. Since partitions are compositions, we can use $(\lambda, \mu)$ to define a seaweed subalgebra of $\mathfrak{sl}(n)$, whose index will be taken to be the definition of the index of the partition pair $(\lambda, \mu)$.
If we let $w(\lambda)$ denote the weight of $\lambda$, then
there are two choices of $\mu$ naturally associated with a given $\lambda$, namely, the trivial partition
where $\mu= w(\lambda$),
and it's conjugate $\mu^C=1^{w(\lambda)}$. Since the values of the index of $(\lambda, w(\lambda))$ and $(\lambda,
1^{w(\lambda)})$ are reliant only on $\lambda$, the index of these partition pairs may be regarded as partition statistics on $\lambda$ alone. Of course, the efficacy of such statistics must be adjudged according to their utility. However, we find that in each of these extremal cases, the index statistic connects to well-established investigations.
In the first case, $(\lambda, 1^{w(\lambda)})$, we find a connection to classical partition theory by establishing that the sequence $\{c^i_n\}_{n=1}^{\infty}$ defined by
$$
c^i_n=|\{\lambda\in\mathcal{P}(n) ~:~{\rm ind }_{1^{w(\lambda)}}(\lambda)=n-i\}|,
$$ for each fixed $i$ is eventually constant -- converging to the number of partitions of $i-1$ into parts of two kinds. See Theorem~\ref{thm:ones}.
In the second case, we consider seaweeds defined by a pair of compositions $(\lambda, w(\lambda))$. The enumeration of these composition pairs, when the index is zero, is of concern to Lie theorists.\footnote{Frobenius algebras form a distinguished class and have been extensively studied from the point of view of invariant theory \textbf{\cite{Ooms}} and are of special interest in deformation and quantum group theory resulting from their connection with the classical Yang-Baxter equation (see \textbf{\cite{G1}} and \textbf{\cite{G2}}).} Recent efforts to enumerate pairs of compositions that define a Frobenius (index zero) seaweed have concentrated on limiting the number of parts in the compositions. For example, Duflo (after the fashion of Coll et al \textbf{\cite{Collar}}), uses certain index-preserving operators on the set of compositions corresponding to a Frobenius seaweed subalgebra of $\mathfrak{sl}(n)$ to show that if $t$ is the number of parts in the defining compositions, then the
number of such compositions is a rational polynomial of degree $\left[\frac{t}{2}\right]$ evaluated at $n$. See \textbf{\cite{df}}, Theorem 1.1 (b). Dufflo's result is existential in nature. However, if compositions are restricted to partitions and a modest limit is placed on the size of the parts -- rather than the number of parts -- the number of such compositions corresponding to a Frobenius seaweed subalgebra of $\mathfrak{sl}(n)$ becomes a periodic function of $n$. See Theorem~\ref{thm:period}.
\bigskip
The organization of the paper is as follows. In Section 2 we develop the definitions and notation for integer partitions and seaweeds.
In Section 3 we use the index theory of seaweeds to define the index of a partition and use this new definition to connect to some well-known classical investigations. We conclude with some open questions.
\section{Preliminaries}\label{sec:prelim}
In Section 2.1 we review standard combinatorial notation. In Section 2.2 we detail the recent index theory of seaweed algebras. Throughout this article, we tacitly assume that all Lie algebras are over the complex numbers.
\subsection{Integer partitions}\label{Intger partitions}
We follow the notation of Andrews \textbf{\cite{andrews}} and adopt the following conventions.
\begin{definition}
A \textit{partition} $\lambda$ of a positive integer $n$ is a finite non-increasing sequence of positive integers $\lambda_1, \lambda_2, \dots, \lambda_m$ such that $n=\sum_{i=1}^{m}\lambda_i$. The $\lambda_i$ are called the \textit{parts} of the partition and $w(\lambda)=n$ is the \textit{weight} of the partition.
\end{definition}
We will often employ the \textit{vector notation} for the partition
$\lambda=(\lambda_1, \lambda_2, \dots, \lambda_m)$. It will sometimes be useful to use a \textit{frequency notation} that makes explicit the number of times a particular integer occurs as a part of a partition. So, if $\lambda=(\lambda_1, \lambda_2, \dots, \lambda_m)$, we alternatively write
$$
\lambda = (1^{f_1}2^{f_2}3^{f_3}\cdots),
$$
\noindent
where exactly $f_i$ of the $\lambda_j$ are equal to $i$.
A graphical representation of a partition, called a \textit{Ferrers diagram}, is helpful to develop the notion of the conjugate of a partition. More formally, the Ferrers diagram of a partition $\lambda=(\lambda_1,\hdots,\lambda_n)$ is a coordinatized set of unit squares in the plane such that the lower left corner of each square will have integer coordinates $(i,j)$ such that $$0\ge i\ge -n+1,0\le j\le \lambda_{|i|+1}-1.$$
The Ferrers diagram of the partition $(4,2,1)$ is illustrated in the left-hand side of Figure 1. The \textit{conjugate} of a partition $\mu$ is the partition $\mu^C$ resulting from exchanging the rows and columns in the Ferrers diagram associated to $\mu$.
\begin{example} The Ferrers diagram of the partition $\lambda= (4,2,1)$
and it's conjugate $\lambda^C= (3,2,1,1)$.
\begin{figure}[H]
$$\begin{tikzpicture}
\node at (0,0) {\Tableau{{ , , , }, { , }, { }}};
\node at (3, 0) {\Tableau{{ , , }, { , }, { }, { }}};
\end{tikzpicture}$$
\caption{Ferrers Diagram of $(4,2,1)$ and $(3,2,1,1)$}\label{fig:421YD}
\end{figure}
\end{example}
\subsection{Seaweed Algebras}\label{sect:swprelim}
In this section, we introduce seaweed algebras in type-A.\footnote{In \textbf{\cite{Panyushev1}}, Panyushev extended the Lie theoretic definition of seaweed algebras to the reductive case. If $\mathfrak{p}$ and $\mathfrak{p}$ are parabolic subalgebras of a reductive Lie algebra $\mathfrak{g}$ such that $\mathfrak{p}+\mathfrak{p'}=\mathfrak{g}$, then $\mathfrak{p}\cap\mathfrak{p'}$ is called a \textit{seaweed subalgebra o}f $\mathfrak{g}$ or simply $seaweed$ when $\mathfrak{g}$ is understood. For this reason, Joseph has elsewhere
\textbf{\cite{Joseph}} called seaweed algebras, \textit{biparabolic}. One can show that type-C and type-B seaweeds, in their standard representations, can be parametrized by a pair of partial compositions of $n$. See \textbf{\cite{CHM}}.} These are seaweed subalgebras of $
\mathfrak{sl}(n)$ -- the set of all $n\times n$ matrices of trace zero. As we will see, such seaweed algebras are naturally defined in terms of two compositions of the positive integer $n$. Recall that a \textit{composition} of $n$ is an unordered partition, which we will denote by $\lambda_1|\lambda_2|\cdots| \lambda_n$ to distinguish it from the ordered case in Definition 1, where there is an order relation on the $\lambda_i's$.
\begin{definition}
If $V$ is an $n$-dimensional vector space with a basis
$\{e_1,\dots, e_n \}$, let $a_1|\dots|a_m$ and $b_1|\dots|b_l$ be two compositions of $n$ and consider the flags
$$
\{0\} \subset V_1 \subset \cdots \subset V_{m-1} \subset V_m =V~~~\text{and}~~~ V=W_0\supset W_1\supset \cdots \supset W_t=\{0\},
$$
where $V_i=\text{span}\{e_1,\dots, e_{a_1+\cdots +a_i}\}$ and $W_j=\text{span}\{e_{b_1+\cdots +b_j+1},\dots, e_n\}$.
\end{definition}
The subalgebra of $\mathfrak{sl}(n)$ preserving these flags is called a \textit{seaweed Lie algebra}, or simply \textit{seaweed}, and is denoted by the symbol
$\displaystyle \frac{a_1|\cdots|a_m}{b_1|\cdots|b_t}$, which we refer to as the \textit{type} of the seaweed. If $b_1=n$, the seaweed is called \textit{maximal parabolic}.
\bigskip
\noindent
\textit{Remark:} The preservation of flags in Definition 2 insures that seaweeds are closed under matrix multiplication, and therefore define an associative algebra, hence also a Lie algebra under the commutator bracket.
\bigskip
The evocative ``seaweed'' is descriptive of the shape of the algebra when exhibited in matrix form.
For example, the seaweed algebra $\frac{2|4}{1|2|3}$ consists of traceless matrices of the form depicted on the left side of Figure \ref{fig:seaweed}, where * indicates the possible non-zero entries from the ground field, which we assume is the complex numbers.
\begin{figure}[H]
$$\begin{tikzpicture}[scale=0.75]
\draw (0,0) -- (0,6);
\draw (0,6) -- (6,6);
\draw (6,6) -- (6,0);
\draw (6,0) -- (0,0);
\draw [line width=3](0,6) -- (0,4);
\draw [line width=3](0,4) -- (2,4);
\draw [line width=3](2,4) -- (2,0);
\draw [line width=3](2,0) -- (6,0);
\draw [line width=3](0,6) -- (1,6);
\draw [line width=3](1,6) -- (1,5);
\draw [line width=3](1,5) -- (3,5);
\draw [line width=3](3,5) -- (3,3);
\draw [line width=3](3,3) -- (6,3);
\draw [line width=3](6,3) -- (6,0);
\draw [dotted] (0,6) -- (6,0);
\node at (.5,5.4) {{\LARGE *}};
\node at (.5,4.4) {{\LARGE *}};
\node at (1.5,4.4) {{\LARGE *}};
\node at (2.5,4.4) {{\LARGE *}};
\node at (2.5,3.4) {{\LARGE *}};
\node at (2.5,2.4) {{\LARGE *}};
\node at (2.5,1.4) {{\LARGE *}};
\node at (2.5,0.4) {{\LARGE *}};
\node at (3.5,2.4) {{\LARGE *}};
\node at (3.5,1.4) {{\LARGE *}};
\node at (3.5,0.4) {{\LARGE *}};
\node at (4.5,2.4) {{\LARGE *}};
\node at (4.5,1.4) {{\LARGE *}};
\node at (5.5,2.4) {{\LARGE *}};
\node at (5.5,1.4) {{\LARGE *}};
\node at (4.5,0.4) {{\LARGE *}};
\node at (5.5,0.4) {{\LARGE *}};
\node at (.5,6.4) {1};
\node at (2,5.4) {2};
\node at (4.5,3.4) {3};
\node at (-0.5,4.9) {2};
\node at (1.5,1.9) {4};
\end{tikzpicture}\hspace{1.5cm}\begin{tikzpicture}[scale=1.3]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (0,0) {};
\node [circle, fill, inner sep=2pt][label=left:$v_1$] (1) at (0,1.8) {};
\node [circle, fill, inner sep=2pt][label=left:$v_2$] (2) at (1,1.8) {};
\node [circle, fill, inner sep=2pt][label=left:$v_3$] (3) at (2,1.8) {};
\node [circle, fill, inner sep=2pt][label=left:$v_4$] (4) at (3,1.8) {};
\node [circle, fill, inner sep=2pt][label=left:$v_5$] (5) at (4,1.8) {};
\node [circle, fill, inner sep=2pt][label=left:$v_6$] (6) at (5,1.8) {};
\draw (1) to[bend left=50] (2);
\draw (3) to[bend left=50] (6);
\draw (4) to[bend left=50] (5);
\draw (2) to[bend right=50] (3);
\draw (4) to[bend right=50] (6);
\end{tikzpicture}$$
\caption{A seaweed of type $\frac{2|4}{1|2|3}$ and its associated meander}
\label{fig:seaweed}
\end{figure}
The \textit{index} of a Lie algebra was introduced by Dixmier \textbf{\cite{Dix}}. Formally, the index of a Lie algebra $\mathfrak{g}$ is defined by
\[{\rm ind } (\mathfrak{g})=\min_{f\in \mathfrak{g^*}} \dim (\ker (B_f)),\]
\noindent where $f$ is a linear form on $\mathfrak{g}$ and $B_f$ is the associated skew-symmetric \textit{Kirillov form} defined by $B_f(x,y)=f([x,y])$ for all $x,y\in\mathfrak{g}$. The index is an important algebraic invariant of the Lie algebra -- though notoriously difficult to compute. However, in \textbf{\cite{dk}}, Dergachev and A. Kirillov developed a combinatorial algorithm to compute the index of a seaweed subalgebra of $\mathfrak{sl}(n)$ by counting the number of connected components of a certain planar graph, called a meander, associated to the seaweed. To construct a meander, let
$\frac{ a_1|\cdots|a_m}{b_1|\cdots|b_t}$ be a seaweed. Now
label the $n$ vertices of our meander as $v_1, v_2, \dots , v_n$ from left to right along a horizontal line. We then place edges above the horizontal line, called top edges, according to $a_1+\hdots+ a_m$ as follows.
Partition the set of vertices into a set partition by grouping together the first $a_1$ vertices, then the next $a_2$ vertices, and so on, lastly grouping together the final $a_m$ vertices. We call each set within a set partition a \textit{block}. For each block in the set partition determined by $a_1+\hdots + a_m$, add an edge from the first vertex of the block to the last vertex of the block, then add an edge between the second vertex of the block and the second to last vertex of the block, and so on within each block. More explicitly, given vertices $v_j,v_k$ in a block of
size $a_i$, there is an edge between them if and only if
$j+k=2(a_1+a_2+\dots+a_{i-1})+a_i+1$.
In the same way, place bottom edges below the horizontal line of vertices according to the blocks in the partition determined by $b_1+\hdots + b_t$. See the right side of Figure \ref{fig:seaweed}.
Every meander consists of a disjoint union of cycles and paths. The main result of \textbf{\cite{dk}} is that the index of a seaweed can be computed by counting the number and type of these components in it's associated meander.
\begin{theorem}\label{thm:dk} \rm{(Dergachev and A. Kirillov, \textbf{\cite{dk}})}
If $\mathfrak{p}$ is a seaweed subalgebra of $\mathfrak{sl}(n)$, then
$${\rm ind } (\mathfrak{p}) =2C + P -1,$$
where $C$ is the number of cycles and $P$ is the number of paths in the associated meander.
\end{theorem}
\noindent
\noindent
\begin{example}
In the example of Figure 1, the meander associated to the seaweed $\frac{2|4}{1|2|3}$ has no cycles and consists of a single path -- so, has index zero, hence is Frobenius.
\end{example}
\bigskip
While Theorem \ref{thm:dk} is an elegant combinatorial result it is difficult to apply in practice. However, Coll et al in \textbf{\cite{Collar}} show that any meander can be contracted or ``wound-down" to the empty meander through a sequence of graph-theoretic moves, each of which is uniquely determined by the structure of the meander at the time of move application. There are five such moves, only one of which affects the component structure of the meander graph and is therefore the only move capable of modifying the index of the meander. Using these winding-down moves the authors in \textbf{\cite{Collar}} established the following index formulas which allow us to ascertain the index directly from the block sizes of the flags that define the seaweed.\footnote{A recent result by Karnauhova and Liebscher \textbf{\cite{Kar}} has established, in particular, that the formulas in Theorems \ref{2parts} and \ref{3parts} are the only nontrivial linear ones that are available in the maximal parabolic case. More specifically, If $m\geq 4$ and $\mathfrak{p}$ is a seaweed of type $\dfrac{a_1|a_2|\cdots|a_m}{n}$, then there do not exist homogeneous polynomials $f_1,f_2 \in \mathbb{Z}[x_1,...,x_m]$, of arbitrary degree, such that the index of $\mathfrak{p}$ is given by $\gcd (f_1(a_1,...,a_m),f_2(a_1,...,a_m))$.}
\begin{theorem}[Theorem 7, \textbf{\cite{Coll2}}]\label{2parts}
A seaweed of type $\dfrac{a|b}{n}$ has index $\gcd (a,b)-1.$
\end{theorem}
\begin{theorem}[Theorem 8,\textbf{ \cite{Coll2}}]\label{3parts}
A seaweed of type $\dfrac{a|b|c}{n}$, or of type $\dfrac{a|b}{c|n-c}$, has index given by $\gcd(a+b,b+c)-1.$
\end{theorem}
Since we will need the explicit winding-down moves in the proof of Theorem~\ref{thm:period} we review the winding-down process.
\begin{lemma}[Winding-down]\label{lem:wd} Given a meander $M$ of type $\dfrac{a_1|a_2|...|a_m}{b_1|b_2|...|b_t}$, create a meander $M'$
by exactly one of the following moves. For all moves except the
Component Elimination move, $M$ and $M'$ have the same index.
\begin{enumerate}
\item {\bf Vertical Flip $(F_v)$:} If $a_1<b_1$, then
$M'$ has type
$\displaystyle\frac{b_1|b_2|...|b_t}{a_1|a_2|...|a_m}$.
\item {\bf Component Elimination $(C(c))$:}
If $a_1=b_1=c$, then $M'$ has type
$\displaystyle\frac{a_2|a_3|...|a_m}{b_2|b_3|...|b_t}.$
\item {\bf Rotation Contraction $(R)$:} If $b_1<a_1<2b_1$, then
$M'$ has type
$\displaystyle \frac{b_1|a_2|a_3|...|a_m}{(2b_1-a_1)|b_2|...|b_t}$.
\item {\bf Block Elimination $(B)$:} If $a_1=2b_1$, then
$M'$ has type
$\displaystyle\frac{b_1|a_2|..|a_m}{b_2|b_3|...|b_t}$.
\item {\bf Pure Contraction $(P)$:} If $a_1>2b_1$, then
$M'$ has type
$\displaystyle
\frac{(a_1-2b_1)|b_1|a_2|a_3|...|a_m}{b_2|b_3|...|b_t}$.
\end{enumerate}
\end{lemma}
\noindent
\begin{example} In this example, the seaweed $\frac{17|3}{10|4|6}$ is wound-down to the empty meander using the moves detailed in Lemma~\ref{lem:wd}.
\begin{figure}[H]
$$\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (10.5,-2){$\frac{17|3}{10|4|6}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\node [circle, fill, inner sep=2pt] (4) at (4,0){};
\node [circle, fill, inner sep=2pt] (5) at (5,0){};
\node [circle, fill, inner sep=2pt] (6) at (6,0){};
\node [circle, fill, inner sep=2pt] (7) at (7,0){};
\node [circle, fill, inner sep=2pt] (8) at (8,0){};
\node [circle, fill, inner sep=2pt] (9) at (9,0){};
\node [circle, fill, inner sep=2pt] (10) at (10,0){};
\node [circle, fill, inner sep=2pt] (11) at (11,0){};
\node [circle, fill, inner sep=2pt] (12) at (12,0){};
\node [circle, fill, inner sep=2pt] (13) at (13,0){};
\node [circle, fill, inner sep=2pt] (14) at (14,0){};
\node [circle, fill, inner sep=2pt] (15) at (15,0){};
\node [circle, fill, inner sep=2pt] (16) at (16,0){};
\node [circle, fill, inner sep=2pt] (17) at (17,0){};
\node [circle, fill, inner sep=2pt] (18) at (18,0){};
\node [circle, fill, inner sep=2pt] (19) at (19,0){};
\node [circle, fill, inner sep=2pt] (20) at (20,0){};
\draw (1) to[bend left] (17);
\draw (2) to[bend left] (16);
\draw (3) to[bend left] (15);
\draw (4) to[bend left] (14);
\draw (5) to[bend left] (13);
\draw (6) to[bend left] (12);
\draw (7) to[bend left] (11);
\draw (8) to[bend left] (10);
\draw (18) to[bend left] (20);
\draw (1) to[bend right] (10);
\draw (2) to[bend right] (9);
\draw (3) to[bend right] (8);
\draw (4) to[bend right] (7);
\draw (5) to[bend right] (6);
\draw (11) to[bend right] (14);
\draw (12) to[bend right] (13);
\draw (15) to[bend right] (20);
\draw (16) to[bend right] (19);
\draw (17) to[bend right] (18);
\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (-2,0){$\overset{R}{\mapsto}$};
\node at (6,-2){$\frac{10|3}{3|4|6}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\node [circle, fill, inner sep=2pt] (4) at (4,0){};
\node [circle, fill, inner sep=2pt] (5) at (5,0){};
\node [circle, fill, inner sep=2pt] (6) at (6,0){};
\node [circle, fill, inner sep=2pt] (7) at (7,0){};
\node [circle, fill, inner sep=2pt] (8) at (8,0){};
\node [circle, fill, inner sep=2pt] (9) at (9,0){};
\node [circle, fill, inner sep=2pt] (10) at (10,0){};
\node [circle, fill, inner sep=2pt] (11) at (11,0){};
\node [circle, fill, inner sep=2pt] (12) at (12,0){};
\node [circle, fill, inner sep=2pt] (13) at (13,0){};
\draw (1) to[bend left] (10);
\draw (2) to[bend left] (9);
\draw (3) to[bend left](8);
\draw (4) to[bend left](7);
\draw (5) to[bend left](6);
\draw (11) to[bend left](13);
\draw (1) to[bend right](3);
\draw (4) to[bend right](7);
\draw (5) to[bend right](6);
\draw (8) to[bend right](13);
\draw (9) to[bend right](12);
\draw (10) to[bend right](11);
\end{tikzpicture}$$
$$\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (-1,0){$\overset{P}{\mapsto}$};
\node at (6,-2){$\frac{4|3|3}{4|6}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\node [circle, fill, inner sep=2pt] (4) at (4,0){};
\node [circle, fill, inner sep=2pt] (5) at (5,0){};
\node [circle, fill, inner sep=2pt] (6) at (6,0){};
\node [circle, fill, inner sep=2pt] (7) at (7,0){};
\node [circle, fill, inner sep=2pt] (8) at (8,0){};
\node [circle, fill, inner sep=2pt] (9) at (9,0){};
\node [circle, fill, inner sep=2pt] (10) at (10,0){};
\draw (1) to[bend left] (4);
\draw (2) to[bend left](3);
\draw (5) to[bend left](7);
\draw (8) to[bend left](10);
\draw (1) to[bend right](4);
\draw (2) to[bend right](3);
\draw (5) to[bend right](10);
\draw (6) to[bend right](9);
\draw (7) to[bend right](8);
\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (-1,0){$\overset{C(4)}{\mapsto}$};
\node at (3.5,-2){$\frac{3|3}{6}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\node [circle, fill, inner sep=2pt] (4) at (4,0){};
\node [circle, fill, inner sep=2pt] (5) at (5,0){};
\node [circle, fill, inner sep=2pt] (6) at (6,0){};
\draw (1) to[bend left](3);
\draw (4) to[bend left](6);
\draw (1) to[bend right](6);
\draw (2) to[bend right](5);
\draw (3) to[bend right](4);
\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (-1,0){$\overset{F_v}{\mapsto}$};
\node at (3.5,-2){$\frac{6}{3|3}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\node [circle, fill, inner sep=2pt] (4) at (4,0){};
\node [circle, fill, inner sep=2pt] (5) at (5,0){};
\node [circle, fill, inner sep=2pt] (6) at (6,0){};
\draw (1) to[bend right](3);
\draw (4) to[bend right](6);
\draw (1) to[bend left](6);
\draw (2) to[bend left](5);
\draw (3) to[bend left](4);
\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}[scale=.4]
\def\node [circle, fill, inner sep=2pt]{\node [circle, fill, inner sep=2pt]}
\node at (-1,0){$\overset{B}{\mapsto}$};
\node at (2,-2){$\frac{3}{3}$};
\node [circle, fill, inner sep=2pt] (1) at (1,0){};
\node [circle, fill, inner sep=2pt] (2) at (2,0){};
\node [circle, fill, inner sep=2pt] (3) at (3,0){};
\draw (1) to[bend right](3);
\draw (1) to[bend left](3);
\end{tikzpicture}
\hspace{1em}
\begin{tikzpicture}[scale=.4]
\node at (-1,0){$\overset{C(3)}{\mapsto}$};
\node at (1,-2){$\emptyset$};
\node at (1,0){$\emptyset$};
\end{tikzpicture}$$
\caption{
Winding down the meander $\dfrac{17|3}{10|4|6}$}
\label{fig:signature}
\end{figure}
\end{example}
\noindent
In what follows, it is helpful to add a sixth
index preserving transformation, $F_h$, called a \textit{horizontal flip} which takes $M$ to $\dfrac{a_m|...|a_2|a_1}{b_m|...|b_2|b_1}$.
\section{The index of a partition}\label{sect:results}
Let $\mathcal{P}(n)$ be the set of integer partitions of a positive integer $n$ and let $\lambda, \mu \in \mathcal{P}(n)$ with
$\lambda = (\lambda_1, \lambda_2, \dots, \lambda_m)$ and
$\mu = (\mu_1, \mu_2, \dots, \mu_t).$
These compositions can be used to define the seaweed
$$\mathfrak{p}(\lambda, \mu)=\frac{\lambda_1| \lambda_2| \dots| \lambda_m}{\mu_1| \mu_2| \dots| \mu_t}.$$
We can then define the \textit{index} of the pair $(\lambda, \mu)$ to be the index of $\mathfrak{p}(\lambda, \mu)$, and we write ind$_{\mu}(\lambda)$. Given $\lambda$ as above, there are two choices for $\mu$ naturally associated with $\lambda$, namely, $w(\lambda)$ and its conjugate $1^{w(\lambda)}$. These yield, respectively, two partition statistics on $\lambda$ defined as
follows:
\begin{eqnarray}\label{index}
\textrm{ind}_{w(\lambda)}(\lambda) =\textrm{ind}\left(\frac{\lambda_1|\hdots|\lambda_m}{w(\lambda)}\right)~~~\textrm{and}~~~
\textrm{ind}_{1^{w(\lambda)}}(\lambda)=\textrm{ind}\left(\frac{\lambda_1|\hdots|\lambda_m}{1|\hdots|1}\right).
\end{eqnarray}
In the first seaweed, the bottom composition is defined by the trivial partition, yielding a maximal parabolic seaweed. In the second case,
the bottom composition consists of $w(\lambda)$ $1's$.
\begin{example} Let $\lambda=(3,2,1)$ in (\ref{index}). An application of Theorem 3 now yields
$$
\textrm{ind}_6(\lambda) =\textrm{ind}\left(\frac{3|2|1}{6}\right)=0~~~\textrm{and}~~~ \textrm{ind}_{1^6}(\lambda)=\textrm{ind}\left(\frac{3|2|1}{1|1|1|1|1|1}\right)=3.
$$
\end{example}
\subsection{All 1's}\label{sect:1s}
In this section we investigate, for fixed $i$ and varying $n$, the sequence of values defined by the number of partitions $\lambda\in\mathcal{P}(n)$ such that ${\rm ind }_{1^{w(\lambda)}}(\lambda)=n-i$. We find that for each $i$, if $c^i_n=|\{\lambda\in\mathcal{P}(n)|{\rm ind }_{1^{w(\lambda)}}(\lambda)=n-i\}|$, then $\{c_n^i\}_{n=i}^{\infty}$ is eventually constant, converging to a well-known classical value $c^i$ (see Theorem~\ref{thm:ones}). The following Table 1, illustrates $c_n^i$ for small values of $n$ and $i$.
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$\boldsymbol{n}\backslash\boldsymbol{i}$ & \textbf{0} & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} & \textbf{7} & \textbf{8} & \textbf{9} \\ \hline
\textbf{1} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
\textbf{2} & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
\textbf{3} & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
\textbf{4} & 0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
\textbf{5} & 0 & 0 & 4 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ \hline
\textbf{6} & 0 & 0 & 3 & 5 & 2 & 1 & 0 & 0 & 0 & 0 \\ \hline
\textbf{7} & 0 & 0 & 0 & 7 & 5 & 2 & 1 & 0 & 0 & 0 \\ \hline
\textbf{8} & 0 & 0 & 0 & 5 & 9 & 5 & 2 & 1 & 0 & 0 \\ \hline
\textbf{9} & 0 & 0 & 0 & 0 & 12 & 10 & 5 & 2 & 1 & 0 \\ \hline
\textbf{10} & 0 & 0 & 0 & 0 & 7 & 17 & 10 & 5 & 2 & 1 \\ \hline
\end{tabular}
\caption{Number of $\lambda\in\mathcal{P}(n)$ with ${\rm ind }_{1^{w(\lambda)}}(\lambda)=i$.}\label{tab:1s}
\end{table}
By coloring partitions, we can better understand the $c^i$'s.
We will use two colors (red and blue), to color the parts of a given partition. When enumerating colored partitions, we will assume that two partitions which are identical, save for their coloring, will be considered different partitions. So, for example the partition of the integer 2 given by $(\bl{1},\bl{1})$ is different from the partition of the integer 2 given by $(\bl{1},\rn{1})$. We also tacitly assume that in a given colored partition all blue parts of a given size precede all red parts of the same size. See Example 9.
\bigskip
\noindent
\textit{Remark: } In the classical literature such (two)-colored partitions are called partitions into parts of two \textit{kinds}. Partitions into two kinds can be found in Guptas' (\textbf{\cite{gupta}},1958) and have recently been connected to other objects as diverse as quandles \textbf{\cite{quandle2,quandle1}}.
\begin{example}
The (two)-colored partitions of 2 are: (\rn{2}), (\bl{2}), (\rn{1},\rn{1}), (\bl{1},\bl{1}), and (\bl{1},\rn{1}).
\end{example}
\noindent
The generating function for the number of (two)-colored partitions of $n$ is well-known and is equal to
$$\prod_{m\ge 1}\frac{1}{(1-x^m)^2}.$$
The following theorem connects the current exposition to classical partition theory.
\begin{theorem}\label{thm:ones}
$$\sum_{i\ge 1}c^ix^{i-1}=\prod_{m\ge 1}\frac{1}{(1-x^m)^2}.$$
\end{theorem}
\begin{proof}
The case $i=1$ is clear since, by Theorem~\ref{thm:dk}, the only $\lambda\in\mathcal{P}(n)$ with ${\rm ind }_{1^{w(\lambda)}}(\lambda)=n-1$ is $\lambda=1^n$. We show that for $i>1$ and $n\ge 3i-3$, there is a bijective correspondence between $\mathcal{M}(i,n)=\{\lambda\in\mathcal{P}(n)~: ~{\rm ind }_{1^{w(\lambda)}}(\lambda)=n-i\}$ and $\mathcal{P}^2(i-1)=\{\text{colored partitions of }i-1\}$. We do this in two steps.
First, let $\mathcal{M}(i-1)$
be the set of partitions $\mu=(\mu_1,\hdots,\mu_m)$ such that $\mu_m>1$ and the meander corresponding to $\mathfrak{p}(\mu,1^{w(\mu)})$ has $i-1$ arcs. Consider the map $\varphi$ which takes $\lambda=(\lambda_1,\hdots,\lambda_m)\in \mathcal{P}^2(i-1)$ to the partition $\varphi(\lambda)=(\mu_1,\hdots,\mu_m)$ defined by
\[\mu_i = \begin{cases}
2\lambda_i+1, & \lambda_i\text{ is blue}; \\
2\lambda_i, & \lambda_i\text{ is red.}
\end{cases}
\]
\noindent
By construction, $\varphi(\lambda)\in \mathcal{M}(i-1)$. Furthermore, $\varphi$ is invertible so $\varphi$ is a bijection between $\mathcal{P}^2(i-1)$ and $\mathcal{M}(i-1)$. Via this correspondence, it is easy to see that the largest partition of $\mathcal{M}(i-1)$ has weight $3i-3$.
By Theorem~\ref{thm:dk}, ${\rm ind }_{1^{w(\lambda)}}(\lambda)$ corresponds to $n$ minus the number of arcs in the meander minus 1. Thus, $\mathcal{M}(i,n)$ consists of partitions $\lambda\in\mathcal{P}(n)$ such that the meander corresponding to $\mathfrak{p}(\lambda,1^{w(\lambda)})$ has exactly $i-1$ arcs. Therefore, if $n\ge 3i-3$, then elements of $\lambda\in \mathcal{M}(i,n)$ can be mapped bijectively to elements of $\mathcal{M}(i-1)$ by the map $\psi$ which removes all parts equal to 1. The required bijection is given by $\psi \circ \varphi$. \end{proof}
\subsection{The maximal parabolic case}\label{sect:n}
As above, let $\lambda = (\lambda_1, \dots, \lambda_m)$ be an element of $\mathcal{P}(n)$.
In this section, we consider the seaweed defined by the pair of compositions $(\lambda, w(\lambda))$. In contrast to the previous section, here we investigate the number of partitions $\lambda$ such that ${\rm ind }_{w(\lambda)}(\lambda)=0$.
We naturally call such partitions, \textit{Frobenius partitions}. The main theorem of this section, Theorem~\ref{thm:period}, remarkably establishes that if $\lambda_i\leq 7$, for $i=1,\dots,m$, then the number of Frobenius partitions is a periodic function of $n$.
We begin with two Lemmas which will be helpful in the proof of Theorem~\ref{thm:period}.
\begin{lemma}\label{lem:sum}
Let $\mathfrak{g}=\frac{a_1|\hdots|a_m}{\sum_{i=1}^ma_i}$ be a seaweed algebra. If there exists $i<j-1$ such that $\sum_{l=1}^ia_l=\sum_{l=j}^ma_l$, then $\mathfrak{g}$ is not Frobenius.
\end{lemma}
\begin{proof}
Applying the winding moves $(F)$ followed by $i$ applications of $(P)$ to the meander corresponding to $\mathfrak{g}$ results in the meander corresponding to the seaweed algebra of type $\frac{b_1|a_i|\hdots|a_1}{a_{i+1}|\hdots|a_j|\hdots|a_m}$ where $b_1=\sum_{l=i+1}^{j-1}a_l>0$; but this meander consists of at least two components, one corresponding to $\frac{b_1}{a_{i+1}|\hdots|a_{j-1}}$, and the other $\frac{a_i|\hdots|a_1}{a_j|\hdots|a_m}$. Thus, by Theorem~\ref{thm:dk}, ${\rm ind }(\mathfrak{g})>0.$
\end{proof}
\begin{lemma}\label{lem:odd}
Let $\mathfrak{g}=\frac{a_1|\hdots|a_m}{\sum_{i=1}^ma_i}$ be a seaweed algebra. If there exists more than two odd $a_i$'s, then $\mathfrak{g}$ is not Frobenius.
\end{lemma}
\begin{proof}
Each odd $a_i$ contributes a vertex of degree 1 to the meander corresponding to $\mathfrak{g}$. Recall that each open path consists of exactly two vertices of degree 1 and no closed paths contains a vertex of degree 1. So, if there are more than two odd $a_i$'s, then the corresponding meander must contain more than one open path and thus, by Theorem~\ref{thm:dk}, ${\rm ind }(\mathfrak{g})>0.$
\end{proof}
Let $\mathcal{P}(n,d)$ be the set of Frobenius partitions $\lambda=(\lambda_1,\hdots,\lambda_m)\in\mathcal{P}(n)$ such that $\lambda_i\le d$ for $1\le i\le m$.
\begin{theorem}\label{thm:period}
If $d\in\{1,2,3,4\}$, then the values of $|\mathcal{P}(n,d)|$ are eventually periodic. More precisely,
\begin{itemize}
\item If $n\ge 3$, $|\mathcal{P}(n,1)|=0$
\item If $n\ge 5$, \[|\mathcal{P}(n,2)| = \begin{cases}
1, & n\text{ odd} \\
0, & n\text{ even}
\end{cases}
\]
\item If $n\ge 13$, \[|\mathcal{P}(n,3)| = \begin{cases}
2, & n\text{ odd} \\
0, & n\text{ even}
\end{cases}
\]
\item If $n\ge 17$, \[|\mathcal{P}(n,4)| = \begin{cases}
4, & n\equiv 1(\text{mod }4) \\
2, & n\equiv 2(\text{mod }4) \\
3, & n\equiv 3(\text{mod }4) \\
0, & n\equiv 0(\text{mod }4) \\
\end{cases}
\]
\end{itemize}
\end{theorem}
\begin{proof} The proof heuristic is described as follows. We consider the possible partitions for each $d\le 4$ -- except for those cases considered in Lemma~\ref{lem:sum} and Lemma~\ref{lem:odd} -- in reverse lexicographic ordering and determine which partitions are Frobenius.
$\mathbf{d=1}$: ${\rm ind }(\frac{1|\hdots|1}{n})=\lfloor\frac{n}{2}\rfloor>0$ for $n\ge 3$. Thus, for $n\ge 3$ we have $|\mathcal{P}(n,1)|=0$.
$\mathbf{d=2,n\ge 5}$: Using the results determined for the case $d=1$, we consider only partitions with $\lambda_1=2$. After applying Lemma~\ref{lem:sum} and Lemma~\ref{lem:odd}, the only partitions remaining are those of the form $(1^12^{f_2})$, which are Frobenius by Theorem 10 of \textbf{\cite{Coll2}}. Thus, there is exactly one such Frobenius partition if and only if the weight is odd.
$\mathbf{d=3,n\ge 13}$: As before, using the results for the cases $d=1$ and $d=2$, we can restrict our attention to partitions with $\lambda_1=3$. After applying Lemma~\ref{lem:sum} and Lemma~\ref{lem:odd}, the only partitions remaining are those of the form $(2^{f_2}3^1)$, which are Frobenius, once again, by Theorem 10 of \textbf{\cite{Coll2}}. Thus, as in the case $d=2$, there is exactly one such Frobenius partition if and only if the weight is odd.
$\mathbf{d=4,n\ge 17}$: Finally, using the results for $d=1,2,3$, we consider only partitions with $\lambda_1=4$. After applying Lemma~\ref{lem:sum} and Lemma~\ref{lem:odd}, we are left with seven cases to consider.
\begin{enumerate}
\item $(3^14^{f_4})$: Partitions of this form are Frobenius by Theorem 10 of \textbf{\cite{Coll2}}.
\item $(3^24^{f_4})$: Applying the sequence of moves $(F_v),(P),(F_h),(R),(R),(B),(F_v),(P),(F_h)$ to the corresponding meander results in the meander for a partition of the form $(1^1f^{f_4})$, which is found to be Frobenius in case 5 below.
\item $(2^14^{f_4})$: Applying the sequence of moves $(F_h),(F_v),(P),(F_v),(F_h),(B)$ inductively to the corresponding meander results in the meander for the seaweed algebra of type $\frac{2}{2}$, which has index 1.
\item $(2^13^14^{f_4})$: Applying the sequence of moves $(F_v),(P),(F_h),(B),(F_v),(R),(B),(F_h)$ to the corresponding meander results in the meander for a partition of the form $(1^14^{f_4})$, which is found to be Frobenius in case 5 below.
\item $(1^14^{f_4})$: Partitions of this form are Frobenius by Theorem 10 of \textbf{\cite{Coll2}}.
\item $(1^12^{f_2}4^{f_4}),f_2\ge 1$: Applying the sequence of moves $(F_v),(P),(F_h),(P)$ to the corresponding meander results in a seaweed algebra of type $\frac{2|1|n-8}{2|\hdots|2|4|\hdots|4}$, which splits into at least two components, $\frac{2}{2}$ and $\frac{1|n-8}{2|\hdots|2|4|\hdots|4}$, and is therefore not Frobenius by Theorem~\ref{thm:dk}.
\item $(1^24^{f_4})$: Applying the sequence of moves $(F_h)(F_v)(P)(P)(F_v)(F_h)(P)(B)(F_h)$ to the corresponding meander results in the meander for a partition of the form $(1^14^{f_4})$, which was found to be Frobenius in case 5 above.
\end{enumerate}
Thus, there are two Frobenius partitions with weight congruent to 1(mod 4); two with weight congruent to 2(mod 4); and one with weight congruent to 3(mod 4).
\end{proof}
\noindent
\textit{Remark:} Similar methods to those used in the proof of Theorem~\ref{thm:period} can be used to establish periodic behavior for $d\in\{5,6,7\}$. In the case of $d=5$, the period is of length 4 -- with values 7,3,5,3 -- while in the cases of $d\in\{6,7\}$, the period jumps to 14. See Example~\ref{ex:period}.
\begin{example}\label{ex:period}
The sequence of values of $|\mathcal{P}(n,d)|$ for $d\in\{5,6,7\}$, along with the value of $n$ at which $|\mathcal{P}(n,d)|$ becomes periodic.
$$d=5,n\ge 21:\text{ }7,3,5,3$$
$$d=6,n\ge 37:\text{ }14,5,9,3,11,5,11,3,12,5,8,3$$
$$d=7,n\ge 41:\text{ }19,9,18,7,19,9,17,7,20,9,17,7$$
\end{example}
\noindent
\textit{Remark:}
At $d=8$ the periodicity stops, which can be seen by considering $|\mathcal{P}(n,8)|$ for $n\equiv 1(\text{mod }8)$ where for $n=8m+1$ we have that ${\rm ind }_{w(\lambda)}(1^14^{2(m-k)}8^k)=0$.
\\*
\section{Conclusion}
Using the index theory of seaweed algebras we advance the notion of the index of a partition pair, by simply defining the index of the latter to be the index of the former. This rather pedestrian definition allows us to describe various statistics on integer partitions. We consider the two extremal cases defined by $(\lambda, 1^{w(\lambda)})$ and $(\lambda, w(\lambda))$. But what about other $\lambda$-based choices for the second composition?
For example one could pair a partition $\lambda=(\lambda_1,\hdots,\lambda_m)$ with its reverse $\text{Rev}(\lambda)=(\lambda_m,\hdots,\lambda_1)$. It follows from a result of \textbf{\cite{collcomp}} that $$|\{\lambda\in \mathcal{P}(n)~:~{\rm ind }_{\text{Rev}(\lambda)}(\lambda)=n-1\}|=d(n).$$ Alternatively, a partition can be paired with its conjugate. In this case, via the same result of \textbf{\cite{collcomp}}, we find that $|\{\lambda\in \mathcal{P}(n)~:~{\rm ind }_{\lambda^C}(\lambda)=n-1\}|$ is equal to twice the number of self-conjugate partitions of $n$.
We might also consider incorporating weighted sums, such as those that appear in Euler's Pentagonal Number Theorem and other Legendre type theorems \textbf{\cite{legendre}}. In such results, partitions $\lambda$ contribute a term of the form $(-1)^{l(\lambda)}q^{w(\lambda)}$ to the weighted sum. One could instead insist that each partition contributes a term of the form $(-1)^{{\rm ind }_{w(\lambda)}(\lambda)}q^{w(\lambda)}$. For example, by restricting to partitions with only odd parts (denoted $\mathcal{P}(n,S_{odd})$) and considering the sets $$e_n=|\{\lambda\in \mathcal{P}(n,S_{odd})~:~{\rm ind }_{w(\lambda)}(\lambda)\text{ is even}\}|\quad o_n=|\{\lambda\in \mathcal{P}(n,S_{odd})~:~{\rm ind }_{w(\lambda)}(\lambda)\text{ is odd}\}|,$$ numerical data suggests the following interesting conjecture.
\begin{conjecture}\label{conj:ws}
\begin{eqnarray}\label{conjecture}
\sum_{n\ge 0}|e_n-o_n|q^n=\prod_{k\ge 1}\frac{1}{1+(-1)^kq^{2k-1}}.
\end{eqnarray}
\end{conjecture}
\noindent
The sequence of coefficients of the product in (\ref{conjecture}) can be found on the Online Encyclopedia of Integer Sequences (``OEIS") as A300574, where this sequence is further conjectured to be nonnegative. If true,
Conjecture~\ref{conj:ws} would not only establish that the sequence A300574 is nonnegative but (\ref{conjecture}) would also provide a combinatorial interpretation.
\bibliographystyle{abbrv}
|
2,869,038,154,840 | arxiv | \section{Introduction}
\label{sec:intro}
Inferring user demographics from social media has been an active research topic, with works focusing on individual tasks regarding individual demographic factors, like gender~\cite{DBLP:conf/icdm/YouBSL14,Alowibdi:2013:EEP:2584692.2584886} and age~\cite{DBLP:conf/icwsm/ZhangHZL16} detection, or multiple demographic factors~\cite{mislove2011understanding, DBLP:conf/icwsm/GoswamiSR09}. A lot of research has been dedicated to solving this problem using textual elements~\cite{mislove2011understanding, DBLP:conf/icwsm/GoswamiSR09,DBLP:conf/icwsm/NguyenGTM13,Dong:2014:IUD:2623330.2623703}. Alternatively, visual content posted on social networks may entail users demographic signature, providing good discriminative elements, while allowing for language independent approaches. We exploit the later while focusing on the task of gender detection.
\footnotetext{https://www.instagram.com/press/}
\begin{table}[]
\label{fig: architecture}
\centering
\begin{tabular}{c|c|c}
\textbf{Profile 1} & \textbf{Profile 2} & \textbf{Profile 3} \\
\midrule
\includegraphics[height=0.15\columnwidth]{profile_images/1_a.jpg} & \includegraphics[height=0.15\columnwidth]{profile_images/1_b.jpg} & \includegraphics[height=0.15\columnwidth]{profile_images/1_c.jpg} \\
\includegraphics[height=0.15\columnwidth]{profile_images/2_a.jpg} & \includegraphics[height=0.15\columnwidth]{profile_images/2_b.jpg} & \includegraphics[height=0.15\columnwidth]{profile_images/3_a.png} \\
\includegraphics[height=0.15\columnwidth]{profile_images/3_b.jpg} & & \includegraphics[height=0.15\columnwidth]{profile_images/3_c.jpg} \\
& & \includegraphics[height=0.15\columnwidth]{profile_images/3_d.jpg} \\
\midrule
$\downarrow$ & $\downarrow$ & $\downarrow$ \\
Female & Male & Female \\
\end{tabular}
\caption{Profile gender inference from a varying number of image postings.}
\end{table}
Gender detection methods that use faces have been shown to be highly effective~\cite{Fan:2014:LCF:2647868.2654960,journals/corr/LiuDBH15,7301352,Schroff_2015_CVPR}.
However, some users do not have a photo of their face as profile picture, and may have photos of idols, celebrities, or friends instead.
Additionally some profile pictures may not be relevant or discriminative. Therefore, it is not reasonable to focus only on this single visual element to classify users' profiles. However, by correlating different types of images (e.g., selfies, photos of visited places, and photos of objects) one may be able to achieve better discrimination. Some pioneering works have already used multiple images~\cite{7177499, DBLP:conf/icdm/YouBSL14} for gender detection. Merler~\emph{et al}.~\cite{7177499} reduced both user profile and posted images, to a set of semantic labels/categories by classifying each image individually and then using the distribution of labels/categories of each user profile for the final classification. You Q.~\emph{et al}.~\cite{DBLP:conf/icdm/YouBSL14} applied visual topic modelling over images of a same category, for a given set of categories. The visual topics distribution of user profile images was then used for classification. Both approaches use proxy methods to encode the set of user profile images, in the sense that the classifier predictions are not based on the canonical form of the content, but on intermediate artifacts. We depart from previous work by considering a multiple instance learning (MIL) framework taking as input features extracted directly from images, which to the best of our knowledge, is the first time such approach is applied to this particular problem.
We leverage on features over a semantic space, extracted from a top performing convolutional neural network, which as opposed to low-level features, are able to reveal semantic discriminative patterns (e.g., concepts). Figure~\ref{fig: architecture} illustrates our approach.
In this work we propose an \emph{exclusively} visual based MIL approach to gender detection by performing inference at the \emph{user profile} granularity, thus using \emph{bags} of images for inference. We model the inference problem using a single instance learning (SIL) approach supported by a majority voting scheme, in which labels are propagated to instances (user images), and a true MIL approach where predictions are performed at the bag level. Different classifiers are considered including standard classifiers for SIL (Na\"{\i}ve Bayes, SVM and Logistic Regression), and MI-SVM classifier~\cite{Andrews:2002:SVM:2968618.2968690} for MIL. Low-level image features (HoC, HoG, GIST) and semantic (deep learning features) space domains are evaluated.
The contributions of this work are threefold: 1) we propose a framework for user gender detection based solely on visual user generated content, considering multiple images when making predictions, to deal with subjectivity and social media content noise. We conduct experiments on a real dataset collected from Instagram and annotated using crowdsourcing.
2) We conduct an extensive analysis and comparison of SIL and MIL approaches where we assess the effectiveness of each approach and classifiers, as well as the importance of considering multiple images.
3) A comparison between high-level (semantic) vs. low-level image representations for the tasks at hand in order to understand its impact on effectiveness.
With our proposed approach, we were able to obtain precision values $>0.825$ for the task of gender detection, using multiple images, with both single and multiple instance learning classifiers. Furthermore, we experimentally confirm the importance of considering multiple images on the effectiveness of each classifier. Semantic (high-level) features outperformed a set of low-level features, confirming their suitability for high-level classes like gender.
\section{Inferring User Profile Gender}
Given a \emph{user-profile} $u_i \in \mathcal{U}$ and the set of images $\mathcal{I}^{u_i} = \{i^{u_i}_1,\ldots,i^{u_i}_j ,\ldots,i^{u_i}_n \} \in u_i$, we are interested in learning a function $f_{gender}:u_i \mapsto L_{gender}$ that given a \emph{user-profile} $u_i$ and the set of classes $L_{gender} = \{female, male\}$, infers the gender $l^{u_i} \in L_{gender}$. Therefore, we frame the learning problem as a two class (\emph{Female} and \emph{Male}) classification problem.
To make the transition towards MIL terminology~\cite{Amores:2013:MIC:2503904.2504011}, from each user profile $u_i$ and its associated images $I^{u_i}$, we define a bag of instances $X^{u_i} = \{\vec{x_1}, \ldots,\vec{x_j},\ldots, \vec{x_n}\}$, where each $\vec{x_j} \in \mathbb{R}^D$ denotes the feature vector of image $i^{u_i}_j$. The respective labels $l^{u_i}$ are defined as the labels of $X^{u_i}$.
\subsection{Deep Semantic Spaces for Demographics}
Visually, gender types are high-level classes. For such classes, traditional low-level features may be too specific and thus miss more high-level patterns. On the other hand, semantic features capture such high-level patterns that may be more adequate for gender detection. Among the large collection of low-level image features available in the literature, in this paper we consider Color Histograms (HoC) and Gradient Histograms (HoG).
We also consider the GIST~\cite{Oliva:2001:MSS:598425.598462} descriptor, which consists of a summarisation of gradient information over different patches of the image, providing, to some extent, the \emph{gist} of the scene, and possibly revealing scene-based patterns.
Recently, deep neural networks have achieved top classification performance in a large variety of tasks. For instance, Convolutional Neural Networks are able to automatically learn representations of images, revealing semantic discriminative patterns. Thus, we leverage on these networks by extracting features from a VGG network~\cite{journals/corr/SimonyanZ14a} that achieved top results in the well-known ImageNET Large Scale Visual Recognition Challenge 2014. The network was trained for detecting semantic concepts (1000 concepts) therefore we expect these extracted \emph{semantic features} to reveal patterns related to semantic concepts (e.g., objects). We posit that this type of features is more suitable for gender detection than low-level features, given that the feature space is more well structured and images are represented in terms of their meaning, allowing classifiers to discriminate between classes at a higher conceptual level, which to some extent, is closer to the way that humans usually accomplish this task.
\subsection{Single Instance Learning Methods}
Images of a given \emph{user-profile} are dependent, in the sense that they are intrinsically related to the profile labels. By disregarding such fact, we arrive at a single instance learning approach, allowing one to use single instance classifiers. In other words, \emph{user-profile} labels are propagated to each instance (user images) and all instances are regarded as independent. More formally, for each profile $u_i$ the set of labels $L^{u_i}$ is propagated to each image $i^{u_i} \in \mathcal{I}^{u_i}$.
Finally, a voting scheme based on individual predictions of each instance is applied to achieve the final profile-level prediction~\cite{Amores:2013:MIC:2503904.2504011}. From a trained classifier $c$ and taking as input an instance $\vec{x_j}$, a prediction $y_{j}$ is obtained. Based on this procedure, a set $y$ is obtained containing the predictions for all the instances of a bag $X^{u_i}$. Then, the following voting scheme is applied:
\begin{equation}
\hat{y^{u_i}} = \argmax_{l\in L_{gender}} \sum^{|X^{u_i}|}_{j=1} \mathbbm{1}_{\hat{y_{j}} = l},\ \ \ with\ \hat{y_{j}} = c(\vec{x_j}),
\end{equation}
which corresponds to choosing the label $l$ with most instances predicted as $l$. In case of a tie, $l$ is randomly sampled from the tied labels $l \in L_{gender}$.
This approach is under the assumption that all instances are discriminative, which may not always be the case. Notwithstanding, if the number of discriminative images is greater than the non-discriminative ones for a profile $u_i$, there is a high probability that the non-discriminative instances will be "silenced" by the voting scheme, giving the SIL approach somewhat the ability to deal with noise. As classifiers, we considered the Na\"{\i}ve Bayes with Gaussian likelihood, SVM and Logistic Regression classifiers.
Both \textbf{SVM} and \textbf{Logistic Regression} (LR) are discriminative classifiers. For both we used the $\ell_2$ norm as penalty.
We consider both linear and (Gaussian) Radial Basis kernels for the SVM classifier.
\subsection{Multiple Instance Learning Methods}
Multiple Instance Learning aims at learning classifiers capable of inferring the label of (unseen) bags of instances, treating bags as a whole and performing the discriminant learning process in bags space.
From now on, we consider that label -1 or 1 may correspond to any of the labels $l \in L_{gender}$. Thus, for negative bags, i.e. $l^{u_i} = -1$, all the instances of $X^{u_i}$ are assumed to be negative (label $-1$). For positive bags, at least one instance of $X^{u_i}$ is a positive example (label $1$) of the underlying concept.
Previously, the MIL approach was shown to yield better performance than the traditional single instance approach, for specific tasks~\cite{Alpaydin:2015:SVM:2791619.2792205}. Additionally, it allows one to explicitly consider all the \emph{user profile} images for classification in an elegant manner. Concretely, extensions to the SVM classifiers for MIL have been proposed~\cite{Andrews:2002:SVM:2968618.2968690, Bunescu:2007:MIL:1273496.1273510, Doran:2014:TEA:2666867.2666935}. These classifiers are interesting because they keep the desirable properties of SVMs. Given that our problem formulation clearly fits in the MIL approach, we apply the MI-SVM classifier~\cite{Andrews:2002:SVM:2968618.2968690} for gender detection.
The MI-SVM classifier generalises the notion of a margin to bags and maximises it directly. Namely, for a given bag $X^{u_i}$, it defines the following functional margin:
\begin{equation}
\label{eq: misvm_hyperplane}
\gamma^{u_i} \equiv l^{u_i} \maxsubscript_{\vec{x_j} \in X^{u_i}} (\langle \vec{w},\vec{x_j} \rangle +b).
\end{equation}
Consequently, the decision function becomes:
\begin{equation}
\label{eq: misvm_fn}
\hat{y^{u_i}} = sgn(\maxsubscript_{\vec{x_j} \in X^{u_i}} (\langle \vec{w_k},\vec{x_j} \rangle +b_k) ),
\end{equation}
where $sgn$ is the sign function. By inspecting equations~\ref{eq: misvm_hyperplane} and~\ref{eq: misvm_fn}, we conclude that for positive bags the \emph{most positive} (from the dot product) instance defines the margin. For negative bags, it is instead defined by the \emph{least negative} instance.
This property is what gives the MI-SVM classifier the ability to "silence" noisy/non-discriminant instances from either negative or positive bags. Either way, this highly depends on the feature space considered, and on its ability to reveal discriminative patterns useful for demographics, more specifically, gender detection.
Finally, using the functional margin definition from equation~\ref{eq: misvm_hyperplane}, the soft-margin SVM classifier for MIL is defined as:
\begin{equation*}
\begin{aligned}
\label{eq: soft_margin}
&\underset{\vec{w},b,\xi}{\text{min}}
\ \frac{1}{2}||\vec{w}||^2 + C \sum_{i=1}^{n} \xi_i\\
\text{s.t.}&\ \ \ \forall u_i \in \mathcal{U}:\ l^{u_i} \maxsubscript_{\vec{x_j} \in X^{u_i}} (\langle \vec{w},\vec{x_j} \rangle +b) \geq 1-\xi_i,\ \xi_i \geq 0,
\end{aligned}
\end{equation*}
where $\xi_i$ are slack variables to address non-linearly separable datasets and $C$ is a regularisation parameter.
This formulation can then be cast as a mixed-integer program, yielding a quadratic optimisation problem that can be solved to optimality. We refer the reader to~\cite{Andrews:2002:SVM:2968618.2968690} for details regarding the derivation of the dual objective function and optimisation procedure. With bags of only 1 element, the MI-SVM classifier reduces to the single instance SVM classifier. Although in our experiments only the linear kernel is considered, this formulation can naturally accommodate other kernels.
\section{Experimental Results}
To evaluate the effectiveness of the proposed methods we conduct a set of experiments on a crawled public Instagram user accounts dataset.
\textbf{Instagram Data.}
We crawled a dataset from Instagram using a breadth-first search strategy over its social network graph starting with a set of seeds, using Instagram's JSON API. We obtained a collection that contains a total of 738,540 images from 56,678 user profiles.
Manual inspection of the collection revealed that overall the photos obtained are of good quality, making them suitable for feature extraction and consequently classification, or more specifically, for the task of gender detection.
\textbf{Labels.} To get ground truth labels for gender detection we resorted to crowd sourcing.
Thus, we created am annotation task in which we show 12 photos from a single user profile and asked human workers to identify the gender and account type (individual/non-individual) of a total of 450 user profiles, by looking at images. Workers were provided with user's biographies. An option "Cannot make a guess" was provided.
It should be noted these 450 profiles were randomly selected from the crawled dataset, where profiles with less than 12 images were excluded a priori. In profiles identified as "Not Personal", the gender question is not shown.
For every pair $\langle profile,question\rangle$ we enforced an agreement $>=0.6$, given that we asked for 3 judgements per question (at least 2 workers needed to agree). For those pairs in which agreement was less than the defined threshold or the agreement was on the "Cannot make a guess" option, the label was excluded (we assume that for those cases it was impossible to know from the posted images and user biography).
\textbf{Protocol.} User profiles in which agreement was not achieved either on profile type and gender questions were excluded. All profiles labelled as "non-individual" type were also excluded.
Train and test splits were created at the profile granularity, i.e., different splits do not share images from the same user, are $80\%$ and $20\%$, respectively.
For configurations in which the number of images used for a given profile, for classification, is less than 12, user images are sampled using an uniform distribution without replacement. To increase the robustness of the results we take the average of 10 runs for all the metrics.
\subsection{Results and Discussion}
To evaluate each classifier's performance, we use weighted precision (P) and accuracy (A) metrics. Weighted precision takes into account label imbalance by first computing the precision of each label and then taking the average weighted by the number of instances of each label.
\subsubsection{Gender Detection}
We posed the task of gender detection as a two class classification problem. Thus, we considered the Na\"{\i}ve Bayes classifier with Gaussian likelihood (NB-Gaussian), Logistic Regression with $\ell_2$ (Log. Reg. $\ell_2$), SVM with linear kernel (SVM-linear), SVM with Gaussian RBF kernel (SVM-RBF) and MI-SVM with linear kernel (MISVM-linear) classifiers. After filtering the dataset as described in the previous section, we obtain 273 \emph{user-profiles} with $\approx 63\%$ and $\approx 37\%$ being \emph{female} and \emph{male} profiles respectively.
We trained each classifier with different bag sizes (number of user images considered from one profile), namely with 1, 2, 5, 10 and 12 instances per bag. To determine hyper-parameters we applied a 5-fold Cross Validation procedure, maximising precision. In particular, we followed a grid-search procedure to select each classifier's final parameters over the full development set (e.g. SVM RBF gamma, distance function, inverse of regularisation strength coefficient, error term penalty, etc.). Thus, each experimental result (i.e. point in figure 2) involved 560 validation experiments.
\begin{table}[t]
\caption{Best results achieved of each classifier and respective bag sizes for Gender detection.}
\centering
\label{fig: table_gender_best}
\begin{tabular}{lccc}
\toprule
Classifier & $P$ & $A$ & $|X^{u_i}|$\\
\midrule
NB-Gaussian & 0.870 & 0.855 & 12\\
Log. Reg. $\ell_2$ & 0.863 & 0.855 & 5\\
SVM-linear & \textbf{0.911} & \textbf{0.909} & 12 \\
SVM-RBF & 0.825 & 0.764 & 5\\
MISVM-linear& 0.838 & 0.836 & 12 \\
\bottomrule
\bottomrule
\end{tabular}
\end{table}
In table~\ref{fig: table_gender_best} we report the configuration in which the classifier achieved the best results (in terms of precision). At a first glance it is clear that in their top performing configuration, all the classifiers achieved high precision values ($>0.825$) and accuracy ($>0.76$). Moreover, except for Log. Reg. $\ell_2$ and SVM-RBF, top performance was achieved using 12 user images. Even for Log. Reg. $\ell_2$ and SVM-RBF, top performance was achieved with $|X^{u_i}| = 5$. This confirms our hypothesis that better performances can be achieved by considering multiple images, instead of only one.
Another interesting aspect is that despite the fact that the MIL classifier, MISVM-linear, achieved good performances overall and achieving better results than SVM-RBF, the remaining SIL classifiers outperformed it. We believe this is a consequence of the $max$ operation on the MI-SVM decision function (equation~\ref{eq: misvm_fn}) which makes the classifier decision based only on one instance. On the other hand, the majority vote scheme decides based on all the instances of the bag, given equal importance to all instances, contributing to its robustness.
To assess the importance of considering multiple images for this task, we plotted in figure~\ref{fig: plot_precision_gender} the performance of each classifier as we vary the bag size $|X^{u_i}|$ on the previous experiment. As the bag size increases the trend is that overall, better results are achieved. It is clear that by using more than one image all classifiers become more effective. Additionally, as more images are used for each decision, the better classifiers handle noise originating from social media.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{plots/precision_gender_True}
\caption{Impact of $|X^{u_i}|$ on each classifier, for gender detection.}
\label{fig: plot_precision_gender}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{plots/features_performance_gender}
\caption{Impact of different feature spaces on discrimination of high level classes (gender).}
\label{fig: plot_precision_features}
\end{figure}
\subsubsection{Feature Spaces}
To evaluate the impact of different feature spaces, we trained all the classifiers for the task of gender detection with the set of image features previously described: HoC, HoG, GIST and VGG-16 (features from VGG net with 16 layers). The results of the experiment are depicted in figure~\ref{fig: plot_precision_features}. We can see that with semantic features (VGG-16) all the classifiers achieved better results, with HoC being the second best feature overall except when used with SVM-RBF and MISVM-linear classifiers, in which both got the lowest results of the experiment. With HoG and GIST features, classifier's performance was very similar. Clearly, semantic features are able to reveal more discriminative patterns for the task at hand, thus resulting in better performance, confirming our initial intuition.
\section{Conclusion}
In this paper we addressed the task of gender detection on social media profiles, using solely visual user generated content. Multiple images from user-profiles are considered when making predictions, allowing for better disambiguation of the underlying target patterns. The gender detection task was modelled under both single and multiple instance learning approaches. We leverage on semantic features, extracted from a top-performing convolutional neural network, trained for object classification, such that classification is performed at a conceptually high level, whose effectiveness for this particular task could be verified in our experiments.
Moreover, we concluded that both single and multiple instance learning approaches turned out to be very effective using semantic features. The importance of considering multiple images for prediction was confirmed in our experiments, given that all classifiers were more effective with multiple images.
Given the promising results obtained, this work can now complement highly effective methods for gender detection that require faces on image profiles and that are based on the textual modality.
\section{Acknowledgements}
This work has been partially funded by the H2020 ICT project COGNITUS with the grant agreement No 687605 and by the project NOVA LINCS Ref. UID/CEC/04516/2013. We also gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU used for this research.
\bibliographystyle{IEEEbib}
|
2,869,038,154,841 | arxiv | \section*{Introduction}
The determination of the nuclear incompressibility $K$ is still a matter of
debate, despite a remarkable number of works on the subject$^{1)}$. In the
present contribution, we present self-consistent calculations of the nuclear
collective modes associated with a compression and expansion of the nuclear
volume, namely the isoscalar giant monopole and dipole resonances (ISGMR and
ISGDR, respectively). In fact, we share the point of view that the most
reliable way to extract information on $K$ is to perform that kind of
calculations, having as the only phenomenological input a given effective
nucleon-nucleon interaction, and choose the value of $K$ corresponding to
the force which can reproduce the experimental properties of the compression
modes in finite nuclei.
The ISGMR, or ``breathing mode'', is excited by the operator
$\sum_{i=1}^A r_i^2$ and it has been identified in many isotopes along the
chart
of nuclei already two decades ago. However, this systematics has never
allowed an unambigous determination of $K$$^{1)}$. This was one of the motivations
for the recent experimental program undertaken at the Texas A\&~M Cyclotron
Institute, which has allowed the extraction of experimental data for the
ISGMR of better quality as compared to the past, by means of the analysis
of the results of inelastic scattering of 240 MeV $\alpha$-particles. We
refer to other contributions in these proceedings for reports on these
experimental data$^{2)}$. Monopole strength functions turn
out to be quite fragmented for nuclei lighter than $^{90}$Zr. For nuclei
like $^{208}$Pb, $^{144}$Sm, $^{116}$Sn and $^{90}$Zr, however, one is able
to identify a single peak which, together with a high-energy
extended tail, exhausts essentially all the monopole
Energy Weighted Sum Rule (EWSR). These medium-heavy
nuclei are, therefore, those suited for the extraction of information about
the nuclear incompressibility and we concentrate ourselves on them in the
present work.
The ISGDR is excited by the operator $\sum_i r_i^3 Y_{10}$ and corresponds
to a compression of the nucleus along a definite direction, so that it has
been called sometimes the ``squeezing mode''. Although some first indication
about the energy location of this resonance dates back to the beginning of
the eighties, a more clear indication about its strength distribution in
$^{208}$Pb has been reported only recently$^{3)}$. Measurements have been
done also for other nuclei, namely $^{90}$Zr, $^{116}$Sn and $^{144}$Sm (as
in the case of the giant monopole resonance). There is some expectation that
the study of this mode can help to shed some light on the problem of nuclear
incompressibility. Actually, at first sight this compressional mode seems to
provide us with a new problem. A
simple assumption like the scaling model (illustrated for the present
purposes in Ref.$^{4)}$) would lead to two different values of the
finite-nucleus incompressibility $K_A$ if applied to the ISGMR and the ISGDR
with the input of their experimental energies. The hydrodynamical model
gives two results which are closer$^{4)}$ but which still make us
wonder about the validity of methods based on extracting $K_A$ and
extrapolating it to large values of $A$, for the determination of $K$. This
points again to the necessity of reliable microscopic calculations of
the compressional modes, in order to reproduce the experimental data and
extract the value of $K$ from the properties of the force which is used.
Our calculations are performed within the framework of
self-consistent Hartree-Fock (HF) plus Random-Phase Approximation (RPA).
We use effective forces of Skyrme type$^{5-7)}$
and we look at their predictions for the properties of ISGMR and ISGDR. The
parametrizations we employ span a large range of values for $K$ (from 200 MeV to
about 350 MeV). In particular, we focus mainly on two original
aspects: firstly, we look at the effects of pairing correlations in
open-shell nuclei; secondly, we study if the
picture obtained at mean-field level is altered by the inclusion of the
coupling of the giant resonances to more complicated nuclear configurations.
This inclusion
is necessary if one wishes to understand theoretically all the
contributions to the resonance width and may shift the resonance centroid.
About the first aspect, it is well known that pairing correlations are
important in general to explain the properties of ground states and
low-lying excited states in open-shell nuclei.
Since we wish to see how these
correlations affect in particular the compressional modes,
we take them into account by extending the HF-RPA
approach to a quasi-particle RPA (QRPA) on top of a HF-BCS calculation.
About the second aspect,
we recall that if
we start from a description of the giant resonance as a superposition of
one particle-one hole (1p-1h) excitations, in their damping process we must
take care of the coupling with states of 2p-2h character. They are in fact
known to play a major role and give rise to the spreading width
$\Gamma^\downarrow$ of the giant resonance which is usually a quite
large fraction of the total width. Within mean field theories, only the
width associated with the resonance fragmentation
(Landau width) and the escape width
$\Gamma^\uparrow$ are included (the latter, provided that 1p-1h
configurations with the particle in the continuum are considered).
In the past, we have developed a theory in which all the contributions to the
total width of giant resonances are consistently treated and we have obtained
satisfactory results when applying it to a number of cases. In
particular, we will recall what has been obtained$^{8)}$ for the
case of the ISGMR in $^{208}$Pb. We
also report about a new calculation for the ISGDR in
the same nucleus.
\section*{Formalism: a brief survey}
For all nuclei we consider, we solve the HF equations on a radial mesh
and, in the case of the open-shell isotopes, we solve HF-BCS equations.
A constant pairing gap $\Delta$ is introduced (for
neutrons in the case of $^{116}$Sn and for protons in the case of
$^{90}$Zr and $^{144}$Sm), and at each HF iteration the quasi-particle
energies, the occupation factors and the densities to be input at the
next iteration are determined accordingly. $\Delta$ is obtained from
the binding energies of the neighboring nuclei$^{9)}$. The states
included in the solution of the HF-BCS equations are those below a cutoff
energy given by $\lambda_{HF}+8.3$ MeV ($\lambda_{HF}$ being the HF Fermi
energy), in analogy with the procedure of Ref.$^{10)}$.
Using the above self-consistent mean fields
we work out the RPA or
QRPA equations (respectively on top of HF or HF-BCS), in their matrix form.
Discrete positive energy states are obtained by diagonalizing the mean
field on a harmonic oscillator basis and they are used to build the 1p-1h (or
2 quasi-particles) basis coupled to $J^\pi$=0$^+$ or 1$^-$. The dimension
of this basis is chosen in such a way that more than 95\% (typically 97-99\%)
of the appropriate EWSR is exhausted in the RPA or QRPA calculation. More
details, especially on the way the QRPA equations are implemented, will be
given in Ref.$^{11)}$.
As mentioned in the previous section, in the case of $^{208}$Pb we perform
also calculations that go beyond this simple discrete RPA. This is done
along the formalism described in Ref.$^{12)}$, which is recalled here only very
briefly.
We label by $Q_1$ the space of discrete 1p-1h configurations in which the RPA
equations are solved. To account for the escape width $\Gamma^\uparrow$
and spreading width $\Gamma^\downarrow$ of the giant resonances, we build two
other orthogonal subspaces $P$ and $Q_2$. The space $P$ is made of particle-hole
configurations where the particle is in an unbound state orthogonal to
all the discrete single-particle levels; the space $Q_2$ is built with the
configurations which are known to play a major role in the damping process
of giant resonances: these configurations are 1p-1h states coupled to a
collective vibration. Using the projection operator formalism one can easily
find that the effects of coupling the subspaces $P$ and $Q_2$ to $Q_1$ are
described by the following effective Hamiltonian acting in the $Q_1$ space:
\begin{eqnarray}
{\cal H} (E) \equiv Q_1 H Q_1 & + & W^\uparrow(E)
+ W^\downarrow(E) \nonumber \\
= Q_1 H Q_1 & + & Q_1 H P {\textstyle 1 \over \textstyle
E - PHP + i\epsilon} P H Q_1 \nonumber \\
\ & + & Q_1 H Q_2 {\textstyle 1 \over \textstyle
E - Q_2 H Q_2 + i\epsilon}
Q_2 H Q_1, \nonumber
\label{H_eff}\end{eqnarray}
where $E$ is the excitation energy. For each value of $E$ the RPA equations
corresponding to this effective, complex Hamiltonian ${\cal H} (E)$ are
solved and the resulting sets of eigenstates enable us to calculate all
relevant quantities, in particular the strength function associated with a
given operator. To evaluate the matrix elements of $W^\downarrow$,
we calculate the collective phonons with the same effective
interaction used for the giant resonance we are studying (within RPA),
and we couple these phonons with the 1p-1h components of the giant resonance
by using their energies and transition densities.
\section*{Results for the isoscalar monopole resonance}
As recalled in the introduction, Youngblood {\it et al.}$^{2)}$ have
recently measured the ISGMR strength distribution with fairly good precision,
in the nuclei $^{90}$Zr, $^{116}$Sn, $^{144}$Sm and $^{208}$Pb. In
their work, they also compare the experimental centroid energies with the
calculations of Blaizot {\it et al.}$^{13)}$ performed
by using RPA and employing the finite-range Gogny effective interaction: a
value of the nuclear incompressibility $K$ = 231 MeV is deduced. In the following,
we denote as centroid energy the ratio $E_0\equiv m_1/m_0$ ($m_0$ and $m_1$ being
the non-energy-weighted and energy-weighted sum rules, respectively).
If we try to compare the experimental values with calculations done at the
same RPA level but using the zero-range Skyrme effective interactions, we
can infer a different conclusion with respect to the value of $K$. Among
the Skyrme type interactions, the parametrization which gives probably the
best account of the experimental centroid energies in the nuclei studied
by the authors of Ref.$^{2)}$, is the SGII force$^{6)}$.
The results are shown in
Table 1. The force SGII is characterized by a value of the nuclear
incompressibility $K$ = 215 MeV: since it reproduces very well the ISGMR
centroid energy in $^{208}$Pb, and it slightly overestimates those in the
other isotopes, one would conclude that $K$ is of the order of or slightly
less than 215 MeV.
This conclusion is inferred by means of simple RPA. It is of course legitimate
to wonder if calculations beyond this simple approximation could lead to
different values of the nuclear incompressibility. We first consider the effect
of pairing correlations. In the case of $^{116}$Sn, the centroid energy of
17.18 MeV obtained with the SGII force in RPA, becomes 17.19 MeV if one
turns to QRPA. A very small shift is found also when other forces are
used (for instance, with the recently proposed
SLy4 force$^{7)}$, one
obtains 17.51 MeV and 17.59 MeV for RPA and QRPA, respectively)
and when other nuclei are considered. In general,
although we know that pairing correlations play a crucial role not only to
explain the ground-state of open-shell nuclei but also their low-lying excited
states, it appears that they do not affect so much the giant resonances like
the ISGMR (or ISGDR, anticipating results of the next section) which
lie at relatively high excitation energy compared to the pairing gap $\Delta$.
Civitarese {\it et al.}$^{14)}$ found also small shifts (of the order
of 100-150 keV) for the ISGMR and ISGQR when pairing correlations are
taken into account: this shift is larger than that obtained in the present
work, but it is the result of a different (non
self-consistent) model.
The present conclusion for the nuclear incompressibility is therefore similar
to that obtained by Hamamoto {\it et al.}$^{15)}$, since they find that
the Skyrme interaction which provides the best results for the ISGMR is the
SKM$^*$ parametrization and this is very similar to SGII (the associated
nuclear incompressibility being 217 MeV).
Our study, however, is done in a more general framework
since we have analyzed also the role of pairing correlations.
If we finally consider the results of calculations beyond mean field$^{8)}$
(which include not only the continuum coupling but also the coupling with the
2p-2h type states) performed
for $^{208}$Pb we find that
it is also possible to reproduce rather well the total width of the
ISGMR, which is around 3 MeV. This width is actually in
large part a consequence of fragmentation (or Landau damping): at least three
states share, at the level of RPA, the resonance strength, but continuum as
well as 2p-2h couplings are able to give to each peak the correct width so that
the overall lineshape coincides with the experimental findings. We stress that
the coupling with the 2p-2h type states is also responsible
for a downward shift of the ISGMR centroid and peak energies, which is of the
order of 0.5 MeV. One may argue that this affects the extraction of the
value of $K$ from theoretical calculations. Actually, since the value of $K$
associated with a given force is obtained by a calculation of nuclear matter
at the mean field level, it is legitimate to draw conclusions about $K$ from
the comparison with the experiment of the
ISGMR results for finite nuclei obtained again at the mean field level. But
the fact that a given force is able to account for the ISGMR linewidth enforces
our confidence about its reliability. And it would be of course legitimate
either, to compare the centroid energies obtained after 2p-2h coupling with
experiment provided the value of $K$ associated with the force is calculated
by including the same couplings at the nuclear matter level. No such calculations
in nuclear matter have been done so far, to our knowledge.
\begin{table}[here]
\caption{Experimental and theoretical values of the centroid energies $E_0\equiv
m_1/m_0$ for the ISGMR and ISGDR. The theoretical values are obtained with
the Skyrme-RPA approach, using the SGII interaction.
All values are in MeV.}
\vspace{0.3cm}
\begin{tabular}{|c|r|r|r|r|}
\hline
& \multicolumn{2}{|c|}{ISGMR} & \multicolumn{2}{|c|}{ISGDR} \\
\cline{2-5}
& Exp. & Theory & Exp. & Theory \\
\hline
$^{90}$Zr & 17.9 & 19.1 & 26.2 & 27.1 \\
$^{116}$Sn & 16.0 & 17.2 & 23.0 & 26.3 \\
$^{144}$Sm & 15.3 & 16.2 & 24.2 & 25.4 \\
$^{208}$Pb & 14.2 & 14.1 & 20.3 & 24.1 \\
\hline
\end{tabular}
\label{table1}
\end{table}
\vspace{0.2cm}
\section*{Results for the isoscalar dipole resonance}
A peculiar feature of calculations of this giant resonance is the appearance of
a spurious state in the calculated spectrum.
When diagonalizing the RPA matrix on a 1$^-$ basis, we expect to see among
all states
the spurious state at zero energy corresponding to the center-of-mass motion
and we expect as well that it exhausts the whole strength associated with the
operator $\sum_i r_i Y_{10}$. Due to a lack of complete self-consistency
(some part of the residual interaction, like the two-body spin-orbit and Coulomb
forces, are usually neglected in the RPA because their effect should be
rather small) and to numerical inaccuracies, this is not the case. The
spurious state
comes out in practice
at finite energy and its wave function does not
overlap completely with that of the exact center-of-mass motion: as a
consequence, the remaining RPA eigenstates are not exactly orthogonal to the
true spurious state, and their spurious component must be projected out.
This is not difficult if RPA is done in the discrete p-h space.
In any case,
it can be shown$^{16)}$ that this projection procedure is equivalent to
replacing the $\sum_i r_i^3 Y_{10}$ operator with $\sum_i (r_i^3-\eta r_i)
Y_{10}$, $\eta$ being ${5\over 3}\langle r^2 \rangle$. Once this projection is
done, we find that a substantial amount of $r^3 Y_{10}$ strength still remains
in the 10 - 15 MeV region, in addition to the strength in the 20 - 25 MeV region.
The data do not show any strength in the lower region, i.e., experimental centroid
energies correspond to the energy region above 15 MeV. Therefore, to have a
meaningful comparison with experiment we will refer from now on to theoretical
centroid energies calculated in the interval 15 - 40 MeV.
In Table 1 we show the ISGDR centroid energies obtained with the SGII
force. Especially in the case of $^{208}$Pb and $^{116}$Sn, it can be noticed
that RPA calculations tend to overestimate the value of the centroid energy,
the discrepancy being less severe in the other two cases. One may wonder
if this is a special feature of SGII, although this force has been said to
behave rather well for the monopole case. Fig. 1 shows that this is not the
case: the centroid energies obtained in RPA with a number of different Skyrme
parametrizations are plotted as a function of their incompressibility $K$, and
it can be noticed that all forces systematically overestimate the experimental
values of the centroid energies.
Gogny
interactions have also been used to study the ISGDR in this and other
nuclei$^{17)}$, but they also predict too large centroid energies.
The same can be said about relativistic models like relativistic RPA$^{18)}$
or time-dependent relativistic mean field$^{19)}$.
We may conclude that the case of the
ISGDR in $^{208}$Pb is a kind of exception among the giant resonances studied
within the self-consistent HF-RPA approach, as usually one never finds such
large discrepancies between theory and experiment ($\sim$ 4-5 MeV).
We finally address the question whether the coupling with more complicated
configurations, which has been seen to be responsible of a downward shift of the
resonance energy, can diminish this discrepancy in the case of $^{208}$Pb. We
have done for the ISGDR a calculation of the type described above in the monopole case.
The resulting strength function, which includes continuum and
2p-2h coupling, is depicted in Fig. 2. One can see that the total
width of the resonance is quite large, and the theoretical value of about 6
MeV compares well with the experimental result which is about 7 MeV$^{20)}$.
Although the resonance lineshape is accounted for by theory, the
downward shift with respect to the RPA result is only about 1 MeV. The
fundamental problem why the ISGDR energy in $^{208}$Pb cannot be reproduced by
theoretical models still remains.
\section*{Conclusion}
In this paper, we have considered the isoscalar monopole and dipole resonances
in a number of nuclei and we have tried to reproduce their properties by means of
HF-RPA, or QRPA on top of HF-BCS, or more sophisticated approach which takes care of
the continuum properly and of the coupling with
nuclear
configurations which are more complicated than the simple 1p-1h. In general, we
have found that the effect of pairing correlations is quite small as these
resonances lie at high energy with respect to the pairing gap $\Delta$.
Concerning the RPA results, the situation looks different in the case of monopole
and dipole. In the former case, the Skyrme-type force SGII is able to reproduce
well the centroid energy in $^{208}$Pb, and it slightly overestimates this energy
in other medium-heavy nuclei which have been measured accurately in recent
experiments. This would allow us to extract a value of the nuclear incompressibility
around 215 MeV. In the case of the ISGDR, however, the same Skyrme force
overpredicts this centroid energy in $^{208}$Pb by about 4 MeV. Other parametrizations
of Skyrme type cannot do better, and the problem is not solved if one turns
to Gogny interactions or to relativistic models. Therefore, although in other nuclei
this discrepancy between theory and experiment can be less than in the case of
$^{208}$Pb, we can say that the ``squeezing mode'', which could be taken as a
further probe of the nuclear incompressibility besides the well-known ``breathing''
monopole oscillation, is challenging us with a new problem.
Calculations beyond mean field do not change substantially our conclusions about the
centroid energies. However,
we stress that these calculations are necessary for a proper
account of the giant resonances lineshape, and in fact in our case they have been
able to reproduce the total width of the ISGMR in $^{208}$Pb, and also of the
ISGDR although its centroid is overestimated.
We would like to thank Umesh Garg for stimulating discussions and
for communicating experimental data prior to publication, and also Jean-Paul
Blaizot for useful discussions.
\begin{figure}[h
\caption{RPA centroid energies of ISGDR calculated in $^{208}$Pb with various
Skyrme interactions, as a function of compression modulus $K$. }
\end{figure}
\begin{figure}[h
\caption{Calculated ISGDR strength distribution of operator $(r^3 - \eta r) Y_{10}$
including damping effects of particle-hole-plus-phonon coupling (see text). The
nucleus is $^{208}$Pb. }
\end{figure}
\section*{References}
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Phys.\ Rev.\ {\bf C50}, 1496 (1994).
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\end{document}
|
2,869,038,154,842 | arxiv | \section{Introduction}
A general class of fundamental problems in physics can be described as an impurity particle interacting with a quantum reservoir. This includes Anderson's orthogonality catastrophe \cite{Anderson1967}, the Kondo effect \cite{Kondo1964}, lattice polarons in semiconductors, magnetic polarons in strongly correlated electron systems and the spin-boson model \cite{Leggett1987}. The most interesting systems in this category can not be understood using a simple perturbative analysis or even self-consistent mean-field (MF) approximations. For example, formation of a Kondo singlet between a spinful impurity and a Fermi sea is a result of multiple scattering processes \cite{Anderson1961} and its description requires either a renormalization group (RG) approach \cite{Wilson1975} or an exact solution \cite{Andrei1980,Wiegmann1983}, or introduction of slave-particles \cite{Read1983}. Another important example is a localization delocalization transition in a spin bath model, arising due to "interactions" between spin flip events mediated by the bath \cite{Leggett1987}.
While the list of theoretically understood non-perturbative phenomena in quantum impurity problems is impressive, it is essentially limited to one dimensional models and localized impurities. Problems that involve mobile impurities in higher dimensions are mostly considered using quantum Monte Carlo (MC) methods \cite{Prokofev1998,Gull2011,Anders2011}. Much less progress has been achieved in the development of efficient approximate schemes. For example a question of orthogonality catastrophe for a mobile impurity interacting with a quantum degenerate gas of fermions remains a subject of active research \cite{Rosch1999,Knap2012}.
\begin{figure}[b!]
\centering
\epsfig{file=FIG1, width=0.5\textwidth}
\caption{By applying a rf-pulse to flip a non-interacting (left inset) into an interacting impurity state (right inset) a Bose polaron can be created in a BEC. From the corresponding rf-spectrum the polaron groundstate energy can be obtained. In the main plot we compare polaronic contributions to the energy $E_\text{p}$ (as defined in Eq.\eqref{eq:EpDef}) predicted by different models, as a function of the coupling strength $\alpha$. Our results (RG) are compared to Gaussian variational calculations \cite{Shchadilova2014}, MC calculations by Vlietinck et al. \cite{Vlietnick2014}, Feynman variational calculations by Tempere et al. \cite{Tempere2009} and MF theory. We used the standard regularization scheme to cancel the leading power-law divergence of $E_\text{p}$. However, to enable comparison with the MC data, we did \emph{not} regularize the UV log-divergence reported in this paper. Hence the result is sensitive to the UV cutoff chosen for the numerics, and we used the same value $\Lambda_0=2000 / \xi$ as in \cite{Vlietnick2014}. Other parameters are $M/m=0.263158$ and $P=0$.}
\label{fig:FIG1}
\end{figure}
Recent experimental progress in the field of ultracold atoms brought new interest in the study of impurity problems.
Feshbach resonances made it possible to realize both Fermi \cite{Schirotzek2009,Nascimbene2009,Koschorreck2012,kohstall2012metastability,Zhang2012,Massignan_review}
and Bose polarons \cite{Catani2012,Fukuhara2013} with tunable interactions between the impurity and host atoms. Detailed information
about Fermi polarons was obtained using a rich toolbox available in these experiments. Radio frequency (rf) spectroscopy was used to measure the polaron binding energy and to observe the transition between the
polaronic and molecular states \cite{Schirotzek2009}. The effective mass of Fermi polarons was studied using measurements of
collective oscillations in a parabolic confining potential \cite{Nascimbene2009}. Polarons in a Bose-Einstein condensate (BEC) received
less experimental attention so far although polaronic effects have been observed in nonequilibrium dynamics
of impurities in 1d systems \cite{Palzer2009,Catani2012,Fukuhara2013}.
The goal of this paper is two-fold. Our first goal is to present a new theoretical technique for analyzing
a common class of polaron problems, the so-called Fr\"ohlich polarons. We develop a unified approach that can describe
polarons all the way from weak to strong couplings. Our second goal is to apply this method to the problem of impurity atoms immersed in a BEC. We focus on calculating the polaron binding energy and effective mass, both of which can
be measured experimentally. Considering a wide range of atomic mixtures with tunable interactions \cite{Chin2010} and very different mass
ratios available in current experiments \cite{Egorov2013,Spethmann2012,Pilch2009,Lercher2011,mccarron2011dual,Catani2008,Wu2012a,Park2012,Schreck2001,truscott2001observation,Shin2008,Bartenstein2005,Roati2002,Ferlaino2006,Ferlaino2006err,Inouye2004,Scelle2013,Hadzibabic2002,Stan2004,Schuster2012,schmid2010dynamics},
we expect that many of our predictions can be tested in the near future. In particular we discuss that currently available technology should make it possible to realize intermediate coupling polarons.
Previously the problem of an impurity atom in a superfluid Bose gas has been studied theoretically using the weak coupling
MF ansatz \cite{BeiBing2009,Shashi2014RF}, the strong coupling approximation \cite{Cucchietti2006,Sacha2006,Bruderer2007,Bruderer2008,Casteels2011}, the variational Feynman path integral approach \cite{Tempere2009}, and the numerical diagrammatic MC simulations \cite{Vlietnick2014}.
The four methods predicted sufficiently different polaron binding energies in the regimes of intermediate and strong interactions, see FIG.\ref{fig:FIG1}. While the MC result can be considered as the most reliable of them, the physical insight gained from this approach is limited.
Our new method builds upon earlier analytical approaches by considering fluctuations on top of the MF state and including correlations between different modes using the RG approach. We verify the accuracy of this method by demonstrating excellent agreement with the MC results \cite{Vlietnick2014} at zero momentum and for intermediate interaction strengths.
Our method provides new insight into polaron states at intermediate and strong coupling by showing the importance of entanglement between phonon modes at different energies. A related perspective on this entanglement was presented in Ref. \cite{Shchadilova2014}, which developed a variational Gaussian wavefunction for Fr\"ohlich polarons. Throughout the paper we will compare our RG results to the results computed with this variational correlated Gaussian wavefunction approach. In particular, we use our method to calculate the effective mass of polarons, which is a subject of special interest for many physical applications and remains an area of much controversy. At the end, we also comment on extensions of our approach to non-equilibrium problems.
This paper is organized as follows. We introduce the Fr\"ohlich Hamiltonian in Section \ref{sec:Model} and discuss how it can be used to describe mobile impurity atoms interacting with the phonons of a BEC. In Section \ref{sec:RevMF} we review MF results for the Fr\"ohlich polaron and formulate a Hamiltonian that describes fluctuations on top of the MF state. We discuss the solution of this Hamiltonian using an RG approach in Section \ref{sec:RGana}. In Section \ref{sec:regularization} we investigate cutoff dependencies of the polaron energy and describe how they should be properly regularized. Section \ref{sec:Results} provides a detailed discussion of our results for both the polaron binding energy and the effective mass. We show that the effective mass should be a much better probe of beyond MF aspects of the system. In Section\ref{Supp:ExperimentalConsiderations} we discuss possible experimental realizations and challenges, before closing with an outlook in Section \ref{sec:outlook}.
\section{Fr\"ohlich Hamiltonian}
\label{sec:Model}
The Fr\"ohlich Hamiltonian represents a generic class of models in which a single quantum mechanical particle interacts with the phonon reservoir of the host system. In particular it can describe the interaction of an impurity atom with the Bogoliubov modes of a BEC \cite{Cucchietti2006,Bruderer2007,Tempere2009}. In this section we review this model in both its original form and following the Lee-Low-Pines (LLP) transformation into the impurity reference frame. The goal of the LLP transformation is to use conservation of the total momentum to eliminate the impurity degrees of freedom, at the cost of introducing interactions between phonon modes.
Our starting point is the Fr\"ohlich Hamiltonian describing the interaction between an impurity atom and phonon modes of the BEC ($\hbar=1$)
\begin{eqnarray}
\H_{\rm FROL} &=& \H_{\rm PHON} + \H_{\rm IMP} +\H_{\rm INT},
\nonumber\\
\H_{\rm PHON} &=& \int_k d^dk ~ \omega_k \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}},
\nonumber\\
\H_{\rm IMP} &=& \frac{P^2}{2M},
\nonumber\\
\H_{\rm INT} &=& \int_{|k|<\Lambda_0} d^d k~ V_k (\hat{a}^\dagger_{\vec{k}} + \a_{-\vec{k}}) e^{i \vec{k} \cdot \vec{R}}.
\label{H_frolich}
\end{eqnarray}
Here $M$ denotes the impurity mass and $m$ the mass of the host bosons, $\a_{\vec{k}}$ is the annihilation operator of the Bogoliubov phonon excitation in a BEC with momentum $\vec{k}$, $\vec{P}$ and $\vec{R}$ are momentum and position operators of the impurity atom, $d$ is the dimensionality of the system and $\Lambda_0$ is a high momentum cutoff needed for regularization.
The dispersion of phonon modes of the BEC and their interaction with the impurity atom are given by the standard Bogoliubov expressions \cite{Tempere2009}
\begin{eqnarray}
\omega_k &=& c k \sqrt{1+\frac{1}{2} \xi^2 k^2},
\nonumber\\
V_k &=& \sqrt{n_0} (2 \pi)^{-3/2} g_\text{IB} \l \frac{(\xi k)^2}{2 + (\xi k)^2} \r^{1/4},
\end{eqnarray}
with $n_0$ being the BEC density and $\xi = \l 2 m g_\text{BB} n_0 \r^{-1/2}$ the healing (or coherence) length and $c=\l g_\text{BB} n_0 / m \r^{1/2}$ the speed of sound of the condensate. Here $g_{\rm IB}$ denotes the interaction strength between the impurity atom with the bosons, which in the lowest order Born approximation is given by $g_{\rm IB}=2\pi a_{\rm IB}/m_{\rm red}$, where $a_{\rm IB}$ is the scattering length and $m_{\rm red}^{-1}=M^{-1}+m^{-1}$ is the reduced mass of a pair consisting of impurity and bosonic host atoms. Similarly, $g_\text{BB}$ is the boson-boson interaction strength. The analysis of the UV divergent terms in the polaron energy will require us to consider a more accurate cutoff dependent relation between $g_{\rm IB}$ and the scattering length $a_{\rm IB}$ (see Eq. (\ref{eq:LSE}) below).
When we calculate the energy of the impurity atom in the BEC we need to consider the full expression $E_{\rm IMP}= E_{\rm IB}^0 + E_{\rm B}$, where $E_{\rm IB}^0 = g_{\rm IB} n_0$ is the MF interaction energy of the impurity with bosons from the condensate, and $E_\text{B} = \langle \H_\text{FROL} \rangle_\text{gs}$ is the groundstate energy of the Fr\"ohlich Hamiltonian. From now one we will call $E_{\rm IB}^0$ the impurity-condensate interaction energy and $E_{\rm B}$ the polaron binding energy. As we discuss below only $E_{\rm IMP}$ is physically meaningful and can be expressed in a universal cutoff independent way using the scattering length $a_{\rm IB}$. Precise conditions under which one can
use the Fr\"ohlich model to describe the impurity BEC interaction, and parameters of the model for specific cold atoms mixtures are discussed in Sec. \ref{Supp:ExperimentalConsiderations}. We point out that the Fr\"ohlich type Hamiltonians \eqref{H_frolich} are relevant for many systems besides BEC-impurity polarons. Its original and most common use is in the context of electrons coupled to crystal lattice fluctuations in solid state systems \cite{Froehlich1954}. Another important application area is for studying doped quantum magnets, in which electrons and holes are strongly coupled to magnetic fluctuations. Motivated by this generality of the model \eqref{H_frolich} we will analyze it for a broader range of parameters than may be relevant for the current experiments with ultra cold atoms.
The Hamiltonian \eqref{H_frolich} describes a translationally invariant system. It is convenient
to perform the Lee-Low-Pines (LLP) transformation \cite{Lee1953} that separates the system into decoupled sectors of conserved total momentum,
\begin{eqnarray}
\hat{U} &=& e^{i \hat{S}} \hspace{1cm} \hat{S}= \vec{R} \cdot \int d^d k ~ \vec{k} \hat{a}^\dagger_k \a_k
\label{LLP transformation}
\\
\H_{\rm LLP}&=& \hat{U}^\dagger \H_\text{FROL} \hat{U} = \frac{1}{2 M} \Bigl( \vec{P} - \int d^d k~\vec{k} \hat{a}^\dagger_k \a_k \Bigr)^2 +
\nonumber \\
& & + \int d^d k ~ \left[ \omega_k \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} + V_k \l \hat{a}^\dagger_{\vec{k}} + \a_{-\vec{k}} \r \right].
\label{H_LLP}
\end{eqnarray}
The transformed Hamiltonian \eqref{H_LLP} does no longer contain the impurity position operator $\vec{R}$. Thus $\vec{P}$ in equation
(\ref{H_LLP}) is a conserved net momentum of the system and can be treated as a $\mathbb{C}$-number (rather than an operator).
Alternatively, the transformation \eqref{LLP transformation} is commonly described as going into the impurity frame, since the term describing
boson scattering on the impurity in \eqref{H_LLP} is obtained from the corresponding term in \eqref{H_frolich} by setting $\vec{R}=0$.
The Hamiltonian (\ref{H_LLP}) has only phonon degrees of freedom but they now interact with each other. This can be understood physically as a phonon-phonon interaction, mediated by an exchange of momentum with the impurity atom. This impurity-induced interaction between phonons in \eqref{H_frolich} is proportional to $1/M$. Thus in our subsequent analysis of the polaron properties, which is based on the LLP transformed Fr\"ohlich Hamiltonian, we will consider $1/M$ as controlling the interaction strength.
\section{Review of the Mean Field Approximation}
\label{sec:RevMF}
In this section we briefly review the MF approach to the polaron problem, which provides an accurate description of the system in the weak coupling regime. We discuss how one should regularize the MF interaction energy, which is UV divergent for $d \geq 2$. To set the stage for subsequent beyond MF analysis of the polaron problem, we derive the Hamiltonian that describes fluctuations around the MF state.
The MF approach to calculating the ground state properties of \eqref{H_LLP} is to consider a variational wavefunction in which all phonons are taken to be in a coherent state \cite{Lee1953}. The MF variational wavefunction reads
\begin{eqnarray}
| \psi_{\rm MF} \rangle = e^{\int d^3 \vec{k} ~ \alpha^\text{MF}_{\vec{k}} \hat{a}^\dagger_{\vec{k}} - \text{h.c.} } \ket{0} = \prod_{\vec{k}} \ket{\alpha_{\vec{k}}}.
\label{Psi_MF}
\end{eqnarray}
It becomes exact in the limit of an infinitely heavy (i.e. localized) impurity.
Energy minimization with respect to the variational parameters $\alpha_{\vec{k}}$ gives
\begin{eqnarray}
\alpha_{\vec{k}}^\text{MF} = - \frac{V_k}{ \Omega^\text{MF}_{\vec{k}}} = - \frac{V_k}{\omega_k + \frac{k^2}{2M} - \frac{\vec{k}}{M} \cdot \l \vec{P} - \vec{P}^\text{MF}_\text{ph} \r},
\label{alpha_k}
\end{eqnarray}
where $\vec{P}^\text{MF}_\text{ph}$ is the momentum of the system carried by the phonons. It has to be determined self-consistently from the solution \eqref{alpha_k},
\begin{eqnarray}
\vec{P}^\text{MF}_\text{ph} = \int d^d k ~ \vec{k} |\alpha^\text{MF}_{\vec{k}}|^2.
\label{Theta_MF}
\end{eqnarray}
The MF character of the wave function (\ref{Psi_MF}) is apparent from the fact that it is a product of wave functions for individual phonon modes. Hence it contains neither entanglement nor correlations between different modes. The only interaction between modes is through the selfconsistency equation (\ref{Theta_MF}).
Properties of the MF solution have been discussed extensively in Refs. \cite{Lee1953,Devreese2013,Shashi2014RF}. Here we reiterate only one important issue
related to the high energy regularization of the MF energy \cite{Tempere2009}. In $d \geq 2$ dimensions the expression for the MF energy,
\begin{eqnarray}
E_{\rm B}^\text{MF} = \frac{P^2}{2 M} - \frac{\l P_\text{ph}^{\text{MF}} \r^2 }{2M} - \int_{|k|<\Lambda_0} d^d k ~ \frac{V_k^2}{\Omega^\text{MF}_{\vec{k}}},
\label{EBmeanfield}
\end{eqnarray}
is UV divergent as the high momentum cutoff $\Lambda_0$ is sent to infinity. In order to regularize this expression we recall that the physical energy of the impurity is a sum of $E^0_{\rm IB}$ and the polaron binding energy $E_\text{B}$. If we use the leading order Born approximation to express $g_{\rm IB} = 2\pi a_{\rm IB}/m_{\rm red}$, we observe that the MF polaron energy has contributions starting with the second order in $a_{\rm IB}$. Consistency requires that the impurity-condensate interaction energy $E_{\rm IB}^0 = g_{\rm IB} n_0$ is computed to order $a_{\rm IB}^2$. The Lippman-Schwinger equation provides the relation between the microscopic interaction $g_{\rm IB}$, the cutoff $\Lambda_0$ and the physical scattering length $a_{\rm IB}$ \cite{Pethick2008},
\begin{align}
a_\text{IB} &= \frac{g_\text{IB}}{2 \pi} m_\text{red} - \frac{g_{\text{IB}}^2}{(2\pi)^4} m_\text{red} \int_{|k|<\Lambda_0} d^d k ~ \frac{2 m_\text{red}}{k^2}.
\label{eq:LSE}
\end{align}
To second order in $a_{\rm IB}$ one has
\begin{equation}
E^0_{\rm IB}= \frac{2 \pi a_{\rm IB} n_0}{m_{\rm red}} + \frac{a_\text{IB}^2 n_0}{\pi m_\text{red}} \int_{|k|<\Lambda_0} d^dk ~ \frac{1}{k^2}.
\label{EIB02ndorder}
\end{equation}
Now we recognize that in the physically meaningful impurity energy $E_\text{IMP} = E_\text{IB}^0 + E_\text{B}^\text{MF}$ the UV divergence cancels between $E_{\rm IB}^0$ and
$E_{\rm B}^{\rm MF}$.
Thus separation of the impurity energy into the impurity-condensate interaction $E_\text{IB}^0=n_0 g_\text{IB}$ and the binding energy $E_\text{B}$ is not physically meaningful. A decomposition of the impurity energy in orders of the scattering length $a_\text{IB}$, on the other hand, is well defined,
\begin{equation}
E_{\rm IMP} = \frac{2 \pi a_{\rm IB} n_0}{m_{\rm red}} + E_\text{p}(a_\text{IB}^2) ,
\label{eq:EpDef}
\end{equation}
where the second-order term is referred to as the polaronic energy of the impurity \cite{Tempere2009}. To characterize the polaronic energy, it is convenient to introduce a dimensionless coupling constant,
\begin{equation}
\alpha = 8 \pi n_0 a_\text{IB}^2 \xi,
\end{equation}
highlighting the analogy e.g. to the solid state Fr\"ohlich model.
The main shortcoming of the MF ansatz \eqref{Psi_MF} is that it discards correlations between phonons with different energies and at different momenta. The goal of this paper is to develop a method that allows us to go beyond the MF solution (\ref{Psi_MF}) and include correlations between modes. In the following we demonstrate how this can be accomplished, and discuss physical consequences of phonon correlations. To simplify the subsequent discussion we perform a unitary transformation that shifts the phonon variables in Eq. \eqref{H_LLP} by the amount corresponding to the MF solution,
\begin{eqnarray}
\hat{V} &=& \exp \left[ \begingroup\textstyle \int\endgroup d^d k ~ \alpha_{\vec{k}}^\text{MF} \hat{a}^\dagger_{\vec{k}} - {\rm h.c.} \right]
\nonumber\\
\tilde{\cal H}_{\rm LLP} &=& \hat{V}^\dagger \H_{\rm LLP} \hat{V}
\nonumber \\
&=& E_\text{B}^\text{MF} + \int_k ~ \Omega^\text{MF}_{\vec{k}} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} +
\int_{kk'} \frac{A_{\vec{k} \vec{k}'}}{2} :\hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{k}'}:. \qquad
\label{H_fluct}
\end{eqnarray}
Here we used $A_{\vec{k} \vec{k}'} = \frac{\vec{k} \cdot \vec{k}'}{M}$ and $\hat{\Gamma}_{\vec{k}} := \alpha^\text{MF}_{\vec{k}} ( \a_{\vec{k}} + \hat{a}^\dagger_{\vec{k}} ) + \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}}$, $: ... :$ stands for normal-ordering and we introduced the short-hand notation $\int_k = \int_{|k|<\Lambda_0} d^d k$. The absence of terms linear in $\a_{\vec{k}}$ in the last equation reflects the fact that $\alpha_{\vec{k}}^\text{MF}$ correspond to the MF (saddle point) solution. We emphasize that (\ref{H_fluct}) is an exact representation of the original Fr\"ohlich Hamiltonian, where the operators $\a_{\vec{k}}$ describe quantum fluctuations around the MF polaron.
\section{RG Analysis}
\label{sec:RGana}
\begin{table}[b]
\begin{tabular}{ p{5cm} | p{2.4cm} }
operator & scaling ($\Lambda \xi \ll 1$) \\[1ex]
\hline
$\a_{\vec{k}}$ & $\Lambda^{-(d+1)/2}$ \\ [0.8ex]
\hline
$ \int_{|k|<\Lambda} d^d k ~ d^d k' ~ \frac{\vec{k} \cdot \vec{k}'}{2 M} \alpha^\text{MF}_{\vec{k}} \alpha^\text{MF}_{\vec{k}'} ~ \a_{\vec{k}} \a_{\vec{k}'}$ & $\Lambda^d$ \\[1ex]
$ \int_{|k|<\Lambda} d^d k ~ d^d k' ~ \frac{\vec{k} \cdot \vec{k}'}{2 M} \alpha^\text{MF}_{\vec{k}} ~ \hat{a}^\dagger_{\vec{k}'} \a_{\vec{k}'} \a_{\vec{k}} $ & $\Lambda^{d/2}$ \\[1ex]
$ \int_{|k|<\Lambda} d^d k ~ d^d k' ~ \frac{\vec{k} \cdot \vec{k}'}{2 M} \hat{a}^\dagger_{\vec{k}'} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} \a_{\vec{k}'}$ & $\Lambda^0=1$
\end{tabular}
\caption{Dimensional analysis is performed by power-counting of the different terms describing quantum fluctuations around the MF polaron state. We fixed the scaling dimension of $\a_{\vec{k}}$ such that $\int_{|k|<\Lambda} d^dk~ \Omega_{\vec{k}} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} \stackrel{!}{\sim} \Lambda^0$ is scale-invariant.}
\label{tab:dimAn}
\end{table}
In this section we provide the RG solution of the Hamiltonian \eqref{H_fluct}, describing quantum fluctuations on top of the MF polaron state. We begin with a dimensional analysis of different terms in \eqref{H_fluct} in the long wavelength limit, which establishes that only one of the interaction terms is marginal and all others are irrelevant. Then we derive the RG flow equations for parameters of the model, including the expression for the polaron binding energy.
Our approach to the RG treatment of the model (\ref{H_fluct}) is similar to the "poor man's RG" in the context of the Kondo problem. We use Schrieffer-Wolff type transformation to integrate out high energy phonons in a thin shell in momentum space
near the cutoff, $\Lambda -\delta \Lambda < k < \Lambda$, using $1/\Omega_{\vec{k}}$ as a small parameter ($\Omega_{\vec{k}} = \Omega_{\vec{k}}^\text{MF}$ being the frequency of phonons in the thin momentum-shell). This transformation renormalizes the effective Hamiltonian for the low energy phonons. Iterating this procedure we get a flow of the effective Hamiltonian with the cutoff parameter $\Lambda$
\footnote{For readers not familiar with the RG methods
we note that this procedure is conceptually similar to the Born-Oppenheimer approximation for the problem of interacting ions and electrons in a molecule. Since ions are much heavier (and thus slower) than electrons, one can consider them as static and calculate the ground state of electrons for a given configuration of ions. Then the ground state energy of the electrons becomes an effective potential for the ions. In our case, fast degrees of freedom are phonons in the shell that we are integrating out, and slow degrees of freedom are phonons at lower energy.}.
To analyze whether the system flows to strong or weak coupling in the long wavelength limit $| \vec{k} | \xi \ll 1$ we consider scaling dimensions of different operators in \eqref{H_fluct}. We fix the dimension of $\a_{\vec{k}}$ using the condition that
$
\int_{|k|<\Lambda} d^dk~ \Omega_{\vec{k}} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}}
$ is scale invariant. In the long wavelength limit phonons have linear dispersion $\Omega_{\vec{k}} \propto |\vec{k}|$, which requires scaling of $\a_{\vec{k}}$ as $\Lambda^{-\frac{d+1}{2}}$. Scaling dimensions of different contributions to the interaction part of the Hamiltonian \eqref{H_fluct} is shown in table \ref{tab:dimAn}.
We observe that, as the cutoff scale tends to zero, most terms are irrelevant and only the quartic term
$\int_{kk'} \frac{\vec{k} \cdot \vec{k}'}{2 M} ~ \hat{a}^\dagger_{\vec{k}'} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} \a_{\vec{k}'}$ is marginal. As we demonstrate below, this term is marginally irrelevant, i.e. in the process of RG flow the impurity mass $M$ flows to large values. This feature provides justification for doing the RG perturbation expansion with $1/M$ as the interaction parameter. Also, an irrelevance of the interaction under the RG flow physically means that at least slow phonons in the system
are Gaussian. This provides an insight why the variational correlated Gaussian wavefunctions \cite{Shchadilova2014} are applicable for the Fr\"ohlich Hamiltonian under consideration.
To facilitate the subsequent discussion of the RG procedure we write a generalized form of Eq. \eqref{H_fluct} that allows for the additional terms in the effective Hamiltonian which will be generated in the process of the RG,
\begin{multline}
\tilde{\cal H}_{\rm RG}(\Lambda) = E_\text{B}+ \int_{|k|<\Lambda} d^d k~ \l \Omega_{\vec{k}} \hat{a}^\dagger_{\vec{k}} \a_{\vec{k}} + W_{\vec{k}} ( \hat{a}^\dagger_{\vec{k}} + \a_{\vec{k}} ) \r \\
+\frac{1}{2} \int_{|k|,|k'|<\Lambda} d^d k ~ d^d k' ~ k_\mu \mathcal{M}_{\mu \nu}^{-1} k_\nu' : \hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{k}'} :.
\label{H_RG}
\end{multline}
Note that the interaction is now characterized by a general tensor $\mathcal{M}^{-1}_{\mu\nu}$ \footnote{The indices $\mu=x,y,z,...$ label cartesian coordinates and they are summed over when occurring twice.}, where the anisotropy originates from the total momentum of the polaron $\vec{P} = P \vec{e}_x$, breaking the rotational symmetry of the system. Due to the cylindrical symmetry of the problem, the mass tensor has the form $\mathcal{M}= \text{diag} ( \mathcal{M}_\parallel , \mathcal{M}_\perp, \mathcal{M}_\perp,... )$, and we will find different flows for the longitudinal and the transverse components of the mass tensor. While $\mathcal{M}$ can be interpreted as the (tensor-valued) renormalized mass of the impurity, it should not be confused with the mass of the polaron. The first line of Eq.\eqref{H_RG} describes the diagonal quadratic part of the renormalized phonon Hamiltonian. It is also renormalized compared to the original expression in Eq.\eqref{H_fluct},
\begin{equation}
\Omega_{\vec{k}} = \omega_k + \frac{1}{2} k_\mu \mathcal{M}_{\mu \nu}^{-1} k_\nu - \frac{\vec{k}}{M} \cdot \l \vec{P} - \vec{P}_\text{ph} \r ,
\label{eq:Ok}
\end{equation}
where the momentum carried by the phonon-cloud, $\vec{P}_\text{ph}$, acquires an RG flow, describing corrections to the MF result $\vec{P}_\text{B}^\text{MF}$. In addition there is a term linear in the phonon operators, weighted by
\begin{equation}
W_{\vec{k}} = \left[ \l \vec{P}_{\text{ph}} - \vec{P}_{\text{ph}}^\text{MF} \r \cdot \frac{\vec{k}}{M} + \frac{k_\mu k_\nu}{2} \l \mathcal{M}_{\mu \nu}^{-1} - \frac{\delta_{\mu \nu}}{ M} \r \right] \alpha_{\vec{k}}^\text{MF}.
\label{eq:defWk}
\end{equation}
By comparing Eq.\eqref{H_RG} to Eq.\eqref{H_fluct} we obtain the initial conditions for the RG, starting at the original UV cutoff $\Lambda_0$ where $ \tilde{\cal H}_{\rm RG}(\Lambda_0) = \tilde{\cal H}_{\rm LLP}$,
\begin{equation}
\mathcal{M}_{\mu \nu}(\Lambda_0) = \delta_{\mu \nu} M, \quad \vec{P}_\text{ph}(\Lambda_0) = \vec{P}_\text{ph}^\text{MF}, \quad E_\text{B}(\Lambda_0)=E_\text{B}^\text{MF}.
\end{equation}
\begin{figure*}[t]
\centering
\epsfig{file=FIG2, width=0.9\textwidth}
\caption{Typical RG flows of the (inverse) renormalized impurity mass $\mathcal{M}^{-1}$ (a) and the excess phonon momentum $P_\text{ph}-P_\text{ph}^\text{MF}$ along the direction of the system momentum $P$ (b). Results are shown for different coupling strengths $\alpha$ and we used parameters $M/m=0.3$, $P/Mc=0.5$ and $\Lambda_0=20 / \xi$ in $d=3$ dimensions.}
\label{fig:FIG2}
\end{figure*}
We now separate phonons into "fast" ones with momenta $\vec{p}$ and "slow" ones with momenta $\vec{k}$, according to $\Lambda -\delta \Lambda < | \vec{p} | < \Lambda$ and $ | \vec{k} | \leq \Lambda -\delta \Lambda$. Then the Hamiltonian \eqref{H_RG} can be split into
\begin{eqnarray}
\tilde{\cal H}_{\rm RG}(\Lambda) &=& \H_\text{S} + \H_\text{F} +\H_\sf
\nonumber\\
\H_{\rm F} &=& \int_\text{F} d^d p~ \left[ \Omega_{\vec{p}} \hat{a}^\dagger_{\vec{p}} \a_{\vec{p}} + W_{\vec{p}} ( \hat{a}^\dagger_{\vec{p}} + \a_{\vec{p}} ) \right]
\nonumber\\
{\cal H}_{\rm MIX} &=& \int_\text{S} d^d k \int_\text{F} d^d p ~ k_\mu \mathcal{M}_{\mu \nu}^{-1} p_\nu' ~ \hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{p}},
\label{Hsplit}
\end{eqnarray}
where we use the short-hand notations $\int_\text{F} d^dp = \int_{\vec{p}~ \text{fast}} d^dp$ and $\int_\text{S} d^dk = \int_{\vec{k}~ \text{slow}} d^dk$. The slow-phonon Hamiltonian $\H_\text{S}$ is given by Eq.\eqref{H_RG} except that all integrals only go over slow phonons, $\int_{|\vec{k}|<\Lambda} d^dk \rightarrow \int_\text{S} d^dk$.
In $\H_\text{F} $ we do not have a contribution due to the interaction term since it would be proportional to $\delta \Lambda^2$ and we will consider the limit $\delta \Lambda \rightarrow 0$. We can obtain intuition into the nature of the transformation needed to decouple fast from slow phonons, by observing that for the fast phonons the Hamiltonian \eqref{Hsplit} is similar to a harmonic oscillator in the presence of an external force (recall that $\hat{\Gamma}_{\vec{p}}$ contains only linear and quadratic terms in $\a_{\vec{p}}^{(\dagger)}$). This external force is determined by the state of slow phonons. Thus it is natural to look for the transformation as a shift operator for the fast phonons,
\begin{eqnarray}
\hat{W}_{\rm RG} = \exp \l \int_\text{F} d^3 \vec{p} ~ \left[ \hat{F}_{\vec{p}}^\dagger \a_{\vec{p}} - \hat{F}_{\vec{p}} \hat{a}^\dagger_{\vec{p}} \right] \r,
\end{eqnarray}
with coefficients $\hat{F}_{\vec{p}}$ depending on the slow phonons only, i.e. $[\hat{F}_{\vec{p}},\a_{\vec{p}}^{(\dagger)}]=0$. One can check that taking
\begin{widetext}
\begin{multline}
\hat{F}_{\vec{p}} = \frac{1}{\Omega_{\vec{p}}} \left[
W_{\vec{p}} + \alpha^\text{MF}_{\vec{p}} p_\mu \mathcal{M}_{\mu \nu}^{-1}
\int_\text{S} d^d k ~ k_\nu \hat{\Gamma}_{\vec{k}} \right] -
\frac{1}{\Omega_{\vec{p}}^2} \left[ \alpha^\text{MF}_{\vec{p}} p_\mu \mathcal{M}_{\mu \nu}^{-1} \int_\text{S} d^d k ~ \Omega_{\vec{k}}^\text{MF} k_\nu \alpha^\text{MF}_{\vec{k}} \l \hat{a}^\dagger_{\vec{k}} - \a_{\vec{k}} \r + \right. \\ \left. +
\l W_{\vec{p}} + \alpha^\text{MF}_{\vec{p}} p_\mu \mathcal{M}_{\mu \nu}^{-1}
\int_\text{S} d^d k ~ k_\nu \hat{\Gamma}_{\vec{k}} \r p_\sigma \mathcal{M}_{\sigma \lambda}^{-1} \int_\text{S} d^d k ~ k_\lambda \hat{\Gamma}_{\vec{k}} \right]
\label{eq:Fresult}
\end{multline}
eliminates non-diagonal terms in $\a_{\vec{p}}^{(\dagger)}$ up to second order in $1/\Omega_{\vec{p}}$.
After the transformation we find
\begin{eqnarray}
\hat{W}^\dagger_{\rm RG} &\tilde{{\cal H}}_{\rm RG}&(\Lambda) \hat{W}_{\rm RG} =
\H_\text{S} + \delta \H_\text{S}
+ \delta E_0 + \int_\text{F} d^d p ~ \l \Omega_{\vec{p}} + \Delta \hat{\Omega}_{\vec{p}} \r \hat{a}^\dagger_{\vec{p}} \a_{\vec{p}},
\label{Transformed_H}
\end{eqnarray}
\end{widetext}
\begin{eqnarray}
\Delta \hat{\Omega}_{\vec{p}} &=& p_\mu \mathcal{M}_{\mu \nu}^{-1} \int_\text{S} d^d k ~ k_\nu \hat{\Gamma}_{\vec{k}},
\label{DeltaOmegaSlow}
\\
\delta \H_\text{S} &=& - \int_\text{F} d^d p ~ \frac{1}{\Omega_{\vec{p}}} \left[ W_{\vec{p}} + \alpha_{\vec{p}}^\text{MF} \Delta \hat{\Omega}_{\vec{p}} \right]^2,
\label{dHs}
\end{eqnarray}
\begin{eqnarray}
\delta E_0 &=& \frac{1}{2} \int_\text{F} d^d p ~ p_\mu \l \mathcal{M}^{-1}_{\mu \nu} - \frac{\delta_{\mu \nu}}{M} \r p_\nu |\alpha_{\vec{p}}^\text{MF}|^2,
\end{eqnarray}
which is valid up to corrections of order $1/\Omega_{\vec{p}}^2$ or $\delta \Lambda^2$. The last equation describes a change of the zero-point energy $\delta E_0$ of the impurity in the potential created by the phonons, and it is caused by the RG flow of the impurity mass. To obtain this term we have to carefully treat the normal-ordered term $:\hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{k}'}:$ in Eq.\eqref{H_RG}
\footnote{The following relation is helpful to perform normal-ordering, $:\hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{k}'}: = \hat{\Gamma}_{\vec{k}} \hat{\Gamma}_{\vec{k}'} - \delta \l \vec{k} - \vec{k}' \r \left[ \hat{\Gamma}_{\vec{k}} + |\alpha^\text{MF}_{\vec{k}} |^2 \right]$.}.
We will show later that this contribution to the polaron binding energy is crucial because it leads to a UV divergence in $d\geq 3$ dimensions.
From the last term in Eq.\eqref{Transformed_H} we observe that the ground state $\ket{\text{gs}}$ of the Hamiltonian is obtained by setting
the occupation number of high energy phonons to zero, $ \bra{\text{gs}} \hat{a}^\dagger_{\vec{p}} \a_{\vec{p}} \ket{\text{gs}}=0$. Then from Eq.\eqref{dHs} we read off the change in the Hamiltonian for the low energy phonons. From the form of the operator $\Delta \hat{\Omega}_{\vec{p}}$ in Eq.\eqref{DeltaOmegaSlow} one easily shows that the new Hamiltonian $\H_\text{S}+\delta \H_\text{S}$ is of the universal form $\tilde{{\cal H}}_{\rm RG}$, but with renormalized couplings. Thus we derive the following flow equations
for the parameters in $\tilde{{\cal H}}_{\rm RG} (\Lambda)$,
\begin{eqnarray}
\frac{\partial \mathcal{M}_{\mu \nu}^{-1} }{\partial \Lambda} &=& 2 \mathcal{M}_{\mu \lambda}^{-1} \int_\text{F} d^{d-1} p ~ \frac{ | \alpha_{\vec{p}}^\text{MF} |^2 }{\Omega_{\vec{p}}} p_\lambda p_\sigma ~\mathcal{M}_{\sigma \nu}^{-1},
\label{eq:gsFlowM}
\\
\frac{\partial P_{\text{ph}}^\mu }{\partial \Lambda} &=& - 2 \mathcal{M}_{\mu \nu}^{-1} \int_\text{F} d^{d-1} p ~ \Big[ \l \vec{P}_\text{ph}^\text{MF} - \vec{P}_{\text{ph}} \r \cdot \vec{p} +
\nonumber \\
&& + \frac{1}{2} p_\sigma \l \delta_{\sigma \lambda} - M \mathcal{M}_{\sigma \lambda}^{-1} \r p_\lambda \Big] \frac{| \alpha_{\vec{p}}^\text{MF} |^2 }{\Omega_{\vec{p}}} p_\nu.
\label{eq:gsFlowQ}
\end{eqnarray}
Here we use the notation $\int_\text{F} d^{d-1} p$ for the integral over the $d-1$ dimensional surface defined by momenta of length $|\vec{p}| = \Lambda$. The energy correction to the binding energy of the polaron beyond MF theory, $E_\text{B} = E_\text{B}^\text{MF} + \Delta E_\text{B}^{\rm RG}$, is given by
\begin{multline}
\Delta E_\text{B}^{\rm RG}= - \int_{|k| < \Lambda_0} d^d k \left\{ \frac{1}{\Omega_{\vec{k}} } \left| W_{\vec{k}} \right|^2 + \right. \\
\left. + \frac{1}{2} |\alpha_{\vec{k}}^\text{MF} |^2 k_\mu \left[ \frac{\delta_{\mu \nu}}{M} - \mathcal{M}_{\mu \nu}^{-1}(\Lambda=k) \right] k_\nu \right\}.
\label{Energy_renormalization}
\end{multline}
Note that, in this expression, we evaluated the renormalized impurity mass $\mathcal{M}_{\mu \nu}(\Lambda=k)$ at a value of the running cut-off $\Lambda=k$ given by the integration variable $k=|\vec{k}|$. Similarly, it is implicitly assumed that $\vec{P}_\text{ph}(\vec{k})$ and $\mathcal{M}(\vec{k})$ appearing in the expressions for $W_{\vec{k}}$ and $\Omega_{\vec{k}}$, see Eqs.\eqref{eq:defWk} and \eqref{eq:Ok}, are evaluated at $\Lambda=k$.
FIG.\ref{fig:FIG2} shows typical RG flows of $\mathcal{M}_{\mu\nu}$ and $P_\text{ph}-P_\text{ph}^\text{MF}$. For $\Lambda \lesssim 1 / \xi$ we observe quick convergence of these coupling constants. One can see comparison of the MC and RG calculations \cite{Vlietnick2014} for the polaron binding energy at momentum $P=0$ in FIG.\ref{fig:FIG1}. The agreement is excellent for a broad range of interaction strengths. We will discuss these results further in Sec.\ref{sec:Results}.
\section{Regularizing cutoff dependence}
\label{sec:regularization}
In a three dimensional system ($d=3$), examination of the binding energy of the polaron $E_{\rm B}$ shows that it has logarithmic UV divergence, $E_\text{B} \sim - \log(\Lambda_0 \xi)$, even after applying the regularization procedure described in Sec.\ref{sec:RevMF}. In FIG.\ref{fig:FIGlogDiv} we show results from the RG (and from our variational calculation introduced in \cite{Shchadilova2014}) and compare them to MC calculations \cite{Vlietnick2014}. While the predicted overall energy scale differs somewhat, a logarithmic divergence can be identified in all three cases \footnote{In their paper \cite{Vlietnick2014}, Vlietinck et al. claim that the polaron energy is UV convergent. When plotting their data on a logarithmic scale as done in our FIG. \ref{fig:FIGlogDiv}, we think that this claim is not justified. Over a range of almost two decades we observe a clearly linear slope. Only for the largest values of the UV cutoff there are some deviations, which are on the order of the statistical errorbars, however.}.
Similar log-divergences related to vacuum fluctuations are well known from the Lamb-shift in quantum electrodynamics and have been predicted in relativistic polaron models \cite{Volovik2014} as well. In this section we derive the form of the log-divergence analytically using the RG, and discuss how the binding energy can be regularized. We focus on the zero momentum polaron $P=0$ and the experimentally most relevant case of $d=3$ dimensions, but the extensions to finite polaron momentum $P \neq 0$ or $d \neq 3$ are straightforward.
\begin{figure}[t]
\centering
\epsfig{file=FIG3, width=0.45\textwidth}
\caption{The polaronic contribution to the ground state energy $E_\text{p}$ (as defined in Eq.\eqref{eq:EpDef}) in $d=3$ dimensions is shown as a function of the UV momentum cutoff $\Lambda_0$ in logarithmic scale. We compare our results (RG - solid, variational \cite{Shchadilova2014} - dash-dotted) to MF theory as well as to predictions by Vlietinck et al. \cite{Vlietnick2014} (diagrammatic MC - squares, Feynman - dashed). The data shows a logarithmic UV divergence of the polaron energy. Parameters are $M/m=0.263158$, $P=0$ and $\alpha=3$.}
\label{fig:FIGlogDiv}
\end{figure}
To check whether the corrections $\Delta E_\text{B}^{\rm RG}$ to the MF polaron binding energy are UV divergent, we need the asymptotic form of the RG flow for the impurity mass $\mathcal{M}(\Lambda=k)$ at large momenta. For $P=0$ the flow equation \eqref{eq:gsFlowM} is separable, and from its exact solution we obtain the asymptotic behavior at the beginning of the RG flow (i.e. for high energies),
\begin{eqnarray}
\frac{M}{\mathcal{M}(\Lambda)} = 1-\frac{32}{3} \frac{n_0 a_\text{IB}^2 m_{\rm red}}{M} \frac{1}{\Lambda} + \mathcal{O}(\Lambda^{-2}),
\label{Masympt}
\end{eqnarray}
where we set $\Lambda_0 =\infty$. Using this result in Eq.\eqref{Energy_renormalization}, simple power-counting shows that the contribution from the zero-point energy of the impurity
\begin{equation}
\Delta E_0^{\rm RG} = - \int_{0}^{\Lambda_0} dk ~ 4 \pi k^2 \frac{|\alpha^\text{MF}_k|^2}{2M} k^2 \l 1- \frac{M}{\mathcal{M}(\Lambda=k)} \r
\label{DeltaE0divergent}
\end{equation}
becomes logarithmically UV divergent (recall that $\alpha^\text{MF}_ k\sim 1/ k^2$ at high momenta $k \gg \xi^{-1}$). The first term $-\int d^3k ~ |W_{\vec{k}}|^2/\Omega_{\vec{k}}$ in Eq.\eqref{Energy_renormalization}, on the other hand, is UV convergent because $W_k \sim k^{-1}$. From Eqs.\eqref{Masympt} and \eqref{DeltaE0divergent} we derive the following form of the UV divergence,
\begin{equation}
\Delta E_{\text{UV}}^{\rm RG} = - \frac{128}{3} \frac{m_{\rm red}}{M^2} n_0^2 a_\text{IB}^4 \log \l \Lambda_0 \xi \r.
\label{eq:EUVRG}
\end{equation}
We find that the slope predicted by this curve is in excellent agreement with the MC data shown in FIG.\ref{fig:FIGlogDiv}.
To regularize this divergence we again need to return to the impurity-condensate interaction energy $E^0_{\rm IB}$. When taking this energy to be $g_{\rm IB} n_0$
we understand that $g_{\rm IB}$ stands for the low energy part of the impurity-boson scattering amplitude, which we need to find from the Lippman-Schwinger equation
$
T= V +VGT
$.
Here $T$ denotes the $T$-matrix, $V$ is the scattering potential and $G$ is the free impurity propagator. In the case of a two particle interaction in vacuum, $G$ is taken as $-\int_{k<\Lambda_0} d^dk \frac{2m_{\rm red}}{k^2} \ket{k} \bra{k}$, and as discussed in Sec.\ref{sec:RevMF} we can
restrict ourselves to the second order in $V$ such that $T= V+VGV$. An important change in the polaronic problem compared to vacuum is that condensate atoms interact with an impurity atom when the latter is dressed by the polaronic cloud. Hence when calculating the impurity-condensate interactions, the $k$-dependence of the impurity mass (due to the RG) should be taken into account. This is achieved by calculating the $k \rightarrow 0$ limit of the scattering amplitude $f_{k \to 0} \stackrel{!}{=} - a_\text{IB}$ from the Lippmann-Schwinger equation with the dressed propagator $G^* = -\int_{k<\Lambda_0} d^dk \frac{2m^*_{\rm red}(k)}{k^2} \ket{k} \bra{k}$, where $(m_{\rm red}^*(k))^{-1} = M^{-1} + \mathcal{M}^{-1}(k)$. A comparison to the MF case Eq.\eqref{eq:LSE}, where the impurity mass does not flow, shows that there is an additional contribution to the impurity-condensate interaction $E^0_{\rm IB}$,
\begin{eqnarray}
\Delta E^0_{\rm IB} = \frac{4 a_\text{IB}^2 n_0}{m_{\text{red}}} \int^{\Lambda_0} dk ~ \l \frac{m_{\text{red}}^*(k)}{m_{\text{red}}} - 1 \r.
\label{eq:deltaE0IB}
\end{eqnarray}
By using the asymptotic solution \eqref{Masympt} in the last equation, together with the definition of $m_{\text{red}}^*(k)$, it is easy to check that the asymptotic behavior of $\Delta E^0_{\rm IB}$ is the same as $\Delta E_{\text{UV}}^{\rm RG}$, see Eq.\eqref{eq:EUVRG}, but with the opposite sign. Hence the resulting impurity energy is UV convergent. The final expression for the impurity energy, that is now free of all UV divergencies, is given by
\begin{eqnarray}
E_{\rm IMP} = \underbrace{E^\text{MF}_\text{B}}_{\rm Eq.\eqref{EBmeanfield}} + \underbrace{E_\text{IB}^0}_{\rm Eq.\eqref{EIB02ndorder}} + \underbrace{\Delta E^{\rm RG}_\text{B}}_{\rm Eq.\eqref{Energy_renormalization}} + \underbrace{\Delta E^0_{\rm IB}}_{\rm Eq.\eqref{eq:deltaE0IB}}.
\label{eq:EIMPfinal}
\end{eqnarray}
Let us recapitulate the meaning of all these terms: $E^\text{MF}_\text{B}$ describes the binding energy of the polaron at the MF level; $E_\text{IB}^0$ is the impurity-condensate interaction, and it regularizes the power-law divergence of the MF binding energy; $\Delta E^{\rm RG}_\text{B}$ is the correction to the polaron binding energy due to the RG; $\Delta E^0_{\rm IB}$ originates from the impurity-condensate interaction when taking into account the $k$-dependence of the impurity mass, including a correction that regularizes the log-divergence of the RG binding energy corrections.
\section{Results: polaron energy and effective mass}
\label{sec:Results}
\begin{figure}[t]
\centering
\epsfig{file=FIG4, width=0.47\textwidth}
\caption{The impurity energy $E_{\rm IMP}(\alpha)$, which can be measured in a cold atom setup using rf-spectroscopy, is shown as a function of the coupling strength $\alpha$. Our prediction from the RG is given by the solid black line, representing the fully regularized impurity energy from Eq.\eqref{eq:EIMPfinal}. We compare our results to MF theory (dashed). Note that, although MF yields a strict upper variational bound on the binding energy $E_\text{B}$, the MF impurity energy $E_{\rm IMP}$ is below the RG prediction because the impurity-condensate interaction $E_\text{IB}^0$ was treated more accurately in the latter case. We used parameters $M/m=0.26316$, $\Lambda_0=2000 / \xi$, $P=0$ and set the BEC density to $n_0=\xi^{-3}$.}
\label{fig:FIGenergy}
\end{figure}
Now we use the general formalism that we developed in the previous sections to calculate the polaron energy and the effective mass. We find an effective mass that agrees with the MF result for weak coupling and crosses over smoothly to the strong coupling regime. We also compare our analysis to the strong coupling Landau-Pekkar theory.
The fully renormalized impurity energy Eq.\eqref{eq:EIMPfinal} (calculated from the RG) is shown in FIG.\ref{fig:FIGenergy} as a function of the coupling strength $\alpha$. The resulting energy is close to, but slightly above, the MF energy \footnote{{This result might be surprising at first glance, because MF polaron theory relies on a variational principle and thus yields an upper bound for the groundstate energy. However, this bound holds only for the binding energy $E_\text{B}$, defined as the groundstate energy of the Fr\"ohlich Hamiltonian, and not for the entire impurity Hamiltonian including the condensate-impurity interaction.}}.
This is a consequence of the regularization for the log-divergence introduced in the last section, as can be seen by comparing to curves (MC, RG, variational) where only the power-law divergence was regularized. These curves, on the other hand, are in excellent agreement with each other. We thus conclude that the large deviations observed in FIG.\ref{fig:FIG1} of beyond MF theories (MC, RG, variational) from MF are merely an artifact of the logarithmic UV divergence which was not properly regularized. We note that this also explains -- at least partly -- the unexpectedly large deviations of Feynman's variational approach from the numerically exact MC results, reported in \cite{Vlietnick2014}, since Feynman's model does not capture the log-divergence (see FIG.\ref{fig:FIGlogDiv}).
Our results have important implications for experiments. The relatively small difference in energy between MF and (properly regularized) RG in FIG.\ref{fig:FIGenergy} demonstrates that a measurement of the impurity energy alone does not allow to discriminate between uncorrelated MF theories and extensions thereof (like RG or MC). Yet such a measurement would still be significant as a consistency check of our regularization scheme. To find smoking gun signatures for beyond MF behavior, other observables are required such as the effective polaron mass, which we discuss next.
\begin{figure}[t]
\centering
\epsfig{file=FIG5, width=0.5\textwidth}
\caption{The polaron mass $M_\text{p}$ (in units of $M$) is shown as a function of the coupling strength $\alpha$. We compare our results (RG) to Gaussian variational \cite{Shchadilova2014} and MF calculations, strong coupling theory \cite{Casteels2011} and Feynman's variational path-integral approach (data taken from Tempere et al.\cite{Tempere2009}). We used parameters $M/m=0.26$, $\Lambda_0=200 / \xi$ and set $P/Mc=0.01$.}
\label{fig:FIGpolaronMass}
\end{figure}
In FIG.\ref{fig:FIGpolaronMass} we show the polaron mass calculated using several different approaches. In the weak coupling limit $\alpha \to 0$ the polaron mass can be calculated perturbatively in $\alpha$, and the lowest-order result is shown in FIG.\ref{fig:FIGpolaronMass}. We observe that in this limit, many approaches follow the same line which asymptotically approaches the perturbative result (as $\alpha \to 0$). One exception is the strong coupling Landau-Pekkar approach, which only yields a self-trapped polaron solution beyond a critical value of $\alpha$ \cite{Casteels2011}. The second exception is Feynman's variational approach, for which we show data by Tempere et al. \cite{Tempere2009} in FIG.\ref{fig:FIGpolaronMass}. The discrepancy observed for small $\alpha$ is surprising since generally Feynman's approach is expected to become exact in the weak coupling limit \cite{Feynman1955,Devreese2013}.
For larger values of $\alpha$, MF theory sets a lower bound for the polaron mass. Naively this is expected, because MF theory does not account for quantum fluctuations due to couplings between phonons of different momenta. These fluctuations require additional correlations to be present in beyond MF wavefunctions, like e.g. in our RG approach, which should lead to an increased polaron mass. Indeed, for intermediate couplings $\alpha \gtrsim 1$ the RG, as well as the variational approach, predict a polaron mass $M_\text{p} > M_\text{p}^\text{MF}$ which is considerably different from the MF result \cite{Shashi2014RF}.
Before proceeding with further discussion of the results, we provide a few specifics on how we calculate the polaron mass. We employ a semi-classical argument to relate the average impurity velocity to the polaron mass $M_\text{p}$ and obtain
\begin{equation}
\frac{M}{M_\text{p}} = 1 - \frac{P_\text{ph}(0)}{P}, \quad P_\text{ph}(0)=\lim_{\Lambda \rightarrow 0} P_\text{ph}(\Lambda).
\label{eq:MpRG}
\end{equation}
The argument goes as follows. The average polaron velocity is given by $v_\text{p} = P / M_\text{p}$. The average impurity velocity $v_\text{I}$, which by definition coincides with the average polaron velocity $v_\text{I} = v_\text{p}$, can be related to the average impurity momentum $P_\text{I}$ by $v_\text{I} = P_\text{I}/M$. Because the total momentum is conserved, $P = P_\text{ph} + P_\text{I}$, we thus have $P / M_\text{p} = v_\text{p} = v_\text{I} = (P - P_\text{ph})/M$. Because the total phonon momentum $P_\text{ph}$ in the polaron groundstate is obtained from the RG by solving the RG flow equation in the limit $\Lambda \to 0$, we have $P_\text{ph} = P_\text{ph}(0)$ as defined above, and Eq.\eqref{eq:MpRG} follows. We note that in the MF case this result is exact and can be proven rigorously, see \cite{Shashi2014RF}.
In FIG.\ref{fig:FIGpolaronMass} we present another interesting aspect of our analysis, related to the nature of the cross-over \cite{Gerlach1988,Gerlach1991} from weak to strong coupling polaron regime. While Feynman's variational approach predicts a sharp transition, the RG and variational results show no sign of any discontinuity. Instead they suggest a smooth cross-over from one into the other regime, as expected on general grounds \cite{Gerlach1988,Gerlach1991}. It is possible that the sharp crossover obtained using Feynman's variational approach is an artifact of the limited number of parameters used in the variational action. It would be interesting to consider a more general class of variational actions \cite{GiamarchiEAD}.
\begin{figure}[t]
\centering
\epsfig{file=FIG6, width=0.45\textwidth} $\quad$
\caption{The inverse polaron mass $M/M_\text{p}$ is shown as a function of the coupling strength $\alpha$, for various mass ratios $M/m$. We compare MF (dashed) to RG (solid) results. The parameters are $\Lambda_0=2000 / \xi$ and we set $P/Mc=0.01$ in the calculations.}
\label{fig:FIGpolaronMassLogLog}
\end{figure}
In FIG.\ref{fig:FIGpolaronMass} we calculated the polaron mass in the strongly coupled regime, where $\alpha \gg 1$ and the mass ratio $M/m=0.26$ is small. It is also instructive to see how the system approaches the integrable limit $M \to \infty$ when it becomes exactly solvable \cite{Shashi2014RF}. FIG.\ref{fig:FIGpolaronMassLogLog} shows the (inverse) polaron mass as a function of $\alpha$ for different mass ratios $M/m$. For $M \gg m$, as expected, the corrections from the RG are negligible and MF theory is accurate. When the mass ratio $M/m$ approaches unity, we observe deviations from the MF behavior for couplings above a critical value of $\alpha$ which depends on the mass ratio. Remarkably, for very large values of $\alpha$ the mass predicted by the RG follows the same power-law as the MF solution, with a different prefactor. This can be seen more clearly in FIG.\ref{fig:FIGpolaronMassCfSC}, where the case $M/m=1$ is presented. This behavior can be explained from strong coupling theory. As shown in \cite{Casteels2011} the polaron mass in this regime is proportional to $\alpha$, as is the case for the MF solution. However prefactors entering the weak coupling MF and the strong coupling masses are different.
To make this more precise, we compare the MF, RG and strong coupling polaron masses for $M/m=1$ in FIG.\ref{fig:FIGpolaronMassCfSC}. We observe that the RG smoothly interpolates between the weak coupling MF and the strong coupling regime. While the MF solution is asymptotically recovered for small $\alpha \to 0$ (by construction), this is not strictly true on the strong coupling side. Nevertheless, the observed value of the RG polaron mass in FIG.\ref{fig:FIGpolaronMassCfSC} at large $\alpha$ is closer to the strong coupling result than to the MF theory.
\begin{figure}[t]
\centering
\epsfig{file=FIG7, width=0.48\textwidth}
\caption{The polaron mass $M_\text{p}/M$ is shown as a function of the coupling strength for an impurity of mass $M=m$ equal to the boson mass. We compare the asymptotic perturbation and strong coupling theories with MF and RG, which can be formulated for all values of the coupling strength. We used parameters $\Lambda_0=200 / \xi$ and $P/Mc=0.01$.}
\label{fig:FIGpolaronMassCfSC}
\end{figure}
Now we return to the discussion of the polaron mass for systems with a small mass ratio $M/m<1$. In this case FIG.\ref{fig:FIGpolaronMassLogLog} suggests that there exists a large regime of intermediate coupling, where neither strong coupling nor MF theory can describe the qualitative behavior of the polaron mass. This is demonstrated in FIG.\ref{fig:FIGpolaronMass}, where our RG approach predicts values for the polaron mass midway between MF and strong coupling, for a wide range of couplings. In this intermediate-coupling regime, the impurity is constantly scattered on phonons, leading to strong correlations between them.
Thus measurements of the polaron mass rather than the binding energy should be a good way to discriminate between different theories describing the Fr\"ohlich polaron at intermediate couplings. Quantum fluctuations manifest themselves in a large increase of the effective mass of polarons, in strong contrast to the predictions of the MF approach based on the wavefunction with uncorrelated phonons. Experimentally, both the quantitative value of the polaron mass, as well as its qualitative dependence on the coupling strength can provide tests of our theory. The mass of the Fermi polaron has successfully been measured using collective oscillations of the atomic cloud \cite{Nascimbene2009}, and we are optimistic that similar experiments can be carried out with Bose polarons in the near future.
\section{Possible experimental realizations}
\label{Supp:ExperimentalConsiderations}
In this Section we discuss conditions under which the Fr\"ohlich Hamiltonian can be used to describe impurities in ultra cold quantum gases. We also present typical experimental parameters and show that the intermediate coupling regime $\alpha \sim 1$ can be reached with current technology. Possible experiments in which the effects predicted in this paper could be observed are also discussed.
To derive the Fr\"ohlich Hamiltonian Eq. \eqref{H_frolich} for an impurity atom immersed in a BEC \cite{Tempere2009,Bruderer2007}, the Bose gas is described in Bogoliubov approximation, valid for weakly interacting BECs. Then the impurity interacts with the elementary excitations of the condensate, which are Bogoliubov phonons. In writing the Fr\"ohlich Hamiltonian to describe these interactions, we included only terms that are linear in the Bogoliubov operators. This implicitly assumes that the condensate depletion $\Delta n$ caused by the impurity is much smaller than the original BEC density, $\Delta n / n_0 \ll 1$, giving rise to the condition \cite{Bruderer2007}
\begin{equation}
|g_\text{IB}| \ll 4 c \xi^{2}.
\label{eq:condBogoFroh}
\end{equation}
To reach the intermediate coupling regime of the Fr\"ohlich model, coupling constants $\alpha$ larger than one $\alpha \gtrsim 1$ are required (for mass ratios $M/m \simeq 1$ of the order of one). This can be achieved by a sufficiently large impurity-boson interaction strength $g_\text{IB}$, which however means that condition \eqref{eq:condBogoFroh} becomes more stringent. Now we discuss under which conditions both $\alpha \gtrsim 1$ and Eq. \eqref{eq:condBogoFroh} can simultaneously be fulfilled. To this end we express both equations in terms of experimentally relevant parameters $a_\text{BB}$ (boson-boson scattering length), $m$ and $M$ which are assumed to be fixed, and we treat the BEC density $n_0$ and the impurity-boson scattering length $a_\text{IB}$ as experimentally tunable parameters. Using the first-order Born approximation result $g_\text{IB} = 2 \pi a_\text{IB} / m_\text{red}$ Eq.\eqref{eq:condBogoFroh} reads
\begin{equation}
\epsilon := 2 \pi^{3/2} \l 1 + \frac{m}{M} \r a_\text{IB} \sqrt{a_\text{BB} n_0} \stackrel{!}{\ll} 1,
\label{eq:epsExp}
\end{equation}
and similarly the polaronic coupling constant can be expressed as
\begin{equation}
\alpha = 2 \sqrt{2 \pi} \frac{a_\text{IB}^2 \sqrt{n_0}}{\sqrt{a_\text{BB}}}.
\label{eq:alphaExp}
\end{equation}
Both $\alpha$ and $\epsilon$ are proportional to the BEC density $n_0$, but while $\alpha$ scales with $a_\text{IB}^2$, $\epsilon$ is only proportional to $a_\text{IB}$. Thus to approach the strong coupling regime $a_\text{IB}$ has to be chosen sufficiently large, while the BEC density has to be small enough in order to satisfy Eq.\eqref{eq:epsExp}. When setting $\epsilon = 0.3 \ll 1$ and assuming a fixed impurity-boson scattering length $a_\text{IB}$, we find an upper bound for the BEC density,
\begin{multline}
n_0 \leq n_0^\text{max} = 4.9 \times 10^{15} \text{cm}^{-3} \times \l 1 + m/M \r^{-2} \\
\times \l \frac{a_\text{IB} / a_0}{100} \r^{-2} \l \frac{a_\text{BB} / a_0}{100} \r^{-1}, $\qquad$
\label{eq:rhoMax}
\end{multline}
where $a_0$ denotes the Bohr radius. For the same fixed value of $a_\text{IB}$ the coupling constant $\alpha$ takes a maximal value
\begin{equation}
\alpha^{\text{max}} = 0.3\times \frac{\sqrt{2}}{\pi} \l 1 + m/M \r^{-1} \frac{a_\text{IB}}{a_\text{BB}}
\end{equation}
compatible with condition \eqref{eq:condBogoFroh}.
\begin{table}[t]
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{c||c|c|c|c|c|c}
\hline
$a_\text{Rb-K}/a_0$ & 284. & 994. & 1704. & 2414. & 3124. & 3834. \\
$\alpha^\text{max}_\text{Rb-K}$ & 0.26 & 0.91 & 1.6 & 2.2 & 2.9 & 3.5 \\
$n_0^{\text{max}} [10^{14} \text{cm}^{-3}]$ & 2.8 & 0.23 & 0.078 & 0.039 & 0.023 & 0.015 \\
\hline
$a_\text{Rb-Cs}/a_0$ & ~650. & 1950. & 3250. & 4550. & 5850. & 7150. \\
$\alpha^\text{max}_\text{Rb-Cs}$ & 0.35 & 1.0 & 1.7 & 2.4 & 3.1 & 3.8 \\
$n_0^{\text{max}} [10^{14} \text{cm}^{-3}]$ & 0.18 & 0.02 & 0.0073 & 0.0037 & 0.0022 & 0.0015 \\
\hline
\end{tabular}
\caption{Experimentally the impurity-boson scattering length $a_\text{IB}$ can be tuned by more than one order of magnitude using a Feshbach-resonance. We consider two mixtures($~^{87}\text{Rb} - ~^{41} \text{K}$, top and $~^{87}\text{Rb} - ~^{133} \text{Cs}$, bottom) and show the maximally allowed BEC density $n_0^\text{max}$ along with the largest achievable coupling constant $\alpha^\text{max}$ compatible with the Fr\"ohlich model, using different values of $a_\text{IB}$.}
\label{tab:alphaMaxNmax}
\end{table}
Before discussing how Feshbach resonances allow to reach the intermediate coupling regime, we estimate values for $\alpha^{\text{max}}$ and $n_0^\text{max}$ for typical background scattering lengths $a_\text{IB}$. Despite the fact that these $a_\text{IB}$ are still rather small, we find that keeping track of condition \eqref{eq:epsExp} is important. To this end we consider two experimentally relevant mixtures, (i) $~^{87}\text{Rb}$ (majority) -$~^{41}\text{K}$ \cite{Catani2008,Catani2012} and (ii) $~^{87}\text{Rb}$ (majority) -$~^{133}\text{Cs}$ \cite{mccarron2011dual,Spethmann2012}. For both cases the boson-boson scattering length is $a_\text{BB}=100 a_0$ \cite{Chin2010,Egorov2013} and typical BEC peak densities realized experimentally are $n_0=1.4 \times 10^{14} \text{cm}^{-3}$ \cite{Catani2008}. In the first case (i) the background impurity-boson scattering length is $a_\text{Rb-K}=284 a_0$ \cite{Chin2010}, yielding $\alpha_\text{Rb-K} = 0.18$ and $\epsilon=0.21<1$. By setting $\epsilon=0.3$ for the same $a_{\text{Rb-K}}$, Eq.\eqref{eq:rhoMax} yields an upper bound for the BEC density $n_0^\text{max}=2.8 \times 10^{14} \text{cm}^{-3}$ above the value of $n_0$, and a maximum coupling constant $\alpha^{\text{max}}_{\text{Rb-K}} = 0.26$. For the second mixture (ii) the background impurity-boson scattering length $a_\text{Rb-Cs}=650 a_0$ \cite{mccarron2011dual} leads to $\alpha_{\text{Rb-Cs}} = 0.96$ but $\epsilon = 0.83 < 1$. Setting $\epsilon=0.3$ for the same value of $a_{\text{Rb-Cs}}$ yields $n_0^\text{max} = 0.18 \times 10^{14} \text{cm}^{-3}$ and $\alpha_{\text{Rb-Cs}}^{\text{max}}=0.35$. We thus note that already for small values of $\alpha \lesssim 1$, Eq.\eqref{eq:epsExp} is \emph{not} automatically fulfilled and has to be kept in mind.
The impurity-boson interactions, i.e. $a_\text{IB}$, can be tuned by the use of an inter-species Feshbach resonance \cite{Chin2010}, available in a number of experimentally relevant mixtures \cite{Park2012,Pilch2009,Ferlaino2006,Ferlaino2006err,Inouye2004,Stan2004,Schuster2012}. In this way, an increase of the impurity-boson scattering length by more than one order of magnitude is realistic. In Table \ref{tab:alphaMaxNmax} we show the maximally achievable coupling constants $\alpha^\text{max}$ for several impurity-boson scattering lengths and imposing the condition $\epsilon < 0.3$. We consider the two mixtures from above ($~^{87}\text{Rb} - ~^{41} \text{K}$ and $~^{87}\text{Rb} - ~^{133} \text{Cs}$), where broad Feshbach resonances are available \cite{Catani2012,Ferlaino2006,Ferlaino2006err,Pilch2009}. We find that coupling constants $\alpha \sim 1$ in the intermediate coupling regime can be realized, which are compatible with the Fr\"ohlich model and respect condition \eqref{eq:condBogoFroh}. The required BEC densities are of the order $n_0 \sim 10^{13} \text{cm}^{-3}$, which should be achievable with current technology. Note that when Eq.\eqref{eq:condBogoFroh} would not be taken into account, couplings as large as $\alpha \sim 100$ would be possible, but then $\epsilon \sim 8 \gg 1$ indicates the importance of the phonon-phonon scatterings neglected in the Fr\"ohlich model.
\section{Conclusion and Outlook}
\label{sec:outlook}
In conclusion, we developed a new method to describe Fr\"ohlich polarons at arbitrary values of the coupling constant. We applied it to analyze the Bose polaron describing impurities immersed in a BEC, which can be realized using mixtures of ultra cold quantum gases. For sufficiently small BEC densities and large interaction strengths, the intermediate coupling regime can be reached with current technology.
We included quantum fluctuations on top of the MF polaron using an RG approach. Our method predicts polaron energies which are in excellent agreement with numerical diagrammatic MC results and deviate considerably from MF theory. We showed that correlations between phonon modes at different momenta give rise to a logarithmic UV divergence of the polaron energy. Our RG analysis allowed us to understand the origin of this divergence and introduce a corresponding regularization scheme. Furthermore we applied our method to calculate the effective mass of the polaron, which shows a smooth cross-over from weak to strong coupling regime. For impurities lighter than the bosons, we identified an extended regime of intermediate coupling strengths. We point out quantitative agreement between our RG analysis and results based on variational correlated Gaussian wavefunctions in Ref. \cite{Shchadilova2014}. In the future we will also explore polaron physics at larger momenta, where in particular the transition from subsonic- to supersonic regime is poorly understood.
Our predictions can be tested in current experiments with ultra cold atomic mixtures \cite{Palzer2009,Catani2012,Fukuhara2013,Spethmann2012,Scelle2013}. In particular we suggest to measure the effective mass of the polaron, which can provide smoking gun signatures for beyond MF behavior. The effective mass can experimentally be measured from the interaction-induced frequency shift of the impurity in a harmonic trapping potential \cite{Nascimbene2009}. Alternatively polaron properties can be measured using radio-frequency (rf) spectroscopy \cite{Schirotzek2009}, see inset of FIG.\ref{fig:FIG1}. The rf-spectrum directly yields the energy of the impurity immersed in the BEC, and a momentum-resolved variant allows a direct measurement of the polaron dispersion relation.
Experiments with ultra cold atoms are well suited to investigate non-equilibrium dynamics of polarons \cite{Palzer2009,Catani2012,Fukuhara2013}. Our RG method can be generalized to describe dynamical polaron problems. This allows us, for instance, to calculate the full rf-spectrum of impurities immersed in a BEC, for which considerable deviations are found from MF results \cite{Shashi2014RF} as will be shown in a forthcoming publication. Moreover the problem of polaron formation can be investigated, as well as dynamics of impurities close to the subsonic-supersonic transition. These questions become particularly interesting in the presence of a lattice potential, where MF theory predicts a phase-transition at the edges of the Brillouin zone \cite{Grusdt2013BOprep}.
\section*{Acknowledgements}
We acknowledge useful discussions with I. Bloch, S. Das Sarma, M. Fleischhauer, T. Giamarchi, S. Gopalakrishnan, W. Hofstetter, M. Oberthaler, D. Pekker, A. Polkovnikov, L. Pollet, N. Prokof'ev, R. Schmidt, V. Stojanovic, L. Tarruell, N. Trivedi, A. Widera and M. Zwierlein. We are indebted to Aditya Shashi and Dmitry Abanin for invaluable input in the initial phase of the project. F.G. is a recipient of a fellowship through the Excellence Initiative (DFG/GSC 266) and is grateful for financial support from the "Marion K\"oser Stiftung". Y.E.S. and A.N.R. thank the Dynasty foundation for financial support. The authors acknowledge support from the NSF grant DMR-1308435, Harvard-MIT CUA, AFOSR New Quantum Phases of Matter MURI, the ARO-MURI on Atomtronics, ARO MURI Quism program.
|
2,869,038,154,843 | arxiv | \section{Introduction}
All groups in this paper are finite.
A group of prime power order is called a primary group.
Let~$H$ be a subgroup of a group~$G$.
The permutizer of~$H$ in $G$ is the subgroup generated
by all cyclic subgroups of~$G$ which permute with $H$, i.\,e.
$$
P_G(H)=\left\langle x\in G\mid \langle x\rangle H=H\langle x\rangle
\right\rangle.
$$
The permutizer $P_G(H)$ contains the normalizer $N_G(H)$, see~\cite[p.~26]{Wein1982}.
X.~Liu and Y.~Wang~\cite{LiuWang05} proved that
a group $G$ has a Sylow tower of supersoluble type
if $P_G(X)=G$ for every Sylow subgroup~$X$ of~$G$.
A.\,F.~Vasil'ev, V.\,A.~Vasil'ev and T.\,I.~Vasil'eva~\cite{VVV2014}
described the structure of a group~$G$ in which $P_Y(X)=Y$
for every Sylow subgroup~$X$ of~$G$ and every subgroup~$Y\ge X$.
They proposed the following notation.
\begin{definition}\label{dper}
A subgroup~$H$ of a group~$G$ is
$(1)$ {\sl permutable} in~$G$ if $P_G(H)=G$;
$(2)$ {\sl strongly permutable} in~$G$ if $P_U(H)=U$
for every subgroup~$U$ of~$G$ such that~$H\le U\le G$.
\end{definition}
We note that a quasinormal subgroup is strongly permutable.
In the symmetric group~$S_n$, $n\in \{3,4,6\}$,
a Sylow 2-subgroup is strongly permutable, but it is not quasinormal.
A.\,F~Vasil'ev, V.\,A.~Vasil'ev and V.\,N.~Tyutyanov~\cite{VVT2010}
proposed the following notation.
\begin{definition}
Let $\mathbb{P}$ be the set of all primes.
A subgroup $H$ of a group $G$ is called
$\mathbb{P}$\nobreakdash-\hspace{0pt}{\sl subnormal} in~$G$
if there is a subgroup chain
$$
H=H_0\le H_1\le \ldots \le H_n=G \eqno (1)
$$
such that $|H_i:H_{i-1}|\in \mathbb{P}\cup \{1\}$ for every $i$.
\end{definition}
The class of all groups with $\mathbb{P}$-subnormal Sylow subgroups
is denoted by~$\rm{w}\mathfrak U$ and the class of all groups
with $\mathbb{P}$-subnormal primary cyclic subgroups is denoted by~$\rm{v}\mathfrak U$.
These classes are quite well studied~\cite{VVT2010,mkRic,Mon2016SMJ}.
In particular, these classes are subgroup-closed saturated formations.
In a soluble group, a $\mathbb{P}$\nobreakdash-\hspace{0pt}subnormal
Hall subgroup (in particular, a Sylow subgroup)
is strongly permutable~\cite[3.8]{VVV2014}.
We prove that in a soluble group, the converse is true, see Proposition~\ref{l4}.
As a result we obtain new criteria for the supersolubility of a group
and also~\cite[Theorem A]{czl}: {\sl a group~$G\in\mathrm{w}\mathfrak{U}$
if and only if every Sylow subgroup is $\mathbb{P}$-subnormal
or strongly permutable in~$G$.}
In the following theorem, we enumerate all simple non-abelian groups
with a $\mathbb{P}$-subnormal or strongly permutable Sylow subgroup.
\begin{theorem
Let $G$ be a simple non-abelian group
and let $R$ be a Sylow $r$-subgroup of $G$.
$(1)$ If $R$ is $\mathbb{P}$-subnormal in~$G$,
then $r=2$ and $G$ is isomorphic to $L_2(7)$, $L_2(11)$
or $L_2(2^m)$ and~$2^m+1$ is a prime.
$(2)$ If $R$ is strongly permutable and $\mathbb{P}$-subnormal in~$G$,
then $r=2$ and $G\cong L_2(7)$.
\end{theorem}
For a primary cyclic subgroups we prove two following theorems.
\begin{theorem
If all primary cyclic subgroups of a group $G$ are
strongly permutable, then~$G$ is supersoluble.
\end{theorem}
\begin{theorem
If every primary cyclic subgroup of a group $G$ is
$\mathbb{P}$\nobreakdash-\hspace{0pt}subnormal or strongly permutable,
then~$G\in \rm{v}\mathfrak U$.
\end{theorem}
\section{Preliminaries}
Let $G$ be a group. We use $\pi (G)$ to denote the set of all prime
devisors of $|G|$.
If $r$ is a maximal element of $\pi(G)$, then we write
$r=\mathrm{max}\ \pi(G)$.
By $H\leq G$ ($H< G$, $H\lessdot \,G$, $H\lhd G$) we denote a
(proper, maximal, normal)
subgroup $H$ of $G$.
We use the GAP system~\cite{gap} to build examples.
Note that GAP package Permut~\cite{gap_perm} is especially useful
for testing subgroups permutability.
\begin{lemma}\label{KPProp}
Let $H$ and $L$ be subgroups of a group $G$
and let $N$ be a normal subgroup of~$G$.
$(1)$ If $H$ is $\mathbb{P}$-subnormal in~$G$,
then $H\cap N$ is $\mathbb{P}$-subnormal in~$N$
and $HN/N$ is $\mathbb{P}$-subnormal in~$G/N$~\textup{\cite[Lemma~3]{mkRic}}.
$(2)$ If $H$ is $\mathbb{P}$-subnormal in a soluble group $G$
and $U\le G$, then $H\cap U$ is $\mathbb{P}$-subnormal
in~$U$~\textup{\cite[Lemma~4\,(1)]{mkRic}}.
$(3)$ If $H\leq L$, $H$ is $\mathbb{P}$-subnormal in $L$
and $L$ is $\mathbb{P}$-subnormal in $G$,
then $H$ is $\mathbb{P}$-subnormal in $G$~\textup{\cite[Lemma~3]{mkRic}}.
\end{lemma}
\begin{lemma}[{\cite[Lemma~2\,(2)]{mkRic}}]\label{UPsn}
Every subgroup of a supersoluble group is $\mathbb{P}$-subnormal.
\end{lemma}
\begin{lemma}\label{KPMax}
Let $H$ be a $\mathbb{P}$-subnormal subgroup of a group $G$.
If $r=\mathrm{max}\ \pi(G)$, then $O_r(H)\leq O_r(G)$.
\end{lemma}
\begin{proof}
By the hypothesis, there is a subgroup chain
\[
H=H_0< H_1< H_2<\ldots< H_n=G
\]
such that for every $i$, $|H_{i}:H_{i-1}|\in\mathbb{P}$.
Assume that $O_r(H)\leq O_r(H_{i-1})$, we prove $O_r(H)\leq O_r(H_{i})$.
Let $H_{i-1}=A$ and $H_{i}=B$.
If $A$ is normal in $B$, then $O_r(A)$ is subnormal in $B$,
and $O_r(H)\leq O_r(A)\leq O_r(B)$.
If $A$ is not normal in $B$, then $|B:A|=q\in\mathbb{P}$ and $A=N_B(A)$.
Consider the representation of $B$ by permutations on the
right cosets of~$A$~\cite[I.6.2]{hup}.
Note that $B/A_B$ is isomorphic to a subgroup
of the symmetric group $S_q$ and $|B/A_B:A/A_B|=q$.
Since $|S_q|=q!=(q-1)!q$ and $|B/A_B|$ divides $|S_q|$,
we get $|A/A_B|$ divides $(q-1)!$. As $q\in\pi(B)\subseteq\pi(G)$
and $r=\mathrm{max}\ \pi(G)$, we have $q\leq r$ and $A/A_B$ is an $r^\prime $-group.
Therefore $O_r(A)\leq A_B$. Since $O_r(A)$ is normal in $A_B$,
we get $O_r(A)$ is subnormal in $B$ and $O_r(H)\leq O_r(A)\leq O_r(B)$.
Hence $O_r(H)\leq O_r(G)$ by induction.
\end{proof}
The following lemma contains permutable and strongly permutable subgroups
properties we need.
\begin{lemma}\label{PerProp}
Let $H$ be a subgroup of a group $G$
and let $N$ be a normal subgroup of~$G$.
$(1)$~If $H$ is (strongly) permutable in $G$,
then $HN/N$ is (strongly) permutable in $G/N$
\textup{\cite[lemma 3.2.\,(1),(4)]{VVV2014}}.
$(2)$~If $N \leq H$, then $H$ is (strongly) permutable in $G$
if and only if $H/N$ is (strongly) permutable in $G/N$.
$(3)$~If $H$ is strongly permutable in $G$ and $H\le U$,
then $H$ is strongly permutable in $U$.
\end{lemma}
\begin{proof}
$(2)$ This statement is known for permutable subgroups~\cite[lemma 3.2\,(3)]{VVV2014}.
We prove it for strongly permutable subgroups.
If $H$ is a strongly permutable subgroup of $G$,
then in view of Statement~$(1)$, $H/N$ is strongly permutable in $G/N$.
Conversely, let $H/N$ be strongly permutable in $G/N$
and let $A$ be a subgroup of $G$ containing $H$.
Then $P_{A/N}(H/N)=A/N$. In view of~\cite[Lemma~3.6]{VVV2014},
$P_{A/N}(H/N)=P_A(H)/N$ and $P_A(H)=A$,
hence $H$ is strongly permutable in $G$.
$(3)$~This is evident in view of Definition~\ref{dper}\,(2).
\end{proof}
\begin{lemma}\label{PG=NG}
Let $r=\mathrm{max}\ \pi(G)$
and let $R$ be a Sylow $r$-subgroup of a group $G$.
Then $N_G(R)=P_G(R)$. In particular,
if $R$ is permutable in $G$, then $R$ is normal in $G$.
\end{lemma}
\begin{proof}
Let $x\in G$ and $R\langle x\rangle=\langle x\rangle R$.
It is clear $\langle x\rangle=\langle x_1\rangle\times\langle x_2\rangle$,
where
$\langle x_1\rangle$ is a Sylow $r$-subgroup of $\langle x\rangle$ and
$\langle x_2\rangle$ is a Hall $r^\prime$-subgroup of $\langle x\rangle$.
In view of~\cite[VI.4.7]{hup}, $R=R\langle x_1\rangle$,
therefore $R\langle x\rangle=R\langle x_2\rangle $.
Now, all Sylow $r^\prime $-subgroups of $R\langle x\rangle$ is cyclic.
As $r=\mathrm{max}\ \pi(R\langle x\rangle)$,
it implies that $R$ is normal in~$R\langle x\rangle$ by \cite[IV.2.7]{hup},
and $\langle x\rangle\leq N_G(R)$.
Since $P_G(R)$ is generated by elements $x$ such that
$R\langle x\rangle=\langle x\rangle R$, we conclude $P_G(R)=N_G(R)$.
\end{proof}
\begin{lemma}\label{SylTower}
If every Sylow subgroup of a group $G$ is
$\mathbb{P}$-subnormal or permutable,
then $G$ has a Sylow tower of supersoluble type.
\end{lemma}
\begin{proof}
Let $R$ be a Sylow $r$-subgroup of a group $G$ for $r=\mathrm{max}\ \pi(G)$.
If $R$ is $\mathbb{P}$-subnormal in $G$,
then $R$ is normal in $G$ by Lemma~\ref{KPMax}.
If $R$ is permutable in $G$, then $R$ is normal in $G$ by Lemma~\ref{PG=NG}.
Hence $R$ is normal in~$G$.
Let $\overline{Q}$ be a Sylow $q$-subgroup of $\overline{G}=G/R$.
Then $\overline{Q}=QR/R$ for a Sylow $q$-subgroup $Q$ of~$G$.
If $Q$ is $\mathbb{P}$-subnormal in $G$,
then $\overline{Q}$ is $\mathbb{P}$-subnormal in $\overline{G}$
in view of Lemma~\ref{KPProp}\,(1).
If $Q$ is permutable in $G$, then $\overline{Q}$ is permutable in $\overline{G}$
by Lemma~\ref{PerProp}\,(1).
Thus, the hypothesis holds for $\overline{G}$,
and by induction, $\overline{G}$ has a Sylow tower of supersoluble type.
Therefore $G$ also has a Sylow tower of supersoluble type.
\end{proof}
\begin{lemma}[{\cite[Lemma~2.1]{QQW2013}}]\label{MC}
Let $M$ be a maximal subgroup of a soluble group $G$,
and assume that $G=MC$ for a cyclic subgroup $C$.
Then $|G : M|$ is a prime or $4$.
Also, if $|G : M| = 4$, then $G/M_G = S_4$.
\end{lemma}
We will also repeatedly use the following statement.
\begin{lemma}[{\cite[Lemma~2.2]{mkRic}}]\label{PrimitiveGr}
Let $\mathfrak{F}$ be a saturated formation and let $G$ be a group.
Suppose that $G\notin\mathfrak{F}$ but $G/N \in \mathfrak{F}$
for any normal subgroup $N$ of $G$, $N\neq 1$. Then $G$ is a primitive group.
\end{lemma}
\section{Groups with permutable and $\mathbb{P}$-subnormal Sylow subgroups}
\setcounter{theorem}{0}
\begin{proposition}\label{l4}
Let $G$ be a soluble group and let $H$ be a Hall subgroup.
Then $H$ is $\mathbb{P}$-subnormal in $G$ if and only if
$H$ is strongly permutable in $G$.
\end{proposition}
\begin{proof}
Let $H$ be $\mathbb{P}$-subnormal in $G$.
According to~\cite[3.8]{VVV2014}, $H$ is strongly permutable in $G$.
For completeness, we give the proof of this statement.
We use induction on the order of $G$.
Since $H$ is $\mathbb{P}$-subnormal in $G$,
there is a maximal subgroup $M$ of $G$ such that $H\leq M$,
$|G:M|\in\mathbb{P}$ and $H$ is $\mathbb{P}$-subnormal in $M$.
By induction, $H$ is strongly permutable in $M$ and
$M=P_M(H)\leq P_G(H)$. Since $M$ is a maximal subgroup of $G$,
we assume $P_G(H)=M$. Suppose that $M_G\neq 1$ and
$L$ is a minimal normal subgroup of $G$ that is contained in $M_G$.
According to Lemma~\ref{KPProp}\,(1), $HL/L$ is $\mathbb{P}$-subnormal in $G/L$,
and by induction, $HL/L$ is permutable in $G/L$.
Hence $HL$ is permutable in $G$ in view of Lemma~\ref{PerProp}\,(3).
Since $G$ is soluble, we conclude $L$ is an elementary abelian $q$-group
for some $q\in\pi(G)$.
If $q\in\pi(H)$, then $HL=H$ and $H$ is permutable in $G$, a contradiction.
Therefore we can assume that $q\notin \pi(H)$.
Since $HL$ is permutable in $G$, then $P_G(HL)=G$ and
there is $x\in G\setminus M$ such that
$\langle x\rangle HL=HL\langle x\rangle=A$.
Suppose that $A$ is a proper subgroup of $G$.
As $H$ is $\mathbb{P}$-subnormal in $G$,
by Lemma~\ref{KPProp}\,(2), it follows that
$H$ is $\mathbb{P}$-subnormal in $A$,
and by induction, $H$ is permutable in $A$.
Therefore $A=P_A(H)\leq P_G(H)=M$ and $x\in M$, a contradiction.
Hence $G=\langle x\rangle HL$.
If $L\leq\Phi(G)$, then $G=\langle x\rangle H$ and
$x\in P_G(H)=M$, a contradiction.
Consequently, $L$ is not contained in $\Phi (G)$ and
there is a maximal subgroup $K$ of $G$
that does not contain $L$.
In that case, $G=LK$ and we can assume $H\leq K$.
By induction, $H$ is permutable in $K$ and $K=P_K(H)\leq P_G(H)=M$.
Hence we get $M=K$ and $L\leq K$, a contradiction.
Thus, $M_G=1$ and $G$ is a primitive group.
Consequently, $G=N\rtimes M$, where $N=F(G)$ is
a unique minimal normal subgroup of $G$.
Since $|G:M|\in\mathbb{P}$, we deduce
$N$ is a cyclic subgroup and $N\leq P_G(H)=M$, a contradiction.
Thus $H$ is permutable in $G$, and in view of Lemma~\ref{KPProp}\,(2),
$H$ is strongly permutable in~$G$.
Conversely, let $H$ be a Hall strongly permutable subgroup
of a soluble group $G$. Using induction on the order of $G$
we prove that $H$ is $\mathbb{P}$-subnormal in $G$.
Let $H\leq M\lessdot G$. By Lemma~\ref{PerProp}\,(3),
$H$ is strongly permutable in $M$, and by induction,
$H$ is $\mathbb{P}$-subnormal in $M$.
If $|G:M|\in\mathbb{P}$, then $H$ is $\mathbb{P}$-subnormal in $G$
by Lemma~\ref{KPProp}\,(3).
Hence we can assume that $|G:M|\notin\mathbb{P}$,
in particular, $M$ is not normal in $G$.
According to Lemma~\ref{PerProp}\,(1), $G/M_G$ contains
a strongly permutable Hall subgroup $HM_G/M_G$.
As $HM_G\leq M$, we obtain that $H$ is $\mathbb{P}$-subnormal in $HM_G$
by induction. If $M_G\neq 1$, then $HM_G/M_G$ is $\mathbb{P}$-subnormal in $G/M_G$
by induction. By Lemma~\ref{KPProp}\,(1), $HM_G$ is $\mathbb{P}$-subnormal in $G$,
and $H$ is $\mathbb{P}$-subnormal in $G$ in view of Lemma~\ref{KPProp}\,(3).
Therefore we can assume that $M_G=1$.
Since $G$ is soluble, we get $G=N\rtimes M$,
$N=F(G)=C_G(N)=O_p(G)$ is a unique minimal normal subgroup in $G$.
Let $HN<G$. By induction, $HN/N$ is $\mathbb{P}$-subnormal in $G/N$,
and $H$ is $\mathbb{P}$-subnormal in $G$.
Finally, we consider the case, when $H=M$ is a Hall subgroup.
By the hypothesis, there is $x\in G\setminus H$
such that $\langle x\rangle H=H\langle x\rangle=G$.
Let $\langle x\rangle=\langle x_1\rangle \times\langle x_2\rangle$,
where $\langle x_1\rangle$ is a $p$-subgroup and
$\langle x_2\rangle$ is a $p^\prime $-subgroup.
Since $N$ is a normal Sylow $p$-subgroup of $G$,
we conclude $\langle x_1\rangle\leq N$ and $|x_1|=p$.
According to~\cite[VI.4.6]{hup}, $H=\langle x_2\rangle H$.
Now, $G=\langle x\rangle H=\langle x_1\rangle H$ and $|G:H|=p$.
\end{proof}
\begin{pcorollary}\cite[Theorem~A]{czl}\label{th4}
If every Sylow subgroup of a group~$G$ is
$\mathbb{P}$-subnormal or strongly permutable,
then $G\in\mathrm{w}\mathfrak{U}$.
Conversely, if $G\in\mathrm{w}\mathfrak{U}$,
then every Sylow subgroup is $\mathbb{P}$-subnormal
and strongly permutable in~$G$.
\end{pcorollary}
\begin{proof}
Assume that every Sylow subgroup of $G$ is $\mathbb{P}$-subnormal or strongly permutable.
By Lemma~\ref{SylTower}, $G$ has a Sylow tower of supersoluble type,
which means that $G$ is soluble. Hence by Proposition~\ref{l4},
every strongly permutable Sylow subgroup of $G$
is $\mathbb{P}$-subnormal and $G\in\mathrm{w}\mathfrak{U}$.
Conversely, let $G\in\mathrm{w}\mathfrak{U}$.
By the definition of $\mathrm{w}\mathfrak{U}$,
every Sylow subgroup is $\mathbb{P}$-subnormal in~$G$.
Since~$G$ is soluble, according to Proposition~\ref{l4},
every Sylow subgroup is strongly permutable in~$G$.
\end{proof}
\begin{pcorollary}\label{c42}
Let $G$ be a group. The following statements are equivalent.
$(1)$ $G$ is supersoluble.
$(2)$ Every Hall subgroup of $G$ is $\mathbb{P}$-subnormal or strongly permutable.
$(3)$ Every Hall subgroup of $G$ is $\mathbb{P}$-subnormal or permutable.
\end{pcorollary}
\begin{proof}
$(1)\Rightarrow (2)$: If $G$ is supersolvable,
then by Lemma~\ref{UPsn}, every subgroup in $G$ is $\mathbb{P}$-subnormal,
and every Hall subgroup is $\mathbb{P}$-subnormal.
Since~$G$ is soluble, we conclude that every Hall subgroup
is strongly permutable in~$G$ in view of Proposition~\ref{l4}.
$(2)\Rightarrow (3)$: It is evident since every strongly permutable subgroup is permutable.
$(3)\Rightarrow (1)$:
Assume that every Hall subgroup of a group $G$ is $\mathbb{P}$-subnormal or permutable.
By Lemma~\ref{SylTower}, $G$ has a Sylow tower of supersoluble type,
and for $r=\mathrm{max}\ \pi(G)$, a Sylow $r$-subgroup $R$ is normal in $G$.
Let $N$ be a normal subgroup of $G$, $N\neq 1$,
and let $\overline{H}$ be a Hall $\pi$-subgroup of
$\overline{G}=G/N$ for $\pi\subseteq\pi(G)$.
Then $\overline{H}=HN/N$ for a Hall $\pi$-subgroup $H$ of $G$.
If $H$ is $\mathbb{P}$-subnormal in $G$,
then $\overline{H}$ is $\mathbb{P}$-subnormal in $\overline{G}$
by Lemma~\ref{KPProp}\,(1). If $H$ is permutable in $G$,
then $\overline{H}$ is permutable in $\overline{G}$ by Lemma~\ref{PerProp}\,(1).
Thus the hypothesis holds for $\overline{G}$,
and by induction, $\overline{G}\in\mathfrak{U}$.
As $\mathfrak{U}$ is a subgroup-closed satuarted formation,
we deduce $G$ is a primitive group by Lemma~\ref{PrimitiveGr}.
According to properties of primitive groups, $\Phi(G)=1$,
$G=R\rtimes M$, $R=F(G)$ is a minimal normal subgroup in $G$, $|R|>r$,
$M$ is a maximal subgroup in $G$, $M\in\mathfrak{U}$.
Note that $M$ is a Hall subgroup. If $M$ is $\mathbb{P}$-subnormal
in $G$, then $|G:M|=r=|R|$, a contradiction.
Suppose that $M$ is permutable in $G$, i.\,e. $P_G(M)=G$.
In that case, there is $x\in G\setminus M$ such that $G=M\langle x\rangle$.
In view of Lemma~\ref{MC}, $|G:M|=4$,
but $r=\mathrm{max}\ \pi(G)$. Hence $G$ is a $2$-group.
\end{proof}
\begin{pcorollary}\label{BiU}
If every Sylow subgroup of a biprimary group $G$ is
$\mathbb{P}$-subnormal or permutable,
then $G$ is supersoluble.
Conversely, in a supersoluble biprimary group
every Sylow subgroup is $\mathbb{P}$-subnormal
and strongly permutable.
\end{pcorollary}
According to Proposition~\ref{l4}, for a Hall subgroup of a soluble group,
$\mathbb{P}$-subnormality and strongly permutability are equivalent.
In simple groups, this is not true.
\begin{example}
In $L_2(8)$, a Hall $\{2,7\}$-subgroup is strongly permutable,
but it is not $\mathbb{P}$-subnormal.
\end{example}
\begin{example}
In $L_2(9)$, a Sylow $2$-subgroup is strongly permutable,
but it is not $\mathbb{P}$-subnormal.
\end{example}
\begin{example}
In $L_2(5)$, a Sylow 2-subgroup is $\mathbb{P}$-subnormal,
but it is not permutable.
\end{example}
\begin{theorem}\label{th0}
Let $G$ be a simple non-abelian group
and let $R$ be a Sylow $r$-subgroup of $G$.
$(1)$ If $R$ is $\mathbb{P}$-subnormal in~$G$,
then $r=2$ and $G$ is isomorphic to $L_2(7)$, $L_2(11)$
or $L_2(2^m)$ and~$2^m+1$ is a prime.
$(2)$ If $R$ is strongly permutable and $\mathbb{P}$-subnormal in~$G$,
then $r=2$ and $G\cong L_2(7)$.
\end{theorem}
\begin{proof}
Since~$R$ is $\mathbb{P}$-subnormal in~$G$,
there is a subgroup chain
$$
R=H_0\le H_1\le H_2\le\ldots\le H_{n-1}=H\le H_n=G
$$
such that $|H_{i+1}:H_i|\in\mathbb{P}$.
It is clear that~$R$ is $\mathbb{P}$-subnormal in~$H$.
Let $|G:H|=p$. Since $H_G=1$,
the representation of~$G$ on the set of left cosets
by~$H$ is exactly of degree $p$~\cite[I.6.2]{hup} and~$G$
is ismorphic to a subgroup of the symmetric group~$S_p$ of order~$p$
for $p=\mathrm{max}\ \pi(G)$, $H$ is a Hall $p^\prime$-subgroup of $G$.
From Lemma~\ref{KPMax} it follows that a Sylow $p$-subgroup of~$G$
is not $\mathbb{P}$-subnormal in~$G$, therefore~$r<p$.
Since the unit subgroup is $\mathbb{P}$-subnormal in~$R$,
the unit subgroup is $\mathbb{P}$-subnormal in~$H$ and in~$G$.
According to \cite{kaz}, \cite[p. 342]{cs}, $G$ is
isomorphic to one of the following groups
$$
L_2(7), \ L_2(11), \ L_3(3), \ L_3(5), \ L_2(2^m),
2^m+1 \mbox{ is prime}.
$$
Let $G\cong L_2(7)$. Then $|G|=2^3\cdot 3\cdot 7$, $p=7$, $H\cong S_4$.
In $S_4$, a Sylow $2$-subgroup is $\mathbb{P}$-subnormal,
a Sylow $3$-subgroup is not $\mathbb{P}$-subnormal, therefore~$r=2$.
In $L_2(7)$, there are two conjugate classes that are isomorphic to $S_4$.
Since all Sylow 2-subgroups are conjugate,
we get $R$ is contained in two non-conjugate subgroups~$A\le G$ and~$B\le G$
that are isomorphic to~$S_4$. Since~$A=RC_3$, $B=RC_3^x$, $x\in G$,
we obtain $G=\langle A,B\rangle \le P_G(R)$. So $R$ is permutable in~$G$.
If~$R<U<G$, then $U\cong S_4$, therefore $R$ is strongly permutable in~$G$.
Let $G\cong L_2(11)$. Then $|G|=2^2\cdot 3\cdot 5\cdot 11$,
$p=11$, $H\cong L_2(5)$. In $L_2(5)$,
only a Sylow 2-subgroup is $\mathbb{P}$-subnormal,
therefore~$r=2$. But $R$ is not permutable in $H\cong L_2(5)$,
hence $R$ is not strongly permutable in~$G\cong L_2(11)$.
Let $G\cong L_3(3)$. Then $|G|=2^4\cdot 3^3\cdot 13$,
$p=13$ and $H\cong M_9:S_3\cong C_3^2:GL_2(3)$.
Since $|H|=2^4\cdot 3^3$ and~$H$ is not $3$-closed,
we have~$r\ne 3$ by Lemma~\ref{KPMax} and~$r=2$.
But a Sylow $2$-subgroup $R$ is not $\mathbb{P}$-subnormal in~$H$ according to~\cite{atl}. Therefore in~$G\cong L_3(3)$, there are no $\mathbb{P}$-subnormal Sylow subgroups.
Let $G\cong L_3(5)$. Then $|G|=2^5\cdot 3\cdot 5^3 \cdot 31$,
$p=31$ and $H\cong C_5^2:GL_2(5)$. Since $|H|=2^5\cdot 3\cdot 5^3$
and~$H$ is not $5$-closed, we get~$r\ne 5$ by Lemma~\ref{KPMax} and~$r\in \{2,3\}$.
But a Sylow $2$--subgroup and $3$-subgroup
are not $\mathbb{P}$-subnormal in~$H$ according to~\cite{atl}.
Therefore in~$G\cong L_3(5)$, there are no $\mathbb{P}$-subnormal Sylow subgroups.
Let $G\cong L_2(2^m)$, where $2^m+1$ is prime.
Then $|G|=2^m(2^m-1)(2^m+1)$, $p=2^m+1$ and $H=N_G(Q)\cong C_2^m:C_{2^m-1}$,
$Q\cong C_2^m$ is a Sylow $2$-subgroup of ~$G$.
Since $Q$ is $\mathbb{P}$-subnormal in~$H$,
we deduce $Q$ is $\mathbb{P}$-subnormal in~$G$.
Suppose that $\langle g\rangle Q=Q\langle g\rangle$ for some~$g\in G$.
Then $\langle g\rangle Q\le N_G(Q)$ according to~\cite[II.8.27]{hup}.
Hence~$P_G(Q)=N_G(Q)$ and a Sylow $2$-subgroup of $G\cong L_2(2^m)$
is $\mathbb{P}$-subnormal in~$G$, but it is not permutable in~$G$.
Suppose that~$r\ne 2$. Then $R\le N_G(Q)=H$ and~$R$ is $\mathbb{P}$-subnormal
in~$RQ$ by Lemma~\ref{KPProp}\,(2), since~$R$ is $\mathbb{P}$-subnormal in~$H$
and~$H$ is soluble. By Lemma~\ref{KPMax}, $R$ is normal in~$RQ$ and~$R\le C_G(Q)$,
which is impossible in~$G\cong L_2(2^m)$.
\end{proof}
\begin{corollary} [{\cite[Theorem 2.1]{km2020umj}}]\label{c01}
If a Sylow $r$-subgroup of a group $G$ is $\mathbb{P}$-subnormal and~$r>2$,
then~$G$ is $r$-soluble.
\end{corollary}
\begin{proof}
By Theorem~\ref{th0}\,(1), $G$ is not simple.
Let~$N$ be a normal subgroup of~$G$, $1\ne N\ne G$.
Then~$R\cap N$ is a Sylow $r$-subgroup of~$N$
and~$R\cap N$ is $\mathbb{P}$-subnormal in~$N$
in view of Lemma~\ref{KPProp}\,(1).
By induction, $N$ is $r$-soluble.
Note that $RN/N$ is a Sylow $r$-subgroup of~$G/N$
and~$RN/N$ is $\mathbb{P}$\nobreakdash-\hspace{0pt}subnormal in~$G/N$
in view of Lemma~\ref{KPProp}\,(1). By induction, $G/N$ is $r$-soluble.
Therefore~$G$ is $r$-soluble.
\end{proof}
\begin{corollary} [{\cite[Corollary 2.1.1]{km2020umj}}]\label{c02}
If a Sylow 3-subgroup and Sylow 5\nobreakdash-\hspace{0pt}subgroup of a group $G$
is $\mathbb{P}$-subnormal, then~$G$ is soluble.
\end{corollary}
\begin{proof}
By Corollary~\ref{c01}, $G$ is 3-soluble and 5-soluble.
Hence $G$ has a normal series, in which factors are 3-groups,
5\nobreakdash-\hspace{0pt}groups or $\{3,5\}^\prime$\nobreakdash-\hspace{0pt}groups.
Since $\{3,5\}^\prime$-groups are soluble~\cite[Theorem, p.18]{gor},
we conclude $G$ is soluble.
\end{proof}
\section{Groups with permutable and $\mathbb{P}$-subnormal primary cyclic subgroups}
Let $\mathfrak{F}$ be a class of groups.
A group $G$ is called a minimal non-$\mathfrak{F}$-group if
$G\notin\mathfrak{F}$ but every proper subgroup of $G$ belongs to $\mathfrak{F}$.
Minimal non-$\mathfrak{N}$-groups are also called Schmidt groups.
We remind the properties of Schmidt groups and minimal non-supersoluble groups
we need.
\begin{lemma}[{\cite[Theorem~1.1, 1.2, 1.5]{umk},\cite[Theorem~3]{Ball_Sch}}]\label{ls1}
Let $S$ be a Schmidt group. Then the following statements hold.
$(1)$~$S=P\rtimes Q$, where $P$ is a normal Sylow $p$-subgroup
and $Q$ is a non-normal Sylow $q$-subgroup, $p$ and $q$ are different primes and
\ \ $(1.1)$~if $P$ is abelian, then $P$ is elementary abelian of order~$p^m$,
where $m$ is the order of $p$ modulo $q$;
\ \ $(1.2)$~if $P$ is not abelian, then $Z(P)=P'=\Phi (P)$ and $|P/Z(P)| = p^m$;
\ \ $(1.3)$~if $p>2$, then $P$ has the exponent~$p$;
for $p=2$, the exponent of~$P$ is not more than~$4$.
\ \ $(1.4)$~$Q=\langle y\rangle$ is a cyclic subgroup and $y^q\in Z(S)$.
$(2)$~$G$ has exactly two classes of conjugate maximal subgroups
\[
\{P\times \langle y^q\rangle\},~~ \{\Phi (P)\times \langle x^{-1}yx\rangle~|~x\in P\setminus \Phi (P)\}.
\]
\end{lemma}
\begin{lemma}[{\cite{D66,BallR07}}]\label{minu}
Let $G$ be a minimal non-supersoluble group.
Then the following statements hold.
$(1)$~$G$ is soluble and $|\pi(G)| \leq 3$;
$(2)$~If $G$ is not a Schmidt group,
then $G$ has a Sylow tower of supersolvable type;
$(3)$~$G$ has a unique normal Sylow subgroup $P$ and $P = G^\mathfrak{U}$;
$(4)$~$|P/\Phi(P)| > p$ and $P/\Phi(P)$ is a minimal normal subgroup of $G/\Phi(P)$;
$(5)$~If $\Phi(G)=1$, then $O_{p^\prime}(G)=1$
and~$Q$ is either nonabelian of order $q^3$ and exponent $q$,
or $Q$ is a cyclic $q$-group,
or $Q$ is a $q$-group with a cyclic subgroup of index~$q$,
or $Q$ is a supersoluble Schmidt group.
\end{lemma}
\begin{lemma}[{\cite[Lemma~1]{M95}}]\label{ls2}
Let $S=P\rtimes Q$ be a supersoluble Schmidt group.
Then $P=\langle x\rangle$ is a normal subgroup of order~$p$,
$Q=\langle y\rangle$ is a cyclic subgroup of order~$q^b$,
where $q$ divides $p-1$.
\end{lemma}
\begin{lemma}\label{lq}
Let $G=P\rtimes Q$ be a Schmidt group, $Q=\langle y\rangle$.
If $x\in G$ and $|y|$ does not divide $|x|$,
then $x\in P\times \langle y^q\rangle$.
\end{lemma}
\begin{proof}
Let $\langle x\rangle=\langle x_1\rangle\times\langle x_2\rangle$,
where $\langle x_1\rangle$ is a Sylow $p$-subgroup and
$\langle x_2\rangle$ is a Sylow $q$-subgroup of $\langle x\rangle$.
Since $P$ is normal in $G$, $\langle x_1\rangle\le P$.
Let $\langle x_2\rangle\leq Q^g=\langle y^g\rangle$, $g\in G$.
As $|y|$ does not divide $|x_2|$, we conclude $\langle x_2\rangle<\langle y^g\rangle$
and $x_2\in \langle (y^g)^q\rangle$. But $y^q\in Z(G)$, therefore
\[
(y^g)^q=\underbrace{y^g\cdot y^g\cdot \ldots \cdot y^g}_{q}=g^{-1}y^q g=y^q
\]
and $x_2\in\langle y^q\rangle$.
Consequently, $\langle x\rangle\leq P\times \langle y^q\rangle$.
\end{proof}
\begin{lemma}\label{ls}
$(1)$ In a supersoluble Schmidt group, every subgroup is strongly permutable.
$(2)$ Let $G=P\rtimes Q$ be a non-supersoluble Schmidt group.
Then the following statements hold.
\ \ $(2.1)$ $Q$ is not permutable and $N_G(Q)=P_G(Q)=\Phi (P)\times Q\lessdot \, G$.
\ \ $(2.2)$ If $H\leq P$ and $P_G(H)=G$, then either $H=P$ or $H\leq \Phi(G)$.
\ \ $(2.3)$ Every primary permutable subgroup is normal in~$G$,
and so it is strongly permutable in~$G$.
$(3)$ In Schmidt group~$G$, every subgroup of prime order and
every cyclic subgroup of order~4 is strongly permutable if and only if $G$ is supersoluble.
\end{lemma}
\begin{proof}
In view of Lemma~\ref{ls1}\,(2), maximal subgroups of $G$ are reduced to
$N_G(Q^g)=\Phi(P)\times Q^g$, $g\in G$ and $P\times \langle y^q\rangle\lhd G$,
$\langle y\rangle =Q$.
$(1)$ Let $G=P\rtimes Q$ be a supersoluble Schmidt group.
Then $|P|=p$ and $q$ divides $p-1$, where $|Q|=q^b$ by Lemma~\ref{ls2}.
It is clear that $P$ and $Q$ are strongly permutable in~$G$. Let $Q_1\lessdot \, Q$.
Note that $P\times Q_1$ is cyclic and normal in $G$. Hence all subgroups of $P\times Q_1$
is normal in $G$ and strongly permutable.
$(2)$ Let $G=P\rtimes Q$ be a non-supersoluble Schmidt group.
$(2.1)$ Suppose that there is $x\in G\setminus N_G(Q)$ such that
$\langle x \rangle Q=Q\langle x \rangle$.
Let $\langle x\rangle=\langle a\rangle\langle b\rangle$,
where $\langle a\rangle$ is a Sylow $p$-subgroup and
$\langle b\rangle$ is a Sylow $q$-subgroup of $\langle x\rangle$.
Since $\langle b\rangle Q=Q$ according to~\cite[VI.4.7]{hup},
\[
\langle x \rangle Q=\langle a\rangle Q=\langle a\rangle \rtimes Q,
\ a\in N_G(Q)=\Phi(P)\times Q,
\]
So, $x\in N_G(Q)$ and $N_G(Q)=P_G(Q)\lessdot \,G$.
$(2.2)$ Assume that $H\leq P$ and $P_G(H)=G$. Since $P$ is normal in $G$, then $P_G(P)=G$.
Let
\[
H<P, \ x_i\in P_G(H), \ \langle x_i\rangle H=H\langle x_i\rangle,
\ i=1,\ldotp,n,
\ P_G(H)=\langle x_1,x_2,\ldots , x_n\rangle .
\]
If $|Q|$ does not divide $|x_i|$ for every~$i$,
then $P_G(H)\leq P\times \langle y^q\rangle<G$ by Lemma~\ref{lq},
a contradiction with the hypothesis. Therefore there is $x_j$ such that
$\langle x_j\rangle H=H \langle x_j\rangle$ and $|Q|$ divides $|x_j|$.
Hence $\langle x_j\rangle=\langle u\rangle \times Q^g$ for some $u\in P$ and $g\in G$.
Since $\langle x_j\rangle <G$,
\[
\langle x_j\rangle=\langle u\rangle \times Q^g\le
\Phi (P) \times Q^g.
\]
Hence $u\in\Phi(P)$. If $\langle x_j\rangle H=G$, then $\langle u\rangle H=P$ and $H=P$,
a contradiction. So, $\langle x_j\rangle H<G$ and
$\langle x_j\rangle H\leq \Phi(P)\times Q^g$, hence $H\leq\Phi(P)$.
Since $P$ is normal in $G$, we conclude $H\leq\Phi(P)\leq \Phi(G)$.
$(3)$ Suppose that in a Schmidt group~$G=P\rtimes Q$, every cyclic subgroup
of prime order and cyclic subgroup of order~4 is strongly permutable. Statement~$(2.2)$
implies $|P|=p$ and~$G$ is supersoluble.
Conversely, if~$G$ is a supersoluble Schmidt group, then by Statement~$(1)$,
every subgroup of prime order and every cyclic subgroup of order~4 is strongly permutable.
\end{proof}
\begin{lemma}\label{expp}
Let $H$ be a $p$-group of exponent~$p$ and $x\notin Z(H)$.
Then $N_H(\langle x\rangle)=P_H(\langle x\rangle)$ and
$\langle x\rangle$ is not permutable in $H$.
\end{lemma}
\begin{proof}
It is clear that $N_H(\langle x\rangle)\leq P_H(\langle x\rangle)$. Choose
$y\in H\setminus \langle x\rangle$ such that $\langle x\rangle\langle y\rangle=\langle y\rangle\langle x\rangle$.
Since $H$ is a $p$-group of exponent~$p$, we get $|\langle x\rangle\langle y\rangle|=p^2$.
Consequently, $H$ is abelian and
$\langle x\rangle\langle y\rangle=\langle x\rangle\times\langle y\rangle$,
and so $y\in N_H(\langle x\rangle)$ and $N_H(\langle x\rangle)=P_H(\langle x\rangle)$.
As $x\notin Z(H)$, we have $H\neq N_H(\langle x\rangle)=P_H(\langle x\rangle)$
and $\langle x\rangle$ is not permutable in $H$.
\end{proof}
\begin{theorem}\label{tc}
If all primary cyclic subgroups of a group $G$ are
strongly permutable, then~$G$ is supersoluble.
\end{theorem}
\begin{proof}
We use induction on the group order.
In view of Lemma~\ref{PerProp}\,(3) and by induction,
all proper subgroups of $G$ are supersoluble.
Hence $G=P\rtimes S$ is a minimal non-supersoluble group,
$P=G^\mathfrak{U}$. By Lemma~\ref{ls}\,(3), $G$ is not a Schmidt group,
therefore $G$ has a Sylow tower of supersoluble type and
$P$ is a Sylow $p$-subgroup of $G$ for $p=\max \ \pi(G)$.
In particular, $p>2$ and all nontrivial elements in $P$ are of order~$p$
by Lemma~\ref{minu}.
From Lemma~\ref{expp} it follows that $P$ is an elementary abelian $p$-subgroup,
and by Lemma~\ref{minu}, $P$ is a minimal normal subgroup in~$G$.
Assume that $N$ is a normal subgroup of~$G$, $N\neq 1$,
and $V/N$ is a cyclic $t$-subgroup, $t\in\pi(G)$.
Let $U$ be a subgroup of least order such that $U\leq V$, $UN=V$.
Then $U\cap N\leq\Phi(U)$, $V/N=UN/N\cong U/U\cap N$,
therefore $U$ is a cyclic $t$-subgroup. By the hypothesis,
$U$ is strongly permutable in $G$, and by Lemma~\ref{PerProp}\,(1),
$V/N$ is strongly permutable in $G/N$. By induction, $G/N$ is supersoluble,
hence $\Phi(G)=1$. From Lemma~\ref{minu}\,(5) it follows that $S$ is
either a cyclic primary group or supersoluble Schmidt group,
and~$O_{p^\prime}(G)=1$.
Let $S$ be either a cyclic primary group or $|\pi(S)|=2$.
In that case, all Sylow subgroups of $S$ are cyclic.
Assume that $A\leq P$, $|A|=p$ and $g\in G\setminus N_G(A)$ such that
$\langle g\rangle A\leq G$. Since $G=PS$, we conclude $g=bx$, $b\in P$, $x\in S$
and $\langle g\rangle=\langle b\rangle\times\langle x\rangle$.
If $x=1$, then $g=b\in P\leq N_G(A)$, a contradiction.
If $b=1$, then $g=x$ and $\langle g\rangle A=A\rtimes \langle g\rangle$,
since $\langle g\rangle A$ is $p$-closed. So, $g\in N_G(A)$, a contradiction.
Thus, $b\neq 1$, $x\neq 1$, $S^b\neq S$, $x^b=x\in S\cap S^b=D\neq 1$.
If $S$ is abelian, then $D\lhd \langle S,S^b\rangle=G$, $D\leq O_{p'}(G)=1$,
a contradiction. Therefore $S$ is not abelian and $S=Q\rtimes R$
is supersoluble Schmidt group by Lemma~\ref{minu} in view of~$\Phi(G)=1$.
Now, $|Q|=q$, $R=\langle y\rangle$ is an $r$-subgroup, $r$ divides $q-1$ and $y^r\in Z(S)$.
If $q$ divides $|D|$, then $Q\leq D$ and $Q\lhd\langle S,S^b\rangle =G$, a contradiction.
So, $D$ is an $r$-subgroup. If $y^r\neq 1$, then $D_1=D\cap \langle y^r\rangle\neq 1$ and
$D_1\lhd \langle S,S^b\rangle =G$, a contradiction. Consequently,
$y^r=1$, $D=\langle y\rangle=R$ and $|R|=r$.
Since $D=\langle x\rangle$, we get $\langle x\rangle=R\leq N_G(\langle b\rangle)$.
If $\langle b\rangle\lhd PQ$, then $N_G(\langle b\rangle)\geq\langle PQ,R\rangle=G$
and $\langle b\rangle\lhd G$, a contradiction. Hence $\langle b\rangle$
is not normal in $PQ$ and there is $u\in PQ\setminus N_{PQ}(\langle b\rangle)$
such that $\langle b\rangle\langle u\rangle=\langle u\rangle\langle b\rangle\leq PQ$.
Let $u=cf$, $c\in P$, $f\in Q$. If $c=1$, then
$\langle b\rangle\langle u\rangle=\langle b\rangle\rtimes\langle f\rangle$
and $u=f\in N_{PQ}(\langle b\rangle)$, a contradiction. If $f=1$,
then $u=c\in P\leq N_{PQ}(\langle b\rangle)$, a contradiction.
Consequently,
$c\neq 1$, $f\neq 1$ and $\langle u \rangle=\langle c\rangle\times \langle f\rangle$, $\langle f\rangle=Q$, $Q=Q^c\leq S\cap S^c$ and $S^c\neq S$, since $c\in P$,
$c\notin N_G(S)=S$. But now $Q\lhd \langle S,S^c\rangle =G$,
a contradiction.
Finally, we consider the case when $S=R$ is noncyclic Sylow $r$-subgroup of $G$.
Assume that in $S$, there is a cyclic subgroup $Z$ of index~$r$.
Let $A\leq P$ such that $|A|=p$ and let $g\in G\setminus N_G(A)$ such that
$\langle g\rangle A\leq G$. Since $G=PR$, we deduce $g=bx$, $b\in P$, $x\in R$ and
$\langle g\rangle=\langle b\rangle\times\langle x\rangle$.
If $b=1$, then $g=\langle x\rangle$ and $\langle g\rangle A=A\rtimes \langle g\rangle$,
since $\langle g\rangle A$ is $p$-closed and $A$ is a Sylow $p$-subgroup
of $\langle g\rangle A$. Hence $g\in N_G(A)$, a contradiction.
If $x=1$, then $g=b\in P\leq N_G(A)$. This contradicts with the choice of~$g$.
So, $b\neq 1$, $x\neq 1$, $R^b\neq R$, $x^b=x\in R\cap R^b$.
If $\langle x_1\rangle=\langle x\rangle\cap Z\neq 1$,
then $\langle x_1\rangle\lhd R$. Since $x_1^b=x_1\in Z^b\lhd R^b$,
we get $\langle x_1\rangle\lhd R^b$.
Therefore $\langle x_1\rangle\lhd \langle R,R^b\rangle=G$.
This contradicts with $R_G=1$.
So, $\langle x\rangle\cap Z=1$ and $R=Z\rtimes\langle x\rangle$,
$|x|=r$, $x\in C_G(\langle b\rangle)\leq N_G(\langle b\rangle)$.
If $\langle b\rangle\lhd PZ$, then $\langle b\rangle\lhd G$,
a contradiction. Thus, $N_{PZ}(\langle b\rangle)<PZ$
and there is $u\in PZ\setminus N_{PZ}(\langle b\rangle)$
such that $\langle u\rangle\langle b\rangle\leq G$.
Since $u\in PZ$, we conclude $\langle u\rangle=\langle c\rangle\times\langle y\rangle$,
$\langle c\rangle\leq P$, $\langle y\rangle\leq Z$. Verification shows that $c\neq 1$,
$y\neq 1$. From $N_G(Z)=R$ it follows that $Z^c\neq Z$ and $y^c=y\in Z\cap Z^c$.
Now, $\langle y\rangle\lhd \langle R,R^c\rangle=G$, a contradiction.
If $S$ is not an abelian group of order~$r^3$ and exponent~$r$,
then by Lemma~\ref{expp}, $S$ contains a nonpermutable cyclic primary subgroup,
which contradicts with the choice of~$G$.
\end{proof}
\begin{theorem}\label{tcp}
If every primary cyclic subgroup of a group $G$ is
$\mathbb{P}$\nobreakdash-\hspace{0pt}subnormal or strongly permutable,
then~$G\in \rm{v}\mathfrak U$.
\end{theorem}
\begin{proof}
We use induction on the group order.
Let $N$ be a normal subgroup of a group $G$, $N\neq 1$,
and let $\langle a\rangle$ be a cyclic primary subgroup of $N$.
By the choice of $G$, $\langle a\rangle$ is $\mathbb{P}$-subnormal
or strongly permutable in $G$. If $\langle a\rangle$ is
$\mathbb{P}$-subnormal in $G$, then by Lemma~\ref{KPProp}\,(1),
$\langle a\rangle$ is $\mathbb{P}$-subnormal in $N$.
If $\langle a\rangle$ is strongly permutable in $G$,
then by Lemma~\ref{PerProp}\,(3), $\langle a\rangle$ is strongly permutable in $N$.
Now assume that $A/N$ is a cyclic $t$-subgroup, $t\in\pi(G)$.
Let $B$ be a subgroup of least order such that $B\leq A$, $BN=A$.
Then $B\cap N\leq\Phi(B)$, $A/N=BN/N\cong B/B\cap N$,
hence $B$ is a cyclic $t$-subgroup. By the choice of $G$,
$B$ is $\mathbb{P}$-subnormal or strongly permutable in $G$.
As $A/N=BN/N$, according to Lemma~\ref{KPProp}\,(1) and Lemma~\ref{PerProp}\,(1),
$A/N$ is $\mathbb{P}$-subnormal or strongly permutable in $G$.
Thus the hypothesis holds for all normal subgroups of $G$
and all quotients subgroups.
Suppose that $G$ is a simple group. If every primary cyclic subgroup of $G$ is
strongly permutable, then $G$ is supersoluble by Theorem~\ref{tc}.
Consequently, $G$ contains a cyclic primary subgroup $A$ such that
$A$ is $\mathbb{P}$-subnormal in $G$. Since the unit subgroup is
$\mathbb{P}$-subnormal in~$A$, then it is $\mathbb{P}$-subnormal in~$G$.
According to~\cite{kaz}, \cite[p. 342]{cs},
$G$ is isomorphic to one of the following groups.
\[
L_2(7), \ L_2(11), \ L_3(3), \ L_3(5), \ L_2(2^m),
2^m+1 \mbox{ is prime}.
\]
In every of these groups, a Sylow $r$-subgroup $R$ is cyclic for $r=\max \ \pi(G)$.
By the choice of $G$, $R$ is $\mathbb{P}$-subnormal or strongly permutable in $G$.
If $R$ is $\mathbb{P}$-subnormal in $G$, then by Lemma~\ref{KPMax},
$R$ is normal in $G$. If $R$ is strongly permutable in $G$,
then in view of Lemma~\ref{PG=NG}, $R$ is normal in $G$.
Consequently, $R$ is normal in $G$ and $G$ is not a simple group, a contradiction.
Thus in $G$, there is a normal subgroup $N$, $N\neq 1$, and by induction,
$G/N\in\mathrm{v}\mathfrak{U}$ and $N\in\mathrm{v}\mathfrak{U}$.
Hence $G$ is soluble. In view of Lemma~\ref{KPProp}\,(2) and by induction,
every proper subgroup of $G$ belongs to~$\mathrm{v}\mathfrak{U}$ and
$G$ is a minimal non-$\mathrm{v}\mathfrak{U}$-group.
According to~\cite[Theorem~B\,(4)]{mkRic},
$G$ is a biprimary minimal non-supersoluble group
in which non-normal Sylow subgroups are cyclic.
Hence $G=R\rtimes Q$ is a group such that a Sylow $r$-subgroup $R$ is normal in $G$
and a Sylow $q$-subgroup $Q$ is cyclic and $\mathbb{P}$-subnormal or strongly permutable
in~$G$ by the choice of~$G$. By Corollary~\ref{BiU},
$G\in \mathfrak{U}\subseteq \mathrm{v}\mathfrak{U}$.
\end{proof}
\begin{example}
In $A_4$, every subgroup of order~2 is $\mathbb{P}$-subnormal,
but it is not permutable.
\end{example}
\begin{example}
In $L_2(7)$, every subgroup of order~3 is permutable,
but it is not $\mathbb{P}$-subnormal.
\end{example}
|
2,869,038,154,844 | arxiv | \section{The statistical hadronization model}
The main idea of the SHM is that hadrons are emitted from regions at statistical
equilibrium. No hypothesis is made about how
statistical equilibrium is achieved; this can be a direct consequence of the
hadronization process~\cite{beca_bari}. In a single collision event, there might
be several clusters with different collective momenta, different
overall charges and volumes. However, Lorentz-invariant quantities like particle
multiplicities are independent of clusters' momenta.
Depending on the system size, statistical analysis can be done in different
statistical ensembles, which have been studied
carefully by different groups and methods suitable for relativistic nucleus-nucleus
collisions are well established~\cite{kerabeca}.
In this work, small systems with net baryon number less than 10 are
calculated in the $BSQ$-canonical ensemble taking into account exact conservation of
$B$, $S$ and $Q$ charges. For the systems with the number of participants ($N_\mathrm{P}$)
between 10 and 100, exact conservation of
strangeness only is taken into account while larger systems are treated
grand-canonically (GC).
In spite of the appropriate ensemble, theoretical multiplicities are calculated within the
main version
of the statistical model by fitting temperature $T$, scaling volume $V$,
and strangeness suppression factor $\gamma_S$~\cite{gammas}.
Additionally, one needs to introduce a chemical potential for all charges that are
treated in a GC manner. Usually the baryon chemical potential is taken as a free fit parameter,
while the
strangeness chemical potential is fixed by assuming net strangeness neutrality
in the system and the chemical potential for electric charge by assuming that $Q/B$ in the
system equals $Z/A$ of the colliding nuclei.
The overall
multiplicity to be compared with the data, is calculated as the sum of primary
multiplicity and the contribution from the decay of heavier hadrons:
$\langle n_j \rangle = \langle n_j \rangle^{\mathrm{primary}} +
\sum_k \mathrm{Br}(k\rightarrow j) \langle n_k \rangle$,
where the branching ratios are taken from the latest issue of the Review of Particle
Physics \cite{pdg}.
\begin{table}[!ht]
\begin{footnotesize}
\vspace{0.3cm}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Parameters & SHM($\gamma_S$) & SHM($\gamma_S$) & SHM(TC) & SHM($\gamma_S$) & SHM(TC) \\
\hline
& p-p 158$A$ GeV & \multicolumn{2}{|c|}{C-C 158$A$ GeV} & \multicolumn{2}{|c|}{Si-Si 158$A$ GeV} \\
\hline
$T$ (MeV) & 177.3$\pm$5.2 & 165.7$\pm$4.1 & 170$\pm$10 & 163.0$\pm$4.7 & 162.0$\pm$7.6 \\
$\mu_B$ (MeV) & & 248.1$\pm$12.5 & & 245.5$\pm$11.0 & 234.4$\pm$22.5 \\
$\gamma_S$ & 0.445$\pm$0.020 & 0.575$\pm$0.042 & 1.0 (fixed) & 0.664$\pm$0.050 & 1.0 (fixed) \\
V' & 0.128$\pm$0.005 & 0.84$\pm$0.05 & 0.23$\pm$0.03 & 2.10$\pm$0.13 & 0.91$\pm$0.11 \\
$\langle N_c \rangle$ & & & 6$\pm$0.4 & & 11.4$\pm$1.8 \\
\hline
$\chi^2$/dof & 13.0/7 & 3.4/4 & 5.8/5 & 7.6/4 & 1.0/4\\
\hline
\end{tabular}
\end{footnotesize}
\caption{Summary of fitted parameters (V'=$VT^3 \, \exp[-0.7 \, {\rm GeV}/T]$) at top SPS beam energy in
the framework of the SHM($\gamma_S$) model and SHM(TC).
\vspace{-0.2cm}\label{parameters}}
\end{table}
\section{Experimental data set and analysis results}
The experimental data consists of measurements made by the NA49
collaboration in central p-p, C-C, Si-Si and Pb-Pb collisions at beam momenta
158$A$ GeV~\cite{NA49pp_all,prel_CCSiSi,Af2002mx,Mischke2003,Anticic2004,Friese2002,Alt2004}.
The analysis has been carried out by looking for the minimum of the
$\chi^2 = \sum_i \frac{(n_i^{\rm exp} - n_i^{\rm theo})^2}{\sigma_i^2}$.
The fitted parameters within the main scheme SHM($\gamma_S$) are shown in Table~\ref{parameters}.
A major result of these fits is that $\gamma_S$ is monotonically increasing
as a function of the number of participating nucleons,
and significantly smaller than 1 in all cases (see Fig.\ref{gsls}),
thus strangeness seems to be under-saturated with respect
to a completely chemically equilibrated hadron gas. This confirms previous
findings \cite{PbPb_last,beca01,cley,cley_systemsize}.
\subsection{{\bf Superposition of NN collisions with an equilibrated fireball}}
In the Two Component model (SHM(TC)), first introduced in~\cite{PbPb_last}, the observed
hadron production is taken as the superposition of two components: one originated from a large
fireball at complete chemical equilibrium at freeze-out, with $\gamma_S=1$, and another
component from single nucleon-nucleon collisions.
Since it is known that in NN collisions strangeness is strongly suppressed,
the idea is to ascribe the observed under-saturation of strangeness in heavy
ion collisions to the NN component.
With the simplifying assumption of disregarding
subsequent inelastic collisions of particles produced in those primary NN collisions,
the overall hadron multiplicity can be written then as
$ \langle n_j \rangle = \langle N_c \rangle \langle n_j \rangle_{NN} +
\langle n_j \rangle_V,$
where $\langle n_j \rangle_{NN}$ is the average multiplicity of the $j^{\rm th}$
hadron in a single NN collision, $\langle N_c \rangle$ is the mean number of
single NN collisions and $\langle n_j
\rangle_V$ is the average multiplicity of hadrons emitted from the equilibrated
fireball.
To estimate $\langle n_j \rangle_{NN}$ in np and nn collisions, the
parameters (see Table \ref{parameters}) of the statistical model determined in pp
are retained and the
initial quantum numbers are changed accordingly. Theoretical multiplicities have
been calculated in the canonical ensemble, which is described in detail in
ref.~\cite{becapt}.
This model was seen to be able to reproduce the experimental particle multiplicities
measured in the Pb-Pb collisions at 158$A$ GeV~\cite{PbPb_last,jamaica_proc}.
For the Si-Si system, $T$, $V$, $\mu_B$ of the central fireball and $\langle N_c \rangle$
were fitted using the S-canonical ensemble in the central fireball.
The fit quality is significantly improved compared to the main version of the statistical
model, if the number of
"single" NN collisions is about 11 with a 16\% uncertainty.
The central fireball produced in the C-C system needs to be analyzed in the
$BSQ$-canonical ensemble
taking into account the actual proton-neutron configurations. The Two Component model can
be used to describe the C-C system as well, if the total baryon number in the central
fireball is $B=N_p+N_n\approx4$. Even though fit quality is slightly worse than with the
main version of the statistical model, experimental multiplicities are well reproduced
with the SHM(TC) also.
\begin{figure}[!ht]
$\begin{array}{c@{\hspace{0.3in}}c}
\epsfxsize=3.5in
\epsffile{gs_vs_Np.eps} & \hspace*{-3cm}
\epsfxsize=3.5in
\epsffile{ls_vs_Np.eps} \\
\end{array}$
\caption{The strangeness non-equilibrium parameter $\gamma_S$ and the corresponding
$\lambda_S$ factor as a function of the number of participating nucleons in the collision system
(Pb-Pb from~\cite{PbPb_last}).
Both lines are of the functional form
$f(\mathrm{N}_\mathrm{P})=A-{\rm e}^{-B\mathrm{N}_\mathrm{P}}$ and are plotted to guide the eye.
Whenever the resulting $\chi^2/dof > 1$ in the fit, errors have been re-scaled by factor
$\sqrt{\chi^2/dof}$, for details see~\cite{pdg,PbPb_last}.
From left to right:
p-p, C-C, Si-Si and Pb-Pb.\label{gsls}}
\end{figure}
\section{System size dependence and conclusions}
Our fit results show non-trivial system size dependence of chemical equilibration
and characteristic thermal parameters of the source in ultra-relativistic heavy ion
collisions.
The chemical freeze-out of all the different
systems, C-C, Si-Si and Pb-Pb with beam momenta 158$A$ GeV seem to happen at similar
chemical state, all of them at $\mu_B\approx$250 MeV. Such weak system size dependence of
the baryon chemical potential has been already reported earlier~\cite{cley_systemsize}.
Systems with few participating
nucleons seem to decouple at slightly higher temperature than heavy systems, but in general
the chemical freeze-out temperature as well as the baryon chemical potential are
determined mostly by the beam energy, not by the number of participants, and thus
C-C, Si-Si and Pb-Pb with the same beam momenta, seem not
to follow the chemical freeze-out curve (see Fig.~\ref{tmu_trho}), but show more
complex system size dependence,
an interplay of the initial beam energy and the number of participants in the system.
The chemical equilibration of strangeness, on the other hand, seems to be strongly dependent
on the number of participants.
Going from small to large systems, the $\gamma_S$ parameter increases monotonically
from 0.45 in the p-p to
0.9 in the Pb-Pb with the same beam momenta, and thus strangeness seems to
be out of equilibrium in all collision systems studied at SPS. The Wroblewski variable
$\lambda_S = 2 \langle {\rm s}\bar{\rm s}\rangle/(\langle {\rm u}\bar{\rm u}\rangle+\langle {\rm d}\bar{\rm d}\rangle)$, the estimated
ratio of newly produced strange quarks to u and d quarks at primary hadron level,
features very similar suppression in strangeness production, see Fig.~\ref{gsls}.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.9,angle=0]{chemFOcurve4.eps}
\end{center}
\caption{Chemical freeze-out points in the [$\mu_B-T$] plane in various heavy ion
collisions. Full round dots~\cite{PbPb_last} refer to the Au-Au at 11.6 and
Pb-Pb collisions at 40, 80, 158$A$ GeV, whilst the square dot
has been obtained by applying the statistical model to the preliminary
$\pi^+$, $\pi^-$, $K^+$, $K^-$ and $N_\mathrm{P}$ multiplicities~\cite{dieterSQM2004}
in the Au-Au collisions at $\sqrt s_{NN} = 200$ GeV.
The hollow round dots refer to the C-C and Si-Si collisions at 158$A$ GeV.
Whenever the resulting $\chi^2/dof > 1$ in the fit, errors have been re-scaled by factor
$\sqrt{\chi^2/dof}$, for details see~\cite{pdg,PbPb_last}.
}\label{tmu_trho}
\end{figure}
\section*{References}
|
2,869,038,154,845 | arxiv | \section{Introduction}
Numerical simulations play a major role among
studies of the glass transition
since, unlike in experimental works,
the individual motion of a large number of particles
can be followed at all times~\cite{hans}.
Computer simulations usually study Newtonian dynamics (ND) by solving
a discretized version of Newton's equations
for a given interaction between particles~\cite{at}.
Although this is the most appropriate
dynamics to study molecular liquids, it can be interesting
to consider alternative dynamics that are not deterministic,
or which do not conserve the energy.
In colloidal glasses and physical gels, for instance,
the particles undergo Brownian motion arising
from collisions with molecules in the solvent, and a stochastic
dynamics is more appropriate~\cite{at}. Theoretical considerations
might also suggest the study of different sorts of dynamics
for a given interaction between particles, for instance to assess
the role of conservation laws~\cite{previous,I,II}
and structural information~\cite{mct,mctcolloid}.
Of course, if a given dynamics satisfies detailed balance with respect to
the Boltzmann distribution, all structural quantities remain unchanged,
but the resulting dynamical behaviour might be very different.
In this paper, we study the relaxation dynamics
of a commonly used theoretical model for silica
using Monte Carlo simulations and compare the results with previous ND
studies for both the averaged dynamical behaviour and
the spatially heterogeneous dynamics of this system.
Several papers have studied in detail the influence of the chosen
microscopic dynamics on the dynamical behaviour in a simple
glass-former, namely a binary mixture of Lennard-Jones
particles~\cite{hans,KA}.
Gleim, Kob and Binder~\cite{gleim} studied Stochastic Dynamics
where a friction term and a random noise
are added to Newton's equations, the amplitude of both terms
being related by a fluctuation-dissipation theorem.
Szamel and Flenner~\cite{szamel2} used Brownian
dynamics, in which there are no momenta,
and positions evolve with a Langevin dynamics.
Berthier and Kob~\cite{ljmc} employed Monte Carlo dynamics,
where the potential energy between two configurations
is used to accept or reject a trial move.
The equivalence between these three stochastic dynamics
and the originally studied ND was established at the level of the averaged
dynamical behaviour, except at very short times where differences
are indeed expected. However, important
differences were found when dynamic fluctuations
were considered, even in the
long-time regime comprising the alpha-relaxation~\cite{I,II,ljmc}.
Silica, the material studied in the present work,
is different from the previously considered Lennard-Jones
case in many aspects which all motivate our Monte Carlo study of
the Beest, Kramer and van Santen (BKS) model for
silica~\cite{beest90}.
First, the BKS model was devised to represent a real material, making
our conclusions more directly applicable to experiments. Second,
the temperature evolution of relaxation times
is well-described by a simple Arrhenius law at low temperatures, typical
of strong glass-formers, which are commonly believed
to belong to a somewhat different class of materials, so that
qualitative differences might be expected with more fragile,
super-Arrheniusly relaxing materials.
Third, the onset of slow dynamics in fragile materials
is often said to be accurately described by the application
of mode-coupling theory, at least over
an intermediate window of 2 to 3 decades of relaxation
times~\cite{mct}. Mode-coupling theory (MCT) formulates in particular a series
of quantitative predictions regarding the time, spatial, and temperature
dependences of dynamic correlators. In the case of
silica melts, previous analysis reported evidence in favour of a narrower
temperature regime where MCT can be applied, but the test
of several theoretical predictions was either
seriously affected, or even made impossible
by the presence of strong short-time thermal vibrations
related to the Boson peak in this material~\cite{hk}.
These vibrations affect
the time dependence of the correlators much more strongly
in silica than in Lennard-Jones systems, which constitutes
a fourth difference between the two systems. Using
Monte Carlo simulations we shall therefore be able to
revisit the MCT analysis performed in Refs.~\cite{hk}.
Fifth, while detailed analysis of dynamic heterogeneity
is available for fragile materials, a comparatively smaller
amount of data is available for strong materials~\cite{I,II,vogel,teboul},
and we shall
therefore investigate issues that have not been addressed in previous
work.
The paper is organized as follows. In Sec.~\ref{mc} we give
details about the simulation technique and compare its efficiency
to previously studied dynamics. In Sec.~\ref{results} we present
our numerical results about the averaged dynamics of silica in
Monte Carlo simulations, while Sec.~\ref{dh}
deals with aspects related to dynamic heterogeneity.
Finally, Sec.~\ref{conclusion} concludes the paper.
\section{Simulating silica using Monte Carlo dynamics}
\label{mc}
Our aim is to study a non-Newtonian dynamics of the glass-former
silica, SiO$_2$.
Therefore, we must first choose a reliable model to describe the
interactions in this two-component system made of
Si and O atoms, and then design a specific stochastic
dynamics which we require to be efficient and to
yield the same static properties as Newtonian dynamics.
Various simulations have shown that a reliable pair potential
to study silica in computer simulations is the one
proposed by BKS~\cite{beest90,hk,vogel,teboul,bks_sim,bks_sim2,voll}.
The functional form of the BKS potential is
\begin{equation}
\phi_{\alpha \beta}^{\rm BKS}(r)=
\frac{q_{\alpha} q_{\beta} e^2}{r} +
A_{\alpha \beta} \exp\left(-B_{\alpha \beta}r\right) -
\frac{C_{\alpha \beta}}{r^6},
\label{pp}
\end{equation}
where $\alpha, \beta \in [{\rm Si}, {\rm O}]$ and $r$ is the distance
between the atoms of type $\alpha$ and $\beta$. The values of the
constants $q_{\alpha}, q_{\beta}, A_{\alpha \beta}, B_{\alpha \beta}$,
and $C_{\alpha \beta}$ can be found in Ref.~\cite{beest90}. For the
sake of computational efficiency the short range part of the potential
was truncated and shifted at 5.5~\AA. This truncation also has the
benefit of improving the agreement between simulation and experiment
with regard to the density of the amorphous glass at low temperatures.
The system investigated has $N_{\rm Si} = 336$ and $N_{\rm O}=672$
atoms in a cubic box with fixed size $L=24.23$~\AA, so
that the density is $\rho = 2.37$ g/cm$^3$, close to the
experimentally measured density at atmospheric pressure
of 2.2~g/cm$^3$~\cite{density}.
Once the pair interaction is chosen, we have to decide what stochastic dynamics
to implement. Previous studies in a Lennard-Jones system concluded
that among Monte Carlo (MC), Stochastic Dynamics (SD)
and Brownian Dynamics (BD),
MC was by far the most efficient algorithm because
relatively larger incremental steps can be used while
maintaining detailed balance, which is impossible for
SD and BD where very small discretized timesteps are needed
to maintain the fluctuation-dissipation relation between
noise and friction terms~\cite{ljmc}. Given the generality of this argument,
it should carry over to silica, and we decided to implement MC
dynamics to the BKS model. An additional justification for our choice stems
from unpublished results by Horbach and Kob who performed preliminary
investigations of the SD of BKS silica~\cite{phdjurgen}.
Using a friction term similar
in magnitude to the one used in Lennard-Jones simulations was however not
enough to efficiently suppress short-time elastic vibrations.
Using an even larger friction term would probably
damp these vibrations, but would also make the simulation
impractically slow.
A standard Monte Carlo dynamics~\cite{at}
for the pair potential in Eq.~(\ref{pp}) should proceed as follows.
In an elementary MC move,
a particle, $i$, located at the position ${\bf r}_i$
is chosen at random. The energy cost, $\Delta E_i$, to move
particle $i$ from position ${\bf r}_i$ to a new,
trial position ${\bf r}_i + \delta {\bf r}$ is evaluated,
$\delta {\bf r}$ being a random vector comprised in a cube
of linear length $\delta_{\rm max}$ centered around the origin.
The Metropolis acceptance rate, $p = {\rm min}
(1, e^{-\beta \Delta E_i /k_B})$, where
$\beta =1/T$ is the inverse temperature,
is then used to decide whether the move is accepted or rejected.
In the following, one Monte Carlo timestep represents
$N=N_{\rm Si} + N_{\rm O}$
attempts to make such an elementary move, and timescales
are reported in this unit. Temperature will be expressed in Kelvin.
Monte Carlo simulations can of course be made even more efficient by
implementing for instance swaps between particles, or using
parallel tempering. The dynamical behaviour, however, is then strongly
affected by such non-physical moves
and only equilibrium thermodynamics can be studied. Since
we want to conserve a physically realistic
dynamics, we cannot use such improved schemes.
An additional difficulty with Eq.~(\ref{pp}) as compared to
Lennard-Jones systems is the Coulombic interaction in the first term.
Such a long-range interaction means that the evaluation of the energy
difference $\Delta E_i$ needed to move particle $i$ in an elementary
MC step requires a sum over every particles $j \neq i$, and over
their repeated images due to periodic boundary conditions.
Of course the sum
can be efficiently evaluated using Ewald summations techniques,
as is commonly employed in ND simulations~\cite{at}. We note, however,
that Ewald techniques are better suited for ND than for MC since
in ND the positions and velocities of all particles are simultaneously
updated so that the Ewald summation is performed once to update
all particles.
In MC simulations,
each single move requires its own Ewald summation, and this remains
computationally very costly.
For the BKS potential
in Eq.~(\ref{pp}) it was recently shown that a simple truncation
can be performed which makes the range of the Coulombic interaction term
finite~\cite{carre}.
Detailed ND simulations have shown that in the range of temperatures
presently accessible to computer experiments,
no difference can be detected between
the finite range and the infinite range versions of the BKS potential
for a wide variety of static and dynamic properties.
Therefore we build on this work and make the replacement~\cite{carre}
\begin{equation}
\frac{1}{r} \to \left( \frac{1}{r} - \frac{1}{r_c} \right)
+ \frac{1}{r_c^2} (r-r_c), \quad {\rm for} \,\, r \leq r_c,
\end{equation}
while $1/r \to 0$ for $r>r_c$. This amounts to smoothly truncating
the potential at a finite range, $r_c$, maintaining both
energy and forces continuous at the cutoff $r=r_c$.
The physical motivation for this form of the truncation was given
by Wolf~\cite{wolf}, and discussed in several
more recent papers~\cite{carre,wolf2}.
Following Ref.~\cite{carre}, we fix $r_c=10.14$~\AA.
Once the potential is truncated, MC simulations become
much more efficient, and much simpler to implement. Furthermore,
this will allow us to perform
detailed comparisons of the dynamics of BKS model of silica
where the Wolf truncation is used for both ND and MC in the
very same manner, so that any difference between the two sets of data can be
safely assigned to the change of microscopic dynamics alone, while reference
to earlier work done using Ewald summations is still quantitatively
meaningful.
\begin{figure}
\begin{center}
\psfig{file=deltainset.ps,width=8.5cm}
\end{center}
\caption{\label{delta} Self-intermediate scattering function
for silicon, Eq.~(\ref{self}), at $T=4000$~K and $|{\bf q}|=1.7$~\AA$^{-1}$
for various values of $\delta_{\rm max}$.
Inset: The evolution of the relaxation time with
$\delta_{\rm max}$ unambiguously defines an optimal value
$\delta_{\rm max} \approx 0.65$~\AA~
for efficient Monte Carlo simulations.}
\end{figure}
The one degree of freedom that remains to be fixed is $\delta_{\rm max}$,
which determines the average lengthscale of elementary moves.
If chosen too small, energy costs are very small and most of the moves
are accepted, but the dynamics is very slow because it takes a long
time for particles to diffuse over the long distances needed to relax
the system.
On the other hand too large displacements will on average be very costly
in energy
and acceptance rates can become prohibitively small. We seek a compromise
between these two extremes by monitoring the
dynamics at a moderately low temperature, $T=4000$~K, for several
values of $\delta_{\rm max}$.
As a most sensitive indicator of the relaxational behaviour, we measure the
contribution from the specie $\alpha$ ($\alpha=$ Si, O) to the
self-intermediate scattering function defined by
\begin{equation} F_s(q,t) =
\left\langle \frac{1}{N_\alpha} \sum_{j=1}^{N_\alpha} e^{i {\bf q}
\cdot [{\bf r}_j(t) - {\bf r}_j(0)]} \right\rangle.
\label{self}
\end{equation}
We make use of rotational invariance to
spherically average over wave vectors of comparable magnitude,
and present results for $|{\bf q}|=1.7$~\AA$^{-1}$,
which is the location of the
pre-peak observed in the static structure factor
$S(q)$ of the liquid. This corresponds to the typical (inverse) size of the
SiO$_4$ tetrahedra.
In Fig.~\ref{delta} we present our results for $\delta_{\rm max}$
values between 0.3 and 1.0~\AA.
As expected we find that relaxation is slow both at small and large
values of $\delta_{\rm max}$, and most efficient for intermediate values.
Interestingly we also note that the overall
shape of the self-intermediate
scattering function does not sensitively depend
on $\delta_{\max}$ over this wide range. We can therefore safely fix
the value of $\delta_{\max}$ based on an efficiency criterium alone.
We define a typical relaxation time, $\tau_\alpha$, as
\begin{equation}
F_s(q,\tau_\alpha) = \frac{1}{e},
\label{deftaualpha}
\end{equation}
and show its $\delta_{\rm max}$ dependence
in the inset of Fig.~\ref{delta}. A clear minimum is observed
at the optimal value of $\delta_{\max} \approx 0.65$~\AA, which
we therefore use throughout the rest of this paper.
This distance corresponds
to a squared displacement of 0.4225~\AA$^2$, which is very close
to the plateau observed at intermediate times in the
mean-squared displacements (see Fig.~\ref{fs} below).
This plateau can be taken
as a rough measurement of the ``cage'' size for the particles, so that
MC simulations are most efficient when the cage is most quickly explored.
This argument and the data in Fig.~\ref{fs}
suggest that the location of the minimum should only be a weak function of
temperature, but we have not verified this point in detail. Therefore
we keep the value of $\delta_{\max}$ constant at all temperatures.
An alternative would be to optimize it at each $T$ and
then carefully rescale timescales between runs at different temperatures.
What about the relative efficiency between MC and ND? If we compare
the relaxation measured at $T=4000$~K, we find
$\tau_\alpha \sim 13400$ Monte Carlo steps, while
$\tau_\alpha \sim 4.7$~ps for ND. When using a discretized timestep of
1.6~fs, this means that, when counting in number of integration
timesteps, MC dynamics is $\approx 5$ times slower than
ND. This result contrasts with the results obtained in
a Lennard-Jones mixture where MC dynamics was about
2 times faster than ND~\cite{ljmc}. We attribute this relative loss
of efficiency to the existence of strong bonds between Si and O atoms
in silica, which have no counterparts in Lennard-Jones systems.
It is obvious that strong bonds are very hard to relax when using
sequential Monte Carlo moves, as recently
discussed in Ref.~\cite{steve}.
We have performed simulations at temperatures between
$T=6100$~K and $T=2750$~K, the latter being smaller than
the fitted mode-coupling temperature, $T_c=3330$~K~\cite{hk}.
For each temperature we have simulated 3 independent samples
to improve the statistics. Initial configurations
were taken as the final configurations obtained from
previous work performed with ND~\cite{carre}, so that production
runs could be started immediately. For each sample, production
runs lasted at least $15\tau_\alpha$ (at $T=2750$~K),
much longer for higher temperatures, so that statistical errors
in our measurements are fairly small. We have performed a few runs for a larger
number of particles, namely $N=8016$ particles, to investigate finite
size effects which are known to be relevant in silica~\cite{hka,fss,fss2},
and the results
will be discussed in Sec.~\ref{results}.
\section{Analysis of averaged two-time correlators}
\label{results}
In this section we report our results about the time behaviour
of averaged two-time correlators, we compare the Monte Carlo
results to Newtonian dynamics, and we perform a quantitative
mode-coupling analysis of the data.
\subsection{Intermediate scattering function and mean-squared displacements}
\begin{figure}
\psfig{file=fsqtsi.ps,width=8.5cm}
\psfig{file=fsqto.ps,width=8.5cm}
\psfig{file=msd.ps,width=8.5cm}
\caption{\label{fs} Top: Self-intermediate scattering function,
Eq.~(\ref{fs}), for $|{\bf q}|=1.7$~\AA$^{-1}$ and
temperatures $T=6100$, 4700, 4000, 3580, 3200, 3000, and 2750~K
(from left to right).
Bottom: Mean-squared displacement, Eq.~(\ref{msd}), for the same
temperatures in the same order.}
\end{figure}
The self-intermediate scattering function, Eq.~(\ref{self}),
is shown in Fig.~\ref{fs}
for temperatures decreasing from $T=6100$~K down to $T=2750$~K
for Si and O atoms at $|{\bf q}| = 1.7$~\AA$^{-1}$.
These curves present well-known features. Dynamics at high temperature
is fast and has an exponential nature. When temperature is decreased
below $T \approx 4500$~K, a two-step decay, the slower being strongly
non-exponential, becomes apparent. Upon decreasing
the temperature further,
the slow process dramatically slows down by about 4 decades,
while clearly conserving an almost temperature-independent,
non-exponential shape, as already reported for ND~\cite{hk}.
We also find that the first process, the decay towards
a plateau, slows down considerably when decreasing temperature, although
less dramatically than the slower process.
The fastest process, called `critical decay' in the language
of mode-coupling theory~\cite{mct}, is not observed when using
ND, because it is obscured by strong thermal vibrations
occurring at high frequencies (in the THz range). Clearly, no such vibrations
are detected in the present results which demonstrates
our first result: MC simulations very efficiently suppress
the high-frequency oscillations observed with ND.
Although the plateau seen in $F_s(q,t)$ is commonly
interpreted as `vibrations of a particle within a cage',
the data in Fig.~\ref{fs} discard this view. From direct
visualization of the particles' individual dynamics
it is obvious that vibrations take place in just a few MC
timesteps, while the decay towards the plateau can be
as long as $10^4$ time units at the lowest temperatures studied here.
This decay is therefore necessarily more complex,
most probably cooperative in nature. This interpretation
is supported by recent theoretical studies where a plateau
is observed in two-time correlators of lattice models where local
vibrations are indeed completely absent~\cite{bethe}.
A detailed atomistic description
of this process has not yet been reported, but would
indeed be very interesting.
Next, we study the mean-squared displacement
defined as
\begin{equation}
\label{msd}
\Delta^2 r(t) = \frac{1}{N_\alpha} \sum_{i=1}^{N_\alpha} \left\langle
|{\bf r}_i(t) - {\bf r}_i(0) |^2 \right\rangle,
\end{equation}
and we present its temperature evolution in Fig.~\ref{fs},
for both Si and O atoms. The evolution of $\Delta^2 r(t)$
mirrors that of the self-intermediate scattering function,
and the development of a two-step relaxation process
is clear from these figures.
Because we are studying a stochastic dynamics,
displacements are diffusive at both short and long timescales.
This constitutes an obvious, expected difference between
ND and MC simulations: data clearly cannot match at very small
times. The goal of the present study is therefore to determine
whether the dynamics quantitatively match at times where the
relaxation is not obviously ruled by short-time ballistic/diffusive
displacements.
The plateau observed in $F_s(q,t)$ now translates into a strongly
sub-diffusive
regime in the mean-squared displacements separating the two diffusive
regimes. At the lowest temperature studied, when $t$ changes by three decades
from $2\times10^2$ to $2\times 10^5$, the mean-squared displacement
of Si
changes by a mere factor 4.6 from 0.16 to 0.074.
Particles are therefore nearly arrested for several decades of times,
before eventually entering the diffusing regime where
the relaxation of the structure of the liquid takes place.
\subsection{Comparison to Newtonian dynamics}
The previous subsection has shown that the Monte Carlo dynamics
of silica is qualitatively similar to the one reported for ND,
apart at relatively short times where the effect of thermal vibrations
is efficiently suppressed and the dynamics is diffusive instead
of ballistic. We now compare our results more
quantitatively with the dynamical behaviour observed using ND.
\begin{figure}
\hspace*{-0.2cm}
\psfig{file=tau.ps,width=8.5cm}
\caption{\label{comp}
Temperature evolution of the alpha-relaxation time
$\tau_\alpha(T)$ for silicon (squares) and oxygen (circles),
and inverse self-diffusion constant for silicon (up triangles)
and oxygen (down triangles), vertically shifted
for clarity. Open symbols are for ND (times rescaled by
$t_0=0.31$ fs) closed symbols for MC.
Full lines are Arrhenius fits below $T\approx 3700$~K with
activation 5.86, 5.60, 5.43, and 4.91~eV (from top to bottom).
An Arrhenius fit for high temperatures is also presented
for $D$(O) with activation energy 2.76~eV.
The dashed line is a power law fit, $\tau_\alpha \sim
(T-T_c)^{-\gamma}$, with $T_c=3330$~K and $\gamma=2.35$.}
\end{figure}
To this end, we compare first the temperature evolution of the
relaxation times, $\tau_\alpha(T)$, defined in Eq.~(\ref{deftaualpha}),
in Fig.~\ref{comp}. Here, we use a standard
representation where an Arrhenius slowing down over a
constant energy barrier $E$, with an attempt frequency $1/\tau_0$,
\begin{equation}
\tau_\alpha = \tau_0 \exp \left( \frac{E}{k_B T} \right),
\label{arrhenius}
\end{equation}
appears as a straight line. To compare both sets of data we rescale
the ND data by a common factor, $t_0 = 0.31$~fs, which
takes into account the discretization timestep and the
efficiency difference discussed in the previous section; $t_0$ will
be kept constant throughout this paper.
We find that the temperature
evolution of the alpha-relaxation time measured in MC simulations
is in complete quantitative agreement with the
one obtained from ND, over the complete temperature range.
This proves that Monte Carlo techniques can be applied
not only to study static properties of silica, but also its
long-time dynamic properties.
In Fig.~\ref{comp} we also show the temperature evolution
of the self-diffusion constant, defined from the long-time limit
of the mean-squared displacement as
\begin{equation}
D = \lim_{t \to \infty} \frac{\Delta^2 r(t)}{6 t}.
\end{equation}
The behaviour of the (inverse)
diffusion constant is qualitatively very close
to the one of the alpha-relaxation time, and we again find
that ND and MC dynamics yield results in full quantitative agreement.
As expected for silica, we find that at low temperatures below
$T \approx 3700$~K, relaxation timescales and diffusion constant
change in an Arrhenius fashion described by Eq.~(\ref{arrhenius}).
We find, however, that the observed activation energies
display small variations between different observables, from
5.86~eV for $\tau_\alpha$(Si) to 4.91~eV for $1/D$(0).
These values compare well with previous analysis~\cite{hk},
and with experimental
findings~\cite{silicaexp}.
From Fig.~\ref{comp}, it is clear that Arrhenius behaviour is
obeyed below $T \approx 3700$~K only, while the data bend up
in this representation for higher temperatures. This behaviour
was interpreted in terms of a fragile to strong behaviour of the
relaxation timescales in several papers~\cite{hk,ftos1,ftos2},
despite the fact that
fragility is usually defined experimentally by considering data
on a much wider temperature window close to the experimental
glass transition.
To rationalize these findings, Horbach
and Kob analyzed the data using mode-coupling theory
predictions~\cite{hk}.
In particular they suggest to fit the temperature
dependence of $\tau_\alpha$ as
\begin{equation}
\tau_\alpha \sim (T-T_c)^{-\gamma},
\label{taumct}
\end{equation}
with $T_c \approx 3330$~K and $\gamma \approx 2.35$.
This power law fit is also
presented in Fig.~\ref{comp} as a dashed line. Its domain
of validity is of about 1 decade, which is significantly less
than for more fragile materials with super-Arrhenius behaviour
of relaxation timescales~\cite{KA}.
It is interesting to note that a simpler
interpretation of this phenomenon could be that this behaviour is nothing
but a smooth crossover from a non-glassy, homogeneous, high-temperature
behaviour to a glassy, heterogeneous, low temperature
behaviour, as found in simple models of strong glass-forming
liquids~\cite{bg}.
In Fig.~\ref{comp}, we implement this simpler scenario by fitting high
temperature data with an Arrhenius law, as is sometimes
done in the analysis of experimental data~\cite{gilles}.
Such a fit works nicely for
high temperatures, from $T=6100$ to 4700~K, but breaks down
below $T \approx 4000$~K.
A physical interpretation for this high-temperature Arrhenius behaviour
was offered in Ref.~\cite{heuer}.
This shows that analyzing silica dynamics
in terms of a simple crossover occurring around 4000~K between two simple
Arrhenius law is indeed a fair description of the data which does not require
invoking a more complex fragile to strong crossover being rationalized by the
existence of an avoided mode-coupling singularity.
\begin{figure}
\psfig{file=dec.ps,width=8.5cm}
\caption{\label{comp2}
Decoupling data for oxygen and silicon. We plot the
product $D \tau_\alpha$ taken from the data shown
in Fig.~\ref{comp} and normalize the product
by its value at $T_o=4700$~K such that deviations
from 1 indicates non-zero decoupling.
Open symbols are for ND, closed symbols for MC. Decoupling
is similar for both types of dynamics.}
\end{figure}
The difference found above for the activation energies describing
$\tau_\alpha$ and $1/D$ for both species
implies that these quantities, although both
devised to capture the temperature evolution of
single particle displacements have slightly different
temperature evolutions and are not proportional to one another.
This well-known feature implies the existence of
a ``decoupling'' between translational
diffusion and structural relaxation in silica,
as documented in previous papers~\cite{hk}.
In Fig.~\ref{comp2} we report the temperature evolution
of the product $D(T) \tau_\alpha(q,T)$ which is a pure constant
for a simply diffusive particle where $\tau_\alpha(q,T) = 1/(q^2 D)$.
We normalize this quantity by its value at $T_o=4700$~K, so
that any deviations from 1 indicates a non-zero
decoupling~\cite{berthier,epl}.
As expected we find that the product is not a constant,
but grows when temperature decreases. Remarkably, although this
quantity is a much more sensitive probe of the dynamics of the liquid,
its temperature evolution remains quantitatively similar for both ND and
MC dynamics. This shows that equivalence of the dynamics between
the two algorithms holds at the level of the complete distribution
of particle displacements, even for those tails that
are believed to dictate the observed decoupling.
In Sec.~\ref{dh}, we shall explore in more detail the
heterogeneous character of the dynamics of silica,
closely related to the decoupling discussed here. It is however
interesting to try and infer the amount of decoupling
predicted for silica at temperatures close to the experimental
glass transition, $T_g \approx 1450$~K. The glass transition temperature
of BKS silica deduced from extrapolation of viscosity measurements
is close to the experimental one,
$T_g^{\rm BKS} \approx 1350$~K~\cite{hk}. Extrapolating the
data in Fig.~\ref{comp2} down to 1400~K predicts
a decoupling of about 40 for Si dynamics, about 7 for O dynamics.
The difference between Si and O dynamics was recently
explained in Ref.~\cite{heuer}, where
it was noted that oxygen diffusion is in fact possible with no
rearrangement of the tetrahedral structure of silica involved.
Moreover, it is interesting to note that the amount of decoupling
found here is smaller than experimental findings in fragile materials
close to their glass transition~\cite{decouplingexp},
but is nonetheless clearly
different from zero. This suggests that
even strong materials display
dynamically heterogeneous dynamics, but its effect seems
less pronounced than in more fragile materials.
Theoretically, an identical temperature evolution of the
alpha relaxation timescale for MC and ND
is an important prediction of mode-coupling theory~\cite{mct} because
the theory uniquely predicts the dynamical behaviour from static
density fluctuations. Gleim {\it et al.} argue that their finding
of a quantitative agreement between SD and ND in a Lennard-Jones
mixture is a nice confirmation
of this non-trivial mode-coupling prediction~\cite{gleim}.
Szamel and Flenner~\cite{szamel2}
confirmed this claim using BD, and argued further that
even deviations from mode-coupling predictions are identical,
a statement that was extended to below the mode-coupling
temperature by Berthier and Kob~\cite{ljmc}.
In the present work we extend these findings to the case of silica
over a large range of temperatures, which goes far beyond
the temperature regime where MCT can be applied.
Therefore, we conclude that such an independence of the
glassy dynamics of supercooled liquids to their
microscopic dynamics, although
predicted by MCT, certainly has a much wider domain of validity
than the theory itself. Finally, we note
that the deviations from MCT predictions observed in Fig.~\ref{comp}
cannot be attributed to coupling to currents
which are expressed in terms of particle velocities.
In our MC simulations we have no velocities,
so that avoiding the mode-coupling singularity is not due to
the hydrodynamic effects pointed out in Ref.~\cite{previous}
(see Ref.~\cite{more} for more recent theoretical viewpoints).
\begin{figure}
\psfig{file=fs.ps,width=8.5cm}
\caption{\label{fss} Self-intermediate scattering function
for fixed $T$ and $q=1.7$~\AA$^{-1}$, obtained in MC and MD simulations
for two system sizes. The time axis in MD data is rescaled
by $t_0 = 0.31$ fs
to obtain maximum overlap with MC results, and
the same factor is used for the two sizes.
Larger systems relax faster and
the amplitude of this finite size effect is the
same for both dynamics.}
\end{figure}
The last comparison to ND we want to discuss concerns the study
of finite size effects. It was shown that the long-time dynamics
of silica is fairly sensitive to system size, and there are
detectable differences when the number of particles is changed from
1000 to 8000~\cite{hka,fss}.
Such a large effect is not observed in more fragile
materials~\cite{KA}.
It was suggested that short-time thermal vibrations, stronger
in silica than in simpler models, are responsible for this system
size dependence~\cite{hka,LW}.
Therefore, it could be expected that by efficiently
suppressing these vibrations finite size effects should be reduced.
But this is not what happens. In Fig.~\ref{fss}, we show
self-intermediate scattering functions measured at $T=3580$~K
and $|{\bf q}|=1.7$~\AA$^{-1}$ in both ND and MC for two system sizes,
$N=1008$ and $N=8016$ particles. Such data have been presented for
ND before~\cite{hka}, and our results agree with these earlier data.
The amplitude of the vibrations observed for $t/t_0 = {\tilde t}
\approx 10^3$
is smaller and the long time dynamics is faster when
$N$ is larger. For MC we find that high-frequency vibrations
and the corresponding finite size effects are indeed suppressed,
but the finite size effect for long-time relaxation, somewhat surprisingly,
survives in our MC simulations, and can therefore not be attributed
to high-frequency thermal vibrations. Recent studies
of the vibration spectrum and elastic properties at $T=0$
of amorphous media have suggested the existence of large-scale
structures~\cite{JL}: these objects are potential
candidates to account for the size effect found at long times.
It should then be
explained how these spatial structures affect the long time dynamics,
and why a finite size simulation box
at the same time affects the absolute
value of the alpha-relaxation timescale but leaves unchanged
many of its detailed properties~\cite{fss2,heuer2}.
\subsection{Mode-coupling analysis of dynamic correlators}
We now turn to a more detailed analysis of the shape and
wave vector dependences of two-time correlation functions,
revisiting in particular
the mode-coupling analysis performed by Horbach and Kob
in Refs.~\cite{hk}. They argue that MCT can generally be applied
to describe their silica data, and attribute most of
the deviations that they observe to short-time thermal vibrations
supposedly obscuring the ``true'' MCT behaviour.
We are therefore in a position to verify if their hypothesis
is correct.
When applied to supercooled liquids, MCT formulates a series of detailed
quantitative predictions regarding the time, wave vector, and temperature
dependences of two-time dynamical correlators close to the mode-coupling
singularity. In particular, MCT predicts that correlation functions
should indeed decay in the two-step manner reported in Fig.~\ref{fs}.
Moreover, for intermediate times corresponding to the plateau observed
in correlation functions, an approximate
equation can be derived which
describes the correlator close enough to the plateau~\cite{mct}.
The following
behaviour is then predicted,
\begin{equation}
F_s(q,t) \approx f_q + h_q F(t),
\label{betacorr}
\end{equation}
where $F(t)$ is the so-called $\beta$-correlator which
is independent of the wave vector, and whose shape
depends on a few parameters: the reduced distance from the
mode-coupling temperature, $\epsilon = |T-T_c|/T_c$, and a parameter
describing the MCT critical exponents, $\lambda$. Once $\lambda$
is known various exponents $(a,b,\gamma)$ are known, which describe, in
particular, the short-time behaviour of $F(t)$ when
$F_s(q,t)$ approaches the plateau, $F(t) \sim t^{-a}$, and
its long-time behaviour when leaving the plateau,
$F(t) \sim t^b$. The exponent $\gamma$
was introduced in Eq.~(\ref{taumct})
and describes the temperature evolution
of the relaxation time $\tau_\alpha$.
\begin{figure}
\psfig{file=fact.ps,width=8.5cm}
\psfig{file=fact2.ps,width=8.5cm}
\caption{\label{beta}
Test of the factorization property, Eq.~(\ref{facteq})
using $F_s(q,t)$ from Si and O dynamics, and wave vectors between
0.8 and 4 \AA$^{-1}$, for $T=3580$~K and 3000~K.
The data do not show collapse for times
$t'<t<t''$, and factorization does not work very well.}
\end{figure}
Several properties follow from Eq.~(\ref{betacorr}).
If one works at fixed temperature and varies the wave vector,
the following quantity,
\begin{equation}
R(t) \equiv \frac{\phi(t)-\phi(t')}{\phi(t'')-\phi(t')}
\approx \frac{F(t)-F(t')}{F(t'')-F(t')},
\label{facteq}
\end{equation}
where $\phi(t)$ stands for a two-time correlation function, should
become independent of $q$.
In Eq.~(\ref{facteq}), $t'$ and $t''$ are two arbitrary times taken
in the plateau regime. This is called the ``factorization property''
in the language of MCT. We follow Ref.~\cite{hk} and show in Fig.~\ref{beta}
the function $R(t)$ in
Eq.~(\ref{facteq}) using self-intermediate scattering functions
for different $q$ and for different species (Si and O)
at fixed temperatures, $T=3580$~K and $T=3000$~K, choosing
times comparable to those reported in Ref.~\cite{hk}, namely
$t''=82$ and $t'=760$ for $T=3580$~K, and
$t''=1360$ and $t'=66700$ for $T=3000$~K.
Although the factorization property seemed to hold
quite well in the ND data,
this is no more the case for our MC data, and $R(t)$ retains
a clear $q$ dependence between $t'$ and $t''$: no collapse of $R(t)$
can be seen in the regime $t'<t<t''$ in Fig.~\ref{beta}.
The reason is clear from Fig.~\ref{fs}: due to
thermal vibrations, the intermediate plateau was very flat in ND,
but it has much
more structure in our MC data. It was therefore easier to collapse
the ND data in this regime than the present MC data for which
a better agreement might have been expected.
In the case of the factorization property, the presence
of thermal vibrations
in fact favours a positive reading of the data, which become much
less convincing when these vibrations are suppressed.
Gleim and Kob had reached an opposite conclusion in the case of
a Lennard-Jones system~\cite{gleim-kob}. They found
that suppressing vibrations made the
mode-coupling analysis of the beta-relaxation more
convincing, suggesting that MCT describes
the Lennard-Jones system more accurately than silica.
\begin{figure}
\psfig{file=beta.ps,width=8.1cm}
\caption{\label{beta2} Self-intermediate scattering function
at fixed $T=3580$~K and various wave vectors,
$q=0.8$, 1.2, 1.7, 2.4, 3.2, and 4~\AA$^{-1}$ (from right to left).
Dashed lines show fits at intermediate times using Eq.~(\ref{betacorr}).
The inset shows the $q$-dependence of the fitting parameters
$h_q$ and $f_q$. Note that the time domains over which the fits
apply shift with $q$.}
\end{figure}
Next we perform a test of the theory which had not been possible
with ND data. We investigate in detail if
the behaviour predicted by Eq.~(\ref{betacorr}) is correct for both
short and long times. This test is not possible using ND because
the approach to the plateau is mainly ruled by thermal vibrations
(see for instance the ND data presented in Fig.~\ref{fss}).
In Fig.~\ref{beta2} we show that a ``critical decay'' does indeed
show up when thermal vibrations are overdamped and no oscillations
can be seen. To check in more detail if this behaviour is indeed
in quantitative agreement with the MCT predictions,
we fit the $F_s(q,t)$ data at $T=3580$~K, i.e. slightly
above $T_c=3330$~K,
for several wave vectors $q$ using the
$\beta$-correlator obtained from numerical integration
of the mode-coupling equation.
To get the fits shown in Fig.~\ref{beta2} we have to fix
the distance to the mode-coupling temperature $\epsilon$ and
the value $\lambda = 0.71$ both taken from Ref.~\cite{hk}, and yielding
$a=0.32$, $b=0.62$ and $\gamma=2.35$.
Additionally we have to adjust the microscopic timescale.
Moreover, for each wave vector we have to fix $h_q$ and $f_q$ which
respectively correspond to the amplitude of the $\beta$-correlator
and the height of the plateau in $F_s(q,t)$.
Finally, there are two additional ``hidden'' free parameters
in each of these fits: the somewhat arbitrarily chosen boundaries of
the time domain where the fitting function describes the data.
We then get the fits shown with dashed lines in Fig.~\ref{beta2},
which are of a quality comparable to the ones
usually found in the MCT literature~\cite{mct}.
The parameters $h_q$ and $f_q$ are also
shown in the inset of Fig.~\ref{beta2}, and behave
qualitatively as in similar studies. Inspection of
Fig.~\ref{beta2} reveals that the use of such freedom to fit the data
allows a qualitatively correct description of the data, although
clearly the time domain over which each wave vector is fit
systematically shifts when $q$ changes, and we could
not simultaneously fit the data at both small and large $q$ by fixing
the time interval of the fit. This failure is consistent with the
above finding that the factorization property is not satisfied.
Therefore we conclude that
MCT provides a qualitatively correct description of our data
in the plateau regime, with no satisfying quantitative
agreement, even in the absence of short-time thermal vibrations.
One has therefore to argue that the data are taken too far from
the transition for MCT to quantitatively apply to silica. However, since it
is not possible to get data closer to the transition (recall that the
transition does not exist), the domain of validity of the theory
then would become vanishingly small.
\begin{figure}
\psfig{file=fsqtcoll.ps,width=8.5cm}
\psfig{file=sqrt.ps,width=8.5cm}
\caption{\label{TTS} Top: Test of time-temperature superposition,
Eq.~(\ref{tts}).
The dashed line is a stretched exponential function
with $\beta=0.87$. Superposition holds at large rescaled times, but
fails in the $\beta$-regime because the plateau height
increases when $T$ decreases. Bottom: extracted plateau height
as a function of temperature fitted with a linear dependence (full line)
and with a square root singularity, Eq.~(\ref{sqrt}) (dashed line).
Open symbols are for ND, closed symbols for MC. No singular behaviour
of $f_q$ is visible in either set of data.}
\end{figure}
We now turn to longer timescales and show
in Fig.~\ref{TTS} a test of the time-temperature superposition
prediction of the theory which states that correlators
at fixed $q$ but different
temperatures should scale as~\cite{mct}
\begin{equation}
F_s(q,t) \approx {\cal F}_q \left( \frac{t}{\tau_\alpha(q)} \right),
\label{tts}
\end{equation}
where ${\cal F}_q(x) \approx f_q \exp (- x^{\beta(q)})$ and for times in
the $\alpha$-regime.
When high temperatures outside the glassy regime
are discarded Eq.~(\ref{tts}) works correctly when
the scaling variable $t/\tau_\alpha$ is not too small, but
fails more strongly in the late $\beta$-regime. Scaling in the
$\beta$-regime is often
one of the most successful prediction of MCT, see e.g. Ref.~\cite{KA}.
In the present case, it could be argued to fail
because we are collapsing data at temperatures which
are both above and below $T_c$. Indeed, below $T_c$
scaling in the $\beta$-regime is not expected anymore
because the height of the plateau, $f_q$ in Eq.~(\ref{betacorr}),
now becomes a temperature dependent quantity,
with the following predicted singular behaviour~\cite{mct}:
\begin{equation}
f_q(T) = f_q(T_c) + \alpha \sqrt{T_c - T}, \quad T \leq T_c,
\label{sqrt}
\end{equation}
while $f_q(T \geq T_c) = f_q(T_c)$.
The non-analytic behaviour of $f_q$ at $T_c$ is a further
characteristic feature of the mode-coupling singularity.
Since we can easily take data for $T<T_c$ which are arguably not
influenced by thermal vibrations, we can directly check for the presence
of the square-root singularity, Eq.~(\ref{sqrt}). This is done in
the bottom panel of Fig.~\ref{TTS},
where we also show data obtained from ND simulations.
That the latter are strongly influenced by thermal vibrations is clear,
since they systematically lie below the MC data and have a stronger
temperature dependence close to $T_c$.
However, even the MC data clearly indicate that $f_q(T)$ is better described
by a non-singular function of temperature, compatible
with the simple linear behaviour expected to hold at
very low temperatures.
The temperature dependence of the plateau height therefore explains
why time temperature superposition does not hold in the
late $\beta$-regime, but the linear temperature behaviour
indicates that there is no clear sign, from our data,
of the existence of a ``true'' underlying singularity at $T_c$.
\section{Dynamic heterogeneity}
\label{dh}
Having established the ability of MC simulations
to efficiently reproduce the averaged
slow dynamical behaviour observed in ND simulations, we now turn
to the study of the dynamic fluctuations around the average dynamical
behaviour, i.e. to dynamic heterogeneity.
Dynamic fluctuations can be studied through a
four-point susceptibility, $\chi_4(t)$, which
quantifies the strength of the spontaneous fluctuations
around the average dynamics by their variance,
\begin{equation}
\chi_4(t) = N_\alpha \left[ \langle f_s^2({\bf q}, t)
\rangle - F_s^2({ q}, t) \right],
\label{chi4lj}
\end{equation}
where
\begin{equation}
f_s({\bf q},t) = \frac{1}{N_\alpha} \sum_{i=1}^{N_\alpha}
\cos ({\bf q}
\cdot [{\bf r}_i(t) - {\bf r}_i(0)] ),
\end{equation}
represents
the real part of the instantaneous value of
the self-intermediate scattering function,
so that $F_s({ q},t) = \langle f_s({\bf q},t) \rangle$.
As shown by Eq.~(\ref{chi4lj}),
$\chi_4(t)$ will be large if run-to-run fluctuations
of the self-intermediate scattering functions
averaged in large but finite volume,
are large. This is
the case when the local dynamics becomes
spatially correlated, as already discussed in several
papers~\cite{FP,silvio2,glotzer,lacevic,toni,mayer,berthier}.
\begin{figure}
\psfig{file=out6100.ps,width=5.3cm}
\psfig{file=out3580.ps,width=5.3cm}
\psfig{file=out3000.ps,width=5.3cm}
\caption{\label{snap} Snapshots of dynamic heterogeneity at $T=6100$,
3580 and 3000~K (from top to bottom). The snapshot presents
particles which, in a particular run at a particular temperature
have been slower than the average, and have therefore
large, positive values of $\delta f_i({\bf q},t \approx \tau_\alpha)$
defined in Eq.~(\ref{snapeq}). Light colour is used for Si, dark for O.
Slow particles tend to cluster in space
on increasingly larger lengthscales when $T$ decreases.}
\end{figure}
What $\chi_4(t)$ captures is information on the spatial
structure of the spontaneous fluctuations of the dynamics around their
average. We define
$f_i({\bf q},t) = \cos ({\bf q}
\cdot [{\bf r}_i(t) - {\bf r}_i(0)] )$, the contribution
of particle $i$ to the instantaneous value of $F_s(q,t)$,
and
\begin{equation}
\delta f_i({\bf q},t) = f_i({\bf q},t) - F_s(q,t),
\label{snapeq}
\end{equation}
its fluctuating
part. Then $\chi_4(t)$ can be rewritten in the suggestive form,
\begin{equation}
\chi_4(t) = \rho \int d{\bf r} \left\langle
\sum_{i,j} \delta f_i({\bf q},t)
\delta f_j({\bf q},t) \delta ( {\bf r} - [{\bf r}_i(0) - {\bf r}_j(0)] )
\right\rangle ,
\label{volume}
\end{equation}
where subtleties related to the exchange between thermodynamic limit
and thermal average are discussed below.
Therefore $\chi_4(t)$ is the volume integral of the spatial
correlator between local fluctuations of the dynamical behaviour
of the particles. It gets larger when the spatial range of these
correlations increases.
To get a feeling of how
these fluctuations look like in real space,
we present snapshots at different temperatures
in Fig.~\ref{snap}. To build these snapshots
we show, for a given run at a given temperature, those particles
for which the fluctuating quantity $\delta f_i({\bf q},t \approx
\tau_\alpha)$, is
positive and larger than a given threshold, which we choose
close to 1/2 for graphical convenience (this leads to about
1/3 of the particles being shown, and clearer snapshots).
The shown particles are therefore slower than the average
for this particular run. The evolution of the snapshots between 6100~K and
3000~K clearly reveals the tendency for slow particles to cluster in space,
revealing the growth of the lengthscale of kinetic heterogeneities.
We should note, however, that the clusters shown here are not
macroscopic objects even at the lowest temperature studied.
Moreover, similar snapshots in Lennard-Jones
systems reveal more clearly the tendency we seek to
illustrate~\cite{berthier}.
We interpret this as a further qualitative indication that dynamic
heterogeneity is less pronounced in this Arrheniusly relaxing
material than in more fragile Lennard-Jones systems.
We turn to more quantitative measures of dynamic heterogeneity and
show the time dependence of the dynamic susceptibility
$\chi_4(t)$ obtained from our MC simulations for various temperatures
in Fig.~\ref{chi4}. Similar data are obtained for Si and O,
and we only present the former.
As predicted theoretically
in Ref.~\cite{toni} we find that $\chi_4(t)$ presents
a complex time evolution, closely related to
the time evolution of the self-intermediate scattering function.
Overall, $\chi_4(t)$ is small at both small and large
times when dynamic fluctuations are small. There is therefore
a clear maximum observed for times comparable to
$\tau_\alpha$, where fluctuations are most pronounced. The position
of the maximum then shifts to larger times when
temperature is decreased, tracking the alpha-relaxation timescale.
The most important physical information revealed by these curves is the
fact that the amplitude of the peak grows when the temperature
decreases. This is direct evidence, recall Eq.~(\ref{volume}),
that spatial correlations grow
when the glass transition is approached.
\begin{figure}
\begin{center}
\psfig{file=chi4.ps,width=8.5cm}
\psfig{file=chi43.ps,width=8.5cm}
\end{center}
\caption{\label{chi4} Top: Four-point susceptibility, Eq.~(\ref{chi4lj}),
for the same temperatures as in Fig.~\ref{fs}, decreasing from
left to right. The low temperature data at $T=2750$~K
are fitted with two power laws shown as
dashed lines with exponents $0.3$ and $0.92$ at short and large times,
respectively. The envelope of the maxima is fit with an exponent 0.285.
Bottom: temperature evolution of the maxima in various
dynamic susceptibilities.}
\end{figure}
The two-step decay of the self-intermediate scattering function
translates into a two-power law regime for $\chi_4(t)$
approaching its maximum. We have fitted these power laws,
$\chi_4(t) \sim t^a$, followed by $\chi_4(t) \sim t^b$ with the
exponents $a =0.3$ and $b=0.92$ in Fig.~\ref{chi4}. We
have intentionally used the notation $a$ and $b$ for these
exponents which are predicted, within mode-coupling theory,
to be equal to the standard exponents also describing
the time dependence of intermediate scattering functions~\cite{toni,II}.
Our findings are in reasonable agreement with
values for $a$ and $b$ discussed above, although the $b$-value
is about 50\% too large. Moreover, a two-power law regime
is only observed for $T<T_c$, where MCT does not apply anymore.
We note that the $b$-value is predicted to be $b=2$
from the perspective of modelling strong glass-formers using
kinetically constrained models with Arrhenius behaviour~\cite{steve2};
this prediction is clearly incorrect for BKS silica~\cite{toni,II}.
We then focus on the amplitude of the dynamic susceptibility
at its maximum and follow its temperature evolution in Fig.~\ref{chi4}.
As suggested by the snapshots shown in Fig.~\ref{snap}, we confirm
that spatial correlations increase when $T$ decreases, as $\chi_4$ gets
larger at low temperatures. The temperature evolution
of the peak was discussed in Ref.~\cite{II}. Both MCT
and kinetically constrained models
strongly overestimate the temperature evolution
of $\chi_4$ at its peak value, as emphasized already in Ref.~\cite{II}.
Finally, we note that the typical values
observed for the peak of $\chi_4$ at low temperatures are
significantly smaller than those observed for more fragile
Lennard-Jones systems, suggesting once more that dynamic
heterogeneity is less pronounced in strong glass-forming
materials.
This comparison is also useful to discuss the possibility
of finite size effects on the present $\chi_4$ data. If computed
in a simulation box which is too small, the dynamic
susceptibility takes values that are too small~\cite{fssprl}.
Our data indicate that no saturation of the maximum value
of $\chi_4(t)$ is reached, and the values we find
of smaller than the ones found in a Lennard-Jones system with
a comparable system size and for which a detailed
search for possible finite size effects was performed~\cite{I}.
We believe therefore that our results are not affected
by finite size effects.
We then compare these results to the ones obtained using
Newtonian dynamics in the same system.
In that case, care must be taken of the order
at which the thermodynamic limit and the thermal average are taken in
Eq.~(\ref{volume}). Indeed when ND is used, the dynamics strictly conserves
the energy during the simulation and thermal averages are then performed
in the microcanonical ensemble, and $\chi_4^E$ is measured.
To measure $\chi_4^T$ in the canonical ensemble for ND, an additional
contribution must be added, which takes into account
the amount of spontaneous fluctuations which are due to
energy fluctuations~\cite{science},
\begin{equation}
\chi_4^{T}(t) - \chi_4^{E}(t) = \frac{T^2}{c_V} \left(
\frac{\partial F_s({ q},t)}{\partial T} \right)^2 \equiv \frac{T^2}{c_V}
\chi_T(t),
\label{chiT}
\end{equation}
where $c_V$ is the constant volume specific heat expressed in
$k_B$ units.
The results for $\chi_4^T$ and $\chi_4^E$ obtained from ND,
and the difference term in Eq.~(\ref{chiT}),
are all presented in Fig.~\ref{chi4}.
We find that the MC results for $\chi_4$ lie closer to the
microcanonical results obtained from ND, while the
canonical fluctuations are significantly larger, due to the
large contribution of the right hand side in Eq.~(\ref{chiT}).
This is at first sight contrary to the intuition
that MC simulations are thermostatted and should be
a fair representation of canonical averages in ND.
But this is not what happens. As discussed
in Refs.~\cite{I,II}, a major role is played by
conservation laws for energy and density when
dynamic fluctuations are measured.
In the case of energy conservation the mechanism can
be physically understood as follows. For a rearrangement
to take place in the liquid,
the system has to locally cross an energy barrier. If dynamics conserves the
energy, particles involved in the rearrangement must borrow
energy to the neighboring particles. This `cooperativity'
might be unnecessary if energy can be locally supplied
to the particles by an external heat bath, as in MC simulations.
Conservation laws, therefore, might
induce dynamic correlations between particles and dynamic fluctuations
can be different when changing from Newtonian, energy conserving
dynamics to a stochastic, thermostatted dynamics.
With hindsight, this is
not such a surprising result. The specific heat, after all,
also behaves differently in different statistical ensembles.
The ensemble dependence and dependence upon the microscopic dynamics
of dynamic susceptibilities in supercooled liquids
are the main subjects of two recent papers~\cite{I,II}.
Our results for silica quantitatively agree
with the theoretical analysis they contain, and with
the corresponding numerical
results obtained in Lennard-Jones systems.
There is an experimentally extremely relevant consequence of
these findings~\cite{science,cecile}.
As shown in
Fig.~\ref{chi4}, the difference
between the microcanonical and canonical values of the dynamic
fluctuations in ND represents in fact the major
contribution to $\chi_4^{T}$, meaning
that the term $\chi_4^{E}$ can be safely neglected in Eq.~(\ref{chiT}).
Since the right hand side of (\ref{chiT}) is
more easily accessible in an experiment than $\chi_4^T$ itself,
Eq.~(\ref{chiT}) opens the possibility of an experimental
estimate of the four-point susceptibility. This finding,
and its experimental application to supercooled glycerol and
hard sphere colloids,
constitute the central result of Ref.~\cite{science}, while more data
are presented in Ref.~\cite{cecile}.
\section{Conclusion}
\label{conclusion}
We have implemented a standard Monte Carlo algorithm to study the
slow dynamics of the well-known BKS model for silica in the temperature
range from 6100~K to 2750~K. Our results clearly establish
that Monte Carlo simulations can be used to study the dynamics
of silica because quantitative agreement is found with results
from Newtonian dynamics for the same potential, apart at very short times
where thermal vibrations are efficiently suppressed by the Monte Carlo
algorithm. The agreement between the two dynamics
is by no means trivial and constitutes an important result of the present
study. This suggests that Monte Carlo simulations
constitute a useful and efficient tool to study also the nonequilibrium aging
dynamics of glass-forming liquids, a line of research initiated in
Ref.~\cite{berthier2}.
Since dynamical correlations are not affected by short-time vibrations,
we have been able to revisit the mode-coupling analysis
initially performed in Ref.~\cite{hk}. We find that mode-coupling theory
accounts for the qualitative features of the data quite well, but
the detailed, quantitative predictions made by the theory were shown
to fail: correlation functions close to the plateau
do not follow the behaviour predicted for the MCT $\beta$-correlator,
time-temperature superposition only holds at very large times but fails
at smaller times because the plateau in correlation function is strongly
temperature dependent, a dependence which does not follow the
singular behaviour predicted by MCT. Moreover, the temperature
regime where the theory can supposedly be applied is found to be
at most 1 decade when only
the temperature evolution of relaxation timescales
is considered. Furthermore, we have argued that the motivation
to analyze silica data in terms of MCT, a ``fragile to strong'' crossover,
can in fact be more simply accounted for in terms
of a crossover between two distinct Arrhenius regimes occurring close to
$T \approx 4000$~K. Overall these results suggest
a negative answer to the question: is there
any convincing evidence of an avoided
mode-coupling singularity in silica?
We have finally analyzed dynamic heterogeneity in silica. We find that
the dynamics is indeed spatially heterogeneous, and spatial
correlation of the local dynamical behaviour was shown to increase
when temperature decreases. We also found that all indicators
of dynamically heterogeneous dynamics such as decoupling and four-point
dynamic susceptibilities, suggest that the effects are less pronounced
in silica than in more fragile glass-forming materials, but do not
seem qualitatively different.
The most natural interpretation is that
strong and fragile materials in fact belong to
the same class of materials, where the effects of dynamic heterogeneity
could become less pronounced, but definitely non-zero,
for materials with lower fragility.
This suggests that it could be incorrect to
assume that strong materials belong to a different universality class
from fragile ones, as studies of kinetically constrained models
with different fragilities would suggest~\cite{steve2,KCM},
and they should rather seat at the end
of the spectrum of fragile systems. It seems however similarly incorrect
to consider that strong materials are ``trivial'' because an Arrhenius
behaviour can be explained from simple thermal activation over a fixed
energy barrier corresponding to a local, non-cooperative event. Our results
show that this is not a correct representation of the physics of strong
glass-formers either. Convincingly incorporating
fragility into current theories of the glass transition while
simultaneously giving it a microscopic interpretation
remains therefore an important challenge.
\begin{acknowledgments}
This work emerged from collaboration with G. Biroli, J.-P. Bouchaud,
W. Kob, K. Miyazaki, and D. Reichman~\cite{I,II}.
D. Reichman suggested to revisit the MCT analysis of BKS silica,
W. Kob helped analyzing and interpreting the results, and A. Heuer
made useful comments on the preprint.
This work has been supported in part by Joint Theory Institute at
Argonne National Laboratory and the University of Chicago.
\end{acknowledgments}
|
2,869,038,154,846 | arxiv | \section{Introduction}\label{SecIntro}
We consider the isoperimetric problem for hypersurfaces, $\Sigma$, embedded in a slab, $M=[0,d]\times\mathbb{R}^n\subset\mathbb{R}^{n+1}$, such that $\partial\Sigma\subset\partial M$. The free boundary critical points of the area function under a volume constraint are the constant mean curvature (CMC) hypersurfaces that meet $\partial M$ orthogonally (we will refer to this as the free boundary condition). It is further known that these hypersurfaces must be rotationally symmetric, \cite{Athanassenas87,Pedrosa99}, thus they are the Delaunay hypersurfaces, consisting of catenoids, spheres, cylinders, unduloids, and nodoids \cite{Delaunay41,Hsiang81}. We will reject the catenoids and nodoids as the former cannot satisfy the boundary conditions, while the later can only satisfy them after leaving the slab.
To investigate whether these hypersurfaces minimise the area (locally under a volume constraint), we introduce the concept of stability. A constant mean curvature hypersurface $\Sigma\subset\mathbb{R}^{n+1}$ is stable if the functional
\begin{equation}\label{StabilityGen}
J_{\Sigma}(u)=\frac{\int \|\nabla u\|_{\Sigma}^2-|A|^2u^2\,d\mu_{\Sigma}}{\int u^2\,d\mu_{\Sigma}}
\end{equation}
is non-negative for all functions $u$ on $\Sigma$ such that $\int u\,d\mu_{\Sigma}=0$, where $|A|$ is the magnitude of the second fundamental form of $\Sigma$. When the functional is non-negative for all functions $u$, we call the hypersurface strictly stable. This functional is the second variation of area under the volume constraint and as such any solution to the isoperimetric problem is stable.
Spheres are known to be stable hypersurfaces \cite{Barbosa84} and, in fact, the only stable compact orientable immersion, while half spheres in the slab are also stable by the same reasoning \cite{Athanassenas87}. A cylinder is stable if and only if its radius is greater than or equal to $\frac{d\sqrt{n-1}}{\pi}$, this was proved in the $n=2$ case in \cite{Athanassenas87,Vogel87} and in general dimensions in \cite{Hartley13,Souam18}, with the former paper doing so in with respect to the stronger condition of dynamic stability of the volume preserving mean curvature flow. The stability of unduloids is a more complicated topic. Pedrosa and Ritor\'e, \cite{Pedrosa99}, proved that if $2\leq n\leq 7$ then all unduloids are unstable (they also proved that nodoids are unstable), and that the near spherical unduloids are unstable. However, they also showed in that paper that if $n\geq9$ there exist stable unduloids. Their proof of existence was based on a comparison of the area of a degenerate half sphere (where its apex touches a boundary of the slab) with the area of the cylinder of the same volume. If $n\geq9$ then the area of the half sphere is smaller, meaning the cylinder does not solve the isoperimetric problem at this volume. Since the half sphere is degenerate it does not either, hence an unduloid must solve the isoperimetric problem at this volume and hence be stable. More recent results \cite{Hartley16,Li18} considered the near cylindrical unduloids and showed that the near cylindrical unduloids of half period are stable if and only if $n\geq11$, with the former paper again doing so in the context of dynamic stability. Any unduloid with over a half a period is unstable, as it is no longer a graph over a boundary component of $S$.
These results have two major gaps. Do their exist stable unduloids of dimension eight? And what characteristic determines the stability of unduloids? In this paper we provide a positive answer to the first question, while giving an answer to the second subject to some conditions.
The paper is organised as follows. In Section \ref{SecFamily} we introduce a family of unduloids and with this are able to state our main theorem. In Section \ref{SecStable} we consider the functional (\ref{StabilityGen}) and recast the concept of stability in terms of an operator. In Section \ref{SecNull} we examine the null space of this operator and in Section \ref{SecVary} we consider how a zero eigenvalue will vary along the family of unduloids, this allows us to prove the main theorem. Sections \ref{SecFamilyApp}, \ref{SecStableApp}, and \ref{PsiSec} contain the details of some technical calculations.
\textbf{Acknowledgement}: The author would like to thank Professor Frank Morgan for his question regarding the isoperimetric problem, which prompted the inclusion of Remark \ref{MorganRem}.
\section{A Family of Rotationally Symmetric CMC Hypersurfaces}\label{SecFamily}
In this section we give a representation of unduloids in terms of a profile curve and use the characteristics of this family to state the main stability theorem. We start by defining a couple of intermediate functions that will appear in our family. The first is a function of the parameter that is used to set the period of the hypersurface:
\[
Q(t):=\left\{\begin{array}{cc}\frac{1-t^{n-1}}{1-t^n}, & t\in\mathbb{R}_+\backslash\{1\},\\ \frac{n-1}n, & t=1,\end{array}\right.
\]
note that $Q$ is continuously differentiable and strictly decreasing for $t>0$ with
\[
Q'(t)=\left\{\begin{array}{cc}-\frac{t^{n-2}\left(t^n-nt+n-1\right)}{\left(1-t^n\right)^2}, & t\in\mathbb{R}_+\backslash\{1\},\\ -\frac{n-1}{2n}, & t=1,\end{array}\right.
\]
and $Q(t^{-1})=tQ(t)$, so $tQ(t)$ is strictly increasing. The second function we define is the gradient of the profile curve:
\[
R(x;t):=\left\{\begin{array}{cc}\frac{1}{|1-t|}\sqrt{\left(\frac{(1-(1-t)x)^{n-1}}{1-Q(t)+Q(t)(1-(1-t)x)^n}\right)^2-1}, & t\in\mathbb{R}_+\backslash\{1\},\\ \sqrt{(n-1)(1-x)x}, & t=1,\end{array}\right.
\]
where $0\leq x\leq1$. We again note that this function is continuous for $t>0$ and satisfies $R(0;t)=R(1;t)=0$ for all $t>0$, along with $R(1-x;t^{-1})=tR(x;t)$. From these functions we further define
\begin{equation}\label{Defzeta}
\zeta(x;t):=\left\{\begin{array}{cc}\int_x^1R(\tilde{x};t)^{-1}\,d\tilde{x}, & 0\leq x<1,\\ 0, & x=1,\end{array}\right.
\end{equation}
\begin{equation}\label{DefP}
P(t):=\zeta(0;t)=\int_0^1R(x;t)^{-1}\,dx,
\end{equation}
and let $\zeta^{-1}(y;t)$ be the inverse of $\zeta$ with respect to its first variable, that is $\zeta^{-1}(\zeta(x;t);t)=x$ for all $x$ and $t$. Also note that $P(t^{-1})=t^{-1}P(t)$. Throughout the paper we will use subscripts to denote differentiation of functions with respect to that variable.
We can now define the family of CMC hypersurfaces, we save the calculations for Section \ref{SecFamilyApp}.
\begin{proposition}\label{FamCMC}
The two parameter family of functions
\[
u(z;r,t):=\frac{Q(r)d}{P(t)Q(t)}\left(1-(1-r)\zeta^{-1}\left(\frac{Q(t)P(t)z}{Q(r)d};r\right)\right),
\]
for $r,t>0$ and $z\in\left[0,\frac{Q(r)P(r)d}{Q(t)P(t)}\right]$ define a two parameter family of rotationally symmetric CMC hypersurfaces with mean curvature
\[
\eta(t):=\frac{nQ(t)P(t)}{d},
\]
and such that
\[
u_z(0;r,t)=u_z\left(\frac{Q(r)P(r)d}{Q(t)P(t)};r,t\right)=0.
\]
\end{proposition}
We will consider the finite unduloids of length $d$ and fixed period $2d$, these are given by the profile curve
\[
v(z;t):=u(z;t,t)=\frac{d}{P(t)}\left(1-(1-t)\zeta^{-1}\left(\frac{P(t)z}{d};t\right)\right),
\]
have a mean curvature $\eta(t)$, and have an $(n+1)$-enclosed volume $V(t):=Vol(v(\cdot;t))$.
\begin{remark}
Note that $\eta(t^{-1})=\eta(t)$. It was this function that was used in \cite{Pedrosa99} to prove their instability results, although they used a different parameterisation and considered the family of with the same mean curvature (not same period). In the notation of this paper, they proved that if $\eta'(t)>0$ for some $t\in(0,1)$ then $v(\cdot;t)$ defines an unstable unduloid.
\end{remark}
The main theorem of this paper is:
\begin{theorem}\label{stability}
Let $n\geq 2$ be such that if $t\in(0,1)$ is such that $V'(t)=0$ then $V''(t)\neq0$ and $\eta'(t)<0$. Let $t_0\in(0,1]$, if $V'(t_0)<0$ the CMC unduloid defined by $v(\cdot;t_0)$ is unstable. While when $V'(t_0)>0$ the CMC unduloid defined by $v(\cdot;t_0)$ is stable.
\end{theorem}
\begin{remark}
This also covers the $t_0>1$ case by symmetry of the hypersurfaces under the transformation $t\to t^{-1}$. That is, if $t_0>1$ and $V'(t_0)<0$, then the CMC unduloid defined by $v(z;t_0)$ is stable, while if $V'(t_0)>0$ it is unstable.
\end{remark}
\begin{remark}
It is conjectured that the conditions $V''(t)\neq0$ and $\eta'(t)<0$ whenever $t\in(0,1)$ is such that $V'(t)=0$, are satisfied in all dimensions $n\geq2$, however the relevant integral bounds have not been obtained. In fact, we have the following conjecture which is stronger than the second condition.
\end{remark}
\begin{conjecture}
The function $\xi(t):=\eta(t)^{n+1}V(t)$ is strictly decreasing on $(0,1)$ for $n\geq2$.
\end{conjecture}
\begin{corollary}\label{Dim8}
There are stable CMC unduloids of dimension eight.
\end{corollary}
\begin{proof}
In Figure \ref{Fig1} the functions $\frac{V'(t)}{1-t}$ (blue), $V''(t)$ (orange), and $\xi'(t)$ (green), normalised so that the maximum of their absolute values over the domain $(0,1)$ is $1$, for $n=8$ are plotted. $V'(t)$ has been divided by $1-t$ to remove the zero at $t=1$, which is not relevant to the discussion.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{8_Dim_Unduloid_Characteristics_Plot.pdf}
\caption{Plots of dimension eight unduloid characteristics: $\frac{V'(t)}{1-t}$ (blue), $V''(t)$ (orange), and $\xi'(t)$ (green).}\label{Fig1}
\end{figure}
From this it is clear that at a point $t_0\in(0,1)$ where $V'(t_0)=0$ we have $\xi'(t_0)<0$ and $V''(t_0)\neq0$. Thus the conditions of Theorem \ref{stability} are met, since $\eta'(t_0)=\frac{\xi'(t_0)}{9\eta(t_0)^9V(t_0)}<0$. Further we see that there exists an open interval $I\subset(0,1)$ such that $V'(t)>0$ on $I$ and hence the unduloids $v(\cdot;t)$ for $t\in I$ are stable.
\end{proof}
\begin{remark}
We note that through a change of variables we can write the function $V(t)$ as:
\begin{align*}
V(t)=&\omega_n\int_0^dv(z;t)^n\,dz\\
=&\frac{\omega_n d^n}{P(t)^n}\int_0^d\left(1-(1-t)\zeta^{-1}\left(\frac{P(t)z}{d};t\right)\right)^n\,dz\\
=&\frac{\omega_n d^n}{P(t)^n}\int_1^0\left(1-(1-t)x\right)^n\frac{-d}{P(t)}R(x;t)^{-1}\,dx\\
=&\frac{\omega_n d^{n+1}}{P(t)^{n+1}}\int_0^1\left(1-(1-t)x\right)^nR(x;t)^{-1}\,dx.
\end{align*}
Also, note that $V(t^{-1})=V(t)$ and so $V'(1)=0$.
\end{remark}
\begin{remark}\label{MorganRem}
Despite these unduloids being stable, computations show that they do not in fact solve the isoperimetric problem. For a particular $t\in(0,1)$ the area of a cylinder with the same volume as $v(;,t)$ is given by $SA_c(t)=n\omega_n^{\frac1n}V(t)^{\frac{n-1}n}d^{\frac1n}$ and the area of the half sphere of the same volume is $SA_s(t)=2^{\frac{-1}{n+1}}(n+1)\omega_{n+1}^{\frac{1}{n+1}}V(t)^{\frac{n}{n+1}}$ (provided $\frac{2V(t)}{\omega_{n+1}}\leq d^{n+1}$). By subtracting the area of the unduloid:
\begin{align*}
SA_u(t)=&n\omega_n\int_0^dv(z;t)^{n-1}\sqrt{1+v_z(z;t)^2}\,dz\\
=&\frac{n\omega_n d^{n-1}}{P(t)^{n-1}}\int_0^d\left(1-(1-t)\zeta^{-1}\left(\frac{P(t)z}{d};t\right)\right)^{n-1}\sqrt{1+(1-t)^2R\left(\zeta^{-1}\left(\frac{P(t)z}{d};t\right);t\right)^2}\,dz\\
=&\frac{n\omega_nd^n}{P(t)^n}\int_0^1(1-(1-t)x)^{n-1}\sqrt{R(x;t)^{-2}+(1-t)^2}\,dx,
\end{align*}
from these areas we can plot the difference for dimension eight, see Figure \ref{Fig2}.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{AreaDifferences.pdf}
\caption{Differences in area for a cylinder (blue) and half sphere (orange) compared to unduloids with the same volume in dimension eight with $d=1$.}\label{Fig2}
\end{figure}
The condition for a valid half sphere, $\frac{2V(t)}{\omega_{n+1}}\leq d^{n+1}$, is always satisfied. This shows that any unduloid in dimension eight has a larger area than the corresponding cylinder and/or half sphere of the same volume.
\end{remark}
\section{The Stability Operator for CMC Hypersurfaces}\label{SecStable}
We will now recast the problem of stability in terms of an operator applicable to our situation. We only consider the case where $\Sigma$ is rotationally symmetric, around an axis in the $z$ direction, with free boundary and defined by a profile curve $v:[0,d]\to\mathbb{R}_+$. With this assumption the functional (\ref{StabilityGen}) becomes
\[
J_v(u)=\frac{\int_{\SS^{n-1}}\int_0^d \left(\frac{1}{1+v_z^2}u_z^2+\frac{1}{v^2}\|\tilde{\nabla} u\|_{\SS^{n-1}}^2 -|A|^2u^2\right)v^{n-1}\sqrt{1+v_z^2}\,dz\,d\mu_{\SS^{n-1}}}{\int_{\SS^{n-1}}\int_0^d u^2v^{n-1}\sqrt{1+v_z^2}\,dz\,d\mu_{\SS^{n-1}}},
\]
for $u:[0,d]\times\SS^{n-1}\to\mathbb{R}$ that satisfies $\int_{\SS^{n-1}}\int_0^d uv^{n-1}\sqrt{1+v_z^2}\,dz\,d\mu_{\SS^{n-1}}=0$ and the free boundary condition $u_z|_{z=0}=u_z|_{z=d}=0$.
We perform the function substitution $w=u\sqrt{1+v_z^2}$ and create the functional $\tilde{J}_v(w)=J_v\left(\frac{w}{\sqrt{1+v_z^2}}\right)$, for functions $w$ satisfying $\int_{\SS^{n-1}}\int_0^d wv^{n-1}\,dz\,d\mu_{\SS^{n-1}}=0$ and the free boundary condition. The functional is given by (see Lemma \ref{Jtil})
\[
\tilde{J}_v(w)=\frac{\int_{\SS^{n-1}}\int_0^d \left(\|\nabla w\|_{\Sigma}^2-\frac{(n-1)w^2}{v^2} \right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}{\int_{\SS^{n-1}}\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}.
\]
Due to the symmetry of $v(\cdot;t)$ we only need to prove this is positive on rotationally symmetric functions, see the reasoning \cite{Vogel87} (for example). The functional acting on rotationally symmetric functions is:
\begin{align*}
\tilde{J}_v(w)=&\frac{\int_0^d\left(\frac{w_z^2}{(1+v_z^2)^{\frac32}}-\frac{(n-1)w^2}{v^2\sqrt{1+v_z^2}}\right)v^{n-1}\,dz}{\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz}\\
=&\frac{\int_0^d\left(-\frac{w_{zz}}{(1+v_z^2)^{\frac32}}+\frac{3v_zv_{zz}w_z}{(1+v_z^2)^{\frac52}}-\frac{(n-1)v_zw_z}{v(1+v_z^2)^{\frac32}}-\frac{(n-1)w}{v^2\sqrt{1+v_z^2}}\right)wv^{n-1}\,dz}{\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz}\\
=&\frac{\int_0^d DH(v)[w]wv^{n-1}\,dz}{\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz},
\end{align*}
where $H$ is the mean curvature functional on rotationally symmetric functions:
\[
H(u)=\frac{-u_{zz}}{(1+u_z^2)^{\frac32}}+\frac{n-1}{u\sqrt{1+u_z^2}}.
\]
Since we are only considering functions, $w$, satisfying $\int_0^d wv^{n-1}\,dz=0$, this can also be written as
\[
\tilde{J}_v(w)=\frac{\int_0^d \left(DH(v)[w]-\frac{\int_0^d DH(v)[w]v^{n-1}\,dz}{\int_0^d v^{n-1}\,dz}\right)wv^{n-1}\,dz}{\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz},
\]
so $\tilde{J}_v$ is positive on functions satisfying $\int_0^d wv^{n-1}\,dz=0$ and the free boundary condition if and only if
\[
A(v)[w]:=DH(v)[w]-\frac{\int_0^d DH(v)[w]v^{n-1}\,dz}{\int_0^d v^{n-1}\,dz}
\]
is positive definite on the same space of functions. However, since $A(v)$ maps back into functions with weighted mean zero, this is equivalent to proving all its eigenvalues are greater than zero.
\begin{proposition}\label{EquivStable}
A rotationally symmetric CMC hypersurface defined by the function $v$ is stable if the operator $A(v):X\to X$ only has positive eigenvalues, where $X:=\{w:[0,d]\to\mathbb{R}:\int_0^dwv^{n-1}\,dz=0,\ w_z|_{z=0}=w_z|_{z=d}=0\}$.
\end{proposition}
\section{Null Space of the Stability Operator}\label{SecNull}
In order the determine the stability of unduloids in our family, it is necessary to determine the critical cases, that is at which $t$ does the operator $\tilde{A}(t):=A(v(\cdot;t))$ have a zero eigenvalue when acting on the space of functions:
\[
X_t:=\{w:[0,d]\to\mathbb{R}:\int_0^dw(z)v(z;t)^{n-1}\,dz=0,\ w_z|_{z=0}=w_z|_{z=d}=0\}.
\]
A zero eigenvalue of $\tilde{A}(t)$ means that the linearised mean curvature operator is constant, so we first consider this situation without restricting to functions in $X_t$.
\begin{theorem}\label{ConsLinMC}
Let $w$ satisfy $DH(v(\cdot;t_0))[w]=C$ for some $t_0>0$ and $C\in\mathbb{R}$. We have three cases:
\begin{itemize}
\item If $\eta'(t_0)\neq0$, there exists $\alpha,\beta\in\mathbb{R}$ such that
\[
w(z)=\alpha u_r(z;t_0,t_0)+\beta u_z(z;t_0,t_0)+\frac{C}{\eta'(t_0)}u_t(z;t_0,t_0).
\]
\item If $\eta'(t_0)=0$ and $t_0\neq1$, let $k\geq2$ be the first $k$ such that $\eta^{(k)}(t_0)\neq0$, then there exists $\alpha,\beta\in\mathbb{R}$ such that
\[
w(z)=\alpha u_r(z;t_0,t_0)+\beta u_z(z;t_0,t_0)+\frac{C}{\eta^{(k)}(t_0)}u_{t^{k}}(z;t_0,t_0).
\]
\item If $t_0=1$, there exists $\alpha,\beta\in\mathbb{R}$ such that
\[
w(z)=\alpha \cos\left(\frac{\pi z}{d}\right)+\beta \sin\left(\frac{\pi z}{d}\right) -\frac{Cd^2}{\pi^2}.
\]
\end{itemize}
\end{theorem}
\begin{proof}
We start by considering the $\eta'(t_0)\neq0$ case. By differentiating the equation $H(u(z;r,t))=\eta(t)$ with respect to each of the variables we obtain
\begin{equation}\label{HuDerivs}
\begin{array}{cc}
DH(u(z;r,t))[u_z(z;r,t)]=0, & DH(u(z;r,t))[u_r(z;r,t)]=0,\\
\multicolumn{2}{c}{DH(u(z;r,t))[u_t(z;r,t)]=\eta'(t),}
\end{array}
\end{equation}
so at $r=t=t_0$ we have
\begin{equation}\label{HuDerivs0}
\begin{array}{cc}
DH(v(z;t_0))[u_z(z;t_0,t_0)]=0, & DH(v(z;t_0))[u_r(z;t_0,t_0)]=0,\\
\multicolumn{2}{c}{DH(v(z;t_0))[u_t(z;t_0,t_0)]=\eta'(t_0),}
\end{array}
\end{equation}
so the statement follows from standard linear second order DE theory provided $u_z(z;t_0,t_0)$ and $u_r(z;t_0,t_0)$ are linearly independent. To see this is the case we note that $u_z(0;t_0,t_0)=0$ and $u_z(z;t_0,t_0)\equiv0$ if and only if $t_0=1$, which is excluded from this case. Therefore the functions are linearly independent if $u_r(0;t_0,t_0)\neq0$. We have $u(z;r,t)=\frac{nQ(r)}{\eta(t)}\left(1-(1-r)\zeta^{-1}\left(\frac{\eta(t)z}{nQ(r)};r\right)\right)$, so
\begin{align*}
u_r(z;r,t)=&\frac{nQ'(r)}{\eta(t)}\left(1-(1-r)\zeta^{-1}\left(\frac{\eta(t)z}{nQ(r)};r\right)\right)+\frac{nQ(r)}{\eta(t)}\zeta^{-1}\left(\frac{\eta(t)z}{nQ(r)};r\right)\\
&+\frac{(1-r)Q'(r)z}{Q(r)}\zeta^{-1}_y\left(\frac{\eta(t)z}{nQ(r)};r\right)-\frac{n(1-r)Q(r)}{\eta(t)}\zeta^{-1}_r\left(\frac{\eta(t)z}{nQ(r)};r\right),
\end{align*}
and hence by using $\zeta^{-1}(0;r)=1$ (and therefore $\zeta^{-1}(0;r)=0$)
\[
u_r(0;r,t)=\frac{n(rQ'(r)+Q(r))}{\eta(t)}=\frac{n(rQ(r))'}{\eta(t)}>0.
\]
Next we consider the case where $\eta'(t_0)=0$ and $t_0\neq1$. We have
\begin{align*}
u_t(z;r,t)=&-\frac{nQ(r)\eta'(t)}{\eta(t)^2}\left(1-(1-r)\zeta^{-1}\left(\frac{\eta(t)z}{nQ(r)};r\right)\right)-\frac{(1-r)\eta'(t)z}{\eta(t)}\zeta^{-1}_y\left(\frac{\eta(t)z}{nQ(r)};r\right)\\
=&-\frac{\eta'(t)}{\eta(t)}\left(\frac{nQ(r)}{\eta(t)}\left(1-(1-r)\zeta^{-1}\left(\frac{\eta(t)z}{nQ(r)};r\right)\right)+(1-r)z\zeta^{-1}_y\left(\frac{\eta(t)z}{nQ(r)};r\right)\right),
\end{align*}
and hence $u_t(z;t_0,t_0)\equiv0$, in fact $u_{t^i}(z;t_0,t_0)\equiv0$ for all $i=1,\ldots,k-1$. By now differentiating the final equation of (\ref{HuDerivs}) again with respect to $t$ we obtain
\[
D^2H(u(z;r,t))[u_t(z;r,t),u_t(z;r,t)]+DH(u(z;r,t))[u_{tt}(z;r,t)]=\eta''(t)
\]
Therefore $DH(v(z;t_0))[u_{tt}(z;t_0,t_0)]=\eta''(t_0)$ and it is clear that
\[
DH(v(z;t_0))[u_{t^i}(z;t_0,t_0)]=\eta^{(i)}(t_0),
\]
for all $i=1,\ldots,k$. In particular, $DH(v(z;t_0))[u_{t^k}(z;t_0,t_0)]=\eta^{(k)}(t_0)\neq0$. The proof of linear independence of $u_z(z;t_0,t_0)$ and $u_r(z;t_0,t_0)$ is the same as in the previous case, so we obtain the conclusion.
Lastly, we consider the case when $t_0=1$. In this case we use that $v(z;1)=\frac{d}{P(1)}$ to write out the differential equation for $w$:
\[
DH\left(\frac{d}{P(1)}\right)[w]=-w''(z)-\frac{(n-1)P(1)^2}{d^2}w(z)=C,
\]
which, using that $P(1)=\int_0^1\frac{1}{\sqrt{(n-1)(1-x)x}}\,dx=\frac{\pi}{\sqrt{n-1}}$, is easily solved to give the conclusion.
\end{proof}
\begin{remark}
We note that in the $t_0=1$ case, $\zeta(x;1)=\frac{2}{\sqrt{n-1}}\arcsin(\sqrt{1-x})$ and hence $\zeta^{-1}(y;1)=\cos^2\left(\frac{\sqrt{n-1}y}{2}\right)=\frac{1}{2}+\frac{1}{2}\cos(\sqrt{n-1}y)$. Therefore
\[
u_r(z;1,1)=\frac{nQ'(1)}{\eta(1)}+\frac{nQ(1)}{\eta(1)}\zeta^{-1}\left(\frac{\eta(1)z}{nQ(1)};1\right)=\frac{d\sqrt{n-1}}{2\pi}\cos\left(\frac{\pi z}{d}\right).
\]
Also $u_{tt}(z;1,1)\equiv-\frac{d^2\eta''(1)}{\pi^2}=-\frac{(n^2-10n+10)\sqrt{n-1}d}{48\pi}\neq0$, so the second case holds with $k=2$ and $u_z(z;t_0,t_0)$ replaced with $\sin\left(\frac{\pi z}{d}\right)$.
\end{remark}
\begin{corollary}\label{NullA}
If $V'(t_0)\neq0$ then $\textrm{Null}(\tilde{A}(t_0))$ is empty, otherwise it is one dimensional and $\textrm{Null}(\tilde{A}(t_0))=\textrm{span}\{v_t(z;t_0)\}$.
\end{corollary}
\begin{proof}
We consider the three cases found in Theorem \ref{ConsLinMC} separately and determine when there is a non-trivial solution in $X_t$.
When $\eta'(t_0)\neq0$ we have $w(z)=\alpha u_r(z;t_0,t_0)+\beta u_z(z;t_0,t_0)+\frac{C}{\eta'(t_0)}u_t(z;t_0,t_0)$ for some $\alpha,\beta,C\in\mathbb{R}$ and so
\[
w'(z)=\alpha u_{rz}(z;t_0,t_0)+\beta u_{zz}(z;t_0,t_0)+\frac{C}{\eta'(t_0)}u_{tz}(z;t_0,t_0).
\]
Using that $u_z(0;r,t)=0$ for any $r,t>0$ we therefore have $w'(0)=\beta u_{zz}(0;t_0,t_0)$, and $u_{zz}(0;t_0,t_0)=\left(\frac{(n-1)P(t_0)}{dt_0}-\eta(t_0)\right)=\eta(t_0)\left(\frac{n-1}{nt_0Q(t_0)}-1\right)$ using the CMC equation. So $u_{zz}(0;t_0,t_0)=0$ if and only if $t_0=1$ (using that $tQ(t)$ is increasing), which is excluded from this case, and hence we require $\beta=0$. Next we use that since $u_z\left(\frac{\eta(r)d}{\eta(t)};r,t\right)=0$ we have
\begin{equation}\label{Dud1}
u_{zz}\left(\frac{\eta(r)d}{\eta(t)};r,t\right)\frac{\eta'(r)d}{\eta(t)}+u_{zr}\left(\frac{\eta(r)d}{\eta(t)};r,t\right)=0,
\end{equation}
and
\begin{equation}\label{Dud2}
-u_{zz}\left(\frac{\eta(r)d}{\eta(t)};r,t\right)\frac{\eta(r)\eta'(t)d}{\eta(t)^2}+u_{zt}\left(\frac{\eta(r)d}{\eta(t)};r,t\right)=0,
\end{equation}
to conclude that
\[
w'(d)=-\frac{\alpha\eta'(t_0)d}{\eta(t_0)}u_{zz}(d;t_0,t_0)+\frac{Cd}{\eta(t_0)}u_{zz}(d;t_0,t_0)=\frac{d(C-\alpha\eta'(t_0))}{\eta'(t_0)}u_{zz}(d;t_0,t_0).
\]
We use that $u_{zz}(d;t_0,t_0)=\frac{(n-1)P(t_0)}{d}-\eta(t_0)=\eta(t_0)\left(\frac{n-1}{nQ(t_0)}-1\right)$, which again is zero only in the excluded $t_0=1$ case (since $Q(t)$ is decreasing), to conclude that $\alpha=\frac{C}{\eta'(t_0)}$. Hence $w(z)=\frac{C}{\eta'(t_0)}\left(u_{r}(z;t_0,t_0)+u_{t}(z;t_0,t_0)\right)=\frac{C}{\eta'(t_0)}v_{t}(z;t_0)$ with $C\neq0$. We now calculate the weighted integral:
\[
\int_0^d\frac{C}{\eta'(t_0)}v_t(z;t_0)v(z;t_0)^{n-1}\,dz=\frac{C}{n\omega_n\eta'(t_0)}V'(t_0),
\]
and hence we require $V'(t_0)=0$ for the null space to be non-empty, in which case a spanning function is $v_{t}(z;t_0)$.
Next we consider when $\eta'(t_0)=0$ and $t_0\neq1$, so
\[
w(z)=\alpha u_r(z;t_0,t_0)+\beta u_z(z;t_0,t_0)+\frac{C}{\eta^{(k)}(t_0)}u_{t^k}(z;t_0,t_0).
\]
As above we use that $w'(z)=\alpha u_{rz}(z;t_0,t_0)+\beta u_{zz}(z;t_0,t_0)+\frac{C}{\eta^{(k)}(t_0)}u_{t^kz}(z;t_0,t_0)$ to obtain $w'(0)=\beta u_{zz}(0;t_0,t_0)$ and conclude that $\beta=0$. By differentiating (\ref{Dud2}) another $k-1$ times and using that $\eta^{(i)}(t_0)=0$ for all $i=1,\ldots,k-1$ we obtain
\[
-u_{zz}(d;t_0,t_0)\frac{\eta^{(k)}(t_0)d}{\eta(t_0)}+u_{zt^k}(d;t_0,t_0)=0,
\]
and hence by also using (\ref{Dud1}) we obtain
\[
w'(d)=-\frac{\alpha\eta'(t_0)d}{\eta(t_0)}u_{zz}(d;t_0,t_0)+\frac{Cd}{\eta(t_0)}u_{zz}(d;t_0,t_0)=\frac{Cd}{\eta(t_0)}u_{zz}(d;t_0,t_0).
\]
Therefore $C=0$, in which case $w(z)=\alpha u_r(z;t_0,t_0)$ with $\alpha\neq0$. However, since $u_t(z;t_0,t_0)\equiv0$ we can also write this as $w(z)=\alpha v_t(z;t_0)$ and obtain $\int_0^dw(z)v(z;t_0)^{n-1}\,dz=\frac{\alpha}{n\omega_n} V'(t_0)$, so we again require $V'(t_0)=0$ for the null space to be non-empty, in which case the spanning function is $v_{t}(z;t_0)$.
Finally, we consider the case $t_0=1$. In this case
\[
w(z)=\alpha \cos\left(\frac{\pi z}{d}\right)+\beta \sin\left(\frac{\pi z}{d}\right) -\frac{Cd^2}{\pi^2}.
\]
From $w'(z)=\frac{\pi}{d}\left(-\alpha \sin\left(\frac{\pi z}{d}\right)+\beta\cos\left(\frac{\pi z}{d}\right)\right)$, we again see that $w'(0)=0$ if and only if $\beta=0$, but now $w'(d)=0$ gives no further condition. We consider the weighted integral, using that $v(z;1)=\frac{d\sqrt{n-1}}{\pi}$:
\begin{align*}
\int_0^dw(z)v(z;1)^{n-1}\,dz=&\left(\frac{d\sqrt{n-1}}{\pi}\right)^{n-1}\int_0^d\alpha\cos\left(\frac{\pi z}{d}\right)-\frac{Cd^2}{\pi^2}\,dz\\
=&-\frac{Cd^{n+2}(n-1)^{\frac{n-1}2}}{\pi^{n+1}},
\end{align*}
and hence we require $C=0$. Therefore, the null space is one dimensional with spanning function $\cos\left(\frac{\pi z}{d}\right)$, since $V'(1)=0$ and $\cos\left(\frac{\pi z}{d}\right)=\frac{2\pi\alpha}{d\sqrt{n-1}}v_t(z;1)$ we complete the theorem.
\end{proof}
\begin{remark}
From the formula:
\[
\frac{\int A(v)[w]wv^{n-1}\,dz}{\int w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz}=J_v(w)=\frac{\int\left(\frac{w_z^2}{(1+v_z^2)^{\frac32}}-\frac{(n-1)w^2}{v^2\sqrt{1+v_z^2}}\right)v^{n-1}\,dz}{\int w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz},
\]
it is easily seen that all eigenvalues of $\tilde{A}(t)$ are real and form a sequence going to infinity. It also shows that they are greater than or equal to $\frac{-(n-1)}{\min_{z\in[0,d]}v(\cdot;t)^2}$. This lower bound and Chapter 4 Theorem 3.16 in Kato \cite{Kato76} ensures that for any eigenvalue $\lambda_j(t)$ that changes sign, it must do so by moving through $0$.
\end{remark}
\section{Variation of a Zero Eigenvalue}\label{SecVary}
In this section we determine how an eigenvalue of $\tilde{A}$ that becomes zero, varies in a neighbourhood of its zero value. To do this we need to alter the operators so that they are all defined on the same domain. To this end we introduce the projections
\[
W(t)[u]:=u(z)-\frac{\int_0^du(z)v(z;t)^{n-1}\,dz}{\int_0^dv(z;t)^{n-1}\,dz},
\]
which are bijections between the projected spaces $X_t$, that is $W(t):X_r\to X_t$ is a bijection for any $r,t>0$. Also note that $W(r)\circ W(t)=W(r)$. Now we define the family of operators $B(t):X_1\to X_1$ given by:
\[
B(t)=W(1)\circ \tilde{A}(t)\circ W(t)=W(1)\circ DH(v(\cdot;t))\circ W(t),
\]
and note that since $\tilde{A}(t)=W(t)\circ DH(v(\cdot;t))$, we have $\tilde{A}(t)=W(t)\circ B(t)\circ W(1)$.
\begin{lemma}\label{SpecB}
$B(t)$ has the same spectrum as $\tilde{A}(t)$.
\end{lemma}
\begin{proof}
This follows directly from the formulas above. If $u\in X_t$ is an eigenfunction of $\tilde{A}(t)$, then $W(1)[u]\in X_1\backslash\{0\}$ is an eigenfunction of $B(t)$ with the same eigenvalue and similarly if $u\in X_1$ is an eigenfunction of $B(t)$, then $W(t)[u]\in X_t\backslash\{0\}$ is an eigenfunction of $\tilde{A}(t)$ with the same eigenvalue.
\end{proof}
By Corollary \ref{NullA} any zero eigenvalue of $B(t)$ has multiplicity $1$. To see that it is in fact simple we ensure that a spanning vector, $W(1)[v_t(\cdot;t)]$, is not in the range. This is not quite trivial since $B(t)$ is not self adjoint. However, a solution $\bar{w}\in X_1$ to the differential equation $W(1)[DH(v(z;t_0))[W(t_0)[\bar{w}]]]=W(1)[v_t(z;t_0)]$ exists if and only if $W(t_0)[DH(v(z;t_0))[W(t_0)[\bar{w}]]]=v_t(z;t_0)$ (using that $v_t(\cdot;t_0)\in X_{t_0}$). The operator $W(t_0)\circ DH(v(\cdot;t_0))\circ W(t_0)$ is self adjoint on $X_{t_0}$ and $v_t(\cdot;t_0)$ is an element of its null space, so the differential equations have no solutions. Therefore, any zero eigenvalue of $B(t)$ is simple.
To prove the existence of continuously differentiable eigenvalues and eigenfunctions we will use Proposition I.7.2 of \cite{Kielhofer12}. For this we need to find a function for which $B(t)$ is the linearisation. We define $\bar{v}(\cdot;t)=W(1)[v(\cdot;t)]$ and start with a lemma that we prove in Section \ref{PsiSec}.
\begin{lemma}\label{PsiExist}
For each $t>0$ there exist open neighbourhoods of $\bar{v}(\cdot;t)$, $U_{t}\subset X_1$, and $t$, $V_{t}\subset\mathbb{R}$, and a function $\psi_{t}:U_{t}\times V_{t}\to X_t$ such that
\begin{itemize}
\item $W(1)[\psi_{t}(\bar{u},r)]=\bar{u}$ for all $(\bar{u},r)\in U_{t}\times V_{t}$,
\item $\omega_n\int_0^d\psi_{t}(\bar{u},r)^n\,dz=V(r)$ for all $(\bar{u},r)\in U_{t}\times V_{t}$, and
\item $W(1)[u]=\bar{u}$ and $\int_0^du(z)^n\,dz=V(r)$ for $(u,\bar{u},r)\in R(\psi_{t})\times U_{t}\times V_{t}$ if and only if $u=\psi_{t}(\bar{u},r)$.
\end{itemize}
In particular, there exists a neighbourhood of $t$, $I_t\subset\mathbb{R}$, such that $\psi_{t}(\bar{v}(\cdot;r),r)=v(\cdot;r)$ for all $r\in I_t$.
Furthermore,
\[
D_{\bar{u}}\psi_{t}(\bar{u},r)[\bar{w}]=\bar{w}-\frac{\int_0^d\bar{w}(z)\psi_{t}(\bar{u},r)(z)^{n-1}\,dz}{\int_0^d\psi_{t}(\bar{u},r)(z)^n\,dz},
\]
for all $(\bar{u},r)\in U_{t}\times V_{t}$ and $\bar{w}\in X_1$. In particular, $D_{\bar{u}}\psi_{t}(\bar{v}(\cdot;r),r)[\bar{w}]=W(r)[\bar{w}]$ for all $\bar{w}\in X_1$ and $r\in I_t$.
\end{lemma}
We now define the function $F_t:U_t\times V_t\to X_1$ by $F_{r,t}(\bar{u},r)=W(1)[H(\psi_{t}(\bar{u},r))]$. It is easily seen that $F_t(\bar{v}(\cdot;r),r)=0$ and $D_{\bar{u}}F_t(\bar{v}(\cdot;r),r)[\bar{w}]=B(t)[\bar{w}]$. Therefore by Proposition I.7.2 of \cite{Kielhofer12} the zero eigenvalue will vary smoothly with $r$ in a neighbourhood of $t$. That is, let $t_0>0$ be such that $V'(t_0)=0$ then there exists an open neighbourhood of $t_0$, $J_{t_0}\subset I_{t_0}$, and a smoothly varying family of functions $w:X_1\times J_{t_0}\to\mathbb{R}$, such that $\lambda\in C^{\infty}(J_{t_0},\mathbb{R})$, $\lambda(t_0)=0$, $w(\cdot;t_0)=0$ and
\[
B(t)[\bar{v}_t(z;t_0)+w(z;t)]=\lambda(t)(\bar{v}_t(z;t_0)+w(z;t)).
\]
We can also take $\int_0^d\bar{v}_t(z;t_0)w(z;t)\,dt=0$ for all $t\in J_{t_0}$.
\begin{lemma}\label{Dla}
\[
\lambda'(t_0)=\frac{-V''(t_0)\eta'(t_0)}{n\omega_n\int_0^dv_t(z;t_0)^2v(z;t_0)^{n-1}\,dz}
\]
\end{lemma}
\begin{proof}
We replace $B(t)$ with $W(1)\circ DH(v(\cdot;t))\circ W(t)$ and use that $W(t)[\bar{v}_t(z;t_0)]=W(t)[v_t(z;t_0)]$:
\[
W(1)[DH(v(z;t))[W(t)[v_t(z;t_0)+w(z;t)]]]=\lambda(t)(\bar{v}_t(z;t_0)+w(z;t)).
\]
Calculating its $t$ derivative at $t=t_0$ gives:
\begin{align*}
W&(1)[D^2H(v(z;t_0))[v_t(z;t_0),v_t(z;t_0)]]\\
&+W(1)[DH(v(z;t_0))[W'(t_0)[v_t(z;t_0)]+W(t_0)[w_t(z;t_0)]]]=\lambda'(t_0)\bar{v}_t(z;t_0).
\end{align*}
We apply $W(t_0)$ to the equation to obtain
\begin{align*}
W&(t_0)[D^2H(v(z;t_0))[v_t(z;t_0),v_t(z;t_0)]]\\
&+W(t_0)[DH(v(z;t_0))[W'(t_0)[v_t(z;t_0)]+W(t_0)[w_t(z;t_0)]]]=\lambda'(t_0)v_t(z;t_0).
\end{align*}
Next we use that since $H(v(z;t))=\eta(t)$, then $DH(v(z;t))[v_t(z;t)]=\eta'(t)$ and $D^2H(v(z;t)[v_t(z;t),v_t(z;t)]+DH(v(z;t))[v_{tt}(z,t)]=\eta''(t)$, to obtain
\begin{equation}\label{EvalueEqD}
W(t_0)[DH(v(z;t_0))[W'(t_0)[v_t(z;t_0)]+W(t_0)[w_t(z;t_0)]-v_{tt}(z;t_0)]]=\lambda'(t_0)v_t(z;t_0),
\end{equation}
where we used that $W(t_0)[\eta''(t_0)]=0$.
We now consider $V(t)=\omega_n\int_0^dv(z;t)^n\,dz$. From this we see that
\[
V'(t)=n\omega_n(I-W(t))[v_t(z;t)]\int_0^dv(z;t)^{n-1}\,dz,
\]
and
\begin{align*}
V''(t)=&n\omega_n\left(\left(I-W(t)\right)[v_{tt}(z;t)]-W'(t)[v_t(z;t)]\right)\int_0^dv(z;t)^{n-1}\,dz\\
&+n(n-1)\omega_n(I-W(t))[v_t(z;t)]\int_0^dv_t(z;t)v(z;t)^{n-2}\,dz.
\end{align*}
Hence
\[
V''(t_0)=n\omega_n\left(\left(I-W(t_0)\right)[v_{tt}(z;t_0)]-W'(t_0)[v_t(z;t_0)]\right)\int_0^dv(z;t_0)^{n-1}\,dz,
\]
and
\[
W'(t_0)[v_t(z;t_0)]=(I-W(t_0))[v_{tt}(z;t_0)]-\frac{V''(t_0)}{n\omega_n\int_0^dv(z;t_0)^{n-1}\,dz}.
\]
Substituting this into (\ref{EvalueEqD}) gives
\[
W(t_0)[DH(v(z;t_0))[W(t_0)[w_t(z;t_0)-v_{tt}(z;t_0)-\frac{V''(t_0)}{n\omega_n\int_0^dv(z;t_0)^{n-1}\,dz}]]]=\lambda'(t_0)v_t(z;t_0).
\]
Multiplying by $v_t(z;t_0)v(z;t_0)^{n-1}$ and integrating gives:
\begin{align*}
\lambda'(t_0)=&\frac{\int_0^dDH(v(z;t_0))[W(t_0)[w_t(z;t_0)-v_{tt}(z;t_0)]]v_t(z;t_0)v(z;t_0)^{n-1}\,dz}{\int_0^dv_t(z;t_0)^2v(z;t_0)^{n-1}\,dz}\\
&-\frac{V''(t_0)\int_0^dDH(v(z;t_0))[1]v_t(z;t_0)v(z;t_0)^{n-1}\,dz}{n\omega_n\int_0^dv(z;t_0)^{n-1}\,dz\int_0^dv_t(z;t_0)^2v(z;t_0)^{n-1}\,dz},
\end{align*}
where the projections vanish due to $v_t(z;t_0)\in X_{t_0}$. Using the self adjointness of $DH(v(\cdot;t))$ (with respect to the weight $v(\cdot;t_0)$, for functions satisfying the free boundary condition) and
\[
DH(v(z;t_0))[v_t(z;t_0)]=\eta'(t_0),
\]
this becomes:
\begin{align*}
\lambda'(t_0)=&\frac{\eta'(t_0)\int_0^dW(t_0)[w_t(z;t_0)-v_{tt}(z;t_0)]v(z;t_0)^{n-1}\,dz}{\int_0^dv_t(z;t_0)^2v(z;t_0)^{n-1}\,dz}\\
&-\frac{\eta'(t_0)V''(t_0)\int_0^dv(z;t_0)^{n-1}\,dz}{n\omega_n\int_0^dv(z;t_0)^{n-1}\,dz\int_0^dv_t(z;t_0)^2v(z;t_0)^{n-1}\,dz},
\end{align*}
and the result follows since $W(t_0)$ projects into $X_{t_0}$.
\end{proof}
We can now prove the main theorem.
\par{\noindent\textit{Proof of Theorem 2.3. }}
We start by labeling the critical points of $V$ as $t_0=1$, $t_1$, $t_2$, etc. in decreasing order, i.e. $V'(t_i)=0$ and $0<t_{i+1}<t_i$ for all $i$.
Suppose at some $\tau\in(t_{i+1},t_{i})$ the eigenvalues of $\tilde{A}(\tau)$ are all positive and $V'(\tau)>0$. Both these things must remain true for $\tilde{A}(t)$ for all $t\in(t_{i+1},t_i)$. Therefore immediately after the critical point at $t_{i+1}$, $V'(t)>0$ and, since we assume no critical points of $V$ are degenerate, $t_{i+1}$ must be a minimum of $V$ and hence $V''(t_{i+1})>0$. From Lemma \ref{Dla} and since we assume $\eta'(t_0)<0$ this means that the critical eigenvalue (with multiplicity one) is increasing at this point and hence is negative immediately prior to $t_{i+1}$. Thus for $t\in(t_{i+2},t_{i+1})$ we have that $V'(t)<0$ and $\tilde{A}(t)$ has a single negative eigenvalue.
Next suppose at some $\tau\in(t_{i+1},t_{i})$, $\tilde{A}(\tau)$ has a single negative eigenvalue, the rest are strictly positive, and $V'(\tau)<0$. These things must remain true for $\tilde{A}(t)$ for all $t\in(t_{i+1},t_i)$. Therefore immediately after the critical point $t_{i+1}$, $V'(t)<0$ and, since we assume no critical points of $V$ are degenerate, $t_{i+1}$ must be a maximum of $V$ and hence $V''(t_{i+1})<0$. From Lemma \ref{Dla} and since we assume $\eta'(t_0)<0$ this means that the critical eigenvalue (with multiplicity one) is decreasing at this point and hence is positive immediately prior to $t_{i+1}$. Thus for $t\in(t_{i+2},t_{i+1})$ we have that $V'(t)>0$ and $\tilde{A}(t)$ has only positive eigenvalues.
As these two cases alternate, we have covered all cases if we can show that on some interval $(a,1)$ one of them is true. However, this was proved in \cite{Hartley16}. In fact, there it was shown that for $2\leq n\leq10$, $V(t)$ has a local minimum and the critical eigenvalue has a local maximum at $t=1$, so that there exists $\epsilon>0$ such that for $t\in(1-\epsilon,1)$, $\tilde{A}(t)$ has a single negative eigenvalue and $V'(t)<0$. While for $n\geq11$, $V(t)$ has a local maximum and the critical eigenvalue has a local minimums at $t=1$, so that there exists $\epsilon>0$ such that for $t\in(1-\epsilon,1)$, $\tilde{A}(t)$ has only positive eigenvalues and $V'(t)>0$. By Proposition \ref{EquivStable}, stability of the hypersurface defined by $v(\cdot;t)$ follows from $\tilde{A}(t)$ having only positive eigenvalues and it is also clear that if $\tilde{A}(t)$ has negative eigenvalues then the hypersurface is unstable. Hence we have proved the result. \hfill $\Box$
\begin{remark}
In \cite{Hartley16} the details are given for the operator which is the negative of $\tilde{A}(t)$, hence the eigenvalue signs are switched. Also, instead of using $V(t)$, the eigenvalue formulas are written in terms of the function $\frac{C}{V(t)^{\frac{1}{n}}}$. Finally, the family of height functions is parameterised by the different parameter $s=\frac{1-t}{1+t}$.
\end{remark}
\section{Mean Curvature Calculation for the Family of Hypersurfaces}\label{SecFamilyApp}
Here we prove Proposition \ref{FamCMC}. When $r=1$, $u(z;1,t)=\frac{(n-1)d}{nP(t)Q(t)}$ and the proposition follows trivially. We now assume $r\neq1$ and note that
\[
\zeta^{-1}_y(y;t)=\frac{1}{\zeta_x(\zeta^{-1}(y;t);t)}=-R(\zeta^{-1}(y;t);t).
\]
Therefore
\[
u_z(z;r,t)=(1-r)R\left(\zeta^{-1}\left(\frac{Q(t)P(t)z}{Q(r)d};r\right);r\right).
\]
Using $\zeta^{-1}(0;r)=1$ and $\zeta^{-1}(P(r);r)=0$ we obtain $u_z(0;r,t)=(1-r)R(1;r)=0$ and $u_z\left(\frac{Q(r)P(r)d}{Q(t)P(t)};r,t\right)=(1-r)R(0;r)=0$.
Next define $S(x;r):=\frac{(1-(1-r)x)^{n-1}}{1-Q(r)+Q(r)(1-(1-r)x)^n}$, and take its $x$-derivative:
\[
S_x(x;r)=-(1-r)\left(\frac{n-1}{1-(1-r)x}-nQ(r)S(x;r)\right)S(x;r).
\]
Therefore, from $R(x;r)=\frac{1}{|1-r|}\sqrt{S(x;r)^2-1}$ we have:
\begin{align*}
R_x(x;r)=&\frac{S(x;r)S_x(x;r)}{|1-r|\sqrt{S(x;r)^2-1}}\\
=&\frac{-S(x;r)^2}{(1-r)R(x;r)}\left(\frac{n-1}{1-(1-r)x}-nQ(r)S(x;r)\right)\\
=&-\frac{(1-r)^2R(x;r)^2+1}{(1-r)R(x;r)}\left(\frac{n-1}{1-(1-r)x}-nQ(r)\sqrt{(1-r)^2R(x;r)^2+1}\right).
\end{align*}
So that, with $x=\zeta^{-1}\left(\frac{Q(t)P(t)z}{Q(r)d};r\right)$, we have
\begin{align*}
u_{zz}(z;r,t)=&\frac{Q(t)P(t)\left((1-r)^2R(x;r)^2+1\right)}{Q(r)d}\left(\frac{n-1}{1-(1-r)x}-nQ(r)\sqrt{(1-r)^2R(x;r)^2+1}\right)\\
=&\frac{Q(t)P(t)\left(u_z(z;r,t)^2+1\right)}{Q(r)d}\left(\frac{n-1}{\frac{Q(t)P(t)}{Q(r)d}u(z;r,t)}-nQ(r)\sqrt{u_z(z;r,t)^2+1}\right)\\
=&\left(u_z(z;r,t)^2+1\right)\left(\frac{n-1}{u(z;r,t)}-\frac{nQ(t)P(t)}{d}\sqrt{u_z(z;r,t)^2+1}\right).
\end{align*}
The result now follows from the formula for $H(u)$ for rotationally symmetric functions.
\section{Calculation of the Stability Functional}\label{SecStableApp}
\begin{lemma}\label{Jtil}
\[
\tilde{J}_v(w)=\frac{\int_{\SS^{n-1}}\int_0^d \left(\|\nabla w\|_{\Sigma}^2-\frac{(n-1)w^2}{v^2} \right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}{\int_{\SS^{n-1}}\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}
\]
\end{lemma}
\begin{proof}
To perform the substitution $u=\frac{w}{\sqrt{1+v_z^2}}$ in $J_v(u)$, we start by calculating the $\frac{u_z^2u^{n-1}}{\sqrt{1+v_z^2}}$ term:
\begin{align*}
\frac{u_z^2u^{n-1}}{\sqrt{1+v_z^2}}=&\left(\frac{w_z^2}{1+v_z}-\frac{2v_zv_{zz}ww_z}{(1+v_z^2)^2}+\frac{v_z^2v_{zz}^2w^2}{(1+v_z^2)^3}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\\
=&\frac{\partial}{\partial z}\left(\frac{-v^{n-1}v_zv_{zz}w^2}{(1+v_z^2)^{\frac52}}\right)+\left(\frac{w_z^2}{1+v_z}+\frac{v_z^2v_{zz}^2w^2}{(1+v_z^2)^3}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\\
&+\left(\frac{(n-1)v_z^2v_{zz}w^2}{v(1+v_z^2)^2}+\frac{v_{zz}^2w^2}{(1+v_z^2)^2}+\frac{v_zv_{zzz}w^2}{(1+v_z^2)^2}-\frac{5v_z^2v_{zz}^2w^2}{(1+v_z^2)^3}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}.
\end{align*}
Now we differentiate $\frac{-v_{zz}}{(1+v_z^2)^{\frac32}}+\frac{n-1}{v\sqrt{1+v_z^2}}=constant$ with respect to $z$ to obtain:
\[
\frac{-v_{zzz}}{(1+v_z^2)^{\frac32}}+\frac{3v_zv_{zz}^2}{(1+v_z^2)^{\frac52}}-\frac{(n-1)v_z}{v^2\sqrt{1+v_z^2}}-\frac{(n-1)v_zv_{zz}}{v(1+v_z^2)^{\frac32}}=0,
\]
so that
\begin{align*}
\frac{u_z^2u^{n-1}}{\sqrt{1+v_z^2}}=&\frac{\partial}{\partial z}\left(\frac{-v^{n-1}v_zv_{zz}w^2}{(1+v_z^2)^{\frac52}}\right)+\left(\frac{w_z^2}{1+v_z}-\frac{v_z^2v_{zz}^2w^2}{(1+v_z^2)^3}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\\
&+\left(\frac{v_{zz}^2w^2}{(1+v_z^2)^2}-\frac{(n-1)v_z^2w^2}{v^2(1+v_z^2)}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\\
=&\frac{\partial}{\partial z}\left(\frac{-v^{n-1}v_zv_{zz}w^2}{(1+v_z^2)^{\frac52}}\right)+\left(\frac{w_z^2}{1+v_z}+\frac{v_{zz}^2w^2}{(1+v_z^2)^3}-\frac{(n-1)v_z^2w^2}{v^2(1+v_z^2)}\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}.
\end{align*}
Using that for a rotationally symmetric hypersurface $|A|^2=\frac{n-1}{v^2(1+v_z^2)}+\frac{v_{zz}^2}{(1+v_z^2)^3}$,
\begin{align*}
\tilde{J}_v(w)=&\frac{\int_{\SS^{n-1}}\int_0^d \left(\frac{w_z^2}{1+v_z}-\frac{(n-1)(v_z^2+1)w^2}{v^2(1+v_z^2)}+\frac{1}{v^2}\|\tilde{\nabla} w\|_{\SS^{n-1}}^2\right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}{\int_{\SS^{n-1}}\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}\\
=&\frac{\int_{\SS^{n-1}}\int_0^d \left(\|\nabla w\|_{\Sigma}^2-\frac{(n-1)w^2}{v^2} \right)\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}{\int_{\SS^{n-1}}\int_0^d w^2\frac{v^{n-1}}{\sqrt{1+v_z^2}}\,dz\,d\mu_{\SS^{n-1}}}.
\end{align*}
\end{proof}
\section{Existence of the Inverse Projection}\label{PsiSec}
In this section we prove Lemma \ref{PsiExist}. Fix $t>0$ and define $\Phi:X_1\times\mathbb{R}\times X_t\to X_1\times\mathbb{R}$ such that
\[
\Phi(\bar{u},r,u)=\left(W(1)[u]-\bar{u},\omega_n\int_0^du(z)^n\,dz-V(r)\right)
\]
and consider it's linearisation with respect to $u$:
\[
D_u\Phi(\bar{u},r,u)[w]=\left(W(1)[w],n\omega_n\int_0^dw(z)u(z)^{n-1}\,dz\right)
\]
If we evaluate at $(\bar{v}(\cdot;t),t,v(\cdot;t))$, noting that $\Phi(\bar{v}(\cdot;t),t,v(\cdot;t))=(0,0)$, we obtain
\[
D_u\Phi(\bar{v}(\cdot;t),t,v(\cdot;t))[w]=\left(W(1)[w],n\omega_n\int_0^dv(z;t)^{n-1}\,dz(I-W(t))[w]\right).
\]
If $w$ is in the null space of $D_u\Phi(\bar{v}(\cdot;t),t,v(\cdot;t))$ then:
\[
W(1)[w]=0\ \text{and}\ (I-W(t))[w]=0.
\]
Therefore $w=(I-W(t))[w]+W(t)[w]=0+W(t)[W(1)[w]]=W(t)[0]=0$. So, $D_u\Phi(\bar{v}(\cdot;t),t,v(\cdot;t))$ is a Banach space isomorphism for any $t>0$ and the existence of $\psi_t$ follows from the implicit function theorem.
The formula for the derivative follows by taking the derivative of $\tilde{\Phi}_t(\bar{u},r)=\Phi(\bar{u},r,\psi_t(\bar{u},r))=(0,0)$ with respect to $\bar{u}$. That is:
\[
\left(W(1)[D_{\bar{u}}\psi_t(\bar{u},r)[\bar{w}]]-\bar{w},n\omega_n\int_0^dD_{\bar{u}}\psi_t(\bar{u},r)[\bar{w}]\psi_t(\bar{u},r)^{n-1}\,dz\right)=(0,0).
\]
By the first condition $D_{\bar{u}}\psi_t(\bar{u},r)[\bar{w}]=\bar{w}+C$, and substitution into the second condition gives
\[
\int_0^d\bar{w}\psi_t(\bar{u},r)^{n-1}\,dz+C\int_0^d\psi_t(\bar{u},r)^{n-1}\,dz=0,
\]
resulting in the formula given.
\bibliographystyle{plain}
|
2,869,038,154,847 | arxiv | \section{Introduction} \label{one}
Much attention has been given in recent years to the classification
of curves in projective space. Here we define a curve to be a
purely one-dimensional locally Cohen-Macaulay (i.e. without
embedded points) closed subscheme of $ \pso_k$, the projective
three-space over an algebraically closed field $k$. Our goal in
this paper is to answer all the interesting questions about curves
contained in surfaces of degree two. We succeed, with one notable
exception.
Curves in the nonsingular quadric surface $Q$ and in the quadric cone
$Q_0$ are well-known \cite{AG} (Exercises III 5.6 and V 2.9). Curves
in the union of two planes $H_1 \cup H_2$ were studied in \cite{hgd},
section 5. The main contribution of this paper is the systematic
study of curves contained in a double plane $2H$.
We are also interested in flat families of curves, so that we can
identify the irreducible components of the Hilbert scheme. Families
of curves whose general member lies on a nonsingular quadric $Q$
and whose special member lies on a cone $Q_0$ or the union of
two planes $H_1 \cup H_2$ were studied in \cite{zeuthen}. In this paper
we will identify the irreducible components of the scheme
$H_{d,g} (2H)$ of curves of degree $d$ and genus $g$ lying in a
double plane, and specializations between them. The question we have
not been able to answer is what families are there whose general
member lies on a nonsingular quadric and whose special member lies in
$2H$?
For example, we do not know if there is a family specializing from
four skew lines on a quadric to a curve contained in the double plane.
Now let us describe the contents of the current paper. To each
curve $C$ contained in the double plane $2H$, we assign a triple
$\{Z,Y,P\}$ consisting of a zero-dimensional closed subscheme $Z$
in the (reduced) plane $H$, and two curves $Y,P$ in $H$. Roughly
speaking, $P$ with embedded points $Z$ is the intersection of $C$ with
$H$,
while $Y$ is the residual intersection (Proposition 2.1).
Conversely to each such triple, satisfying certain conditions, we
can associate curves $C$ in $2H$ (Proposition 3.1). The numerical
invariants of $C$ can be computed in terms of $Z,Y,P $ (section~\ref{five}).
Thus the triple $\{Z,Y,P \}$ of data in the plane $H$ is
an effective tool for studying curves $C$ in the double plane.
We can translate information about a flat family of
curves in $2H$ to families of triples in $H$ (section \ref{four}). This
allows us to identify the irreducible components of the Hilbert
scheme $H_{d,g} (2H)$. They are given by triples of integers
$(z,y,p)$ representing the length of $Z$ and the degrees of
$Y$ and $P$ (Theorem~\ref{components}).
We also discuss the Rao module of these curves (section \ref{five}) and
liaison and biliaison equivalence classes (section \ref{six}). Finally,
generalizing a technique of Nollet \cite{nthree}, we show the existence of
flat families joining the various irreducible components of
$H_{d,g} (2H)$, and conclude that $H_{d,g} (2H)$ is connected
(Theorem~\ref{connect}).
Our purpose in studying curves in the double plane
was to treat one very special case in view of the more
general problem whether the Hilbert scheme $H_{d,g}$ of all
locally Cohen-Macaulay curves in $\pso$ is connected for
all $d$,$g$. Because of our connectedness result (\ref{connect}), it
would be sufficient for that more general problem to show that any
curve in $\pso$ can be connected by a sequence of flat families to a
curve in 2H.
We would like to thank E. Ballico for encouraging us to write this paper.
\section{The triple associated to a curve in 2H} \label{two}
We first recall the notion of ``residual scheme''
(\cite{fulton}, 9.2.8, and \cite{PS} ). Suppose $T \subseteq W$
are closed subschemes of an ambient scheme $V$. The residual scheme
$R$ of $T$ in $W$ is the closed subscheme of $V$ with ideal sheaf
$$
\ideal{R} = (\ideal{W} : \ideal{T}).
$$
Intuitively, $R$ is equal to $W$ minus $T$. It does not depend on the
ambient scheme. We will need this notion in the following two cases.
If $T$ is a Cartier divisor in $V$, then
$\ideal{R} \ideal{T} = \ideal{W}$. On the other hand, suppose
that $H$ is an effective Cartier divisor on $V$, and $T$ is the scheme
theoretic intersection of $W$ and $H$. In this case, we say
(cf. \cite{hhdg}, p.176) that $R$ is the residual scheme to the
intersection of $W$ with $H$.
We have
$$
\ideal{R} \ideal{H} = \ideal{W} \cap \ideal{H}
$$
hence an exact sequence:
\begin{equation} \label{res-int}
0 \rrr
\ideal{R}(-H) \rrr \ideal{W} \rrr
\ideal{W \cap H,H} \rrr 0.
\end{equation}
Let $H$ be a plane in $\pso_{k}$,
defined by $h=0$. Let $C$ be a
curve contained in the scheme $2H$ (``the double plane'') defined by
the equation $h^2 = 0$. To the curve $C$ we will associate a triple
$T(C) = \{Z,Y,P \}$ where $Z$ is a zero-dimensional subscheme of $H$,
and $Y$,$P$ are curves in $H$, with $Z \subseteq Y \subseteq P$.
First consider the scheme-theoretic intersection $C \cap H$. This will
be a one dimensional subscheme of $H$, possibly with embedded points.
So we can write
$$
\ideal{C \cap H,H} = \ideal{Z,H} (-P)
$$
where $Z$ is zero-dimensional and $P$ is a curve.
In fact, the inclusion
$\ideal{C \cap H,H} \hookrightarrow \coo_{H}$ defines a global
section of the invertible sheaf $\Hom (\ideal{C \cap H,H}, \coo_{H})$
whose scheme of zeros is the effective Cartier divisor $P \subset H$,
and $Z$ is the residual scheme of $P$ in $C \cap H$.
Next we let $Y$ be the residual scheme to the
intersection of $C$ with $H$:
it is a curve in $\pso$. By the discussion above we have an exact sequence
\begin{equation} \label{first}
0 \rrr
\ideal{Y,\pso}(-1) \stackrel{h}{\rrr} \ideal{C,\pso} \rrr
\ideal{C \cap H,H} \rrr 0.
\end{equation}
Since $C$ is contained in $2H$, $Y$ will be contained in
$H$. Furthermore, since $P$ is the largest curve in $H$ which is
contained in $C$, it is clear that $Y \subseteq P$. Note that $C$ is
contained in the reduced plane $H$ if and only if $Y$ is empty.
Now we use the inclusion $P \subseteq C \cap H$ to create a diagram
as follows
\begin{equation} \label{second}
\begin{CD}
&&0 && 0 && 0&& \\
&&@VVV @VVV @VVV \\
0 @>>> \ideal{Y,\pso} (-1) @>h>> \ideal{C, \pso} @>>>
\ideal{C \cap H,H} @>>> 0 \\
&&@VVV @VVV @VVV \\
0 @>>> \coo_{\pso} (-1) @>h>> \ideal{P,\pso} @>>> \ideal{P,H} @>>> 0\\
&&@VVV @VV{u}V @VVV \\
0 @>>> \coo_{Y} (-1) @>>> \scrl @>>> \coo_{Z} (-p) @>>> 0 \\
&&@VVV @VVV @VVV \\
&&0 && 0 && 0&& \\
\end{CD}
\end{equation}
where $p$ is the degree of $P$.
Since $C$ is locally Cohen-Macaulay, the ideal sheaf $\ideal{C,\pso}$
has depth $\geq 2$ at every closed point. Therefore the sheaf $\scrl$,
defined as the quotient $\ideal{P} / \ideal{C}$, will have depth $\geq
1$. Multiplying a section of $\scrl$ by a local section of $\ideal{Y}$
will give something zero outside of $Z$. But $\scrl$ can have no section
supported at a point because of its depth. So we see that $\scrl$ is an
$\coo_{Y}$-module. Because $Y$ is a Gorenstein curve, $\scrl$ is a
reflexive sheaf of rank one \cite{hgd}, 1.6, and then by \cite{hgd}, 2.8 and 2.9,
$\scrl$ is
of the form $\scrl(D-1)$ for some effective generalized divisor $D$ on
$Y$. Then by \cite{hgd}, 2.10, the quotient of $\scrl$ by $\coo_{Y}
(-1)$ is
$$
\coo_{Z} (-p) \cong \omega_{D} \otimes \dual{\omega}_{Y} (-1),
$$
so
$$
\coo_{Z} (-p) \cong \omega_{D} (2-y)
$$
where $y$ is the degree of $Y$.
From this we conclude several things. First $Z=D$, so that $Z
\subseteq Y$. Secondly, $\coo_{Z}$ is locally isomorphic to
$\omega_{Z}$, so $Z$ is a Gorenstein scheme. Since $Z$ is of
codimension two in $H$, it follows from the theorem of Serre
\cite{eisenbud}, Corollary 21.20, that
$Z$ is a locally complete intersection in $H$. Summing up,
\bp \label{1.1}
To each curve $C$ in $2H$, we
can associate a triple $T(C) = \{Z,Y,P \}$ where $Z \subseteq Y
\subseteq P \subset H$, and $Z$ is a locally complete
intersection zero-dimensional subscheme, and $Y$,$P$ are curves. If we
denote by $g$ the arithmetic genus of $C$, by $d$,$y$,$p$ the degrees
of $C$,$Y$,$P$ respectively, and by $z$ the length of $Z$, then
\begin{align*}
d&= y+p \\
g&= \frac{1}{2} (y-1)(y-2) + \frac{1}{2} (p-1)(p-2) +y-z-1.
\end{align*}
\ep
The computation of the degree and genus of $C$ is straightforward from
the exact sequence~(\ref{first}).
\br
The curve $C$ is arithmetically Cohen-Macaulay if and only if $Z$ is
empty. From the exact sequence~(\ref{first}) we obtain an exact sequence
$$
0 \xrightarrow{} \mbox{H}^{1}_{*} (\ideal{C})
\xrightarrow{} \mbox{H}^{1}_{*} ( \ideal{Z,H}(-p))
\xrightarrow{} \mbox{H}^{2}_{*} (\ideal{Y} (-1) )
$$
where, for a a coherent sheaf $\caf$ on $\pso$,
we let
$$
\mbox{H}^{i}_{*} ( \caf) = \bigoplus_{n \in \Z}
\mbox{H}^{i} ( \caf (n) ).
$$
If $Z$ is empty, the middle term is zero, and that forces the first
term, which is the Rao module $M_{C}$ of $C$ to be zero also. Hence
$C$ is ACM.
Conversely, if $M_{C} = 0$, then we look at the particular twist
$$
0 \xrightarrow{}
\mbox{H}^{1} (\ideal{Z,H} (-1)) \xrightarrow{}
\mbox{H}^{2} (\ideal{Y} (p-2)).
$$
The last group is isomorphic to $\mbox{H}^{1} (\coo_{Y} (p-2))$,
which is zero because $p \geq y$. Hence
$\mbox{length} \, Z = \dim \mbox{H}^{1} (\ideal{Z,H} (-1))=0$,
and $Z$ is
empty.
\er
\section{Existence of curves with a given triple} \label{three}
Suppose given $\{Z,Y,P\}$ as in \ref{1.1}. Let $\scrl$ be the sheaf
$\coo_{Y} (Z-1)$ associated to the generalized divisor $Z$ on $Y$ as
in \cite{hgd}, 2.8. Then we will show how to construct a surjective
map $u: \ideal{P,\pso} \rrr \scrl$, so as to define $\ideal{C,\pso} =
\mbox{Ker} \, u$ as in the diagram~(\ref{second}). Note that $C$ will
be locally Cohen-Macaulay and pure dimensional because at closed
points $\scrl$ has depth one and $\ideal{P, \pso}$ has depth two.
The map $u$ needs to be compatible with the existing inclusions
$\coo_{\pso} (-1) \hookrightarrow \ideal{P,\pso}$ and $\coo_{Y} (-1)
\hookrightarrow \scrl$. Furthermore, $\scrl$ is an $\coo_{Y}$-module,
so $u$ must factor through $\ideal{P, \pso} \otimes \coo_{Y}$. Tensor
the middle line of the diagram~(\ref{second}) with $\coo_{Y}$. Then it
will be sufficient to define a surjective map $\bar{u}$ to complete
the diagram
\begin{equation} \label{third}
\begin{CD}
0 @>>> \coo_{Y} (-1) @>>> \ideal{P}/{\ideal{Y} \ideal{P}}
@>>> \coo_{Y} (-p) @>>> 0 \\
&& @VVV @VV\bar{u}V @VVwV \\
0 @>>> \coo_{Y} (-1) @>>> \scrl @>>> \omega_{Z} (2-y) @>>> 0
\end{CD}
\end{equation}
Since $P$ is a complete intersection in $\pso$, the top
row, which is the restriction to $Y$ of the conormal sequence of $P$,
splits. Thus to find $\bar{u}$, we need only find a map $v: \coo_{Y}
(-p) \rrr \scrl$ whose image in $ \omega_{Z} (2-y)$ is surjective. Since
$Z$ is a locally complete intersection in $H$, the sheaf $\omega_{Z}$
is generated by a single element at every point, so any sufficiently
general $w: \coo_{Y} (-p) \rrr \omega_{Z} (2-y)$ will be
surjective. To show that $w$ lifts to $v$, think of $v \in
\mbox{H}^{0} (Y, \scrl (p))$. Then we have an exact sequence
$$
0 \xrightarrow{} \mbox{H}^{0} (\coo_{Y} (p-1))
\xrightarrow{} \mbox{H}^{0} ( \scrl (p))
\xrightarrow{} \mbox{H}^{0} (\omega_{Z} (2+p-y) )
\xrightarrow{} \mbox{H}^{1} (\coo_{Y} (p-1))
$$
Now $p \geq y$, so $p-1 > y-3$ hence $\mbox{H}^{1} (\coo_{Y}
(p-1))=0$. Thus any surjective $w$ lifts to $v$ and gives the desired
$\bar{u}$. We can now define $C$ by setting $\ideal{C,\pso} =
\mbox{Ker} \, u$. Since $\scrl \cong \ideal{P}/\ideal{C}$ has depth
$\geq 1$ at closed points, $C$ is locally Cohen-Macaulay, and it is
clear from our construction that $T(C) = \{Z,Y,P \}$.
Finally, we claim that the construction above gives a one to one
correspondence between curves $C \subset 2H$
with $T(C) = \{Z,Y,P \}$ and global sections $ v \in \mbox{H}^{0} (Y,
\scrl (p))$ whose image in $\mbox{H}^{0} ( \omega_{Z} (2+p-y))$
generates $\omega_{Z} (2+p-y)$ at every point. Note that it is enough
to show that a surjective $\bar{u}$ fitting in the diagram~(\ref{third})
is determined by its kernel. Now, if $\bar{u}_{1}$ and $\bar{u}_{2}$
have the same kernel and fit in the diagram~(\ref{third}), then their
difference factors through a morphism $\omega_{Z} (2-y) \rrr \scrl$
which must be zero, thus $\bar{u}_{1} = \bar{u}_{2}$. So we have:
\bp \label{2.1}
For each triple $\{Z,Y,P \}$ as in~\ref{1.1} there exists a curve $C
\subset 2H$ with $T(C) = \{Z,Y,P \}$. The set of such $C$ is
parametrized by an open subset of the vector space ${\it H}^{0}
(Y,\scrl (p))$, which has dimension
$$
h^{0} ( \scrl (p)) = z + (p-1)y + 1 - \frac{1}{2} (y-1)(y-2).
$$
\ep
\section{Families of curves in the double plane} \label{four}
In order to understand the Hilbert scheme of curves in the double
plane, we must carry out the constructions of section \ref{two} and
\ref{three} for flat families. Consider a base scheme $S$, and let $\ccc
\subset 2H \times S$ be a family of curves of degree $d$ and genus
$g$, i.e. a closed subscheme, flat over $S$, whose fibre over each
point $s \in S$ is a locally Cohen-Macaulay curve $C_{s}$ in $2H$ of
given degree and genus. The functor which to each $S$ associates the
set of such flat families is represented by a Hilbert scheme which we
denote by $H_{d,g}(2H)$.
If $\ccc \subset 2H \times S$ is a flat family as above, the
intersection $ \ccc \cap (H \times S)$ need not be flat. For example,
if $S$ is integral,
flatness of $ \ccc \cap (H \times S)$ is equivalent to local constancy
of the integers $z$,$y$,$p$ associated to the fibres $C_{s}$ as in
section~\ref{two} above. Applying Mumford's flattening stratification
to $ \ccc \cap (H \times S)$, where $\ccc$ is the universal family
over $H_{d,g}(2H)$, we find that the scheme $H_{d,g}(2H)$ is
stratified by locally closed subschemes $H_{z,y,p} (2H)$ representing
families for which $ \ccc \cap (H \times S)$ is flat and the fibres
correspond to $z$,$y$,$p$ as above.
We claim that the procedures of section~\ref{two} and \ref{three}
above can be relativized over a base scheme $S$, to show that a flat
family $\ccc \subset 2H \times S$ with the condition that $ \ccc \cap
(H \times S)$ is also flat gives rise to a triple $\{Z,Y,P \}$ where
$Z \subseteq Y \subseteq P $ are closed subschemes of $H \times S$,
flat over $S$, and where the fibres of $Z$ are zero-dimension locally
complete intersection subschemes of $H$, and the fibres of $Y$ and
$P$ are curves in $H$. Conversely, given any such triple $\{Z,Y,P
\}$, there are families of curves giving rise to this triple, and the
resulting curves are parametrized by a global section of the sheaf
$\scrl (P)$, where $\scrl$ is the sheaf
$\Hom (\ideal{Z,Y} , \coo_{Y}(- 1))$ and $P$ is the divisor on $H
\times S$ corresponding to $P$.
Thus there is a natural map from the scheme $H_{z,y,p} (2H)$ to the
Hilbert flag scheme $D_{z,y,p} (H)$ which parametrizes triples of flat
families $Z \subseteq Y \subseteq P $ in $H \times S$, flat over
$S$, as above, whose fibres $Z_{s}$ are locally complete intersection
of length $z$, and $Y_{s}$ and $P_{s}$ are curves of degree $y$ and
$p$ respectively. The fibres of $H_{z,y,p} (2H)$ over $D_{z,y,p} (H)$
are open subsets of the vector spaces
$\mbox{H}^{0} (Y_{s}, \scrl_{s} (p))$, which have constant dimension, and
thus $H_{z,y,p} (2H)$ appears as an open subscheme of a geometric
vector bundle over $D_{z,y,p} (H)$.
The functorial details to justify all this are standard, if rather
lengthy, so we leave them to the reader and content ourselves with
summarizing the results in the following proposition.
\bp \label{vect}
The Hilbert scheme $H_{d,g} (2H)$ of curves in $2H$ of degree $d$ and
genus $g$ is stratified by locally closed subschemes $H_{z,y,p} (2H)$
corresponding to families of curves whose integers $(z,y,p)$
associated by \ref{1.1} are constant.
The scheme $H_{z,y,p}$ has a natural map to the Hilbert flag scheme
$D_{z,y,p}(H)$ of triples $\{Z,Y,P \}$ as in~\ref{1.1}, and this map makes
$H_{z,y,p} (2H)$ into an open subset of a geometric vector bundle
$\cae$ over $D_{z,y,p} (H)$, where $\cae$ is locally free of rank
$$z + (p-1)y + 1 - \frac{1}{2} (y-1)(y-2).$$
\ep
Next, we study the Hilbert flag scheme $D_{z,y,p} (H)$.
\bp
Given integers $z$,$y$,$p$ with $z \geq 0$, $p \geq y \geq 1$, the
Hilbert flag scheme $\bar{D}_{z,y,p} (H)$ of closed subschemes
$Z \subseteq Y \subseteq P $ in $H$, with $Z$ of dimension zero and
length $z$, and $Y$, $P$ curves of degree $y$,$p$ respectively, is
irreducible and generically smooth of dimension
$$
z + \frac{1}{2} \, y (y+3) + \frac{1}{2} \, (p-y) (p-y+3).
$$
\ep
\begin{proof}
Since $Y \subseteq P$, we can write $P=Y+W$, where $W$ is an effective
divisor, i.e. a curve, of degree $p-y$, which can be chosen
independently of $Z$,$Y$. So
$$\bar{D}_{z,y,p}(H) = D_{z,y} (H) \times D_{p-y}(H), $$ where $D_{p-y}$ is
a projective space of dimension $\frac{1}{2} (p-y)(p-y+3)$
parametrizing curves of degree $p-y$. Thus we reduce to considering
$D_{z,y} (H)$. This is the non-trivial part, which was proved by Brun
and Hirschowitz \cite{brun-hirsch}, proposition 3.2. The main ideas of
their proof are as follows.
We regard $D_{z,y}$ as a closed subscheme
of $M \times K$, where $M= Hilb^{z} (H)$ is the scheme of zero
dimensional closed subschemes of $H$ of length $z$, and $K$ is a
projective space of dimension $\frac{1}{2} y (y+3)$ parametrizing
curves of degree $y$ in $H$. Let $Z$ and $Y$ denote the universal
families over $M$ and $K$ respectively. It is a theorem of Fogarty \cite{hl},
example 4.5.10,
that $M$ is smooth irreducible of dimension $2z$.
Brun and Hirschowitz first show that
$D_{z,y}$ is the zero scheme of a section of the rank $z$ vector
bundle $\Hom (\ideal{Y} (y), \coo_{Z} (y))$ on $M \times K$, and so
has codimension at most $z$ at every point. Thus the dimension of
$D_{z,y}$ is at least $z + \frac{1}{2} y (y+3)$ at every point. Next,
observe that $D_{z,y}$ contains an open subset $U$ corresponding to
divisors of degree $z$ on smooth curves, that is smooth irreducible of
dimension $z +\frac{1}{2} y(y+3)$. To complete the proof, they use the
theorem of Brian{\c{c}}on \cite{brian} that the punctual Hilbert
scheme of zero dimensional schemes of length $z$ at a point has
dimension $z-1$. This shows that the fibre of $D_{z,y}$ over any curve
$Y$ has total dimension at most $z$. Thus the dimension of $D_{z,y}
\setminus U$ must be strictly less than $z+ \frac{1}{2} y (y+3)$, and
hence it is contained in the closure of $U$.
\end{proof}
\bc \label{dim}
Suppose $z \geq 0$ and $p \geq y \geq 1$, or $z=y=0$ and $p \geq 1$.
Then the scheme $H_{z,y,p} (2H)$ is
irreducible and generically smooth of dimension
$$ 2z + \frac{1}{2} y (y+1)+\frac{1}{2} p (p+3).$$
\ec
\section{Irreducible components of $H_{d,g} (2H)$} \label{fours}
We can now describe the irreducible components of $H_{d,g} (2H)$.
\bt \label{components}
Let $d$ and $g$ be integers, with $d \geq 1$.
$H_{d,g} (2H)$ is non-empty if and only if either
$g = \frac{1}{2} (d-1)(d-2)$, or $d \geq 2$ and
$g \leq \frac{1}{2} (d-2)(d-3)$.
The irreducible components of $H_{d,g}$ are the closures
$\bar{H}_{z,y,p}$ of the subschemes $H_{z,y,p}$ defined in
section~\ref{four}, where $(z,y,p)$ varies in the set
of triples of nonnegative integers satisfying the following conditions:
$p \geq 1$, $p \geq y$ ,$z=0$ if $y=0$, and
\begin{align*}
p &= d-y \\
z &= \frac{1}{2} (y-1)(y-2) + \frac{1}{2} (p-1)(p-2) +y-g-1.
\end{align*}
\et
\begin{proof}
By corollary \ref{dim}, $\bar{H}_{z,y,p}$ is
irreducible of dimension $2z + \frac{1}{2} y (y+1)+\frac{1}{2} p (p+3)$.
To see
that $\bar{H}_{z,y,p}$ is an irreducible component, it is enough to
observe that its generic point is not contained in a different
$\bar{H}_{z',y',p'}$. Now this is clear, since the support of the
generic point of $\bar{H}_{z,y,p}$ is the union of two smooth plane
curves of degrees $y$ and $p-y$ respectively.
By propositions~\ref{1.1} and \ref{2.1}, as a topological space
$H_{d,g}$ is the disjoint union of its subschemes $H_{z,y,p}$, where
$z$,$y$,$p$ are nonnegative integers satisfying
$p \geq 1$, $p \geq y$, $z=0$ if $y=0$, and
\begin{align*}
p &= d-y \\
z &= \frac{1}{2} (y-1)(y-2) + \frac{1}{2} (p-1)(p-2) +y-g-1.
\end{align*}
It remains to determine the triples $(z,y,p)$ satisfying these
conditions. We must have
$$z=\frac{1}{2} (d-2)(d-3)-g - (y-1)(d-y-2).$$
We now impose the conditions on $z$, $y$ and $p$. The conditions
$p \geq y$ and $p \geq 1$ translate into $y \leq d/2$ and $d-y -1
\geq 0$. As $y$ and $z$ must be nonnegative, we have
$$\frac{1}{2} (d-1)(d-2) -g = y(d-y-1)+z \geq 0$$
with equality if and only if $y=z=0$ or $z=0$, $y=1$ and
$d=2$. In all other cases we have $y \geq 1$ and either $d-y >1$, or
$y=1$,$d=2$ and $z \geq 1$, hence
$$\frac{1}{2} (d-2)(d-3) -g = (y-1)(d-y-2)+z \geq 0.$$
\end{proof}
\br \label{rcomponents}
How many irreducible components does $H_{d,g}$ have ?
If $d \neq 2$ and $g=1/2(d-1)(d-2)$, $H_{d,g}=H_{0,0,d}$ is
irreducible. $H_{2,0}$ has two irreducible components, namely
$\bar{H}_{0,1,1}$ and $\bar{H}_{0,0,2}$.
Suppose $d \geq 2$ and $g \leq 1/2 (d-2)(d-3)$ with $(d,g) \neq
(2,0)$. Let $y_{M}$ be largest integer $n$ in
the closed interval $[1,d/2]$ such that $(n-1)(d-n-2) \leq
1/2 (d-2) (d-3) - g$. The irreducible components of $H_{d,g}$
are in one to one correspondence with the integers $y$ in the closed
interval $[1,y_{M}]$.
\er
\section{Numerical invariants} \label{five}
Out of the exact sequence $(\ref{first})$ one immediately computes the
postulation of the curve $C$ in terms of the two integers $y$ and $p$
and of the postulation of $Z$. However to compute the dimensions of
the first and second cohomology modules of $\ideal{C} (n)$ one needs
some additional argument. For example, one could use the results on
liaison of the next section. Here we take a different approach.
Choose a point $R$ of $\pso \setminus H$, and let $\pi: 2H \rrr H$ be
the morphism induced by the projection from $R$. $\pi$ is a finite and
flat morphism, and $\pi_{*} (\coo_{2H}) = \coo_{H} \oplus \coo_{H} (-1)$.
Now given any other point $R'$ in $\pso \setminus H$, there is an
automorphism of $\pso$ which sends $R$ to $R'$ and fixes every point
of $H$. It follows that for any coherent sheaf of $\coo_{2H}$-modules
$\caf$, the isomorphism class of the sheaf $\pi_{*} \caf$ is
independent of the choice of the point $R$.
\bp \label{cae}
Let $\pi: 2H \rrr H$ be
the morphism induced by the projection from $R$.
Let $C$ be a curve in $2H$. Then
$\cae = \pi_{*} (\ideal{C,2H})$ is a locally free $\coo_{H}$-module
of rank two. Furthermore, the Rao module
\begin{em}
$\mbox{H}^{1}_{*} (\pso, \ideal{C})$
\end{em}
is isomorphic to
\begin{em}
$\mbox{H}^{1}_{*} (H, \cae)$
\end{em} as a module over the
homogeneous coordinate ring of $\pso$.
\ep
\begin{proof}
We use the fact proven in \ref{1.1} that the sheaf of
$\coo_{Y}$-modules $\scrl$,
defined as the quotient $\ideal{P} / \ideal{C}$ has depth $\geq
1$ at closed points. It follows from this and the exact sequence
$$
0 \rrr \pi_{*} (\ideal{C,2H}) \rrr
\pi_{*} (\ideal{P,2H}) \rrr \scrl \rrr 0
$$
that $\pi_{*} (\ideal{C,2H})$ is locally free of rank two provided
$\pi_{*} (\ideal{P,2H})$ is locally free of rank two . But this is
clear as we have an exact sequence
$$
0 \rrr \coo_{H}(-1) \rrr
\pi_{*} (\ideal{P,2H}) \rrr \ideal{P,H} \cong \coo_{H} (-p) \rrr 0.
$$
Finally, from the exact sequence $(\ref{first})$
we see that the Rao module of $C$ is annihilated by the equation of
$H$, hence it is isomorphic to
$\mbox{H}^{1}_{*} (H, \pi_{*} (\ideal{C,2H}) )$ as a modules over
$\mbox{H}^{0}_{*} (\pso, \coo_{\pso} )$.
\end{proof}
\bc
Let $C$ be a curve of degree $d$ in the double plane, with Rao module
$M_{C}$. Let $M_{C}^{*}$ denote the $k$-dual module to $M_{C}$ (see
\cite{MDP}, 0.1.7 p.20). Then $M_{C}^{*} \cong M_{C} (d-2)$, and in
particular $h^{1} (\ideal{C} (n))=h^{1} (\ideal{C} (d-2-n)$ for all
integers $n$.
\ec
\begin{proof}
Let $M=M_{C} \cong \mbox{H}^{1}_{*} (H, \cae)$. Since $\cae$ is a rank
two vector bundle on $H$, we have $\dual{\cae} \isom \cae (-c_{1})$
where $c_{1}$ denotes the first Chern class of $\cae$. By Serre
duality we have $M^{*} \cong M(-c_{1}-3)$. Now from the exact
sequence
\begin{equation} \label{caeseq}
\exact{\coo_{H} (-y-1)}{\cae}{\ideal{Z,H}(-p)}.
\end{equation}
we compute $c_{1} = -p-y-1=-d-1$, and we are done.
\end{proof}
\br
If we apply $\pi_{*}$ to the exact sequence $(\ref{first})$, we obtain
an extension class in $\mbox{Ext}^{1}_{H} (\ideal{Z,H}
(-p),\coo_{H}(-y-1))$, hence an element $w$ in $\mbox{H}^{0}
(\omega_{Z} (2+p-y))$. In the notation of section~\ref{three}, one
verifies that $w$ is the image of the global section $v$ of $\scrl
(p)$ corresponding to $C$, under the map
$\mbox{H}^{0} (Y, \scrl (p)) \rrr \mbox{H}^{0}
(Y, \omega_{Z} (2+p-y))$ coming from diagram~\ref{third}.
\er
\br
We may ask which locally free sheaves $\cae$ arise from this
construction. Let us say that a locally free sheaf $\caf$ is normalized if
it has a global section, while $\caf (-1)$ has no section. Let $\caf$
be a locally free normalized $\coo_{H}$-module of rank two. Then one
may check that there exists a curve $C \subset 2H$ such that
$\pi_{*} (\ideal{C,2H})$ is
isomorphic to $\caf(n)$ for some integer $n$, if and only if $c_{1}
\caf \leq 1$. In particular, all unstable locally free sheaves of rank
two arise from this construction.
\er
\bc \label{cohomology}
Let $C$ be a curve in $2H$ with $T(C)= \{Z,Y,P\}$. Then
\begin{align*}
h^{0} ( \ideal{C} (n)) &=
h^{0} ( \coo_{\pso} (n-2)) +
h^{0} ( \coo_{H} (n-y-1)) +\\
& + h^{0} ( \ideal{Z,H} (n-p))
\end{align*}
and
\begin{align*}
h^{2} (\ideal{C} (n)) & =
h^{0} (\coo_{\pso} (-n-4)) - h^{0} ( \coo_{\pso} (-n-2)) + \\
& + h^{0} (\coo_{H} (p-3-n)) +
h^{0} (\ideal{Z,H} (y-2-n)).
\end{align*}
\ec
\begin{proof}
The formula for the postulation follows immediately from the exact
sequence $(\ref{first})$. To obtain the formula for $h^{2}$ we
compute $h^{2} \cae (n)$ using Serre duality:
$$
h^{2} \cae (n) = h^{0} \cae (p+y-2-n) = h^{0} \coo_{H} (p-3-n) +
h^{0} \ideal{Z,H} (y-2-n).
$$
\end{proof}
We can also determine the postulation character of a curve $C$ in
$2H$. Recall \cite{MDP} for any numerical function $f(n)$ one defines
the difference function $\df f(n) = f(n) - f(n-1)$. If $C$ is a curve
in $\pso$, one defines its {\em postulation character} $\gmc$ by
$$
\gmc (n) = \df^{3} ( h^{0} ( \ideal{C} (n)) - h^{0} ( \coo_{\pso} (n)) ).
$$
Similarly, for a zero-dimensional closed subscheme $Z \subset \ptwo$
we define its postulation character
$$
\gmz (n) = \df^{2} ( h^{0} ( \ideal{Z} (n)) - h^{0} ( \coo_{\ptwo} (n)) ).
$$
By analogy with the case of ACM curves \cite{MDP}, I.2 and V.1.3, one
shows easily the following:
\bp \label{gamma}
Let $Z$ be a zero dimensional closed subscheme of $\ptwo$ of degree
$z$, and let $s$ be the least degree of a curve containing $Z$. Then
$$
\gmz (n)
= \left\{
\begin{array}{ll}
0 & \mbox{ if \;\;\; $ n < 0$},
\\
-1
&
\mbox{ if \;\;\; $ 0 \leq n < s$},
\\
a_{n} \geq 0
&
\mbox{ if \;\;\; $ n \geq s$}.
\end{array}
\right.
$$
Furthermore, $\gmz$ is a character, so $\sum_{n \geq s} a_{n} = s$,
and we can determine the degree z by
$$
z= \sum_{n \geq s} n a_{n} - \frac{1}{2} s (s-1).
$$
Conversely, given integers $s \geq 1$ and $a_{n} \geq 0$ for $n \geq
s$ there exists a reduced zero-dimensional closed subscheme $Z \subset \ptwo$
with postulation character as above.
\ep
Now using (\ref{cohomology}) we can compute the postulation character
of $C$. We find
\bt \label{gammac}
Let $C$ be a curve in $2H$ with associated triple $\{Z,Y,P \}$. Then
there is an integer $s$ with $0 \leq s \leq y$ and there are integers
$b_{n}$ for $n \geq p+s$ such that $\sum b_{n} = s$, and the
postulation character $\gmc$ is the sum of the following functions:
$-1$ in degree $0$, $-1$ in degree $1$, $1$ in degree $y+1$, $1$ in
degree $p+s$, and $\df \beta$, where
$$
\beta (n) = \left\{
\begin{array}{ll}
0 & \mbox{ if \;\;\; $ n < p+s$}
\\
b_{n}
&
\mbox{ if \;\;\; $ n \geq p+s$}.
\end{array}
\right.
$$
Conversely, given integers $0 \leq s \leq y \leq p$ and $b_{n} \geq 0$
for $n \geq p+s$ with $\sum b_{n} =s$, there exists a curve $C$ in
$2H$ with postulation character as described.
\et
\br
In the notation of (\ref{gammac}), the degree $z$ of $Z$ is determined
by
$$
z = \sum_{n \geq p+s} (n-p) b_{n} - \frac{1}{2} s (s-1).
$$
\er
\br
One can see easily that the possible postulation characters of curves
on surfaces of degree $2$ other than $2H$ form a subset of the
postulation characters described in (\ref{gammac}). Thus we have found
all postulation characters of curves in surfaces of degree $2$. This
gives some hope that one day it will be possible to determine all
possible postulation characters of curves in $\pso$.
\er
\section{Liaison} \label{six}
In this section we use notation and terminology of \cite{hgd}, section
4.
The following proposition describes the behaviour of the triple
$T(C) = \{Z,Y,P\}$ under liaison.
\bp
Let $C$ be a curve in $2H$ with $T(C)= \{Z,Y,P\}$.
Suppose $S$ is a surface containing $C$ and meeting $H$ properly.
Let $Q= H \cap S$. Then the curve $D$ linked to $C$ by the complete
intersection $2H \cap S$ has triple $T(D)=\{Z,Q-P,Q-Y\}$; in particular,
$Z \subset Q-P$.
Conversely, if $Q \subset H$ is a curve containing $P$ such that
$Z \subset Q-P$, then $C$ is linked by the complete intersection of
$2H$ with some surface $S$ to a curve $D$ with $T(D)=\{Z,Q-P,Q-Y\}$.
In particular, $Z$ and $P-Y$ are
invariant under liaison on $2H$.
\ep
\begin{proof}
Suppose first $C$ is linked to $D$ by the complete intersection $E= 2H
\cap S$, and let $T(D)=\{Z',Y',P' \}$. Let $Q$ be the intersection of
$S$ with the reduced plane $H$. We claim that $Z'=Z$, $Y'=Q-P$ and
$P'=Q-Y$.
By \cite{hgd} page 317 the ideal sheaf of $D$ in $2H$ is
$\Hom_{\coo_{2H}} ( \ideal{C,2H}, \ideal{E,2H})$ with its natural
embedding in $\Hom_{\coo_{2H}} ( \ideal{E,2H}, \ideal{E,2H}) \cong
\coo_{2H}$. By \cite{AG} exercises III 6.10 and 7.2, applying the functor
$\Hom_{\coo_{2H}} ( -, \ideal{E,2H})$ to the exact sequence
$$
0 \rrr
\ideal{Y,H}(-1) \stackrel{h}{\rrr} \ideal{C,2H} \rrr
\ideal{Z,H} (-P) \rrr 0
$$
we obtain a long exact sequence
$$
0 \rrr
\ideal{Q-P,H}(-1) \stackrel{h}{\rrr} \ideal{D,2H} \rrr
\ideal{Q-Y,H} \rrr \omega_{Z} (p-q+2) \rrr 0.
$$
As $\omega_{Z} (p-q+2) \cong \coo_{Z} (y-q)$, the kernel of the last
map is $ \ideal{Z,H}(Y-Q)$, and our claim follows.
Conversely, suppose that $Q$ is a curve in $H$ containing $P$, and
such that $Z \subset Q-P$. The exact sequence~(\ref{first}) tells us that
there exists a surface $S$ containing $C$ but not $H$, whose
intersection with $H$ is $Q$. By what we have just shown,
$T(D)=\{Z,Q-P,Q-Y\}$.
\end{proof}
\bc
Let $C$ be a curve in $2H$ with $T(C)= \{Z,Y,P\}$, and let $Y'$ be a
curve in $H$ containing $Z$. Let $y$ and $y'$ be the degrees of $Y$
and $Y'$ respectively. There is curve $D$ obtained from $C$
by an elementary biliaison of height $y'-y$ on $2H$ with $T(D)=
\{Z, Y',Y'+P-Y \}$.
\ec
\begin{proof}
Use the above proposition with $Q=P+Y'=(Y'+P-Y) + Y$.
\end{proof}
\bc
Let $C$ be a curve in $2H$ with $T(C)= \{Z,Y,P\}$. Suppose that $C$ is
not arithmetically Cohen-Macaulay, that is, $Z$ is not empty. Then $C$
is minimal in its biliaison class if and only if $Y$ has minimal
degree among curves in $H$ containing $Z$.
\ec
\begin{proof}
If $h^{0} (H, \ideal{Z,H} (\deg Y -1) > 0$, by the previous corollary
there is a curve obtained from $C$ by an elementary biliaison of
negative height, hence $C$ is not minimal.
If $h^{0} (H, \ideal{Z,H} (\deg Y -1) = 0$, then by
corollary~\ref{cohomology} we have
$$h^{2} (\pso, \ideal{C} (1)) - 2 h^{2} (\pso, \ideal{C} ) +
h^{2} (\pso, \ideal{C} (-1) ) \leq 1.$$ Now it follows from \cite{MDP}
Proposition~III.3.5 that $C$ is minimal: see \cite{sch-tran} Corollary~4.4.
\end{proof}
\section{Connectedness of the Hilbert scheme} \label{seven}
\bt \label{connect}
The Hilbert scheme $H_{d,g} (2H)$ is connected.
\et
\begin{proof}
If $g=\frac{1}{2} (d-1)(d-2)$ and $d\neq 2$, $H_{d,g}$ is irreducible by
theorem~\ref{components}.
To handle the case $d=2$ and $g=0$, we fix homogeneous
coordinates $[x:y:z:w]$ on $\pso$, so that $x=0$ is an equation for
$H$, and we look at the family of curves $C_{t}$ in $\pso \times Spec \,
k[t]$ defined by the global ideal
$$
I = < x^{2}, xy, y^{2}, x+ty >.
$$
For $t\neq 0$, $C_{t}$ is in $H_{0,1,1}$, while $C_{0}$ belongs to
$H_{0,0,2}$. It now follows from remark~\ref{rcomponents} that
$H_{2,0}$ is connected.
If $g < \frac{1}{2}(d-1)(d-2)$, by remark~\ref{rcomponents} we have set
theoretically
$$
H_{d,g} = \bigcup_{1 \leq y \leq y_{M}} H_{z_{y},y,d-y}
$$
where
$$
z_{y} = \frac{1}{2} (d-2) (d-3) - g - (y-1)(d-y-2).
$$
In particular, $H_{d,g}$ is irreducible for $d \leq 3$, and for
$d \geq 4$ the theorem is a consequence of the following
proposition.
\end{proof}
\bp \label{pconnect}
If $y \geq 2$, $p \geq y$ and $r \geq 0$,
there is a curve in $H_{r,y,p}$
specializing to one in $H_{r+p-y,y-1,p+1}$.
\ep
\begin{proof}
Suppose the claim is true when $y=2$. Then
by adding $y-2$ times a plane section (i.e. by performimg an
elementary biliaison of height $y-2$ on $2H$, see \cite{hmdp1}
proposition~1.6) we find that the claim is true for all $y \geq 2$,
and we are done.
We now construct a family of curves in $H_{r,2,p}$
specializing to one in $H_{r+p-2,1,p+1}$. This has already been done
by Nollet in \cite{nthree}, example 3.10, in the case $p=2$ and $r=1$,
and his construction, generalizes without any major
modification (Nollet noticed this independently \cite{nolletp}).
Here are the details.
As above we fix homogeneous coordinates $[x:y:z:w]$ on $\pso$, so that
$x=0$ is
an equation for $H$. We let $Y$ and $P$ denote respectively the double
line $x=y^{2}=0$ and the multiple line $x=y^{p}=0$. Let $Z$ have equations
$x=y=f=0$ where $f$ is a form of degree $r = \deg Z$ in $k[z,w]$.
To give a curve $C$ in $2H$ with $T(C) = \{Z,Y,P \}$ is by
proposition~\ref{2.1} the same as giving a morphism $v :
\ideal{Z,H} (-p) \rrr \coo_{Y} (-1)$ whose image in in $\mbox{H}^{0}
(\omega_{Z} (p))$ generates $\omega_{Z} (p)$ at every point. This
amounts to choosing forms $s$ and $g$ in $k[z,w]$, of degrees $p-1$ and
$r+p-2$ respectively, and such that $g$ and $f$ have no common zeros
on the line $x=y=0$. The corresponding morphism $\ideal{Z,H} (-p) \rrr
\coo_{Y} (-1)$ sends $y$ to $ys$ and $f$ to $fs+yg$. For the corresponding
curve $C$ we have:
$$
I_{C} = \,\, < x^{2}, xy^{2}, y^{p+1}+xys, xfs+xyg+y^{p}f >.
$$
Now we look at the family of curves in $\pso \times Spec \, k[t]$
obtained by flattening the ideal generated by $x^{2}, A, B ,C$ where
\begin{align*}
A & = xy^{2} \\
B & = ty^{p+1}-xyz^{p-1} \\
C & = xyw^{r+p-2}-tz^{r} (ty^{p}-xz^{p-1}). \\
\end{align*}
For $t \neq 0$ we obtain a curve $C$ as above with $s=-\frac{1}{t} z^{p-1}$,
$g=w^{r+p-2}$ and $f =-t^{2} z^{r}$. To see what happens at $t=0$, we
set
\begin{align*}
D & = \frac{1}{t} (w^{r+p-2} B+z^{p-1}C )=y^{p+1}w^{r+p-2}+xz^{r+2p-2}+tF \\
E & = \frac{1}{t} (z^{p-1} A + y B) = y^{p+2}. \\
\end{align*}
It follows that the ideal of the limit scheme $C_{0}$ contains the
ideal
$$
J= \,\, < x^{2}, xy^{2}, xyz^{p-1}, xyw^{r+p-2},y^{p+2},
y^{p+1}w^{r+p-2}+xz^{r+2p-2} >.
$$
The saturation of $J$ is the ideal
$$
I = \,\,< x^{2}, xy ,y^{p+2}, y^{p+1}w^{r+p-2}+xz^{r+2p-2} >.
$$
But this is the homogeneous ideal of a curve $D$ in the double plane:
$Y(D)$ is the line $x=y=0$, $P(D)$ has equations $x=y^{p+1}=0$, and
$Z(D)$ is defined on $Y(D)$ by the equation $w^{r+p-2}=0$. Hence $D$
belongs to $H_{r+p-2,1,p+1}$. In particular, $D$ has the same degree
and genus as $C_{0}$. So we must have $D=C_{0}$, and this finishes the
proof.
We remark that in this family the zero dimensional scheme $Z$
associated to $C_{t}$ is supported at the point $[0:0:1:0]$ for
$t \neq 0$, and at the point $[0:0:0:1]$ for $t=0$ !
\end{proof}
\section{Extremal and subextremal curves}
Given a curve $C$, the function $\rho_{C}: \Z \rrr \Z$ defined by
$\rho_{C} (n) =h^{1} \ideal{C} (n)$
is called the Rao function of $C$. It is the Hilbert function of the
Rao module $M_{C} = \mbox{H}^{1}_{*} (\pso, \ideal{C})$
of $C$.
\bt[\cite{bounds},\cite{subextremal}] \label{bounds}
Let $C \subset \pso$ be a curve of degree $d$ and arithmetic
genus $g$. Then
\begin{enumerate}
\item
$C$ is planar if and only if $g = \frac{1}{2} (d-1)(d-2)$.
\item
If $C$ is not planar, then $d \geq 2$, $g \leq \frac{1}{2} (d-2)(d-3)$
and
$$
\rho_{C} (n) \leq \rho^{E}_{d,g} (n) \;\;\; \mbox{for all $n \in \Z$}
$$
where
$$
\rho^{E}_{d,g} (n) =
\left\{
\begin{array}{ll}
\mbox{\em max} (0, \rho^{E}_{d,g} (n+1)-1) &
\mbox{ if \;\;\; $ n \leq -1$},
\\
\frac{1}{2} (d-2)(d-3) - g
&
\mbox{ if \;\;\; $ 0 \leq n \leq d-2$},
\\
\mbox{\em max} (0, \rho^{E}_{d,g} (n-1)-1)
&
\mbox{ if \;\;\; $ n \geq d-1$}.
\end{array}
\right.
$$
\end{enumerate}
\et
\bd
A curve $C \subset \pso$ of degree $d$ and genus $g$ is called {\em
extremal} if
it is not planar and $\rho_{C} = \rho^{E}_{d,g}$.
\ed
\br
The definition of an
extremal curve in \cite{extremal},
\cite{subextremal} is different from ours, since
they require the curve not to be ACM, but we allow the
ACM case if $\rho^E_{d,g} =0$.
\er
\br
There exist extremal curves (for example in $2H$, see below) if $d=2$
and $g \leq -1$ and if $d \geq 3$ and $g \leq \frac{1}{2} (d-2)(d-3)$.
\er
The following characterization of extremal curves is essentially
contained in \cite{extremal} and \cite{subextremal}.
\bp \label{extremal}
Let $C$ be a curve of degree $d \geq 2$ and genus $g$.
The following are equivalent:
\begin{enumerate}
\item
$C$ is extremal;
\item
either $C$ is a minimal curve contained in the union of two distinct
planes, or $C$ is obtained from a plane curve by an elementary
biliaison of height one on a quadric surface;
\item
either $C$ is ACM with $(d,g) \in \{ (3,0),(4,1) \} $ and $C$ is
contained in an integral quadric surface, or $C$ is a non-planar
curve which contains a plane curve of degree $d-1$.
\end{enumerate}
\ep
\begin{proof}
Looking at \cite{bounds}, \cite{subextremal} one sees that $C$ is
extremal if and only if it is not planar and $h^{0} \ideal{C} (2) \geq
2$. From this we deduce that 1 implies 3. Curves $C$ in the union of two
distinct planes $H_1 \cup H_2$ are studied in \cite{hgd}, section 5:
if $C \cap H_{1}$ has degree $d-1$, then $C$ is either minimal or
ACM. From this we see that 3 implies 2.
Finally, we prove 2 implies 1. If $C$ is obtained from a plane curve
by an elementary biliaison of height one on a quadric surface, then
one computes $g = \frac{1}{2} (d-2)(d-3)$ and so $C$ is extremal.
If $C$ is a minimal curve in $H_1 \cup H_2$, then by \cite{hgd},
section 5, adding a suitable number
(namely, $\frac{1}{2}(d-2)(d-3)-g-1 $) of plane sections to $C$ we
obtain a curve linearly equivalent to the
disjoint union of two plane curves. Thus the Rao module of $C$ is
isomorphic to $R/(h,k,f,g) \, (a-1)$ where $a=\frac{1}{2} (d-2)(d-3)-g$,
$R$ is the homogeneous coordinate ring of $\pso$, $h$, $k$ are the
equations of $H_{1}$ and $H_{2}$, and $f$,$g$ are forms of degrees $a$,
$d+a-2$ respectively, having no common zeros along the line
$h=k=0$. Hence $\rho_{C} = \rho^{E}_{d,g}$.
\end{proof}
\bt[\cite{subextremal}] \label{subbounds}
Let $C \subset \pso$ be a curve of degree $d$ and genus $g$, which is
neither planar nor extremal. Then $d \geq 3$ and
\begin{enumerate}
\item
$g \leq \frac{1}{2} (d-3)(d-4) + 1$; if equality holds, then $d \geq
5$ and $C$ is ACM;
\item
if $d \geq 4$, then
$$
\rho_{C} (n) \leq \rho^{S}_{d,g} (n) \;\;\; \mbox{for all $n \in \Z$}
$$
where
$$
\rho^{S}_{d,g} (n) =
\left\{
\begin{array}{ll}
\mbox{\em max} (0, \rho^{S}_{d,g} (n+1)-1) &
\mbox{ if \;\;\; $ n \leq 0$},
\\
\frac{1}{2} (d-3)(d-4) + 1 - g
&
\mbox{ if \;\;\; $ 1 \leq n \leq d-3$},
\\
\mbox{\em max} (0, \rho^{S}_{d,g} (n-1)-1)
&
\mbox{ if \;\;\; $ n \geq d-2$}.
\end{array}
\right.
$$
\end{enumerate}
\et
\bd
A curve $C \subset \pso$ of degree $d \geq 4$ and genus $g$ is called
{\em subextremal} if it is neither planar nor extremal and $\rho_{C} =
\rho^{S}_{d,g}$. Again, this differs from the terminology of
\cite{subextremal} in that we include among subetxremal curves those
ACM curves of degree $d \geq 5$ which have genus $1/2 (d-3)(d-4) +1$.
\ed
\br
There exist subextremal curves (for example in $2H$, see below) if $d=4$
and $g \leq 0$ and
if $d \geq 5$ and $g \leq \frac{1}{2} (d-3)(d-4)+1$.
\er
\bp \label{subextremal}
Let $C \subset \pso$ be a curve of degree $d \geq 4$ and genus $g$.
The following are equivalent:
\begin{enumerate}
\item
$C$ is subextremal;
\item
$C$ is obtained from an extremal curve by an elementary biliaison of
height one on a quadric surface;
\item
either $C$ is ACM and $(d,g) \in \{(5,2),(6,4) \}$, or $C$ is a
divisor of type $(1,3)$ on a smooth quadric surface, or there is a plane $H$
such that $\ideal{C \cap H,H} = \ideal{Z,H} (2-d)$ with $Z$ zero-dimensional
contained in a line, and the residual scheme $Res_{H} (C)$ to the
intersection of $C$ with $H$ is a plane curve of degree two.
\end{enumerate}
\ep
\begin{proof}
If $C$ is not ACM, the equivalence of $1$ and $2$ is the content of
Theorem~2.14 in \cite{subextremal}. The same proof works for the ACM case
as well (cf. \cite{subextremal}, Lemma~2.5).
We claim that $2$ implies $3$: suppose $C$ is obtained from an
extremal curve $B$ by an elementary biliaison of height one on a
quadric surface $F$.
If there is a plane $H$ whose intersection with $B$ has degree $d-3 =
\deg B -1 $ and if $F$ contains $H$ as a component, then
$\ideal{C \cap H,H} = \ideal{Z,H} (2-d)$ with $Z$ zero-dimensional
contained in a line, and $Res_{H} (C)$ is a plane curve.
Otherwise, by proposition~\ref{extremal} either $B$ is an ACM curve with
$(d,g) \in \{(3,0),(4,1) \}$, or $B$ is a divisor of type $(0,2)$ on
the smooth quadric surface $F$. So $C$ is either ACM with
$(d,g) \in \{(5,2),(6,4) \}$, or $C$ is a divisor of type $(1,3)$ on
$F$.
Finally, we show that $3$ implies $1$. The special cases are clear, so
we may assume that there is a plane $H$
such that $\ideal{C \cap H,H} = \ideal{Z,H} (2-d)$ with $Z$ zero-dimensional
contained in a line, and $Y = Res_{H} (C)$ is a plane curve of degree two.
Then there is an exact sequence:
\begin{equation} \label{last}
0 \rrr
\ideal{Y,\pso}(-1) \stackrel{h}{\rrr} \ideal{C,\pso} \rrr
\ideal{Z,H}(2-d) \rrr 0
\end{equation}
from which we deduce that $ \mbox{length} \, Z
=\frac{1}{2}(d-3)(d-4)+1-g$, that $C$ is contained in a unique quadric
surface and that $\rho_{C} (n) = \rho^{S}_{d,g} (n)$ for $n \geq
1$. Since the Rao function of a curve of degree $d$ contained in a
quadric surface is symmetric around $\frac{d-2}{2}$, $C$ is subextremal.
\end{proof}
As a corollary, we now identify extremal and subextremal curves in
a smooth quadric surface and in a double plane, leaving the case of
the quadric cone and of the union of two distinct planes to the reader.
\bc
Let $C$ be an effective divisor of type $(a,b)$ on the smooth quadric surface
$Q$, with $a \leq b$. Then
\begin{enumerate}
\item
$C$ is planar if and only if $(a,b) \in \{(0,1), (1,1) \}$;
\item
$C$ is extremal if and only if $(a,b) \in \{(0,2), (1,2), (2,2) \}$;
\item
$C$ is subextremal if and only if $(a,b) \in \{ (1,3), (2,3), (3,3) \}$.
\end{enumerate}
\ec
\bc
Let $C$ be a curve in the double plane $2H$ with associated triple
$\{Z,Y,P\}$. Then
\begin{enumerate}
\item
$C$ is planar if and only if either $y=0$, or $p=y=1$ and $z=0$;
\item
$C$ is extremal if and only if either $y=1$ and $p \geq 2$, or $y=p=1$ and
$z \geq 1$, or $y=p=2$ and $z=0$;
\item
$C$ is subextremal if and only if $Z$ is contained in a line and
either $y=2$, $p \geq 3$, or $y=p=2$ and $z \geq 1$, or $y=p=3$ and $z=0$.
\end{enumerate}
\ec
The following proposition has been proven independently by Nollet
\cite{nolletp}; see also \cite{samir}, especially Proposition~4.15.
\bp
Let $H$ be a plane in $\pso$.
For every $d \geq 5$ and $g \leq \frac{1}{2} (d-3)(d-4)+1$
(resp. $d=4$ and $g \leq 0$) the closure of the family of subextremal curves
in $H_{d,g} (2H)$ contains an extremal curve. In particular, for $d
\geq 4$ and $g \geq \frac{1}{2} (d-3)(d-4)+1$ the Hilbert scheme
$H_{d,g} (\pso)$ is connected.
\ep
\begin{proof}
The first statement follows from proposition~\ref{pconnect}. For $d
\geq 4$, $g > \frac{1}{2} (d-3)(d-4)+1$ and for $(d,g)=(4,1)$ the
Hilbert scheme $H_{d,g} (\pso)$ is irreducible \cite{extremal},
section 5.
For $d \geq 5$ and $g = \frac{1}{2} (d-3)(d-4)+1$, $H_{d,g} (\pso)$
has two irreducible components, namely the closure of the family of
subextremal curves and the family of extremal curves \cite{samir}.
Hence it is connected by the first statement.
\end{proof}
|
2,869,038,154,848 | arxiv | \section{Introduction}
Many-body phenomena in the strong coupling regime of light--matter
interactions, in particular Bose-Einstein condensation, have attracted
considerable attention in recent
years\cite{deng10a,kavokin_book11a}. The photon, one of the intrinsic
components of the polariton, has an inevitable interaction with the
environment, which leads to its decay and often affects the dynamic in
a non-negligible way. A well--studied example of non--equilibrium
quantum transition takes place in microcavity~\cite{imamoglu96a}. A
semiconductor microcavity with embedded low dimensional structures
provides a unique laboratory to study a variety of quantum
phases. This advantage finds its existence due to a composite boson:
the \emph{exciton-polariton}~\cite{hopfield58a,weisbuch92a}, a quantum
superposition of light and matter with not-only fermionic but also a
photonic component. The polaritonic side of the light--matter
coupling has stimulated research in both fundamental and applied
fields. With regard to the applicability, polaritons promise devices
with remarkable upgrades as compared to their semiconductor
counterparts, from which polariton
lasers~\cite{imamoglu96a,christopoulos07a,daskalakis13a,azzini11a} and
polariton transistors~\cite{amo10a,ballarini13a,anton14a,gao15a} are
the most obvious examples. Concerning fundamental aspects, polaritons
cover immense areas in physics, including Bose-Einstein Condensation
(BEC)~\cite{kasprzak06a,deng07a,balili07a,lai07a},
superfluidity~\cite{carusotto04a,wouters10b}, spin Hall
effect~\cite{kavokin05b}, superconductivity~\cite{laussy10a} and
Josephson--effects
effects~\cite{lagoudakis10a,abbarchi13a,voronova15b,Rahmani16a}.
The simplest description for the kinetic of a polariton gas is
provided by semiclasical Boltzmann equations~\cite{snoke10b}. This
approach has been used widely by many
authors~\cite{tassone99a,porras02a,malpuech02a,doan05a,hartwell10a,maragkou10a}. Due
to the fast photon leakage from the microcavity, polaritons have a
short lifetime, which keeps the polaritonic system in nonequilibrium
regime. Therefore the system should be pumped to compensate the losses
of polaritons. The rate equation for condensation kinetics of
polaritons is:
\begin{align}
\partial_tn_\mathbf{k}=&p_\mathbf{k}-\frac{n_\mathbf{k}}{\tau_\mathbf{k}}+\frac{\partial n_\mathbf{k}}{\partial t}\vert_{lp-lp}+\frac{\partial n_\mathbf{k}}{\partial t}\vert_{lp-ph}\,,\label{eq:3}
\end{align}
where~$p_\mathbf{k}$~is the pumping term,
and~$(\tau_\mathbf{k})^{-1}$~describes the particle decay rate. The
two next terms in Eq.~(\ref{eq:3}) account for polariton--polariton
and polariton--phonon scattering rates, respectively. We refer to
Appendix~\ref{app:Boltzmanneq} for detailed calculations of scattering
rates. To illustrate the formation of the condensate in polaritonic
system, we present a typical result of numerical simulation of the
Boltzmann equation in
Fig~\ref{fig:marjul7120100CEST2015115sd8}(b). Initially, polaritons
are introduced incoherently in exciton--like region of the lower
polariton dispersion~(Fig.~\ref{fig:marjul7120100CEST2015115sd8}(a)),
and then relax quickly, except near the exciton--photon resonance. In
this region, the polariton density of states is reduced and the
photonic contribution to the polariton is increased, which results in
polariton accumulations in the bottleneck
region~\cite{tassone97a}. This effect is shown as the peak in the
curve in Fig.~\ref{fig:marjul7120100CEST2015115sd8}(b). To make the
population degenerate, that is to overcome the bottleneck effect, one
needs to take into account the action of both polariton--polariton and
polariton--phonon processes, as the only polariton--phonon scattering
mechanism arises pilled up population in non--zero state. With both
scattering processes, then Bose stimulation effectively amasses
polaritons in the ground state.
While it is a simple, though extremely time consuming, simulation,
Boltzmann equations exclude the quantum aspect of the dynamic, to wit,
it does not consider the effect of coherence. One then needs to
upgrade the formalism to include both quantum and non--equilibrium
aspects of the dynamic. A powerful and widely used method that allows
such an exact treatment is provided by the Keldysh functional integral
approach~\cite{kamenev_book11a,sieberer16a}. This method has been
applied to study the driven open system including polaritons in
microcavities~\cite{szymanska06a,szyma07a,proukakis13a,Dunnett16a,pavlovic13a},
glassy and supperradiant phase of ultracold atoms in optical
cavity~\cite{buchholda13,Torre13a}, photon condensations in dye filled
optical cavity~\cite{leeuwa13}, etc. It also allows to explore the
Bose--Hubbard model with time-dependent hopping~\cite{kennetta11}, and
non--equilibrium Bosonic Josephson oscillations~\cite{trujilloa09}, among others.
In this paper, a quantum field theory for polariton internal degrees
of freedoms in dissipative regime using Keldysh functional method is
developed. The need for such an approach in describing the polariton
relaxation is motivated mainly by the polariton specificity of
providing two types of lifetimes~\cite{dominici14a}: one for the bare
states (exciton and photon), the other for the dressed states (upper
polariton)~\cite{skolnick00a}. In the latter case, excitons and
photons are removed in a correlated way from the system. This effect
has attracted attention recently in particular as it can lead to
interesting applications proper to polaritons~\cite{colas15a}. While
this polariton lifetime can be described in a linear regime from
standard master equation approaches, more general scenarios involving,
e.g., interactions, get out of reach of this description as the nature
of the dressed state changes dynamically with the state of the
system. A description of the dissipation therefore needs being made at
a more fundamental level than through the standard Lindblad
phenomenological form.
While photon and exciton are considered as independent quantum field
interacting through the Rabi energy, to model the decay, we assume two
independent excitonic and photonic baths, which are present in most
light--matter coupled systems. In this text, we restrict our analysis
to the linear (Rabi) regime (no interaction). The interaction
certainly has important effects, such as
self--trapping~\cite{raghavan99a}, and the optical parametric
oscillator regime~\cite{Dunnett16a}. However, it can be found that
even in the non-interacting regime of the dynamic, some aspects of
nonlinearity emerge from the exciton-photon
coupling~\cite{rubo03a,voronova15a,Rahmani16a}. Therefore, we first
attack the problem of polariton dynamics in the Keldysh formalism in
the simpler case of non-interacting particles, as a basis for more
elaborate and involved studies. At such, we derive the mean--field
equations of motions in the photon-exciton
basis~\cite{hampa15,elistratova16,voronova15a}. In particular, we show
that the polaritonic internal dynamic satisfies the Josephson
criterion of coherent flow.
This paper is organised as follows. In Sec.~\ref{sec:two} we present
the polaritonic Hamiltonian and how to turn it into a dissipative
system. This includes the Hamiltonian for coupling both bare and
dressed fields baths. The Keldysh technique is introduced in
Sec.~\ref{sec:three}, where the mean-field solutions and fluctuation
actions are also presented. In Sec.~\ref{sec:four}, we represent the
internal dynamic on the Paria sphere~\cite{Rahmani16a} (dynamically
renormalized Bloch sphere), and discuss on the stability of the
solution. Conclusions are presented in Sec.~\ref{sec:five}.
\begin{figure}[t]
\centering
\includegraphics[width=.7\linewidth]{fig1.pdf}
\caption{(a)~Dispersion of polariton when the detuning at~$k=0$~is
zero ($\epsilon_a(0)=\epsilon_b(0)$). Strong coupling between
photon (in dash-dark green) and exciton (in dashed red) results
the lower polariton (denoted by Lp and in blue) and the upper
polariton (denoted by Up and in light green). The energy splitting
at~$k=0$~is~$2g$. In the regime of weak pumping, polaritons are
accumulated at the bottleneck region (B.N.). By increasing the
pumping strength, the B.N. is relaxed and the condensate
forms. The relaxation of polaritons due to interactions with other
polariton or phonons is fast in exciton--like region, but is slow
in photon--like region. (b)~Distribution function of
polariton. The initial non--degenerate distribution~(red dashed
line)~evolves to a degenerate distribution (blue dashed--dot
line), with both polariton--polariton and polariton--phonon
interactions. Considering the phonon--polariton processes only,
the distribution remains non--degenerate (gray dashed--dot
line). Using parameters for simulation are taken form
Ref.~\cite{doan05a}.}
\label{fig:marjul7120100CEST2015115sd8}
\end{figure}
\section{Polariton Hamiltonian}
\label{sec:two}
The strong coupling between photon and exciton fields in a
semiconductor microcavity results in a quasiparticle with very
peculiar properties: the polariton. Denoting the photon and exciton
field operators by~$a_\mathbf{k}$~and~$b_\mathbf{k}$~respectively,
then the Hamiltonian describing the internal coupling between the two
fields is given by:
\begin{subequations}
\begin{align}
\label{eq:Hc}
H_c=&H_0+H_{Rabi}\,,\\
\label{eq:H0}
H_0=&\sum_\mathbf{k}(\epsilon_a(k)a^\dagger_\mathbf{k} a_\mathbf{k}+\epsilon_b(k)b^\dagger_\mathbf{k} b_\mathbf{k})\,,\\
\label{eq:HR}
H_{Rabi}=&\sum_\mathbf{k}g(a^\dagger_\mathbf{k}b_\mathbf{k}+b^\dagger_\mathbf{k}a_\mathbf{k})\,,
\end{align}
\end{subequations}
where~$\epsilon_a$~and~$\epsilon_b$~are the cavity photon and quantum
well exciton dispersion given respectively by:
\begin{subequations}
\begin{align}\label{eq:photondispersion}
\epsilon_a(k)=&\frac{\hbar c}{\sqrt{\varepsilon}}\sqrt{k^2_\perp+k^2}\,,\\
\epsilon_b(k)=&\epsilon^{ex}(0)+\frac{\hbar^2k^2}{2m_{ex}}\,,
\end{align}
\end{subequations}
with~$\epsilon^{ex}(0)=2m_{ex}e^4/\varepsilon^2\hbar^2$~as the 2D
exciton binding energy. In Eq.~(\ref{eq:HR}), $g$~shows the strength of
coupling between photon and exciton fields and in the regime of strong
coupling, it is referred to as the Rabi energy.
Diagonalising the Hamiltonian in Eq.~(\ref{eq:Hc}) leads to the new
bosonic dressed modes: the lower~($L_\mathbf{k}$)~and
upper~($U_\mathbf{k}$)~polariton. Then the~$H_c$~takes the diagonal
form:
\begin{align}\label{eq:indiagonalform}
H_c=\sum_\mathbf{k}\epsilon_lL_\mathbf{k}^\dagger L_\mathbf{k}+\epsilon_uU_\mathbf{k}^\dagger U_\mathbf{k}\,,
\end{align}
with~$\epsilon_{\underset{l}{u}}=\frac12(\epsilon_a+\epsilon_b\pm\sqrt{(\epsilon_a-\epsilon_b)^2+(2g)^2}$.
Note that for zero detuning ($\epsilon_a(0)=\epsilon_b(0)$), the
splitting the two polariton branches is~$2g$. The dispersion in zero
detuning is shown in Fig.~(\ref{fig:marjul7120100CEST2015115sd8}).
Such a transformation from bare states (photon and exciton) to dressed
states (upper and lower polariton) is done through the operator
relation:
\begin{subequations}
\begin{align}
L_\mathbf{k}=&A(k)a_\mathbf{k}+B(k)b_\mathbf{k}\,,\\
U_\mathbf{k}=&B(k)a_\mathbf{k}-A(k)b_\mathbf{k}\,,
\end{align}
\end{subequations}
where~$A(k)$~and~$B(k)$~are the so called Hopfield
coefficients~\cite{hopfield58a}, given by~\cite{ciuti03a,laussy04b}:
\begin{subequations}
\begin{align}
B(k)=\frac{1}{\sqrt{1+(\frac{g}{\epsilon_{lp}-\epsilon_c})^2}}\,,\\
A(k)=-\frac{1}{\sqrt{1+(\frac{\epsilon_{lp}-\epsilon_c}{g})^2}}\,,
\end{align}
\end{subequations}
Due to photon leakage from the microcavity, the polariton has a short
lifetime. To consider the dynamic in dissipative regime, one should
begin from a microscopic view of the mechanism underling dissipation,
namely to model the \emph{environmental interaction} by coupling the
system to a bath. Here the Hamiltonian for the undamped system is
given in Eq.~(\ref{eq:H0}), while the baths are modeled as a
collection of harmonic oscillators:
\begin{align}\label{Hbath}
H_{bath}=&\sum_{\mathbf{p}}\omega_\mathbf{p}^{ph}r^\dagger_\mathbf{p}r_\mathbf{p}+\sum_{\mathbf{p}}\omega_\mathbf{p}^{ex}c^\dagger_\mathbf{p}c_\mathbf{p}\,,
\end{align}
with~$\omega_\mathbf{p}^{ph(ex)}$~as the dispersion of the photonic
(excitonic) bath, and corresponding creation and annihilation
operators~$r^\dagger_\mathbf{p}(c^\dagger_\mathbf{p})$~and~$r_\mathbf{p}(c_\mathbf{p})$,
respectively. It is assumed that each bath is in thermal equilibrium
and unaffected by the behavior of the system. The bath--system
interaction can be described through:
\begin{align}
\label{eq:Hdey}
H_{Dec}=\sum_{\mathbf{k,p}}[(F_\mathbf{k}^\mathbf{p}a^\dagger+R_\mathbf{k}^\mathbf{p}b^\dagger)r+(S_\mathbf{k}^\mathbf{p}a^\dagger+G_\mathbf{k}^\mathbf{p}b^\dagger)c]+\mathrm{H.c.}\,,
\end{align}
where~$\mathrm{H.c.}$~stands for Hermitian conjugate. Parameters in
Eq.~(\ref{eq:Hdey})~are related to the coupling between polaritonic
system and baths which are defined as:
\begin{align}
F_\mathbf{k}^\mathbf{p}&\equiv \Gamma_{\mathbf{k},ph}^\mathbf{p}+B\Gamma_{\mathbf{k},u}^\mathbf{p}\,,\hspace{5mm}R_\mathbf{k}^\mathbf{p}\equiv -A\Gamma_{\mathbf{k},u}^\mathbf{p}\nonumber\,,\\
G_\mathbf{k}^\mathbf{p}&\equiv \Gamma_{\mathbf{k},ex}^\mathbf{p}-A\Gamma_{\mathbf{k},u}^\mathbf{p}\,,\hspace{5mm}S_\mathbf{k}^\mathbf{p}\equiv B\Gamma_{\mathbf{k},u}^\mathbf{p}\nonumber\,,
\end{align}
where~$\Gamma_{\mathbf{k},ph(ex)}^\mathbf{p}$~shows the coupling
strength of the photonic (excitonic) component of the polarion to the
photonic (excitonic) bath. The polariton can also decay through its
upper branch, which is modeled via direct coupling to the both baths
with coupling strength of~$\Gamma_{\mathbf{k},u}$. Deriving all the
needed Hamiltonian we find for our final Hamiltonian:
\begin{align}
\label{eq:Hamiltonian+in+Linear+regime}
H=&H_c+H_{Dec}+H_{bath}\,.
\end{align}
\section{Functional representation of polaritons}
\label{sec:three}
In this section, we present the functional approach to the internal
dynamic of polaritons. Any equilibrium many--body theory involves
adiabatic switching on of \emph{interaction} at a distant past
($t=-\infty$), and off at a distant future ($t=\infty$). The state of
the system at these two reference times is the ground state of the
non--interacting system. Then any correlation function in the
interaction representation can be averaged with respect to a known
ground state of the non--interacting Hamiltonian.
The postulate of independence of the reference states from the details
of switching on and off the interaction breaks in non--equilibrium
condition, as the system evolves to an unpredictable state. However,
one needs to know the final state. It was Schwinger~\cite{scwa60}'s
suggestion that the final state to be exactly is the same as that of
the initial time. Then the theory can evolve along a two--branch
closed time contour with a forward and backward direction.
The central quantity in the functional integral method is the
partition function of the system that can be written as a Gaussian
integral over the bosonic fields of~$\phi,\bar{\phi}$:
\begin{align}\label{eq:partionfun}
Z=&N~\int D[\bar{\phi},\phi]~e^{iS[\bar{\phi},\phi]}\,,
\end{align}
where~$N$~is the normalization constant and~$S$~is the action, which
carries the dynamical information. In the Keldysh formalism, the
bosonic field~$\phi$~is split into two
components~$\phi^+$~and~$\phi_-$, which reside on the forward and
backward part of the time contour. Then the field are rotated to the
Keldysh basis defined as:
\begin{align}\label{eq:keldyshrotation}
\phi_{\underset{q}{cl}}=&\frac{1}{\sqrt{2}}(\phi_+\pm\varphi_-)\,,
\end{align}
where the~$+(-)$~sign stands for~$cl(q)$. Here~$cl(q)$~stands for
classical (quantum) component of the field.
Corresponding to the terms in the
Hamiltonian~(\ref{eq:Hamiltonian+in+Linear+regime}), the actions take
the following components in Keldysh space:
\begin{align}\label{eq:S0}
S_0=&\Delta_\mathbf{k}^t[\Psi_\mathbf{k}^\dagger(i\partial_t-\epsilon_a)\sigma_1^K\Psi_\mathbf{k}+\Phi_\mathbf{k}^\dagger(i\partial_t-\epsilon_b)\sigma_1^K\Phi_\mathbf{k}]\,,
\end{align}
\begin{align}\label{eq:SR}
S_{Rabi}=&-\Delta_\mathbf{k}^{t}g~[\Phi_\mathbf{k}^\dagger\sigma_1^K\Psi_\mathbf{k}+\Psi_\mathbf{k}^\dagger\sigma_1^K\Phi_\mathbf{k}]\,,
\end{align}
\begin{align}\label{eq:SDec}
S_{Dec}^{r}=&-\Delta_\mathbf{k,p}^{t}~[(F_\mathbf{k}^\mathbf{p}\Psi^\dagger_\mathbf{k}+R_\mathbf{k}^\mathbf{p}\Phi^\dagger_\mathbf{k})\sigma^K_1X_{R,\mathbf{p}}+\mathrm{h.c.}]\,,
\end{align}
\begin{align}\label{eq:SDecwx}
S_{Dec}^{c}=&-\Delta_\mathbf{k,p}^{t}~[(S_\mathbf{k}^\mathbf{p}\Psi^\dagger_\mathbf{k}+G_\mathbf{k}^\mathbf{p}\Phi^\dagger_\mathbf{k})\sigma^K_1X_{C,\mathbf{p}}+\mathrm{h.c.}]\,,
\end{align}
\begin{align}\label{eq:Sbath}
S_{bath}=&\Delta_\mathbf{p}^{t}~\sum_{j=R,C}X_{j,\mathbf{p}}^\dagger(i\partial_t-\omega_\mathbf{p}^j)\sigma_1^KX_{j,\mathbf{p}}\nonumber\\=&\Delta_\mathbf{p}^{t}~\sum_{j=R,C}X_{j,\mathbf{p}}^\dagger C_{j}^{-1}X_{j,\mathbf{p}}\,,
\end{align}
where we use~$\Delta_\mathbf{k}^{t}$~as an abbreviation
for~$\sum_\mathbf{k}\int_{-\infty}^\infty~dt$, and
superscript~$r$~and~$c$~refer to excitonic and photonic baths,
respectively. Other notations are summarised in Table~\ref{tab:one}.
\begin{table}[b]
\centering\caption{Fields and matrices in Keldysh space}
\label{tab:one}
\begin{tabular}{@{}|c|c|c|c|@{}}\hline
\multicolumn{4}{@{}|c|@{}}{\rule[-0.125cm]{0mm}{0.5cm}%
\mbox{Fields in Keldysh space}}\\
\hline
\mbox{Photonic}&
\mbox{Ecxitonic} & \mbox{Photonic Bath} & \mbox{Excitonic Bath}\\ \hline
$\Psi_\mathbf{k}$
&$\Phi_\mathbf{k}$&$X_{c,\mathbf{k}}$&$X_{r,\mathbf{k}}$\\
\hline$\left( \begin{array}{c}
\psi_{cl}\\\psi_{q}
\end{array}\right)_\mathbf{k}$&$\left( \begin{array}{c}
\varphi_{cl}\\\varphi_{q}
\end{array}\right)_\mathbf{k}$&$\left( \begin{array}{c}
x_{cl}\\x_{q}
\end{array}\right)_\mathbf{k}$&$\left( \begin{array}{c}
y_{cl}\\y_{q}
\end{array}\right)_\mathbf{k}$
\\ \hline
\multicolumn{4}{@{}|c|@{}}{\rule[-0.125cm]{0mm}{0.5cm}%
\mbox{Pauli matrices in Keldysh space}}\\
\hline
\mbox{$\sigma_K^0$}&\mbox{$\sigma_K^1$}&
\mbox{$\sigma_K^2$} & \mbox{$\sigma_K^3$}\\ \hline
$\left( \begin{array}{cc}
1&0\\0&1
\end{array}\right)$
&$\left( \begin{array}{cc}
0&1\\1&0
\end{array}\right)$&$\left( \begin{array}{cc}
0&-i\\i&0
\end{array}\right)$&$\left( \begin{array}{cc}
1&0\\0&-1
\end{array}\right)$\\\hline
\end{tabular}
\end{table}
The bath action in Eq.~(\ref{eq:Sbath}) is in the standard form in
Keldysh space: it contains a quadratic form of the fields with a
matrix which is the inverse of a
correlator~$C_j$~with~$j=\left\lbrace r,c\right\rbrace $. One can
show that:
\begin{align}
C_j\equiv&\left( \begin{array}{cc}
C^K_i&C^R_j\\
C^A_j&0
\end{array}\right)_{t,t'}\nonumber\\
=&\left( \begin{array}{cc}
-if_je^{-i\omega_p^j(t-t')}&-i\Theta(t-t')e^{-i\omega_p^j(t-t')}\\
i\Theta(t-t')e^{-i\omega_p^j(t-t')}&0
\end{array}\right)\,,
\end{align}
where~$\Theta(t)$~is the Heaviside step function and~$f_j$~is the
distribution function of the bath.
As the bath coordinates appear in a quadratic form, they can be
integrated out to reduce the degree of freedom to photon and exciton
fields only. We follow the procedure described in
Refs.~\cite{kamenev_book11a,szyma07a}. Employing the properties of
Gaussian integration, the decay bath eliminating leads to two
effective actions:
\begin{subequations}
\begin{align}
\label{eq:SDecwithoutbath1}
S_{Dec}^{r}=&\Delta_\mathbf{k,p}^{t,t'}~(F_\mathbf{k}^\mathbf{p}\Psi^\dagger_\mathbf{k}+R_\mathbf{k}^\mathbf{p}\Phi^\dagger_\mathbf{k})_{t}~\mathcal{C}^{-1}_r(t-t')~(F_\mathbf{k}^\mathbf{p}\Psi_\mathbf{k}+R_\mathbf{k}^\mathbf{p}\Phi_\mathbf{k})_{t'}\,,\\
\label{eq:SDecwithoutbath11}
S_{Dec}^{c}=&\Delta_\mathbf{k,p}^{t,t'}~(S_\mathbf{k}^\mathbf{p}\Psi^\dagger_\mathbf{k}+G_\mathbf{k}^\mathbf{p}\Phi^\dagger_\mathbf{k})_{t}~\mathcal{C}^{-1}_c(t-t')~(S_\mathbf{k}^\mathbf{p}\Psi_\mathbf{k}+G_\mathbf{k}^\mathbf{p}\Phi_\mathbf{k})_{t'}\,,
\end{align}
\end{subequations}
where~$\mathcal{C}^{-1}_{j}(t-t')=-\sigma_1^K~[C_{j}(t-t')]~\sigma_1^K$~and~$j=\left\lbrace
r,c\right\rbrace$.
Straightforward matrix multiplication shows that
the~$\mathcal{C}^{-1}$~correlator has the causality structure, given
by:
\begin{align}
\mathcal{C}^{-1}_j(t-t')\equiv&\left( \begin{array}{cc}
0&\mathcal{C}^A_j\\
\mathcal{C}^R_j&\mathcal{C}^K_j
\end{array}\right)_{t,t'}\,.
\end{align}
To proceed further, we make some simplifying assumptions about the
baths. Firstly, we assume that all modes of the systems are coupled to
their baths with the same strength, i.e,
$\Gamma_j(p)\equiv\Gamma_{\mathbf{k},j}^\mathbf{p}$. Besides, it is
assumed that the bath is in the Markovian limit, where the density of
state for the baths and the coupling between the system and baths are
constant. In the following, we restrict our analysis to these
assumptions. More details including non--Markovian cases are presented
in Appendix~\ref{app:freqencydependence}.
Denoting the decay action
as~$S_{Dec}=\sum_{i=r,c}S_{bath}^i+S_{Dec}^i$, one gets the components
of the~$S_{D}$~as (see Appendix~\ref{app:freqencydependence}):
\begin{align}\label{eq:sDecay1}
S_{Dec}^1=&\sum_\mathbf{k}\int~dt\{ \gamma_1(k)\Psi^\dagger_\mathbf{k}\sigma_2^K\Psi_\mathbf{k}\\\nonumber&+2i(\gamma_1(k))\int dt'\bar{\psi}_q(t)(f_c+f_d)_{t-t'}\psi_q(t')\}\,,
\end{align}
\begin{align}\label{eq:sDecay2}
S_{Dec}^2=&\sum_\mathbf{k}\int~dt\{ \gamma_2(k)\Phi^\dagger_\mathbf{k}\sigma_2^K\Phi_\mathbf{k}\\\nonumber&+2i(\gamma_2(k))\int dt'\bar{\varphi}_q(t)(f_c+f_d)_{t-t'}\varphi_q(t')\}\,,
\end{align}
\begin{align}\label{eq:sDecay3}
S_{Dec}^3=&\sum_\mathbf{k}\int~dt\{ \gamma_3(k)\Psi^\dagger_\mathbf{k}\sigma_2^K\Phi_\mathbf{k}\\\nonumber&+2i(\gamma_3(k))\int dt'\bar{\psi}_q(t)(f_c+f_d)_{t-t'}\varphi_q(t')\}\,,
\end{align}
\begin{align}\label{eq:sDecay4}
S_{Dec}^4=&\sum_\mathbf{k}\int~dt\{ \gamma_3(k)\Phi^\dagger_\mathbf{k}\sigma_2^K\Psi_\mathbf{k}\\\nonumber&+2i(\gamma_3(k))\int dt'\bar{\varphi}_q(t)(f_c+f_d)_{t-t'}\psi_q(t')\}\,,
\end{align}
where we
define~$\gamma_1=B^2\gamma_u^2+(\gamma_a+B\gamma_u)^2$,~$\gamma_2=A^2\gamma_u^2+(\gamma_b-A\gamma_u)^2$,~$\gamma_3=\gamma_u(B(\gamma_b-A\gamma_u)-A(\gamma_a+B\gamma_u))$,
and~$\sigma^{K}_2$~is given in Table~\ref{tab:one}. One notes that
with decay for the upper polariton, the decay action has the
weightings of excitonic and photonic Hopfield coefficients; moreover,
even without bare field couplings~$\gamma_a$~and~$\gamma_b$, the decay
in the upper branch is enough to remove the photon and exciton fields
both independently and in a correlated way.
\subsection{Mean field solutions}
Having integrated out the bath degree of freedom, the action appears
in its final form as: $S=S_0+S_R+S_{Dec}$. We can then obtain the
equations of motions from the saddle point condition on the action:
\begin{align}
\frac{\partial S}{\partial\bar{\psi}_{i=q,cl}}=0\,,\hspace{5mm}\frac{\partial S}{\partial\bar{\varphi}_{i=q,cl}}=0\,.
\end{align}
which yields:
\begin{subequations}
\begin{align}
\label{eq:equationmotion1}
\frac{\partial S}{\partial\bar{\psi}_q}=&(i\partial_t-\epsilon_a+i\gamma_1)\psi_{cl}-(g-i\gamma_3)\varphi_{cl}\nonumber\\&+2i\gamma_1\int~dt'(f_c+f_d)(t-t')\psi_q(t')\nonumber\\&+2i\gamma_3\int~dt'(f_c+f_d)(t-t')\varphi_q(t')\,,\\
\label{eq:equationmotion2}
\frac{\partial S}{\partial\bar{\varphi}_q}=&(i\partial_t-\epsilon_b+i\gamma_2)\varphi_{cl}-(g-i\gamma_3)\psi_{cl}\nonumber\\&+2i\gamma_2\int~dt'(f_c+f_d)(t-t')\varphi_q(t')\nonumber\\&+2i\gamma_3\int~dt'(f_c+f_d)(t-t')\psi_q(t')\,,\\
\label{eq:equationmotion3}
\frac{\partial S}{\partial\bar{\psi}_{cl}}=&(i\partial_t-\epsilon_a-i\gamma_1)\psi_{q}-(g+i\gamma_3)\varphi_{q}\,,\\
\label{eq:equationmotion4}
\frac{\partial S}{\partial\bar{\varphi}_{cl}}=&(i\partial_t-\epsilon_b-i\gamma_2)\varphi_{q}-(g+i\gamma_3)\psi_{q}\,.
\end{align}
\end{subequations}
One notices that
Eqs.(\ref{eq:equationmotion3}-\ref{eq:equationmotion4}) are satisfied
by~$\varphi_q=\bar{\varphi}_q=0$~and~$\psi_q=\bar{\psi}_q=0$,
irrespective of what the classical
components,~$\varphi_0\equiv\varphi_{cl}$~and~$\varphi_0\equiv\varphi_{cl}$,
are. Then, under these conditions,
Eqs.~(\ref{eq:equationmotion1}-\ref{eq:equationmotion2}) lead to:
\begin{align}\label{eq:coupledequations1}
\partial_t\psi_{0}=&(-i\epsilon_a-\gamma_1)\psi_{0}+(-ig-\gamma_3)\varphi_{0}\,,
\end{align}
\begin{align}\label{eq:coupledequations2}
\partial_t\varphi_{0}=&(-i\epsilon_b-\gamma_2)\varphi_{0}+(-ig-\gamma_3)\psi_{0}\,.
\end{align}
These equations provide an extreme limit for the action~$S$ and
describe the system of two coupled-equations of motions in the mean
field analysis.
\subsection{Fluctuations of the action}
By separating the fluctuations from the mean field:
\begin{align}
\psi=\psi_0+\delta\psi_{cl}\,,\hspace{5mm}\psi_q=\delta\psi_q\,,\\
\varphi=\varphi_0+\delta\varphi_{cl}\,,\hspace{5mm}\varphi_q=\delta\varphi_q\,,
\end{align}
for the actions defined in
Eqs.~(\ref{eq:S0})--(\ref{eq:SR})~and~(\ref{eq:sDecay1})--(\ref{eq:sDecay4}),
then employing the Fourier transform, the fluctuating action takes the
form:
\renewcommand\arraystretch{1.3}
\begin{align}\label{eq:DeltaS}
\Delta S=\int~d\omega\sum_\mathbf{k}\Delta\mathcal{T}_\mathbf{k}^\dagger\left( \begin{array}{c|c}
0&[\mathcal{F}^{-1}]^A\\\hline
[\mathcal{F}^{-1}]^R&[\mathcal{F}^{-1}]^K
\end{array}\right) \Delta\mathcal{T}_\mathbf{k}\,,
\end{align}
where the superscripts~$A$,~$R$, and~$K$~stand for the advanced,
retarded and Keldysh components of the inverse Green function,
respectively. The fluctuation vector has the form of:
\begin{align}
\Delta\mathcal{T}_\mathbf{k}^\dagger\equiv(CL,Q)^T
\end{align}
with
\begin{align}
CL\equiv\left( \begin{array}{c}
\delta\bar{\psi}_{cl}(\omega)\\
\delta\psi_{cl}(-\omega)\\
\delta\bar{\varphi}_{cl}(\omega)\\
\delta\varphi_{cl}(-\omega)
\end{array}\right)_\mathbf{k}\quad\text{and}\quad Q\equiv\left( \begin{array}{c}
\delta\bar{\psi}_{q}(\omega)\\
\delta\psi_{q}(-\omega)\\
\delta\bar{\varphi}_{q}(\omega)\\
\delta\varphi_{q}(-\omega)
\end{array}\right)_\mathbf{k}\,.
\end{align}
Further, the off--diagonal matrix elements in Eq.~(\ref{eq:DeltaS})
have the following relations~\cite{kamenev_book11a}:
\begin{align}\label{eq;relationamonggreenF1}
[\mathcal{F}^{-1}]^{R(A)}~\mathcal{F}^{R(A)}=1\,,\\
\mathcal{F}^A=(\mathcal{F}^R)^\dagger\,,
\end{align}
while for the diagonal element one finds:
\begin{align}\label{eq;relationamonggreenF2}
\mathcal{F}^K=\mathcal{F}^RF-F\mathcal{F}^A
\end{align}
where ~$F$~is referred to as the distribution function of the
system~\cite{kamenev_book11a}. Having found the Green functions of the
dynamic, one can decide about the stability of the solution by
studying the retarded Green function, namely by
solving~$\det([\mathcal{F}^{-1}]^R(\omega_r))=0$, where~$\omega_r$~is
the pole of the retarded Green
function. If~$\mathrm{Im}[\omega_r]<0$, then the proposed solution is
stable.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{fig3.pdf}
\caption{(a) The internal dynamic of polariton in a photon--like
point on the Paria sphere, when detuning between exciton and
photon fields is zero. The~$\rho_p$~direction shows the
orientation of the upper--lower polariton states, while
the~$\rho$~shows the direction of exciton--photon states. The red
point shows the initial point. (b)~The same as (a) but for an
exciton--like point. (c)~and~(d)~shows the population dynamic in
photon and exciton like point respectively. Using parameters
are:~$\gamma_a=0.2g$,~$\gamma_b=0.02g$,~$\gamma_u=0.3g$.}
\label{fig:marjul7120100CEST20151158}
\end{figure*}
\section{Discussion}\label{sec:four}
To analyse the equations of motions, we start from
Eqs.~(\ref{eq:coupledequations1}) and~(\ref{eq:coupledequations2}),
and restrict calculations to two points in reciprocal space: the space
center ($k=0$), that is a photon--like point, and an exciton--like
point at~$k\sim\sqrt{2m_{ex}\epsilon_{ex}^{0}}$. In the absence of
exciton and photon detuning at~$k=0$, the coupling fields are at
resonance, and the exciton and photon have the same weight of~$1/2$ in
the polariton. However, bare states at~$k\neq0$ are positively
detuned, which provides an intrinsic detuning in the internal dynamic
of polaritons.
Introducing~$\rho_\mathbf{k}\equiv\frac12(|\psi_0|^2-|\varphi_0|^2)$ as
the population imbalance
and~$N_\mathbf{k}\equiv(|\psi_0|^2+|\varphi_0|^2)$ as the total field
population in state~$\mathbf{k}$, one can show that the
Eqs.~(\ref{eq:coupledequations1}) and~(\ref{eq:coupledequations2}) are
in the form of Josephson equations, that is:
\begin{subequations}
\label{eq:inrhoandsigma}
\begin{align}
\partial_t(\rho/N)_\mathbf{k}=&-\sqrt{1-4(\rho/N)_\mathbf{k}^2}(g\sin(\sigma_\mathbf{k})-\gamma_3\cos(\sigma_\mathbf{k})(\rho/N)_\mathbf{k})\nonumber\\&-(\gamma_1+\gamma_2)(1-4(\rho/N)_\mathbf{k}^2)\,,\\
\partial_t\sigma_\mathbf{k}=&-\delta_\mathbf{k}\nonumber\\&+\frac{1}{\sqrt{1-4(\rho/N)_\mathbf{k}^2}}(4g\cos(\sigma_\mathbf{k})(\rho/N)_\mathbf{k}+\gamma_3\sin(\sigma_\mathbf{k}))\,,
\end{align}
\end{subequations}
where~$\delta_\mathbf{k}=\epsilon_a-\epsilon_b$ stands for detuning
between the states, and~$\sigma_\mathbf{k}=\arg[\psi_0^\ast\varphi_0]$
is the relative phase between bare states. With such a representation
of the dynamic, each polaritonic state in~$\mathbf{k}$~has an
intrinsic internal Josephson--like dynamic, when the relative phase
drives the population difference. Recently, the same equations of
motions were reported by one of the authors, but for an effective
two-level system and in a different
formalism~\cite{Rahmani16a,Rahmani16wolfram}. One notices that the
phase difference between the two coupled fields, with a definite phase
in each field, is crucial to drive the internal dynamic. This is the
case when two or more condensates are coupled, through for example
Josephson junctions. However, here we do not take any assumption about
the condensate phase, and this is left to the initial condition to
define a clear phase for each field; then as long as the polaritonic
system is initially prepared by a source of specific phase, the
internal dynamic follows the Josephson dynamic, and the coherence
oscillates between the fields.
An example of the dynamic is shown in
Fig.~(\ref{fig:marjul7120100CEST20151158}). Here we adopt the Paria
sphere~\cite{Rahmani16wolfram} to observe the dynamic in a three
dimensional representation. Two directions are indicated on the
sphere: the~$\vec\rho$~direction, that shows the direction for
exciton--photon states, and~$\vec\rho_p$, which shows the orientation
of lower--upper polariton states. Starting from an initial point (the
red point in Fig.~(1)), the dynamic goes toward a fixed point (when
the sphere is kept normalized). In the zero detuning case,
(Fig~(\ref{fig:marjul7120100CEST20151158}--a)), the laboratory basis
is orthogonal to the dressed state basis,
with~$\vec\rho\perp\vec\rho_p$, and the relative phase remains in an
oscillatory mode. Going toward an exciton--like point in
Fig~(\ref{fig:marjul7120100CEST20151158}--b), the two directions are
not orthogonal, that shows the final state has a more exciton
weight. At the same time, one can see a switching from the running
mode to the oscillatory mode in relative phase, which is mediated by
decay. We also show the dynamic of bare state populations in
photon--like (Fig.~(\ref{fig:marjul7120100CEST20151158}--c)) and
exciton--like(Fig.~(\ref{fig:marjul7120100CEST20151158}--d)) points of
the indirect space. We set the initial conditions to have more
population in the photon field. At zero detuning, both fields are
oscillating in the same trend, as the decay affects the dynamic
equivalently; however, by increasing the detuning, decay affects the
field (in this example the photon field) that has the more
population. In other words, by increasing the detuning, bare states
become decoupled and each field loses its coupling to other fields
while the coupling to the bath is yet active.
\begin{figure}[b]
\centering
\includegraphics[width=.7\linewidth]{fig33.pdf}
\caption{(a) damped oscillations in population imbalance mediated by
decay in the upper polariton. Corresponding relative phase is
shown in (b). Parameters used
are:~$\delta=3g,~\gamma_a=\gamma_b=.2g,~\gamma_u=.6g,~\rho(0)=-0.3N,~\sigma(0)=\pi$.}
\label{fig:marjul7120100CEST2015115sd8s}
\end{figure}
The equations of motions in their Josephson representation bring a new
variant for the dynamics of polariton. Careful inspection in
Eqs.~(\ref{eq:inrhoandsigma}) shows that the relative phase is driving
the population in two ways: one is the well--studied Josephson dynamic
with the term proportional to~$g\sin(\sigma_\mathbf{k})$ in the
equations form Ref.~\cite{raghavan99a}. The other comes from the term
proportional to~$\gamma_3\cos(\sigma_\mathbf{k})(\rho/N)_\mathbf{k}$,
which existence is related to the upper polariton decay, and is
specific to this phenomenon. Such a peculiar aspect of the dynamic
holds even for the case of disconnected fields which are correlatively
coupled to a bath. A particular example of the internal dynamic
mediated only by polariton decay is shown in
Fig.~(\ref{fig:marjul7120100CEST2015115sd8s}). As the relative phase
is in the running mode, the population imbalance exhibit damped
oscillations toward a fixed point.
In any dynamical system, the stability of the solution in the steady
state is the most important property. In normalized coordinates, the
fixed points of the dynamic have a finite values, even in the
dissipative regime~\cite{Rahmani16a}. For a given fixed point, the
stability condition is determined through the inverse retarded Green
function in Eq.~(\ref{eq:DeltaS}), which reads:
\begin{widetext}
\begin{align}\label{InverseRetardedGreen}
[\mathcal{F}^{-1}]^R(\omega,\mathbf{k})=\frac{1}{2}\left( \begin{array}{cccc}
\omega-\epsilon_a+i\gamma_1&0&-g+i\gamma_3&0\\
0&-\omega-\epsilon_a-i\gamma_1&0&-g-i\gamma_3\\
-g+i\gamma_3&0&\omega-\epsilon_b+i\gamma_2&0\\
0&-g-i\gamma_3&0&-\omega-\epsilon_b-i\gamma_2
\end{array}\right)\,.
\end{align}
\end{widetext}
One notices how coupling between photon and exciton fields are
reflected as the coupling between quantum and classical parts of the
fields in Keldysh space, which makes the retarded matrix
non--diagonal. More interestingly is the appearance of the
term~$\gamma_3$, which has the same weight in the dynamic as the
coupling-constant~$g$. This is the direct consequence of the decay in
the upper polariton branch, as it removes bare fields in a correlated
fashion.
Solving~$[\det(\mathcal{F}^{-1}]^R(\omega_r)=0$, one finds the energies
of the dressed states in the dissipative regime as:
\begin{align}
\omega_r^{\pm}=&\frac12[\epsilon_a+\epsilon_b-i(\gamma_1+\gamma_2)\nonumber\\&\pm\sqrt{(\delta_\mathbf{k}-2ig-\gamma_+)(\delta_\mathbf{k}+2ig-\gamma_-)}]\,,
\end{align}
where~$\delta_\mathbf{k}=\epsilon_a-\epsilon_b$ is the detuning, and~$\gamma_\pm=\pm2\gamma_3+i(\gamma_1-\gamma_2)$. The version with no dissipation has the familiar form~\cite{elistratova16}:
\begin{align}
\omega_r^{\pm}=&\frac12(\epsilon_a+\epsilon_b\pm\sqrt{\delta_\mathbf{k}+4g^2})\,,
\end{align}
that is the result from a pure Hamiltonian picture (see the notes
after Eq.~\ref{eq:indiagonalform}). For zero detuning, one gets:
\begin{align}
\omega_r^{\pm}=&\frac12[\epsilon_a+\epsilon_b-i(\gamma_1+\gamma_2)\nonumber\\&\pm\sqrt{(-2ig-\gamma_+)(+2ig-\gamma_-)}]\,.
\end{align}
For~$\gamma_3=0$, the term under the square root can go from real to
imaginary, namely, it happens when~$g<(\gamma_1-\gamma_2)/2$, which is
considered as the criterion for strong to weak coupling
transition~\cite{laussy08a,kavokin_book11a}. However,
for~$\gamma_3\neq0$, we see clearly that the term under the square
root remains imaginary, which breaks the criterion of strong-to-weak
coupling transition at zero detuning. The imaginary parts
of~$\omega^\pm_r$ determine the stability of the
solutions. Straightforward calculations lead to:
\begin{align}
\mathrm{Im}[\omega_r^\pm]=-(\gamma_1+\gamma_2)-\sqrt{\frac{-\mathbb{IM}+\sqrt{\mathbb{IM}^2+\mathbb{RE}^2}}{2}}\,,
\end{align}
%
where~$\mathbb{IM}=-2\delta_\mathbf{k}(\gamma_1-\gamma_2)-8g\gamma_3$~and~$\mathbb{RE}=\delta_\mathbf{k}^2-(\gamma_1-\gamma_2)^2+4(g^2-\gamma_3^2)$. Clearly,
it can be seen that the imaginary part of the~$\omega_r^\pm$, for
given parameters of the system, always remains negative, which results
in stable solutions. One direct consequence of such stability is the
resistance of the system against phase transitions, which is the case
in presence of interactions and pumping. Clearly, the combination of
decay and interaction makes the dynamics richer and their full effects
will be discussed in future works.
\section{Conclusion}
\label{sec:five}
In conclusion, we study the internal dynamic of polariton in the
Keldysh functional approach, when fields are removed from both bare
and dressed states. In the linear Rabi regime, the coupled equations
of motions are local in reciprocal space, and the intrinsic detuning
between bare states works as an intrinsic potential affecting the
dynamic. It is shown also that the equations of motions are in the
form of Josephson equations, but that the upper-polariton lifetime
(correlated decay of the dressed state) brings a peculiar feature in
the dynamics, namely, it mediates an internal dynamic between the bare
states. This would happen even if the bare states would be decoupled
(although then the origin for their correlated decay would be less
clear on physical grounds). Considering the retarded Green functions,
we show that the dynamic in the Rabi regime is stable, and the
criterion of strong coupling is fragile in presence of an upper
polariton decay.
\begin{acknowledgments}
We thanks F.P. Laussy for having suggested some aspects of this
problem and for discussions.
\end{acknowledgments}
|
2,869,038,154,849 | arxiv | \section{Introduction}
Conventional time series analysis is heavily dependent on the assumption of stationarity. But this assumption is unsatisfactory for modelling many real-life phenomena that exhibit seasonal behaviour. Seasonal variations in the mean of time series data can be easily removed by a variety of methods, but when the variance varies with the season, the use of periodic time series models is suggested. In order to model periodicity in autocorrelations, a class of Periodic Autoregressive Moving Average (PARMA) models was introduced. The PARMA$(p,q)$ system is given by:
\begin{eqnarray}\label{parma}
X_n-\sum\limits_{j=1}^pb_j(n)X_{n-j}=\sum_{i=0}^{q-1}a_i(n)\xi_{n-i},
\end{eqnarray}
where $n\in Z$ and the coefficients $\{b_j(n)\}_{j=1}^{p}$ and $\{a_i(n)\}_{i=0}^{q-1}$ are nonzero sequences periodic in $n$ with the same period $T$ while the innovations $\{\xi_n\}$ are independent Gaussian random variables.
For such PARMA models, the covariance function is a tool for describing the dependence structure of the time series and we can recall that the sequence given by (\ref{parma}) is periodically correlated, more precisely the covariance function $Cov(X_n,X_{n+k})$ is $T-$periodic in $n $ for every $k$ (see \cite{gladyshev, makms91, ww}).
As the coefficients in (\ref{parma}) are periodic, it is obvious that the class of PARMA models is an extension of the class of commonly used ARMA models.
Due to their interesting properties, PARMA systems have received much attention in the literature and turned out to be an alternative to the conventional stationary time series as they allow for modelling many phenomena in various areas, e.g., in hydrology (\cite{vec1}), meteorology (\cite{blo}), economics (\cite{bmww, parpag}) and electrical engineering (\cite{garfra}).
The assumption of normality for the observations seems not to be reasonable in the number of applications, such as signal processing, telecommunications, finance, physics and chemistry, and heavy-tailed distributions seem to be more appropriate, see e.g. \cite{mit}. An important class of distributions in this context is the class of $\alpha$-stable (stable) distributions because it is flexible for data modelling and includes the Gaussian distribution as a special case. The importance of this class of distributions is strongly supported by the limit theorems which indicate that the stable distribution is the only possible limiting distribution for the normed sum of independent and identically distributed random variables. Stable random variables have found many practical applications, for instance in finance (\cite{mit}), physics (\cite{janwer}), electrical engineering (\cite{stuckkleiner}).
PARMA models with symmetric stable innovations combine the advantages of classical PARMA models and stable distributions -- they offer an alternative for modelling periodic time series with heavy tails. However, in this case the covariance function is not defined and thus other measures of dependence have to be used. The most popular are the covariation and codifference presented in \cite{st, nw, n}.
In this paper we consider a special case of stable PARMA models, i.e. PARMA(1,1) systems with symmetric $\alpha$-stable innovations. In this case we rewrite the equation (\ref{parma}) in a simple way, i.e.:
\begin{eqnarray}\label{parma12}
X_n-b_nX_{n-1}=a_n\xi_{n},
\end{eqnarray}
where $n\in Z$ and the coefficients $\{b_n\}$ and $\{a_n\}$ are nonzero periodic sequences with the same period $T$ while the innovations are independent symmetric $\alpha$-stable (S$\alpha$S for short) random variables given by the following characteristic function:
\begin{eqnarray}\label{innovations}
E\exp(i\theta \xi_n)=\exp(-\sigma^{\alpha}|\theta|^{\alpha}), ~0<\alpha\leq2,
\end{eqnarray}
where $\sigma$ denotes the scale parameter. Let us define $P=b_1 b_2\dots b_T$ and $B_r^s=\prod_{j=r}^sb_j$, with the convention $B_r^s=1$ if $r>s$.
As the covariance function is not defined for stable random vectors, in Section \ref{s_mofdep} we present two other measures of dependence that can be used for symmetric stable time series -- the covariation and the codiffrence. In Section \ref{s_bsol} we discuss the necessary conditions for existence of the bounded solution of PARMA(1,1) systems for $1<\alpha\leq2$ and we note that results obtained in \cite{ww} for PARMA systems with Gaussian innovations can be extended to the case of stable innovations. The covariation and the codifference for PARMA(1,1) models with stable innovations are studied in Section \ref{s_depstr} and the asymptotic relation between these two measures of dependence for the considered models is examined there. We find it interesting to illustrate theoretical results and thus in Section \ref{s_ex} we give an example of PARMA(1,1) systems with S$\alpha$S innovations and illustrate the periodicity of the considered measures of dependence and the asymptotic relation between them.
\section{Measures of dependence for stable time series}\label{s_mofdep}
Let $X$ and $Y$ be jointly S$\alpha$S and let $\Gamma$ be the spectral measure of the random vector $(X,Y)$ (see for instance \cite{st}). If $\alpha<2$ then the covariance is not defined and thus other measures of dependence have to be used. The most popular measures are: the {\bf covariation $CV(X,Y)$} of $X$ on $Y$ defined in the following way:
\begin{eqnarray}
CV(X,Y)=\int_{S_2}s_1s_2^{<\alpha-1>}\Gamma(d{\bf s}), \quad 1<\alpha\leq2,
\end{eqnarray}
where ${\bf s}=(s_1,s_2)$ and the signed power $z^{<p>}$ is given by $z^{<p>}=|z|^{p-1}\bar{z}$, and the {\bf codifference $CD(X,Y)$} of $X$ on $Y$ defined for $0<\alpha\leq2$:
\begin{eqnarray}
CD(X,Y)=\ln E\exp\{i(X-Y)\}-\ln E\exp\{iX\}-\ln E\exp\{-iY\}.
\end{eqnarray}
Properties of the considered measures of dependence one can find in \cite{st}. Let us only mention here that, in contrast to the codifference, the covariation is not symmetric in its arguments. Moreover, when $\alpha=2$ both measures reduce to the covariance, namely
\begin{eqnarray}\label{covcvcd}
Cov(X,Y)=2CV(X,Y)=CD(X,Y).
\end{eqnarray}
The covariation induces a norm on the linear space of jointly S$\alpha$S random variables $S_{\alpha}$ and this norm is equal to the scale parameter, see \cite{st}. Hence throughout this paper the norm (so-called covariation norm) $||X||_{\alpha}$ is defined by $||X||_{\alpha}=(CV(X,X))^{1/\alpha}$, for a S$\alpha$S random variable $X$ and $\alpha>1$. The sequence $\{X_n\}_{n \in Z}$, is bounded in a space $S_{\alpha}$ with norm $||.||_{\alpha}$ if $\sup_{n \in Z}||X_n||_{\alpha}^{\alpha}<\infty$. Moreover, in this paper we write $X=Y$ in $S_{\alpha}$ if and only if $||X-Y||_{\alpha}=0$.
If it is possible to transform the sequence $\{X_n\}$ to the moving average representation $X_n=\sum_{j=-\infty}^{\infty}c_j(n)\xi_{n-j}$, where the innovations $\{\xi_n\}$ are independent S$\alpha$S random variables with parameters $\sigma$ and $1<\alpha \leq 2$, then both the covariation and the codifference can be expressed in terms of the coefficients $c_j(n)$ (see \cite{nwylom}):
\begin{eqnarray*}\label{lemat11}
CV(X_n, X_m)=\sigma^{\alpha}\sum_{j=-\infty}^{\infty}c_j(n)c_{m-n+j}(m) ^{<\alpha-1>},
\end{eqnarray*}
\begin{eqnarray*}\label{lemat12}
CD(X_n, X_m)=\sigma^{\alpha}\sum_{j=-\infty}^{\infty}\left(|c_j(n)|^{\alpha}+|c_{m-n+j}(m)| ^{\alpha}-|c_j(n)-c_{m-n+j}(m)|^{\alpha}\right).
\end{eqnarray*}
\section{Bounded solution of stable PARMA(1,1) system}\label{s_bsol}
In this section we consider PARMA(1,1) system given by (\ref{parma12}) with the S$\alpha$S innovations for $1<\alpha<2$ and we investigate when the bounded solution exists.
Let us assume that $|P|<1$. In this case we show that the considered PARMA(1,1) system has a bounded solution given by the formula:
\begin{eqnarray}\label{forma1}
X_n=\sum\limits_{s=0}^{\infty}B^{n}_{n-s+1}a_{n-s}\xi_{n-s}.
\end{eqnarray}
Provided that every $s\in Z$ can be represented as $s=NT+j$ for some $N=0,1,\dots$ and $j=0,1,\dots, T-1$, we have
\[||X_n||^{\alpha}_{\alpha}=\sigma^{\alpha}\sum\limits_{s=0}^{\infty}\left|B^{n}_{n-s+1}a_{n-s}\right|^{\alpha}=
\sigma^{\alpha}\sum\limits_{N=0}^{\infty}\sum\limits_{j=0}^{T-1}\left|B^{n}_{n-NT-j+1}a_{n-NT-j}\right|^{\alpha}.\]
Now it sufficies to notice that by periodicity of coefficients \mbox{$a_{n-NT-j}=a_{n-j}$} and $B^{n}_{n-NT-j+1}=P^NB^{n}_{n-j+1}$ and that $\sum\limits_{N=0}^{\infty}|P|^{N\alpha}=1/(1-|P|^{\alpha})$. Thus
\[||X_n||^{\alpha}_{\alpha}=\frac{\sigma^{\alpha}}{1-|P|^{\alpha}}\sum\limits_{j=0}^{T-1}\left|B^{n}_{n-j+1}a_{n-j}\right|^{\alpha}
\leq \frac{\sigma^{\alpha}K}{1-|P|^{\alpha}}<\infty,\]
where $K$ is a real constant, $K=\max_{s=1,\dots,T}\{\sum_{j=0}^{T-1}\left|B^{s}_{s-j+1}a_{s-j}\right|^{\alpha}\}$. This implies that $\sup_{n\in Z}||X_n||^{\alpha}_{\alpha}<\infty$ and hence $\{X_n\}$ given by (\ref{forma1}) is bounded. Moreover, it is easy to check that $\{X_n\}$ given by (\ref{forma1}) satisfies equation (\ref{parma12}).
To prove the converse, let us note that iterating the equation (\ref{parma12}) yields:
\[X_{n}=B^{n}_{n-k+1}X_{n-k}+\sum_{s=0}^{k-1}B^{n}_{n-s+1}a_{n-s}\xi_{n-s}.\]
For each $n\in Z$ and $k=NT+j$ by periodicity of the coefficients we have $B_{n-k+1}^n=P^N B_{n-j+1}^n$. Therefore, if $|P|<1$ and $\{X_n\}$ is a bounded solution of (\ref{parma12}) then
\begin{eqnarray*}\label{lim}
\lim_{k\rightarrow\infty}||X_n-\sum_{s=0}^{k-1}B_{n-s+1}^{n}a_{n-s}\xi_{n-s}||_{\alpha}=\lim_{k\rightarrow\infty}||X_{n-k}B^{n}_{n-k+1}||_{\alpha}=0.
\end{eqnarray*}
This means that the bounded solution of the system considered in this section is given by
\[X_n= \lim_{k\rightarrow\infty}\sum_{s=0}^{k-1}B^{n}_{n-s+1}a_{n-s}\xi_{n-s}=\sum_{s=0}^{\infty}B_{n-s+1}^{n}a_{n-s}\xi_{n-s}.\]
In a similar manner, it can be shown, that if $|P|>1$, then the bounded solution of the system under study is given by
\begin{eqnarray}\label{forma2}
X_n=-\sum\limits_{s=1}^{\infty}\frac{a_{n+s}}{B^{n+s}_{n+1}}\xi_{n+s}.
\end{eqnarray}
It is worth pointing out here that the solution for the PARMA(1,1) system with S$\alpha$S innovations takes the same form of the moving average as in the case of Gaussian innovations obtained in \cite{ww} and it reduces to well-known formula for ARMA models in case of constant coefficients.
\section{Dependence structure of stable PARMA(1,1) models}\label{s_depstr}
Let us consider PARMA(1,1) system given by (\ref{parma12}) with S$\alpha$S innovations for $1<\alpha\leq2$. Moreover, we will restrict our attention to the case $|P|<1$ because this case is more important for applications -- as shown in Section \ref{s_bsol}, the considered system has a bounded solution $\{X_n\}$ in the linear space of jointly S$\alpha$S random variables with norm $||.||_{\alpha}$ given by the causal moving average representation (\ref{forma1}).
In order to investigate the dependence structure of stable PARMA(1,1) models we will first rewrite $X_n$ as
$X_n=\sum_{s=-\infty}^{\infty}c_s(n)\xi_{n-s}$, where
\begin{eqnarray}\label{cspmn1}
c_s(n)=
\left\{\begin{array}{ll} 0, & \mbox{if}~ s<0,\\
B^{n}_{n-s+1}a_{n-s},& \mbox{if}~ s\geq 0,
\end{array}\right.
\end{eqnarray}
and then using results presented in Section \ref{s_mofdep} we will find formulas for the covariation and the codifference. We will also study periodicity of these measures and their asymptotic relation.
{\bf The covariation.}
If $n\geq m$, then for $1<\alpha \leq 2$ we have
\[CV(X_n,X_{m})=\sigma^{\alpha}\sum\limits_{j=n-m}^{\infty}B^n_{n-j+1}a_{n-j}\left(B^{m}_{n-j+1}a_{n-j}\right)^{<\alpha-1>}.\]
It is easy to notice that $z\cdot z^{<\alpha-1>}=|z|^{\alpha}$ and $B^{n}_{n-j+1}=B^{n}_{m+1}B^{m}_{n-j+1}$. Therefore, for $s=j-n+m$ we obtain
\[CV(X_n,X_{m})=\sigma^{\alpha}B_{m+1}^{n}\sum\limits_{s=0}^{\infty}\left|B_{m-s+1}^{m}a_{m-s}\right|^{\alpha}.\]
As every $s\in Z$ can be represented as $s=NT+j$ for some $N=0,1,\dots$ and $j=0,1,\dots, T-1$, we have
\[CV(X_n,X_{m})=\sigma^{\alpha}B_{m+1}^{n}\sum\limits_{N=0}^{\infty}\sum\limits_{j=0}^{T-1}\left|B_{m-NT-j+1}^{m}a_{m-NT-j}\right|^{\alpha}.\]
Now it suffices to notice that by periodicity of coefficients \mbox{$a_{m-NT-j}=a_{m-j}$} and $B^{m}_{m-NT-j+1}=P^NB^{m}_{m-j+1}$ and that $\sum\limits_{N=0}^{\infty}|P|^{N\alpha}=1/(1-|P|^{\alpha})$. Thus
\begin{eqnarray}\label{cv1}
CV(X_n,X_{m})=\sigma^{\alpha}\frac{B_{m+1}^{n}}{1-|P|^{\alpha}}\sum\limits_{j=0}^{T-1}\left|B_{m-j+1}^{m}a_{m-j}\right|^{\alpha}.
\end{eqnarray}
If $n<m$, then the covariation is given by
\[CV(X_n,X_{m})=\sigma^{\alpha}\sum\limits_{j=0}^{\infty}B^{n}_{n-j+1}a_{n-j}\left(B^{m}_{n-j+1}a_{n-j}\right)^{<\alpha-1>}.\]
In this case $B^{n}_{n-j+1}=B^{m}_{n-j+1}/B^{m}_{n+1}$ which results in formula
\[CV(X_n,X_{m})=\frac{\sigma^{\alpha}}{B_{n+1}^{m}}\sum\limits_{j=0}^{\infty}\left|B_{n-j+1}^{m}a_{n-j}\right|^{\alpha}.\]
The simple calculation similar like in the previous case leads us to the result
\begin{eqnarray}\label{cv2}
CV(X_n,X_{m})=\frac{\sigma^{\alpha}}{B_{n+1}^{m}(1-|P|^{\alpha})}\sum\limits_{j=0}^{T-1}\left|B_{n-j+1}^{m}a_{n-j}\right|^{\alpha}.
\end{eqnarray}
As the sequences $\{a_n\}$ and $\{b_n\}$ are periodic in $n$ with period $T$, the covariation is periodic in $n$ and $m$ with the same period, indeed $CV(X_n,X_m)=CV(X_{n+T},X_{m+T})$.
{\bf The codifference.}
For $1<\alpha \leq 2$ and $n\geq m$ we have to calculate
\[\sum\limits_{j=n-m}^{\infty}\left(\left|B^n_{n-j+1}a_{n-j}\right|^{\alpha}+\left|B^{m}_{n-j+1}a_{n-j}\right|^{\alpha}-\left|B^n_{n-j+1}a_{n-j}-B^m_{n-j+1}a_{n-j}\right|^{\alpha}\right).\]
Observe that there are $a_{n-j}$ and $B^{m}_{n-j+1}$ (as $B^{n}_{n-j+1}=B^{m}_{n-j+1}B^{n}_{m+1}$) in each part of the above formula. Therefore we can write
\[CD(X_n,X_{m})=\sigma^{\alpha}\left(1+|B_{m+1}^n|^{\alpha}-|1-B_{m+1}^n|^{\alpha}\right)\sum\limits_{j=n-m}^{\infty}|B_{n-j+1}^{m}a_{n-j}|^{\alpha},\]
that can be transformed to
\[CD(X_n,X_{m})=\sigma^{\alpha}\left(1+|B_{m+1}^n|^{\alpha}-|1-B_{m+1}^n|^{\alpha}\right)\sum\limits_{s=0}^{\infty}|B_{m-s+1}^{m}a_{m-s}|^{\alpha}.\]
And now it is sufficient to notice that the sum $\sum_{s=0}^{\infty}|B_{m-s+1}^{m}a_{m-s}|^{\alpha}$ has been already calculated for the covariation. So we obtain
\begin{eqnarray}\label{cd1}
CD(X_n,X_{m})=\sigma^{\alpha}\frac{1+|B_{m+1}^n|^{\alpha}-|1-B_{m+1}^n|^{\alpha}}{1-|P|^{\alpha}}\sum\limits_{j=0}^{T-1}|B_{m-j+1}^{m}a_{m-j}|^{\alpha}.
\end{eqnarray}
It is not necessary to calculate the codifference for $n<m$ as this measure is symmetric in its arguments, i.e. for every $n,m\in Z$ we have $CD(X_n,X_{m})=CD(X_m,X_{n})$.
By the assumption, the coefficients $\{a_n\}$ and $\{b_n\}$ are periodic in $n$ with period $T$. Thus it is not difficult to notice that
$CD(X_n,X_m)=CD(X_{n+T},X_{m+T})$, which means that the codifference is periodic in $n$ and $m$ with the period $T$.
{\bf Asymptotic relation between CV and CD.}
Using formulas (\ref{cv1}), (\ref{cv2}) and (\ref{cd1}) and two simple facts that hold for $1<\alpha<2$, $0<|P|<1$ and any real constant $a\neq0$:
\[\lim\limits_{m\rightarrow\infty}\frac{1+|aP^m|^{\alpha}-\left|1-aP^m\right|^{\alpha}}{aP^m}=\alpha,\]
\[\lim\limits_{m\rightarrow\infty}\frac{aP^m\left(1+\left|aP^m\right|^{\alpha}-\left|1-aP^m\right|^{\alpha}\right)}{\left|aP^m\right|^{\alpha}}=0,\]
it can be proved that for the considered PARMA(1,1) systems with S$\alpha$S innovations the following relations hold for each $n\in Z$ and $1<\alpha<2$:
\begin{eqnarray}\label{glowne11}
\lim\limits_{k\rightarrow \infty}\frac{CD(X_n,X_{n-k})}{CV(X_n,X_{n-k})}=\lim\limits_{k\rightarrow \infty}\frac{CD(X_{n+k},X_{n})}{CV(X_{n+k},X_{n})}=\alpha,
\end{eqnarray}
\begin{eqnarray}\label{glowne12}
\lim\limits_{k\rightarrow \infty}\frac{CD(X_{n-k},X_{n})}{CV(X_{n-k},X_{n})}=\lim\limits_{k\rightarrow \infty}\frac{CD(X_{n},X_{n+k})}{CV(X_{n},X_{n+k})}=0.
\end{eqnarray}
The interesting point is that, because of asymmetry of the covariance, we obtained quite different asymptotic results. Moreover, (\ref{glowne11}) extends the results obtained in \cite{nw} for ARMA models and it reduces to (\ref{covcvcd}) for $\alpha=2$.
\section{Example}\label{s_ex}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9cm]{NW_fig1} \caption[The realizations of PARMA(1,1) model with S$\alpha$S innovations for $\alpha=2$ (top panel), $\alpha=1.7$ (middle panel) and $\alpha=1.4$ (bottom panel).]{}\label{fig_real}
\end{center}
\end{figure}
In order to illustrate our theoretical results let us consider PARMA(1,1) model with S$\alpha$S innovations with $\sigma=1$, where the coefficients are given by
\begin{eqnarray*}
b_n = \left\{
\begin{array}{ll} \displaystyle
0.5 &\mbox{if} ~ n=1,4,7,\dots,\\
\displaystyle 1.6,& \mbox{if} ~ n=2,5,8,\dots,\\
\displaystyle 0.4,& \mbox{if} ~ n=3,6,9,\dots,\\
\end{array}
\right. \quad a_n = \left\{
\begin{array}{ll} \displaystyle
1& \mbox{if} ~ n=1,4,7,\dots,\\
\displaystyle 2,& \mbox{if} ~ n=2,5,8,\dots,\\
\displaystyle 0.003,& \mbox{if} ~ n=3,6,9,\dots.\\
\end{array}
\right.
\end{eqnarray*}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9cm]{NW_fig2} \caption[The codifference $CD(X_n,X_{n+k})$ (top panel) and the covariation $CV(X_n,X_{n+k})$ (bottom panel) for PARMA(1,1) model with S$\alpha$S innovations for $\alpha=1.7$ (solid line) and $\alpha=1.4$ (dotted line) vs $ n=3,4,\dots,20$ for $k=5$.]{}\label{fig_miary}
\end{center}
\end{figure}
It is clear that the coefficients are periodic with $T=3$ and in this case $P=0.32$. Therefore we are allowed to use formulas obtained in Sections \ref{s_bsol} and \ref{s_depstr}. We first want to demonstrate how the parameter $\alpha$ influences the behaviour of the time series, so we plot 1000 realizations of the considered model for $\alpha=2$, $\alpha=1.7$ and $\alpha=1.4$, see Figure \ref{fig_real}. It is easy to notice that the smaller $\alpha$ we take, the greater values of the time series can appear (property of heavy-tailed distributions). Next, we want to show the dependence structure of the considered model, especially periodicity of measures of dependence. In order to do this we plot the codifference $CD(X_n,X_{n+k})$ and the covariation $CV(X_n,X_{n+k})$ for $\alpha=1.7$ and $\alpha=1.4$ in case of $k=5$, see Figure \ref{fig_miary}. Although the behaviour of the measures of dependence depends on the parameter $\alpha$, one can observe that both measures are periodic with the same period $T=3$.
Finally, let us illustrate the asymptotic relation between the covariation and the codifference that is studied in Section \ref{s_depstr}. Figure \ref{fig_asym} contains plots of the functions $\frac{CD(X_{n+k},X_{n})}{\alpha CV(X_{n+k},X_{n})}$, $\frac{CD(X_{n},X_{n-k})}{\alpha CV(X_{n},X_{n-k})}$, $\frac{CD(X_{n},X_{n+k})}{\alpha CV(X_{n},X_{n+k})}$ and $\frac{CD(X_{n-k},X_{n})}{\alpha CV(X_{n-k},X_{n})}$ for $ k=0,1,\dots,40 $, $n=50$ and $\alpha=1.7$ and $\alpha=1.4$.
According to the theoretical results, the first two quotients tend to 1 and the next two tend to 0 as $k$ increases.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=9cm]{NW_fig3} \caption[The plots of the functions $\frac{CD(X_{n+k},X_{n})}{\alpha CV(X_{n+k},X_{n})}$ (the first panel), $\frac{CD(X_{n},X_{n-k})}{\alpha CV(X_{n},X_{n-k})}$ (the second panel), $\frac{CD(X_{n},X_{n+k})}{\alpha CV(X_{n},X_{n+k})}$ (the third panel) and $\frac{CD(X_{n-k},X_{n})}{\alpha CV(X_{n-k},X_{n})}$ (the fourth panel) vs $ k=0,1,\dots,40 $ for $n=50$, $\alpha=1.7$ (solid line) and $\alpha=1.4$ (dotted line).]{}\label{fig_asym}
\end{center}
\end{figure}
|
2,869,038,154,850 | arxiv | \section{Introduction}
Globular clusters are ideal sites for the formation of binary systems
hosting a compact object thanks to the frequent dynamical interaction
caused by their dense environment \citep{1997A&ARv...8....1M}. Low mass
X-ray binaries (LMXB) formed by a neutron star (NS) that accretes
matter lost by a companion, low mass star are particularly favored, as
stellar encounters may cause the lower mass star of a binary to be
replaced by an heavier NS \citep{Verbunt.etal:87}. Some of the densest
and most massive globular clusters have the highest predicted rates of
stellar interactions and host a numerous population of LMXBs
\citep{Heinke.etal:03b}.
Terzan 5 is a compact, massive cluster at a distance of 5.5 kpc
\citep{2007A&A...470.1043O} which hosts at least three
stellar populations with different iron abundances; the observed
chemical pattern suggests that it was much more massive in the past,
so to be able to hold the iron rich ejecta of past supernova
explosions \citep{2009Natur.462..483F,2013ApJ...779L...5O},
and \citep[][]{ferraro2016}. Terzan 5
has the highest stellar interaction rate than any cluster in the
Galaxy \citep{Verbunt.etal:87,Heinke.etal:03a,Bahramian:2013}.
This reflects into the largest population known of millisecond radio pulsars
\citep[34;][]{Ransom.etal:05, Hessels.etal:06}, and in at least 50
X-ray sources, including a dozen likely quiescent LMXBs
\citep{Heinke.etal:06b}. The populations of millisecond radio pulsars
and LMXBs are linked from an evolutionary point of view, as mass
accretion in a LMXB is expected to speed up the rotation of a NS down
to a spin period of a few milliseconds \citep{alpar1982}. This link
was confirmed by the discovery of accreting millisecond pulsars
\citep[AMSPs;][]{wijnands1998}, and by the observations of binary
millisecond pulsars swinging between a radio pulsar and an accretion
disk state on time scales that can be as short as weeks
\citep{archibald2009,Papitto.etal:13,Bassa.etal:14}. Globular clusters like Terzan 5
are preferential laboratories to study the relation between these two
classes of sources.
Many LMXBs are X-ray transients; they show outbursts lasting typically
a few-weeks and characterized by a high X-ray luminosity ($L_{x}
\approx 10^{36}-10^{38} $~{~erg~s$^{-1}$}), while for most of the time they
are found in X-ray quiescence ($L_X\approx10^{31} -
10^{33}$~{~erg~s$^{-1}$}). X-ray transient activity has been frequently
observed from Terzan 5 since 1980s
\citep{Makishima.etal:81,Warwick.etal:88,Verbunt.etal:95} and ten
outbursts have been detected ever since \citep[see, e.g., Table~1
in][]{degenaar2012}. The large number of possible counterparts in
the cluster complicates the identification of the transient
responsible for each event when a high spatial resolution X-ray (or
radio) observation was not available. As a consequence, only three
X-ray transients of Terzan 5 have been securely identified, {EXO~1745--248}
\citep[Terzan 5 X--1, active in 2000, 2011 and
2015][]{Makishima.etal:81,2000IAUC.7454....1M,Heinke.etal:03a,2012PASJ...64...91S,tetarenko2016},
IGR J17480--2446 \citep[Terzan 5 X--2, active in
2010;][]{2011A&A...526L...3P,2011MNRAS.414.1508M} and Swift
J174805.3--244637 \citep[Terzan 5 X--3, active in
2012;][]{2014ApJ...780..127B}.
The first confirmed outburst observed from {EXO~1745--248} took place in 2000,
when a \emph{Chandra} observation could pin down the location of the
X-ray transient with a sub-arcsecond accuracy
\citep{Heinke.etal:03a}. The outburst lasted $\sim100$~d, showing a
peak of X-ray luminosity\footnote{Throughout this paper we evaluate
luminosities and radii for a distance of 5.5 kpc, which was
estimated by \citet{2007A&A...470.1043O} with an uncertainty of 0.9
kpc. There is also an determination from \citet{Valenti.etal:07} for the distance (5.9kpc) consistent within errors with Ortolani's distance.} $\sim6\times10^{37}$~{~erg~s$^{-1}$} \citep[][]{degenaar2012}. The
X-ray spectrum was dominated by thermal Comptonization in a cloud with
a temperature ranging between a few and tens of keV
\citep{Heinke.etal:03a,Kuulkers.etal:03}; a thermal component at
energies of $\approx 1$ keV, and a strong emission line at energies
compatible with the Fe K-$\alpha$ transition were also present in the
spectrum. More than 20 type-I X-ray bursts were observed, in none of
which burst oscillations could be detected \citep{galloway2008}. Two
of these bursts showed evidence of photospheric radius expansion, and
were considered by \citet{ozel2009} to draw constraints on the mass
and radius of the NS. A second outburst was observed from {EXO~1745--248} in
2011, following the detection of a superburst characterized by a decay
timescale of $\approx 10$ hr \citep{2012MNRAS.426..927A,
Serino.etal:12}. The outburst lasted $\approx20$~d, reaching an
X-ray luminosity of $9\times10^{36}$~{~erg~s$^{-1}$}. \citet{degenaar2012}
found a strong variability of the X-ray emission observed during
quiescence between the 2000 and the 2011 outburst, possibly caused by
low-level residual accretion.
A new outburst from Terzan 5 was detected on 2015 March, 13
\citep{2015ATel.7240....1A}. It was associated to {EXO~1745--248} based on the
coincidence between its position \citep{Heinke.etal:06b} and the
location of the X-ray source observed by Swift XRT
\citep{2015ATel.7247....1L} and of the radio counterpart detected by
the Karl G. Jansky Very Large Array
\citep[VLA;][]{2015ATel.7262....1T}, with an accuracy of 2.2 and 0.4
arcsec, respectively. The outburst lasted $\approx100$ d and attained
a peak X-ray luminosity of $10^{38}$~{~erg~s$^{-1}$}, roughly a month into
the outburst \citep{tetarenko2016}. The source performed a transition
from a hard state (characterized by an X-ray spectrum described by a
power law with photon index ranging from 0.9 to 1.3) to a soft state
(in which the spectrum was thermal with temperature of
$\approx2$--$3$~keV) a few days before reaching the peak flux
\citep{2015ATel.7430....1Y}. The source transitioned back to the hard
state close to the end of the outburst. \citet{tetarenko2016} showed
that throughout the outburst the radio and X-ray luminosity correlated
as $L_R \propto L_X^{\beta}$ with $\beta=1.68^{+0.10}_{-0.09}$,
indicating a link between the compact jet traced by the radio emission
and the accretion flow traced by the X-ray output. The optical
counterpart was identified by \citet{ferraro2015}, who detected the
optical brightening associated to the outburst onset in {\it Hubble
Space Telescope} images; the location of the companion star in the
color-magnitude diagram of Terzan 5 is consistent with the main
sequence turn-off. We stress that the HST study suggests that {EXO~1745--248}
is in an early phase of accretion stage with the donor expanding and
filling its Roche lobe thus representing a prenatal stage of a millisecond
pulsar binary. This would make more interesting the study of this
source as well as linking what we stated above regarding rotation-powered
MSPs and AMSPs.
Here we present an analysis of the X-ray properties of {EXO~1745--248}, based
on an {\it XMM-Newton} observation performed $\approx10$ days into the
2015 outburst, when the source was in the hard state.
The main goal of this paper is to observe at a better statistics the region of spectrum
around the iron line. Then adding the broad-band coverage allowed by {\it INTEGRAL} observations,
we are able to study the possible associated reflection features and give a definite
answer on the origin of the iron line.
We also make use of additional monitoring observations of the source
carried out with {\it INTEGRAL} during its 2015 outburst to spectroscopically
confirm the hard-to-soft spectral state transition displayed by EXO~1745-248
around 57131~MJD \citep[as previously reported by][]{tetarenko2016}.
We stress out that this transition was observed by {\it Swift}. In order to
understand the physical properties of this state, we performed an observation
with {\it XMM-Newton } allowing more sensitive and higher resolution data.
We focus in Sec.~\ref{sec:spectrum} on the shape of the X-ray spectrum and in
Sec.~\ref{sec:timing} on the properties of the temporal variability, while
an analysis of the X-ray bursts observed during the considered
observations is presented in Sec.~\ref{sec:burst}.
\section{Observations and Data Reduction}
\label{sec:reduc}
\subsection{XMM-Newton}
{\it XMM-Newton} observed {EXO~1745--248} for 80.8 ks starting on 2015, March 22 at 04:52
(UTC; ObsId 0744170201). Data were reduced using the SAS (Science
Analysis Software) v.14.0.0.
The EPIC-pn camera observed the source in timing mode to achieve a
high temporal resolution of 29.5~$\mu$s and to limit the effects of
pile-up distortion of the spectral response during observations of
relatively bright Galactic X-ray sources. A thin optical blocking
filter was used. In timing mode the imaging capabilities along one of
the axis are lost to allow a faster readout. The maximum number of
counts fell on the RAWX coordinates 36 and 37. To extract the source
photons we then considered a 21 pixel-wide strip extending from
RAWX=26 to 46. Background photons were instead extracted in the region ranging
from RAWX=2 to RAWX=6. Single and double events were retained. Seven
type-I X-ray bursts took place during the {\it XMM-Newton} observation with a
typical rise time of less than 5~s and a decay e-folding time scale
ranging from 10 to 23~s. In order to analyze the {\it persistent}
(i.e. non-bursting) emission of {EXO~1745--248} we identified the start time of
each burst as the first 1 s-long bin that exceeded the average
count-rate by more than 100~ s$^{-1}$, and removed from the analysis a
time interval spanning from 15~s before and 200~s after the burst
onset. After the removal of the burst emission, the mean count rate
observed by the EPIC-pn was 98.1~s$^{-1}$. Pile-up was not expected to
affect significantly the spectral response of the EPIC-pn at the
observed {\it persistent} count rate (Guainazzi et al. 2014; Smith et
al. 2016)\footnote{http://xmm2.esac.esa.int/docs/documents/CAL-TN-0083.pdf,
http://xmm2.esac.esa.int/docs/documents/CAL-TN-0018.pdf}. To check
the absence of strong distortion we run the SAS task {\it epatplot},
and obtained that the fraction of single and double pattern events
falling in the 2.4--10 keV band were compatible with the expected value
within the uncertainties. Therefore, no pile-up correction method was
employed. The spectrum was re-binned so to have not more than three
bins per spectral resolution element, and at least 25 counts per
channel.
The MOS-1 and MOS-2 cameras were operated in Large Window and Timing
mode, respectively. At the count rate observed from {EXO~1745--248} both
cameras suffered from pile up at a fraction exceeding $10\%$ and were
therefore discarded for further analysis.
We also considered data observed by the Reflection Grating
Spectrometer (RGS), which operated in Standard Spectroscopy mode. We
considered photons falling in the first order of diffraction. The same
time filters of the EPIC-pn data analysis were applied.
\section{INTEGRAL}
\label{sec:integral}
We analyzed all {\it INTEGRAL} \citep{w03} available data collected in the direction
of {EXO~1745--248} during the source outburst in 2015. These observations included both
publicly available data and our proprietary data in AO12 cycle.
The reduction of the {\it INTEGRAL} data was performed using the standard Offline
Science Analysis (OSA) version 10.2 distributed by the ISDC \citep{courvoisier03}.
{\it INTEGRAL} data are divided into science windows (SCW), i.e. different pointings
lasting each $\sim 2-3$\,ks. We analyzed data from the IBIS/ISGRI \citep{ubertini03,lebrun03},
covering the energy range 20-300~keV energy band, and from the two
JEM-X monitors \citep{lund03}, operating in the range 3-20~keV.
As the source position varied with respect to the aim point of the
satellite during the observational period ranging from 2015 March 12 at 19:07
(satellite revolution 1517) to 2015 April 28 at 11:40 UTC (satellite
revolution 1535), the coverage provided by IBIS/ISGRI was generally much
larger than that of the two JEM-X monitors due to their smaller field of view.
As the source was relatively bright during the outburst, we extracted a lightcurve
with the resolution of 1 SCW for both IBIS/ISGRI and the two JEM-X units.
This is shown in Fig.~\ref{fig:intlc}, together with the monitoring observations
provided by Swift/XRT (0.5-10~keV). The latter data were retrieved from the
Leicester University on-line analysis tool \citep{evans09} and used only to
compare the monitoring provided by the Swift and {\it INTEGRAL} satellites.
We refer the reader to \citet{tetarenko2016} for more details on the Swift
data and the corresponding analysis. In agreement with the results discussed
by these authors, also the {\it INTEGRAL} data show that the source underwent a
hard-to-soft spectral state transition around 57131~MJD. In order to prove
this spectral state change more quantitatively, we extracted two sets
of {\it INTEGRAL} spectra accumulating all data before and after this date for
ISGRI, JEM-X1, and JEM-X2.
Analysis of broad-band {\it INTEGRAL} spectra for both hard and soft state is reported
in Sect. \ref{sec:integral_spec}.
We also extracted the ISGRI and JEM-X data by using only the observations
carried out during the satellite revolution 1521, as the latter partly overlapped
with the time of the {\it XMM-Newton} observation. The broad-band fit of the combined
quasi-simultaneous {\it XMM-Newton} and {\it INTEGRAL} spectrum of the source is discussed in Sect.~3.2.
We removed from the data used to extract all JEM-X and ISGRI spectra mentioned
above the SCWs corresponding to the thermonuclear bursts detected by {\it INTEGRAL}.
These were searched for by using the JEM-X lightcurves collected with 2~s
resolution in the 3-20~keV energy band. A total of 4 bursts were clearly detected
by JEM-X in the SCW 76 of revolution 1517 and in the SCWs 78, 84, 94 of
revolution 1521. The onset times of these bursts were 57094.24535~MJD, 57104.86423~MJD,
57104.99993~MJD, and 57105.25787~MJD, respectively. None of these bursts
were significantly detected by ISGRI or showed evidence for a photospheric
radius expansion. Given the limited statistics of the two JEM-X monitors during
the bursts we did not perform any refined analysis of these events.
\begin{figure}
\centering
\includegraphics[width=6cm,angle=-90]{intlc_new.eps}
\caption{Lightcurve of the 2015 outburst displayed by {EXO~1745--248} as observed by IBIS/ISGRI and JEM-X on-board {\it INTEGRAL}.
For completeness, we report also the lightcurve
obtained from Swift/XRT and published previously by \citet{tetarenko2016}.
The hard-to-soft spectral state transition of {EXO~1745--248} around 57131~MJD discussed by \citet{tetarenko2016} is marked with a dashed vertical line in the above plots (around this date the
count-rate of the source in the IBIS/ISGRI decreases significantly, while it
continues to raise in JEM-X).}
\label{fig:intlc}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=6cm,angle=-90]{hard_fig.eps}
\includegraphics[width=6cm,angle=-90]{soft_fig.eps}
\caption{The broad-band spectrum of {EXO~1745--248} as observed by {\it INTEGRAL} in the hard (top)
and soft (bottom) state (ISGRI data are in black, JEM-X1 data in red, and JEM-X2
data in green). For both states the best fit to the spectrum was obtained with
an absorbed cut-off power-law model (see text for details).
The residuals from the best fits are shown in the bottom panels of the upper and
lower figure.}
\label{fig:intspe}
\end{figure}
\section{Spectral Analysis}
\label{sec:spectrum}
Spectral analysis has been performed using XSPEC v.12.8.1
\citep{Arnaud:96}.
Interstellar absorption was described by the TBAbs component.
For each fit we have used element abundances from \citet{2000ApJ...542..914W}.
The uncertainties on the parameters quoted in the following are evaluated
at a 90\% confidence level.
\subsection{Hard and soft INTEGRAL spectra}
\label{sec:integral_spec}
The broad-band {\it INTEGRAL} spectrum of the
source could be well described by using a simple absorbed power-law model
with a cut-off at the higher energies (we fixed in all fits the absorption
column density to the value measured by {\it XMM-Newton} in Sect.~3.2,
i.e. $N_{\rm H}$ = 2.02$\times$10$^{22}~$ cm$^{-2}$).
In the hard state ($\chi^2_{\rm red}/d.o.f.=1.2/21$), we measured a
power-law photon index $\Gamma$=1.1$\pm$0.1 and a cut-off energy of
23$\pm$2~keV.
The source X-ray flux was (2.9$\pm$0.2)$\times$10$^{-9}$~erg~cm$^{-2}$~s$^{-1}$
in the 3-20~keV energy band, (1.0$\pm$0.1)$\times$10$^{-9}$~erg~cm$^{-2}$~s$^{-1}$
in the 20-40~keV energy band, and (6.1$\pm$0.3)$\times$10$^{-10}$~erg~cm$^{-2}$~s$^{-1}$
in the 40-100~keV energy band. The effective exposure time was of 123~ks
for ISGRI and 75~ks for the two JEM-X units.
In the soft state ($\chi^2_{\rm red}/d.o.f.=1.3/17$), we measured a power-law
photon index $\Gamma$=0.6$\pm$0.2 and a cut-off energy of 3.8$\pm$0.5~keV.
The source X-ray flux was (9.5$\pm$0.5)$\times$10$^{-9}$~erg~cm$^{-2}$~s$^{-1}$
in the 3-20~keV energy band, (1.6$\pm$0.3)$\times$10$^{-10}$~erg~cm$^{-2}$~s$^{-1}$
in the 20-40~keV energy band, and (1.1$\pm$0.5)$\times$10$^{-12}$~erg~cm$^{-2}$~s$^{-1}$
in the 40-100~keV energy band. The effective exposure time was of 32~ks for
ISGRI and 20~ks for the two JEM-X units.
The two broad-band spectra and the residuals from the best fits are shown
in Fig.~\ref{fig:intspe}.
\subsection{The 2.4--10 keV EPIC-pn spectrum}
\label{sec:epnspec}
We first considered the spectrum observed by the EPIC-pn at energies
between 2.4 and 10 keV (see top panel of Fig.~\ref{fig:epn}), as a
soft-excess probably related to uncertainties in the redistribution
calibration affected data taken at lower energies (see the discussion
in Guainazzi et
al. 2015\footnote{http://xmm2.esac.esa.int/docs/documents/CAL-TN-0083.pdf},
and references therein). Interstellar absorption was described by the
TBAbs component \citep{2000ApJ...542..914W} the
photoelectric cross sections from \citet{Verner.etal:96} with the hydrogen column
density fixed to $N_H=2\times10^{22}$~cm$^{-2}$, as indicated by the
analysis performed with the inclusion of RGS, low energy data (see
Sec.~\ref{sec:broad}). The spectral continuum was dominated by a hard,
power-law like component with spectral index $\Gamma\simeq2$, which we
modeled as thermal Comptonization of soft photons with
$kT_{in}\simeq1.3$~keV, by using the model \textsc{nthcomp}
\citep{1996MNRAS.283..193Z,1999MNRAS.309..561Z}. As the electron
temperature fell beyond the energy range covered by the EPIC-pn, we
fixed such parameter to 37~keV, as suggested by the analysis of data
taken by INTEGRAL at higher energies (see Sec.~\ref{sec:broad}). We
modeled the strong residuals left by the Comptonization model at low
energies with a black-body with effective temperature
$kT_{th}\simeq0.6$ keV and emission radius
$R_{th}\simeq5.5\,d_{5.5}$~km, where $d_{5.5}$ is the distance to the
source in units of 5.5~kpc. The addition of such a component was
highly significant as it decreased the model reduced chi-squared from
47.9 to 26.5 for the two degrees of freedom less, out of 122.
The F-test gives a probability of chance improvement of $\sim 1.3\times10^{-15}$
for the addition of the blackbody component. In this case the use of the F-test is justified by the fact that the
model is additive \citep[see][]{Orlandini.etal:12}.
Even after the addition of a thermal component, the quality of the
spectral fit was still very poor mainly because of residuals observed
at energies of the Fe K-$\alpha$ transition (6.4--7 keV; see middle
panel of Fig.~\ref{fig:epn}). The shape of this emission complex is
highly structured and one emission line was not sufficient to provide
an acceptable modeling. We then modeled the iron complex using three
Gaussian features centered at energies
$E_1\simeq6.75^{+0.02}_{-0.03}$, $E_2\simeq6.48^{+0.03}_{-0.01}$ and
$E_3\simeq7.12^{+0.04}_{-0.07}$ keV. These energies are compatible
with K-$\alpha$ transition of ionized Fe XXV, K-$\alpha$ and K-$\beta$
of neutral or weakly ionized Fe (I-XX), respectively. The ionized iron
line is relatively broad ($\sigma_1=0.24\pm0.03$~keV) and strong
(equivalent width $EW_1=62.0\pm0.02$~eV), while the others are weaker
and have a width lower than the spectral resolution of the
instrument. In order to avoid correlation among the fitting
parameters, we fixed the normalization of the K-$\beta$ transition of
weakly ionized iron to one tenth of the K-$\alpha$. The addition of
the three Fe emission lines decreased the model $\chi^2$ to 266 for
114 degrees of freedom. In this case, for the addition of the
iron lines, the F-test probability results to be $\sim 2.7\times10^{-16}$.
Three more emission lines were required at
lower energies, $E_4=2.74^{+0.01}_{-0.03}$, $E_5=3.30(3)$ and
$E_6=3.94^{+0.05}_{-0.06}$ keV, compatible with K-$\alpha$ transitions
of S XVI, Ar XVIII, and Ca XX (or XIX), respectively. The significance
of these lines has been evaluated with an F-test, giving probabilities
of $3\times10^{-4}$, $9\times10^{-6}$ and $7.7\times10^{-8}$,
respectively, that the improvement of the fit $\chi^2$ obtained after
the addition of the line is due to chance. The chi-squared of the
model (dubbed Model I in Table~\ref{tab:epn}) is $\chi^2=154.5$ for
106 degrees of freedom.
\footnotesize
\begin{table*}[!h]
\caption{
Best fit parameters for the models used to fit the spectrum of EXO 1745--248.
In details, Model I is given by tbabs * (bbody + Gaussian(E${_1}$) + Gaussian(E${_2}$) + Gaussian(E${_3}$) + Gaussian(E${_4}$) + Gaussian(E${_5}$) + Gaussian(E${_6}$) + nthComp) - Model II: tbabs * (bbody + Diskline(E${_1}$) + Gaussian(E${_2}$) + Gaussian(E${_3}$) + Diskline(E${_4}$) + Diskline(E${_5}$) + Diskline(E${_6}$)+nthComp) - Model II$*$: the same as Model II but applied to the broad band spectrum
- Model III: tbabs * (bbody + Diskline(E${_1}$) + Gaussian(E${_2}$) + Gaussian(E${_3}$) + Diskline(E${_4}$) + Diskline(E${_5}$) + rdblur*rfxconv*nthComp) - Model IV: tbabs * (bbody + Diskline(E${_1}$) + Gaussian(E${_2}$) + Gaussian(E${_3}$) + Diskline(E${_4}$) + Diskline(E${_5}$) + Diskline(E${_6}$) + rdblur*pexriv+nthComp). Where tbabs describes the photoelectric absorption, bbody is the blackbody emission and nthComp describes the primary Comptonization spectrum. The component rdblur describes the smearing effects, while rfxconv, pexriv are different reflection models (see text for details). }
\centering
\begin{tabular}{clccccc}
\hline\hline
& & \multicolumn{2}{c}{EPIC-pn (2.4--11~keV)} & \multicolumn{3}{c}{Broadband (0.35--180~keV)}\\
Component & Parameter & Model I & Model II & Model II* & Model III & Model IV \\
\hline
tbabs &N$_{H}\,$($\times10^{22}$ cm$^{-2}$) &
(2.0) & (2.0) & $2.02\pm0.04$ & $2.13\pm0.05$ & $2.06\pm0.05$ \\
\noalign{\medskip}
bbody &$kT_{th}$ (keV) & 0.58$^{+0.03}_{-0.06}$ & 0.64$^{+0.04}_{-0.02}$ &
0.63$\pm$0.04&$0.73\pm0.03$ \\
bbody &R$_{bb}\,$ ($d_{5.5}$~km) &
5.5$^{+0.8}_{-0.4}$ & $4.6\pm0.2$ & $4.5\pm0.5$ & $3.8\pm0.2$ & $4.4\pm0.4$ \\
\noalign{\medskip}
nthComp &$\Gamma$ & 2.06$^{+0.08}_{-0.12}$ & $2.02^{+0.19}_{-0.09}$& $1.93\pm0.07$& $1.89\pm0.08$& $1.90\pm0.05$\\
nthComp &$kT_{e}$ (keV) & (37.0) & (37.0)& 37.2$^{+6.9}_{-5.1}$& $40^{+7}_{-5}$&33.6$^{+5.7}_{-4.4}$\\
nthComp &$kT_{in}$ (keV) & 1.33$^{+0.06}_{-0.14}$ & $1.3\pm0.1$& $1.27\pm0.06$&$1.34\pm0.07$&1.25$^{+0.08}_{-0.04}$\\
nthComp &R$_{w}$ ($d_{5.5}$~km) & $1.6\pm0.3$ & $1.5\pm0.3$ & $2.8\pm0.3$ & $2.4\pm0.3$ & $2.5\pm0.4$\\
nthComp & $F_{Comp}$ & $0.86\pm0.02$ &$0.86^{+0.03}_{-0.02}$ &$0.96\pm0.08$ &$0.85\pm0.08$ & $0.83\pm0.08$\\
\noalign{\medskip}
Diskl/rdblur &$\beta_{\mbox irr}$ & ... & $-2.44^{+0.04}_{-0.06}$ & $-2.44\pm0.07$& $-2.24\pm0.07$ & $-2.43\pm0.05$ \\
Diskl/rdblur &$R_{in}$ ($R_g$) & ... & $20^{+4}_{-6}$ & $20\pm6$& $<8.5$& $18.3^{+3.9}_{-6.2}$\\
Diskl/rdblur &$R_{out}$ ($R_g$) & ... & $(10^7)$ &$(10^7)$& $(10^7)$& $(10^7)$\\
Diskl/rdblur &$i$ ($^{\circ}$) & ... & $37^{+2}_{-3}$ & $37\pm3$& $38\pm1$& $37.2^{+2.1}_{-1.7}$ \\
\noalign{\medskip}
reflection &$\Omega_r/2\pi$ & ... & ... & ... & $0.22\pm0.04$ \\
reflection &$log\xi$ & ... & ... & ... & $2.70\pm0.07$ & $2.39^{+0.41}_{-0.27}$\\
reflection &$T_{disk}$ (k) & ... & ... & ... & ... & $(10^6)$ \\
reflection & Norm ($\times 10^{-2})$ & ... & ... & ... & ... & $0.86\pm0.55$ \\
\noalign{\medskip}
Gauss/Diskl &$E_1$ (keV) &$6.75^{+0.02}_{-0.03}$ & $6.75\pm0.02$ & $6.74\pm0.02$& ...& $6.75\pm0.02$ \\
Gauss/Diskl &$\sigma_1$ (keV) &$0.24^{+0.03}_{-0.02}$ & ... & ... & ... \\
Gauss/Diskl &$N_1$ &$6.0^{+0.7}_{-0.5}$ & $6.6^{+0.6}_{-0.4}$ & $7.1\pm0.1$& ...& $6.7^{+0.2}_{-0.4}$\\
Gauss/Diskl &${EW}_1$ (eV)&$62.0\pm0.02$ & $68.2\pm0.04$ & $72.9\pm2.5$ & ... & $68.6\pm2.4$ \\
\noalign{\medskip}
Gaussian &$E_2$ (keV) & $6.48^{+0.03}_{-0.01}$ & $6.50\pm0.01$ & $6.50\pm0.02$ & $6.49\pm0.02$& $6.49\pm0.02$\\
Gaussian &$\sigma_2$ (keV) & (0.0) & (0.0) & (0.0)& (0.0)& (0.0)\\
Gaussian &$N_2$ & $2.8\pm0.3$& $3.2\pm0.2$ & $3.2\pm0.2$& $2.4\pm0.2$& $3.2\pm0.3$\\
Gaussian &${EW}_2$ (eV)&$26.8\pm0.02$ & $31.6\pm0.2$ & $31.3\pm1.4$& $23.1\pm1.9$& $31.1\pm1.9$ \\
\noalign{\medskip}
Gaussian &$E_3$ (keV) &$7.12_{-0.07}^{+0.04}$ & $7.09\pm0.07$& (7.06) & (7.06)& (7.06) \\
Gaussian &$\sigma_3$ (keV) & (0.0) & (0.0) & (0.0) & (0.0) & (0.0)\\
Gaussian &$N_3$ & ($N_2/10$) & ($N_2/10$)& ($N_2/10$)& ($N_2/10$)& ($N_2/10$)\\
Gaussian &${EW}_3$ (eV)&$3.1\pm0.1$ & $3.6\pm0.1$ &$3.5\pm0.7$ &$2.7\pm0.8$& $3.5\pm0.9$ \\
\noalign{\medskip}
Gauss/Diskl &$E_4$ (keV) & $2.74^{+0.01}_{-0.03}$ & $2.68\pm0.03$ & $2.67\pm0.03$ & $2.67^{+0.01}_{-0.02}$& $2.67\pm0.03$\\
Gauss/Diskl &$\sigma_4$ (keV) & (0.0) & ... & ... & ...& ...\\
Gauss/Diskl &$N_4$ & $1.0^{+0.2}_{-0.1}$ & $2.0\pm0.4$ & $2.3\pm0.4$ &$1.2\pm0.4$&$2.2\pm0.3$ \\
Gauss/Diskl &${EW}_4$ (eV)&$3.8\pm0.2$ & $7.5\pm0.4$ & $8.4\pm1.1$& $4.2\pm0.9$& $8.0\pm1.3$ \\
\noalign{\medskip}
Gauss/Diskl &$E_5$ (keV) & $3.30\pm0.03$ & $3.29\pm0.02$ & $3.27\pm0.04$& $3.28\pm0.03$ & $3.29\pm0.03$ \\
Gauss/Diskl &$\sigma_5$ (keV) & $0.13^{+0.04}_{-0.02}$ & ... & ... & ...& ... \\
Gauss/Diskl &$N_5$ & $2.5^{+0.6}_{-0.7}$ & $2.1\pm0.3$ & $2.1\pm0.3$& $1.7^{+0.2}_{-0.5}$& $1.8^{+0.3}_{-0.5}$\\
Gauss/Diskl &${EW}_5$ (eV) & $11.5\pm0.1$ & $9.2\pm0.1$ & $9.5\pm1.2$ & $7.2\pm1.1$& $8.8\pm1.1$ \\
\noalign{\medskip}
Gauss/Diskl &$E_6$ (keV) & $3.94^{+0.05}_{-0.06}$ & $3.96\pm0.02$ & $3.96\pm0.05$ & $4.01\pm0.05$& $3.96\pm0.05$ \\
Gauss/Diskl &$\sigma_6$ (keV) & $0.26^{+0.10}_{-0.07}$ &... & ... & ...& ...\\
Gauss/Diskl &$N_6$ &$2.8^{+1.8}_{-0.9}$ & $1.6\pm0.3$& $2.21\pm0.05$& $1.2\pm0.4$& $1.5^{+0.1}_{-0.3}$ \\
Gauss/Diskl &${EW}_6$ (eV)& $15.4\pm0.2$ & $8.5\pm0.1$ & $8.3\pm0.9$ & $6.3\pm1.9$& $8.1\pm1.2$ \\
\noalign{\medskip}
\noalign{\medskip}
&$Flux$ & $9.34\pm0.01$ & $9.23\pm0.03$ & $26\pm3$ & $28\pm3$ & $26\pm3$\\
\noalign{\medskip}
&$\chi^2$ (d.o.f.) & 1.457 (106) & 1.338 (106) & 1.152 (1083)& 1.173 (1083)& 1.1487 (1081)\\
&p$_{\mbox{null}}$ & $1.5\times10^{-3}$ & $1.1\times10^{-2}$ &$3.6\times10^{-4}$ & $6.1\times10^{-5}$& $4.6\times10^{-4}$\\
\noalign{\medskip}
\hline
\label{tab:epn}
\end{tabular}
\tablefoot{Fluxes are unabsorbed and expressed in units of
$10^{-10}$~erg~cm$^{-2}$~s$^{-1}$. $F_{nthComp}$ is the flux in the Comptonization
component expressed as a fraction
of total flux. For the fits of the EPIC-pn spectrum alone (second and third column)
the fluxes are evaluated in the 0.5--10 keV energy band, while these are calculated in the
0.5--100 keV range for the broadband spectrum (from the fourth to the seventh
column). The normalization of the lines are expressed in units of
$10^{-4}$~ph~cm$^{-2}$~s$^{-1}$.}
\end{table*}
\normalsize
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{EPN_spectrum.eps}}
\caption{Spectrum observed by the EPIC-pn between 2.4 and 10 keV
together with the best fitting black body (red dashed line) and
Comptonization (blue dashed line) component of Model II listed in
Table~\ref{tab:epn} (top panel). Residuals obtained when the six
emission features at energies $E_1=6.75$ (green solid line),
$E_2=6.48$ (red solid line), $E_3=7.12$ (blue solid line),
$E_4=2.74$ (cyan), $E_5=3.30$ (magenta), $E_6=3.94$ (yellow) are
removed from Model II (middle panel). The model is not fitted after
the line removal, so the residuals are plotted for an illustrative
purpose, only. Residuals left by Model II are plotted in the bottom
panel.}
\label{fig:epn}
\end{figure}
The broadness of the $6.75$ keV Fe XXV line suggests reflection of
hard X-rays off the inner parts of the accretion disk as a plausible
origin. We then replaced the Gaussian profile with a relativistic
broadened \textsc{diskline} profile \citep{Fabian.etal:89}. The three
emission lines found between 2.4 and 4 keV have high ionization states
and probably originate from the same region. We then modeled them
with relativistic broadened emission features as well, keeping the
disk emissivity index, $\beta_{irr}$, and the geometrical disk
parameters (the inner and outer disk radii, $R_{in}$ and $R_{out}$,
and inclination, $i$) tied to the values obtained for the Fe XXV
line. As the spectral fit was insensitive to the outer disc radius
parameter, we left it frozen to its maximum value allowed
($10^{7}$~R$_g$, where $R_g=GM/c^2$ is the NS gravitational
radius). Modeling of the neutral (or weakly ionized) narrow Fe lines
at $\simeq6.5$ and $7.1$ keV with a Gaussian profile was maintained.
We found that the energy of the lines were all consistent within the
uncertainties with those previously determined with Model I. The
parameters of the relativistic lines indicate a disk extending down
to $R_{in}=20_{-6}^{+4}$~R$_g$ with an inclination of
$i=(37\pm2)^{\circ}$ and an emissivity index of
$\beta=-2.44^{+0.04}_{-0.06}$ (see column dubbed Model II of
Tab.~\ref{tab:epn} for the whole list of parameters). Modeling of the
spectrum with these broad emission lines decreased the fit $\chi^2$ to
141.8, for 106 degrees of freedom, which translates
into a probability of $p_{null}=10^{-2}$ of obtaining a value of the
fit $\chi^2$ as large or larger if the data are drawn from such a
spectral model. Figure~\ref{fig:epn} shows the observed spectrum, the
residuals with and without the inclusion of the emission lines. The
model parameters are listed in the third column of
Table~\ref{tab:epn}.
\subsection{The 0.35--180 keV XMM-Newton/INTEGRAL broadband spectrum}
\label{sec:broad}
In order to study the broadband spectrum of {EXO~1745--248} we fitted
simultaneously the spectra observed by the two RGS cameras (0.35--2.0
keV) and the EPIC-pn (2.4--10 keV) on-board {\it XMM-Newton}, together with the
spectra observed by the two JEM-X cameras (5--25 keV) and ISGRI
(20-180 keV) on board {\it INTEGRAL} during the satellite revolution 1521,
which partly overlapped with the {\it XMM-Newton} pointing. We initially
considered Model II, in which the continuum was modeled by the sum of
a Comptonized and a thermal component, the lines with energies
compatible with ionized species were described by a relativistic
broadened disk emission lines, and the K-$\alpha$ and K-$\beta$
lines of neutral (or weakly) ionized iron were modeled by a Gaussian
profile. For the broadband spectrum we decided to fix the energy
of the K$\beta$ iron line to its rest-frame energy of $7.06$ keV to make
the fit more stable.
The inclusion of the RGS spectra at low energies yielded a
measure of the equivalent hydrogen column density
$N_H=(2.02\pm0.04)\times10^{22}$~cm$^{-2}$. At the high energy end of
the spectrum, the ISGRI spectrum constrained the electron temperature
of the Comptonizing electron population to
$kT_e=37^{+7}_{-5}$~keV. The other parameters describing the continuum
and the lines were found to be compatible with those obtained from the
modeling of the EPIC-pn spectra alone. The model parameters are
listed in the fourth column of Table~\ref{tab:epn}, dubbed Model II*.
In order to entertain the hypothesis that the broad emission lines are
due to reflection of the primary Comptonized spectrum onto the inner
accretion disk, we replaced the Fe XXV broad emission line described
as {\textsc disklines} in Model II* with a self-consistent model
describing the reflection off an ionized accretion disk.
We convolved the Comptonized component describing the main source of
hard photons, \textsc{nthComp}, with the disk reflection model
\textsc{rfxconv} \citep{2011MNRAS.416..311K}.
We further convolved the
\textsc{rfxconv} component with a relativistic kernel
(\textsc{rdblur}) to take into account relativistic distortion of the
reflection component due to a rotating disc. Because the
\textsc{rfxconv} model does not include Ar and Ca transitions
and does not give a good modeling of the S line, leaving clear residuals at $\sim 2.7$ keV,,
we included three \textsc{diskline} components for them, linking the
parameters of the \textsc{rdblur} component to the corresponding
smearing parameters of the \textsc{disklines} , according to the
hypothesis that all these lines originate from the same disk region \citep[see,
e.g.][]{disalvo2009,egron2013,disalvo2015}.
The best fit with this model (dubbed Model III, see sixth column of
Table~\ref{tab:epn}) was slightly worse ($\chi^{2}$
/dof =1271/1083) than for Model II* ($\chi^{2}$
/dof =1248/1083). According to the reflection model, the solid angle
($\Omega_r/2\pi$) subtended by the reflector as seen from the
illuminating source was 0.22$\pm$ 0.04. The logarithm of the
ionization parameter of the disc was $\simeq$ 2.7, which could well
explain the ionization state of the Fe XXV, S XVI, Ar XVIII and Ca XX
(or XIX) emission lines observed in the spectrum. The inclination
angle of the system was found to be consistent with 37$^{\circ}$. The
broadband continuum and the line parameters were not significantly
changed by the introduction of the reflection model. The six instruments
spectrum, Model III and residuals are plotted in Fig.~\ref{fig:rfxconv}.
This figure shows a clear trend of residuals between 10 and
a 30 keV (consistent with a possible reflection hump), which our
reflection models cannot describe well.
\begin{figure*}
\centering
\includegraphics[scale=0.35, angle=-90]{fit3_diskline_exo_broad_NEW_PATH.eps}
\includegraphics[scale=0.35, angle=-90]{fit3_rfxconv_exo_mod_new.eps}
\includegraphics[scale=0.35, angle=-90]{new_fit3_disklines_exo_pexriv_NEW_PATH.eps}
\caption{Broadband spectra, models and residuals in units of $\sigma$ with respect to Model II*, Model III and Model IV are plotted in the top panel, middle panel and lower panel, respectively. RGS1 (red), RGS2 (green), EPIC-pn (black), JEMX1 (blue), JEMX2 (cyan) and ISGRI (magenta) spectra. }
\label{fig:rfxconv}
\end{figure*}
To test independently the significance of the Compton hump and absorption
edges, constituting the continuum of the reflection component,
we also tried a different reflection model, namely \texttt{pexriv}
\citep{Magdziarz.etal:95}, which describes an exponentially cut off
power law spectrum reflected from ionized material. We fixed the disk
temperature to the default value $ 10^{6}$ K, and the value on
reflection fraction to 0.22, that is the best value found in Model
III. We also tied parameters describing the irradiating power-law
(photon index and energy cut-off) to those indicated by the
\texttt{nthComp} component. As the iron emission is not included in
the \texttt{pexriv} model, we added a \texttt{diskline} centered at
6.75 keV. The results of the fit are reported in the sixth column
of Table~\ref{tab:epn}, labeled 'Model IV'. The parameters describing
the irradiating continuum and the reflection component are compatible
with those obtained with \textsc{rfxconv}, and the fit $\chi^2$
slightly improved with respect to Model III ($\Delta\chi^2=28.6$ for
two degrees of freedom less), while is compatible with the results
obtained with Model II*.
Since the width of the Gaussian lines describing the K$\alpha$
and K$\beta$ lines from neutral or mildly ionized iron were always compatible
with 0, we fixed the width of these lines to be null in all our fits.
This is going to be an acceptable assumption because the upper limits
calculated for the width of the iron K$\alpha$ line at 6.5 keV are
$\sigma < 0.022$ keV and $\sigma < 0.040$ keV,
for Model III and Model IV , respectively.
\section{Temporal analysis}
\label{sec:timing}
The {\it persistent} (i.e, non-bursting) emission observed during the
XMM-Newton EPIC-pn observation was highly variable, with a sample
fractional rms amplitude of 0.33. A portion of the {\it
persistent} light curve is shown in Fig.~\ref{fig:perslc} for
illustrative purposes. To study the power spectrum of the aperiodic
variability we performed a fast Fourier transform of 32-s long
intervals of the 0.5--10 keV EPIC-pn time series with 59~$\mu$s time
resolution (corresponding to a Nyquist maximum frequency of
8468~Hz). We averaged the spectra obtained in the various intervals,
re-binning the resulting spectrum as a geometrical series with a ratio
of 1.04. The Leahy normalized and white noise subtracted average power
spectrum is plotted in Fig.~\ref{fig:pds}. The spectrum is dominated
by a flicker noise component described by a power law,
$P(\nu)\propto\nu^{-\alpha}$, with $\alpha=1.05(1)$, slightly
flattening towards low frequencies. In order to search for kHz quasi
periodic oscillations already observed from the source at a frequency
ranging from 690 to 715~Hz \citep{mukherjee2011,barret2012}, we
produced a power density spectrum over 4 s-long intervals to have a
frequency resolution of 0.25~Hz, and averaged the spectra extracted
every 40 consecutive intervals. No oscillation was found within a
3-$\sigma$ confidence level upper limit of 1.5\% on the rms variation.
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{EPN_lcurve.eps}}
\vskip 1cm
\caption{Snapshot of the 0.5-10 keV {\it persistent} light curve
observed by the EPIC-pn on-board XMM-Newton. Counts were binned in
32 s-long intervals.}
\label{fig:perslc}
\end{figure}
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{EPN_pds.eps}}
\caption{Leahy normalized power density spectrum evaluated averaging
spectra computed over 8~s-long intervals of the EPIC-pn observation,
and re-binning the resulting spectrum as a geometrical series with
ratio equal to 1.04. A white noise level equal to 1.99(1) has been
subtracted. The solid line represents a power law,
$P(\nu)=\nu^{-\alpha}$ with index $\alpha=1.05$.}
\label{fig:pds}
\end{figure}
In order to search for a coherent signal in the light curve obtained
by the EPIC-pn, we first reported the observed photons to the Solar
system barycenter, using the position RA=17$^h$~48$^m$~05.236,
DEC=-24$^{\circ}$~46'~47.38" reported by \citet{Heinke.etal:06b} with
an uncertainty of 0.02" at 1-$\sigma$ confidence level). We performed
a power density spectrum on the whole $t_{pds}=77.5$~ks exposure,
re-binning the time series to a resolution equal to eight times the
minimum ($t_{res}=2.3\times10^{-4}$~s, giving a maximum frequency of
$\nu_{Ny}=2117$~Hz). After taking into account the number of
frequencies searched, $N_f\simeq\nu_{Ny}\,t_{pds}=1.64\times10^8$,
we could not find any significant signal with an
upper limit at 3-$\sigma$ confidence level of 0.5\% on the amplitude
of a sinusoidal signal, evaluated following \citet{1994ApJ...435..362V}.
The orbital period of {EXO~1745--248} is currently unknown. On the
spectral properties, \citet{Heinke.etal:06b} suggested it might be
hosted in an ultra-compact binary ($P_{orb}<<1$~d). Based on empirical
relation between the V magnitude of the optical counterpart, the X-ray
luminosity and the orbital period, \citet{ferraro2015} estimated a
likely range for the orbital period between 0.1 and 1.3~d. As the
orbital period is likely of the same order of the length of the exposure
of the observation considered, or shorter, the orbital motion will induce
shifts of the frequency of a coherent signal that hamper any
periodicity search. We then performed a search on shorter time intervals,
with a length ranging from 124 to 5500~s. The data acquired during
type-I X-ray bursts was discarded. No signal was detected at a confidence
level of 3-$\sigma$, with an upper limit ranging between 14\% and 2\%, with
the latter limit relative to the longer integration time.
In order to improve the sensitivity to signals affected by the unknown
binary orbital motion, we applied the quadratic coherence recovery
technique described by \citet{1991ApJ...379..295W} and
\citet{1994ApJ...435..362V}. We divided the entire light curve in
time intervals of length equal to $\Delta t=495$~s. In each of the
intervals the time of arrival of X-ray photons $t_{arr}$ were
corrected using the relation $t'=\alpha t_{arr}^2$; the parameter
$\alpha$ was varied in steps equal to $\delta\alpha=(2\nu_{Ny}\Delta
t^2)^{-1}=9.6\times10^{-10}$~s$^{-1}$ to cover a range between
$\alpha_{max}=1.7\times10^{-8}$~s$^{-1}$ and
$\alpha_{min}=-\alpha_{max}$. The width of the range is determined by
a guess on the orbital parameters of the system that would be optimal
for an orbital period of $12$~h, a donor star mass of
$M_2=0.3$~M$_{\odot}$, a NS spin period of $P=3$~ms, and a donor to NS
mass ratio of $q=0.2$ (see Eq.~14 of \citealt{1991ApJ...379..295W}).
This method confirmed the lack of any significant periodic signal,
with an 7\% on the sinusoidal
amplitude. We also considered a shorter time interval of
$\Delta~t=247$~s, and still obtained no detection within a 3$\sigma$
c.l. upper limit of 10.5\%.
We also searched for burst oscillations in the seven events observed
during the {\it XMM-Newton} exposure. To this aim, we produced power density
spectra over intervals of variable length, ranging from 2 to 8~s, and
time resolution equal to that used above
($t_{res}=2.3\times10^{-4}$~s). No significant signal was detected in
either of the bursts, with 3$\sigma$ c.l. upper limit on the signal
amplitude of the order of $\simeq20$ and $\simeq10\%$ for the
shorter and longer integration times used, respectively.
\section{Type I X-ray bursts}
\label{sec:burst}
Seven bursts took place during the {\it XMM-Newton} observation, with a
recurrence time varying between $t_{rec}=2.5$ and 4 hours (see
Table~\ref{tab:bursts}). The bursts attained a peak 0.5--10 keV
EPIC-pn count rate ranging from 1100 to 1500 counts/s (see top panel
of Fig.~\ref{fig:burst} where we plot the light curve of
the second burst seen during the {\it XMM-Newton} exposure). Such values exceed the
EPIC-pn telemetry limit ($\approx$450 counts/s; ), and data overflows
occurred close to the burst maximum. The burst rise takes place in
less than $\approx5$~s, while the decay could be approximately modeled
with an exponential function with an e-folding time scale ranging
between 10 and 23~s.
In order to analyze the evolution of the spectral shape during
the bursts, we extracted spectra over time intervals of length ranging
from 1 to 100 s depending on the count rate.
In order to minimize the
effect of pile up, which becomes important when the count rate
increases above a few hundreds of counts per second, we removed the
two brightest columns of the EPIC-pn chip (RAWX=36-37). Background was
extracted considering the {\it persistent emission} observed between
600 and 100~s before the burst onset. The resulting spectra were
modeled with an absorbed black-body, fixing the absorption column to
the value found in the analysis of the {\it persistent} emission
($N_H=2\times10^{22}$~cm$^{-2}$). The evolution of the temperature and
apparent radius observed during the second burst, the one with the
highest peak flux seen in the {\it XMM-Newton} observation, are plotted in the
middle and bottom panels of Fig.~\ref{fig:burst}, respectively. The
temperature attained a maximum value of $\approx{3.5}~$keV and then
decreased steadily, confirming the thermonuclear nature of the
bursts. The estimated apparent extension of the black-body emission
remained always much lower than any reasonable value expected for the
radius of a standard neutron star ($\gtrsim$8-13~km). The maximum flux
attained a value of $3.8(7) \times 10^{-8}$ {~erg~cm$^{-2}$~s$^{-1}$} (see Table~\ref{tab:bursts}),
which translate into a luminosity of
$1.4(2)\times10^{38}\,d_{5.5}^2$~{~erg~s$^{-1}$}. This value is both lower
than the Eddington limit for a NS and cosmic abundance
($1.76\times10^{38}\,(M/1.4\,M_{\odot})$~{~erg~s$^{-1}$}) and the luminosity
attained during the two bursts characterized by photospheric radius
expansion reported by \citet[][$L_{\rm pre}\simeq
2.2\times10^{38}\,d_{5.5}^2$~{~erg~s$^{-1}$}]{galloway2008}. Similar properties were
observed also in the other bursts and we concluded that photospheric
radius expansion did not occur in any of the bursts observed by {\it XMM-Newton}.
In addition, bursts characterized by photospheric radius expansion often
show a distinctive spectral evolution after the rise, characterized by a dip of
the black-body temperature occurring when the radius attains its maximum value,
while the flux stays at an approximately constant level (see, e.g., Fig. 2 of
Galloway et al. 2008). Although the minimum time resolution of our spectral
analysis (1s) is limited to by the available photon statistics, such a
variability pattern does not seem to be present in the observed evolution
of the burst parameters (see Fig.~7). Combined to the relatively low X-ray
luminosity attained at the peak of the bursts, we then conclude that it is
unlikely that photospheric radius expansion occurred in any of the bursts
observed by {\it XMM-Newton}.
\begin{figure}
\resizebox{\hsize}{!}
{\includegraphics{EPN_burst_new.eps}}
\caption{0.5-10 keV light curve of the second burst observed by the
EPIC-pn, which begun on $T_2=57103.41516$~MJD (top panel). The
central and bottom panels show the temperature and apparent radius
of the black body used to model the time-resolved spectra,
respectively. The radius is evaluated for a distance of 5.5 kpc.
Errors are reported with a 90\% confidence.}
\label{fig:burst}
\end{figure}
Table~\ref{tab:bursts} lists the energetics of the seven bursts
observed by {\it XMM-Newton}. The persistent flux was evaluated by fitting the
spectrum observed from 500~s after the previous burst onset, and 50~s
before the actual burst start time, using Model I (see
Table~\ref{tab:epn}). We measured the fluence $\mathcal{F}$ by summing
the fluxes observed in the different intervals over the duration of
each burst. We also evaluated the burst timescale as the ratio
$\tau=\mathcal{F}/F_{\rm peak}$ \citep{1988MNRAS.233..437V}. The
rightmost column of table~\ref{tab:bursts} displays the parameter
$\alpha$, defined as the ratio between the persistent integrated
flux and the burst fluence \citep[$\alpha=c_{\rm bol}F_{\rm
pers}t_{rec}/\mathcal{F}$; see, e.g.,][]{galloway2008}, where
$c_{\rm bol}$ is a bolometric correction factor that we estimated from
the ratio between the flux observed in the 0.5--100 keV and the
0.5--10 keV band with Model II* and II, respectively (see
Table~\ref{tab:epn}), $c_{\rm bol}=2.8\pm0.3$. We evaluated values of
$\alpha$ ranging between 50 and 110, with an average $<\alpha>=82$.
\begin{table*}[!h]
\caption{Properties of the type-I X-ray bursts observed by XMM-Newton. \label{tab:bursts}}
\centering
\begin{tabular}{lccccccc}
\hline\hline
No. & Start time (MJD) & $t_{rec}$~(s) & $F_{\rm pers}$ & $F_{\rm peak}$ & $\mathcal{F}$ & $\tau$ (s) & $\alpha$\\
\hline
I & 57103.26624 & ... & 0.99(2) & 17(2) & 38(3) & $22.7\pm3.4$ & \\
II & 57103.41516 & 12866 & 0.955(7) & 38(7) & 40(6) & $10.5\pm2.6$ & $86\pm16$ \\
III & 57103.56912 & 13303 & 0.924(6) & 18(2) & 31(3) & $16.5\pm3.0$ & $111\pm12$ \\
IV & 57103.67557 & 9197 & 0.91(1) & 21(3) & 42(5) & $20.4\pm3.8$ & $56\pm6$ \\
V & 57103.84017 & 14221 & 0.868(4) & 29(4) & 44(5) & $18.8\pm2.8$ & $79\pm9$ \\
VI & 57103.96830 & 11071 & 0.929(5) & 24(3) & 38(4) & $16.2\pm2.8$ & $76\pm11$ \\
VII & 57104.10384 & 11710 & 0.922(4) & 22(3) & 37(4) & $17.1\pm3.1$ & $82\pm12$ \\
\hline
\end{tabular}
\tablefoot{The 0.5--10 keV persistent flux $F_{pers}$ and the burst
peak flux $F_{peak}$ are unabsorbed and expressed in units of
$10^{-9}$~{~erg~cm$^{-2}$~s$^{-1}$}. The bolometric fluence $\mathcal{F}$ is
unabsorbed and given in units of $10^{-8}$~erg~cm$^{-2}$. The burst
decay timescale was evaluated as $\tau=\mathcal{F}/F_{pers}$, the
parameter $\alpha$ as $c_{bol}F_{pers}t_{rec}/\mathcal{F}$ with
$c_{bol}=2.8\pm0.3$. }
\end{table*}
\section{Discussion}
We analyzed quasi-simultaneous {\it XMM-Newton} and {\it INTEGRAL} observations of the transient LMXB
{EXO~1745--248} in the massive globular cluster Terzan 5, carried out when the
source was in the hard state, just after it went into outburst in
2015, with the aim to characterize its broad-band spectrum and its
temporal variability properties.
We also made use of all additionally available INTEGRAL data collected
during the outburst of the source in 2015
to spectroscopically confirm its hard-to-soft state transition occurred around 57131~MJD.
This transition was firstly noticed
by \citet{tetarenko2016} using the source lightcurves extracted from Swift/BAT, Swift/XRT, and MAXI.
\subsection{The combined XMM-Newton and INTEGRAL spectrum}
We modeled the spectrum observed simultaneously by {\it XMM-Newton} and {\it INTEGRAL}
to study the X-ray emission from the source in the energy range
0.8--100 keV. We estimated an unabsorbed total luminosity ($0.5-100$ keV
energy range) of $\approx 1 \times
10^{37}\,d_{5.5}^2$~{~erg~s$^{-1}$}. The continuum was well described by a
two-component model, corrected by the low-energy effects of
interstellar absorption. The best-fit value of the equivalent hydrogen
column density, $N_H$, is $(2.02\pm0.05) \times 10^{22}$~cm$^{-2}$,
slightly lower than the estimate of interstellar absorption towards
Terzan 5 given by \citet[][]{2014ApJ...780..127B},
$N_H=(2.6\pm0.1)\times 10^{22}$~cm$^{-2}$. The two-component
continuum model consist of a quite hard Comptonization component,
described by the \texttt{nthComp} model, with electron temperature
$kT_e \sim 40$ keV, photon index $\Gamma \simeq 1.8-2$ and seed-photon
temperature of about 1.3 keV, and of a soft thermal component
described by a black-body with temperature $kT \sim 0.6-0.7$ keV. The
Comptonization component contributed to more than 90 per cent of the
flux observed during the observations considered, clearly indicated
that the source stayed in the hard state.
Assuming a spherical geometry for both the black-body and the
seed-photon emitting regions, and ignoring any correction factor
due to color temperature corrections or boundary conditions, we found a
radius of the black-body emitting region of about $R_{bb}=3.5-5$ km and a
radius of the seed-photon emitting region of about $R_w=2-3$ km.
Given these modest extensions, it is likely that the surfaces of seed
photons are related to hot spots onto the neutron star surface.
The latter was calculated using the relation reported by
\citet{1999A&A...345..100I}, assuming an optical depth of the
Comptonization region, $\tau=2.2\pm0.3$, evaluated using the relation
between the optical depth, the temperature of the Comptonizing
electrons and the asymptotic power-law index given by
\citet{1987ApJ...319..643L}.
A similar spectral shape was found during the 2000 outburst of {EXO~1745--248}
observed by {\it Chandra} and {\it RXTE} \citep{Heinke.etal:03a}. In
that case the continuum model consisted of a multicolor disk
black-body, characterized by an inner temperature of $kT = 0.6 - 1.2$
keV and an inner disk radius of $r_{in}/d_{10}(\cos{i})^{0.5} = 4.3 -
9.2$ km, and a Comptonization component, described by the
\texttt{comptt} model, characterized by a seed photon temperature of
$kT_{0} = 1.2 - 1.7$ keV and radius $R_{W}$ = 3.1 - 6.7 km, an
electron temperature of $kT_{e} = 9.8 - 10.7$ keV, and an optical
depth $\tau=8$. The Comptonization spectrum was softer during the
Chandra/RXTE observations than during the XMM-Newton/INTEGRAL
observation analyzed here, and the 0.1-100 keV luminosity was $L_X
\approx 6.6 \times 10^{37}$ erg/sec, higher by about a factor 6 than
during our observation. Such a softening of the Comptonization spectrum
with increasing luminosity is in agreement with the results presented
by \citet{tetarenko2016}
for the 2015 outburst using Swift/XRT data (see their Table~1) and our
findings in Sect.~\ref{sec:integral} by using the INTEGRAL monitoring data.
Thanks to the large effective area and the moderately-good
energy-resolution of the EPIC-pn, we could detect several emission
features in the spectrum of EXO~1745-248. Most of the emission
features are broad and identified with K$\alpha$ transitions of highly
ionized elements. These are the $2.6-2.7$ keV line identified as S XVI
transition (H-like, expected rest frame energy 2.62 keV), the 3.3 keV
line identified as Ar XVIII transition (H-like, expected rest-frame
energy 3.32 keV), the $3.96-4.1$ keV line identified as Ca XIX or Ca
XX transition (He or H-like, expected rest-frame energy $3.9$ and
$4.1$ keV, respectively), and the 6.75 keV line identified as Fe XXV
(He-like) transition (expected rest-frame energy 6.7 keV). The Gaussian
width of the Fe XXV line we observed from {EXO~1745--248}, $\sigma_1 =
0.24^{+0.03}_{-0.02}$ keV, is compatible with the width of the Fe line
detected during the 2000 outburst \citep{Heinke.etal:03a}. The widths
of the low energy lines are compatible with being about half the width
of the iron line, in agreement with the expectations from Doppler or
thermal Compton broadening, for which the width is proportional to the
energy. Therefore all these lines are probably produced in the same
emitting region, characterized by similar velocity dispersion or
temperature (i.e., the accretion disk).
The fitting of the iron line appears, however, much more complex and
puzzling than usual. At least two components are needed to fit the
iron emission feature because of highly significant residuals still
present after the inclusion in the model of a broad Gaussian. We
fitted these residuals using another Gaussian centered at $\sim 6.5$
keV (therefore to be ascribed to neutral or mildly ionized iron) which
appears to be much narrower than the previous component (its width is
well below the energy resolution of the instrument and compatible with
0). Driven by a small residual still present at $\sim 7$ keV and by
the expectation that the 6.5-keV $K\alpha$ transition should be
accompanied by a 7.1-keV $K\beta$ transition, we also added to
the model a narrow Gaussian centered at $\sim 7.1$ keV, which we
identify with the K$\beta$ transition of neutral or mildly ionized
iron. Note that the flux ratio of the K$\beta$ transition to the
K$\alpha$ transition reaches its maximum of $0.15-0.17$ for Fe VIII,
while it drops to less than 0.1 for charge numbers higher than Fe X-XI
\citep[see][]{palmeri03}. This suggest that these components originate
from low-ionization iron (most probably Fe I-VIII) and come from a
different region, plausibly farther from the ionizing central engine,
with respect to the other broad and ionized emission lines.
In the hypothesis that the width of the broad lines is due to Doppler
and relativistic smearing in the inner accretion disk, we fitted these
lines in the EPIC-pn spectrum using relativistic broadened
disk-lines instead of Gaussian lines (see Model II and II* in
Table~\ref{tab:epn}). We obtained a slight improvement of the fit.
According to this model we obtained the emissivity index
of the disk, $\propto r^{\beta}$ with $\beta \sim -2.4$, the inner
radius of the disk, $R_{in} \sim 14-24$ R$_g$, and the inclination
angle of the system, $\sim 37^\circ$.
Although we have hints for the inclination angle to be relatively
low (e.g., lack of intrinsic absorption, dip activity or eclipses), it is
worth noting that values of the inclination angle of the system derived from
spectral fitting of the reflection component may rely on the assumed geometry
of the disk-corona system and therefore uncertainties on this parameter may
be underestimated.
Taking advantage from the broad-band coverage ensured by the almost
simultaneous XMM-Newton and INTEGRAL spectra, we also attempted to use
a self-consistent reflection model, which takes into account both the
discrete features (emission lines and absorption edges, as well as
Compton broadening of all these features) and the Compton scattered
continuum produced by the reflection of the primary Comptonized
spectrum off a cold accretion disk (Model III in Table~\ref{tab:epn}).
However, we could not obtain a statistically significant improvement
of the fit with respect to the disklines model. All
the parameters were similar to those obtained with the diskline
model. The only change in the smearing parameters we get using the
reflection model instead of disklines is in the value of the inner
disk radius, which is now constrained to be $< 8.5$ R$_g$. The
reflection component required a ionization parameter of $\log \xi \sim
2.7$, consistent the high ionization degree of the broad lines, and a
reflection fraction (that is the solid angle subtended by the
reflector as seen from the corona, $\Omega / 2 \pi$) of about 0.22. A
non significant improvement in the description of the spectrum ($\Delta
\chi^2 \simeq -5$ for the addition of two parameters) was obtained
when using {\texttt pexriv} to model the reflection continuum (Model
IV, see Table~\ref{tab:epn} with respect to best fit model (Model II*
in Tab~\ref{tab:epn}).
The observation analyzed here were then not sufficient to ascertain with
statistical significance whether a reflection continuum is present in the
spectrum.
The smearing parameters of the reflection component were similar
to what we find for other sources. The emissivity index of the disk,
$\sim -2.5$, the inner radius of the disk, about 30 km or below 13 km,
according to the model used for the reflection component, as well as
the inclination with respect to the line of sight, $35-40^\circ$, are
similar to the corresponding values reported in literature for
many other sources \citep[see e.g.][and references
therein]{disalvo2015}. For instance, in the case of atoll LMXB 4U
1705--44 the inner disk radius inferred from the reflection component
lay around $14-17$ R$_g$ both in the soft and in the hard state,
changing very little (if any) in the transition from one state to the
other \citep{disalvo2009,egron2013,disalvo2015}. In the case of 4U
1728--34, caught by {\it XMM-Newton} in a low-luminosity (most probably hard)
state, the inner disk radius was constrained to be $14-50$ R$_g$
\citep{egron2011}. Even in the case of accreting millisecond pulsars
(AMSPs), which are usually found in a hard state and for which we
expect that the inner disk is truncated by the magnetic field, inner
disk radii in the range $6-40$ R$_g$ were usually found \citep[see,
e.g.][]{papitto2009,cackett2009,papitto2010,papitto2013,pintore2016,king2016}.
Also, the reflection fraction inferred from the {\texttt rfxconv}
model, $\Omega/2\pi \sim 0.22$, although somewhat smaller than what is
expected for a geometry with a spherical corona surrounded by the
accretion disk ($\Omega/2\pi \sim 0.3$), is in agreement with typical
values for these sources. Values of the reflection fraction below or
equal 0.3 were found in a number of cases
\citep[e.g.][]{disalvo2015,degenaar2015,pintore2015,pintore2016,ludlam2016,chiang2016}.
More puzzling is the high ionization parameter required from the broad
emission lines, $\log \xi \sim 2.7-2.8$, where $\xi = (L_X / (n_e
r^2)$ is the ionization parameter, $L_X$ is the bolometric luminosity
of the central source and $n_e$ and $r$ are the electron density in
the emitting region and the distance of the latter from the central
source, respectively. This high value of the ionization parameter is
quite usual in the soft state, while in the hard state a lower
ionization is usually required, $\log \xi < 2$. This was clearly evident in the
hard state of 4U 1705—44 \citep{disalvo2015}, although in that case
the luminosity was $\sim 6 \times 10^{36}$ ergs/s, about a factor 2
below the observed luminosity of EXO~1745-248 during the observations
analysed here.
Perhaps the most unusual feature of this source is the
simultaneous presence in its spectrum of a broad ionized iron line and
at least one narrow, neutral or mildly ionized iron line, both in
emission and clearly produced in different regions of the
system. Sometimes, in highly inclined sources, broad iron emission
lines were found together with highly ionized iron lines in
absorption, clearly indicating the presence of an out-flowing disk wind
\citep[see, e.g., the case of the bright atoll source GX 13+1;][and
references therein]{pintore2014}. In the case of 4U 1636—536,
\citet{pandel2008} tentatively fitted the very broad emission feature
present in the range $4-9$ keV with a combination of several K$\alpha$
lines from iron in different ionization states. In particular they
fitted the iron complex with two broad emission lines with centroid
energies fixed at 6.4 and 7 keV, respectively. However, to our
knowledge, there is no other source with a line complex modeled by
one broad and one (or two) narrow emission features, as the one showed by
EXO~1745-248. While a natural explanation for the broad, ionized
component is reflection in the inner rings of the accretion disk, the
narrow features probably originate from illumination an outer region
in which the motion of the emitting material is much slower, as well
as the corresponding ionization parameter.
Future observation with instruments with a higher spectral resolution will be
needed to finely deconvolve the line shape, and firmly assess the
origin of each component.
\subsection{Temporal variability}
The high effective area of the EPIC-pn on board {\it XMM-Newton}, combined with
its $\mu$s temporal resolution, make it the best instrument currently
flying to detect coherent X-ray pulsations, and in particular those
with a period of few milliseconds expected from low magnetic field NS
in LMXBs. We performed a thorough search for periodicity in the
EPIC-pn time series observed from {EXO~1745--248}, but found no significant
signal. The upper limits on the pulse amplitude obtained range from
$2$ to $15\%$ depending on the length of the intervals considered, the
choice of which is a function of the unknown orbital period, and on
the application of techniques to minimize the decrease of sensitivity
to pulsations due to the orbital motion. Such upper limits are of the
order, and sometimes lower than the amplitudes usually observed from
AMSPs \citep[see, e.g.,][]{2012arXiv1206.2727P}. Though not excluding
the possibility of low amplitude pulsations, the non detection of a
signal does not favor the possibility that {EXO~1745--248} hosts an observable
accreting millisecond pulsar (AMSP). This is also hinted by the
significantly larger peak luminosity reached by {EXO~1745--248} during its
outbursts ($\sim 7\times10^{37}$~{~erg~s$^{-1}$}) with respect to AMSPs
($\approx\mbox{few}\times10^{36}$~{~erg~s$^{-1}$}). Together with the long
outburst usually shown (t$\sim100$~d), such a large X-ray luminosity
suggests that the long term accretion rate of {EXO~1745--248} is more than ten
times larger than in AMSPs. A larger mass accretion rates is though to
screen the NS magnetic field \citep{2001ApJ...557..958C}, possibly
explaining why ms pulsations are observed only from relatively faint
transient LMXBs.
At the moment of writing this paper, the orbital parameters of {EXO~1745--248}
were not known. In agreement with \citet{galloway2008} and
\citet{tetarenko2016},
we have reported a relatively long time scale of the X-ray bursts’ decay,
indicating presence of Hydrogen and hence providing evidence against an
ultra-compact nature of this system.
Recently, \citet{ferraro2015} showed that the location
of the optical counterpart of {EXO~1745--248} in the color-magnitude diagram of
Terzan 5 is close to the cluster turnoff, and is compatible with a
0.9~M$_\odot$ sub-giant branch star if it belongs to the low
metallicity population of Terzan 5. In such a case the mass transfer
would have started only recently. The orbital period would be
$\sim0.9$ days and the optimal integration time to perform a search
for periodicity $\sim920\,(P_s/3\mbox{ms})^{1/2}$~s, where $P_s$ is
the spin period of the putative pulsar (when not performing an
acceleration search; see Eq.~21 in \citet{1991ApJ...368..504J},
evaluated for a sinusoidal signal and an inclination of
37$^{\circ}$). The upper limit on the signal amplitude we obtained by
performing a signal search on time intervals of this length is 5\%.
A useful comparison can be made considering the only accreting pulsar
known in Terzan 5, IGR~J17480--2446, a NS spinning at a period of
90~ms, hosted in a binary system with an orbital period of 21.3~hr
\citep{2011A&A...526L...3P}. Its optical counterpart in quiescence
also lies close to the cluster turnoff
\citep{2012A&A...547A..28T}. The relatively long spin period of this
pulsar and its relatively large magnetic field compared to AMSP, let
\citet{2012ApJ...752...33P} to argue that the source started to
accrete and spin-up less than a few $10^{7}$~yr, and was therefore
caught in the initial phase of the mass transfer process that could
possibly accelerate it to a spin period of few milliseconds. When the
IGR~J17480--2446 was found in a hard state, X-ray pulsations were
observed at an amplitude of 27 per cent, decreasing to a few per cent
after the source spectrum became softer and cut-off at few keV
\citep{2012MNRAS.423.1178P}. The upper limit on pulsations obtained
assuming for {EXO~1745--248} similar parameters than IGR~J17480--244 is $2\%$,
of the order of the amplitude of the weaker pulsations observed from
IGR~J17480--244.
On the other hand, if the companion star belongs to the metal-rich
population of Terzan 5, it would be located in the color-magnitude
diagram at a position where companions to redback millisecond pulsars
are found \citep{ferraro2015}. In such a case a spin period of few
millisecond would be expected for the NS, and upper limits ranging
from 5 to 15\% on the pulse amplitude would be deduced from the
analysis presented here, depending on the orbital period. For
comparison, the redback transitional ms pulsar IGR J18245--2452 in the
globular cluster M28 showed pulsations with amplitude as high as 18\%,
that were easily detected in an {\it XMM-Newton} observation of similar length
than the one presented here
\citep{2013Natur.501..517P,2014A&A...567A..77F}. This further suggests
that {EXO~1745--248} is unlikely an observable accreting pulsar, unless its
pulsations are weak with respect to similar systems and/or it belongs
to a very compact binary system. Neither a search for burst
oscillations yielded to a detection, with an upper limit of
$\approx 10\%$ on the pulse amplitude, and therefore the
spin period of the NS in {EXO~1745--248} remains undetermined.
\subsection{Type-I X-ray bursts}
Seven type-I X-ray bursts were observed during the 80~ks {\it XMM-Newton}
observation presented here, with a recurrence time varying from 2.5 to
4 hours. None of the bursts showed photospheric radius expansion, and
all the bursts observed had a relatively long rise time ($\sim2$--5~s)
and decay timescale ($\tau=15$--$23$~s, except the second, brightest
burst which had $\tau\simeq10$~s). Bursts of pure helium are
characterized by shorter timescales ($\tau<10$~s) and we deduce that a
fraction of hydrogen was probably present in the fuel of the bursts we
observed. More information on the fuel composition can be drawn from
the ratio between the integrated persistent flux and the burst
fluence, $\alpha$. This parameter is related to the ratio between the
efficiency of energy conversion through accretion onto a compact
object ($GM_*/R_*$) and thermonuclear burning
($Q_{\rm}=1.6+4$<X>~MeV~nucleon$^{-1}$, where <X> is the abundance of
hydrogen burnt in the burst), {$\alpha=44(Q_{\rm
nuc}/4.4$~MeV~nucleon$^{-1})^{-1}$} for a 1.4~$M_{\odot}$ NS with
a radius of 10~km \citep[see Eq.~6 of][and references
therein]{galloway2008}. The observed values of $\alpha$ range from
50 to 100, with an average of 82, indicating that hydrogen fraction in
the bursts was $<X>\approx0.2$. Mass accretion rate should have then
been high enough to allow stable hydrogen burning between bursts, but
part of the accreted hydrogen was left unburnt at the burst onset and
contributed to produce a longer event with respect to pure helium
bursts. Combined hydrogen-helium flashes are expected to occur for
mass accretion rates larger than $\simeq 0.1\,\dot{m}_{\rm Edd}$
\citep[for solar metallicity, lower values are expected for low
metallicity,][]{2004ApJS..151...75W}, where $\dot{m}_{\rm Edd}$ is
the Eddington accretion rate per unit area on the NS surface
($8.8\times10^4$~g~cm$^{-2}$~s, or $1.3\times10^{-8}${~M$_{\odot}$~yr$^{-1}$}
averaged over the surface of a NS with a radius of 10~km). The
persistent broadband X-ray luminosity of {EXO~1745--248} during the
observations considered here indicates a mass accretion rate of
$8.5\times10^{-10}\,d_{5.5}^2\simeq0.05\,\dot{M}_{\rm Edd}${~M$_{\odot}$~yr$^{-1}$}
for a 1.4~M$_{\odot}$ NS with a 10 km radius, lower than the above
threshold not to exhaust hydrogen before the burst onset. A low
metallicity could help decreasing the steady hydrogen burning rate and
leave a small fraction of hydrogen in the burst fuel.
The seven bursts observed during the {\it XMM-Newton} observation analyzed here
share some of the properties of the 21 bursts observed by {\it RXTE}
during the 2000 outburst before the outburst peak, such as the decay
timescale, $\tau\approx25$~s, and the peak and persistent flux $F_{\rm
peak}=(3$--$19)\times10^{-9}${~erg~cm$^{-2}$~s$^{-1}$}, $F_{\rm
pers}=(1$--$5)\times10^{-9}${~erg~cm$^{-2}$~s$^{-1}$} and the absence of
photospheric radius expansion \citep[see Table 10 and appendix A31
in][]{galloway2008}. However those bursts showed recurrence times
between 17 and 49 minutes, and correspondingly lower values of
$\alpha=20$--$46$ with respect to those observed here. The observation
of frequent, long bursts and infrequent, short bursts at similar X-ray
luminosity made \citet{galloway2008} classify {EXO~1745--248} as an {\it
anomalous} burster. The observations presented here confirm such
a puzzling behavior for {EXO~1745--248}.
We note that 4 additional type-I bursts were detected by {\it INTEGRAL} during
the monitoring observations of {EXO~1745--248}.
As discussed in Sect.~2.2 we did not perform a spectroscopic analysis
of these events due to the limited statistics of the two JEM-X units
and the lack of any interesting detection in ISGRI which could have
indicated the presence of a photospheric radius expansion phase.
\begin{acknowledgements}
Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA, and with INTEGRAL, an ESA project with instruments and science data
centre funded by ESA member states, and Poland, and with the participation of Russia and the USA.
AP acknowledges support via an EU Marie Sklodowska-Curie Individual Fellowship under contract No. 660657-TMSP-H2020-MSCA-IF-2014, as well as fruitful discussion with the international team on “The disk-magnetosphere interaction around transitional millisecond pulsars” at ISSI (International Space Science Institute), Bern. We also acknowledges financial support from INAF ASI contract I/037/12/0.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,154,851 | arxiv | \section{Introduction}
\label{sec:intro}
\subsection{Motivation}
\label{subsec:motivation}
A hyperspectral image is a three-dimensional (3D) image cube comprised of a collection of two-dimensional (2D)
images (slices), where
each 2D image is captured at a specific wavelength.
Hyperspectral images allow us to analyze spectral information about each spatial point in a scene, and thus can help us identify different materials that appear
in the scene~\cite{Heinz2001}. Therefore, hyperspectral imaging has applications
to areas such as medical imaging~\cite{Schultz2001,Panasyuk2007}, remote sensing~\cite{Schaepman2009}, geology~\cite{Kruse2003},
and astronomy~\cite{Hege2004}.
Conventional spectral imagers include whisk broom scanners, push broom scanners~\cite{Brady2009,Eismann2012}, and spectrometers~\cite{Gat2000}.
In whisk broom scanners, a mirror reflects light onto a single detector, so that one pixel of data is collected at a time;
in push broom scanners, an image cube is captured with one focal plane array (FPA) measurement per spatial line of the scene; and in spectrometers, a set of optical bandpass filters are tuned in steps in order to scan the scene.
The disadvantages of these techniques are that
({\em i}) data acquisition takes a long time, because they require scanning a number of zones linearly in proportion to the desired spatial and spectral resolution; and
({\em ii}) large amounts of data are acquired and must be stored and transmitted.
For example, for a megapixel camera ($10^6$ pixels) that captures a few hundred spectral bands ($>100$ spectral channels) at 8 or 16 bits per frame, conventional spectral imagers demand roughly 10 megabytes per raw spectral image, and thus require space on the order of gigabytes for transmission or storage, which exceeds existing streaming capabilities.
To address the limitations of conventional spectral imaging techniques, many spectral imager sampling schemes based on compressive sensing~\cite{DonohoCS,CandesRUP,BaraniukCS2007} have been proposed~\cite{Gehm2007,Yuan2015Side,August2013Hyper}.
The coded aperture snapshot
spectral imager (CASSI)~\cite{Gehm2007,Wagadarikar2008,Arguello2011,Wagadarikar2008single} is a popular compressive spectral imager and acquires
image data from different wavelengths simultaneously.
In CASSI, the voxels of a scene are first coded by an aperture, then dispersed by a dispersive element, and finally detected by a 2D FPA. That is, a 3D image cube is suppressed and measured by a 2D array, and thus CASSI acquires far fewer measurements than those acquired by conventional spectral imagers, which significantly accelerates the imaging process.
In particular, for a data cube with spatial resolution of $N\times M$ and $L$ spectral bands, conventional spectral imagers collect $MNL$ measurements. In contrast, CASSI collects measurements on the order of $M(N+L-1)$. Therefore, the acquisition time, storage space, and required bandwidth for transmission in CASSI are reduced.
On the other hand, because
the measurements from CASSI are highly compressive, reconstructing 3D image cubes from CASSI measurements becomes challenging.
Moreover, because of the massive size of 3D image data, it is desirable to develop fast reconstruction algorithms in order to realize real time acquisition and processing.
Fortunately, it is possible to reconstruct the 3D cube from the 2D measurements according to the theory of compressive sensing~\cite{DonohoCS,CandesRUP,BaraniukCS2007}, because the 2D images from different wavelengths are highly correlated, and the 3D image cube is sparse in an appropriate transform domain, meaning that only a small portion of the transform coefficients have large values.
Approximate message passing (AMP)~\cite{DMM2009} has recently become a popular algorithm that solves compressive sensing problems, owing to its promising performance and efficiency. Therefore, we are motivated to investigate how to apply AMP to the CASSI system.
\subsection{Related work}
\label{subsec:relatedWork}
Several algorithms have been
proposed to reconstruct image cubes from measurements acquired by CASSI.
First, the reconstruction problem for the CASSI system can be solved by $\ell_1$-minimization. In Arguello and Arce~\cite{Arguello2014}, gradient projection for sparse reconstruction (GPSR)~\cite{GPSR2007} is utilized to solve for the $\ell_1$-minimization problem, where the sparsifying transform is the Kronecker product of a 2D wavelet transform and a 1D discrete cosine transform (DCT).
Besides using $\ell_1$-norm as the regularizer, total variation is a popular alternative;
Wagadarikar et al.~\cite{Wagadarikar2008} employed total variation~\cite{Chambolle2004,Chan2005} as the regularizer in the
two-step iterative shrinkage/thresholding (TwIST) framework~\cite{NewTWIST2007}, a modified and fast version of standard iterative shrinkage/thresholding.
Apart from using the wavelet-DCT basis,
one can
sparsify image cubes by dictionary learning~\cite{Yuan2015Side}, or using Gaussian mixture models~\cite{Rajwade2013}.
An interesting idea to improve the reconstruction quality of the dictionary learning based approach is to use a standard image with red, green, and blue (RGB) components of the same scene as side information~\cite{Yuan2015Side}. That is, a coupled dictionary is learned from the joint datasets of the CASSI measurements and the corresponding RGB image.
We note in passing that using color sensitive RGB detectors directly as the FPA of CASSI is another way to improve the sensing of spectral images, because spatio-spectral coding can be attained in a single snapshot without requiring extra optical elements~\cite{Rueda2015}.
Despite the good results attained with the algorithms mentioned above, they all need manual tuning of some parameters, which may be time consuming. In GPSR and TwIST,
the optimal regularization parameter could be different in reconstructing different image cubes.
In dictionary learning methods, although the parameters can be learned automatically by methods such as Markov Chain Monte Carlo, the learning process is usually time consuming. Moreover,
the patch size and the number of dictionary atoms in dictionary learning methods must be chosen carefully.
\subsection{Contributions}
\label{sec:contrib}
In this paper, we develop a robust and fast reconstruction algorithm for the CASSI system using approximate message passing (AMP)~\cite{DMM2009}. AMP is an iterative algorithm that can apply image denoising at each iteration.
Previously, we proposed a 2D compressive imaging reconstruction algorithm, AMP-Wiener~\cite{Tan_CompressiveImage2014}, where an adaptive Wiener filter was applied as the image denoiser within AMP.
Our numerical results showed that AMP-Wiener outperformed the prior art
in terms of both reconstruction quality and runtime.
The current paper extends AMP-Wiener to reconstruct 3D hyperspectral images from the CASSI system, and we call the new approach ``AMP-3D-Wiener."
Because the matrix that models the CASSI system is highly sparse, structured, and ill-conditioned,
applying AMP to the CASSI system becomes challenging.
For example, ({\em i}) the noisy image cube that is obtained at each AMP iteration contains non-Gaussian noise; and ({\em ii}) AMP encounters divergence problems, i.e., the reconstruction error may increase with more iterations.
Although it is favorable to use a high-quality denoiser within AMP, so that the reconstruction error may decrease faster as the number of iteration increases, we have found that in such an ill-conditioned imaging system, applying aggressive denoisers within AMP causes divergence problems.
Therefore, besides using standard techniques such as damping ~\cite{Rangan2014ISIT,Vila2014} to encourage the convergence of AMP, we modify the adaptive Wiener filter and make it robust to the ill-conditioned system model.
There are existing denoisers that may outperform the modified adaptive Wiener filter in a single step denoising problem. However, the modified adaptive Wiener filter fits into the AMP framework and allows AMP to improve over successive iterations.
Our approach is applied in nature, and the convergence of AMP-3D-Wiener is tested numerically.
We simulate AMP-3D-Wiener on several settings where complementary random coded apertures (see details in Section IV-A) are employed. The numerical results show that AMP-3D-Wiener reconstructs 3D image cubes with less runtime and higher quality than other compressive hyperspectral imaging reconstruction algorithms such as GPSR~\cite{GPSR2007} and TwIST~\cite{Wagadarikar2008,NewTWIST2007} (Figure~\ref{fig.iter_Psnr}), even when the regularization parameters in GPSR and TwIST have already been tuned.
These favorable results provide AMP-3D-Wiener major advantages over GPSR and TwIST.
First,
when the bottleneck is the time required to run the reconstruction algorithm,
AMP-3D-Wiener can provide the same reconstruction quality in 100 seconds that the other algorithms provide in 450 seconds (Figure~\ref{fig.iter_Psnr}).
Second, when the bottleneck is the time required for signal acquisition by CASSI hardware, the improved reconstruction quality could allow to reduce the number of shots taken by CASSI by as much as a factor of $2$ (Figure~\ref{fig.shots_Psnr}).
Finally, the reconstructed image cube can be obtained by running AMP-3D-Wiener only once, because AMP-3D-Wiener does not need to tune any parameters. In contrast, the regularization parameters in GPSR and TwIST need to be tuned carefully, because the optimal values of these parameters may vary for different test image cubes. In order to tune the parameters for each test image cube, we run GPSR and TwIST many times with different parameter values, and then select the ones that provide the best results.
The remainder of the paper is arranged as follows. We review
CASSI in Section~\ref{sec:CASSI}, and describe our AMP based compressive hyperspectral imaging reconstruction algorithm in Section \ref{sec:Algo}.
Numerical results are presented in Section \ref{sec:NumSim}, while Section~\ref{sec:disc} concludes.
\section{Coded Aperture Snapshot Spectral Imager (CASSI)}
\label{sec:CASSI}
\subsection{Mathematical representation of CASSI}
\label{subsec:MathCASSI}
The coded aperture snapshot spectral imager (CASSI)~\cite{Wagadarikar2008single} is a compressive spectral imaging system
that collects far fewer measurements than traditional spectrometers. In CASSI, ({\em i}) the 2D spatial information of a scene is coded by an aperture, ({\em ii}) the coded spatial projections are spectrally shifted by a dispersive element, and ({\em iii}) the coded and shifted projections are detected by a 2D FPA. That is, in each coordinate of the FPA, the received projection is an integration of the coded and shifted voxels over all spectral bands at the same spatial coordinate. More specifically, let $f_0(x,y,\lambda)$ denote the voxel intensity of a scene at spatial coordinate $(x,y)$ and at wavelength $\lambda$, and let $T(x,y)$ denote the coded aperture. The coded density $T(x,y)f_0(x,y,\lambda)$ is then spectrally shifted by the dispersive element along one of the spatial dimensions. The energy received by the FPA at coordinate $(x,y)$ is therefore
\begin{equation}
g(x,y) = \int_\Lambda T(x,y-S(\lambda))f_0(x,y-S(\lambda),\lambda)d\lambda,
\label{eq:contInt}
\end{equation}
where $S(\lambda)$ is the dispersion function induced by the prism at wavelength $\lambda$. Suppose we take a scene of spatial dimension $M$ by $N$ and spectral dimension $L$, i.e., the dimension of the image cube is $M\times N\times L$, and the dispersion is along the second spatial dimension $y$, then the number of measurements captured by the FPA will be $M(N+L-1)$. If we approximate the integral in~\eqref{eq:contInt} by a discrete summation and vectorize the 3D image cube and the 2D measurements, then we obtain a matrix-vector form of~\eqref{eq:contInt},
\begin{equation}
{\bf g} = {\bf H}{\bf f_0}+{\bf z},
\label{eq:CASSI}
\end{equation}
where ${\bf f_0}$ is the vectorized 3D image cube of dimension $n=MNL$, vectors ${\bf g}$ and ${\bf z}$ are the measurements and the additive noise, respectively,
and the matrix ${\bf H}$ is an equivalent linear operator that models the integral in~\eqref{eq:contInt}.
In this paper, we assume that the additive noise ${\bf z}$ is independent and identically distributed (i.i.d.) Gaussian.
With a single shot of CASSI, the number of measurements is~$m=M (N+L-1)$, whereas $K$ shots will yield $m=K M(N+L-1)$ measurements.
The matrix H in~\eqref{eq:CASSI} accounts for the effects of the coded aperture and the dispersive element. A sketch of this matrix is depicted in Figure~\ref{fig.1a} when $K=2$ shots are used. It consists of a set of diagonal patterns that repeat in the horizontal direction, each time with a unit downward shift, as many times as the number of spectral bands. Each diagonal pattern is the coded aperture itself after being column-wise vectorized. Just below, the next set of diagonal patterns are determined by the coded aperture pattern used in the subsequent shot. The matrix H will thus have as many sets of diagonal patterns as FPA measurements.
Although ${\bf H}$ is sparse and highly structured, the restricted isometry property~\cite{Candes05a} still holds, as shown by Arguello and Arce~\cite{Arguello2012}.
\begin{figure}[t]
\vspace*{-5mm}
\subfigure[The matrix ${\bf H}$ for standard CASSI]{
\includegraphics[width=80mm]{Fig1}
\label{fig.1a}
}
\subfigure[The matrix ${\bf H}$ for higher order CASSI]{
\includegraphics[width=89mm]{Fig1b}
\label{fig.1b}
}
\vspace*{0mm}
\caption{\small\sl
The matrix ${\bf H}$ is presented for $K=2,M=N=8$, and $L=4$. The circled diagonal patterns that repeat horizontally correspond to the coded aperture pattern used in the first FPA shot. The second coded aperture pattern determines the next set of diagonals. In (a) standard CASSI, each FPA shot captures $M(N+L-1)=88$ measurements; in (b) higher order CASSI, each FPA shot captures $M(N+L+1)=104$ measurements.
}
\label{fig.1}
\end{figure}
\subsection{Higher order CASSI}
\label{subsec:HighCASSI}
Recently, Arguello et al.~\cite{Arguello2013higher} proposed a higher order model to characterize the CASSI system with greater precision, and improved the quality of the reconstructed 3D image cubes. In the standard CASSI system model, each cubic voxel in the 3D cube contributes to exactly one measurement in the FPA. In the higher order CASSI model, however, each cubic voxel is shifted to an oblique voxel because of the continuous nature of the
dispersion, and therefore the oblique voxel contributes to more than one measurement in the FPA. As a result,
the matrix ${\bf H}$ in~\eqref{eq:CASSI} will have multiple diagonals as shown in Figure~\ref{fig.1b}, where there are sets of $3$ diagonals for each FPA shot, accounting for the voxel energy impinging into the neighboring FPA pixels.
In this case, the number of measurements with $K=1$ shot of CASSI will be $m=M(N+L+1)$, because each diagonal entails the use of $M$ more pixels
(we refer readers to~\cite{Arguello2013higher} for details).
In Section~\ref{sec:NumSim}, we will provide promising image reconstruction results for this higher order CASSI system. Using the standard CASSI model, our proposed algorithm produces similar advantageous results over other competing algorithms.
\section{Proposed Algorithm}
\label{sec:Algo}
The goal of our proposed algorithm is to reconstruct the image cube ${\bf f_0}$ from its compressive measurements ${\bf g}$, where the matrix ${\bf H}$ is known.
In this section, we describe our algorithm in detail. The algorithm employs ({\em i}) approximate message passing (AMP)~\cite{DMM2009}, an iterative algorithm for compressive sensing problems, and ({\em ii}) adaptive Wiener filtering, a hyperspectral image denoiser that can be applied within each iteration of AMP.
\subsection{Image denoising in scalar channels}
\label{subsec:scalarChannel}
Below we describe that the linear imaging system model in~\eqref{eq:CASSI} can be converted to a 3D image denoising problem in scalar channels. Therefore, we begin by defining scalar channels, where the noisy observations~${\bf q}$ of the image cube~${\bf f_0}$
obey
\begin{equation}
{\bf q=f_0+v},
\label{eq:scalar}
\end{equation}
and ${\bf v}$ is the additive noise vector. Recovering ${\bf f_0}$ from ~${\bf q}$ is known as a 3D image denoising problem.
\subsection{Approximate message passing}
\label{subsec:AMP}
{\bf Algorithm framework:}
AMP~\cite{DMM2009} has recently become a popular algorithm for solving signal reconstruction problems in linear systems as
defined in~\eqref{eq:CASSI}.
The AMP algorithm proceeds iteratively according to
\begin{align}
{\bf f}^{t+1}&=\eta_t({\bf H}^T{\bf r}^t+{\bf f}^t)\label{eq:AMPiter1},\\
{\bf r}^t&={\bf g}-{\bf Hf}^t+\frac{1}{R}{\bf r}^{t-1}
\langle\eta_{t-1}'({\bf H}^T{\bf r}^{t-1}+{\bf f}^{t-1})\rangle\label{eq:AMPiter2},
\end{align}
where~${\bf H}^T$ is the transpose of ${\bf H}$, $R=m/n$ represents the measurement rate, $\eta_t(\cdot)$ is a denoising function at the $t$-th iteration, $\eta_t'({\bf s})=\frac{\partial}{\partial {\bf s}}\eta_t({\bf s})$, and~$\langle{\bf u}\rangle=\frac{1}{n}\sum_{i=1}^n u_i$
for some vector~${\bf u}=(u_1,u_2,\ldots,u_n)$.
We will explain in Section~\ref{subsec:deriv} how ${\bf f}^t$ and ${\bf r}^t$ are initialized.
The last term in~\eqref{eq:AMPiter2} is called the ``Onsager reaction term"~\cite{Thouless1977,DMM2009} in statistical physics.
This Onsager reaction term helps improve the phase transition (trade-off between the measurement rate and signal sparsity) of the reconstruction process over existing iterative thresholding algorithms~\cite{DMM2009}.
In the~$t$-th iteration, we obtain the estimated image cube~${\bf f}^t$ and the residual~${\bf r}^t$.
We highlight that the vector~${\bf H}^T{\bf r}^t+{\bf f}^t$ in~\eqref{eq:AMPiter1} can be regarded as a noise-corrupted version of~${\bf f_0}$ in the~$t$-th iteration with noise variance~$\sigma_t^2$, and therefore~$\eta_t(\cdot)$ is a 3D image denoising function that is performed on a scalar channel as in~\eqref{eq:scalar}. Let us denote the equivalent scalar channel at iteration~$t$ by
\begin{equation}
{\bf q}^t = {\bf H}^T{\bf r}^t +{\bf f}^t= {\bf f_0} + {\bf v}^t,
\label{eq:scalar_t}
\end{equation}
where the noise level $\sigma^2_t$ is estimated by~\cite{Montanari2012},
\begin{equation}
\widehat{\sigma}^2_t=\frac{1}{m}\sum_{i=1}^m (r^t_i)^2,\label{eq:sigma_t}
\end{equation}
and~$r^t_i$ denotes the $i$-th component of the vector~${\bf r}^t$ in~\eqref{eq:AMPiter2}.
{\bf Theoretical properties:}
AMP can be interpreted as minimizing a Gaussian approximation of the Kullback-Leibler divergence~\cite{Cover06} between the estimated and the true posteriors subject to a first order and a second order moment matching constraints between ${\bf f_0}$ and ${\bf Hf_0}$~\cite{Rangan2013ISIT}. If the measurement matrix ${\bf H}$ is i.i.d. Gaussian and the empirical distribution of ${\bf f_0}$ converges to some distribution on $\mathbb{R}$, then the sequence of the mean square error achieved by AMP at each iteration converges to the information theoretical minimum mean square error asymptotically~\cite{Bayati2011}.
Moreover,
if the matrix is i.i.d. random,
then the noise in the scalar channel~\eqref{eq:scalar} can be viewed as asymptotically i.i.d. Gaussian~\cite{DMM2009,Montanari2012,Krzakala2012probabilistic}.
\subsection{Damping}
We have discussed in Section~\ref{subsec:AMP} that many mathematical properties of AMP hold for the setting where the measurement matrix is i.i.d. Gaussian. When the measurement matrix is not i.i.d. Gaussian, such as the highly structured matrix ${\bf H}$ defined in~\eqref{eq:CASSI}, AMP may encounter divergence issues.
A standard technique called ``damping"~\cite{Rangan2014ISIT,Vila2014} is frequently employed to solve for the divergence problems of AMP, because it only increases the runtime modestly.
Specifically, damping is an extra step within AMP iterations. In~\eqref{eq:AMPiter1}, instead of updating the value of~${\bf f}^{t+1}$ by the output of the denoiser~$\eta_t({\bf H}^T{\bf r}^{t}+{\bf f}^{t})$, we assign a weighted average of~$\eta_t({\bf H}^T{\bf r}^{t}+{\bf f}^{t})$ and~${\bf f}^t$ to~${\bf f}^{t+1}$ as follows,
\begin{equation}
{\bf f}^{t+1} = \alpha\cdot\eta_t({\bf H}^T{\bf r}^{t}+{\bf f}^{t})+(1-\alpha)\cdot{\bf f}^t,
\label{eq:damping1}
\end{equation}
for some constant~$0<\alpha\le1$. Similarly, after obtaining~${\bf r}^t$ in~\eqref{eq:AMPiter2}, we add an extra damping step that updates the value of ${\bf r}^t$ to be $\alpha\cdot {\bf r}^t + (1-\alpha) \cdot {\bf r}^{t-1}$, where the value of $\alpha$ is the same as that in~\eqref{eq:damping1}.
AMP has been proved~\cite{Rangan2014ISIT} to converge with sufficient damping,
under the assumption that the prior of ${\bf f_0}$ is i.i.d. Gaussian with fixed means and variances throughout all iterations,
and the amount of damping depends on the condition number of the matrix ${\bf H}$.
Note that other AMP variants~\cite{Swamp2014,Vila2014,RanganADMMGAMP2015} have also been proposed in order to encourage convergence for a broader class of measurement matrices.
\begin{comment}
in our modified algorithm AMP-3D-Wiener, we propose a simpler version of adaptive Wiener filter as described in Section~\ref{subsec:Wiener} to stabilize the estimation of the prior distribution of ${\bf f_0}$.
Although we do not have justifications for convergence of AMP-3D-Wiener at this point, we find in our simulations that AMP-3D-Wiener converges for all tested hyperspectral image cubes with moderate amount of damping.
\end{comment}
\subsection{Adaptive Wiener filter}
\label{subsec:Wiener}
We are now ready to describe our 3D image denoiser, which is the function~$\eta_t(\cdot)$ in the first step of AMP iterations in~\eqref{eq:AMPiter1}.
{\bf Sparsifying transform:}
Recall that in 2D image denoising problems, a 2D wavelet transform is often performed, and some shrinkage function is applied to the wavelet coefficients in order to suppress noise~\cite{Donoho1994,Figueiredo2001}. The wavelet transform based image denoising method is effective, because natural images are usually sparse in the wavelet transform domain, i.e., there are only a few large wavelet coefficients and the rest of the coefficients are small. Therefore, large wavelet coefficients are likely to contain information about the image, whereas small coefficients are usually comprised mostly of noise, and so it is effective to denoise by shrinking the small coefficients toward zero and suppressing the large coefficients according to the noise variance. Similarly, in hyperspectral image denoising, we want to find a sparsifying transform such that hyperspectral images have only a few large coefficients in this transform domain.
Inspired by Arguello and Arce~\cite{Arguello2014},
we apply a wavelet transform to each of the 2D images in a 3D cube, and then apply a discrete cosine transform (DCT) along the spectral dimension, because the 2D slices from different wavelengths are highly correlated. That is, the sparsifying transform ${\bf\Psi}$ can be expressed as a Kronecker product of a DCT transform ${\bf \Phi}$ and a 2D wavelet transform ${\bf W}$, i.e., ${\bf\Psi=\Phi\otimes W}$, and it can be shown that ${\bf\Psi}$ is an orthonormal transform.
Let ${\bf\theta}_{\bf q}^t$ denote the coefficients of ${\bf q}^t$ in this transform domain, i.e., ${\bf\theta}_{{\bf q}}^t={\bf\Psi q}^t$.
Our 3D image denoising procedure will be applied to the coefficients ${\bf\theta}_{\bf q}^t$.
Besides 2D wavelet transform and 1D DCT, it is also possible to sparsify 3D image cubes by dictionary learning~\cite{Yuan2015Side} or Gaussian mixture models~\cite{Rajwade2013}. Moreover, using an endmember mixing matrix~\cite{Martin2015} is an alternative to DCT for characterizing the spectral correlation of 3D image cubes. In this work, we focus on a 2D wavelet transform and 1D DCT as the sparsifying transform, because it is an efficient transform that does not depend on any particular types of image cubes, and an orthonormal transform that is suitable for the AMP framework.
{\bf Parameter estimation in the Wiener filter:}
In our previous work~\cite{Tan_CompressiveImage2014} on compressive imaging reconstruction problems for 2D images,
one of the image denoisers we employed was an adaptive Wiener filter in the wavelet domain, where the variance of each wavelet coefficient was estimated from its neighboring coefficients within a $5\times 5$ window, i.e., the variance was estimated locally.
As an initial attempt, we applied the previously proposed AMP-Wiener to the reconstruction problem in the CASSI system defined in~\eqref{eq:CASSI}. More specifically, the previously proposed adaptive Wiener filter is applied to the noisy coefficients $\theta_{\bf q}^t$.
Unfortunately, AMP-Wiener encounters divergence issues for the CASSI system even with significant damping such as $\alpha=0.01$ in~\eqref{eq:damping1}. AMP-Wiener diverges, because it is designed for the setting where the measurement matrix is i.i.d. Gaussian, whereas the measurement matrix ${\bf H}$ defined in~\eqref{eq:CASSI} is highly structured and not i.i.d., and we found in our numerical experiments that the scalar channel noise ${\bf v}^t$ in~\eqref{eq:scalar_t} is not i.i.d. Gaussian. On the other hand, because the Wiener filter allows to conveniently calculate the Onsager term in~\eqref{eq:AMPiter2}, we are motivated to keep the Wiener filter strategy, although the scalar channel \eqref{eq:scalar_t} does not contain i.i.d. Gaussian noise.
Seeing that estimating the coefficient variance from its neighboring coefficients (a $3\times 3$ or $5\times 5$ neighboring window) does not produce reasonable reconstruction for the CASSI system, we modify the local variance estimation to a global estimation within each wavelet subband.
The coefficients $\widehat{\bf\theta}_{\bf f}^t$ of the estimated (denoised) image cube ${\bf f}^t$ are obtained by Wiener filtering,
which can be interpreted as the conditional expectation of ${\bf\theta}_{\bf f}$ given ${\bf\theta}_{\bf q}^t$ under the assumption of Gaussian prior and Gaussian noise,
\begin{eqnarray}
\widehat{\bf\theta}_{{\bf f},i}^t
&=&\frac{\max\{0,\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2\}}{(\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2)+\widehat{\sigma}_t^2}\left(\theta_{{\bf q},i}^t-\widehat{\mu}_{i,t}\right)+\widehat{\mu}_{i,t}\nonumber\\
&=&\frac{\max\{0,\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2\}}{\widehat{\nu}_{i,t}^2}\left(\theta_{{\bf q},i}^t-\widehat{\mu}_{i,t}\right)+\widehat{\mu}_{i,t},
\label{eq:Wiener}
\end{eqnarray}
where ${\bf\theta}_{{\bf q},i}^t$ is the $i$-th element of~${\bf\theta}_{\bf q}^t$, and $\widehat{\mu}_{i,t}$ and $\widehat{\nu}_{i,t}^2$ are the empirical mean and variance of ${\bf \theta}_{{\bf q},i}^t$ within an appropriate wavelet subband, respectively. Taking the maximum between 0 and $(\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2)$ ensures that if the empirical variance $\widehat{\nu}_{i,t}^2$ of the noisy coefficients is smaller than the noise variance $\widehat{\sigma}_t^2$, then the corresponding noisy coefficients are set to 0. After obtaining the denoised coefficients~$\widehat{\bf\theta}_{\bf f}^t$, the estimated image cube in the $t$-th iteration satisfies ${\bf f}^t={\bf \Psi}^{-1}\widehat{\bf\theta}_{\bf f}^t={\bf\Psi}^T\widehat{\bf\theta}_{\bf f}^t$. Therefore, the adaptive Wiener filter as a denoiser function~$\eta_t(\cdot)$ can be written as
\begin{eqnarray}
{\bf f}^{t+1} &=&\eta_t({\bf q}^t)\nonumber\\
&=& \boldsymbol\Psi^T\left(\max\{{\bf 0},\widehat{\bf V}_t - \widehat{\sigma}_t^2{\bf I}\}\widehat{\bf V}_t^{-1}
\left(\boldsymbol\Psi{\bf q}^t - \widehat{\boldsymbol\mu}_t\right) + \widehat{\boldsymbol\mu}_t\right),\nonumber\\
\label{eq.etaWiener}
\end{eqnarray}
where {\bf 0} is a zero matrix, $\widehat{\bf V}_t$ is a diagonal matrix with $\widehat{\nu}_{i,t}^2$ on its diagonal, ${\bf I}$ is the identify matrix, $\widehat{\boldsymbol\mu}_t$ is a vector that contains $\widehat{\mu}_{i,t}$, and $\max\{\cdot,\cdot \}$ is operating entry-wise.
We apply this modified adaptive Wiener filter within AMP, and call the algorithm ``AMP-3D-Wiener." We will show in Section~\ref{sec:NumSim} that only a moderate amount of damping is needed for AMP-3D-Wiener to converge.
\subsection{Derivative of adaptive Wiener filter}
\label{subsec:deriv}
The adaptive Wiener filter described in Section~\ref{subsec:Wiener} is applied in~\eqref{eq:AMPiter1} as the 3D image denoising function $\eta_t(\cdot)$. The following step in~\eqref{eq:AMPiter2} requires~$\eta'_t(\cdot)$, i.e., the derivative of~$\eta_t(\cdot)$. We now show how to obtain~$\eta'_t(\cdot)$.
It has been discussed~\cite{Tan_CompressiveImage2014} that when the sparsifying transform is orthonormal, the derivative calculated in the transform domain is equivalent to the derivative in the image domain.
According to~\eqref{eq:Wiener}, the derivative of the Wiener filter in the transform domain with respect to $\widehat{\bf\theta}_{{\bf q},i}^t$ is~$\max\{0,\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2\}/\widehat{\nu}_{i,t}^2$.
Because the sparsifying transform ${\bf\Psi}$ is orthonormal, the Onsager term in~\eqref{eq:AMPiter2} can be calculated efficiently
as
\begin{equation}
\langle\eta'_t({\bf q}^t)\rangle = \frac{1}{n} \sum_{i\in \mathcal{I}}\frac{\max\{0,\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2\}}{\widehat{\nu}_{i,t}^2},
\label{eq:Onsager}
\end{equation}
where $\mathcal{I}$ is the index set of all image cube elements, and the cardinality of $\mathcal{I}$ is $n=MNL$.
We focus on image denoising in an orthonormal transform domain and apply Wiener filtering to suppress noise, because it is convenient to obtain the Onsager correction term in~\eqref{eq:AMPiter2}. On the other hand, other denoisers that are not wavelet-DCT based can also be applied within the AMP framework. Metzler et al.~\cite{Metzler2014}, for example, proposed to utilize a block matching and 3D filtering denoising scheme (BM3D)~\cite{Dabov2007} within AMP for 2D compressive imaging reconstruction, and run Monte Carlo~\cite{Ramani2008} to approximate the Onsager correction term. However, the Monte Carlo technique is accurate only when the scalar channel~\eqref{eq:scalar_t} is Gaussian. In the CASSI system model~\eqref{eq:CASSI}, BM4D~\cite{Maggioni2013} may be an option for the 3D image denoising procedure. However, because the matrix~${\bf H}$ is ill-conditioned, the scalar channel~\eqref{eq:scalar_t} that is produced by AMP iterations~(\ref{eq:AMPiter1},\ref{eq:AMPiter2}) is not Gaussian, and thus the Monte Carlo technique fails to approximate the Onsager correction term.
Having completed the description of AMP-3D-Wiener, we summarize AMP-3D-Wiener in Algorithm~\ref{algo:amp_wiener}, where $\widehat{\bf f}_\text{AMP}$ denotes the image cube reconstructed by AMP-3D-Wiener.
Note that in the first iteration of Algorithm~\ref{algo:amp_wiener}, initialization of ${\bf q}^0$ and $\widehat{\sigma}^2_0$ may not be necessary, because ${\bf r}^0$ is an all-zero vector, and the Onsager term is 0 at iteration 1.
\begin{algorithm}[h]
\caption{AMP-3D-Wiener}
\label{algo:amp_wiener}
\textbf{Inputs:} ${\bf g}$, ${\bf H}$, $\alpha$, maxIter\\
{\bf Outputs:} $\widehat{{\bf f}}_\text{AMP}$\\
\textbf{Initialization:} ${\bf f}^1={\bf 0}$, ${\bf r}^{0}={\bf 0}$
\begin{algorithmic}
\For{$t=1:\text{maxIter}$}\\
\begin{enumerate}
\item
${\bf r}^t={\bf g}-{\bf Hf}^t+\frac{1}{R}{\bf r}^{t-1}
\frac{1}{n} \sum_{i=1}^n \frac{\max\{0,\widehat{\nu}_{i,t-1}^2-\widehat{\sigma}_{t-1}^2\}}{\widehat{\nu}_{i,t-1}^2}$
\item
${\bf r}^t = \alpha\cdot{\bf r}^t + (1-\alpha)\cdot{\bf r}^{t-1}$
\item
${{\bf q}}^t={\bf H}^T{\bf r}^t+{\bf f}^t$
\item
$\widehat{\sigma}^2_t=\frac{1}{m}\sum_j ({r}^t_j)^2$
\item
$\theta_{\bf q}^t = {\bf\Psi}{\bf q}^t$
\item
$\widehat{\bf\theta}_{{\bf f},i}^t=\frac{\max\{0,\widehat{\nu}_{i,t}^2-\widehat{\sigma}_t^2\}}{\widehat{\nu}_{i,t}^2}\left(\theta_{{\bf q},i}^t-\widehat{\mu}_{i,t}\right)+\widehat{\mu}_{i,t}$
\item
${\bf f}^{t+1}=\alpha\cdot{\bf\Psi}^T\widehat{\bf\theta}_{\bf f}^t+(1-\alpha) \cdot{\bf f}^{t}$
\end{enumerate}
\EndFor\\
$\widehat{{\bf f}}_\text{AMP}={\bf f}^{\text{maxIter+1}}$
\end{algorithmic}
\end{algorithm}
\begin{figure*}[t]
\setcounter{cnt01}{2}
\vspace*{-5mm}
\hspace*{-10mm}
\includegraphics[width=190mm]{Lego_24}
\vspace*{-15mm}
\caption{\small\sl
The Lego scene.
(The target object presented in the experimental results was not endorsed by the trademark owners and it is used here as fair use to illustrate the quality of reconstruction of compressive spectral image measurements. LEGO is a trademark of the LEGO Group, which does not sponsor, authorize or endorse the images in this paper. The LEGO Group. All Rights Reserved.
http://aboutus.lego.com/en-us/legal-notice/fair-play/.)
}
\label{fig.Lego}
\setcounter{figure}{\value{cnt01}}
\end{figure*}
\section{Numerical Results}
\label{sec:NumSim}
In this section, we provide numerical results where we compare the reconstruction quality and runtime of AMP-3D-Wiener,
gradient projection for sparse reconstruction (GPSR)~\cite{GPSR2007}, and two-step iterative shrinkage/thresholding (TwIST)~\cite{Wagadarikar2008,NewTWIST2007}.
In all experiments, we use the same coded aperture pattern for AMP-3D-Wiener, GPSR, and TwIST.
In order to quantify the reconstruction quality of each algorithm, the peak signal to noise ratio (PSNR) of each 2D slice in the reconstructed cubes is measured.
The PSNR is defined as the ratio between the maximum squared value of the ground truth image cube~${\bf f_0}$ and the mean square error of the estimation~$\widehat{\bf f}$, i.e.,
\begin{equation*}
\text{PSNR}=10\cdot\log_{10}\left(\frac{\max_{x,y,\lambda}\left(f^2_{0,(x,y,\lambda)}\right)}{\sum_{x,y,\lambda}\left(\widehat{f}_{(x,y,\lambda)}-f_{0,(x,y,\lambda)}\right)^2}\right),
\end{equation*}
where $f_{(x,y,\lambda)}$ denotes the element in the cube ${\bf f}$ at spatial coordinate $(x,y)$ and spectral coordinate $\lambda$.
In AMP, the damping parameter~$\alpha$ is set to be 0.2.
Recall that increasing the amount of damping helps prevent the divergence of AMP-3D-Wiener, and that the divergence issue can be identified by evaluating the values of $\widehat{\sigma}_t^2$ from~\eqref{eq:sigma_t}. We select 0.2 as the damping parameter value, because 0.2 is the maximum damping value such that AMP-3D-Wiener converges in all the image cubes we test.
The divergence issues of AMP-3D-Wiener can be detected by evaluating the value of $\widehat{\sigma}_t^2$ obtained by~\eqref{eq:sigma_t} as a function of iteration number $t$. Recall that $\widehat{\sigma}_t^2$ estimates the amount of noise in the noisy image cube~${\bf q}^t$ at iteration $t$. If AMP-3D-Wiener converges, then we expect the value of $\widehat{\sigma}_t^2$ to decrease as $t$ increases. Otherwise, we know that AMP-3D-Wiener diverges.
The choice of damping mainly depends on the structure of the imaging model in~\eqref{eq:CASSI} but not on the characteristics of the image cubes, and thus the value of the damping parameter~$\alpha$ need not be tuned in our experiments.
To reconstruct the image cube~${\bf f_0}$, GPSR and TwIST minimize objective functions of the form
\begin{equation}
\widehat{\bf f}=\arg\min_{\bf f} \frac{1}{2}\|{\bf g-Hf}\|_2^2 + \beta\cdot\phi({\bf f}),
\label{eq:Obj}
\end{equation}
where $\phi(\cdot)$ is a regularization function that characterizes the structure of the image cube ${\bf f_0}$, and $\beta$ is a regularization parameter that balances the weights of the two terms in the objective function. In GPSR, $\phi({\bf f}) = \|{\bf \Psi f}\|_1$; in TwIST, the total variation regularizer is employed,
\begin{eqnarray}
\phi({\bf f}) &=& \sum_{\lambda=1}^L\sum_{x=1}^M\sum_{y=1}^N
\bigg((f(x+1,y,\lambda)-f(x,y,\lambda))^2 \nonumber\\
&&+ (f(x,y+1,\lambda)-f(x,y,\lambda))^2\bigg)^{1/2}.
\end{eqnarray}
Note that the role of the $\ell_1$-norm of the sparsifying coefficients in GPSR is to impose the overall sparsity of the sparsifying coefficients, whereas the total variation in TwIST encourages spatial smoothness in the reconstructed image cubes.
The implementation of GPSR is downloaded from ``http://www.lx.it.pt/~mtf/GPSR/," and the implementation of TwIST is downloaded from ``http://www.disp.duke.edu/projects/CASSI/experimentaldata/\\
index.ptml."
The value of the regularization parameter~$\beta$ in~\eqref{eq:Obj} greatly affects the reconstruction results of GPSR and TwIST, and must be tuned carefully.
We select the optimal values of $\beta$ for GPSR and TwIST manually, i.e., we run GPSR and TwIST with $5-10$
different values of $\beta$, and select the results with the highest PSNR.\footnote{As an example, we simulate GPSR with many different values for $\beta$, and obtain that for $\beta=1\cdot 10^{-5},2\cdot 10^{-5},3\cdot 10^{-5},4\cdot 10^{-5},5\cdot 10^{-5},6\cdot 10^{-5}$, and $7\cdot 10^{-5}$, the corresponding PSNRs of the reconstructed cubes are $31.25$ dB, $32.30$ dB, $32.82$ dB, $32.99$ dB, $33.02$ dB, $33.09$ dB, and $33.06$ dB. Therefore, we select $\beta=6\cdot 10^{-5}$ for this specific image cube. We follow the same procedure to select the optimal $\beta$ values for each test image cube.}
The typical value of the regularization parameter for GPSR is between $10^{-5}$ and $10^{-4}$, and the value for TwIST is around 0.1.
We note in passing that the ground truth image cube is not known in practice, and estimating the PSNR
obtained using different $\beta$ may be quite involved and require oracle-like information when using GPSR and TwIST.
Reweighted $\ell_1$-minimization~\cite{Candes2008} does not need regularization parameter tuning, and has been shown to outperform $\ell_1$-minimization by Candes et al.~\cite{Candes2008}. However, the existing reweighted $\ell_1$-minimization implementations require either QR decomposition~\cite{Meyer} of the measurement matrix ${\bf H}$ or the null space of ${\bf H}$,
which requires ${\bf H}$ to be expressed as a matrix. That said, ${\bf H}$ is a very large matrix, and we implement it as a linear operator.
Therefore, implementing the reweighted $\ell_1$-minimization that is applicable to the system model in \eqref{eq:CASSI} is beyond the scope of this paper, and the reweighted $\ell_1$-minimization is not included in our simulation results.
There exist other hyperspectral image reconstruction algorithms based on dictionary learning~\cite{Rajwade2013,Yuan2015Side}.
In order to learn a dictionary that represents a 3D image, the image cube needs to be divided into small patches, and the measurement matrix~${\bf H}$ also needs to be divided accordingly. Dividing the measurement matrix into smaller patches is convenient for the standard CASSI model (Figure~\ref{fig.1a}), because there is a one-to-one correspondence between the measurement matrix and the image cube, i.e., each measurement is a linear combination of only one voxel in each spectral band. In higher order CASSI, however, each measurement is a linear combination of multiple voxels in each spectral band.
Therefore,
it is not straightforward to modify these dictionary learning methods to the higher order CASSI model described in Section~\ref{subsec:HighCASSI}, and we do not include these algorithms in the comparison.
\subsection{Test on ``Lego" image cube}
\label{subsec:LegoTest}
The first set of simulations is performed for the scene shown in Figure~\ref{fig.Lego}.
This data cube was acquired using a wide-band Xenon lamp as the illumination source, modulated by a visible monochromator spanning the spectral range between $448$ nm and $664$ nm, and each spectral band has $9$ nm width. The image intensity was captured using a grayscale CCD camera, with pixel size $9.9$ $\mu$m, and 8 bits of intensity levels. The resulting test data cube has $M \times N = 256 \times 256$ pixels of spatial resolution and $L = 24$ spectral bands.
{\bf Setting 1:} The measurements~${\bf g}$ are captured with $K=2$ shots.
The coded aperture in the first shot is generated randomly with 50\% of the aperture being opaque, and
the coded aperture in the second shot is the complement of the aperture in the first shot.
The measurement rate with two shots is $m/n=KM(N+L+1)/(MNL)\approx0.09$.
Moreover, we add Gaussian noise with zero mean to the measurements. The signal to noise ratio (SNR) is defined as $10\log_{10}(\mu_g/\sigma_\text{noise})$~\cite{Arguello2014}, where $\mu_g$ is the mean value of the measurements ${\bf Hf_0}$ and $\sigma_\text{noise}$ is the standard deviation of the additive noise~${\bf z}$. In Setting 1, we add measurement noise such that the SNR is 20 dB.
We note in passing that
the complementary random coded apertures are binary, and can be implemented through photomask technology or emulated by a digital micromirror device (DMD). Therefore, the complementary random coded apertures are feasible in practice~\cite{Arguello2014}.
Moreover,
the complementary random coded apertures ensure that in the matrix ${\bf H}$ in~\eqref{eq:CASSI}, the norm of each column is similar, which is suitable for the AMP framework.
However, it is a limitation of the current AMP-3D-Wiener that the complementary random coded apertures must be employed,
otherwise, AMP-3D-Wiener may diverge.
Let us now evaluate the numerical results for Setting 1.
Figure~\ref{fig.iter_Psnr} compares the reconstruction quality of AMP-3D-Wiener, GPSR, and TwIST within a certain amount of runtime.
Runtime is measured on a Dell OPTIPLEX 9010 running an Intel(R)
CoreTM i7-860 with 16GB RAM, and the environment is Matlab R2013a.
In Figure~\ref{fig.iter_Psnr},
the horizontal axis represents runtime in seconds, and the vertical axis is the averaged PSNR over the 24 spectral bands.
Although the PSNR of AMP-3D-Wiener oscillates during the first few iterations, which may be because the matrix~${\bf H}$ is ill-conditioned, it becomes stable after 50 seconds and reaches a higher level when compared to the PSNRs of GPSR and TwIST at 50 seconds.
After 450 seconds, the average PSNR of the cube reconstructed by AMP-3D-Wiener (solid curve with triangle markers) is 26.16 dB, while the average PSNRs of GPSR (dash curve with circle markers) and TwIST (dash-dotted curve with cross markers) are 23.46 dB and 25.10 dB, respectively. Note that in 450 seconds, TwIST runs roughly 200 iterations, while AMP-3D-Wiener and GPSR run 400 iterations.
Figure~\ref{fig.spectral_Psnr} complements Figure~\ref{fig.iter_Psnr} by illustrating the PSNR of each 2D slice in the reconstructed cube separately. It is shown that the cube reconstructed by AMP-3D-Wiener has $2-4$ dB higher PSNR than the cubes reconstructed by GPSR and $0.4-3$ dB higher than those of TwIST for all 24 slices.
\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
\includegraphics[width=80mm]{iter_vs_Psnr_20dB}
\end{center}
\vspace*{0mm}
\caption{\small\sl
Runtime versus average PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the Lego image cube. Cube size is $M=N=256$, and $L=24$. The measurements are captured with $K=2$ shots using complementary random coded apertures, and the number of measurements is $m=143,872$. Random Gaussian noise is added to the measurements such that the SNR is 20 dB.}
\label{fig.iter_Psnr}
\end{figure}
\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
\includegraphics[width=80mm]{spectral_vs_Psnr_20dB}
\end{center}
\vspace*{0mm}
\caption{\small\sl
Spectral band versus PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the Lego image cube. Cube size is $M=N=256$, and $L=24$. The measurements are captured with $K=2$ shots using complementary random coded apertures, and the number of measurements is $m=143,872$. Random Gaussian noise is added to the measurements such that the SNR is 20 dB.}
\label{fig.spectral_Psnr}
\end{figure}
In Figure~\ref{fig.ImgComp}, we plot the 2D slices at wavelengths $488$ nm, $533$ nm, and $578$ nm in the actual image cubes reconstructed by AMP-3D-Wiener, GPSR, and TwIST. The images in these four rows are slices from the ground truth image cube~${\bf f_0}$, the cubes reconstructed by AMP-3D-Wiener, GPSR, and TwIST, respectively.
The images in columns $1-3$ show the upper-left part of the scene, whereas images in columns $4-6$ show the upper-right part of the scene. All images are of size $128\times128$.
It is clear from Figure~\ref{fig.ImgComp} that the 2D slices reconstructed by AMP-3D-Wiener have better visual quality; the slices reconstructed by GPSR have blurry edges, and the slices reconstructed by TwIST lack details, because the total variation regularization tends to constrain the images to be piecewise constant.
\begin{figure*}[t!]
\setcounter{cnt02}{5}
\vspace*{-5mm}
\hspace*{-10mm}
\includegraphics[width=190mm]{All_3_9_15}
\vspace*{-15mm}
\caption{\small\sl
2D slices at wavelengths $488$ nm, $533$ nm, and $578$ nm in the image cubes reconstructed by AMP-3D-Wiener, GPSR, and TwIST for the Lego image cube. Cube size is $M=N=256$, and $L=24$. The measurements are captured with $K=2$ shots using complementary random coded apertures, and the number of measurements is $m=143,872$. Random Gaussian noise is added to the measurements such that the SNR is 20 dB.
First row: ground truth; second row: the reconstruction result by AMP-3D-Wiener; third row: the reconstruction result by GPSR; last row: the reconstruction result by TwIST.
Columns $1-3$: upper-left part of the scene of size $128\times 128$; columns $4-6$: upper-right part of the scene of size $128\times128$.
}
\label{fig.ImgComp}
\setcounter{figure}{\value{cnt02}}
\end{figure*}
\begin{figure*}[t!]
\setcounter{cnt03}{6}
\vspace*{-0mm}
\hspace*{10mm}
\subfigure[Original image]{
\includegraphics[width=0.4\textwidth]{BCD}
\label{fig.BCD}
}
\subfigure[$x=190,y=50$]{
\includegraphics[width=0.4\textwidth]{sigB}
\label{fig.B}
}\\
\hspace*{10mm}
\subfigure[$x=176,y=123$]{
\includegraphics[width=0.4\textwidth]{sigC}
\label{fig.C}
}
\subfigure[$x=63,y=55$]{
\includegraphics[width=0.4\textwidth]{sigD}
\label{fig.D}
}
\caption{\small\sl
Comparison of AMP-3D-Wiener, GPSR, and TwIST on reconstruction along the spectral dimension of three spatial pixel locations as indicated in (a). The estimated pixel values are illustrated for (b) the pixel B, (c) the pixel C, and (d) the pixel D. }
\label{fig.sig}
\setcounter{figure}{\value{cnt03}}
\end{figure*}
Furthermore, a spectral signature plot analyzes how the pixel values change along the spectral dimension at a fixed spatial location, and we present such spectral signature plots for the image cubes reconstructed by AMP-3D-Wiener, GPSR, and TwIST in Figure~\ref{fig.sig}.
Three spatial locations are selected as shown in Figure~\ref{fig.BCD}, and the spectral signature plots for locations B, C, and D are shown in Figures~\ref{fig.B}--\ref{fig.D}, respectively. It can be seen that the spectral signatures of the cube reconstructed by AMP-3D-Wiener closely resemble those of the ground truth image cube (dotted curve with square markers), whereas there are discrepancies between the spectral signatures of the cube reconstructed by GPSR or TwIST and those of the ground truth cube.
According to the runtime experiment from Setting 1, we run AMP-3D-Wiener with 400 iterations, GPSR with 400 iterations, and TwIST with 200 iterations for the rest of the simulations, so that all algorithms complete within a similar amount of time.
{\bf Setting 2:} In this experiment, we add measurement noise such that the SNR varies from 15 dB to 40 dB, which is the same setting as in Arguello and Arce~\cite{Arguello2014}, and the result is shown in Figure~\ref{fig.SNR_Psnr}. Again, AMP-3D-Wiener achieves more than 2 dB higher PSNR than GPSR, and about 1 dB higher PSNR than TwIST, overall.
{\bf Setting 3:} In Settings 1 and 2, the measurements are captured with $K=2$ shots. We now test our algorithm on the setting where the number of shots varies from $K=2$ to $K=12$ with pairwise complementary random coded apertures.
Specifically, we randomly generate the coded aperture for the $k$-th shot for $k=1,3,5,7,9,11$, and the coded aperture in the $(k+1)$-th shot is the complement of the aperture in the $k$-th shot.
In this setting, a moderate amount of noise (20 dB) is added to the measurements. Figure~\ref{fig.shots_Psnr} presents the PSNR of the reconstructed cubes as a function of the number of shots, and AMP-3D-Wiener consistently beats GPSR and TwIST.
\begin{figure}[t]
\vspace*{-5mm}
\begin{center}
\includegraphics[width=80mm]{SNR_vs_Psnr1}
\end{center}
\vspace*{0mm}
\caption{\small\sl
Measurement noise versus average PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the Lego image cube. Cube size is $M=N=256$, and $L=24$. The measurements are captured with $K=2$ shots using complementary random coded apertures, and the number of measurements is $m=143,872$.}
\label{fig.SNR_Psnr}
\end{figure}
\begin{figure}[h]
\vspace*{-5mm}
\begin{center}
\includegraphics[width=80mm]{shots_vs_Psnr_20dB}
\end{center}
\vspace*{0mm}
\caption{\small\sl
Number of shots versus average PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the Lego image cube. Cube size is $N=M=256$, and $L=24$. The measurements are captured using pairwise complementary random coded apertures. Random Gaussian noise is added to the measurements such that the SNR is 20 dB.}
\label{fig.shots_Psnr}
\end{figure}
\subsection{Test on natural scenes}
\label{subsec:NaturalTest}
Besides the Lego image cube, we have also tested our algorithm on image cubes of natural scenes~\cite{Foster2006}.\footnote{The cubes are downloaded from http://personalpages.manchester.ac.uk/staff/\\
d.h.foster/Hyperspectral$\_$images$\_$of$\_$natural$\_$scenes$\_$04.html and
http://per-\\sonal
pages.manchester.ac.uk/staff/d.h.foster/Hyperspectral$\_$images$\_$of$\_$natural$\_$\\scenes$\_$02.html.}
There are two datasets, ``natural scenes 2002" and ``natural scenes 2004,"
each one with 8 image data cubes. The cubes in
the first dataset have $L=31$ spectral bands with spatial resolution of around $700\times700$, whereas the cubes in
the second dataset have $L=33$ spectral bands with spatial resolution of around $1000\times 1000$.
To satisfy the dyadic constraint of the 2D wavelet, we crop their spatial resolution to be $M=N=512$.
Because the spatial dimensions of the cubes ``scene 6" and ``scene7" in the first dataset are smaller than $512\times 512$, we do not include results for these two cubes.
The measurements are captured with $K=2$ shots, and the measurement rate is $m/n=KM(N+L+1)/(MNL)\approx0.069$ for ``natural scene 2002" and $0.065$ for ``natural scene 2004."
We test for measurement noise levels such that the SNRs are 15 dB and 20 dB.
The typical runtimes for AMP with 400 iterations, GPSR with 400 iterations, and TwIST with 200 iterations are approximately $2,800$ seconds.
The average PSNR over all spectral bands for each reconstructed cube is shown in Tables~\ref{tb:scene2002} and~\ref{tb:scene2004}. We highlight the highest PSNR among AMP-3D-Wiener, GPSR, and TwIST using bold fonts. It can be seen from Tables~\ref{tb:scene2002} and~\ref{tb:scene2004} that AMP-3D-Wiener usually outperforms GPSR by $2-5$ dB in terms of the PSNR, and outperforms TwIST by $0.2- 4$ dB, while TwIST outperforms GPSR by up to 3 dB for most of the scenes.
Additionally, the results of 6 selected image cubes are displayed in Figure~\ref{fig.scene} in the form of 2D RGB images.\footnote{We refer to the tutorial from http://personalpages.manchester.ac.uk/staff/\\david.foster/Tutorial$\_$HSI2RGB/Tutorial$\_$HSI2RGB.html and convert 3D image cubes to 2D RGB images.}
The four rows of images correspond to ground truth, results by AMP-3D-Wiener, results by GPSR, and results by TwIST, respectively. We can see from Figure~\ref{fig.scene} that
the test datasets contain both smooth scenes and scenes with large gradients, and
AMP-3D-Wiener consistently reconstructs better than GPSR and TwIST, which suggests that AMP-3D-Wiener is adaptive to various types of scenes.
\begin{table}[t]
\vspace*{-0mm}
\centering
\begin{tabular}{|c|| c | c | c ||c|c|c|}
\hline
SNR&\multicolumn{3}{c||}{15 dB}&\multicolumn{3}{c|}{20 dB}\\
\hline
Algorithm& AMP & GPSR & TwIST & AMP & GPSR & TwIST \\
\hline
Scene 1& {\bf32.69} & 28.10 & 31.05 & {\bf33.29} & 28.09 & 31.16\\
Scene 2& {\bf26.52} & 24.32 & 26.25 & {\bf26.65} & 24.40 & 26.41\\
Scene 3& {\bf32.05} & 29.33 & 31.21 & {\bf32.45} & 29.55 & 31.54\\
Scene 4& {\bf27.57} & 25.19 & 27.17 & {\bf27.76} & 25.47 & 27.70\\
Scene 5& {\bf29.68} & 27.09 & 29.07 & {\bf29.80} & 27.29 & 29.42\\
Scene 8& {\bf28.72} & 25.53 & 26.24 & {\bf29.33} & 25.77 & 26.46 \\
\hline
\end{tabular}
\caption{\small\sl
Average PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the dataset ``natural scene 2002" downloaded from~\cite{Foster2006}. The spatial dimensions of the cubes are cropped to $M=N=512$, and each cube has $L=31$ spectral bands.
The measurements are captured with $K=2$ shots, and the number of measurements is $m=557,056$.
Random Gaussian noise is added to the measurements such that the SNR is 15 or 20 dB. Because the spatial dimensions of the cubes ``scene 6" and ``scene7" in ``natural scenes 2002" are smaller than $512\times 512$, we do not include results for these two cubes.}
\label{tb:scene2002}
\end{table}
\begin{table}[t]
\vspace*{-0mm}
\centering
\begin{tabular}{|c|| c | c | c ||c|c|c|}
\hline
SNR&\multicolumn{3}{c||}{15 dB}&\multicolumn{3}{c|}{20 dB}\\
\hline
Algorithm& { AMP} & GPSR & TwIST & { AMP} & GPSR & TwIST \\
\hline
Scene 1& {\bf30.48} & 28.43 &30.17 & {\bf30.37} & 28.53 & 30.31 \\
Scene 2& {\bf27.34} & 24.71 &27.03 & {\bf27.81} & 24.87 & 27.35 \\
Scene 3& {\bf33.13} & 29.38 & 31.69 & {\bf33.12} & 29.44 & 31.75\\
Scene 4& {\bf32.07} & 26.99 & 31.69 & {\bf32.14} & 27.25 & 32.08\\
Scene 5& {\bf27.44} & 24.25 & 26.48 & {\bf27.83} & 24.60 & 26.85\\
Scene 6& {\bf29.15} & 24.99 & 25.74 & {\bf30.00} & 25.53 & 26.15 \\
Scene 7& {\bf36.35} & 33.09 & 33.59 & {\bf37.11} & 33.55 & 34.05\\
Scene 8& {\bf32.12} & 28.14& 28.22 & {\bf32.93} & 28.82 & 28.69 \\
\hline
\end{tabular}
\caption{\small\sl
Average PSNR comparison of AMP-3D-Wiener, GPSR, and TwIST for the dataset ``natural scene 2004" downloaded from~\cite{Foster2006}. The spatial dimensions of the cubes are cropped to $M=N=512$, and each cube has $L=33$ spectral bands.
The measurements are captured with $K=2$ shots, and the number of measurements is $m=559,104$.
Random Gaussian noise is added to the measurements such that the SNR is 15 or 20 dB.}
\label{tb:scene2004}
\end{table}
\begin{figure*}[t!]
\setcounter{cnt04}{9}
\vspace*{0mm}
\begin{center}
\includegraphics[width=160mm]{scene}
\end{center}
\vspace*{0mm}
\caption{\small\sl
Comparison of selected image cubes reconstructed by AMP-3D-Wiener, GPSR, and TwIST for the datasets ``natural scene 2002" and ``natural scene 2004." The 2D RGB images shown in this figure are converted from their corresponding 3D image cubes. Cube size is $N=M=512$, and $L=31$ for images in columns $1-2$ or $L=33$ for images in columns $3-6$. Random Gaussian noise is added to the measurements such that the SNR is 20 dB. First row: ground truth; second row: the reconstruction result by AMP-3D-Wiener; third row: the reconstruction result by GPSR; last row: the reconstruction result by TwIST.}
\label{fig.scene}
\setcounter{figure}{\value{cnt04}}
\end{figure*}
\section{Conclusion}
\label{sec:disc}
In this paper, we considered the compressive hyperspectral imaging reconstruction problem for the coded aperture snapshot spectral imager (CASSI) system.
Considering that the CASSI system is a great improvement in terms of imaging quality and acquisition speed over conventional spectral imaging techniques, it is desirable to further improve CASSI by accelerating the 3D image cube reconstruction process. Our proposed AMP-3D-Wiener used an adaptive Wiener filter as a 3D image denoiser within the approximate message passing (AMP)~\cite{DMM2009} framework. AMP-3D-Wiener was faster than existing image cube reconstruction algorithms, and also achieved better reconstruction quality.
In AMP, the derivative of the image denoiser is required, and the adaptive Wiener filter can be expressed in closed form using a simple formula, and so its derivative is easy to compute.
Although the matrix that models the CASSI system is ill-conditioned and may cause AMP to diverge, we helped AMP converge using damping, and reconstructed 3D image cubes successfully.
Numerical results showed that AMP-3D-Wiener is robust and fast, and outperforms gradient projection for sparse reconstruction (GPSR) and two-step iterative shrinkage/thresholding (TwIST) even when the regularization parameters for GPSR and TwIST are optimally tuned. Moreover, a significant advantage over GPSR and TwIST is that AMP-3D-Wiener need not tune any parameters, and thus an image cube can be reconstructed by running AMP-3D-Wiener only once,
which is critical in real-world scenarios.
In contrast, GPSR and TwIST must be run multiple times in order to find the optimal regularization parameters.
{\bf Future improvements: }In our current AMP-3D-Wiener algorithm for compressive hyperspectral imaging reconstruction, we estimated the noise variance of the noisy image cube within each AMP iteration using~\eqref{eq:sigma_t}. In order to denoise the noisy image cube in the sparsifying transform domain, we applied the estimated noise variance value to all wavelet subbands. The noise variance estimation and 3D image denoising method were effective, and helped produce promising reconstruction. However, both the noise variance estimation and the 3D image denoising method may be sub-optimal, because the noisy image cube within each AMP iteration does not contain i.i.d. Gaussian noise, and so the coefficients in the different wavelet subbands may contain different amounts of noise.
On the other hand, in the proposed adaptive Wiener filter, the variances of the coefficients in the sparsifying transform domain were estimated empirically within each wavelet subband, whereas it is also possible to apply Wiener filtering via marginal likelihood or generalized cross validation~\cite{Rasmussen06}.
Therefore, it is possible that the denoising part of the proposed algorithm can be further improved. The study of such denoising methods is left for future work.
In our current AMP-3D-Wiener, the coded apertures must be complementary, because complementary coded apertures ensure that the norm of each column in the matrix~${\bf H}$ in~\eqref{eq:CASSI} is similar, otherwise, AMP-3D-Wiener may diverge. Although using complementary coded aperture has practical importance, it provides more flexibility in coded aperture design when such a complementary constraint can be removed, and the development for AMP-based algorithms without such constraints is left for future work.
Finally, besides reconstructing image cubes from compressive hyperspectral imaging systems, it would also be interesting to investigate problems such as target detection~\cite{Manolakis2003} and unmixing~\cite{Bioucas2012} using compressive measurements from hyperspectral imaging systems. We leave these problems for future work.
\section*{Acknowledgments}
We thank Sundeep Rangan and Phil Schniter for inspiring discussions on approximate message passing; Lawrance Carin, and Xin Yuan for kind help on numerical experiments; Junan Zhu for informative explanations about CASSI systems; Nikhil Krishnan for detailed suggestions on the manuscript; and the reviewers for their careful evaluation of the manuscript.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
2,869,038,154,852 | arxiv |
\section{Related Work}
We surveyed existing studies from two aspects: 1) detection of concept drifts, and 2) visualization of time-series data.
\subsection{Detection of Concept Drifts}
Concept drifts are defined as the changes in the joint distribution between the environment variables (\textit{i.e.}, the time-series data that analysts collected) and target variables(\textit{i.e.}, the labels that analysts want to predict)~\cite{schlimmer1986incremental, gama2014survey}. Approaches for detecting concept drifts are fall in two categories: performance-based approaches and distribution-based approaches. Performance-based approaches detect concept drifts from the abnormal fluctuation of performance indicators, like accuracy. Drift Detection Method (DDM)~\cite{gama2004learning} recognizes an abnormal increase in error rate over certain ranges as a warning or a concept drift occurrence. The ranges are determined by the confidence intervals of the Normal distribution. A variety of statistical test methods can be applied after the study subject transforms into the error rate. For instance, concept drifts are identified by a set of the chi-square test~\cite{nishida2007detecting}. To avoid the influence of the size of the upcoming data, Fisher’s Exact test is chosen~\cite{de2018concept}.
Following the definition of the concept, distribution-based approaches compare data distributions and detect concept drifts by identifying distribution changes. However, calculating distribution similarity is a time-consuming task due to the complexity of distribution characteristics, \textit{i.e.}, extreme values, skewness, variance, etc. Considering an incremental learning process, the problem can be simplified by focusing on the differences. Certain assumptions are made~\cite{dos2016fast} to describe the changes as a series of operations between constant values. Besides, grouping variables is a prominent approach to simplify the density statistics process. The framework~\cite{sethi2016grid} maps variables into a grid space and performs density-based clustering on grid cells. Then the influences caused by the upcoming data can be summarized as the cluster generations or cluster extensions. Taking advantage of \textit{k-nearest neighbor} (KNN), sub-spaces can be constructed~\cite{liu2018accumulating} for a sample set, and density variations are identified with a distance measurement. In summary, distribution-based approaches need to be supported by intricate quantitative evaluation~\cite{lu2018learning}
\subsection{Visualization of Time-series Data}
Identifying patterns from multidimensional time-series data is a comprehensive process. It is thus essential to integrate visualization and data analysis methods for better efficiency. Existing studies achieve this goal from three perspectives: setting target patterns interactively, extracting repeated patterns based on clustering and frequency features, and detect abnormal patterns by leveraging machine learning approaches.
Within a visual interface, analysts can express what they want from the visual analysis system. Thermalplot~\cite{stitz2015thermalplot} supports specify by setting weights for each attribute. Time-varying objects are mapped into a two-dimensional space defined by the degree of interest and corresponding change over time. To explore co-occurrence patterns, COPE~\cite{li2018cope} needs analysts to specify events by setting thresholds for attribute values. The spatiotemporal pattern of similar events can be checked with COPE. It is effective to allow users to gradually narrow their search, especially when they are not sure what they want. TimeNotes~\cite{walker2015timenotes} provide users with a hierarchy time axes. Users are allowed to iteratively select one or more small time range of interest from a large time range by brushing.
When analysts have limited knowledge of datasets, it is necessary to augment the analysis with automatic methods. Temporal Multidimensional Scaling (TMDS)~\cite{jackle2015temporal} discretizes the time dimension by a user-defined sliding window, and projects the multi-dimensional data in each window to one dimension via MDS. After a series of flipping operations, one-dimensional projections are juxtaposed to show temporal patterns.
In addition to dimensionality reduction methods, extracting important periods can also reduce the user's workload. StreamExplorer~\cite{wu2018streamexplorer} employs a subevent detection model to identify important periods from a social stream. Because major events always lead to popular discussions, StreamExplorer recommends users the periods with a large number of tweets to analyze related events. To extract patterns flexibly, TPFlow~\cite{liu2018tpflow} employs a piecewise rank-one tensor decomposition to detect sub-tensors (\textit{i.e.} multidimensional patterns) with a top priority.
If unpredictable events are regarded as abnormal, the high prediction error of automatic models may imply abnormal patterns~\cite{tkachev2019local}. Taking advantage of the same feature, concept drift detection can be applied to locate useful patterns from dynamic environments. Visualization techniques have been used to depict the development of concept drifts~\cite{yao2013concept, demvsar2018detecting}.
Common charts, like line charts~\cite{webb2018analyzing}, scatter plots~\cite{stiglic2011interpretability} and parallel coordinates~\cite{pratt2003visualizing} are employed for this purpose.
On the other hand, model-generating information can convey the characteristics of concept drifts. The time-varying contributions of each attribute value can be visualized for classification~\cite{demvsar2014visualization}. By marking the fluctuations, the occurrences of concept drifts can be easily identified from a micro-level. Also, users are allowed to take an overview from a macro level to assess the importance of each attribute value~\cite{demvsar2014visualization}.
None of these studies can explain why a concept drift is identified by the detection model, which is viral for the final decision.
\section{Problem Definition and Models}
Two models are applied to support our goals. Before introducing the goals and the models, we first explain related definitions.
\subsection{Definitions}
\label{sec:def}
In this work, time-series from multiple sources are presented as temporal \textbf{data records}.
Each data record contains multiple \textbf{environment variables} $X$ and a \textbf{target variable} $y$.
The data records are distributed non-uniformly along the timeline.
We group data records in a unit time segment into a \textbf{batch}.
A batch forms a basic unit for coordinated analysis among different data sources.
We denote the data source and timestamp of a batch as its \textbf{context}.
In machine learning scenarios, records are usually used by the training model, where the target variable is the data label.
Without loss of generality, the target variable is limited to be \textit{binary} in this paper.
It is assumed that there is an underlying relationship between the environment variables and the target variable, e.g., an underlying mapping $y = f(X)$, or a conditional distribution $p(y\mid X)$~\cite{gama2014survey}.
A \textbf{concept} refers to such a relationship.
The changes of concept over time is denoted as \textbf{concept drift}~\cite{widmer1996learning}.
\subsection{Goals}
\label{sec:goa}
Our main goal is a visual analytics tool for identifying and understanding concept drifts in multiple time-series data.
We identify four goals in building such a visual analytics system:
\textbf{G1: Automatic identification of concept drifts and concepts.} It is laborious to browse the time-series data through the entire time span. Moreover, the concept and concept drift are only implicitly embedded in the time-series data. Identifying numerous concepts and concept drifts individually is cumbersome. Therefore, the first goal of our system is to support the automatic identification of stable concept and important concept drifts.
\textbf{G2: Visual representation of concepts.} There is not an explicit definition of concepts. The assumed definitions, including the mapping or the conditional probability, are complicated to describe and understand. Therefore, presenting visualization to disclose the pattern of concepts, \textit{i.e.}, the relationship between the target variable and environment variables is needed.
\textbf{G3: Discrimination of concept drifts in multiple data sources.} With the same target variable and environment variables are collected, concepts drifts could be heterogeneous in different data sources~\cite{hochman2012visualizing}. When an inconsistency occurs among different data sources, analysts should be able to discriminate them and verify the interested concept drift.
\textbf{G4: Interactive specification of concepts.} Concepts extracted from the batches with different contexts may be numerous. Although automatic models can augment the selection process, there is a natural need for interactive exploration and specifications of concepts~\cite{lu2017state, endert2017state}.
\subsection{The Drift Level Index}
\label{sec:det}
Conventional machine learning approaches provide a quantitative description of concept drifts (\textbf{G1}).
The basic idea is that if the concept is stationary over time, the performance of a trained prediction model should be stable or increasing.
Otherwise, the prediction accuracy will decrease and trigger an index of concept drifts.
Therefore, the decrease of the prediction accuracy is a meaningful index for concept drifts.
When a concept drift occurs, the pre-trained prediction model might have a decreasing performance and be less sensitive to subsequent concept drifts.
Therefore, the prediction model should update with the evolving of concepts, and hence the learning process is iterative, \textit{i.e.}, learning parameters for each attribute are updated when a new data record comes. Following the technique presented in~\cite{brzezinski2014combining}, we maintain a set of prediction models and take the one with the highest accuracy in the last verification as the output model. The weakest model in the set is replaced with a newly trained model. Models trained on the new data can learn new characteristics of the label and be adaptable to dynamic environments.
As a result, the prediction process has a high performance and is sensitive to the upcoming concept drifts.
We use $p_i$ to denote the error rate of the prediction model in a sliding window, which ends at the $i$th record.
The sliding window is set to cover $500$ data records in our implementation. The distribution of correct predictions in a sliding window can be regarded as a binomial distribution. Therefore, for each $p_i$, we compute the standard deviation as $s_i = \sqrt{p_i(1-p_i)/n}$. Similar to the work in~\cite{gama2004learning}, we assume that the error rates are distributed normally. The concept drift level can be measured by the confidence levels of corresponding confidence intervals. As discussed above, in a static environment, the error rate of a prediction model is supposed to be decreasing or approximately stable over time. Although the fluctuation of the error rate is normal, the degree of its increasing typically indicates a high probability of the occurrence of a concept drift. As shown in Figure~\ref{fig:ddm}, the probability of a concept drift is computed with the minimum of the error rates $p_{min}$ (after the latest concept drift) and the standard deviation $s_{min}$~\cite{gama2004learning}. With the verified record $i$, the drift level $r_i$ is defined as:
\begin{equation}
r_i = \frac{p_i + s_i - p_{min}}{s_{min}}
\end{equation}
The threshold to determine whether a concept drift appears, \textit{i.e.}, the confirmation level, is set to be $r_i \geq 3$, which implies that the confidence level of a concept drift occurrence exceeds $99\%$~\cite{gama2004learning}. Also, when $r_i \geq 2$ (\textit{i.e.}, the warning level, the corresponding confidence level is over $95\%$), a warning is issued. The drift level $r^t$ for a batch is regarded as the average of $r_i, r_{i+1}, ..., r_j$, where $i, i+1, ..., j$ denote the output sequence from the data records in the batch.
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/ddm.eps}
\caption{Explanation of the drift detection model~\cite{gama2004learning}. }
\label{fig:ddm}
\end{figure}
\subsection{The Consistency Judgment Model}
The similarity between concepts derived from different data sources can be represented by the parameter similarity of the prediction models. However, parameters of the prediction models can not summarize the consistency of concept drifts. To support \textbf{G3}, we need to learn about the response of each data source to the dynamic environment. The dynamic environment can be described by the time-varying drift levels of data sources, which are continuously captured with the training of prediction models. Supposed that a data source has a consistent response with others, the Na\''ive Bayes theory is employed to infer the time segment of the concept drift occurrence. The judgment about whether the detected concept drifts satisfy the inferred results can be concluded.
Considering the differences among data sources, we propose a consistency judgment model that is used for each data source separately. At time $t$, the drift levels of $m$ data sources $d_1,d_2,...,d_i, ...,d_m$ are recorded as $r_1^t, r_2^t,...,r_i^t, ...,r_m^t$. Regarding $x^t_i =[r_1^t,...,r_{i - 1}^t, r_{i + 1}^t, ...,r_m^t] $ as inputs of the judgment model, the corresponding label $y_i^t$ of data source $d_i$, is defined as whether a concept drift occurs in the recent time segment $\Delta t$ (normally as a unit time segment), that is, whether the confirmation level exceeds by one of $r_i^{t - \Delta t}, r_i^{t - \Delta t + 1},..., r_i^{t+\Delta t}$. Based on the Na\"ive Bayes theory, a probability curve can be generated to depict the probabilities of concept drifts over time. The curve segments, where the probability is higher than a user-defined probability threshold $c$, implies the corresponding labels are judged as ``yes''. On the contrary, the labels of the rest time segments are ``no''. Therefore, the time segment $[t_{start}, t_{end}]$ for an occurrence of a concept drift is inferred. If the label $y_i$ is independent with $x_i$, the corresponding probability curve would be flat and has less chance to be higher than the threshold. That is, if the data source $i$ is inconsistent, the consistency judgment model results in zero time segment.
We compare the time points of detected concept drifts with the time segments to verify if a data source drifts in an inconsistent way. When the label is defined, the corresponding concept drifts should appear in the time segment $[t_{start}-\Delta t, t_{end}+\Delta t]$. A data source is regarded as inconsistent with the entire environment at $t$ if its previous concept drift occurs out of the previous time segment. Thus, the larger the parameter $c$ is, the higher the probability that data sources are considered as inconsistent.
\section{System Overview}
\subsection{Design Requirements}
To address the goals mentioned in Section~\ref{sec:goa}, five design requirements are identified to guide system design.
\textbf{DR1: Provide an overview of concept drift occurrences over time.} The drift occurrences over time can help analysts identify the interesting time segment. When the analysts' preferences are unknown, interactive and hierarchical exploration of occurrences along time intervals is needed~\cite{niederer2017taco, walker2015timenotes}
\textbf{DR2: Integrate features of concept drifts from the prediction models.} Concept drifts hinder existing prediction models (the model trained from historical data) from accurate predictions in new environments. The accuracy fluctuation of the prediction model indicates the occurrence of concept drifts~\cite{stiglic2011interpretability, becker2007real}. In addition, parameters of prediction models can reflect the relationship between different inputs and the label, that is, the model's understanding of the concept~\cite{cassidy2014calculating}.
\textbf{DR3: Identify the context of concepts and allow adjustments.} Analysts need to know the context of the analyzed data records. Considering that analysts may miss details or may disagree with the navigation, interactive adjustments for recommended results are needed~\cite{law2016vismatchmaker, liang2017photorecomposer}.
\textbf{DR4: Study the relationship between attributes and labels.} While concepts have not an explicit definition, it is essential to provide a visual explanation. The relationship between labels and attributes are considered to be an important description of concepts~\cite{gama2014survey,ditzler2015learning}.
\textbf{DR5: Compare concepts in different contexts.} Comparing different concepts facilitates the understanding of the evolving concepts and their contexts, \textit{e.g.}, the trends and outliers. There is a need to compare a newly identified concept with previously studied ones and record the identified concepts~\cite{webb2018analyzing}. Comparing concepts related to a concept drift also favors the understanding of the drift.
\subsection{Workflow}
\label{sec:wor}
To derive concepts from multi-source time-series datasets, analysts need to manipulate the data and select contexts.
We design a five-step workflow, as shown in Figure~\ref{fig:ppl}.
We provide an overview of all concept drifts detected from all data sources during the entire time span (see Figure~\ref{fig:ppl}(a), \textbf{DR1}). Analysts may be attracted by time segments when a single data source has interesting patterns (\textit{e.g.} dense occurrences or a periodicity), or multiple data sources need to be compared (\textit{e.g.} abnormal concept drifts). When a time segment is selected, concept drifts from the prediction models are shown (\textbf{DR2}). As shown in Figure~\ref{fig:ppl}(b), the accuracy fluctuation and parameters of prediction models can assist the detection of concept drifts. Next, analysts can specify the context of the concept to be analyzed with the external knowledge of concept drifts, as shown in Figure~\ref{fig:ppl}(c). \textit{ConceptExplorer}{} can recommend the time segment between two adjacent concept drifts according to an analyst-specified time point. The recommended time segments for different data sources may be different, or even inconsistent. \textit{ConceptExplorer}{} assesses the consistency of data sources by the consistency judgment model and recommends the group of data sources with consistent concept drifts. The recommended selection is displayed (\textbf{DR3}). If analysts are not satisfied with the recommendations, they can make flexible adjustments on contexts to support special analysis tasks. To explore the concepts with specified contexts, the relationship between attributes and concepts (\textbf{DR4}, see Figure~\ref{fig:ppl}(d)) are visualized. Analysts can identify and record significant concepts that may be involved in subsequent analysis. The identified concepts can be compared with other concepts (\textbf{DR5}, see Figure~\ref{fig:ppl}(e)).
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/workflow.eps}
\caption{The five-step workflow: (a) Observing the distribution of concept drifts and warnings; (b) Inspecting concept drifts through accuracy fluctuation and parameter changes; (c) Specifying the context of the concept to be analyzed; (d) Analyzing and comparing concepts based on correlations; (e) Identifying interesting concepts. }
\label{fig:ppl}
\end{figure}
\section{ConceptExplorer}
As shown in Figure~\ref{fig:tea}, \textit{ConceptExplorer}{} consists of a data entrance (see Figure~\ref{fig:tea}(a)) and four views. The data entrance lists the label definition, description of data sources, and attributes. The online system is available through the link: \url{http://101.132.126.253/}.
\subsection{The Timeline Navigator View}
As required by \textbf{DR1}, the timeline navigator view presents the entire timeline and the indices of concept drifts from multiple data sources (see Figure~\ref{fig:tea}(b)). Each row corresponds to a data source. \textit{ConceptExplorer}{} assigns a unique color to each data source. Due to the limited horizontal space, the distribution of concept drifts may be dense. \textit{ConceptExplorer}{} employs a ``$\times$'' to mark a concept drift, which can highlight the specific moment by its intersection. Time segments, in which the drift level exceeds a certain value (initialized as the warning level, namely, $2$) are highlighted by ``$-$''. These marks indicate various patterns along the timeline, like dense occurrences, outliers, inconsistency with other data sources, periodicity.
\subsection{The Prediction Model View}
The prediction model view (Figure~\ref{fig:tea}(c)) supports \textbf{DR2}.
\subsubsection{The Accuracy Fluctuation Chart}
The line charts on the left (Figure~\ref{fig:tea}(c)) show the accuracy fluctuation of the prediction models trained by the data from each data source. The occurrences of concept drifts are labeled by ``$\times$'', which is the same as that in the timeline navigator view. In addition, the moments with warnings are encoded by hollow dots. To explain concept drift detection, the accuracy fluctuation chart visualizes the magnitude of the accuracy drop of the time segments whose drift levels are above the warning level (Figure~\ref{fig:str}(a)). Different data sources may issue drift warnings at similar time segments. To avoid misunderstandings caused by overlaps, shifted stripes are employed to highlight warning time segments (Figure~\ref{fig:str}(b)). It can be seen that even when the warning segments of different data sources are staggered, the start and end moments of different time segments can be clearly distinguished. Vertical stripes are used because they can emphasize the height, that is, the magnitude of accuracy drops. The results from the consistency judgment model are also shown in the accuracy fluctuation chart. If a concept drift is detected during a time segment that is not included by the result from the consistency judgment model, we emphasize them by a triangle mark to distinguish from circles representing others (Figure~\ref{fig:str}(c)).
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/stripes.eps}
\caption{The visual designs for explaining the detection model and the consistency judgment model. (a) The explanation corresponds to the formula mentioned in Section~\ref{sec:det}. (b) Strips are shifted to avoid overlaps. (c) Encodings of concept drifts and warnings.}
\label{fig:str}
\end{figure}
\subsubsection{The Projected Parameter View}
The model parameters updated after each batch during the entire training process are projected into a two-dimensional plane using principal components analysis (PCA) based on singular value decomposition (SVD)~\cite{wall2003singular}. The points projected by parameters of the same data source are connected in order to form a curve. The distance between each pair of projected points illustrates the similarity of corresponding model parameters, namely, the concept similarity described by prediction models. Evolution patterns, like intersections and bundles~\cite{bach2016time, hinterreiter2020exploring} can be identified from the formed curves~\cite{ceneda2016characterizing}. The convergence and dispersal of curve segments representing different data sources indicate the agreement and disagreement of related models on the understanding of the concept. The curve segments corresponding to time segments selected in the timeline navigator view is colored by transparency, from which analysts can learn about the temporal order. The view is automatically zoomed in or out so as to fit the curves of the entire training process or the selected time segment within the window, as shown in Figure~\ref{fig:zoom}. With the background of the entire trajectories (Figure~\ref{fig:zoom}(a)), analysts can better measure relative distances. After zooming in, it can be seen (Figure~\ref{fig:zoom}(b)) that curves are not overlapped but with similar directions, that is, data sources have similar drifts. Analysts can drag the handle on the time axis of the accuracy fluctuation chart to move the circles, which highlight the projected parameters corresponding to the same moment.
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/zoom.eps}
\caption{The parameter projections of prediction models running for all data sources. The opacity encodes the time order. (a) The overview of the entire time range. (b) An enlarged part.}
\label{fig:zoom}
\end{figure}
\subsection{The Concept-Time View}
The concept-time view displays the time segments in different data sources that are integrated for concept analysis, as shown in Figure~\ref{fig:tea}(d). To display the sources of the applied data records, as mentioned in \textbf{DR3}, each data source is listed in a row to distinguish different data sources. Then, their data records are divided individually to introduce a specific time. Analysts need to make the trade-off between the number of data records and the clarity of concepts for appropriate adjustments. To facilitate decision-making, the drift level and the size of each batch are encoded with the color and height of the bar, respectively. The batches that compose the data records to be analyzed are highlighted. The number of these batches and the total number of the related data records are counted.
\subsection{The Concept Explanation View}
A correlation matrix (Figure~\ref{fig:tea}(e)) is employed to support \textbf{DR4} because of its representation ability~\cite{suschnigg2020exploration, wang2017utility}. The data source is considered as an attribute to label the context of the data records. For other attributes, the correlation for each batch of data records is quantified by the cosine similarity. The attributes are sorted by the average correlation of the selected batches.
\textit{ConceptExplorer}{} draws correlation matrices for the data source and analyst-specified number of the attributes with the highest correlations subject to the concept.
For each cell, the horizontal and vertical axes of each matrix are defined by two attributes. A square in non-diagonal cells represents a set of data records whose two attributes fall into the value ranges which are encoded with the position of the square. The differences between the number of records with positive labels and those with negative labels are counted for each square. The ratio of the difference of two counts over their sum (\textit{i.e.}, $\frac{\#Positives - \#Negatives}{\#Positives + \#Negatives}$) is encoded in color (ranging from red to blue). When the label distribution in the dataset is nonuniform, analysts can reset the color mapping and encode the percentage difference in all data records in white. In some specific contexts, certain cells may be empty. To distinguish squares without a record, strokes are added in the squares with more than one record. Darker strokes imply that the record number of the square is larger than $5\%$ of the amount of chosen data records. The matrix view exhibits a symmetrical layout. Taking advantage of this feature, the current correlation pattern can be compared with the other one. Each cell on the diagonal presents a pair of histograms (i.e., a grounded histogram for lower-left corner and an inverted histogram for upper-right corner).
\subsection{Interactions}
Following the workflow mentioned in Section~\ref{sec:wor}, \textit{ConceptExplorer}{} supports the following interactions.
\textbf{Navigate by overview.} Analysts first brush a time segment in the timeline navigator view and check related details in the prediction model view and the concept-time view.
\textbf{Inspect concept drifts.} In the accuracy fluctuation chart, the probability threshold $c$ for the consistency judgment model can be defined by a slider. To study why a concept drift is emphasized, analysts can check the inferred time segments for a data source.
\textbf{Specify the context of a concept.} Analysts can specify a timestamp by dragging the handle in the accuracy fluctuation chart. According to the time stamp, the concept-time view shows the recommended time segments and selects the batches in the time segments. If analysts are unsatisfied with the automatically selected batches, they can adjust the selection ranges by dragging the boundaries. Analysts can choose the data records which need to be included in the concept explanation view. Data sources are labeled with ``inconsistent'' and ``consistent''. The set of all consistent data sources is recommended.
\textbf{Identify concepts.} The data records with the specified context are integrated into the correlation matrix. Analysts are allowed to set the number of listed attributes. If analysts are interested in the pattern shown in the lower-left corner of the correlation matrix, i.e., the description of the concept with the current selected context, they can save the screenshot and related concept (see the right of Figure~\ref{fig:tea}(e)) by clicking the ``Identify'' button.
\textbf{Compare concepts.} In subsequent explorations, analysts can change the data in the upper-right corner of the correlation matrix. \textit{ConceptExplorer}{} highlights a pair of squares at symmetrical positions for comparison when one of them is specified by analysts.
\section{Case Studies}
\label{sec:cas}
We present three case studies based on real-world datasets. Various concepts and concept drifts are analyzed to evaluate the effectiveness of \textit{ConceptExplorer}.
\subsection{Beijing Air Quality Forecast}
In this case, we attempt to understand the dominant factors affecting air quality. We employ the air pollutant data~\cite{zhang2017cautionary} from four nationally-controlled air-quality monitoring sites in Beijing, which was collected every hour from March 1st, 2013 to February 28th, 2017 ($34,536$ data records per site). $22$ meteorology-related dimensions are applied to predict if the air quality index (AQI) is higher than $100$ (\textit{i.e.}, worse than mild pollution) after $24$ hours.
The timeline navigator view indicates that concept drifts occurred almost every few days (Figure~\ref{fig:tea}(b), \textbf{DR1}). The drift levels of all data sources are abnormally stable ($<2$, \textit{i.e.}, the warning level) during the week at the end of March 2015, except for data source (DS$1$, \textit{i.e.}, the site at Guanyuan park). We choose a $40$-day time segment around the week. As shown in Figure~\ref{fig:tea}(c), all data sources have a similar fluctuation of the prediction accuracy (\textbf{DR2}). Each one experienced more than one concept drift between March 17th and March 20th. We select two time segments before and after the drift time segment (see the black and blue time segments in Figure~\ref{fig:air_dd}, \textbf{DR3}).
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/air_db.eps}
\caption{Details of three time segments analyzed in the first case.}
\label{fig:air_dd}
\end{figure}
As shown in Figure~\ref{fig:tea}(e), the comparison result of two time segments indicates that their associated concepts have similarities (\textbf{DR5}). For instance, the higher the \textbf{PM10} concentration is, the more records are labeled with poor air quality (\textbf{DR4}). The difference mainly lies in that more high air quality records are observed (\textit{i.e.}, more blue squares in the upper-right corner) after the drift time segment. The records with a low \textbf{Dew Point} (\textit{i.e.}, dew point temperature ($^\circ$C)) are more likely to be labeled with good air quality. Also, the order of the attributes indicates that the dominant pollutant \textbf{PM10} is replaced by \textbf{PM2.5\_day} (\textit{i.e.}, the average PM2.5 concentration ($\mu g/m^3$) in the past $24$ hours) after the concept drift.
The concept drift of DS$1$ occurred on March 26th is identified as inconsistent with others by the consistency judgment model. Besides, the projected parameter trajectory of DS$1$ (see Figure~\ref{fig:air_wpv}) indicates that the parameters of DS$1$ go through a twist that is different from others (\textbf{DR2}). To explore the inconsistent behavior of DS1, the time segments after (the red time segments) the concept drift on March 26th for DS1 is checked (Figure~\ref{fig:air_dd}, \textbf{DR3}) . All cells in the correlation matrix turn into red (see the upper-right corner highlighted by the red dashed line in Figure~\ref{fig:air_ma}), which implies that almost all records are labeled with poor air quality (\textbf{DR4, DR5}). The weather records indicate that there was a sandstorm in Beijing at the end of March. The dominant pollutant during the time segment before the sandstorm (see the blue time segments in Figure~\ref{fig:air_dd}) is PM10, as shown in the lower-left corner highlighted by the blue dashed line of Figure~\ref{fig:air_ma}, which contributes to the inconsistent concept drift.
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.7\columnwidth]{pictures/air_wpv.eps}
\caption{The parameter projection view between March 22nd, 2014 to March 28th, 2014.}
\label{fig:air_wpv}
\end{figure}
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.9\columnwidth]{pictures/air_ds1.eps}
\caption{The correlation matrix compares the two concepts from two time segments of DS1. The lower-left corner (the dashed region in blue) corresponds to the blue time segment in Figure~\ref{fig:air_dd}. The upper-right corner (the dashed region in red) exhibits the sandstorm pattern.}
\label{fig:air_ma}
\end{figure}
Actually, other data sources record the same sandstorm. The dominance of PM2.5 has not been replaced by PM10 before the sandstorm, and thus the concept drift was not triggered by the sandstorm. Instead, the same label with poor air quality make the prediction task simple for the prediction model. The prediction accuracy has risen to $100\%$ in a couple of days. At the beginning of April, the sandstorm ended, and the air quality detected by all data sources improves, which leads to the next concept drift. An expert working in the meteorological bureau told us that the spring sandstorms in Beijing are basically caused by PM2.5. Such pollutants are spread by wind. Therefore, the detection results of sites in different locations have slight differences.
\subsection{Consumption Behaviors of MMORPG Players}
In the second case, we study the dynamics of consumption behaviors in multiplayer online role-playing game (MMORPG) to understand game company's operating strategies. For example, releasing a new role may attract new players to join in the game and consume, which leads to changes in the concept of consumption behaviors. The employed dataset contains player records from three servers ($647,800$ player records from Server17, $702,125$ player records from Server164, and $585,048$ player records from Server230) of a MMORPG from August 16th, 2013 to January 19th, 2014. Three servers were started at different timestamps: Server17, Sever164, and Server230, which are in order of time, that is, players on different servers register for the game at different time periods. For each player, $21$ attributes, like \textbf{equipment} (\textit{i.e.}, the combat effectiveness score of the player's equipment), \textbf{practice} (\textit{i.e.}, the level of practice, improved by learning and improving skills and finishing tasks), are recorded every day. The consumption records for the upcoming week of players form a group of time-series.
We first browse the entire time span to learn about the evolution of consumption behaviors in three servers.
With the threshold of $70\%$, all concept drifts are identified as inconsistent by the consistency judgment model. Besides, the projection of parameters of Server230 is far from those of the parameters of other data sources (see Figure~\ref{fig:net_wpv}, \textbf{DR2}), which implies that the consumption behaviors of the players in Sever230 is quite different from the other two servers. In particular, Server164 has a similar trajectory with Server17 from August 2013 to November 2013.
After that, the trace of Server230 shows a sharp downward turn. The specific time point is further studied. As shown in Figure~\ref{fig:net_ove}(a), the number of players at the moment was doubled---on October 24th, the game operators merged Server230 with Server229 to maintain player engagement. We notice that Server230 has fewer concept drifts than the other two servers before this merge, as shown in Figure~\ref{fig:net_ove}(b). We come up with a hypothesis that the consumption behaviors of players in Server230 affected less by various events than other servers. This phenomenon may be one reason for the operators to merge servers.
\begin{figure} [!ht]
\centering
\includegraphics[width=0.8\columnwidth]{pictures/net_wpv.eps}
\caption{The overview of the parameter projection view. The orange circle denotes the moment when Server230 is merged. }
\label{fig:net_wpv}
\end{figure}
\begin{figure} [!ht]
\centering
\includegraphics[width=0.9\columnwidth]{pictures/net_ove.eps}
\caption{Visualizations of a server merging event of Sever230: (a) the data details view and (b) the timeline navigator view.}
\label{fig:net_ove}
\end{figure}
To verify this hypothesis, an activity held a month earlier than the server merge (see Figure~\ref{fig:net_ove}(b)) is analyzed. The consistency judgment model regards Server230 as inconsistent with the other two servers. We use records from Server17 and Server164 to study players' respond to this event. The time segments before and during the event are selected separately (\textbf{DR3}). As shown on the left of Figure~\ref{fig:net_act}, the right-bottom square changes from gray to blue, which implies that a certain number of players in Server164 (DS2) took the opportunity to update their \textbf{equipment} to the highest level (\textbf{DR4, DR5}). Besides, some high-\textbf{practice} but poorly equipped players in both servers were enthusiastic about the event and consumed virtual currency during the event (see the left-top squares in the two cells on the right of Figure~\ref{fig:net_act}). However, no significant changes are observed in Server230.
We invited a data analysis expert, who was in charge of operating the game, to check our findings. She told us that because of player loyalty, the older the server is, the more the enthusiasm for game events. For newly opened servers, payment peaks occurred mainly at the moment of launching. As for merging servers, she told us that some players created smurfs in servers to collect equipment or provide assistance after merging servers. And the most efficient way to create a high-quality smurf is to consume during events. These observations verify the hypothesis.
\begin{figure} [htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/net_act.eps}
\caption{Patterns before and after the update event. Blue indicates that more players have consumption in the upcoming seven days.}
\label{fig:net_act}
\end{figure}
\begin{figure*} [!ht]
\centering
\includegraphics[width=1.99\columnwidth]{pictures/mov_olv.eps}
\caption{The accuracy fluctuation of the prediction models trained by the data from three data sources (June 15th, 2014 - July 10th, 2014).}
\label{fig:mov_olv}
\end{figure*}
\subsection{Movie Rating Prediction}
To comprehensively learn about the evolution of audience preference on different movies, we study whether the average rating of a movie will increase in the next seven days from three platforms: Rotten Tomatoes~\cite{rottenTomato} (recorded reviews from critics), IMDB (collected from Twitter)~\cite{IMDBtweet}, and MovieLens~\cite{movielens}. By extracting data stamped in the common time segment (from February 28th, 2013 to March 31st, 2015), there are $96174$, $385015$, $1127948$ records from three sources, respectively. Each record includes rating date, rating score, and movie ID. The movie ID is replaced with the movie description~\cite{movie}, like the release \textbf{year}, \textbf{budget}, \textbf{duration}, etc. The training data for the prediction model has $15$ dimensions.
Our analysis starts from the summer vacation because most people have chances to watch movies during this period. We select the segment from June 15th, 2014 to July 10th, 2014 from the timeline navigator view. As shown in Figure~\ref{fig:mov_olv}, the prediction models trained by the data records from different data sources have distinct accuracy fluctuations (\textbf{DR2}).
The consistency judgment model suggests to study three data sources separately. After grouping tests (\textbf{DR3}), we find that the records from Rotten Tomatoes and IMDB (the lower-left corner highlighted by the yellow dashed line) show clearer patterns than those from MovieLens (the upper-right corner highlighted by the brown dashed line, \textbf{DR5}), as shown in Figure~\ref{fig:mov_tf4}. By studying red squares in the lower-left corner, we detect three descriptions corresponding to the movies whose ratings have declined: movies whose release \textbf{year} is 2014, movies with relatively high \textbf{budget}s and \textbf{action} movies (\textbf{DR4}). Moreover, squares corresponding to each intersection of the above descriptions are in conspicuous red. The reason may be that a highly anticipated movie does not meet audience expectations. The rating records turn out that the disappointing movie is \textit{Transformers: Age of Extinction}.
\begin{figure} [!htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/mov_tf4.eps}
\caption{The correlation matrix displays concepts extracted from the segment around June 28th, 2014. The lower-left corner (the dashed region in yellow) shows records from Rotten Tomatoes (DS2) and IMDB (DS3). The upper-right corner (the dashed region in brown) shows those from MovieLens (DS1). Due to the uneven distribution of labels, the color mapping of the correlation matrix is reset to map the average difference to white.}
\label{fig:mov_tf4}
\end{figure}
We further observing the accuracy curves to learn platform characteristics. It can be seen from Figure~\ref{fig:mov_olv} that the accuracy fluctuation of Rotten Tomatoes is more severe than others. Especially, there exists periodic fluctuations in the curve (\textbf{DR2}). Concept drifts appeared once in about a week. The number of arriving data records has the same periodicity, as shown in Figure~\ref{fig:mov_rtw}(a). We select the time segments of the previous week and the next week (\textbf{DR3}). The main difference is identified from movies released in the 1990s and 2000s (see Figure~\ref{fig:mov_rtw}(b), \textbf{DR4}, \textbf{DR5}): new ratings reduce the average ratings of certain old movies. Similar patterns are not found from other data sources. This may be caused by specific recommendations from the Rotten Tomatoes---we notice that there are sections for ``hidden gem movies'' on the Rotten Tomatoes website.
\begin{figure} [htbp]
\centering
\includegraphics[width=0.99\columnwidth]{pictures/mov_rtw.eps}
\caption{The details of data records collected from Rotten Tomatoes in two weeks. (a) The data details view shows two specific time segments. (b) Two cells of the correlation matrix depict the correlation between \textbf{year} and the concept.}
\label{fig:mov_rtw}
\end{figure}
\section{Discussion}
In this section, we summarize the feedback from three experts and discuss the considerations of our approach.
\subsection{Expert Reviews}
We invited three professors in related fields as experts to review our system. The first expert (E$1$) has been working on massive data analysis for twelve years. The other two experts (E$2$ $\&$ E$3$) have at least seven years of experience in visual analytics of time-series data.
A semi-structured interview was conducted with each expert through a remote conference. We first introduced our method and visual design in about 20 minutes. Then, we showed them case studies, during which they were free to ask questions and express opinions. We summarized their feedbacks as follows.
\textbf{Effectiveness.} All experts agree with the effectiveness of our approach. ``The concept drift index can indeed reflect the change process of the transformation data distribution over time to a certain extent,'' E1 commented. E$2$ and E$3$ also appreciated our idea of applying the drift level index. E$3$ said that the index can effectively support interactive exploration of unknown concepts and concept drifts.
\textbf{Scalability.} E1's main concern is whether high-dimensional data, \textit{i.e.}, data with hundreds of dimensions, can be applied in our system. Through system demo, we proved to him that our visual analysis approach is minimally affected by the curse of dimension. Related discussion can be found in Section~\ref{sec:sca}. In summary, he believes that our workflow and system can meet the need for analyzing time-series data and he would like to use our system when he has related analysis requirements. For further extension of our approach, he gave us two suggestions: 1) considering a multi-model hybrid prediction method to enhance the reliability of the concept drift index; 2) recommending concepts or concept drifts based on data features automatically.
\textbf{Visual Designs.} Concerning the interface design, all experts gave positive feedbacks. E$2$ particularly likes the hierarchical abstraction in the timeline navigator view and the prediction model view. E$3$ was impressed by the shifted stripes in the prediction model view.
\textbf{Learn costs.} E$2$ and E$3$ commented that analysts need time to learn before they can use the system. Considering that the definition and visual representation of concept drift are abstract and complex, they agree that it does worth learning costs.
\textbf{Advice.} Considering that the analysis may only involve partial data sources, E$2$ suggested supporting the filter of data sources in the data entrance, which can facilitate analysts to focus on certain data sources. We update the system and allow analysts to control the display of data sources. However, the training of the consistency judgment model can not be completed interactively. Analysts have to reset data sources from the backend to modify the model results.
\subsection{The Navigation of the Drift Level Index}
The drift level index provides analysts with comprehensive navigation of various dynamics of concepts by connecting prediction models and visual analysis of time-series data. However, not all dynamics are identified by the drift level index. As mentioned in Section~\ref{sec:det}, the computation of the drift level index ignores the situations that the accuracy of predictive models is increasing or stable. The understanding of concepts keeps updating with iterations, even when the accuracy does not drop. The dynamics that can not be reflected from the drift level index mainly fall into two categories: improvements during learning processes and slow changes that can be caught by iterations.
Prediction models initialize their understanding of concepts (\textit{i.e.}, parameters) at the beginning of the training process. The subsequent iterations always contribute to a rapid rise of accuracy. The same phenomena appear when the adaptive mechanics (replacing the weakest prediction model) are triggered to stop the accuracy declines caused by concept drifts. In other words, the results of these changes can be observed by inspecting the concepts following related concept drifts. In addition, concepts may evolve slowly. If prediction models can follow the changes by accumulative updates in iterations, no concept drift can be detected. To reveal the imperceptible changes, parameter evolution is monitored by the parameter projection view, which not only provides an overview of the entire learning process but also indicates the accumulative changes.
\subsection{Scalability}
\label{sec:sca}
\subsubsection{Visual Designs}
We discuss the visual scalability issue from the following aspects.
\textbf{Data records.} \textit{ConceptExplorer}{} assists analysts to locate appropriate contexts of concepts step by step, during which no attention needs to be paid on single data records or their attribute values. Because the dynamic features of data records distributed in different contexts are extracted by automatic approaches. The concepts corresponding to the selected contexts are summarized by the differences in the number of records with positive labels and negative ones. Analysts can contribute to qualitative conclusions based on the distribution of differences over an attribute or a pair of attributes, as mentioned in Section~\ref{sec:cas}.
\textbf{Attributes.} Due to the limitation of display space, up to $15$ attributes are shown in the concept explanation view. To provide significant patterns with sufficient spaces, only attributes with top correlations with the label are listed.
\textbf{Data sources.} Shifted stripes are employed to eliminate visual clutter caused by multiple data sources in the accuracy fluctuation chart. The gap width of stripes can be increased to insert more lines, \textit{i.e.}, adapt to more data sources. In addition, the color map that encodes data sources should also be adapted to the increasing number of data sources.
\subsubsection{Computation Time}
The performance of models are tested on a desktop with 16G memory and two Intel Core i7 6700 at 3.4 GHz and 3.41GHz processors (see Table~\ref{tab:time}). Data records from four data sources used in the first case are composed into an $88$-dimensional data source, named Case$1_{mixed}$. It can be seen that the size of data affects the performance of prediction model training. The
computation time of drift level indices is not affected by the data dimension, but is related to the size of sliding windows. In this work, the size of sliding windows is determined according to the update frequency of data records.
\begin{table}[!htbp]
\caption{Average computation time (in milliseconds) of an iteration for the three stages with different combinations of attribute amount. The size of sliding window of drift level index is labeled in brackets.}
\label{tab:time}
\scriptsize
\centering
\begin{tabular}{p{2.5cm}|p{1.3cm}|p{1.5cm}|p{1.3cm}}
\toprule
{Data source name \quad \quad \quad\quad \quad \quad(\#Attribute)}
& {Prediction model} & {Drift level index calculation (size)} & {Consistency judgment model} \\
\midrule
MovieLens ($15$) & $2.532$ & $0.088$ ($1500$) & $0.028$ \\
RottenTomatoes ($15$) & $2.698$ & $0.013$ ($100$) & $0.018$ \\
IMDB ($15$) & $2.542$ & $0.036$ ($500$) & $0.015$\\
Guanyun ($22$) & $3.152$ & $0.009$ ($100$) & $0.034$\\
Tiantan ($22$) & $3.203$ & $0.009$ ($100$) & $0.022$\\
Case$1_{mixed}$ ($88$) & $7.681$ & $0.009$ ($100$) & $0.032$\\
\bottomrule
\end{tabular}
\end{table}
In summary, the time-consuming part of automatic approaches is training prediction models. In the current version of \textit{ConceptExplorer}, training and verifying are completed in the preprocessing stage. With the help of powerful computing clusters or cloud computing, it is possible to extend our system to process massive in real-time data. Hence, our system design and workflow have adequate scalability in terms of data records and attributes.
\section{Conclusion}
In this paper, we propose a visual analysis approach to facilitate the exploration of concept drifts from multi-source time-series data. Analysts are allowed to flexibly identify and compare the concepts with different contexts. The gradually progressive specification of the contexts is navigated by the model-derived drift level index and the consistency judgment model, which correspond to time segments and the set of data sources, respectively. A visual analysis system, \textit{ConceptExplorer}, is designed and implemented.
The effectiveness of \textit{ConceptExplorer}{} is verified through three case studies with various real-world data sets. In addition, positive reviews are received from two experts on related fields. In the future, we plan to improve the concept explanation view to explain the relationship between attributes and the label in a more comprehensive way.
|
2,869,038,154,853 | arxiv | \section{\bf Heron and Heinz Means}
Let $a,b>0$ and $\lambda\in[0,1]$ be real numbers. The following expressions
\begin{equation}\label{eq1}
a\nabla_{\lambda}b=(1-\lambda)a+\lambda b,\;\; a\sharp_{\lambda}b=a^{1-\lambda}b^{\lambda}
\end{equation}
are known as the $\lambda$-weighted arithmetic mean and $\lambda$-weighted geometric mean, respectively. They satisfy the following
\begin{equation}\label{2}
a\sharp_{\lambda}b\leq a\nabla_{\lambda}b
\end{equation}
known as the Young's inequality. Some refinements and reverses of \eqref{2} have been discussed in the literature. In particular, the following result has been proved in \cite{KM},
\begin{equation}\label{2a}
r_{\lambda}\big(\sqrt{a}-\sqrt{b}\big)^2\leq a\nabla_{\lambda}b-a\sharp_{\lambda}b \leq (1-r_{\lambda})\big(\sqrt{a}-\sqrt{b}\big)^2,
\end{equation}
where we set
\begin{equation}\label{2b}
r_{\lambda}:=\min(\lambda,1-\lambda).
\end{equation}
For more refinements and reverses of the Young's inequality, we refer the interested reader to \cite{F,M1,T} and the related references cited therein.
From the previous means we introduce the following expressions
\begin{equation}\label{3a}
K_{\lambda}(a,b)=(1-\lambda)\sqrt{ab}+\lambda\frac{a+b}{2},
\end{equation}
\begin{equation}\label{3b}
HZ_{\lambda}(a,b)=\frac{a^{1-\lambda}b^{\lambda}+a^{\lambda}b^{1-\lambda}}{2}
\end{equation}
known in the literature as the Heron and Heinz means, respectively. They satisfy the following inequalities
\begin{equation}\label{4a}
\sqrt{ab}\leq K_{\lambda}(a,b)\leq\frac{a+b}{2},
\end{equation}
\begin{equation}\label{4b}
\sqrt{ab}\leq HZ_{\lambda}(a,b)\leq\frac{a+b}{2}.
\end{equation}
An inequality between the Heron and Heinz means was proved in \cite{B} and is as follows
\begin{equation}\label{5}
HZ_{\lambda}(a,b)\leq K_{\alpha(\lambda)}(a,b),
\end{equation}
where $\alpha(\lambda)=(2\lambda-1)^2$ for any $\lambda\in[0,1]$. It is easy to check that \eqref{5} is better than the following inequality
$$HZ_{\lambda}(a,b)+r_{\lambda}\big(\sqrt{a}-\sqrt{b}\big)^2\leq\frac{a+b}{2}$$
which has been later obtained in \cite{KM}, where $r_{\lambda}$ is defined by \eqref{2b}.
A reverse of Heinz inequality was recently proved in \cite{KSS} as follows
\begin{equation}\label{5a}
HZ_{\lambda}(a,b)\geq\frac{a+b}{2}-\frac{1}{2}\lambda(1-\lambda)(b-a)\log(b/a).
\end{equation}
Another reversed version of Heinz inequality was already showed in \cite{KM2} and reads as follows
\begin{equation}\label{5b}
\Big(HZ_{\lambda}(a,b)\Big)^2\geq\Big(\frac{a+b}{2}\Big)^2-\frac{1}{2}(1-r_{\lambda})(a-b)^2.
\end{equation}
In \cite{KSS}, the authors mentioned some comments about comparison between \eqref{5a} and \eqref{5b} that we present in the following remark (see page 745).
\begin{remark}
Numerical experiments show that neither \eqref{5a} nor \eqref{5b} is uniformly better than the
other. However, these experiments show that, for most values of $\lambda$ , \eqref{5a} is better than
\eqref{5b} when $a/b$ is relatively small and \eqref{5b} is better when $a/b$ is large.
\end{remark}
The extension of the previous means, from the case that the variables are positive real numbers to the case that the arguments are positive operators, has been investigated in the literature. Let $H$ be a complex Hilbert space and ${\mathcal B}(H)$ be the $\mathbb{C}^*$-algebra of bounded linear operators acting on $H$. We denote by ${\mathcal B}^{+*}(H)$ the open cone of all (self-adjoint) positive invertible operators in ${\mathcal B}(H)$. For $A,B\in {\mathcal B}^{+*}(H)$, the following expressions
\begin{equation*}
A\nabla_{\lambda}B:=(1-\lambda)A+\lambda B=B\nabla_{1-\lambda} A,
\end{equation*}
\begin{equation*}
A\sharp_{\lambda}B:=A^{1/2}\Big(A^{-1/2}BA^{-1/2}\Big)^{\lambda}A^{1/2}=B\sharp_{1-\lambda}A
\end{equation*}
are known as the $\lambda$-weighted operator mean and $\lambda$-weighted geometric operator mean of $A$ and $B$, respectively. For $\lambda=1/2$, they are simply denoted by $A\nabla B$ and $A\sharp B$, respectively. These operator means satisfy the following inequality
\begin{equation}\label{6}
A\sharp_{\lambda}B\leq A\nabla_{\lambda}B,
\end{equation}
which is an operator version of the Young's inequality \eqref{2}. The notation $\leq$ refers here for the L\"{o}wner partial order defined by: $T\leq S$ if and only if $T$ and $S$ are self-adjoint and $S-T$ is positive.
An operator version of \eqref{2a} has also been established in \cite{KM} and reads as follows
\begin{equation}\label{65}
2r_{\lambda}\big(A\nabla B-A\sharp B\big) \leq A\nabla_{\lambda}B -A\sharp_{\lambda}B \leq2(1-r_{\lambda})\big(A\nabla B-A\sharp B\big),
\end{equation}
where $r_{\lambda}$ is defined in \eqref{2b}. In fact, according to the Kubo-Ando theory \cite{KA}, \eqref{65} can be immediately deduced from \eqref{2a}.
By analogy with the scalar case, the Heron and Heinz operator means are, respectively, defined as follows
\begin{equation}\label{7a}
K_{\lambda}(A,B)=(1-\lambda)A\sharp B+\lambda A\nabla B,
\end{equation}
\begin{equation}\label{7b}
HZ_{\lambda}(A,B)=\frac{A\sharp_{\lambda}B+A\sharp_{1-\lambda}B}{2}.
\end{equation}
The following operator inequalities, extending respectively \eqref{4a} and \eqref{4b} for operator arguments, have also been proved in the literature, see \cite{YR} and the related references cited therein.
\begin{equation}\label{8a}
A\sharp B\leq K_{\lambda}(A,B)\leq A\nabla B,
\end{equation}
\begin{equation}\label{8b}
A\sharp B\leq HZ_{\lambda}(A,B)\leq A\nabla B.
\end{equation}
The following refinement of the inequality in \eqref{8a} has been recently obtained in \cite{YR}
\begin{multline}\label{9}
\lambda(1-\lambda)\big(A\nabla B-A\sharp B\big)+A\sharp B\leq K_{\lambda}(A,B)\\
\leq A\nabla B-\lambda(1-\lambda)\big(A\nabla B-A\sharp B\big).
\end{multline}
For more inequalities related to the Heron and Heinz means involving matrix and operator arguments, we refer the reader to \cite{DDF,FFN, I2,Kho, KCMSS, KM, KM2, KSS, LS, Z} and the related references cited therein.
\section{\bf Functional Version}
The previous operator means have been extended from the case that the variables are positive operators to the case that the variables are convex functionals, see \cite{RB}. Let us denote by $\Gamma_0(H)$ the cone of all $f:H\longrightarrow{\mathbb R}\cup\{+\infty\}$ which are convex lower semi-continuous and not identically equal to $+\infty$. Throughout this paper, we use the following notation:
$${\mathcal D}(H)=\Big\{(f,g)\in \Gamma_0(H)\times \Gamma_0(H) :\;\;{\rm dom}\;f\cap{\rm dom}\;g\neq\emptyset\Big\},$$
where ${\rm dom}\;f$ refers to the effective domain of $f:H\longrightarrow{\mathbb R}\cup\{+\infty\}$ defined by
$${\rm dom}\;f=\Big\{x\in H,\;\; f(x)<+\infty\Big\}.$$
If $(f,g)\in {\mathcal D}(H)$ and $\lambda\in(0,1)$, then the following expressions
\begin{equation}\label{10a}
{\mathcal A}_{\lambda}(f,g):=(1-\lambda)f+\lambda g,
\end{equation}
\begin{equation}\label{10b}
{\mathcal
G}_{\lambda}(f,g):=\displaystyle{\frac{\sin(\pi\lambda)}{\pi}\int_{0}^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}}\Big((1-t)f^*+tg^*\Big)^*dt
\end{equation}
are called, by analogy, the $\lambda$-weighted functional arithmetic mean and $\lambda$-weighted functional geometric mean of $f$ and $g$, respectively. For $\lambda=1/2$, they are simply denoted by ${\mathcal A}(f,g)$ and ${\mathcal G}(f,g)$, respectively. Here, the notation $f^*$ refers to the Fenchel conjugate of any $f:H\longrightarrow{\mathbb R}\cup\{+\infty\}$ defined through
\begin{equation}\label{13}
\forall x^*\in H\;\;\;\;\;\;\; f^*(x^*)=\sup_{x\in H}\Big\{\Re e\langle x^*,x\rangle-f(x)\Big\}.
\end{equation}
It is easy to see that
\begin{equation}\label{12}
{\mathcal
G}_{\lambda}(f,g)=\displaystyle{\frac{\sin(\pi\lambda)}{\pi}\int_{0}^1\frac{1}{t\sharp_{\lambda}(1-t)}}{\mathcal H}_t(f,g)dt,
\end{equation}
where
\begin{equation}\label{125}
{\mathcal H}_{\lambda}(f,g):=\Big((1-\lambda)f^*+\lambda g^*\Big)^*
\end{equation}
is the so-called $\lambda$-weighted functional harmonic mean of $f$ and $g$. For $\lambda=1/2$, we simply denote it by ${\mathcal H}(f,g)$.
With this, we can write
$$\forall x^*\in H\;\;\;\;\;\;\; f^*(x^*)=\sup_{x\in{\rm dom}\;f}\Big\{\Re e\langle x^*,x\rangle-f(x)\Big\},$$
provided that ${\rm dom}\;f\neq\emptyset$. As supremum of a family of affine (so convex) functions, $f^*$ is always convex even when $f$ is not.
We extend the previous functional means on the whole interval $[0,1]$ by setting, see \cite{RF}
\begin{equation}\label{eq133}
{\mathcal A}_{0}(f,g)={\mathcal G}_{0}(f,g)={\mathcal H}_{0}(f,g)=f,\;\;{\mathcal A}_{1}(f,g)={\mathcal G}_{1}(f,g)={\mathcal H}_{1}(f,g)=g.
\end{equation}
The previous functional means satisfy the following relationships
\begin{equation}\label{135}
{\mathcal A}_{\lambda}(f,g)={\mathcal A}_{1-\lambda}(g,f),\; {\mathcal H}_{\lambda}(f,g)={\mathcal H}_{1-\lambda}(g,f),\;
{\mathcal G}_{\lambda}(f,g)={\mathcal G}_{1-\lambda}(g,f),
\end{equation}
for any $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$. The two first relationships are immediate while the proof of the third one can be found in \cite{RB}. In particular, if $\lambda=1/2$, the three previous functional means are symmetric in $f$ and $g$.
We have the following double inequality
\begin{equation}\label{14}
{\mathcal H}_{\lambda}(f,g)\leq{\mathcal
G}_{\lambda}(f,g)\leq {\mathcal A}_{\lambda}(f,g),
\end{equation}
whose the right inequality is the functional version of the Young's operator inequality \eqref{6}. Here the symbol $\leq$ denotes the point-wise order defined by, $f\leq g$ if and only if $f(x)\leq g(x)$ for all $x\in H$.
\begin{remark}\label{rem1}
We adopt here the conventions $0\cdot(+\infty)=+\infty$ and $(+\infty)-(+\infty)=+\infty$, as usual in convex analysis. Since our involved functionals $f$ and/or $g$ can take the value $+\infty$ we then mention the following:\\
(i) The relations \eqref{eq133} are not immediate from their related functional means \eqref{10a}, \eqref{10b} and \eqref{125}, respectively.\\
(ii) We must be careful with any proof of functional equality or inequality. As example, the equalities $f-f=0$ and $f-g=-(g-f)$ are not always true. Also, the two inequalities $f\leq g$ and $f-g\leq0$ are not always equivalent whereas $f\leq g$ and $g-f\geq0$ are equivalent.
\end{remark}
The previous functional means are, respectively, extensions of their related operator means in the following sense
\begin{equation}\label{145}
{\mathcal A}_{\lambda}(f_A,f_B)=f_{A\nabla_\lambda B},\; {\mathcal
G}_{\lambda}(f_A,f_B)=f_{A\sharp_{\lambda}B},\; {\mathcal H}_{\lambda}(f_A,f_B)=f_{A!_{\lambda}B},
\end{equation}
where
$$A!_{\lambda}B:=\big((1-\lambda)A^{-1}+\lambda B^{-1}\big)^{-1}$$
stands for the $\lambda$-weighted harmonic operator mean of $A$ and $B$ and the notation $f_A$ refers to the convex quadratic form generated by the positive operator $A$, i.e. $f_A(x)=(1/2)\langle Ax,x\rangle$ for all $x\in H$. This because $f_A^*(x^*)=(1/2)\langle A^{-1}x^*,x^*\rangle$, or in short $f_A^*=f_{A^{-1}}$, for any $A\in{\mathcal B}^{+*}(H)$. We also mention that, since $A$ and $B$ are self-adjoint then, $f_A=f_B$ if and only if $A=B$.\\
For all $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$, we introduce the following expressions
\begin{equation}\label{15}
{\mathcal K}_{\lambda}(f,g)=(1-\lambda){\mathcal G}(f,g)+\lambda{\mathcal A}(f,g)
\end{equation}
and
\begin{equation}\label{16}
{\mathcal{HZ}}_{\lambda}(f,g)=\frac{1}{2}\Big({\mathcal
G}_{\lambda}(f,g)+{\mathcal
G}_{1-\lambda}(f,g)\Big)
\end{equation}
which will be called the Heron functional mean and the Heinz functional mean of $f$ and $g$, respectively. It is clear that ${\mathcal K}_{\lambda}(f,g)$ is symmetric in $f$ and $g$ and, due to the last relation of \eqref{135}, ${\mathcal{HZ}}_{\lambda}(f,g)$ is also symmetric in $f$ and $g$. We mention that we have ${\mathcal{HZ}}_{\lambda}(f,g)={\mathcal{HZ}}_{1-\lambda}(f,g)$ while in general ${\mathcal K}_{\lambda}(f,g)\neq{\mathcal K}_{1-\lambda}(f,g)$, unless $\lambda=1/2$.
By virtue of \eqref{145}, it is not hard to see that, for any $A,B\in{\mathcal B}^{+*}(H)$, we have
\begin{equation}\label{17}
{\mathcal K}_{\lambda}(f_A,f_B)=f_{K_{\lambda}(A,B)},\;\; {\mathcal{HZ}}_{\lambda}(f_A,f_B)=f_{HZ_{\lambda}(A,B)}.
\end{equation}
Furthermore, according to \eqref{14} for $\lambda=1/2$ we immediately deduce that the functional map $\lambda\longmapsto{\mathcal K}_{\lambda}(f,g)$, for fixed $(f,g)\in{\mathcal D}(H)$, is point-wisely increasing in $\lambda\in[0,1]$. This gives the functional version of the operator inequality \eqref{8a} that reads as follows
\begin{equation}\label{18}
{\mathcal G}(f,g)\leq {\mathcal K}_{\lambda}(f,g)\leq{\mathcal A}(f,g),
\end{equation}
which, in its turn, immediately yields \eqref{8a} by virtue of \eqref{145} and \eqref{17}.
Now, a question arises from the above: Is the functional version of \eqref{8b} true when the operator variables $A$ and $B$ are replaced by convex functionals. Precisely, is the following
\begin{equation}\label{19}
{\mathcal G}(f,g)\leq {\mathcal{HZ}}_{\lambda}(f,g)\leq{\mathcal A}(f,g)
\end{equation}
hold for any $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$. An affirmative answer to this latter question will be discussed in the next section.
\begin{remark}
Usually, for proving an operator inequality like \eqref{6}, \eqref{65}, \eqref{8a}, \eqref{8b} and \eqref{9} we start from its analog for scalar case and we then proceed by using the techniques of functional calculus. In this paper, after defining the Heron and Heinz means of two convex functionals, we establish some inequalities involving these functional means. We also obtain new refinements of some known operator inequalities. Our approach is with functional character and the proofs of our theoretical results are short, simple and nice and do not need to use the techniques of functional calculus.
\end{remark}
\section{\bf The Main Results}
We preserve the same notations as in the previous sections. Our first main result is about a refinement of \eqref{18} recited in the following.
\begin{theorem}\label{th1}
For any $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$ there hold
\begin{multline}\label{20}
r\Big({\mathcal A}(f,g)-{\mathcal
G}(f,g)\Big)+{\mathcal
G}(f,g)={\mathcal K}_{r}(f,g)\leq{\mathcal K}_{\lambda}(f,g)\\
\leq{\mathcal K}_{1-r}(f,g)={\mathcal A}(f,g)-r\Big({\mathcal A}(f,g)-{\mathcal
G}(f,g)\Big),
\end{multline}
where we set $r=r_{\lambda}:=\min(\lambda,1-\lambda)$ for the sake of simplicity.
\end{theorem}
\begin{proof}
Since $\lambda\longmapsto {\mathcal K}_{\lambda}(f,g)$, for fixed $(f,g)\in{\mathcal D}(H)$, is point-wisely increasing in $\lambda \in[0,1]$ and $r_{\lambda}\leq\lambda\leq R_{\lambda}=1-r_{\lambda}$ we then immediately deduce the desired inequalities.
\end{proof}
The operator version of Theorem \ref{th1} reads as follows.
\begin{corollary}\label{cor1}
The following operator inequalities
\begin{equation}\label{21}
r_{\lambda}\Big(A\nabla B-A\sharp B\Big)+A\sharp B\leq K_{\lambda}(A,B)\leq A\nabla B-r_{\lambda}\Big(A\nabla B-A\sharp B\Big)
\end{equation}
hold for any $A,B\in{\mathcal B}^{+*}(H)$ and $\lambda\in[0,1]$.
\end{corollary}
\begin{proof}
It is immediate from \eqref{20} when we take $f=f_A$ and $g=f_B$ and we use \eqref{145} and \eqref{17}. Details are simple and therefore omitted here.
\end{proof}
\begin{remark}
(i) It is clear that \eqref{21} refines the double inequality \eqref{8a}. Further, \eqref{21} also refines \eqref{9} since $\lambda(1-\lambda)\leq r_{\lambda}:=\min(\lambda,1-\lambda)$ for any $\lambda\in[0,1]$. Note that \eqref{9} has been proved in \cite{YR} via the techniques of functional calculus.\\
(ii) The operator inequality \eqref{21} has been immediately deduced from \eqref{20} which, in its turn, refines \eqref{18}. Moreover, \eqref{20} has a simple proof and immediately implies \eqref{21} without the need to the techniques of functional calculus.
\end{remark}
Before stating our second main result we need the following lemma.
\begin{lemma}\label{lem1}
Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in(0,1)$. Then we have
\begin{equation}\label{22}
{\mathcal A}_{\lambda}(f,g)-{\mathcal G}_{\lambda}(f,g)=\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{1}{t\sharp_{\lambda}(1-t)}\Big({\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\Big)dt.
\end{equation}
In particular, one has
\begin{equation}\label{225}
{\mathcal A}(f,g)-{\mathcal G}(f,g)=\frac{1}{\pi}\int_0^1\frac{1}{\sqrt{t(1-t)}}\Big({\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\Big)dt.
\end{equation}
\end{lemma}
\begin{proof}
Let $\Gamma$ and $B$ denote the standard special functions Gamma and Beta, respectively. Then we have
$$\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}dt=\frac{\sin(\pi\lambda)}{\pi}B\big(\lambda,1-\lambda\big)
=\frac{\sin(\pi\lambda)}{\pi}\Gamma(\lambda)\Gamma(1-\lambda)=1.$$
This, with the definition of ${\mathcal G}_{\lambda}(f,g)$, yields
$${\mathcal A}_{\lambda}(f,g)-{\mathcal G}_{\lambda}(f,g)=\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}
\Big({\mathcal A}_{\lambda}(f,g)-{\mathcal H}_{t}(f,g)\Big)dt.$$
Now, since ${\mathcal A}_{t}(f,g)=(1-t)f+t g$, with the following
\begin{multline*}
\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}(1-t)dt=\frac{\sin(\pi\lambda)}{\pi}B\big(\lambda,2-\lambda\big)
\\=\frac{\sin(\pi\lambda)}{\pi}\Gamma(\lambda)\Gamma(2-\lambda)
=\frac{\sin(\pi\lambda)}{\pi}(1-\lambda)\Gamma(\lambda)\Gamma(1-\lambda)=1-\lambda
\end{multline*}
and (by similar arguments)
$$\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}t dt=\lambda,$$
we deduce that
\begin{equation*}\label{23}
{\mathcal A}_{\lambda}(f,g)-{\mathcal G}_{\lambda}(f,g)=\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{t^{\lambda-1}}{(1-t)^{\lambda}}
\Big({\mathcal A}_{t}(f,g)-{\mathcal H}_{t}(f,g)\Big)dt,
\end{equation*}
which is the desired result.
\end{proof}
\begin{proposition}
Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in(0,1)$. Then the following equality
\begin{multline}\label{24}
{\mathcal A}(f,g)(x)-{\mathcal{HZ}}_{\lambda}(f,g)(x)\\
=\frac{\sin(\pi\lambda)}{\pi}\int_0^1HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)\Big({\mathcal A}_t(f,g)(x)-{\mathcal H}_t(f,g)(x)\Big)dt
\end{multline}
holds for any $x\in H$.
\end{proposition}
\begin{proof}
If $x\notin{\rm dom}\;f\cap{\rm dom}\;g\neq\emptyset$, i.e. $f(x)=+\infty$ or $g(x)=+\infty$, then the two sides of \eqref{24} are infinite and so \eqref{24} holds. Assume that $x\in{\rm dom}\;f\cap{\rm dom}\;g\neq\emptyset$. First, it is easy to check that
\begin{equation}\label{25}
{\mathcal A}_{\lambda}(f,g)(x)+{\mathcal A}_{1-\lambda}(f,g)(x)=(f+g)(x):=2 {\mathcal A}(f,g)(x).
\end{equation}
Otherwise, from \eqref{22} we can write
\begin{multline}\label{26}
{\mathcal A}_{1-\lambda}(f,g)(x)-{\mathcal G}_{1-\lambda}(f,g)(x)\\
=\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{1}{t\sharp_{1-\lambda}(1-t)}\Big({\mathcal A}_t(f,g)(x)-{\mathcal H}_t(f,g)(x)\Big)dt.
\end{multline}
Adding side to side \eqref{22} and \eqref{26}, with the help of \eqref{16} and \eqref{25}, we deduce
\begin{multline*}
{\mathcal A}(f,g)(x)-{\mathcal{HZ}}_{\lambda}(f,g)(x)\\
=\frac{\sin(\pi\lambda)}{2\pi}\int_0^1\Big(\frac{1}{t\sharp_{\lambda}(1-t)}+
\frac{1}{t\sharp_{1-\lambda}(1-t)}\Big)\Big({\mathcal A}_t(f,g)(x)-{\mathcal H}_t(f,g)(x)\Big)dt,
\end{multline*}
from which the desired result follows after a simple manipulation.
\end{proof}
\begin{remark}
For the sake of clearness for the reader, we mention that according to \eqref{145} and \eqref{17} we immediately deduce that the operator versions of \eqref{22}, \eqref{225} and \eqref{24} are, respectively, given by
$$A\nabla_{\lambda}B-A\sharp_{\lambda}B=\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{1}{t\sharp_{\lambda}(1-t)}\Big(A\nabla_tB-A!_tB\Big)dt,$$
$$A\nabla B-A\sharp B=\frac{1}{\pi}\int_0^1\frac{1}{\sqrt{t(1-t)}}\Big(A\nabla_tB-A!_tB\Big)dt,$$
$$A\nabla B-HZ_{\lambda}(A,B)=\frac{\sin(\pi\lambda)}{\pi}\int_0^1HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)\Big(A\nabla_tB-A!_tB\Big)dt.$$
\end{remark}
We now are in a position to state our second main result as recited in the following.
\begin{theorem}\label{th3}
Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in(0,1)$. Then the following inequalities hold:
\begin{equation}\label{27}
{\mathcal{HZ}}_{\lambda}(f,g)
\leq{\mathcal K}_{\theta(\lambda)}(f,g)\leq{\mathcal A}(f,g),
\end{equation}
where we set $\theta(\lambda)=1-\sin(\pi\lambda)$ for any $\lambda\in(0,1)$.
\end{theorem}
\begin{proof}
First, the right inequality of \eqref{27} follows from the right inequality in \eqref{18}. Note that, by \eqref{14}, we have ${\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\geq0$ for any $(f,g)\in{\mathcal D}(H)$ and $t\in(0,1)$. Now, by \eqref{24} with the left inequality of \eqref{4b}, we can write
$${\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)
\geq\frac{\sin(\pi\lambda)}{\pi}\int_0^1\frac{1}{\sqrt{t(1-t)}}\Big({\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\Big)dt.$$
Thanks to \eqref{225} we then obtain
\begin{equation}\label{28}
{\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\geq\big(\sin(\pi\lambda)\big)\Big({\mathcal A}(f,g)-{\mathcal G}(f,g)\Big).
\end{equation}
To finish the proof, we have to take into account some precautions (see Remark \ref{rem1}). At $x\in H$ such that ${\mathcal K}_{\theta(\lambda)}(f,g)(x)=+\infty$, the left inequality of \eqref{27} is obviously satisfied at $x$. Now, let $x\in H$ be such that ${\mathcal K}_{\theta(\lambda)}(f,g)(x)<+\infty$. By the definition of the Heron functional mean \eqref{15} we deduce that ${\mathcal A}(f,g)(x)<+\infty$ and ${\mathcal G}(f,g)(x)<+\infty$. With this, \eqref{28} yields
$${\mathcal K}_{\theta(\lambda)}(f,g)(x)-{\mathcal{HZ}}_{\lambda}(f,g)(x)\geq0,$$
which, with Remark \ref{rem1} again, means that the left inequality of \eqref{27} is also satisfied at $x$. The proof is completed.
\end{proof}
To give more main results in the sequel, we need some lemmas. The first is recited in the following.
\begin{lemma} \cite{R}. Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in(0,1)$. Then we have
\begin{multline}\label{30}
2r_{\lambda}\Big({\mathcal A}(f,g)-{\mathcal H}(f,g)\Big)\leq {\mathcal A}_{\lambda}(f,g)-{\mathcal H}_{\lambda}(f,g)\\
\leq 2(1-r_{\lambda})\Big({\mathcal A}(f,g)-{\mathcal H}(f,g)\Big),
\end{multline}
where, as already pointed before, $r_{\lambda}:=\min(\lambda,1-\lambda)$.
\end{lemma}
We also need the following lemma that concerns a refinement of the so-called Hermite-Hadamard inequality, see \cite{DR,FA} for instance.
\begin{lemma}\label{lem4}
Let $a,b$ with $a<b$ and $\Phi:[a,b]\longrightarrow{\mathbb R}$ be a convex function. Then, for all $p\in[0,1]$ we have
\begin{equation}\label{301}
\Phi\Big(\frac{a+b}{2}\Big)\leq m(p)\leq\frac{1}{b-a}\int_a^b\Phi(t)dt\leq M(p)\leq\frac{\Phi(a)+\Phi(b)}{2},
\end{equation}
where
$$m(p):=p\Phi\Big(\frac{pb+(2-p)a}{2}\Big)+(1-p)\Phi\Big(\frac{(1+p)b+(1-p)a}{2}\Big)$$
and
$$M(p):=\frac{1}{2}\Big(\Phi\big(pb+(1-p)a\big)+p\Phi(a)+(1-p)\Phi(b)\Big).$$
If $\Phi$ is concave then the inequalities \eqref{301} are reversed.
\end{lemma}
\begin{lemma}\label{lem5}
For $\lambda\in(0,1)$ fixed, let $\Psi_{\lambda}$ be the function defined by
$$\Psi_{\lambda}(t)=\Big(\frac{t}{1-t}\Big)^{\lambda},\;\; t\in[0,1/2].$$
Then $\Psi_{\lambda}$ is concave on $[0,\frac{1-\lambda}{2}]$ and convex on $[\frac{1-\lambda}{2},1/2]$.
\end{lemma}
\begin{proof}
Simple computations lead to, for any $t\in(0,1/2]$,
$$\Psi_{\lambda}^{\prime}(t)=\lambda\frac{t^{\lambda-1}}{(1-t)^{\lambda+1}}$$
and
$$\Psi_{\lambda}^{\prime\prime}(t)=\lambda\frac{t^{\lambda-2}}{(1-t)^{\lambda+2}}(2t+\lambda-1).$$
The desired results follow.
\end{proof}
For the sake of simplicity, we introduce more notation. For $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$ we set
\begin{equation}\label{305}
{\mathcal L}_{\lambda}(f,g):=(1-\lambda){\mathcal H}(f,g)+\lambda{\mathcal A}(f,g).
\end{equation}
It is not hard to see that the following inequalities
\begin{equation}\label{307}
{\mathcal H}(f,g)\leq{\mathcal L}_{\lambda}(f,g)\leq{\mathcal K}_{\lambda}(f,g)\leq{\mathcal A}(f,g)
\end{equation}
hold for any $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$. With this, another main result is recited in what follows.
\begin{theorem}\label{th2}
Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$. Then, for any $p\in[0,1]$, there hold:
\begin{equation}\label{31}
{\mathcal{HZ}}_{\lambda}(f,g)\leq{\mathcal L}_{\gamma_p(\lambda)}(f,g)\leq{\mathcal K}_{\gamma_p(\lambda)}(f,g)\leq{\mathcal A}(f,g),
\end{equation}
where we set
$$\gamma_p(\lambda)=1-\frac{2\sin(\pi\lambda)}{\pi}\Big((1-\lambda){\mathcal M}_{\lambda}(p)+\lambda{\mathcal M}_{1-\lambda}(p)\Big),$$
with
$${\mathcal M}_{\lambda}(p):=\frac{M_{\lambda}(p)+m_{1-\lambda}(p)}{2},$$
$$M_{\lambda}(p):=\frac{1}{2}\left(\Big(\frac{p(1-\lambda)}{2-p(1-\lambda)}\Big)^{\lambda}+(1-p)\Big(\frac{1-\lambda}{1+\lambda}\Big)^{\lambda}\right),$$
and
$$m_{\lambda}(p):=p\left(\frac{2-2\lambda+p\lambda}{2+2\lambda-p\lambda}\right)^{\lambda}+(1-p)
\left(\frac{2-\lambda+p\lambda}{2+\lambda-p\lambda}\right)^{\lambda}.$$
\end{theorem}
\begin{proof}
For $\lambda=0$ or $\lambda=1$, it is clear that \eqref{31} are reduced to equalities. Assume that $\lambda\in(0,1)$. By \eqref{24}, with the left inequality of \eqref{30}, we have
\begin{equation}\label{32}
{\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\geq\frac{2\sin(\pi\lambda)}{\pi}\int_0^1r_t HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)
\Big({\mathcal A}(f,g)-{\mathcal H}(f,g)\Big)dt.
\end{equation}
Since $r_t=\min(t,1-t)$ and the function $t\longmapsto r_t HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)$ is symmetric around $1/2$ then one has
\begin{multline}\label{33}
\int_0^1r_t HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)dt=2\int_0^{1/2}t HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)dt\\
=\int_0^{1/2}\left\{\Big(\frac{t}{1-t}\Big)^{\lambda}+\Big(\frac{t}{1-t}\Big)^{1-\lambda}\right\}dt.
\end{multline}
According to Lemma \ref{lem5}, we write
\begin{equation}\label{34}
\int_0^{1/2}\Big(\frac{t}{1-t}\Big)^{\lambda}dt:=\int_0^{1/2}\Psi_{\lambda}(t)dt=\int_0^{\frac{1-\lambda}{2}}\Psi_{\lambda}(t)dt
+\int_{\frac{1-\lambda}{2}}^{1/2}\Psi_{\lambda}(t)dt.
\end{equation}
By Lemma \ref{lem5} again, with Lemma \ref{lem4}, we have for any $p\in[0,1]$ (after some elementary computations)
$$\int_0^{\frac{1-\lambda}{2}}\Psi_{\lambda}(t)dt\geq\frac{1-\lambda}{2}M_{\lambda}(p),$$
with
\begin{multline}\label{35}
M_{\lambda}(p):=\frac{1}{2}\left(\Psi_{\lambda}\Big(p\frac{1-\lambda}{2}\Big)+(1-p)\Psi_{\lambda}\Big(\frac{1-\lambda}{2}\Big)\right)\\
=\frac{1}{2}\left(\Big(\frac{p(1-\lambda)}{2-p(1-\lambda)}\Big)^{\lambda}+(1-p)\Big(\frac{1-\lambda}{1+\lambda}\Big)^{\lambda}\right)
\end{multline}
and
$$\int_{\frac{1-\lambda}{2}}^{1/2}\Psi_{\lambda}(t)dt\geq\Big(\frac{1}{2}-\frac{1-\lambda}{2}\Big)\;l_{\lambda}(p)=\frac{\lambda}{2}\;m_{\lambda}(p),$$
with
\begin{multline}\label{36}
m_{\lambda}(p):=p\Psi_{\lambda}\left(\frac{p+(2-p)(1-\lambda)}{4}\right)+(1-p)\Psi_{\lambda}\left(\frac{1+p+(1-p)(1-\lambda)}{4}\right)\\
=p\left(\frac{2-2\lambda+p\lambda}{2+2\lambda-p\lambda}\right)^{\lambda}+(1-p)\left(\frac{2-\lambda+p\lambda}{2+\lambda-p\lambda}\right)^{\lambda}.
\end{multline}
With this, \eqref{34} yields
$$\int_0^{1/2}\Big(\frac{t}{1-t}\Big)^{\lambda}dt\geq\frac{1-\lambda}{2}M_{\lambda}(p)+\frac{\lambda}{2}\;m_{\lambda}(p),$$
and so
$$\int_0^{1/2}\Big(\frac{t}{1-t}\Big)^{1-\lambda}dt\geq\frac{\lambda}{2}M_{1-\lambda}(p)+\frac{1-\lambda}{2}\;m_{1-\lambda}(p).$$
These, with \eqref{33}, imply that (after simples manipulations)
$$\int_0^1r_t HZ_{\lambda}\Big(\frac{1}{t},\frac{1}{1-t}\Big)dt\geq(1-\lambda)\left\{\frac{M_{\lambda}(p)+m_{1-\lambda}(p)}{2}\right\}
+\lambda\left\{\frac{M_{1-\lambda}(p)+m_{\lambda}(p)}{2}\right\}$$
which, with \eqref{32}, implies that
$${\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\\
\geq\frac{2\sin(\pi\lambda)}{\pi}\Big((1-\lambda){\mathcal M}_{\lambda}(p)+\lambda{\mathcal M}_{1-\lambda}(p)\Big)\Big({\mathcal A}(f,g)-{\mathcal H}(f,g)\Big).$$
With some precautions, as in the proof of Theorem \ref{th3}, we deduce the desired result.
\end{proof}
The operator versions of the previous functional results can be immediately deduced. For instance we have the following result which is the operator version of Theorem \ref{th2}.
\begin{corollary}
Let $A,B\in{\mathcal B}^{+*}(H)$ and $\lambda\in[0,1]$. Then we have
$$HZ_{\lambda}(A,B)\leq L_{\gamma_p(\lambda)}(A,B)\leq K_{\gamma_p(\lambda)}(A,B)\leq A\nabla B,$$
where $\gamma_p(\lambda)$ is the same as in Theorem \ref{th2} and $L_{\lambda}(A,B)=(1-\lambda)A!B+\lambda A\nabla B$ for any $\lambda\in(0,1)$, with $A!B:=A!_{1/2}B$.
\end{corollary}
Now, in the aim to give lower bounds of ${\mathcal{HZ}}_{\lambda}(f,g)$, we need to introduce another notation. For $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$ we set
\begin{equation}\label{37}
\Theta_{\lambda}(f,g)=\frac{1}{2}\Big({\mathcal H}_{\lambda}(f,g)+{\mathcal H}_{1-\lambda}(f,g)\Big).
\end{equation}
By virtue of the second relation of \eqref{135} we have $\Theta_{\lambda}(f,g)=\Theta_{\lambda}(g,f)$, and by \eqref{14} and \eqref{25}, we have $\Theta_{\lambda}(f,g)\leq{\mathcal A}(f,g)$, for any $\lambda\in[0,1]$.
We need to prove the following lemma.
\begin{lemma}\label{lem7}
Let $(f,g)\in{\mathcal D}(H)$. Then the following assertions hold:\\
(i) The map $t\longmapsto{\mathcal H}_t(f,g)$ is point-wisely convex in $t\in[0,1]$. That is, for all
$x\in H,\; t_1,t_2\in[0,1]$ and $p\in[0,1]$ one has
$${\mathcal H}_{(1-p)t_1+pt_2}(f,g)(x)\leq(1-p){\mathcal H}_{t_1}(f,g)(x)+p{\mathcal H}_{t_2}(f,g)(x).$$
(ii) The map $t\longmapsto{\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)$ is point-wisely concave in $t\in[0,1]$.\\
(iii) The map $t\longmapsto\Theta_t(f,g)$ is also point-wisely convex in $t\in[0,1]$.
\end{lemma}
\begin{proof}
(i) By definition, we have, for all $x\in H$,
$${\mathcal H}_t(f,g)(x):=\Big((1-t)f^*+tg^*\Big)^*(x):=\sup_{x^*\in H}\Big\{\Re e\langle x^*,x\rangle-(1-t)f^*(x^*)-tg^*(x^*)\Big\}.$$
Fixing $x\in H$, we consider the following family of functionals, indexed by $x^*\in H$,
$$t\longmapsto\Re e\langle x^*,x\rangle-(1-t)f^*(x^*)-tg^*(x^*),$$
which are all affine and so convex in $t\in[0,1]$. It follows that $t\longmapsto{\mathcal H}_t(f,g)(x)$ is convex as a supremum of a family of convex functionals.\\
(ii) For any $x\in H$, the map $t\longmapsto{\mathcal A}_{t}(f,g)(x)$ is affine and so concave. This, with (i), implies the desired result.\\
(iii) Since $t\longmapsto{\mathcal H}_t(f,g)(x)$ is convex then so is $t\longmapsto{\mathcal H}_{1-t}(f,g)(x)$, because the real function $t\longmapsto1-t$ is affine. This, with \eqref{37}, implies that $t\longmapsto\Theta_t(f,g)$ is point-wisely convex in $t\in[0,1]$.
\end{proof}
\begin{proposition}
Let $f$ and $g$ be as above. Then, for any $\lambda\in[0,1]$, we have
\begin{equation}\label{38}
{\mathcal H}(f,g)\leq\Theta_{\lambda}(f,g)\leq{\mathcal{HZ}}_{\lambda}(f,g)\leq{\mathcal A}(f,g)
\end{equation}
\end{proposition}
\begin{proof}
For $\lambda=0$ or $\lambda=1$, \eqref{38} are reduced to ${\mathcal H}(f,g)\leq{\mathcal A}(f,g)$. Assume that below $\lambda\in(0,1)$. The two right inequalities of \eqref{38} follow from the left inequality in \eqref{14}, \eqref{16} and \eqref{27}. For proving the left inequality in \eqref{38} we proceed as follows, by using Lemma \ref{lem7},(i),
$$\Theta_{\lambda}(f,g):=\frac{{\mathcal H}_{\lambda}(f,g)+{\mathcal H}_{1-\lambda}(f,g)}{2}\geq
{\mathcal H}_{\frac{\lambda+(1-\lambda)}{2}}(f,g)={\mathcal H}_{1/2}(f,g):={\mathcal H}(f,g).$$
The proof is complete.
\end{proof}
\begin{proposition}\label{pr3}
Let $(f,g)\in{\mathcal D}(H)$. Then the map
$$t\longmapsto\frac{{\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)}{t(1-t)}$$
is point-wisely integrable on $(0,1)$. That is, for any $x\in H$, the integral
\begin{equation}\label{40}
{\mathcal J}(f,g)(x):=\int_0^1\frac{{\mathcal A}_t(f,g)(x)-{\mathcal H}_t(f,g)(x)}{t(1-t)}dt
\end{equation}
exists in ${\mathbb R}\cup\{+\infty\}$. Furthermore, we have
\begin{equation}\label{41a}
{\mathcal J}(f,g)(x)=2\;\int_0^{1/2}\frac{{\mathcal A}(f,g)(x)-\Theta_t(f,g)(x)}{t(1-t)}dt.
\end{equation}
\end{proposition}
\begin{proof}
Because the map $t\longmapsto{\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)$ is point-wisely concave in $t\in[0,1]$ then the map
$$t\longmapsto\frac{{\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)}{t(1-t)}$$
is point-wisely continuous on $(0,1)$ and so its point-wise integral over $(0,1)$ exists in ${\mathbb R}\cup\{+\infty\}$. It is easy to see that (by using the change of variables $t=1-s$)
\begin{equation}\label{41b}
{\mathcal J}(f,g)(x):=\int_0^1\frac{{\mathcal A}_{1-t}(f,g)(x)-{\mathcal H}_{1-t}(f,g)(x)}{t(1-t)}dt
\end{equation}
Adding side to side \eqref{40} and \eqref{41b} we obtain, with the help of \eqref{25},
\begin{equation}\label{41c}
{\mathcal J}(f,g)(x):=\int_0^1\frac{{\mathcal A}(f,g)(x)-\Theta_{t}(f,g)(x)}{t(1-t)}dt.
\end{equation}
The map
$$t\longmapsto\frac{{\mathcal A}(f,g)(x)-\Theta_{t}(f,g)(x)}{t(1-t)}$$
is symmetric around $1/2$, for any $x\in H$, we then deduce \eqref{41a} from \eqref{41c}, so completes the proof.
\end{proof}
If in \eqref{40} we take $f=f_A$ and $g=f_B$, with $A,B\in{\mathcal B}^{+*}(H)$ then we obtain, by using \eqref{145},
\begin{multline}\label{41d}
{\mathcal J}(f_A,f_B)=\int_0^1\frac{{\mathcal A}_t(f_A,f_B)-{\mathcal H}_t(f_A,f_B)}{t(1-t)}dt\\=\int_0^1
\frac{f_{A\nabla_tB}-f_{A!_tB}}{t(1-t)}dt=f_{J(A,B)},
\end{multline}
where $J(A,B)$ is given by
\begin{equation}\label{42}
J(A,B)=\int_0^1\frac{A\nabla_tB-A!_tB}{t(1-t)}dt.
\end{equation}
The operator integral $J(A,B)$ can be exactly computed as recited in the following result.
\begin{proposition}\label{pr5}
With the above, we have
\begin{equation}\label{43}
J(A,B)=(B-A)A^{-1}S(A|B),
\end{equation}
where $S(A|B)$ refers to the relative operator entropy given by
$$S(A|B):=A^{1/2}\log\Big(A^{-1/2}BA^{-1/2}\Big)A^{1/2}.$$
\end{proposition}
\begin{proof}
By Kubo-Ando theory \cite{KA}, we first compute the integral $J(A,B)$ for scalar case. That is, we need to compute the following real integral
$$J(a,b)=\int_0^1\frac{a\nabla_tb-a!_tb}{t(1-t)}dt:=\int_0^1\frac{(1-t)a+tb-\big((1-t)(1/a)+t(1/b)\big)^{-1}}{t(1-t)},$$
for any real numbers $a,b>0$. Simple computation leads to (after all reductions)
$$J(a,b)=\int_0^1\frac{(a-b)^2}{ta+(1-t)b}dt=(a-b)\big(\log a-\log b\big).$$
It follows that, for any $T\in{\mathcal B}^{+*}(H)$ we have
\begin{equation}\label{44}
J(I,T)=(T-I)\log T,
\end{equation}
where $I$ denotes the identity operator of ${\mathcal B}^{+*}(H)$. Since $(A,B)\longmapsto A\nabla_tB$ and $(A,B)\longmapsto A!_tB$ are operator means in the Kubo-Ando sense then we can write, by using \eqref{44} with $T=A^{-1/2}BA^{-1/2}$,
\begin{multline*}
J(A,B)=A^{1/2}J\Big(I,A^{-1/2}BA^{-1/2}\Big)A^{1/2}\\
=A^{1/2}\Big(A^{-1/2}BA^{-1/2}-I\Big)\log\Big(A^{-1/2}BA^{-1/2}\Big)A^{1/2}.
\end{multline*}
The desired result follows after a simple manipulation.
\end{proof}
\begin{remark}
From \eqref{42}, it is immediate that $J(A,B)$ is a positive operator, since $A\nabla_tB\geq A!_tB$. Further, $J(A,B)$ is symmetric in $A$ and $B$. These properties of $J(A,B)$ are not easy to deduce from \eqref{43}.
\end{remark}
Now, we are in a position to state the following main result which gives a lower bound of ${\mathcal{HZ}}_{\lambda}(f,g)$.
\begin{theorem}\label{th5}
Let $(f,g)\in{\mathcal D}(H)$ and $\lambda\in[0,1]$. Then we have
\begin{equation}\label{50}
{\mathcal{HZ}}_{\lambda}(f,g)\\
\geq{\mathcal K}_{\delta(\lambda)}(f,g)-\alpha(\lambda)\frac{\sin(\pi\lambda)}{2\pi}{\mathcal J}(f,g),
\end{equation}
where ${\mathcal J}(f,g)$ was defined in Proposition \ref{pr3} and
$$\alpha(\lambda)=(2\lambda-1)^2,\;\; \delta(\lambda)=1-4\lambda(1-\lambda)\sin(\pi\lambda).$$
\end{theorem}
\begin{proof}
For $\lambda=0$ or $\lambda=1$, \eqref{50} are immediate after a simple discussion as in the proof of Theorem \ref{th3}. We now assume that $\lambda\in(0,1)$. By \eqref{24}, with \eqref{5}, we have
\begin{multline}\label{52}
{\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\\
\leq\frac{\sin(\pi\lambda)}{\pi}\int_0^1K_{\alpha(\lambda)}\Big(\frac{1}{t},\frac{1}{1-t}\Big)
\Big({\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\Big)dt,
\end{multline}
where $\alpha(\lambda)=(2\lambda-1)^2$. By \eqref{3a} we have
$$K_{\alpha(\lambda)}=\big(1-\alpha(\lambda)\big)\frac{1}{\sqrt{t(1-t)}}+\frac{\alpha(\lambda)}{2}\frac{1}{t(1-t)}.$$
With this \eqref{52} yields
\begin{multline*}
{\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\leq4\lambda(1-\lambda)\frac{\sin(\pi\lambda)}{\pi}
\int_0^1\frac{1}{\sqrt{t(1-t)}}\Big({\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)\Big)dt\\
+\frac{\alpha(\lambda)}{2}\frac{\sin(\pi\lambda)}{\pi}
\int_0^1\frac{{\mathcal A}_t(f,g)-{\mathcal H}_t(f,g)}{t(1-t)}dt.
\end{multline*}
According to \eqref{225}, with \eqref{40}, we obtain
$${\mathcal A}(f,g)-{\mathcal{HZ}}_{\lambda}(f,g)\leq4\lambda(1-\lambda)\sin(\pi\lambda)\Big({\mathcal A}(f,g)-{\mathcal G}(f,g)\Big)+
\frac{\alpha(\lambda)}{2}\frac{\sin(\pi\lambda)}{\pi}{\mathcal J}(f,g).$$
We then deduce our desired inequality \eqref{50}, after a simple discussion, as in the proof of Theorem \ref{th3}. Details are routine and therefore omitted here.
\end{proof}
The operator version of Theorem \ref{th5} reads as follows.
\begin{corollary}
Let $A,B\in{\mathcal B}^{+*}(H)$ and $\lambda\in[0,1]$. Then there holds:
\begin{equation}\label{53}
HZ_{\lambda}(A,B)\geq K_{\delta(\lambda)}(A,B)-\alpha(\lambda)\frac{\sin(\pi\lambda)}{2\pi}J(A,B),
\end{equation}
where $\alpha(\lambda)$ and $\delta(\lambda)$ are as in Theorem \ref{th5} and $J(A,B)$ is as in Proposition \ref{pr5}.
In particular, for any $a,b>0$ and $\lambda\in[0,1]$, we have
\begin{multline}\label{54}
HZ_{\lambda}(a,b)\geq\frac{a+b}{2}-2\lambda(1-\lambda)\sin(\pi\lambda)\big(\sqrt{a}-\sqrt{b}\big)^2\\
-(2\lambda-1)^2\;\frac{\sin(\pi\lambda)}{2\pi}(b-a)\log(b/a).
\end{multline}
\end{corollary}
\begin{proof}
Taking $f=f_A$ and $g=f_B$ in \eqref{50}, with the help of \eqref{17} and \eqref{41d}, we obtain \eqref{53}. We then deduce \eqref{54} after simple manipulations. Details are simple and therefore omitted here.
\end{proof}
We now present some comments about comparison between \eqref{5a}, \eqref{5b} and \eqref{54}. First, we mention that, for $\lambda=0$ or $\lambda=1$, \eqref{5a} and \eqref{54}, which are both reduced to an equality, imply \eqref{5b}. For $\lambda=1/2$, \eqref{5b} and \eqref{54} are both reduced to the same equality and yield \eqref{5a}. We now consider the general case. For this purpose, by putting $a=t^2, b=1$ we consider the following function
$$f_{\lambda}(t) = g_{\lambda}(t) (t-1) \log t$$
for $t>0$ and $0\leq \lambda \leq 1$, where
\begin{equation}\label{55}
g_{\lambda}(t)=(t+1)\left\{\left(\lambda(1-\lambda)-\frac{(2\lambda-1)^2\sin(\pi\lambda)}{\pi}\right)
-\frac{2\lambda(1-\lambda) (t-1)\sin(\pi\lambda)}{(t+1)\log t}\right\}.
\end{equation}
We find
$$\lim_{t \to 0} g_{\lambda}(t) = h(\lambda),$$
where
$$h(\lambda) = \lambda(1-\lambda) -\frac{(2\lambda-1)^2 \sin(\pi \lambda)}{\pi}.$$
Here we show $h(\lambda) \geq 0$ for $0\leq \lambda \leq 1$.
We calculate
$$h'(\lambda) = \frac{(1-2\lambda)}{\pi} k(\lambda),$$
where
$$k(\lambda) = \pi -\pi(1-2\lambda)\cos(\pi \lambda) + 4\sin(\pi \lambda).$$
Since $k(\lambda) = k(1-\lambda)$, we show $k(\lambda) \geq 0$ for $0\leq \lambda \leq \frac{1}{2}$. Then we calculate
$$k'(\lambda) =\pi(6\cos(\pi \lambda)+\pi(1-2\lambda)\sin(\pi\lambda)) \geq 0$$
for $0<\lambda \leq \frac{1}{2}$ so that we have $k(\lambda) \geq k(0) =0$. Therefore $h'(\lambda) \geq 0$ for $0\leq \lambda \leq \frac{1}{2}$ and $h'(\lambda) \leq 0$ for $\frac{1}{2} \leq \lambda \leq 1$ and $h(0)=h(1)=0$ which implies $h(\lambda) \geq 0$ for $0 \leq \lambda \leq 1$. In addition, from \eqref{55} we find that $\lim_{t\to\infty}g_{\lambda}(t)=\infty$ by $h(\lambda) \geq 0$.
Therefore we have $f_{\lambda}(t) \geq 0$ for enough large $t$ and enough small $t$ which means the lower bound in \eqref{54} gives better bound than that in \eqref{5a}. Moreover the same results are shown by numerical computations in almost cases for $t>0$ and $0 \leq \lambda \leq 1$.
However, it is not true that $f_{\lambda}(t) \geq 0$ for all $t>0$ and $0 \leq \lambda \leq 1$ in general, since we have counter-examples such as $f_{0.9}(0.75)\simeq -0.0000722089$ and $f_{0.9}(1.5)\simeq -0.000197205$.
We also set the function, the square of R.H.S. in \eqref{54} minus R.H.S. in \eqref{5b} with $a=t$ and $b=1$ by
$
\frac{1}{4}\left(\alpha_{\lambda}(t)+\beta_{\lambda}(t)^2\right)
$
for $t>0$ and $0\leq \lambda \leq 1$. Where
$$\alpha_{\lambda}(t) = 2(1-r_{\lambda})(t-1)^2-(t+1)^2$$
and
$$\beta_{\lambda}(t) =t+1-4\lambda(1-\lambda)(\sqrt{t}-1)^2\sin(\pi \lambda)-\frac{(2\lambda -1)^2(t-1)(\log t) \sin(\pi \lambda)}{\pi}.$$
We easily find the lower bounds both in \eqref{54} and \eqref{5b} are symmetric with respect to $\lambda=\frac{1}{2}$ so that we consider the case $\lambda \in [0,\frac{1}{2}]$. We also find that $\alpha_{\lambda}(t)=2(1-\lambda)(t-1)^2-(t+1)^2$ is decreasing in $\lambda \in [0,\frac{1}{2}]$ for any $t>0$, since $r_{\lambda}=\lambda$.
Numerical computations show $\alpha_{\lambda}(t) +\beta_{\lambda}(t)^2 \geq 0$ for any $t>0$ and $\lambda \in [0,\frac{1}{2}]$ which means our lower bound in \eqref{54} is tighter than that in \eqref{5b}. However we have not found the analytical proof for the inequality $\alpha_{\lambda}(t) +\beta_{\lambda}(t)^2 \geq 0$ for any $t>0$ and $\lambda \in [0,\frac{1}{2}]$ and also not found any counter-examples for the inequality $\alpha_{\lambda}(t) +\beta_{\lambda}(t)^2 \geq 0$.\\
Finally, we state the following remark which may be of interest for the reader.
\begin{remark}
Let ${\mathcal M}(f,g)$ be one of the functional means (depending on $\lambda\in[0,1]$ or not) previously introduced. All theoretical results and inequalities investigated in this paper are still valid for any $f,g:H\longrightarrow{\mathbb R}\cup\{+\infty\}$ such that ${\rm dom}\;f\cap{\rm dom}\;g\neq\emptyset$. As the reader can remark it, the condition $f,g\in\Gamma_0(H)$ was not needed in the proofs. In fact, the condition $f,g\in\Gamma_0(H)$ is needed only when we want to ensure the axiom ${\mathcal M}(f,f)=f$ for any $f\in\Gamma_0(H)$. This because every functional mean was introduced as an extension of its related operator mean, denote it by $m(A,B)$ which, in its turn, should satisfy by definition the idempotent axiom for an operator mean, namely $m(A,A)=A$ for any $A\in{\mathcal B}^{+*}(H)$.
\end{remark}
\section*{\bf Acknowledgment}
The author (S.F.) was partially supported by JSPS KAKENHI Grant Number 16K05257.
\bibliographystyle{amsplain}
|
2,869,038,154,854 | arxiv | \section{Introduction}
\let\thefootnote\relax\footnotetext{Electronic address: {\tt [email protected]}\\Electronic address: {\tt [email protected]}}
A range of cosmological observations indicate that the current expansion rate of the universe is accelerating \cite{Riess:1998cb}. This is accommodated within general relativity by the introduction of the so called cosmological constant $\Lambda$. Viewed as a source of vacuum energy density, it has an equation of state $w =-1$. By taking its energy density to be $\sim 70\%$ of the current critical density, it leads to acceleration in a fashion that is beautifully compatible with current data \cite{Hinshaw:2012aka, Ade:2013uln}. On the one hand, this is another spectacular triumph for general relativity and particle physics, which suggests the appearance of vacuum energy. On the other hand, typical estimates for the value of the vacuum energy are many orders of magnitude larger than the observed value of $\Lambda_{\rm obs}\sim (10^{-3}\,\mbox{eV})^4$. This leads to the problem of why the observed cosmological constant is so small; for a review see Ref.~\cite{Martin:2012bt}.
Many proposals have tried to address this problem, often involving radical modifications to the structure of gravity or quantum field theory.
One class of recent proposals stays within the framework of ordinary field theory, but appeals to the existence of extra dimensions \cite{Brown:2013fba,Brown:2014sba}. An appreciable vacuum energy is included in the higher-dimensional theory and various other sources of energy; fluxes and curvature (for earlier work on flux compactifications, see \cite{Douglas:2006es,Denef:2007pq}, and for some difficulties in achieving a positive cosmological constant in string compactifications, see \cite{Hertzberg:2007ke,Hertzberg:2007wc}). It is then found that upon compactifying to 4 dimensions, the resulting lower-dimensional cosmological constant is arbitrarily small and has an accumulation point at $\Lambda=0$. This appears to be a wonderful solution to the cosmological constant problem. At the same time, it is acknowledged that this solution comes at a price; it predicts the existence of additional arbitrarily light scalars, with masses of order Hubble.
In this paper, we show that the lightness of various scales, completely undermines the whole purported solution.
As we explain, the heart of the cosmological constant problem is to explain why $\Lambda$ is so small despite the presence of various heavy scales (such as the Standard Model fields, and possible heavier fields associated with unification and quantum gravity). It misses the essential problem to merely send all mass scales to zero in a toy four-dimensional theory (in units of the four-dimensional Planck mass). Moreover, we show that in order for the cosmological constant to be small in these models, the fundamental Planck scale is also being sent to zero; thus removing any high energy completely. However the heart of the cosmological constant problem is to explain why $\Lambda$ is small despite the existence of high energy physics, including heavy fields and a high fundamental scale. Related to this, we clarify some issues surrounding the problem in different setups, such as electromagnetism and gravity, and provide a general explanation as to why the moduli masses are naturally of order Hubble and how $\Lambda$ is linked to the higher-dimensional fundamental scale.
Our paper is organized as follows:
In Section \ref{Free} we describe two different notions of fine-tuning.
In Section \ref{E&MGrav} we describe fine-tuning in two different models.
In Section \ref{Comp} we discuss a class of compactification models.
In Section \ref{Mass} we compute the moduli masses and the fundamental Planck mass in this class of models.
Finally, in Section \ref{Disc} we conclude.
We work in units in which $c=1$, but we will keep powers of $\hbar$ to track classical versus quantum effects. We will write the ``mass" couplings in the field theory as $m$, even though $m$ in classical field theory is actually a frequency, and the mass of quanta is $\hbar\,m$.
\section{Notions of Fine-Tuning}\label{Free}
\subsection{Sharp Cutoff Analysis}\label{Sharp}
To set the stage for our later argument, it is useful to study the vacuum energy in free theories. Of course the vacuum energy only has consequences when we include gravitation, so we mean ``free" in the particle sector.
The vacuum energy receives a quantum contribution $\Lambda_{\rm quant}$ from a one loop diagram.
It is well known that this leads to the following vacuum energy
\begin{equation}
\Lambda_{\rm quant}= \pm \,g\,\hbar\! \int\! {d^3k\over(2\pi)^3}\,{\omega_k\over 2}
\end{equation}
where $g$ is the number of degrees of freedom, $+$ is for bosons, $-$ is for fermions, and $\omega_k=\sqrt{k^2+m^2}$ (we allow for massive free particles). It follows that quantum corrections provide a quartic divergence $\Lambda_{\rm quant}\sim \hbar\, m_{\mbox{\tiny{UV}}}^4$, and if we put $m_{\mbox{\tiny{UV}}}=1/\sqrt{\hbar\,G}$ (the Planck frequency) then we have a Planck energy density.
On the other hand, the total contribution to the vacuum energy receives a contribution from the ``bare" or ``classical term" $\Lambda_{\rm bare}$, so that the total vacuum energy is
\begin{equation}
\Lambda = \Lambda_{\rm bare} + \Lambda_{\rm quant}
\end{equation}
Hence in order for $\Lambda$ to be much less than the cutoff density, requires an extreme {\em fine-tuning} between these two contributions. So it appears as though any model which can dynamically produce a very small $\Lambda$, especially if $\Lambda$ can be made arbitrarily small, is a solution of the cosmological constant problem.
Moreover, if we define the momentum integral with a UV cutoff $m_{\mbox{\tiny{UV}}}$ and expand in powers of $m/m_{\mbox{\tiny{UV}}}$ we obtain
\begin{eqnarray}
\Lambda_{\rm quant} &=& \pm\,{g\,\hbar\over(4\pi)^2}\Bigg{[}c_1\,m_{\mbox{\tiny{UV}}}^4+c_2\,m_{\mbox{\tiny{UV}}}^2\, m^2\nonumber\\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+c_3\,m^4\ln\!\left(m^2\overm_{\mbox{\tiny{UV}}}^2\right)+\ldots\Bigg{]}
\label{rhoexp}\end{eqnarray}
where $c_{1,2,3}=\mathcal{O}(1)$ numbers that depend on choice of regularization. This is the most general expansion for a field at one-loop.
The first term provides the usual claim of a quartic contribution to vacuum energy. It suggests that even if $m=0$, there must be tremendous fine-tuning to cancel against $\Lambda_{\rm bare}$ in order for $\Lambda$ to be small.
\subsection{Renormalization Group Analysis}\label{Renormalization}
The above analysis is highly suggestive that there is a quartic sensitivity to the cutoff, requiring significant fine-tuning to avoid a large $\Lambda$.
One could, however, focus on another notion of ``fine-tuning". To explain this, we should recall that the only physical parameters of a theory are the {\em renormalized} couplings, rather than the bare couplings or quantum corrections, which are scheme dependent. Such couplings are defined at some renormalization scale and change according to the renormalization group. By including gravity, this includes the physical cosmological constant $\Lambda$. We can write
\begin{equation}
\Lambda=\Lambda(\mu)
\end{equation}
where $\mu$ is some renormalization scale. Within the framework of local quantum field theory, one physical notion of fine-tuning is that there is a delicate cancellation among renormalized parameters in order to fit the data. In standard renormalization schemes, the quartic and quadratic diverges of eq.~(\ref{rhoexp}) can be absorbed by $\Lambda_{\rm bare}$, while the logarithm, proportional to $\hbar\,m^4$, is associated with an actual flow of the coupling. Further discussion of these issues includes Refs.~\cite{Babic:2001vv,Burgess:2013ara}.
In particular, as we flow from some high scale $\mu_H\gg m$, down to some low scale $\mu_L\ll m$, there is a jump in the renormalized coupling from passing through a mass scale of the order $\sim\hbar\,m^4$. If we pass through several mass scales, denoted $m_i$, the change is roughly
\begin{eqnarray}
\Delta\Lambda & = & \Lambda(\mu_H)-\Lambda(\mu_L)\\
& \sim & {\hbar\over(4\pi)^2}\sum_i(\pm)_i\,g_i\,m_i^4
\end{eqnarray}
where we suppress possible logarithmic and threshold effects.
Hence, in order for $\Lambda_{\rm obs}\approx\Lambda(\mu_L)$ to be very small, there must be some exquisite cancellation between the renormalized coupling at a high scale $\Lambda(\mu_H)$ and the sum and differences of various renormalized masses $\sim\hbar\,m_i^4$.
Conversely, if one investigates theories that are built out of massless or extremely light particles (and no dynamically generated scales due to strong dynamics) then the flow of the renormalized $\Lambda(\mu)$ will be small, and it does not require fine tuning. We will return to this issue when considering a class of compactification models.
\subsection{Summary}
In physical models, we generally study interesting theories with various heavy particles, and the challenge is to explain how the observed cosmological constant is small compared to the quartic power of some scale. From the point of view of the ``sharp cutoff analysis", we compare $\Lambda_{\rm obs}$ to $\hbar\,m_{\mbox{\tiny{UV}}}^4$, and from the point of view of the ``renormalization group analysis", we compare $\Lambda_{\rm obs}$ to $\hbar\,m_i^4$, where $m_i$ is the heaviest mass scale. Often these two scales will not differ too much anyhow, as often heavy fields appear around the cutoff of an effective theory. If we take $m_{\mbox{\tiny{UV}}}\simM_{\rm Pl}$, or we suppose there are particles whose mass is close to $m_i\simM_{\rm Pl}$, then in either point of view we expect a Planck density for $\Lambda$. Moreover, there can be additional classical field contributions to the vacuum energy in interacting theories (for example, from scalar potentials or when gravity is included) that can be large too, but this again depends on the presence of high scales. Let us now further explore this in some important interacting theories.
\section{Vacuum Energy Examples}\label{E&MGrav}
\subsection{Pure Electromagnetism}\label{E&M}
As a warm-up to the gravitational case, we start here by studying the problem of vacuum energy from photons.
We consider the following interacting theory of only massless photons minimally coupled to gravity
\begin{equation}
S = \int\! d^4x\sqrt{-g}\left[-\Lambda-{1\over4}F_{\mu\nu}F^{\mu\nu}+{1\over M^4}(F_{\mu\nu}F^{\mu\nu})^2+\ldots\right]
\label{EM}\end{equation}
As shown in the previous section, introducing a sharp cutoff on the vacuum energy loop integral reveals a vacuum energy contribution $\Lambda_{\rm quant}\sim\hbar\, m_{\mbox{\tiny{UV}}}^4$. However if we focus on the renormalization group flow in a local Lorentz invariant theory, we see that there is no flow from massless fields, and the higher order interaction term does not change this conclusion (at least for energies well below the cutoff). Hence a theory of massless photons does not strictly speaking have any cosmological constant problem from the point of renormalization group flow.
However the interaction term renders the theory non-renormalizable. So one expects new physics to enter at some scale $m_{\rm new}$ satisfying $m_{\rm new}<M/\hbar^{1/4}$ to cure the problem of unitarity violation at frequencies above $m_{\mbox{\tiny{UV}}}\sim M/\hbar^{1/4}$. We expect that the new physics can renormalize the vacuum energy. Dimensional analysis selects a unique form:
\begin{eqnarray}
\Delta\Lambda\sim\hbar\,m_{\rm new}^4
\end{eqnarray}
Of course we know that this theory is UV completed by QED (modulo the Landau pole) with the introduction of the electron with frequency
\begin{equation}
m_{\rm new}=m_{\rm e}\sim\sqrt{\alpha}\,M/\hbar^{1/4}\sim\sqrt{\alpha}\,m_{\mbox{\tiny{UV}}}
\end{equation}
which is parameterically smaller than the cutoff $m_{\mbox{\tiny{UV}}}$. Indeed the electron introduces a renormalization to the vacuum energy of the form $\Delta\Lambda \sim \hbar\,m_{\rm e}^4$, in accord with the above mentioned expectation.
Hence even though the action in eq.~(\ref{EM}) does not by itself generate a large renormalization in vacuum energy, there is a large contribution introduced by the physics associated with its UV completion. This leads to a vacuum energy renormalization that is $\sim32$ orders of magnitude larger than the observed value. Hence a solution to this cosmological constant problem is to invent a mechanism by which $\Lambda\lll \hbar\,m_{\rm e}^4$ in a natural way in the theory with electrons.
\subsection{Pure Gravity}\label{Grav}
Let us now consider the case of pure Einstein gravity with standard action
\begin{equation}
S=\int\! d^4x\sqrt{-g}\left[-\Lambda+M_{\rm Pl}^2\mathcal{R}+\ldots\right]
\end{equation}
where $M_{\rm Pl}^2\equiv1/(16\pi G)$. Here we indicate a tower of higher derivative corrections by the dots. Let us expand around flat space by writing the metric as
\begin{equation}
g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}/M_{\rm Pl}
\end{equation}
Focussing on the curvature term, this leads to an action that is schematically given by
\begin{equation}
S_R \sim \int\!d^4x\left[(\partial h)^2+h(\partial h)^2/M_{\rm Pl}+h^2(\partial h)^2/M_{\rm Pl}^2+\ldots\right]
\end{equation}
This theory leads to a large quantum correction to the bare vacuum energy of the form $\Lambda\sim\hbar\,m_{\mbox{\tiny{UV}}}^4$, as usual.
However, there is not a significant flow in the {\em renormalized} vacuum energy, similar to the case of massless photons. Of course, it is also non-renormalizable and requires a UV completion. The theory requires new physics at a scale $m_{\rm new}<1/\sqrt{\hbar\,G}$ (the Planck frequency). Unlike the case of the interacting photons, here we do not know the form of the new physics to unitarize graviton scattering; it may involve extra dimensions, supersymmetry, strings, and/or other possibilities.
In any case, we expect that the new physics will introduce contributions to shift the vacuum energy. In this case we can have at least two scales: (i) the scale that sets the new physics $m_{\rm new}$ (which might be many new scales), and (ii) (assuming the new physics permits a four-dimensional description in some regime) we also have Newton's constant $G$. This permits the following tower of dimensionally correct possibilities in increasing powers of $\hbar$:
\begin{equation}
\Delta\Lambda \sim M_{\rm Pl}^2m_{\rm new}^2,\,\hbar \, m_{\rm new}^4,\,\hbar^2\,m_{\rm new}^6/M_{\rm Pl}^2,\ldots
\label{rhograv}\end{equation}
The first term involves no powers of $\hbar$; it is a {\em classical} contribution from possible phase transitions, etc. It may or may not arise, depending on how the new physics interacts with gravity. The second term is the leading (1-loop) quantum contribution to the renormalization of $\Lambda$. Assuming new physics enters below the Planck scale, the higher order terms will be sub-dominant. The challenge to solve this cosmological constant problem is to find a mechanism in which $\Lambda\lll M_{\rm Pl}^2 m_{\rm new}^2,\,\hbar \, m_{\mbox{\tiny{UV}}}^4,\ldots$. It would be some significant progress to have a dynamical mechanism wherein $\Lambda$ is naturally much less than the leading few terms, say, even if it is not smaller than the higher order terms.
On the other hand, the absolute worst case scenario is to have a mechanism in which $\Lambda\sim M_{\rm Pl}^2m_{\rm new}^2$ or $\Lambda\gtrsim\hbar\,m_{\mbox{\tiny{UV}}}^4$; this would not represent any progress at all. One might think this was progress if $\Lambda$ was small in the Planck units, due to $m_{\rm new}$ and/or $m_{\mbox{\tiny{UV}}}$ being small, but this is not the real cosmological constant problem. If no Planck scale energies are permitted in the theory, then comparing to the Planck scale is irrelevant. The problem is to be small in terms of the energy scales that arise from the particle sector and in terms of the fundamental cutoff. Since (i) we know that even conventional Standard Model particle physics gives masses $m_{\rm new}\sim m_{t}\sim m_H\sim 100$\,GeV (which is ``new" physics from the low energy pure gravity point of view, even though these degrees of freedom may not be relevant to unitarizing graviton scattering), (ii) it is plausible that there are many heavier particles, such as at some extra dimension ``radion" scale, the GUT scale $m_{\rm new}\sim 10^{16}$\,GeV, and perhaps even heavier particles still, and (iii) the fundamental cutoff $m_{\mbox{\tiny{UV}}}$ must be correspondingly even larger, if one was to obtain $\Lambda\sim M_{\rm Pl}^2m_{\rm new}^2$, or $\Lambda\gtrsim\hbar\,m_{\mbox{\tiny{UV}}}^4$, it would be catastrophically large. Indeed one would anticipate that any purported dynamical solution at least achieves $\Lambda\lll M_{\rm Pl}^2m_{\rm new}^2,\,\hbar \, m_{\mbox{\tiny{UV}}}^4$.
Shortly we will show that in a class of compactification models, $\Lambda\sim M_{\rm Pl}^2m_{\rm new}^2$ and $\Lambda\gtrsim\hbar \,m_{\mbox{\tiny{UV}}}^4$ is generically obtained, which is indeed the worst case scenario and is therefore not a dynamical solution of the real problem.
\section{Compactification Models}\label{Comp}
Consider the following $D$-dimensional action involving gravity and a collection of $p$-form field strengths
\begin{equation}
S = \int\! d^{D}\!x\sqrt{-g_D}\left[-\Lambda_D+M_{*}^{D-2} \mathcal{R}_D-\sum_p |F_p|^2\right]
\label{action}\end{equation}
where we have also included a higher-dimensional cosmological constant $\Lambda_D$ and the fundamental Planck mass is $M_{*}^{D-2}\equiv 1/(16\pi G_D)$. We now illustrate how the four-dimensional theory emerges by compactification.
Here we consider a product manifold in the form $\mathbb{R}^4 \otimes {\cal M}^N $ where $\cal M$ is a $d$-dimensional manifold with a metric $h_{ij}$. This means we are considering $\tilde{d}\equiv N\,d$ extra dimensions.
We write the metric in the form
\begin{eqnarray}
&&ds^2= e^{- d(\Psi(x)-\Psi_0)} g_{\mu \nu}(x) dx^\mu d x^\nu \nonumber\\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sum_a e^{2(\psi_a(x) - \psi_{0a})}h_{(a)ij} dy_{(a)}^{i}dy_{(a)}^j.
\label{metric}\end{eqnarray}
where $a$ is an index that runs from $a=1,\ldots,N$ over each of the internal manifolds of dimension $d$.
We have considered the simple case in which the higher-dimensional metric decomposes into a four-dimensional piece and a compact space with radion moduli $\psi_a=\psi_a(x)$ that only depends on the large dimensions.
Here $\mu,\nu \in\{0,1,2,3\}$ and $x$ is the large dimension co-ordinates,
while $i,j\in\{1,\ldots,d\}$.
In the first term we have, without loss of generality, pulled out a factor of $e^{-d(\Psi(x)-\Psi_0)}$ so that the four-dimensional action is immediately in the Einstein frame, where
\bel{Psi}
\Psi(x)=\sum_a \psi_a(x), \qquad \Psi_0=\sum_a \psi_{0a}~.
\end{equation}
and $\psi_{0a}$ is the value of $\psi_a$ at its stabilized value from compactification.
When integrating the fluxes over the compact space, we assume the fluxes are only functions of the compact co-ordinates, apart from an overall volume dependence. For the flux wrapping around the compact space labelled $(a)$, we have
\begin{equation}
\sum_p|F_p|^2 = \sum_p \mathcal{F}_{p(a)}(y) e^{-2p (\psi_a(x)-\psi_{a0})}
\end{equation}
We can now integrate over the compact space. This leads to 3 quantities that characterize its structure, namely
\begin{eqnarray}
&& \mathcal{V}_{(a)} \equiv \int d^d y_{(a)}\sqrt{h_{(a)}}\\
&& C_{(a)} \equiv \int d^d y_{(a)}\sqrt{h_{(a)}}\,\mathcal{R}_{(a)}\label{Cdef}\\
&& f_{p(a)} \equiv \int d^d y_{(a)}\sqrt{h_{(a)}}\,\mathcal{F}_{p(a)}
\end{eqnarray}
where the first quantity $\mathcal{V}_{(a)}$ is the volume of each compact space, with total volume
\begin{equation}
\mathcal{V}=\prod_a \mathcal{V}_{(a)},
\end{equation}
the second quantity $C_{(a)}$ is a volume integral over the compact space Ricci scalar $\mathcal{R}_{(a)}$, and the third quantity $f_{p(a)}$ is a volume integral over the flux number that threads the compact space. Note that each of these 3 quantities are constant; independent of space and time.
Dropping boundary terms, we obtain the following action in 4 dimensions
\bel{action2}
S = \int d^4x \sqrt{-g_{(4)}} (-V_4+K_E)~,
\end{equation}
where $K_E$ and $V_4$ are the kinetic terms and the four-dimensional potential term given by
\beal{KinetricPotential}
V_4&=&e^{-d (\Psi-\Psi_0)}\Big{[}V_\Lambda - \sum_a V_{C(a)} e^{ - 2 (\psi_a-\psi_{a0})} \nonumber \\
&&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+ \sum_{a,p} V_{p(a)} e^{-2p (\psi_a- \psi_{a0})}\Big{]},\label{V4}\\
K_E&=&M_{\rm Pl}^2 \[{\mathcal R} - \frac {d^2}2 \left( \nabla \Psi \right)^2 - \sum_a d \( \nabla \psi_a \)^2 \],
\end{eqnarray}
where
\bel{defs}
V_{\Lambda}\equiv\Lambda_D\,{\cal V}, \,\,\,\, V_{C(a)}\equiv {C_{(a)} M_{\rm Pl}^2\over\mathcal{V}_{(a)}},\,\,\,\, V_{p(a)}\equiv \frac{f_{p(a)} {\cal V}}{\mathcal{V}_{(a)}},
\end{equation}
Here the four-dimensional Planck mass $M_{\rm Pl}$ is related to the fundamental Planck mass $M_{*}$ by
\begin{equation}
M_{\rm Pl}^2=\mathcal{V}\,M_{*}^{D-2}
\label{mpmps}\end{equation}
The above action involves scalar fields $\psi_a$ that are not canonically normalized. We can switch to a new set of fields $\phi_a$ that are canonically normalized by defining
\bel{diagonalize}
\phi_a\equiv \sqrt{2d} \( \psi_a -\psi_{0a} + b(\Psi-\Psi_0)\)M_{\rm Pl},
\end{equation}
where
\begin{equation}
b\equiv\frac{-2+\sqrt{4 + 2 N d}}{2N}.
\end{equation}
which is defined such that the minima of the potential is at $\phi_a=0$.
In terms of these canonically normalized fields the action takes on the canonical form
\begin{equation}
S=\int d^4x\sqrt{-g}\left[-V(\phi_a)+M_{\rm Pl}^2\mathcal{R}-\sum_a{1\over2}(\partial\phi_a)^2\right]
\end{equation}
The potential function $V$ for the canonically normalized field is simply $V=V_4$, but expressed in different variables.
For a positively curved compact space, $C_{(a)}>0$, or for a negative higher-dimensional cosmological constant, $\Lambda_D<0$, one or both of the first two terms of $V$ is negative and it can compete with the other positive flux terms to lead to a stable vacuum solution, either AdS, Minkowski, or dS, depending on parameters.
In the simplest versions of these models, with only one internal manifold $N=1$ and $\Lambda_D>0$ there can be dS vacua, but no accumulation of vacua as $\Lambda\to 0^+$ in the large flux limit. On the other hand, for $\Lambda_D\leq 0$ there is an accumulation of vacua with $\Lambda\to 0^-$ in the large flux limit.
More interestingly, for $N\geq 2$ an accumulation point can arise for dS vacua as $\Lambda\to 0^+$ (as well as a much more dominant accumulation of vacua as $\Lambda\to 0^-$). This was shown in Ref.~\cite{Brown:2013fba} in the case where the internal manifolds are spheres.
This appears to be a beautiful solution of the cosmological constant problem.
In the next section we will describe a general property of their solutions regarding moduli masses.
\section{Mass Scales}\label{Mass}
\subsection{Moduli Scale}\label{Moduli}
In this class of compactifications, and even for more general classes, the potential energy is a sum of terms involving exponentials of the radion fields $\phi_a$, as seen in eq.~(\ref{V4}). In a general fashion, we can write the potential as
\begin{equation}
V = \sum_q V_q\,\exp\left(- \sum_a \beta^{qa}\,\phi_a/M_{\rm Pl}\right)
\label{potsimp}\end{equation}
Comparing to (\ref{V4}) it is straightforward to read off the value of $V_q$ and $\beta^{qa}$ in this particular class of models. What is important to note is that the coefficients in the exponents $\beta^{qa}$ are $\mathcal{O}(1)$.
The mass of the moduli is given by the eigenvalues of the Hessian matrix of the potential at the minimum $\phi_a=0$. This is
\begin{equation}
H_{ab} = {\partial^2 V\over\partial\phi_a\partial\phi_b}\Bigg{|}_{\phi=0} = \sum_q {\beta^{aq}\beta^{bq}\over M_{\rm Pl}^2}\,V_q
\label{mphi}\end{equation}
Notice the eigenvalues cannot be much larger than the elements of $H_{ab}$, and so we will estimate the typical masses $m_a$ to be of order the typical values of $H_{ab}$.
On the other hand, the cosmological constant is given by
\begin{equation}
\Lambda=V\big{|}_{\phi=0}=\sum_q V_q
\end{equation}
Then assuming the potential does not have any accidental cancellations at its minima $\Lambda=V\big{|}_{\phi=0}$, and recalling that $\beta^{aq}=\mathcal{O}(1)$, we can express $\Lambda$ in terms of $m_a^2$ by comparing (\ref{potsimp}) to (\ref{mphi}), giving
\begin{equation}
\Lambda \sim M_{\rm Pl}^2\,m_a^2
\end{equation}
Then using the Friedmann equation, we have $m_a\sim H$ quite generally, where $H$ is the Hubble parameter. So we see that generically the moduli mass is related to the Hubble scale, which is reasonable on dimensional grounds.
Typically, most vacua are AdS. One can restrict attention to only dS vacua (as was the case in Refs.~\cite{Brown:2013fba,Brown:2014sba}). These dS vacua require a fine-tuning to achieve a special cancellation in the potential between the large $V_\Lambda$ contribution and a curvature contribution, leaving $\Lambda$ especially small (see Section \ref{nongeneric} for more details). However, one can show that a typical dS vacuum has a corresponding special cancellation in the potential, leaving $\Lambda\sim M_{\rm Pl}^2\,m_a^2$ still valid for light moduli (although some moduli can be heavier).
In more complicated models, even if one were to obtain $\Lambda\ll M_{\rm Pl}^2\,m_a^2$, there is no evidence that $\Lambda\ll\hbar \, m_a^4$ can be obtained within this framework.
Hence from the point of view of the renormalization group flow of $\Lambda$, we see that these models do not produce a $\Lambda$ that is smaller than the typical expectation from renormalization. In the next section, we will study whether $\Lambda$ is much smaller than the estimates based on a sharp cutoff.
\subsection{Fundamental Scale}\label{Fund}
\subsubsection{AdS Vacua (generic)}
In this class of models, the higher-dimensional cosmological constant $\Lambda_D$ plays a very important role. In most compactifications, it provides an $\mathcal{O}(1)$ contribution to the four-dimensional vacuum energy. As shown in \cite{Brown:2014sba} most of these vacua are AdS.
For these vacua the four-dimensional vacuum energy is
\begin{equation}
\Lambda\sim -V_{\Lambda}=-\Lambda_D\,\mathcal{V}
\end{equation}
Now suppose we parameterize the higher-dimensional cosmological constant as
\begin{equation}
\Lambda_D = \lambda_D\,M_{*}^{D}
\end{equation}
So if $\lambda_D$ is chosen to be $\lambda_D=\mathcal{O}(1)$ (in units of $\hbar$), then we have a ``naturally" large value for the higher-dimensional cosmological constant, according to the ``sharp cutoff analysis" of Section \ref{Sharp}.
Now, on the one hand, we can eliminate $M_{*}$ in the expression for $\Lambda$, by using eq.~(\ref{mpmps}) leading to
\begin{equation}
\Lambda \sim - \lambda_D\,M_{\rm Pl}^4\left(l_{Pl}^{\tilde{d}}\over\mathcal{V}\right)^{\!2/(\tilde{d}+2)}
\label{LammpV}\end{equation}
where $\tilde{d} = N\,d$.
Since these are large volume compactifications, in the large flux limit, we see that $\Lambda\to0^{-}$, which appears to solve the problem of why the cosmological constant is small (although this is negative).
On the other hand, we can eliminate $\mathcal{V}$ and express $\Lambda$ in terms of $M_{*}$ leading to
\begin{equation}
\Lambda \sim - \lambda_D\,M_{\rm Pl}^2M_{*}^2
\end{equation}
For $\lambda_D$ not too small, this $\Lambda$ is {\em much larger} than even the ``natural" value of $\sim M_{*}^4$ (in units of $\hbar$), since $M_*\llM_{\rm Pl}$ in the large volume limit. Hence this clearly does not address the cosmological constant problem. We see that the only reason $\Lambda\to 0^{-}$ is because $M_{*}\to 0$ which removes all high energy physics trivially.
\subsubsection{dS Vacua (non-generic)}\label{nongeneric}
Alternatively, one can introduce very special choices of flux parameters, so as to fine-tune away such huge contributions to the vacuum energy, and allow for dS vacua. To do this, we need a hierarchy among the internal radii, which leads to a hierarchy among the curvature contributions $V_{C(a)}$. There are two important possibilities:
\begin{enumerate}[{(i)}]
\item $ V_{C(1)} \ll V_{C(2)}\sim \cdots \sim V_{C(N)} $
\item $ V_{C(1)} \gg V_{C(2)}\sim \cdots \sim V_{C(N)} $
\end{enumerate}
In the first case (i), the curvature contribution $\sum_{a=2}^N V_{C(a)}$ is tuned to cancel against the vacuum energy contribution $V_\Lambda$.
The residual four dimensional vacuum energy can be estimated by the residual curvature contribution
\begin{equation}
\Lambda\sim V_{C(1)}
\end{equation}
From eq.~(\ref{Cdef}), the curvature parameter $C_{(1)}$ is roughly $C_{(1)}\sim\mathcal{V}_{(1)}^{1-2/d}$. Expressing $\Lambda$ in terms of $M_{\rm Pl}$ and $\mathcal{V}_{(1)}$ gives
\begin{equation}
\Lambda\simM_{\rm Pl}^4\left(l_{Pl}^{d}\over\mathcal{V}_{(1)}\right)^{\!2/d}
\label{curv4a}
\end{equation}
which shows that indeed the vacuum energy is even smaller than previously in eq.~(\ref{LammpV}), for large volumes $\mathcal{V}_{(1)}$. This is the result of fine-tuning the leading contributions to vanish, and allows $\Lambda\to 0^{+}$ more rapidly, which appears to solve the problem of why the cosmological constant is small and positive. However, we can again eliminate $\mathcal{V}_{(1)}$ and express the result in terms of $M_{*}$, to find
\begin{equation}
\Lambda \sim {M_{*}^4\over\lambda_D^{N-1}}\left(M_{\rm Pl}\overM_{*}\right)^{\!2-{4\overd}}
\end{equation}
We note that for curvature to be present, we obviously need $d\geq 2$. So $\Lambda\gtrsim M_{*}^4$ (in units of $\hbar$)
and is bounded by $\Lambda\lesssim M_{*}^2M_{\rm Pl}^2$ for high $d$.
In the second case (ii), the curvature contribution $V_{C(1)}$ is tuned to cancel against the vacuum energy contribution $V_\Lambda$.
The residual four dimensional vacuum energy can be estimated by the residual curvature contribution
\begin{equation}
\Lambda\sim V_{C(a)},\,\,\,\,\mbox{where}\,\,\,a=2,\ldots,N
\end{equation}
So we can write an expression similar to eq.~(\ref{curv4a}) as
\begin{equation}
\Lambda\simM_{\rm Pl}^4\left(l_{Pl}^{d}\over\mathcal{V}_{(a)}\right)^{\!2/d}
\label{curv4b}
\end{equation}
using the fact that $\mathcal{V}_{(a)}$ are all similar for $a=2,\ldots,N$ in this case. This again says that the vacuum energy tends to $\Lambda\to 0^{+}$ in the large volume limit. We now eliminate the volume dependence to express the result in terms of $M_*$, as we did in case (i), to find
\begin{equation}
\Lambda \sim {M_{*}^4\over\lambda_D^{1/(N-1)}}\left(M_{\rm Pl}\overM_{*}\right)^{\!2-{4\overd(N-1)}}
\end{equation}
Since we need $d\geq2$ for curvature to be present and $N\geq2$ for this cancellation to take place, we again have $\Lambda\gtrsim M_{*}^4$ (in units of $\hbar$) and is bounded by $\Lambda\lesssim M_{*}^2M_{\rm Pl}^2$ for high $d$ or $N$.
So in both cases (i) and (ii) we see that despite having fine-tuned away the leading term to produce dS vacua, the resulting cosmological constant is still not smaller than the estimate based on a simple cutoff.
\section{Conclusions}\label{Disc}
Hence even though there are interesting compactification models \cite{Brown:2013fba,Brown:2014sba} in which the four-dimensional cosmological constant has an accumulation point as $\Lambda\to 0$, it does so only in so far as the mass scales of fundamental physics $m_{\rm new},\,m_{\mbox{\tiny{UV}}}\to 0$. Generally in these models, $\Lambda$ scales as some power of the product of the mass and Planck mass appearing in the four-dimensional theory, or alternatively, as a power of the fundamental Planck scale. This is essentially the worst case scenario from both the renormalization group point of view, and also from the sharp cutoff point of view. This does not address the real cosmological constant problem, wherein we need to explain how $\Lambda$ is incredibly small, despite the presence of high scales of physics.
A general way to see the problem is the following: From the low energy four-dimensional point of view, it cannot matter that there are extra dimensions, or otherwise, in the UV. Effective field theory says that these effects cannot naturally reach down and remove the already large contributions to vacuum energy from known Standard Model physics. In the above toy models, an attempt to do this comes from having the new ``UV" physics scale simply inserted at fantastically low energies, which misses the real problem.
Returning to the structure of eqs.~(\ref{potsimp}) we see that the only possibility would be to allow the potential $V$ to have various {\em accidental} cancellations at its minimum $\Lambda$, while maintaining large mass scales $m_\phi,\,M_{*}$. This could be conceivable in some landscape framework with many fluxes and an exponentially large number of vacua, and is a conceivable solution \cite{Sakharov:1984ir,Bousso:2000xa}. Other directions to address the problem could involve radical alterations to local quantum field theory.
\begin{center}
{\bf Acknowledgments}
\end{center}
We would like to thank Alex Vilenkin and Erick Weinberg for helpful discussions.
This work is supported by the U.S. Department of Energy under cooperative research agreement Contract Number DE-FG02-05ER41360. AM is supported by a grant from National Science Foundation (grant PHY-1213888).
|
2,869,038,154,855 | arxiv | \section{Introduction}
Learning to translate between two image domains is a common problem in computer vision and graphics, and has many potentially useful applications including colorization \cite{pix2pix}, photo generation from sketches \cite{pix2pix}, inpainting \cite{inpainting}, future frame prediction \cite{framepredict}, superresolution \cite{superres}, style transfer \cite{style}, and dataset augmentation. It can be particularly useful when images from one of the two domains are scarce or expensive to obtain (for example by requiring human annotation or modification). \par
Until recently the problem has been posed as a supervised learning problem or a one-to-one mapping, with training datasets of paired images from each domain \cite{pix2pix}. However having access to paired images is a difficult and resource intensive challenge, and so it is helpful to learn to map between unpaired image distributions. Multiple approaches have been successfully applied in solving this task in the recent months \cite{cyclegan,cogan,unit,xgan,stargan}.
Most of the work done in this area deal with translation of images between a single pair of distributions. In this work, we generalize this translation mechanism to multiple pairs. In other words, given a set of distributions which share an underlying joint distribution, we come up with a set of translators that can convert samples from images belonging to any distribution to any other distribution, on which these translators were trained. The effectiveness of these translators is exhibited by considering them as a set of composite functions which can be applied on top of one another.
Further, we explore if these models have the capability of disentanglement of shared and individual components between different distributions. For example, instead of learning to translate from a smiling person that is wearing glasses to a person that is not smiling and not wearing glasses, or a horse in a field on a summer's day to a zebra in a field on a winter's day, we learn to translate from wearing glasses to not wearing glasses, smiling to not smiling, horse to zebra, and summer to winter separately, then compose the results. \par
There are a number of potential advantages to this approach. It becomes possible to learn granular unpaired image to image translations, whilst only having access to either less granular or no labels. It facilitates training on larger datasets since only the marginal, more general labels are required. It gives finer grained control to users of the translation process since they can compose different translation functions to achieve their desired results. Finally, it makes it possible to generate entirely new combinations, by translating to combinations of the marginal distributions that never appeared in the training set.
We also experiment with different training mechanisms to efficiently train models on multiple distributions and show results on decoupled training performing better joint training. Overall, decoupled training followed by some finetuning by joint training produces the best results.
\section{Related Work}
\label{ref:related-work}
\textbf{Generative Adversarial Networks}: Image generation through GAN \cite{gan} and it's several variants such as DCGAN \cite{dcgan} and WGAN \cite{wgan} have been groundbreaking in terms of how realistic the generated samples were. The adversarial loss originally introduced in \cite{gan} has led to creation of a new kind of architecture in generative modelling and subsequently been applied in several areas such as \cite{pix2pix, inpainting, framepredict}. It consists of a generator and a discriminator, wherein the former learns to generate novel realistic samples in order to fool the latter, while the latter's objective is to distinguish between real samples and generated ones. The combined learning objective is to minimize the adversarial loss.
\textbf{Image-to-Image translation}: Supervised image-to-image translation \cite{pix2pix} has achieved outstanding results where the data used for training is available in one-to-one pairs. Apart from adversarial loss, it uses L1 (reconstruction) loss as well, which has now become a common practice in these types of tasks.
Unsupervised methods take samples of images from two distributions and learn to cross-translate between them. This introduces the well known issue of there being infinitely many mappings between the two unpaired image domains \cite{cyclegan,cogan,unit,xgan,stargan} and so further constraints are required to do well on the problem. \cite{cyclegan} introduces the requirement that translations be cycle-consistent; mapping image $x \in X$ to domain $Y$ and back again to $X$ must yield an image that is close to the original. \cite{cogan} takes a different approach, enforcing weight sharing between the early layers of the generators and later layers of the discriminators. \cite{unit} combines these two ideas and models each image domain using a VAE-GAN. \cite{xgan} utilizes reconstruction loss and teacher loss instead of VAE using a pretrained teacher network to ensure the encoder output lies in a meaningful subregion.
To our knowledge, only \cite{stargan} has presented results in generating translations between multiple distribution samples. However, their generator is conditioned on supervised labels.
\section{Method}
\label{ref-method}
Our work broadly builds on the assumption of shared-latent space \cite{cogan}, which theorizes that we can learn a latent code $z$ that can represent the joint distribution $P(x_1, x_2)$, given samples from marginal distributions $P(x_1)$ and $P(x_2)$. The generator or translator consists of an Encoder $E_i$, shared latent space $z$ and a Decoder $G_i$, such that $z = E_1(x_1)$, $z = E_2(x_2)$ and $x_1 = G_1(z)$, $x_2 = G_2(z)$. The composability property of these translators would be then as follows: $x_2 = G_2 \circ E_1(x_1)$ and vice versa. In other words, we want the translator to learn to map similar characteristics of image samples from two distributions to $z$ and then the decoder should learn to disentangle unique characteristics of that distribution on which it is trained and apply that transformation on any given image sample as input.
We extend this framework to learn composite functions for $|N|$ distributions. To formalize, given sets of samples from distributions $N = \{X_1, X_2, ..., X_{|N|}\}$ with an existing and unknown joint distribution $P(X_1, X_2, ..., X_{|N|}) \neq \phi $, we learn a set of composite functions and a shared latent space, such that $x_j = G_j \circ E_i (x_i)$, where $\{i, j\} \in N$. Thus giving us a total of $|N|^2$ unique transformations possible. To approach solving towards this problem, we start with a bottom-up approach and take $|N| = 4$ sets of sample images.
\subsection{Model architecture}
We extend the model proposed by Liu et. al \cite{unit} to learn to simultaneously translate between two pairs of image distributions (making four distributions in total). There are four encoders, four decoders, and four discriminators in our model, one for each image distribution. Additionally, there is a shared latent space following \cite{unit}, consisting of the last layers of the encoders, and first layers of the decoders. See Figure \ref{fig-model} for more detail. \par
\begin{figure}[h]
\includegraphics[width=12cm, height=6cm]{im2im2im.png}
\centering
\caption{Model architecture. Distributions have been selected from CelebA dataset \cite{celeba} having the following unique properties: smiling, not-smiling, eyeglasses, no-eyeglasses.}
\label{fig-model}
\end{figure}
We felt that sharing a latent space, introduced in \cite{cogan}, and used to great effect in \cite{unit} has applications beyond improving pairwise domain translations, and could improve the composability of image translations. Sharing a latent space implements the assumption that there exists a single latent code $z$ from which images in any of the four domains can be recovered \cite{unit}. If this assumption holds, then complex image translations can be disentangled into simpler image translations which learn to map to and from this shared latent code. \par
\subsection{Objective}
We adapted the objective function from \cite{unit}, and benefit from the extensive tuning that the authors carried out. Since we had access to limited computational resources, we kept the same weightings as \cite{unit} on the individual components of loss function for a single pairing.
There are three components to the objective function for each learned translation, making twelve elements in total.
\begin{align}
&\min_{E_1,E_2,E_3,E_4,G_1,G_2,G_3,G_4}\max_{D_1,D_2,D_3,D_4} \mathcal{L} = \\ &\mathcal{L}_{\text{\tiny VAE}_1}(E_1,G_1) +\mathcal{L}_{\text{\tiny GAN}_1}(E_1,G_1,D_1) +\mathcal{L}_{\text{\tiny CC}_1}(E_1,G_1,E_2,G_2)\nonumber\\
+ &\mathcal{L}_{\text{\tiny VAE}_2}(E_2,G_2) + \mathcal{L}_{\text{\tiny GAN}_2}(E_2,G_2,D_2)+\mathcal{L}_{\text{\tiny CC}_2}(E_2,G_2,E_1,G_1)\nonumber\\
+ &\mathcal{L}_{\text{\tiny VAE}_3}(E_3,G_3) +\mathcal{L}_{\text{\tiny GAN}_3}(E_3,G_3,D_3) +\mathcal{L}_{\text{\tiny CC}_3}(E_3,G_3,E_4,G_4)\nonumber\\
+ &\mathcal{L}_{\text{\tiny VAE}_4}(E_4,G_4) + \mathcal{L}_{\text{\tiny GAN}_4}(E_4,G_4,D_4)+\mathcal{L}_{\text{\tiny CC}_4}(E_4,G_4,E_3,G_3)
\end{align}
The VAE loss objective is responsible for ensuring that the model can reconstruct and image from the same domain. That is,
$$G(E(x)) \approx x$$
The adversarial loss objective is responsible for ensuring that the decoder (or generator $G_i$) generates realistic samples when translating from an image lying in domain $X_1$ into domain $X_2$, which is evaluated by the discriminator ($D_i$). Finally, the cycle-consistency component ensures that when the model translates an image from domain $X_1$ to $X_2$ and back to $X_1$ the resulting image is similar to the original. That is,
$$G_1(E_2(G_2(E_1(x))) \approx x $$
We refer readers to \cite{unit} for a full explanation and motivation of these different elements.
\subsection{Training and Inference}
The model is conceptually split into two, with each part responsible for learning to translate between one pair of distributions. Each of the three loss components; reconstruction, GAN, and cycle-consistency is enforced within the pair.
\begin{enumerate}
\item (E1,E2,G1,G2,D1,D2): Learns $f_1:X_1 \Rightarrow X_2$, and $f_2:X_2 \Rightarrow X_1$
\begin{itemize}
\item $f_1(x) = G_2(E_1(x))$
\item $f_2(x) = G_1(E_2(x))$
\end{itemize}
\item (E3,E4,G3,G4,D3,D4): Learns $f_3:X_3 \Rightarrow X_4$, and $f_4:X_4 \Rightarrow X_3$
\begin{itemize}
\item $f_3(x) = G_4(E_3(x))$
\item $f_4(x) = G_3(E_4(x))$
\end{itemize}
\end{enumerate}
The shared latent space between all of the encoders and generators is responsible for ensuring realistic translations to image distributions the model has not seen before. \par
At inference time we complete a "double-loop" through the model. Suppose we had learned the following translations:
\begin{itemize}
\item $f_1$: glasses to no glasses
\item $f_2$: no glasses to glasses
\item $f_3$: smiling to not smiling
\item $f_4$: not smiling to smiling
\end{itemize}
Then to translate from someone who is not smiling and not wearing glasses to smiling and wearing glasses, we do:
\begin{equation}
\begin{aligned}
& \text{not smiling, no glasses} \Rightarrow \text{smiling, no glasses} \Rightarrow \text{smiling, glasses} \\
& \Leftrightarrow f_2(f_4(x)) \\
& \Leftrightarrow G_2(E_1(G_3(E_4(x))))
\end{aligned}
\end{equation}
Contrary to the above approach, it seems straightforward to think that training this model would be done in a joint manner with the objective of minimizing $\mathcal{L}$. However, as $N \rightarrow \infty$, this method will become unscalable. Hence we present the above given training strategy that splits the shared latent space $z$ and trains $\frac{|N|}{2}$ pairs. It must be noted that at this point in the training, there has been no weight sharing between pair 1 ($X_1$ and $X_2$) and pair 2 ($X_3$ and $X_4$).
In Section \ref{ref:expriments} we see that this method results in generating better quality samples at inference time as compared to joint training from scratch. However, we hypothesized that having a shared latent space would improve the translation quality so experimented with training the models in an uncoupled manner first and then jointly training all the models, sharing a latent space, for a few iterations to fine tune. We found that this approach yielded the best results (see Section \ref{ref:expriments}).
\section{Experiments}
\label{ref:expriments}
We conducted all of our experiments using the celebA dataset \cite{celeba}. This dataset consists of 202,599 images each labeled with 40 binary attributes, for example brown hair, smiling, eyeglasses, beard, and mustache \cite{celeba}. These binary attributes naturally lend themselves to composition, making this an ideal dataset to test our proposed model. We focused on translating between glasses, no glasses, smiling, not smiling (experiment $1$), and blonde hair, brown hair, smiling and not smiling (experiment $2$). For each experiment we constructed four datasets, one corresponding to each image distribution with the relevant characteristic. So that we could test our models for their ability to generate combinations of characteristics that did not appear in the training set, we ensured that there were no faces which were smiling and wearing glasses in experiment $1$, and no faces which were smiling with either blonde or brown hair in experiment $2$. \par
We experimented with the following training approaches.
\begin{itemize}
\item \textbf{Four way}: Training the model described in Section \ref{ref-method} from scratch
\item \textbf{Separately Trained (Baseline)}: Following the method and models architectures from \cite{unit}. To compose the image translation we first passed an image through one model, then another.
\item \textbf{Warm start}: First training separate models. Then initializing the model described in Section \ref{ref-method} with the weights from the separately trained models, and continuing to train to fine tune.
\end{itemize}
The baseline model was intended to help test the role of the shared latent space between all four distributions. If the latent space is helpful, the translations of the four way or warm start model should be better than the baseline model.
High quality translations should have the following characteristics. Realism, variety, and the clear presence of the translated feature, distinct from the pre-translated image. To evaluate our models on these criteria we used three evaluation metrics.
\begin{itemize}
\item \textbf{Realism}: qualitative, manual examination of the generated images
\item \textbf{Variety}: Low cycle consistency loss. The lower this loss, the less likely a model is to have mode collapse. If a model experiences mode collapse and translates all example to only a few images, then it will be unable to reconstruct the original image from the translated image well.
\item \textbf{Presence of translated feature}: We trained a 11-layer VGG \cite{vgg} net using original images from the dataset to classify examples into four classes, one for each possible combination of features for each experiment. Then we selected a batch of 100 original images from a single class (e.g. blonde and not smiling, eyeglasses and not smiling), translated them to every other class using our model, and classified them after every translation. If a model is making clear translations, the class they are classified into should change with each translation. To mitigate the fact that our classifier was imperfect, we excluded any images that the classifier was not able to classify correctly.
\end{itemize}
\section{Results}
\textbf{Realism}: Overall the warm start model described in Section \ref{ref:expriments} generated the most visually appealing and coherent double translations (see Figures \ref{fig-eye-smile} and \ref{fig-hair-smile}). The presence of the translated features are clear and generally integrated in a coherent way, with minimal distortions or artifacts. The model is able to successfully handle atypical translations, such as adding glasses when one eye is occluded (see Figure \ref{fig-eye-smile}). The warm start model is significantly better than a joint model trained from scratch. This is clear from Figure \ref{fig-eye-smile}, and we were not able to successfully train a joint model from scratch for experiment 2, which exhibited results at par with the other methods. Interestingly the separately trained models generated reasonably good double translations, particularly in experiment two (Figure \ref{fig-hair-smile}), and was significantly better than a joint model trained from scratch. This suggests that unpaired image to image translation already exhibits some composablility. However, enforcing a shared latent space and fine tuning these models (the warm start training approach) does seem to improve the overall quality of images. This is particularly apparent in the results from experiment 1 (Figure \ref{fig-eye-smile}). These results suggest that the more scalable decoupled training strategy in which $\frac{|N|}{2}$ pairs are trained separately, then fine-tuned through joint training, is also the approach which yields the highest quality results. Though we weren't able to experiment on more than 4 distributions due to time and resource constraints. \par
Finally, what is particularly exciting about these results is that our best model has no problem generating images with combinations of characteristics that never appeared in the training set. In experiment 1, there are no pictures of people wearing glasses and smiling, and yet the model generates high quality images of people smiling and wearing glasses (see right most image in the triplets in Figure \ref{fig-eye-smile}). Similarly for experiment 2 there was no one with either brown or blonde hair that was smiling in the training data.
\begin{figure}[h]
\includegraphics[width=14cm]{eye_smile.png}
\centering
\caption{Selected results from experiment 1 for all three models. For each triplet of images, the image on the left is the original image, selected from the celebA dataset \cite{celeba}. They are all not smiling and not wearing glasses. The center image is the translation to not smiling and wearing glasses. The image on the right is the second translation to smiling and wearing glasses.}
\label{fig-eye-smile}
\end{figure}
\begin{figure}[h]
\includegraphics[width=10cm]{hair_smile.png}
\centering
\caption{Selected results from experiment 2 for the warm start and baseline models. For each triplet of images, the image on the left is the original image, selected from the celebA dataset \cite{celeba}. They are all not smiling and have either blonde or brown hair. The center image is the translation to not smiling and either blonde or brunette, depending on the original hair color. The image on the right is the second translation to smiling.}
\label{fig-hair-smile}
\end{figure}
\textbf{Variety}: Generally, our models were able the reconstruct the original image from the translated image well, suggesting they did not suffer from mode collapse. This is also consistent with what we observed by inspecting the generated images.\par
\textbf{Presence of translated features} (Quantitative Analysis): Figure \ref{ref:clf1} shows our assessment of translation quality. VGG classifier trained on the classes: blonde \& not smiling, brunette \& not smiling, blonde \& smiling, brunette \& smiling with 87\% accuracy is able to separate the translated images into their respective classes very efficiently for the baseline model. For the warmstarted model, the classifier gets somewhat confused between smiling and not smiling images. This makes us think how finetuning the model is distorting generated samples as to give a mixed classification decision. We leave this for future work.
\begin{figure}[h]
\includegraphics[width=14cm]{clf1.png}
\centering
\caption{A batch of 100 blonde \& not smiling images are classified and then translated of which again the correctly classified ones are translated again and so on. Some samples from the batch are displayed here. The label map is as follows- 0: Blonde \& Not Smiling, 1: Brunette \& Not Smiling, 2: Blonde \& Smiling, 3: Brunette \& Smiling.}
\label{ref:clf1}
\end{figure}
\section{Further Work}
The joint models we trained sometimes dropped one of the translation modes, most noticeably translating from not smiling to smiling. We hypothesize that this was because this translation was the most difficult of the four translations. This could potentially be remedied by increasing the contribution to the loss function from this translation. More generally it would be interesting to explore the effect of varying the contribution from the many different loss components more fully. Time and computational resource constraints prevented us from doing this.\par
We constructed four images domains from a single more general domain, celebrity faces \cite{celeba}. This ensured that the domains were fundamentally related. It would be interesting to explore the degree of relatedness between different image domains required to achieve good results. For example, given outdoor scenes labeled with the weather (e.g. snow, sun, rain), and outdoor scenes containing different animals (e.g. horse zebra), could we learn to translate between horses in sunshine to zebras in snow?
\section{Conclusion}
In this work we extend a given model of unpaired image to image translation for handling multiple pairs of distributions. We devise scalable training methods with modified architecture and objective for this kind of model and compare the results of the model through each of these methods. We set qualitative and quantitative evaluation criterion and assess how performance of our model in various training scenarios. Moreover, we show the translation flexibility property of our model by using the translators as stacked composable functions for multi-way translation into novel distributions.
\section*{Acknowledgments}
We are grateful to M. Liu, T. Breuel, and J. Kautz for making their research and codebase publicly available, and to Professor Rob Fergus for his valuable advice.
|
2,869,038,154,856 | arxiv | \section{Introduction}
The interplay between dynamical chiral symmetry breaking and color
confinement in a hot/dense medium has not been sufficiently understood,
and remains one of the central subjects in QCD~\cite{review,qm:fuku}.
The chiral symmetry breaking and its restoration are well characterized
by the quark-antiquark (chiral) condensate, whereas no reliable order
parameter for the confinement-deconfinement phase transition is known.
The Polyakov-loop expectation value, which plays the role of the order
parameter in pure Yang-Mills (YM) theory, is disturbed seriously by
dynamical quarks. Hence, even though the expectation value exhibits an
inflection point at a certain temperature, it is not manifest that
the system undergoes a transition from hadrons to quarks and gluons.
A constructive way to identify the deconfined phase is to explore
various fluctuations associated with conserved charges.
In particular, the kurtosis of net-quark number fluctuations measures
clearly the onset of deconfinement~\cite{R42}.
Recently, other fluctuations more addressing the gluon sector have been
calculated in lattice gauge theory with light quarks~\cite{Lo1,Lo2}, where
two ratios, $R_T = \chi_I/\chi_R$ and $R_A = \chi_A/\chi_R$, are considered
in terms of the susceptibilities associated with the modulus, real and
imaginary parts of the Polyakov loop.
Asymptotic values of those ratios are properly quantified within a
$Z(3)$-symmetric model when there are no dynamical quarks (see
Fig.~\ref{RA}). Once the light flavored quarks are introduced, the $R_T$
becomes much broadened, similarly to the Polyakov loop expectation value.
On the other hand, the $R_A$ retains the underlying center symmetry
fairly well even in full QCD with the physical pion mass. Also, ambiguities
of the renormalization prescription can be avoided to large extent in
the ratio. The $R_A$ thus serves as a better pseudo-order parameter
than the Polyakov loop by itself.
\begin{figure}
\begin{center}
\includegraphics[width = 8cm]{qcd_model_R_T}
\includegraphics[width = 8cm]{qcd_model_R_A}
\caption{
Lattice results of the ratios of the Polyakov loop susceptibilities,
$R_T = \chi_I/\chi_R$ and $R_A = \chi_A/\chi_R$ for pure YM and
$N_f=2+1$ QCD at vanishing chemical potential~\cite{Lo2}.
The temperature is normalized by the critical temperature in pure YM
theory, and by the pseudo-critical temperature for the chiral symmetry
restoration in full QCD.
}
\label{RA}
\end{center}
\end{figure}
In Fig.~\ref{qf}, the $R_A$ is compared with the kurtosis of the quark
number fluctuations. The quark liberation takes place evidently together
with a qualitative changeover in $R_A$. Those abrupt changes in the
Polyakov loop and quark number fluctuations appear in a narrow range
of temperature lying on the pseudo-critical temperature of chiral symmetry
restoration. Therefore, at vanishing chemical potential,
$T_{\rm deconf} \simeq T_{\rm chiral}$ is concluded.
\begin{figure}
\begin{center}
\includegraphics[width = 12cm]{Prezentacja1}
\caption{
The ratio of the Polyakov loop susceptibilities $R_A=\chi_A/\chi_R$
and the kurtosis of net quark number fluctuations. Lattice data points
are taken from~\cite{Lo2,lat}.
}
\label{qf}
\end{center}
\end{figure}
{}From the field theoretical point of view, it remains incomplete to
capture the interplay of such non-perturbative dynamics in a form of
an effective theory. In this contribution, we will briefly review recent
progress in QCD thermodynamics and address the issues to be disentangled.
\section{Low-lying Dirac eigenmodes and confinement}
Spontaneous chiral symmetry breaking is locked to a non-vanishing
spectral density with the zero eigenvalues of the Dirac operator,
known as the Banks-Casher relation~\cite{BC}. An intriguing question
is whether confinement would also be lost if the Dirac zero modes are
artificially removed from a system.
In~\cite{Gattringer,Bruckmann,Synatschke,Gongyo,Doi}, a relation of
the Dirac eigenmodes to the Polyakov loop has been formulated on a lattice,
and their dynamical correlations have been investigated in SU(3) lattice
gauge theory. The expression in a gauge-invariant formalism is found
as~\cite{Gongyo,Doi}
\begin{equation}
\langle L \rangle
= \frac{(2i)^{N_\tau-1}}{12V}\sum_n \lambda_n^{N_\tau-1}
\langle n | \hat{U}_4 | n \rangle\,.
\label{eq:poly}
\end{equation}
Those simulations revealed that there are no particular
modes which crucially affect confinement. In fact, the string tension
extracted from the potential between static quarks is unchanged even
when the low-lying Dirac modes are eliminated, as shown in
Fig.~\ref{dirac:pol}. This apparently indicates that the disappearance
of the chiral symmetry breaking does not dictate deconfinement of quarks.
\begin{figure}
\begin{center}
\includegraphics[width = 10cm]{Fig5}
\caption{
Static quark potential in SU(3) lattice gauge theory~\cite{Gongyo}.
The points with filled circles (squares) represent the results
with (without) removal of the low-lying Dirac modes.
}
\label{dirac:pol}
\end{center}
\end{figure}
The derived analytic relation (\ref{eq:poly}) tells manifestly that the
Dirac zero modes has no role in the Polyakov loop. However, the matrix
elements of the link variable must be affected by the light quarks, so that
the Wilson loop and the quark potential may be significantly modified.
The simulations carried out so far are limited to the quenched theories.
Therefore, a conclusive statement should be postponed until computations
with the dynamical quarks are made in future.
A conventional picture for color confinement is based on the dual
superconductor~\cite{mono1,mono2}. With a particular gauge fixing for
the QCD Lagrangian, so-called Maximum Abelian Gauge~\cite{MAG1,MAG2},
magnetic monopoles naturally emerge and get condensed. As a result,
a linear confinement potential and dynamical chiral symmetry breaking
are generated. In order to accommodate the relation between the Polyakov
loop and Dirac zero eigenmodes into this scenario, one needs to somehow
link the Dirac eigenmodes to (a part of) the monopoles.
\section{Hadrons near chiral symmetry restoration}
Another implication of the chiral symmetry restoration with confinement
is found in the hadron mass spectra with a systematic removal of
the low-lying Dirac modes on a lattice~\cite{GLS}. The masses of several
baryons and mesons with positive and negative parity are summarized
in Fig.~\ref{dirac:mass}.
\begin{figure}
\begin{center}
\includegraphics[width = 12cm]{fitted_masses_baryons_ratios}
\includegraphics[width = 12cm]{fitted_masses_mesons_ratios}
\caption{
Masses of baryons (upper) and mesons (lower) in unit of the $\rho$
meson mass as a function of the truncation level~\cite{GLS}.
}
\label{dirac:mass}
\end{center}
\end{figure}
As increasing the truncation level of the Dirac modes, the masses of parity
partners approach and eventually become degenerate. What is remarkable is
that those hadrons remain quite massive, around 1 GeV for the lowest nucleon
and $m_\rho$ for the lowest vector meson. Furthermore, universal scaling
--- $2m$ for mesons and $3m$ for baryons --- is not observed.
Hence, it is considerably suggestive that those hadrons keep their particle
identities and survive in the chiral restored phase.
\begin{figure}
\begin{center}
\includegraphics[width = 10cm]{PLRS}
\caption{
The ratio of the in-medium nucleon mass to its vacuum value
for the density-dependent omega-nucleon coupling $g_{V_\omega}$~\cite{PLRS}.
}
\label{dilaton}
\end{center}
\end{figure}
Given the lattice observation that $T_{\rm deconf} \simeq T_{\rm chiral}$
at vanishing chemical potential, the chiral restored phase with confinement
might appear at high density. A large hadron mass needs to be saturated by
certain condensates of chirally even operators. A good candidate is
gluon condensates. Not only in matter-free space but also in a medium,
the QCD trace anomaly exists and this is accompanied by a non-vanishing
expectation value of a dilaton field, which is identified with a scalar
glueball~\cite{Schechter}. The in-medium gluon dynamics in the context
of scale symmetry breaking is accommodated in a chiral effective field
theory. In~\cite{PLRS}, the $\rho$ and $\omega$ mesons are shown to interact
with a nucleon differently: the $\rho NN$ coupling runs, whereas the
$\omega NN$ coupling walks in density. An immediate consequence is that
the in-medium nucleon mass reaches a constant around the saturation
density, and stays in higher density, as given in Fig.~\ref{dilaton}.
Note that the original Lagrangian does not have an explicit ``bare'' mass.
Nevertheless, due to the dynamics in dense matter, a chirally-invariant
mass for the nucleon emerges. The above density dependences encoded in
the renormalization group equations are governed by a non-trivial IR fixed
point, dilaton-limit fixed point~\cite{DL1,DL2}. These features remind us of
the modern technicolor models for the Higgs physics beyond the Standard
Model~\cite{techni}.
Higher-dimension operators can also be condensed and contribute to the
hadronic quantities in dense matter. In particular, tetra-quark states
play a crucial role to construct reliable equations of state
for nuclear matter~\cite{Gallas} and near the chiral phase transition via
a mixing to a bilinear quark condensate~\cite{Heinz}.
Also, a novel phase with chiral symmetry breaking on top of the vanishing
chiral condensate is an interesting theoretical option at finite
density~\cite{HST}, where the tetra-quark condensate saturates the pion
decay constant and could yield more critical point(s) in the phase diagram.
\section{Role of higher-lying hadrons}
The in-medium vector spectrum $\rho_V$ is more or less established both
in theory and in dilepton measurements~\cite{RWvH}. Yet, it has not been
clarified how the observed modifications are linked to the (partial) chiral
symmetry restoration. Instead of measuring the in-medium axial-vector
spectrum $\rho_A$ in experiments, which is hopeless, a method to construct
$\rho_A$ using a phenomenologically accepted $\rho_V$ via QCD and Weinberg
sum rules has been proposed~\cite{HR}.
Fig.~\ref{va:t} summarizes the obtained thermal evolution of the spectra.
\begin{figure}
\begin{center}
\includegraphics[width = 14cm]{FTSFs}
\caption{
Vector and axial-vector spectral functions at
various temperatures~\cite{HR}.
}
\label{va:t}
\end{center}
\end{figure}
The $a_1$ meson mass smoothly approaches the $\rho$ mass, and the two
spectral functions become almost on top at a high temperature, indicating
chiral restoration. Not only the lowest vector states, $\rho$ and $a_1$,
but also the second lowest states, $\rho^\prime$ and $a_1^\prime$, are
shown to contribute to the $\rho_{V,A}$ rather significantly as approaching
the restoration temperature. It is intuitively understood since
more hadronic states must be populated toward the chiral phase
transition that cannot be achieved within any conventional perturbative
treatment including just a few numbers of mesonic states.
At non-vanishing chemical potential, it is more involved since charge
conjugation invariance is lost, which leads to a mixing between transverse
$\rho$ and $a_1$ states at tree level:
\begin{equation}
{\mathcal L}_{\rm mix}
= 2C\epsilon^{0\nu\lambda\sigma}
\text{tr}\left[
\partial_\nu V_\lambda\cdot A_\sigma
{}+ \partial_\nu A_\lambda\cdot V_\sigma
\right]\,.
\end{equation}
Their dispersion relations are
modified and the spectral functions do not follow a simple Breit-Wigner
distribution~\cite{HS} (see Fig.~\ref{va:mu}).
Its relevance on observables crucially relies on a mixing strength $C$
which intrinsically depends on density. There are two available numbers:
$C = 1$ GeV at the saturation density $n_0$ from an AdS/QCD model~\cite{DH},
and $C = 0.1$ GeV at $n_0$ from the gauged Wess-Zumino-Witten action in
four dimensions as well as a mean field approximation~\cite{HS}. The former
yields vector meson condensation slightly above $n_0$, which is odd. Therefore,
this is most likely an artifact of the large $N_c$ approximation employed
for the gauge/gravity duality conjecture.
One conceivable reason for this huge difference in C's is that all the
Kaluza-Klein (KK) modes, corresponding to all the vector mesons, contribute
to the dynamics in holographic QCD models. Heavier states can be integrated
out, whereas a naive truncation may provide a different result since
truncated modes carrying the information about the underlying physics are
artificially omitted. In fact, the nuclear potential as a function of
a distance $r$ exhibits a $1/r^2$ dependence when all the KK modes are
considered~\cite{HSS}, while it follows a $1/r$ behavior when a truncation
is made.
\begin{figure}
\begin{center}
\includegraphics[width = 10cm]{spc_n0_lowp}
\caption{
Vector spectral function for $C=1$ GeV~\cite{HS}.
}
\label{va:mu}
\end{center}
\end{figure}
As increasing temperature and density, more hadrons are activated
and eventually change the ground state. It is not straightforward to
deal with many (or all the) hadrons. In holographic QCD approach,
although infinite KK modes are naturally accommodated, a systematic
technique to include $1/N_c$ corrections in a medium is not established yet.
In the standard effective theories in four dimensions, interactions with
the higher-lying hadrons are not completely known. Integrating the heavier
modes out at finite temperature and density is not an easy task either. Those
effective interactions near the phase transition may be to some extent
captured by use of more microscopic computations, e.g. lattice simulations,
Dyson-Schwinger equations and functional approach~\cite{qm:frg}.
\section{Conclusions}
Various fluctuations of conserved changes~\cite{qm:lat} as well as
the ratio of the Polyakov-loop susceptibilities $R_A$ in lattice QCD
consistently indicate that deconfinement takes place in the chiral
crossover region at vanishing chemical potential. Although the kurtosis of
net quark number fluctuations is reasonably quantified in a class of
chiral models with the Polyakov loop~\cite{pnjl}, modifications in
$R_A$ and $R_T$ by the dynamical quarks cannot be explained in the same
framework. Since those models do not posses the dynamical mechanism for
quark confinement, the properties of gluon-oriented quantities are
supposed to be less captured. An effective theory that can better handle
the confinement nature of non-abelian gauge theories is indispensable
to reveal the Polyakov loop fluctuations in the presence of light quarks.
Also, it is vital to bridge the gap between the Dirac zero eigenmodes,
responsible for dynamical chiral symmetry breaking, and the magnetic monopoles.
The role of higher-lying hadrons has been found in low-energy constants and
spectral functions near the QCD phase transition. On a practical level,
it is not yet established to fully accommodate them to effective theories.
Several attempts constrained by the relevant global symmetries lead to
certain non-trivial medium effects. More elaborated and systematic prescription
certainly requires a novel scheme. Functional approaches in terms of quarks
and gluons may provide some benefits in this context.
Heavy-light hadrons, such as charmed mesons, are also good probes for
the quark-gluon dynamics. In dilute nuclear matter, in-medium modifications
of the color-electric and color-magnetic gluons are extracted from the
D and B meson dynamics~\cite{YS}. In increasing density/temperature, those
heavy-light mesons will change their chiral properties, as expected from
the chiral doubling scenario~\cite{cd1,cd2,cd3}. The mass gap between the
chiral partners is around 350 MeV, and this is much bigger than a mass
difference between the charged D mesons in dense matter, $\sim 50$ MeV.
Further theoretical investigations, along with the lattice input~\cite{Burger},
will supply more reliable understanding of heavy-flavor transport properties.
\begin{figure}
\begin{center}
\includegraphics[width = 10cm]{fig2_r}
\caption{
The chiral susceptibility calculated in lattice QCD
for $N_f=2$ and $N_f=2+1+1$~\cite{Burger}.
}
\label{charm}
\end{center}
\end{figure}
\section*{Acknowledgments}
I am grateful for fruitful discussions and correspondence with
K.~Redlich, H.~Suganuma and S.~Sugimoto.
I acknowledge partial support by the Hessian
LOEWE initiative through the Helmholtz International
Center for FAIR (HIC for FAIR), and
by the Polish Science Foundation (NCN) under
Maestro grant 2013/10/A/ST2/00106.
|
2,869,038,154,857 | arxiv | \section{Introduction and main result}
Let $\BC_+=\{z\in\BC: \Im z>0\}$ be the upper half-plane in the
complex plane $\BC$.
We recall that the
classical Hardy space $H^p(\BC_+)$ consists of analytic functions
$f$ in $\BC_+$ such that $\|f\| \defn \Big(
\sup_{y>0}\int_{\BR}|f(x+iy)|^p\,dx \Big)^{1/p} $ is finite.
It is a Banach space for any $p$ as above.
The space $H^\infty(\BC_+)$ is defined as the Banach space
of bounded analytic functions in $\BC_+$.
We refer to the book \cite{Dur} for an account of the
theory of $H^p$ spaces of the upper half-plane and of the unit disc.
Functions in $H^p(\BC_+)$ have non-tangential boundary limit
values on $\BR$, which permits us to identify $H^p(\BC_+)$ with a
closed subspace of $L^p(\BR)$.
We put $H^p=H^p(\BC_+)$, $1\le p\le \infty$.
For any function space $\Psi$, we denote by
$\Psi_\BR$ the set of its real elements and by
$\Psi_r$, $\Psi_{r\times r}$, respectively,
the spaces of $r\times 1$ vector-valued functions and of
$r\times r$ matrix-valued functions with entries in $\Psi$.
If $\CA$ is a scalar or matrix functional algebra, we denote by
$\CG\CA$ the set of all its invertible elements.
Let the natural number $r$ be fixed and let $G\in L^\infty_{r\times r}(\BR)$.
The vector Toeplitz operator $T_G$ with the symbol $G$
acts on the vector Hardy space $\Htwor$ by the formula
\beqn
\label{TG}
T_Gx=P_+\big(G\cdot x\big), \quad x\in\Htwor,
\neqn
here $P_+$ is the orthogonal projection of
$\Ltwor$ onto its closed subspace $\Htwor$.
A (bounded linear) operator $K$ on a Banach space $B$ is called
normally solvable \cite{GK}, \cite{MikhPr} if its image is closed.
$K$ is called a $\Phi_+$ operator (a $\Phi_-$ operator) if it is
normally solvable and $\dim\Ker K<\infty$ ( $\dim\Coker K=\dim B /
\Range K<\infty$, respectively). We denote by $\Phi_\pm(B)$ these
classes of operators on $B$. Operators in $\Phi_+$ and $\Phi_-$
are called semi-Fredholm. Operators in $\Phi(B)=\Phi_-(B)\cap
\Phi_+(B)$ are called Fredholm.
The index of a semi-Fredholm operator is defined by
$$
\Ind K=\dim\Ker K -\dim\Coker K;
$$
its values are integers or $\pm\infty$.
A semi-Fredholm operator is Fredholm if and only if its index is finite.
Fredholm and semi-Fredholm operators have several important
properties. For instance, the product of two $\Phi_\pm$ operators
is again a $\Phi_\pm$ operator, and the formula
$\Ind(K_1K_2)=\Ind(K_1)+\Ind(K_2)$ holds for $K_1$, $K_2$ in
$\Phi_+$ or in $\Phi_-$. We refer to \cite{GK}, \cite{MikhPr} for
detailed expositions of the theory of these classes and for
applications.
We put $\Cb=\Cb(\BR)$ to be the Banach space of all continuous
bounded functions on $\BR$ with the supremum norm. Our
paper is devoted to finding necessary conditions for
semi-Fredholmness and Fredholmness of $T_G$ for the case when $G$
is an $r\times r$ matrix function whose entries are in $\Cb$. Such
questions appear naturally in connection with the Riemann--Hilbert
problem on the real line. This problem appears in many different
situations, such as various problems in mechanics of continuous
media and hydrodynamics \cite{Begehr}, \cite{MonSem},
\cite{AntPopYats}, \cite{Beg_Dai}, \cite{ShargTol}, inverse
scattering method for integrable equations \cite{AblCla}, linear
control theory of systems with delays \cite{CallWi}, convolution
equations and systems on finite intervals (see \cite{BKSpi},
\cite{KSpi_Semifr}) and others. The case of infinite index often
appears in these applications.
First we quote the following well-known result.
\begin{thmA} [see \cite{BSlb}]
The condition $\det G\ge\eps>0$ is necessary for $T_G$ to be
semi-Fredholm.
\end{thmA}
We will always assume this condition to be fulfilled.
For a function $G\in \Cbrr$ which has limits
at $\pm\infty$,
$T_G$ is semi-Fredholm iff it is Fredholm, and
a complete criterion for it is known
(see \cite{BSlb} or \cite{BKSpi}).
In a particular case, when
$G(-\infty)=G(+\infty)$,
$T_G$ is Fredholm if and only if $|\det G|\ge\eps>0$ on
$\BR$, and
\beqn
\label{ind-wind}
\ind T_G=-\wind\det G,
\neqn
where $\wind $ stands for the winding number
(around the origin).
So our main concern is about symbols that
have no limits either at $-\infty$ or at $+\infty$.
Let $\BMO=\BMO_\BR$ be the space of real-valued functions on $\BR$
of bounded mean oscillation. We recall that $\BMO$
consists of those locally integrable functions $f$ on $\BR$ that satisfy
\beqn
\label{defBMO}
\sup_J\frac 1 {|J|}\int \big|f-f_J \big|\le C,
\neqn
where the supremum is taken over all finite subintervals $J$ of
the real line and $f_J=\frac 1 {|J|}\int_J f$ is the mean of $f$
on the interval $J$. It is known \cite{GR} that
if there exist
a constant $C$ and
arbitrary real numbers $f_J$ such that
(\ref{defBMO}) holds for any finite interval
$J$ in $\BR$, then $f$ belongs to $\BMOR$.
We refer to \cite{GR} for an exposition of the theory of these
spaces.
Let $C_+(\BR)$ be the class of real continuous
(nonstrictly) increasing functions on $\BR$, and put
\beqnay
\BMOp= \big\{ u+v: \quad u\in\BMOR, \,v\in C_+(\BR) \big\},
\nonumber \\
\BMOm=
\big\{
u-v: \quad u\in\BMOR, \, v\in C_+(\BR)
\big\}. \nonumber
\neqnay
The main result of \S2 is as follows.
\begin{thm1}
{\it Suppose that $G\in \Cbrr$.
(1) If $T_G\in\Phi_\pm(\Htwor)$, then
$\arg \det
G\in\BMO^\pm_\BR$.
(2) If $T_G\in\Phi(\Htwor)$, then $\arg \det G\in\BMOR$. }
\end{thm1}
In \S3, we introduce a system of mean winding numbers of
$\det G$ and formulate and prove
Theorems 2 and 3 (they will follow from Theorem 1). In
\S4, we discuss some unresolved questions, related with our results.
Our principal motivation comes from the control theory. In a
problem about the complete controllability of delay equations
it turned out to be necessary to estimate the number
\beqn
\inf\, \Big\{
\tau\in \BR: \quad T_{e^{-i\tau x}G(x)} \text{ is onto}
\Big\}\,\defn\,\beta(G)
\label{betaG}
\neqn
in terms of some computable characteristics of a matrix function $G\in \GCbrr(\BR)$.
The number $\beta(G)$ has a meaning
of the least time of complete controllability.
Theorems 2 and 3 permitted us to give
a good estimate of this number.
These results were obtained
jointly by the author and Sjoerd Lunel
and will be published elsewhere.
A great part of the recent book \cite{DybGru} by Dybin and Grudsky
treats scalar and matrix functions that are continuous on the real
line. This book summarized
(and generalized) earlier work by these authors.
Several novel
tools are used, such as the notion of a $u$-periodic function,
where $u$ is an inner function on $\BC_+$, continuous on the real
line. Another tools are a construction of an inner function whose
argument models an arbitrarily given increasing continuous
function and the notion of a generalized factorization with
infinite index. These hard analysis
tools permitted the authors to give a sufficient condition for
semi-fredholmness (see \cite{DybGru}, Theorem 5.10. By applying
this result, Dybin and Grudsky
get complete answers in cases of whirls at
$\pm \infty$ with different asymptotic, such as power, logarithmic
or exponential.
Earlier work on whirled symbols include the works by Govorov
\cite{Gov}, Ostrovsky \cite{Ostrovs}, Monakhov, Semenko (see the
book \cite{MonSem}) and others; the approach of these authors is
based on the theory of analytic functions of completely regular
growth. In various works, the behavior of the property of
Fredholmness under an orientation preserving homeomorphism of
$\BR$ have been studied, see \cite{BGRa}, \cite{DybGru},
\cite{BGSpi2} and others.
Various mean winding numbers were introduced in the work
by Sarason \cite{Sar} for symbols in $\QC$ and by Power \cite{Pow}
for slowly oscillating symbols.
For symbols of these classes, these
mean winding numbers allow one to
formulate nice complete criteria for a Toeplitz operator
to be Fredholm or semi-Fredholm. We remark that a wider $C^*$-algebra of
slowly oscillating functions was considered in a recent paper
by Sarason \cite{Sar3}, where the maximal ideal space
of this algebra was studied.
Necessary and sufficient conditions for a
Toeplitz operator to be Fredholm and semi-Fredholm are also known
if $G$ belongs to various algebras of symbols. For instance,
classes $\PC$ of piecewise continuous symbols, $\QC=L^\infty\cap
\VMO$ of quasicontinuous symbols, and $\PQC=\alg(\PC,\QC)$ have
been studied both in scalar and matrix case.
Another well-studied cases are that of almost periodic and
semi-almost periodic symbols. For matrix symbols of these types, a
great breakthrough has been done recently by B\"ottcher, Karlovich
and Spitkovsky, see \cite{BKSpi}. Among other things,
generalizations of the index formula (\ref{ind-wind}) are known
for these cases (see \cite{BSlb}, \cite{Nik}, \cite{NikOFS}).
We refer to \cite{BKSpi2003} for an alternative approach.
In \cite{Abr, BasFernKarl, BGRa},
other classes of symbols are studied. In \cite{BKS04},
a Fredholm criterion and an index formula are given for vector Toeplitz
operators, whose (matrix) symbols belong to the Banach algebra, generated
by semi-almost periodic matrix functions and slowly
oscillating matrix functions.
See \cite{LSpi} for a connection
with the factorization and the Riemann--Hilbert problem.
For symbols in $\Cbrr(\BR)$ with no other assumptions, our
knowledge is much less complete. We refer to Subsections 2.26 and
4.73 in \cite{BSlb} and to \cite{BG} for several relevant results.
The criterion for surjectivity of a Toeplitz operator
with a nontrivial kernel, given in \cite{HrSarSeip}, can also be
reformulated as a criterion for a Toeplitz operator to belong to
$\Phi_+\sm\Phi$.
Some additional comments will be given at the end of the article.
We refer to \cite{LSpi}, \cite{Nik}, \cite{NikOFS}, \cite{BKSpi} for
systematic expositions of the spectral theory of Toeplitz operators.
It is worth to note that recently, Toeplitz operators with symbols
like ours have been appeared in papers by Baranov, Havin, Makarov,
Mashreghi, Poltoratsky and others in relation with the
Beurling--Malliavin theorem, bases in de Branges spaces and
related topics (see \cite{HavMashr1-2}, \cite{MakPolt2},
\cite{BarHav} and references therein). It seems that the ideas and
methods of these papers can be applied to achieve a better
understanding of semi-Fredholm Toeplitz operators with continuous
symbols at least in the case of scalar symbol $G$.
\textbf{Acknowledgements.} The author expresses his gratitude to
M. Gamal and I. Spitkovsky for valuable comments.
\section{Proof of Theorem 1}
\setcounter{equation}{0}
First we need some facts and definitions.
Let $0<\al<1$. We put
$$
\Lloc=\{f\in \Cb(\BR):f|J\in \Lal(J)\quad \forall J\}; \nonumber
$$
here $J$ runs over all compact intervals in $\BR$ and
$\Lal(J)$ is the H\"older--Lipschitz class on $J$ with the exponent $\al$.
Next, we will need the classes
\begin{align}
C_a&=\{f\in C(\clos \BC_+):f|\BC_+\in H^\infty\}, \nonumber \\
\Aloc&=\{f\in C_a:f|\BR\in \Lloc\}. \nonumber
\end{align}
A function $f$ in $\Llocrr$ or in $\Alocrr$ is invertible
if and only if $|\det f|>\eps>0$ on $\BR$ (or on
$\clos\BC_+$, respectively).
Recall that a function $g$ in $\Hnty$ is called {\it inner} if its
modulus is equal to one a.e. on $\BR\,$. The function $g$ is
called {\it outer} if it has a form $g(z)=\exp(u(z)+iv(z))$,
$$
(u+iv)(z)=\frac1{\pi i}\int_{-\infty}^{\infty} \bigg[
\frac1{t-z}-\frac t{1+t^2} \bigg] \log
k(t)\,dt+is,
$$
where $k>0$ a.e. on $\BR$, $\log k\in L^1(\BR)$, and $s$ is a
real constant. We assume $u$ and $v$ to be real-valued and harmonic in $\BC_+$.
These functions have boundary limit values a.e. on
$\BR$, which satisfy $u|\BR=\log k$ a.e. and $v|\BR=
\CH(u|\BR)$, where $\cal H$ is the Hilbert transform on $\BR$.
Each function $g$ in $\GHnty$ is outer; in this case
$\log k\in L^\infty(\BR)$. We refer to
\cite{Dur}, \cite{GR} for all these (classical) facts.
For each function $g$ in $\GHnty$, $\arg g(z)=s+v(z)$ is
well-defined on $\BC_+$ (up to an additive constant
$2\pi n$). We also see that the function $\arg g(z)$ has boundary
limit values a.e. on $\BR$, which will be denoted as $\arg g(x)$,
$x\in\BR$.
\nin{\bf Definition.} \enspace We define the class
$\Hntys$ as the set of functions
$f\in\Hnty$ that have the form
$$f=g\cdot h,
$$
where $g\in\GHnty$ and $h$ is inner in $\BC_+$ and has a
continuous extension to $\BR$.
A function $h$ is inner of the above type if and only if
it has the form
\beqn
\label{inner cont on R}
h(z)=Ce^{iaz}
\prod_j
\frac
{|z_j^2+1|} {z_j^2+1} \,
\frac {z-z_j} {z-\bar z_j}, \qquad
z\in\BC_+,
\neqn
where $|C|=1$, $a>0$, and $z_j\in\BC_+$, $|z_j|\to\infty$. Take
any positive continuous function $y=\psi(x)$ on $\BR$ such that
the subgraph $\Ga_\psi= \{(x+iy):\, 0<y<\psi(x)\}\subset\BC_+$
does not contain the zeros $z_j$ of $h$. Then $\arg h(z)$ is
well-defined and continuous on $\Ga_\psi\cup\BR$.
\nin{\bf Definition.} \enspace
Let $f\in\Hntys$, and let $g$, $h$, $\Ga_\psi$ be as above.
We define the argument $\arg f$ on $\Ga_\psi\cup \BR$ by
$$
\arg f=\arg g+\arg h.
$$
\nin So for $f\in\Hntys$, the argument $\arg f$ is well-defined on
$\Ga_\psi$ (up to adding $2\pi n$, $n\in \BZ$). It is continuous
on $\Ga_\psi$ and its values on $\BR$
exist almost everywhere in the sense of nontangential limits.
\begin{pro1}
For any $f\in\Hntys$, $\arg f\in\BMOp$.
\end{pro1}
\begin{proof}
For any $f=g\cdot h\in\Hntys$ as above, $\arg g\in\BMOR$ and $\arg
h$ is a continuous increasing function.
\end{proof}
\begin{lem}
\label{lem Hntys}
Let $f\in H^\infty$. Then $f\in H^\infty_*$ if and only if there is a
positive function $\psi\in C(\BR)$ and some $\eps>0$
such that $|f|>\eps$ on the subgraph $\Ga_\psi$.
\end{lem}
\begin{proof}
If $f\in H^\infty_*$, then it is clear that $f$ satisfies the above property.
Conversely, suppose $|f|>\eps>0$ on $\Ga_\psi$, for a certain
positive function $\psi\in C(\BR)$. Let $f=h\cdot g$ be the inner - outer
factorization of $f$, then $g\in {\cal G}H^\infty$. It follows that
the inner function $h=f/g$ satisfies an inequality $|h|>\eps_1>0$
on $\Ga_\psi$, and consequently, it has a form (\ref{inner cont on R}), see
\cite[Chapter 3]{Nik}.
\end{proof}
In many works on Toeplitz operators, the
unit disc $\BD=\{z\in\BC: |z|<1\}$ instead of
the upper half-plane $\BC_+$ is considered. If $G\in
L^\infty_{r\times r}(\BT)$, where $\BT=\partial \BD$ is the unit
circle, then the same formula
(\ref{TG}) defines a Toeplitz operator $\wh T_G$ on $H^2_r(\BD)$
(in this setting, $P_+$ stands for the orthogonal projection of $L^2_r(\BT)$
onto the vector Hardy space $H^2_r(\BD)$).
Let
\beqn
\varphi(z)=\frac{z-i}{z+i}
\neqn
be the conformal mapping of $\BC_+$ onto the unit disc $\BD$.
The formula
\beqn
T_G=W \wh T_{G\circ \phi} W^{-1},
\neqn
where $W: \Htwor(\BD)\to\Htwor$ is the unitary isomorphism, given by
$$
(Wf)(z)=\pi^{-1/2}\,(z+i)^{-1}
\big(f\circ\varphi\big)(z)
$$
shows that each vector Toeplitz operator on $\BC_+$ is unitarily
equivalent to a vector Toeplitz operator on $\BD$, and vice versa, so there
is no difference in the study of Toeplitz operators in these two settings.
The symbols on $\BT$ that correspond to symbols in
$\Cbrr$ by means of this construction
have the only discontinuity at the point $1$.
Notice first of all that each function $G$ in $\GLntyrr(\BT)$
factors (in an essentially unique way) as $G=UG_e$, where
$G_e\in\CG\HntyrrBD$ and $U$ is unitary-valued on $\BT$. Then
$T_G=T_UT_{G_e}$, and $T_{G_e}$ is invertible, so that
Fredholmness or semi-Fredholmness of $T_G$ is equivalent to the
corresponding property of $T_U$. For unitary symbols, the
following results hold, see \cite{ClaGo}, \cite{BSlb}.
\begin{thmB}
Let $U\in\GLntyrr(\BR)$ be unitary-valued. Then
(i) $T_U$ is left-invertible
if and only if $\dist(U, \,\Hntyrr)<1$.
(ii) $T_U$ is invertible if and only if $\dist(U,\, \GHntyrr)<1$.
\end{thmB}
\begin{thmC}
Let $U\in\GLntyrr(\BT)$ be unitary-valued. Then
(i) $\wh T_U\in \Phi_+$ if and only if $\dist(U,\, \Crr(\BT)+\HntyrrBD)<1$.
(ii) $\wh T_U\in \Phi$ if and only if $\dist(U,\, \CG\big(\Crr(\BT)+\HntyrrBD )\big)<1$.
\end{thmC}
We refer to \cite{BSlb}, Section 4.38 for the connection with Fredholmness.
We will also make use of the following properties.
\begin{pro2}
\label{pro2}
\begin{itemize}
\item[(1)] Each selfadjoint matrix function
$K\in\Lntyrr(\BR)$ such that
$K(x)\ge\eps I>0$ on $\BR$ has a factorization
$K(x)=G^*_e(x)G_e(x)$ on $\BR$, where
$G_e\in\GHntyrr$.
This factorization is unique up to multiplying
$G_e$ on the left by a constant unitary matrix.
\item[(2)] If the matrix $K$ (as above) satisfies additionally
$K\circ \phi\in\Lalrr(\BT)$, then $G_e\circ\phi\in {\cal G}\Arr(\clos \BD)$; here
$$
A^\al(\clos \BD)=
\{
f\in C(\clos \BD): f|\BD\in H^\infty(\BD), \;
f|\BT\in{\rm Lip\,}^\al(\BT)
\}.
$$
\end{itemize}
\end{pro2}
For the property (1), see \cite{LSpi}, Theorem 7.9 and
\cite{SzNF}, Proposition 7.1. The proof of
(2) is contained in \cite{Shmu}, \cite{BSlb}.
\begin{lem}
\label{lm185}
Let $G\in\Lntyrr(\BR)$. Then $T_G\in\Phi_+$ if and only if $T_{\phi^nG}$
is left invertible for some integer $n\gews0$.
\end{lem}
\begin{proof} It is more transparent
to work with $\Htwor(\BD)$ instead of
$\Htwor$. Suppose $G=G(z)\in\Lntyrr(\BT)$ and $\wh T_G\in\Phi_+$; we have to
check that there is some
integer $n\gews0$ such that $\wh T_{z^nG}$ is left invertible. By
the assumption, the kernel
$\Ker\wh T_G$ is finite dimensional; let $x_1, \dots x_m\in \Htwor(\BD)$
be its basis. Put
$$
L_n=\{(c_1,\dots c_m)\in\BC^m: G\cdot \sum_j c_jx_j\in z^{-n}\Htwomr\},
\quad n\gews 0,
$$
where $H^2_{-,r}=L^2_r(\BT)\ominus \Htwor(\BD)$.
Then
$
\BC^m=L_0\supset L_1\supset \dots \supset L_n\supset \dots\qquad.
$
Since $\bigcap_0^\infty L_k=0$, one has $L_n=0$ for some $n\gews0$.
If $x\in \Ker \wh T_{z^nG(z)}$, then $x=\sum_{j=1}^m c_jx_j$ for some
coefficients $c_j$ and $z^nGx\in \Htwomr$, which implies that
$c_1=\dots=c_m=0$.
Hence $\Ker \wh T_{z^nG(z)}=0$.
Since $\wh T_{z^nG}=\wh T_G \wh T_{z^n}$ is a $\Phi_+$ operator
with trivial kernel, it follows that it is
left invertible.
\end{proof}
\begin{lem}
\label{intersBMO}
1) Suppose that $u_1, u_2$
are real increasing functions on $\BR$ and $u:=u_1+u_2\in \BMOR$. Then
$u_1,u_2\in \BMOR$.
2) $\BMOm\cap \BMOp=\BMOR$.
\end{lem}
\begin{proof} 1) For any finite interval $J\subset \BR$, one can find a point
$c=c_J\in J$ such that $u(x)\lews u_J$ for $x<c_J$ and
$u(x)\gews u_J$ for $x>c_J$. There exist numbers
$\al_{1J}$, $\al_{2J}$ such that
$u_k(c_J-0)\lews \al_{kJ}\lews u_k(c_J+0)$ for $k=1,2$ and
$\al_{1J}+\al_{2J}=u_J$. Then for any subinterval
$J\subset \BR$,
$$
\int_J |u_1(x)-\al_{1J}|\, dx
+\int_J |u_2(x)-\al_{2J}|\, dx
=\int_J |u(x)-u_J|\, dx \le C|J|,
$$
where $C=\|u\|_{\BMOR}$.
It follows that $u_1,u_2\in\BMOR$.
2) If $h=w_1-v_1=w_2+v_2\in\BMOm\cap\BMOp$, where
$w_1,w_2\in \BMOR$ and
$v_1,v_2\in C_+(\BR)$, then by part 1),
$v_1,v_2\in \BMOR$ because
$v_1+v_2\in \BMOR$.
\end{proof}
The next lemma is not new; in fact, Spitkovsky gives in
\cite[Theorem 2]{Spi2} a more general result. We will give a proof
for completeness.
\begin{lem}
\label{lm186}
\enspace
Suppose that $J$ is a finite open interval
on the real line, $F, G\in\GHntyrr$, and
$F^*F=G^*G$ a.e. on $J$.
Then there exists a neighbourhood $\cal W$ of
$J$ in $\BC$ and a bounded analytic $r\times r$ matrix function
$V$ on $\cal W$
such that $F=VG$ on $\cal W$ (and a.e. on $J$)
and $V$ is unitary valued on $J$.
\end{lem}
\begin{proof}
Put $V=FG^{-1}$, then $F=VG$ on $\BC_+$ and a.e. on $\BR$ and
$V$ is unitary on $J$. We apply the symmetry
principle to $V$. Since $V\in\GHntyrr$, it is easy to prove that
$\widetilde V(z)=V^{*-1}(\bar z)$ is an analytic continuation of $V$ onto
the lower half-plane through the arc $J$.
\end{proof}
\begin{lem}
\label{lm196}
Every matrix function $G\in\Llocrr$
such that
$\inf_\BR |\det G|>0$
has a factorization $G=UG_e$, where $G_e$, $G_e^{-1}\in \Alocrr$
and $U\in\Llocrr$ is unitary valued.
\end{lem}
\begin{proof}
Put $K(x)=G^*(x)G(x)$, then
$K(x)\ge\eps_1 I > 0$ on $\BR$. By the above property (1),
$K$ can be factorized as $K(x)=G_e^*(x)G_e(x)$,
where $G_e\in\GHntyrr$.
Hence $G=UG_e$, where $U\in\Lntyrr$.
Consider a sequence of
matrix functions $K_n$ such that
$K_n(x)=K(x)$ on $[-n,n]$, $K_n(x)\ge \eps_1 I >0$ on $\BR$ and
$K_n\circ \phi$ are Lipschitz on $\BT$. By property (2),
we arrive at functions
$G_{ne}\in\GHntyrr$ such that
$G_{ne}\circ \phi\in\Arr(\BD)$ and
$K_n=G_{ne}^*G_{ne}$ on $\BR$. By
Lemma \ref{lm186}, $G_e=V_nG_{ne}$ on $(-n,n)$,
where $V_n$ are unitary on $(-n,n)$ and
analytic in neighbourhoods of these
intervals. It follows that $G_e\in \Alocrr$.
Therefore $U\in\Llocrr$.
\end{proof}
\begin{lem}
\label{lm188}
Suppose that $H\in\Cbrr(\BR)$ and $\Psi\in \Hnty$.
Then for any finite interval $L$ on the
real line we have
\begin{equation}
\limsup_{y\to0+}\|\Psi(\cdot+iy)-H(\cdot)\|_{\Lntyrr(L)}\lews
\|\Psi-H\|_\infty; \label{AA}
\end{equation}
here
$\|\Psi-H\|_\infty=\|\Psi-H\|_{\Lntyrr(\,\BR\,)}$.
\end{lem}
\begin{proof}
Denote by $H(z)$, $z\in \BC_+$, the harmonic extension of $H$
by means of the Poisson formula. Then for any $y>0$,
$$
\|\Psi(\cdot +iy)-H(\cdot+iy)\|_{\Lntyrr(L)}\lews \|\Phi-H\|_\infty.
$$
Since $H(x+iy)\to H(x)$ as $y\to0+$ uniformly on
compact subsets of the real line, the result follows.
\end{proof}
\begin{proof}[Proof of Theorem 1]
We prove part (1).
Suppose
$G\in \Cbrr$ and
$T_G\in\Phi_+$. We have to prove that
$\arg\det G\in\BMOp$.
By Lemma \ref{lm185}, there is some
$k>0$ such that $T_{G_1}$ is left invertible, where
$G_1=\phi^kG$. Since $\arg\det G=
\arg\det G_1-kr\arg\phi$ and
$\arg\phi\in L^\infty(\BR)\subset\BMOR$,
we have only to prove that
$\arg\det G_1$ is in $\BMOp$.
Let $\|T_{G_1}x\|\gews\eps\|x\|$, $x\in H^2_r$,
where $\eps\gews0$, then
for any $G_2$ with $\|G_1-G_2\|_\infty<\eps$,
$T_{G_2}$ is also left invertible. Take
$G_2=G_1+R$ such that
$G_2\in\Llocrr$ and
$R\in C^b_{r\times r}$ has a small norm
$\|R\|_\infty$:
$\|R\|_\infty<\eps'<\eps$, where $\eps'$ has to be chosen.
Since
$$
\arg\det G_2=\arg\det G_1
+\arg\det (I+G_1^{-1}R),
$$
it follows that
$\arg\det G_2-\arg\det G_1\in L^\infty(\BR)$
if we assume that
$\eps'\cdot\|G_1^{-1}\|_\infty<1$.
So it suffices to consider $G_2$ instead of $G$.
By Lemma \ref{lm196}, we have a factorization
$G_2=UG_{2e}$, where
$U\in\Llocrr$ is unitary valued and
$G_{2e}\in \CG\Alocrr$.
Then
\beqn
T_{G_2}= T_U\, T_{G_{2e}}.
\nn
\neqn
Since $T_{G_{2e}}^{-1}=T_{G_{2e}^{-1}}$,
we conclude that $T_U$ is
left invertible. We apply Theorem B
and arrive at a function $F\in\Hntyrr$
with $\|U-F\|_\infty<1-\eps_0<1$. Put
$F_y(x)=F(x+iy)$, $y>0$,
$L=L_\rho=[-\rho, \rho]$, where
$\rho>0$. By Lemma \ref{lm188},
\begin{equation}
\|I-U(x)^{-1}F_y(x)\|_{\Lntyrr(L_\rho)}=
\big\|U(x)-F_y(x)\big\|_{\Lntyrr(L_\rho)}< 1-\eps_0 \label{AD}
\end{equation}
for $x\in L_\rho$, $y\in(0,\de)$, where
$\de=\de(\rho)>0$.
It follows, in particular, that
there is a graph $y=\psi(x)$ of a
positive function $\psi\in C^b(\BR)$
such that
$$
\|I-U(x)^{-1}F(x+iy)\|<1-\eps_0 \quad \text{for }x+iy\in\Ga_\psi.
$$
It follows that $\arg\det F$ is well defined on $\Ga_\psi$.
By Lemma \ref{lem Hntys}, $\det F$ belongs to $\Hntys$.
One can define a continuous branch of
$\arg\det\big(U(x)^{-1}F(x+iy)\big)$
for
\linebreak
$x+iy\in\Ga_\psi$ so that
$|\arg\det\big(U(x)^{-1}F(x+iy)\big)|<r \pi/ 2$.
Therefore there is a continuous branch of
$\arg\det F(x+iy)$, $x+iy\in\Ga_\psi$ such that its limit values satisfy
$$
\big|
\arg\det F(x)-\arg\det U(x)
\big|
\lews \frac {r \pi} 2 \qquad\text{ a.e. on }\BR.
$$
By Proposition 1, $\arg\det F\in\BMOp$. Hence $\arg\det U\in\BMOp$.
Since $G_{2e}\in\CG\Arrloc(\BC_+)$,
it follows that
$\det G_{2e}\in \GHnty$,
so that
$\arg\det G_{2e}\in\BMOR$.
Finally, we deduce from the formula
$$
\arg\det G_2=
\arg\det U+\arg\det G_{2e}
$$
that $\arg\det G_2\in \BMOp$.
The case when $T_G\in\Phi_-$ is obtained by considering
$G^*$ instead of $G$.
The assertion (2) follows from (1) and Lemma \ref{intersBMO}.
\end{proof}
I. M. Spitkovsky communicated to the author an outline
of an alternative proof of Theorem 1, which is based on some properties of the
transplantation of the algebra $H^\infty(\BD)+C(\BT)$ to the real line.
\section{Mean winding numbers}
\setcounter{equation}{0}
Let $H^1_{\BR}$ be the real Hardy space,
\beqn
H^1_{\BR}=\big\{
u\in L^1_\BR(\BR): \CH u\in L^1_\BR(\BR)
\big\}.
\neqn
We put $\|u\|_{H^1_{\BR}}=\|u\|_{L^1}+\|\CH u\|_{L^1}$.
Consider the cone
$$
\Pi=\{ \eta\in\HoneR: \quad \eta \text{
has a compact support on }\BR, \quad
\int_{-\infty}^x\eta\le0\quad \forall x\in\BR\}.
$$
\begin{thm2}
Let $G$ be a $r\times r$
matrix function in $\Cbrr$.
(1) If $T_G\in\Phi_-(\Htwor)$,
there is a constant $C>0$ such that
for any $\eta$ in ~$\Pi$,
$$
\int_\BR\eta(x)
\big(\arg\det G\big)(x)\,dx\lews C\|\eta\|_{\HoneR}.
$$
(2) If $T_G\in\Phi_+(\Htwor)$
there is a constant $C>0$ such that for any $\eta$ in ~$\Pi$,
$$
\int_\BR\eta(x)
\big(\arg\det G\big)(x)\,dx\gews -C\|\eta\|_{\HoneR}.
$$
\end{thm2}
It is well known that $\int_\BR\eta=0$ for any function $\eta$ in
$\HoneR$, see \cite{GR}, Chapter III.
Hence
the above integrals do not depend on the
additive constant in $\arg\det G$.
As a consequence, we obtain that if $T_G\in\Phi(\Htwor)$, then
\beqn
\bigg|\int_\BR\eta(x)
\big(\arg\det G\big)(x)\,dx\bigg|\lews C\|\eta\|_{\HoneR},
\qquad \eta\in \Pi.
\nn
\neqn
In the scalar case, this inequality follows from
the Widom-Devinatz theorem (Theorem B),
together with the Fefferman duality theorem,
and takes place for all
$\eta \in \HoneR$ (the integral is to be
understood in the sense of the
the duality $\HoneR$ -- $\BMOR$).
\begin{dfn}
Let $\eta\in\Pi$, $\eta\not\equiv0$ be fixed, and let $G\in\Cbrr$.
Define the upper and the lower \textit{mean winding numbers} of
$\det G$ (associated with $\eta$) by
\begin{align*}
\displaystyle
\uw_\eta(G)&=
\underset{T\to+\infty} {\ulim} \;
\sup_{y\in\BR}
\frac1T\int_{\BR}\eta\big(\frac{x-y}T\big)\cdot\arg\det G(x)\,dx, \\
\lw_\eta(G)&=
\underset{T\to+\infty} {\llim} \;
\inf_{y\in\BR}\,
\frac1T\int_{\BR}\eta\big(\frac{x-y}T\big)\cdot\arg\det G(x)\,dx.
\end{align*}
\end{dfn}
\begin{thm3}
(1) If $T_G\in\Phi_+(\Htwor)$, then $\lw{}_\eta(G)\ne-\infty$;
(2) If $T_G\in\Phi_-(\Htwor)$, then $\uw_\eta(G) \ne+\infty$.
\end{thm3}
One can also define simpler characteristics
\begin{equation*}
\displaystyle
\utw_\eta(G)=
\underset{T\to+\infty} {\ulim} \;
\frac1T\int_{\BR}\eta\big(\frac xT\big)\cdot\arg\det G(x)\,dx
\end{equation*}
and the number
$\ltw{}_\eta(G)$, defined as the corresponding lower limit.
One has
$\lw_\eta(G)\le
\ltw{}_\eta(G)\le
\utw_\eta(G)\le
\uw_\eta(G)$, so that Theorem 3 implies the same
assertions for
$\ltw{}_\eta(G)$, $\utw_\eta(G)$.
Consider a scalar $G\in\CbR$, $|G|>\eps>0$ on $\BR$. If $\arg
G$ has finite limits on $\pm\infty$, then
$\utw_\eta(G)=\ltw{}_\eta(G)= K\cdot \arg G\big|_{-\infty}^{+\infty}$,
where $K=\int_0^{+\infty}\eta(x)\,dx$.
One also has
$\uw_\eta(G)=L\cdot \big(\arg G\big|_{-\infty}^{+\infty}\big)_+$,
$\lw_\eta(G)=L\cdot \big(\arg G\big|_{-\infty}^{+\infty}\big)_-$,
where $L=\sup_{y\in\BR}\int_y^{+\infty}\eta$,
$y_+=\max(y,0)$, $y_-=\min(y,0)$.
So in this case all these
winding numbers have a simple sense.
For these symbols, each of the
conditions $T_G\in\Phi_-(\Htwor)$, $T_G\in\Phi_+(\Htwor)$,
$T_G\in\Phi(\Htwor)$ is equivalent to the requirement
$(\arg G)\big|_{-\infty}^{+\infty} \ne\pm\pi, \pm 3\pi, \pm 5\pi$, etc.
(see, for instance, \cite{BSlb} or \cite{GK}, Ch. 9).
\begin{cor1}
Let $\al>0$, and define generalized winding numbers
\begin{align*}
\displaystyle
&\uw_{\eta,\al}(G)=
\underset
{T\to+\infty}
{\ulim}
\sup_{y\in\BR}\;
\frac1{T^{1+\al}}
\int_{\BR}\eta\big(\frac{x-y}T\big)\cdot(\arg\det G)(x)\,dx, \\
&\lw_{\eta,\al}(G)=
\underset
{T\to+\infty}
{\llim}
\inf_{y\in\BR}\;
\frac1{T^{1+\al}}
\int_{\BR} \eta\big(\frac{x-y}T\big) \cdot(\arg\det G)(x)\,dx.
\end{align*}
(1) If $T_G\in\Phi_+(\Htwor)$, then $\lw_{\eta,\al}(G)\gews0$;
(2) If $T_G\in\Phi_-(\Htwor)$, then $\uw_{\eta,\al}(G)\lews0$.
\end{cor1}
This follows immediately from Theorem 3. \hfill $\Box$
In particular, the function $\eta_\al=\frac{1+\al}2
\big(\chi_{[0,1]}-\chi_{[-1,0]}\big)$ is in $\Pi$. The
corresponding upper winding number is given by
\beqn
\uw_\al(G)=
\underset
{T\to+\infty}
{\ulim}\;
\frac{1+\al}{2T^{1+\al}}\;
\sup_{y\in\BR}\;
\bigg[
\int_y^{T+y}
-\int_{y-T}^y \bigg] \; \arg\det G(x)\,dx.
\label{wal}
\neqn
Let us define similarly the lower winding number
$\lw_\al(G)$, by taking $\displaystyle\inf_{y\in\BR}$ and
the corresponding lower limit.
Corollary 1 holds, in particular, for
these characteristics of $G$.
If $n=1$, $G(x)=\exp\big(ik(\sign x)\cdot|x|^\al\big)$,
and $0<\al\le 1$, then
$\uw_\al(G)=\lw{}_\al(G)= k$.
In fact, we could take
instead of $T^{1+\al}$ any function $\rho(T)$ such that
$\rho(T)>0$, $\frac T{\rho(T)}\to0$ as $T\to+\infty$
in the above definitions of generalized winding numbers.
\begin{cor2th3}
Let $G\in \GCarr(\BC_+)$ or
$G\in \GCarr(\BC_-)$,
where $\BC_-=\{z\in\BC:\, \Im z<0\}$. Then for any $\al>0$,
$\uw_\al(G)=\lw{}_\al(G)= 0$.
\end{cor2th3}
Indeed, in both cases $T_G^{-1}=T_{G^{-1}}$, hence
$T_G\in \Phi(H^2_r)$, and we can apply Corollary 1.
\hfill $\Box$
\begin{cor3th3}
Let $G\in \GCbrr(\BR)$, and define
$\uw_1(G)$ by (\ref{wal})
and $\be(G)$ by (\ref{betaG}). Then
$\ds\beta(G)\ge\frac{\uw_1(G)}r$.
\end{cor3th3}
Indeed, if $T_{e^{-i\tau x}G}$ is onto, then it is a
$\Phi_-$ operator, which implies that
\[
\qquad \qquad \qquad \qquad \uw_1(e^{-i\tau x}G)= \uw_1(G) - r\tau \le 0.
\qquad \qquad \qquad \qquad \Box
\]
We remark that if $G$ is a semi-almost periodic $r\times r$
matrix function
such that $G, G^{-1}\in C^b(\BR)$, then $\det G$ is a scalar semi-almost periodic
function, and $\det G$ has almost periodic representatives
$(\det G)_{\pm \infty}$ at $+\infty$ and $-\infty$, respectively
(see \cite{BKSpi}). These representatives, by the Bohr mean motion theorem,
have the form
$$
(\det G)_{\pm \infty}(x)=e^{\kappa_\pm x}e^{g_\pm(x)},
$$
where $\kappa_\pm $ are mean motions of
$\det G(x)$ at $\pm \infty$ and functions
$g_\pm$ are almost periodic. In this case,
\begin{gather*}
\lw{}_1(G)=\min(\kappa_-, \kappa_+), \qquad
\uw_1(G)=\max(\kappa_-, \kappa_+), \\
\ltw{}_1(G)=\utw{}_1(G)=\frac {\kappa_-+\kappa_+}2.
\end{gather*}
If $r=1$, complete criteria of Fredholmness,
as well as the calculation of the Fredholm index are
known since the work by Sarason \cite{Sar2}.
It follows, in particular, that in this case
$\beta(G)=\max(\kappa_-, \kappa_+)$. So
Corollary 3 of Theorem 3 gives an exact estimate
for the case of scalar almost periodic
functions.
The study of the almost periodic and
semi-almost periodic matrix cases depend on the
existence of some special factorizations of $G$.
If these factorizations exist, then complete
criteria for Fredholmness and formulas for
the index are available, see
\cite{KSpi} and \cite{BKSpi}, Ch. 10 and \S19.6.
\begin{proof}[Proof of Theorem 2]
By Theorem 1, it only has to be proved that if $f\in\BMOp$,
then
$$
\int_\BR f(x)\eta(x)\,dx\gews -C\|\eta\|_{H^1_\BR}
\qquad
\text{ for all }\eta\in \Pi.
$$
This inequality follows from the Fefferman duality
$H^1_\BR$ -- $\BMOR$ (see \cite{GR})
in the case when $f\in \BMOR$.
Now let $f$ be nondecreasing, and take any function $\eta\in\Pi$.
Suppose that $\supp \eta\subset I$, where
$I$ is a finite interval. Approximate $f$
in $L^\infty(I)$ by
a sequence of nondecreasing step functions
$\{f_n\}$
of the form
$$
f_n=C_n+\sum \al_{nk}\,\chi_{(-\infty,a_{nk}]},
$$
where $C_n, a_{nk}\in\BR$ and $\al_{nk}$ are negative.
Then
$\int_{\BR}\eta f_n\gews0$ for all $n$,
hence $\int_{\BR}\eta f\gews0$.
We obtain the result by combining these two cases.
\end{proof}
\begin{proof}[Proof of Theorem 3]
Let $\eta_{T,y}(x)=\eta\big(\frac{x-y}T\big)$. Since $\CH(\eta_{T,y})=(\CH\eta)_{T,y}$,
it follows that $\|\eta_{T,y}\|_{H^1_\BR}= T\|\eta\|_{H^1_\BR}$.
So the assertions follow directly from
\linebreak
Theorem 2.
\end{proof}
\section{Some related questions }
\setcounter{equation}{0}
\nin{\bf Problem 1.} \enspace Give a real variable characterization of classes
$\BMOR^\pm$.
The next two questions are certainly known for specialists
for a long time, however, complete answers are not known.
\nin{\bf Problem 2.} \enspace
1) Let $r=1$, and let
$G\in C(\BR)$, $\arg G\in C_+(\BR)$, \linebreak
$\lim _{x\to\pm\infty}\arg G=\pm\infty$.
What additional conditions guarantee that $T_G\in \Phi_+(\Htwor)$?
2) What can be said in this respect for the matrix case $r>1$?
Sufficient conditions for $r=1$
are given in \cite{BG} and in \cite{DybGru},
Theorem 5.10. As it follows from the construction of Lemma 4.9 in
\cite{BG}, there are symbols $G$ of the above type
such that $T_G$ is not semi-Fredholm.
See also \cite{Gov}, Theorem 28.2 and Section 32 for related counter-examples.
The book \cite{DybGru} also contains results about the matrix valued case.
At least for the scalar case, it seems that more complete answers can be found.
\nin{\bf Problem 3.} \enspace Suppose that $T_G\in\Phi_+$.
Can one give some estimates of $\ind T_G$ in terms of some
explicit real variable characteristics of $\arg\det G$?
\nin{\bf Problem 4.} \enspace
Suppose that $\eta_1, \eta_2\in\Pi$.
When can one assert that $\uw_{\eta_1}(G)\ne+\infty$
implies that $\uw_{\eta_2}(G)\ne+\infty$
for all $G\in\Cbrr(\BR)$ with $|\det G|>\eps>0$ on $\BR$?
Is there a ``universal'' function
$\eta_0\in\Pi$ such that for any $G$ as above,
$\uw_{\eta_0}(G)\ne+\infty$
implies that $\uw_{\eta}(G)\ne+\infty$
for all $\eta\in\Pi$?
\vskip-.5cm
|
2,869,038,154,858 | arxiv |
\section{Introduction}
\begin{figure*}
\centering
\includegraphics[width=0.95\textwidth,page=1]{imgs/QG-formal.pdf}
\caption{Model overview.}
\label{fig:model_overview}
\end{figure*}
Knowledge-Based Question Generation (KBQG) is a task that aims to generate natural language questions given a knowledge base (KB).
KBQG has a wide range of applications and has won wide attention in academia and industry.
In this paper, we use a specific knowledge graph (KG), Freebase\cite{bollacker2008freebase}, as KB.
Recent works mainly adopt sequence-to-sequence neural networks to generate questions given a Resource Description Framework \cite{miller1998introduction, Eric2001Introduction} (RDF) graph that is a directed graph composed of triple statements in a knowledge graph.
For a single relation graph, a series of works \cite{elsahar_zero-shot_2018, liu_generating_2019, bi_knowledge-enriched_2020} enriched the input with auxiliary information, equipped the decoder with copy mechanisms, and devised delicate models to improve the fluency or the semantic accuracy of generated questions.
For star-like graph, \cite{kumar_difficulty-controllable_2019} used Transformer encoder while \cite{chen_toward_2022} applied a bidirectional Graph2Seq model to encode the graph structure.
Very recently, JointGT \cite{ke_jointgt_2021} adopted the modern pre-trained language model BART \cite{lewis_bart_2020} to generate questions, achieving state-of-the-art performance.
However, KBQG still faces three major challenges:
\paragraph{1. Absence of complex operations}
The existing KBQG methods mainly generated questions from an RDF graph. However, RDF is a standard model for data interchange on the Web, it's oriented to describe resources not to express constraints on top of them, such as aggregation, superlative, and comparative questions. Compared with the RDF graph, SPARQL has a complete semantic representation that covers all the question types mentioned above. How to generate questions based on SPARQL is a challenge.
\paragraph{2. Low resources}
For expressing the predicates in KG to NL form, traditional supervised methods tended to annotate large-scale SPARQL-question pairs. However, the labor cost is enormous. Besides, KG may contain complex schemas, such as Compound Value Type\footnote{A Compound Value Type is a Type within Freebase which is used to represent data where each entry consists of multiple fields. See \url{https://developers.google.com/freebase/guide/basic_concepts}} (CVT) in Freebase. Each CVT combination has a different meaning, hence it is difficult for annotations to cover all combinations. How to generate questions without sufficient resources is still under-explored to date.
\paragraph{3. Efficient generation}
The advanced generative pre-trained language models (PLMs) have been proven effective in natural language generation (NLG) tasks \cite{lewis_bart_2020, chen_few-shot_2020, ke_jointgt_2021}. Nonetheless, PLMs were trained in NL-to-NL paradigm, but the SPARQL expression is different from the NL form. Therefore, how to leverage the strengths of PLMs to generate high-quality questions matters a lot.
To address the challenges mentioned above, we propose AutoQGS, an auto-prompt approach for low-resource KBQG from SPARQL. Figure \ref{fig:model_overview} shows the overall process.
Firstly, we incorporate SPARQL expression directly as input, which retains the original semantics.
Secondly, we propose a model, auto-prompter, to rephrase the SPARQL to the corresponding NL description, named prompt text.
Auto-prompter combines the strengths of distant supervision \cite{mintz-etal-2009-distant} and the strong generative ability of PLMs.
Specifically, the training data of auto-prompter are subgraphs that could be massively sampled from KB, and the target prompt text is collected from large-scale corpus as well.
Lastly, we explore an efficient question generation method in low-resource scenarios.
Our model significantly outperforms existing state-of-the-art baselines by a large margin, especially in low-resource settings.
The main contributions of this paper are summarized as follows.
\begin{itemize}
\item Put forward to generate questions directly from SPARQL for KBQG task to handle complex operations.
\item Propose AutoQGS, an approach to rephrase RDF graph to prompt text, and generate questions from SPARQL and prompt text.
\item Conduct extensive experiments on two datasets, and the results show that AutoQGS improve the performance observably.
\end{itemize}
\section{Related Work}
KBQG has come a long way in the past decades. In the early time, many works generate questions in template-based approaches.
\cite{song_question_2017} proposed an unsupervised system to collect questions by a search engine from a small number of template-based questions.
\cite{Seyler2015Generating, seyler_knowledge_2017} collect structured triple-pattern query from seed question and use a template-based method to verbalize the structured query.
However, template-based approaches lack flexibility, and the annotation cost is high.
Recent works for KBQG are mainly based on sequence-to-sequence neural networks given a set of subgraphs from knowledge graph (KG) .
\cite{serban_generating_2016} first used a neural network for encoding KG fact triples into natural language questions and generated the 30M Factoid Question-Answer datasets. However, it was trained on a mass of fact-question pairs, which is challenging to collect.
\cite{reddy_generating_2017} proposed an RNN-based model to generate simple questions and corresponding answers by converting all the KG entities to a set of keywords.
\cite{Bao2018Table} and \cite{Bao2019TextGeneration} developed a flexible copying mechanism to alleviate the rare words problem.
For unseen predicates and entity types problem, \cite{elsahar_zero-shot_2018} collected textual contexts in the Wikipedia as auxiliary information and adopted a part-of-speech copy action mechanism to generate questions.
However, \cite{liu_generating_2019} thought the textual contexts were noisy or even wrong. They presented a complicated model that integrates diversified off-the-shelf contexts and devised an answer-aware loss to make sure the generated questions are associated with a definitive answer.
Based on the Transformer \cite{vaswani_attention_2017}, \cite{kumar_difficulty-controllable_2019} proposed an end-to-end neural network-based method for generating complex multi-hop and difficulty-controllable questions over a subgraph in KG.
\cite{bi_knowledge-enriched_2020} focus on semantic drift problem. They proposed to incorporate auxiliary information and word types in generated questions, make the decoder output conditioned on these types, and design a DPT-based evaluator to encourage question structural conformity in a reinforcement learning framework.
Instead of using a set of KG triples, \cite{chen_toward_2022} proposed to apply a bidirectional Graph2Seq model to encode the KG subgraph and target answers, and then generate questions with a node-level coping mechanism.
\cite{ke_jointgt_2021} proposed a pre-trained model called JointGT for KG-to-text generation tasks. They added a structure-aware semantic aggregation module in BART to model the structure of input graphs. They then pre-trained the model in a large-scale corpus, and fine-tuned in question generation task.
In comparison, our approach utilizes advanced generative pre-trained language models to generate questions from SPARQL, rather than either make templates to rephrase questions or generate questions from an KG subgraph.
\section{Approach}
\subsection{Problem Formulation}
This study investigates the task of knowledge-based question generation (KBQG).
Conventional KBQG works generated questions from fact triples in the knowledge graph, which could not express complex operations.
A SPARQL expression is of complete syntax, which is capable of fully formalizing questions with complex operations\footnote{Complex operations are defined as functions beyond the KG predicates.}, e.g., aggregation(\textsf{COUNT}), comparative(\textsf{<,>,<=,>=}), and superlative(\textsf{ORDER BY ?x LIMIT 1}).
Moreover, due to the costly annotation of large-scale SPARQL-question pairs, KBQG from SPARQL under low-resource scenarios urgently needs to be explored.
In this paper, we propose to \textbf{generate question directly from SPARQL for low-resource KBQG} task.
Given an executable SPARQL expression $S$ and a knowledge graph $\mathcal{K}$, our goal is to generate a natural language (NL) question $Q$ that is consistent with the SPARQL.
Given the above definitions, the task can be formalized as learning the distribution $p(Q|S, \mathcal{K})$.
\subsection{Model Overview}
Recently, the generative pre-trained language models (PLMs) typically trained in NL-to-NL paradigm have been proven effective for low-resource generation, e.g., T5\cite{2020t5} and BART\cite{lewis_bart_2020}.
However, generating questions directly from non-NL SPARQL is not friendly to the generative PLMs.
In this paper, we propose AutoQGS, an auto-prompt approach which rephrases SPARQL to NL text automatically, smoothing the transformation from non-NL SPARQL to NL question.
The overall process of AutoQGS is shown in Figure \ref{fig:model_overview}.
Specifically, AutoQGS consists of two procedures, (1) \textbf{auto-prompt} from SPARQL to NL text, and (2) \textbf{question generation} (QG) based on SPARQL and NL prompt text.
Formally, auto-prompt aims to generate prompt text $T$ from SPARQL $S$.
The process of auto-prompt can be formalized as
\begin{equation}
\label{eq:ap_overview}
T = \mathtt{AP}(S, \mathcal{K}; \Theta)
\end{equation}
Afterwards, $S$ and $T$ are concatenated as input\footnote{We replace the machine identifiers of the topic entities in SPARQL with their surface names, but for brevity, we still use the symbol $S$.} of a question generator to produce a question $\hat{Q}$.
The procedure of question generation can be formalized as
\begin{equation}
\label{eq:qg_overview}
\begin{aligned}
Q = \mathtt{QG}(S, T; \Phi)
\end{aligned}
\end{equation}
We will introduce the auto-prompt $\mathtt{AP}(\cdot)$ and the question generation $\mathtt{QG}(\cdot)$ procedures in Section \ref{sec:ap} and Section \ref{sec:qg}, respectively.
\begin{figure*}
\centering
\includegraphics[width=1\textwidth,page=2]{imgs/QG-formal.pdf}
\caption{Overview of auto-prompt training and inference procedures. \protect\footnotemark}
\label{fig:ap_overview}
\end{figure*}
\footnotetext{For brevity, we omit the prefix: \textsf{PREFIX ns: <http://rdf.freebase.com/ns/>}.}
\subsection{Auto-prompt}\label{sec:ap}
Auto-prompt $\mathtt{AP}(\cdot)$ generates prompt text $T$ from SPARQL $S$.
Figure~\ref{fig:ap_overview} shows the overview of auto-prompt.
Given a SPARQL $S$, we first execute it based on $\mathcal{K}$ and obtain an instantiated subgraph $G$ via subgraph sampling defined in Section \ref{subgraph_sampling}.
$G$ can be decomposed into a set of atomic subgraphs. Each $g \in G$ is serialized as input of an auto-prompter and converted into text.
They are then concatenated to construct the prompt text $T$.
Specifically, this procedure is formalized as follows:
\begin{equation}
\label{eq:ap_detail}
T = \mathtt{CONCAT}(\{ \mathtt{BeamSearch_{Top1}}(p_{\Theta}(d_{i}|d_{<i},g)) | g \in G\})
\end{equation}
\begin{equation}
\label{eq:ap_g}
G = f(S, \mathcal{K})
\end{equation}
where function $\mathtt{CONCAT}(\cdot)$ means concatenating all the elements, $\mathtt{BeamSearch_{Top1}}(\cdot)$ is the generation process with beam search (we keep the one with the highest score), $\Theta$ is the learnable parameters of auto-prompter, $d_i$ is the $i$-th predicted token, and function $f(\cdot)$ indicates the graph sampling process.
In this work, the auto-prompter supports both star and chain topologies, and the prompt text will vary as the input entities and their information change.
By contrast, previous works only save the hard matched results for single-relation predicate (e.g. ``\textsf{is birthplace of}'' for ``\textsf{person/place\_of\_birth}'').
\subsubsection{Subgraph Sampling}\label{subgraph_sampling}
Subgraph sampling is a process that aims to find a KG subgraph that can instantiate a given SPARQL based on $\mathcal{K}$.
Firstly, we define an atomic subgraph $g$, $type(g) \in \{ CVT, Single \}$ in Freebase, where $type(\cdot)$ is a function that identifies the type of an atomic subgraph.
$CVT$ is a Compound Value Type (CVT) subgraph and $Single$ is a single-relation one.
A CVT subgraph consist of a central CVT node and its corresponding one-hop edges.
A single-relation subgraph can be formalized to a (subject, predicate, object) triple.
Specifically, we use Virtuoso to store Freebase.
Following Google's instruction\footnote{\url{https://github.com/google/Freebase-wikidata-converter}}, we classify the type of predicates in Freebase into single-relation and CVT.
The difference is that the tail node of a CVT predicate is a CVT node and the entire CVT graph expresses an event, as the predicate ``\textsf{film.film\_character.portrayed\_in\_films}'' shown in Figure \ref{fig:ap_overview} a.
The single-relation one links two named entities, which means a fact.
In the training stage, given a predicate, we sample a set of corresponding atomic subgraphs in the database.
For single-relation predicate, we simply construct ``\textsf{SELECT ?s ?o WHERE \{?s [predicate] ?o.\}}'' to query KG and save all \textsf{(subject name, predicate, object name)} triples returned.
For each CVT predicate, we sample a set of nodes that are the tails of the predicate, then instantiate a graph center at the node for each one in the set.
As shown in Figure \ref{fig:ap_overview} a, the instantiated CVT graph consists of both inside and outside edges in one-hop connection.
For example, in order to sample the graphs of predicate ``\textsf{film.film\_character.portrayed \_in\_films}'', we firstly construct a query, ``\textsf{SELECT ?x WHERE \{?y film.film\_character.portrayed\_in\_films ?x.\}}'', to get corresponding CVT nodes, and then for each node (e.g. \textsf{m.0gxrhxd}), we construct another query, ``\textsf{SELECT ?e1 ?p\_in ?e2 ?p\_out WHERE \{?e1 ?p\_in m.0gxrhxd. m.0gxrhxd ?p\_out ?e2.\}}'', to instantiate the CVT graph.
Due to limited resources, instead of covering the entire predicates in Freebase, we selected a subset of predicates involved in the two datasets mentioned in Section \ref{dataset_preprocess}.
In the inference stage, given a SPARQL expression $S$, we search all variables (e.g. replacing ``\textsf{?x}'' with ``\textsf{?y ?x ?num}'' in the SPARQL) in KG and sample one from the results to instantiate a complete graph $G$.
As shown in Figure \ref{fig:ap_overview} b, a SPARQL may consist of both single-relation and CVT subgraphs.
\subsubsection{Subgraph Serialization}\label{subgraph_serialization}
To explore how to mine the general statements more effectively for different entities with the same predicate, we propose two strategies of serialization.
\paragraph{Entity name} In this setup, we serialize the subgraph with the original entity name. As we know, Wikipedia is used to train PLMs, so that keeping the original name may be a simple but effective way to utilize the knowledge the model has learned.
For single-relation subgraph, the serialization pattern is ``\textsf{<H> [head entity name] <D> [head entity description] <P> [predicate] <T> [tail entity name] <D> [tail entity description]}'', the special tokens <H>, <D>, <P>, and <T> mean the head entity, description, predicate and tail entity, respectively.
For CVT subgraph, we add a special token ``\textsf{R@}'' in front of inside predicates (point to a CVT node) in order to traverse the subgraph in a uniform format. Therefore, the serialization pattern of one edge in the subgraph is ``\textsf{<P> [R@][predicate] <T> [entity name] <D> [entity description]}'', and then we just concatenate all edges together. Figure \ref{fig:ap_overview} gives an example.
\paragraph{Entity type placeholder} In this setup, we replace the entity name with its entity type in both input and target text. Intuitively, the delexicalization will make model focus more on predicate rephrasing.
Previous work \cite{elsahar_zero-shot_2018} picked the entity type that is mentioned most in the first sentence of the entity's Wikipedia article. However, it does not make sense to fix the entity type in all contexts.
Instead, we directly utilize the head and tail entity types contained in a Freebase predicate. The predicate in Freebase is organized as ``\textsf{domain.subject\_type.property}'', where we treat the ``\textsf{property}'' part as the type of object entity.
For example, the part of serialization shown in the Figure \ref{fig:ap_overview} a will be: ``\textsf{<P> [email protected]\_character. portrayed\_in\_films <T> [film\_character] <D> [film\_character] is the fictional character of 1985 film Return to Oz .}''.
In general, the serialization patterns used here are similar to those in the entity name setup.
\subsubsection{Collection of Subgraph Description}
Given a atomic subgraph $g$, the auto-prompter is trained to generate the corresponding subgraph description, denote as $\bar{d}$.
Training the aforementioned auto-prompter needs large-scale labeled SPARQL and NL description pairs, since the SPARQL involves a lot of KB predicates, e.g., ``\textsf{film.film.actor}'', and SPARQL operators, e.g., ``\textsf{ORDER BY ?num DESC LIMIT 1}'' (means ``\textsf{Argmax}'').
For example, given a non-NL SPARQL (a simplified version of the Figure \ref{fig:ap_overview} b) ``\textsf{SELECT DISTINCT ?x WHERE \{m.01d1st film.actor.film ?y. ?y film.performance.film ?x .\}}'' and the name of topic entity \textsf{m.01d1st} ``Nick Cannon'', an annotator needs to understand the SPARQL and write the corresponding NL text ``\textsf{Nick Cannon star in film [?x]}''.
Labelling such large-scale data is impracticable.
Therefore, the motivation here is to make use of large-scale unstructured corpus to fill the gap between predicates and natural language expression.
Different from \cite{elsahar_zero-shot_2018} who use a heuristic string matching rule to find the phrases in corpus by co-occurrence of entity names, we propose a novel soft-generation approach by combining the distant supervision and the ability of modern generative PLMs together.
Specifically, to find the NL descriptions for atomic subgraphs, we rely on the Wikipedia 2018-12-20 dump as the source of text documents.
Each page in Wikipedia has a title and a content that consists of a list of paragraphs. All these fields are re-tokenized by Spacy\footnote{\url{https://spacy.io}} and indexed by Elasticsearch\footnote{\url{https://www.elastic.co/elasticsearch}}.
As mentioned in \cite{riedel_modeling_2010, elsahar_zero-shot_2018}, we believe that the distant supervision assumption has been effective on Wikipedia. For single-relation subgraph $g$, we match sentence $\bar{d}$ in Wikipedia if the subject name and the object name of this triple co-occur in the same sentence.
For CVT subgraph $g$, we find paragraphs $\bar{d}$ in Wikipedia if all entities of this subgraph co-occur in the same paragraph.
In addition, we remove the sentences that do not have any entity in matched paragraphs and drop subgraphs that match nothing.
Furthermore, during the training phase, the entity names in the descriptions are replaced by their types in the entity type placeholder setup, while in another setup, the descriptions remain untouched.
Accordingly, we need to replace the entity types with the corresponding names in the inference phase to get the prompt text.
\subsubsection{Training}
The auto-prompter is based on an advanced generative PLM, BART\cite{lewis_bart_2020}.
Specifically, in order to make the optimization process more stable, we merge and shuffle the two kinds of subgraph as training data, making them evenly distributed in each mini-batch. Then we fine-tune two auto-prompters based on the two serialization strategies mentioned above. The details of the hyper-parameters setting are recorded in Appendix \ref{hyper_setting}.
The auto-prompter is trained with a maximum likelihood objective. Given the training samples $(\bar{d}, g)$, the objective $\mathcal{L}$ is defined as:
\begin{equation}
\mathcal{L} = - \sum_{M} \sum_{i=1}^{|\bar{d}|} \log p_{\Theta} (\bar{d}_i|\bar{d}_{<i}, g)
\end{equation}
where $\bar{d}_i$ is the $i$-th token in $\bar{d}$ and the $M$ is the length of training instances.
\subsubsection{Prompt Text}
The motivation for developing the prompt text is to smooth the transformation from non-NL SPARQL to NL question.
Therefore we rephrase the formal expression in KB to a NL form with the help of PLMs.
As shown in Figure \ref{fig:ap_overview} b, the auto-prompter successfully and correctly generates the relation, ``\textsf{A Very Scholl Gyrls Holla-Day starring Nick Cannon}'', from a relative complex CVT subgraph.
In a word, given a SPARQL expression $S$, Section \ref{subgraph_sampling} instantiates and samples one RDF subgraph $G$.
Section \ref{subgraph_serialization} adds entity descriptions and serializes each $g \in G$.
Lastly, the well-trained auto-prompter is used to generate prompt text $T$ given $G$.
It is hard to evaluate the quality of prompt text by automatic metrics because there is no target text.
Therefore we conduct human evaluations and report the scores in Section \ref{human_eval}.
\subsection{Question Generation}\label{sec:qg}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth,page=3]{imgs/QG-formal.pdf}
\caption{Overview of question generation.}
\label{fig:qg}
\end{figure}
Question generation $\mathtt{QG}(\cdot)$ takes both the SPARQL $S$ and prompt text $T$ as input, and generates the question $\hat{Q}$ with max likelihood logit in the beam search space.
The question generator is implemented based on another BART.
The training and inference processes here are similar to the classic fine-tuning methods of BART.
The overall process is shown in Figure \ref{fig:qg}.
\subsubsection{Data Construction}
We construct a training sample by concatenating $S$ and $T$ together as input and the annotated question as target.
In order to better prompt the question generator, we add ``\textsf{(the [var])}'' behind the corresponding instantiated entity in prompt text for each variable in SPARQL \footnote{For example, add ``\textsf{(the ?x)}'' behind ``\textsf{A Very School Gyrls Holla-Day}''}.
It is worth noting that, following previous works, we convert both input and output to lowercase.
\subsubsection{Training and Inference}
We fine-tune the question generator as a standard sequence-to-sequence model from the input to the output text, as presented in BART \cite{lewis_bart_2020}. Formally, given the training sample $(S, T, \bar{Q})$, the objective $\mathcal{L'}$ is defined as
\begin{equation}
\mathcal{L'} = \sum_{N} p_{\Phi}(\bar{Q}|S, T)= - \sum_{N} \sum_{i=1}^{|\bar{Q}|} \log p_{\Phi} (\bar{Q}_i|\bar{Q}_{<i}, S, T)
\end{equation}
where $\bar{Q}_i$ is the $i$-th token in the target question $\bar{Q}$, $N$ is the length of training samples and $\Phi$ denotes the parameters of the question generator.
We use the standard beam search method for decoding. The specific parameter settings are reported in Appendix \ref{hyper_setting}.
\section{Experiments}
In this section, we conduct extensive experiments to evaluate the effectiveness of our proposed approach.
\subsection{Dataset \& Preprocessing}\label{dataset_preprocess}
\begin{table}
\caption{Statistics of the datasets.}
\label{tab:data_statistics}
\scalebox{0.85}{
\begin{tabular}{lcccc}
\toprule
Dataset & \#Rel & \begin{tabular}[c]{@{}c@{}}\#Instances\\ (Train/Valid/Test)\end{tabular} & Total & Length \\ \hline
WQCWQ1.1 & 931 & 29,909 / 2,529 / 2,529 & 34,967 & 14.12 \\
\quad w/ OPs & 494 & 4,097 / 300 / 343 & 4,740 (13.56\%) & 14.69 \\
\quad w/o OPs & 870 & 25,812 / 2,229 / 2,186 & 30,277 (86.44\%) & 14.03 \\
PathQuestions & 378 & 9,793 / 1,000 / 1,000 & 11,793 & 12.4 \\ \bottomrule
\end{tabular}
}
\end{table}
\paragraph{WQCWQ1.1} WebQuestionsSP \cite{yih_value_2016} and ComplexWebQuestions 1.1 \cite{talmor_web_2018} are widely used question answering datasets that contain natural language questions and corresponding SPARQL queries. Previous works \cite{kumar_difficulty-controllable_2019, ke_jointgt_2021, chen_toward_2022} combined WebQuestionsSP and ComplexWebQuestions (the older version) and convert the SPARQL from \textsf{SELECT} query to \textsf{CONSTRUCT} query to get RDF graphs. However, ComplexWebQuestions is unavailable now, and previous works only released the processed results without original SPARQL expressions, wherein the exact semantics are lost. Therefore, we merge and reprocess these two datasets by ourselves and release a new dataset, WQCWQ1.1. We leave the original training set untouched and randomly divide the validation/test set equally.
\paragraph{PQ} PathQuestions (PQ) is constructed from a question answering dataset \cite{zhou_interpretable_2018}. The main characteristic is that PQ only consists of chain questions, wherein the path between the topic entity and the answer is 2-hop or 3-hop. Compared with WQCWQ1.1, PQ does not contain any complex operation in questions. We used the same data as the existing works \cite{kumar_difficulty-controllable_2019, chen_toward_2022}.
Brief statistics of the datasets are listed in Table \ref{tab:data_statistics}, including the total number of relations, the data split, the subset divided by whether containing complex operations (OPs), and the average length of questions.
\subsection{Implementation}
Our auto-prompter and question generator are both based on pre-trained models BART \cite{lewis_bart_2020}.
We initialize our model weights with the BART-base checkpoint released by HuggingFace's Transformers \cite{wolf_transformers_2020}.
We follow BART to use BytePair Encoding (BPE) vocabulary \cite{radford2019language} with the size of 50,265 and adopt Adam \cite{kingma_adam_2015} as the optimizer.
Since the computational resources are limited, we pick a set of hyper-parameters and train the auto-prompter on unsupervised data for 10 epochs. It took 60 hours on 4 NVIDIA A100 (40GB) GPUs.
For training question generator, we apply different hyper-parameters on each data proportion setting.
More details, including the hyper-parameters and search space settings, are reported in Appendix \ref{hyper_setting}.
\subsection{Baseline Methods}
We choose the following two categories of models as our baselines:
\paragraph{Pre-trained Models} We chose JointGT as the pre-trained baseline. JointGT \cite{ke_jointgt_2021} is a BART-based model for KG-to-text generation. It adopts a structure-aware semantic aggregation module to model the structure of an input graph at each Transformer layer. Afterward, it is pre-trained in large-scale KG-to-text corpora, and then fine-tuned in the downstream tasks, including question generation.
\paragraph{Task-Specific Models without Pre-training} We also adopted the recent task-specific models without pre-training as baselines, including Graph2Seq \cite{chen_toward_2022} and KTG \cite{bi_knowledge-enriched_2020}. Graph2Seq introduces a bidirectional graph encoder to model subgraphs in KG and generate questions by a graph-to-sequence generator with a coping mechanism. KTG proposes a knowledge-enriched, type-constrained, and grammar-guided model with auxiliary information to enrich input.
We report the baseline results directly if they use the same dataset as ours. Otherwise, we implement these baselines using the codes and parameters released by the original papers.
\subsection{Automatic Evaluation Metrics}
Following previous QG works \cite{kumar_difficulty-controllable_2019, bi_knowledge-enriched_2020, ke_jointgt_2021, chen_toward_2022} , we use BLEU-4 (B-4) \cite{papineni_bleu_2001}, METEOR (ME) \cite{denkowski_meteor_2014} and ROUGE-L (R-L) \cite{lin_rouge_2004} as our evaluation metrics. Initially, BLEU-4 and METEOR were designed to evaluate machine translation systems, and ROUGE-L was designed to assess text summarization systems.
\subsection{Experimental Results}
\begin{table*}
\centering
\caption{Results on WQCWQ1.1 and PathQuestions in six data proportion settings. The results marked with $\dagger$, $\ddagger$ and $\sharp$ are re-printed from the references \cite{bi_knowledge-enriched_2020}, \cite{chen_toward_2022} and \cite{ke_jointgt_2021}, respectively.}
\label{tab:main_res}
\scalebox{0.80}{
\begin{tabular}{c | ccc | ccc | ccc | ccc | ccc | ccc}
\toprule
\multicolumn{19}{c}{WQCWQ1.1} \\
\hline
Data Proportion & \multicolumn{3}{c|}{0.1\%} & \multicolumn{3}{c|}{0.5\%} & \multicolumn{3}{c|}{1\%} & \multicolumn{3}{c|}{5\%} & \multicolumn{3}{c|}{10\%} & \multicolumn{3}{c}{100\%} \\ \hline
Model & B-4 & ME & R-L & B-4 & ME & R-L & B-4 & ME & R-L & B-4 & ME & R-L & B-4 & ME & R-L & B-4 & ME & R-L \\ \hline
Graph2Seq & 0.00 & 0.87 & 5.98 & 7.01 & 12.65 & 26.86 & 8.53 & 13.51 & 31.70 & 17.17 & 21.56 & 45.11 & 19.69 & 23.25 & 46.64 & 29.56 & 31.14 & 58.34 \\
JointGT & 6.92 & 18.77 & 32.14 & 11.31 & 23.10 & 38.62 & 13.83 & 24.90 & 42.52 & 20.79 & 28.41 & 49.67 & 25.25 & 30.50 & 54.05 & 32.81 & 34.48 & 60.92 \\ \hline
AutoQGS & \textbf{15.26} & \textbf{22.07} & \textbf{42.97} & \textbf{21.56} & \textbf{27.06} & \textbf{48.70} & \textbf{24.09} & \textbf{28.77} & \textbf{51.04} & \textbf{29.58} & \textbf{32.43} & \textbf{56.90} & 31.81 & \textbf{33.82} & 59.07 & \textbf{36.93} & \textbf{36.63} & \textbf{63.82} \\
AutoQGS-T & 14.43 & 21.45 & 42.59 & 20.40 & 26.09 & 47.53 & 23.18 & 28.11 & 50.40 & 28.86 & 31.83 & 56.13 & \textbf{32.00} & 33.79 & \textbf{59.52} & 36.49 & 36.38 & 63.53 \\
\hline \hline
\multicolumn{19}{c}{PathQuestions} \\
\hline
KTG & \multicolumn{3}{c|}{-} & \multicolumn{3}{c|}{-} & \multicolumn{3}{c|}{-} & \multicolumn{3}{c|}{-} & \multicolumn{3}{c|}{-} & 45.58$\dagger$ & 52.31$\dagger$ & 73.21$\dagger$ \\
Graph2Seq & 1.01 & 4.99 & 12.07 & 2.63 & 10.64 & 41.45 & 17.59 & 18.35 & 51.44 & 43.43 & 31.34 & 67.51 & 42.72 & 32.20 & 67.62 & 61.48$\ddagger$ & 44.57$\ddagger$ & 77.72$\ddagger$ \\
JointGT & 43.15 & \textbf{35.91} & \textbf{69.57} & 51.05 & 41.23 & 73.23 & 51.89 & 42.19 & 73.62 & 55.90 & 43.25 & 74.49 & 57.39 & 43.51 & 75.26 & \textbf{65.89}$\sharp$ & \textbf{48.25}$\sharp$ & \textbf{78.87}$\sharp$ \\ \hline
AutoQGS & \textbf{43.46} & 33.55 & 68.23 & \textbf{56.30} & \textbf{41.95} & \textbf{74.68} & \textbf{58.69} & \textbf{42.48} & \textbf{75.50} & \textbf{61.55} & \textbf{44.81} & \textbf{76.68} & \textbf{60.73} & \textbf{44.92} & \textbf{76.95} & 65.13 & 47.50 & 76.80 \\
\bottomrule
\end{tabular}
}
\end{table*}
\begin{table}
\caption{Average gains on WQCWQ1.1 and PathQuestions over six data proportion settings.}
\label{tab:avg_gain}
\begin{tabular}{l|ccc|ccc}
\toprule
Dataset & \multicolumn{3}{c|}{WQCWQ1.1} & \multicolumn{3}{c}{PathQuestions} \\ \hline
Metrics & B-4 & ME & R-L & B-4 & ME & R-L \\ \hline
Graph2Seq & 12.88 & 12.97 & 17.98 & 29.82 & 18.83 & 22.26 \\
JointGT & 8.06 & 3.44 & 7.43 & 3.43 & 0.15 & 0.63 \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection{Automatic Evaluation}
Table \ref{tab:main_res} shows the detailed evaluation results comparing our proposed models against other state-of-the-art baselines in six data proportion settings, from 0.1\% to 100\%, respectively.
AutoQGS is implemented on the entity name serialization setup by default, and we also report the experimental results on entity type placeholder serialization setup on WQCWQ1.1, denoted as AutoQGS-T.
Since AutoQGS performs better than AutoQGS-T under most settings on WQCWQ1.1, we only report AutoQGS performance hereafter.
As we can see, both AutoQGS and AutoQGS-T outperform every baseline by a large margin on WQCWQ1.1, particularly in few-shot settings.
Table \ref{tab:avg_gain} shows the mean average gains across six settings. AutoQGS generally exceeds Graph2Seq/JointGT by 12.88/8.06 BLEU-4 points in WQCWQ1.1, and 29.82/3.43 in PQ, respectively.
Specifically, for Graph2Seq, the vocab depends on the training data heavily.
As the training instances decrease, the vocabulary becomes smaller and more words become OOV (out of vocabulary), which severely degrades the performance of this model.
Furthermore, our model outperforms JointGT in all six settings except in 0.1\% of PQ.
We speculate that, in the 0.1\% setting, there are only ten training instances (9793*0.001, we take the upper bound), which is too difficult for both JointGT and AutoQGS.
These results verify that AutoQGS can effectively and accurately generate questions based on SPARQL in low-resource scenarios.
\subsubsection{Impact of Complex Operations}
\begin{table}
\centering
\caption{Results on WQCWQ1.1 subsets dividing by whether containing complex operations.}
\label{tab:impact_of_op}
\begin{tabular}{c | ccc | ccc}
\toprule
WQCWQ1.1 & \multicolumn{3}{c|}{w/ OPs} & \multicolumn{3}{c}{w/o OPs} \\ \hline
Model & B-4 & ME & R-L & B-4 & ME & R-L \\ \hline
Graph2Seq & 22.17 & 26.63 & 51.72 & 30.11 & 31.32 & 58.79 \\
JointGT & 24.33 & 30.83 & 54.35 & 35.08 & 35.35 & 62.48 \\
AutoQGS & \textbf{35.75} & \textbf{36.73} & \textbf{62.11} & \textbf{36.03} & \textbf{36.09} & \textbf{63.37} \\
\bottomrule
\end{tabular}
\end{table}
Next, we evaluate how the complex operations affect the performance of our AutoQGS.
We keep the same divisions in WQCWQ1.1 and split data into two subsets by whether containing complex operations.
The statistics are reported in Table \ref{tab:data_statistics}. About 86.59\% of the data do not contain any complex operation, which means that this part is similar to the data used by previous works.
We report the comparison results on both subsets in Table \ref{tab:impact_of_op}.
Results show that AutoQGS performs better in both settings.
The results confirm the point that our model can generate questions from SPARQL better than others.
\subsection{Ablation Test}
\begin{table}
\caption{Ablation tests on WQCWQ1.1}
\label{tab:ablation_test}
\centering
\scalebox{0.95}{
\begin{tabular}{l|lll|lll}
\toprule
Data Proportion & \multicolumn{3}{c|}{0.1\%} & \multicolumn{3}{c}{1\%} \\ \hline
Model & B-4 & \multicolumn{1}{c}{ME} & \multicolumn{1}{c|}{R-L} & \multicolumn{1}{c}{B-4} & \multicolumn{1}{c}{ME} & \multicolumn{1}{c}{R-L} \\ \hline
AutoQGS & 15.26 & 22.07 & 42.97 & 24.09 & 28.77 & 51.04 \\
\quad w/o prompt text & 13.72 & 20.87 & 42.46 & 18.69 & 24.39 & 46.75 \\
\quad re/desc & 11.65 & 19.16 & 40.39 & 21.92 & 26.66 & 48.09 \\ \hline
& \multicolumn{3}{c|}{10\%} & \multicolumn{3}{c}{100\%} \\ \hline
AutoQGS & 31.81 & 33.82 & 59.07 & 36.93 & 36.63 & 63.82 \\
\quad w/o prompt text & 31.07 & 33.08 & 58.79 & 36.18 & 36.19 & 63.57 \\
\quad re/desc & 31.49 & 33.31 & 58.82 & 36.00 & 36.17 & 63.19 \\
\bottomrule
\end{tabular}
}
\end{table}
We conduct ablation tests to investigate the effectiveness of AutoQGS in four representative data proportion settings (0.1\%, 1\%, 10\%, 100\%) by removing the prompt text and replacing the prompt text with topic entity descriptions (\textsf{re/desc}, for short) one at a time.
As shown in Table \ref{tab:ablation_test}, removing the prompt text leads to significant performance reduction in all settings, particularly in 0.1\% and 1\%. The decline is consistent with our purpose as the prompt text is designed to enable the model to perform better in low-resource scenarios.
Similarly, we also observe performance decline by replacing prompt text with topic entity descriptions. This result again validates the effectiveness of prompt text, as it is also developed to mine relations between entities rather than only describe them.
\subsection{Human Evaluation}\label{human_eval}
\begin{table}
\caption{Human evaluations results ($\pm$ standard deviation). Pred and Natural mean the percent of predicates identification and naturalness score (0-5), respectively.}
\label{tab:human_evaluation_both}
\scalebox{0.96}{
\begin{tabular}{lcccc}
\toprule
\multicolumn{5}{c}{auto-prompter} \\ \hline
\multicolumn{1}{l|}{Graph Type} & \multicolumn{2}{c|}{Pred} & \multicolumn{2}{c}{Natural} \\ \hline
\multicolumn{1}{l|}{single-relation} & \multicolumn{2}{c|}{81.85\% (5.4)} & \multicolumn{2}{c}{4.25 (0.18)} \\
\multicolumn{1}{c|}{CVT} & \multicolumn{2}{c|}{71.11\% (7.2)} & \multicolumn{2}{c}{3.66 (0.25)} \\ \hline \hline
\multicolumn{5}{c}{question generator} \\ \hline
\multicolumn{1}{l|}{Data proportion} & \multicolumn{2}{c|}{0.1\%} & \multicolumn{2}{c}{1\%} \\ \hline
\multicolumn{1}{l|}{Model} & Pred & \multicolumn{1}{c|}{Natural} & Pred & Natural \\ \hline
\multicolumn{1}{l|}{JointGT} & 44\% (4.2) & \multicolumn{1}{c|}{2.76 (0.16)} & 72\% (4.9) & 3.72 (0.22) \\
\multicolumn{1}{l|}{AutoQGS} & 56\% (6.5) & \multicolumn{1}{c|}{4.36 (0.19)} & 80\% (5.0) & 4.60 (0.24) \\ \hline
\multicolumn{1}{l|}{} & \multicolumn{2}{c|}{10\%} & \multicolumn{2}{c}{100\%} \\ \hline
\multicolumn{1}{l|}{JointGT} & 80\% (3.5) & \multicolumn{1}{c|}{4.20 (0.30)} & 84\% (3.3) & 4.28 (0.19) \\
\multicolumn{1}{l|}{AutoQGS} & 84\% (4.1) & \multicolumn{1}{c|}{4.68 (0.32)} & 88\% (3.9) & 4.84 (0.16) \\ \hline \hline
\multicolumn{1}{c|}{} & \multicolumn{2}{c|}{Pred} & \multicolumn{2}{c}{Natural} \\ \hline
\multicolumn{1}{l|}{Golden} & \multicolumn{2}{c|}{95\% (2.1)} & \multicolumn{2}{c}{4.80 (0.18)} \\ \bottomrule
\end{tabular}
}
\end{table}
To further evaluate AutoQGS, we conduct two human evaluations on the auto-prompter and question generator results, respectively.
Considering that the goals of the two components are essentially similar, we decide to run the same two criteria following \cite{elsahar_zero-shot_2018}.
\paragraph{Predicates identification} Annotators were asked to estimate whether the generated text expresses all predicates in the given SPARQL (or subgraph) or not.
\paragraph{Naturalness} Annotators were requested to assign each generated text based on the fluency and readability by a score from 1 to 5, where (5) perfectly clear and natural, (3) grammatically correct but seems artificial, and (1) entirely not understandable.
For the evaluation of auto-prompter, we randomly sample 100 atomic subgraphs from the dataset.
For the evaluation of question generator, we randomly sample 100 instances from the test set.
We also collect the outputs from the most competitive baseline, JointGT, for comparison.
All evaluations are done with the help of 3 annotators.
Results in Table \ref{tab:human_evaluation_both} show some critical observations.
First of all, the auto-prompter has the ability to paraphrase predicates consistently and fluently. On the other hand, the question generator achieves remarkable results in all four settings, whereas the naturalness score in the 100\% setting is even higher than that of annotated questions (denoted as Golden).
Generally speaking, our AutoQGS can beat the corresponding baselines in both predicates identification and naturalness.
\subsection{Case Study}
\begin{figure*}
\centering
\includegraphics[width=1\textwidth,page=4]{imgs/QG-formal.pdf}
\caption{Case study}
\label{fig:case}
\end{figure*}
To provide a complete and visual presentation of AutoQGS, we provide a case in Figure \ref{fig:case}.
Steps 1 to 4 display the processes of how to get the prompt text by the auto-prompter.
Firstly, we re-write the SPARQL query (adding ``\textsf{?c ?num}'' in this case) to search all variables in KG and sample one from the results to instantiate a complete subgraph.
Secondly, for each atomic subgraph in the complete subgraph, we serialize it in corresponding pattern.
Afterward, we use the well-trained auto-prompter to get the prompt text of every atomic subgraph.
Finally, in order to explicitly describe the relationship between entities and variables, we add ``\textsf{(the [var])}'' behind the corresponding entities in the prompt text, as shown in Step 4.
The table in Figure \ref{fig:case} demonstrates the quality of generated questions (converted to lowercase).
Compared to JointGT, AutoQGS can generate questions more faithfully and completely.
More importantly, AutoQGS is able to generate questions with complex operations according to the SPARQL accurately.
For example, JointGT fails to express ``\textsf{earliest}'' for ``\textsf{ORDER By ?num LIMIT 1}'' expressed in SPARQL in all settings, whereas ours successfully capture it with only 10\% training data (marked in bold).
On the other hand, our model successfully generate ``\textsf{Australia}'' with the help of prompt text (marked in underscore). It proves that the prompt text successfully supplements information for question generation.
Moreover, in 0.1\% setting, the question generated by ours is much more natural and fluent.
\subsection{Error Analysis}
\begin{table}
\caption{Error analysis}
\label{tab:error_analysis}
\begin{tabular}{l|p{6cm}}
\toprule
SPARQL &
\begin{tabular}[p{6cm}]{@{}l@{}}SELECT DISTINCT ?x WHERE \{\\ \quad "Julius Caesar" people.deceased\_person.place\\ \_of\_death "The Theatre of Pompey"(the ?x) .\\ \}\end{tabular} \\ \hline
Prompt text &
The Theatre of Pompey (the ?x) was built during the reign of Julius Caesar. \\ \hline
Golden &
where was caesar when he \textbf{was stabbed}? \\ \hline
AutoQGS &
where did julius caesar die? \\ \hline \hline
SPARQL &
\begin{tabular}[p{6cm}]{@{}l@{}}SELECT DISTINCT ?x WHERE \{\\ \quad "Chile" location.country.form\_of\_government \\ "Presidential system"(the ?x) .\\ \quad "Presidential system"(the ?x) government.form\\ \_of\_government.countries "Brazil" .\\ \}\end{tabular} \\ \hline
Prompt text &
Chile has a Presidential system (the ?x) with a bicameral legislature. Brazil is a country with a Presidential system (the ?x) \\ \hline
Golden &
what are the government types of chile and brazil? \\ \hline
AutoQGS &
what type of government \textbf{is used} in both brazil and chile? \\
\bottomrule
\end{tabular}
\end{table}
Table \ref{tab:error_analysis} shows some failure cases on the WQCWQ1.1 test set in 100\% data proportion setting.
For convenience, variables in SPARQL are instantiated and put together with their variable names.
The most common mistake for the auto-prompter is that the distant supervision approach sometimes produces descriptions that are not consistent with the facts, as shown in the first case\footnote{In this example, the Theatre of Pompey was built during the latter part of the Roman Republican era by Pompey the Great, not Julius Caesar.}.
On the other hand, one of the frequent error patterns we find out for the question generator is incorrect statement.
For instance, in the second example, the statement ``\textsf{what type of government is used}'' is grammatically correct, but the usage is inappropriate.
Another error pattern is lack of knowledge.
In the first example, given the SPARQL and the prompt text, we still don't know that Caesar was stabbed.
But it is usual to include common sense in a question.
\subsection{Data Augmentation}
Automatically constructing question-answer pairs from knowledge bases is one of the objectives of question generation.
Based on the WQCWQ1.1 dataset, we augment data by replacing topic entities in SPARQL.
Specifically, given a SPARQL, we construct a new one to get a list of different topic entities by replacing the topic entities with variables.
For each set of entities in the result, we construct another SPARQL, in which only the named entity is different from the original one, to form a new sample.
Afterward, we use the best AutoQGS trained in 100\% data to generate the corresponding questions.
In order to multiply the data tenfold, we randomly choose ten instances from querying results for each item in WQCWQ1.1.
Finally, we generate a corpus of 340K question-SPARQL pairs for future research on question generation and question answering.
\section{Conclusion}
In this paper, we propose AutoQGS, an auto-prompt approach for low-resource KBQG from SPARQL.
Firstly, we generate questions directly from SPARQL to handle complex operations.
Secondly, we propose an auto-prompter to rephrase SPARQL into NL prompt text, smoothing the transformation from non-NL SPARQL to NL question with PLMs.
Lastly, we devise a question generator to generate questions given SPARQL and corresponding prompt text.
Experimental results show that our approach achieves state-of-the-art performance especially in low-resource scenarios.
Furthermore, we generate a dataset of 330k factoid complex question-SPARQL pairs for further KBQG research.
\section{Appendix}
\subsection{Hyper-Parameter Setting}\label{hyper_setting}
\begin{table}[htp]
\caption{Hyper-parameter search space. \textsf{[]} indicates that the listed numbers will be traverse one by one. }
\label{tab:hyper_search}
\begin{tabular}{l|c|c}
\toprule
Model & auto-prompter & question generator \\ \hline
Hyper-parameter & Setting & Search Space \\ \hline
Learning Rate & 3e-5 & {[}3e-5, 5e-5, 1e-4{]} \\
Warmup Proportion & 0.1 & {[}0.1, 0.2, 0.3{]} \\
Batch Size & 64 & {[}16, 24, 32, 64{]} \\
Beam Size & 10 & {[}5, 10{]} \\
Length Penalty & - & {[}1.0, 1.2{]} \\
Input Length & 512 & 512 \\
Output Length & 512 & 128 \\
Warmup Epoch & 10 & 50 \\
Early Stop Patience & - & 10 \\
Maximum Gradient Norm & 1.0 & 1.0 \\
Optimizer & Adam & Adam \\
Epsilon (for Adam) & 1e-8 & 1e-8 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htp]
\caption{Best assignments of hyper-parameters for question generator.}
\label{tab:hyper_assignments}
\begin{tabular}{c|cccccc}
\toprule
Dataset & \multicolumn{6}{c}{WQCWQ 1.1} \\ \hline
Data Proportion & 0.1\% & 0.5\% & 1\% & 5\% & 10\% & 100\% \\ \hline
Warmup Proportion & 0.2 & 0.1 & 0.1 & 0.2 & 0.2 & 0.2 \\
Batch Size & 16 & 16 & 32 & 32 & 32 & 64 \\
Beam Size & 10 & 10 & 10 & 10 & 10 & 10 \\
Length Penalty & 1.2 & 1.0 & 1.2 & 1.0 & 1.0 & 1.0 \\ \hline \hline
Dataset & \multicolumn{6}{c}{Pathquestions} \\ \hline
Data Proportion & 0.1\% & 0.5\% & 1\% & 5\% & 10\% & 100\% \\ \hline
Warmup Proportion & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 \\
Batch Size & 16 & 16 & 16 & 24 & 24 & 24 \\
Beam Size & 10 & 10 & 10 & 5 & 5 & 5 \\
Length Penalty & 1.2 & 1.2 & 1.2 & 1.0 & 1.0 & 1.0 \\
\bottomrule
\end{tabular}
\end{table}
We provided the detailed settings of hyper-parameters for training the auto-prompter and the question generator in Table \ref{tab:hyper_search}.
The table include the hyper-parameter settings for training auto-prompter and the hyper-parameter search space for training question generator.
We implement our models base on Huggingface’s Transformers \cite{wolf_transformers_2020}.
For auto-prompter, we train the model on unsupervised data for 10 epochs.
For question generator, We set \textsf{Warmup Epoch}, which means the expected training rounds, to control warmup steps.
The hyper-parameter search space and the best assignments are listed in Table \ref{tab:hyper_search}. We go through each combination of hyper-parameters in every few-shot settings to find the optimal one. The selection criterion is BLEU-4 on the validation set.
|
2,869,038,154,859 | arxiv | \section{Introduction}
\IEEEPARstart{T}{he} field of robotics has made remarkable progress in providing diverse sets of robotic platforms with different physical properties, sensor configurations, and locomotion capabilities (e.g., climbing, running, or flying). Thus, developing new planning algorithms that can be ubiquitously applied to a team of heterogeneous robots is a worthwhile endeavor, and applicable to a wide range of tasks from search and rescue to space exploration. However, for multi-agent planners to be used in unknown and uncertain environments, they should consider complex robot dynamics, uncertainty from imperfect exteroceptive and proprioceptive sensor measurements, update uncertainty when robots are in communication range, avoid obstacle collisions, and address desired multi-agent behavior.
One common approach is to fully address some but not all of the described requirements while assuming the rest can either be satisfied in future work, or can be combined with other existing methods. However, combining multiple approaches towards a unified motion planning framework satisfying all requirements can be nontrivial, requiring close examination of the overall feasibility and performance of such a complex system. Thus, in this work, we will examine the feasibility and performance of an end-to-end motion planning framework that addresses the above requirements termed as $\lq$Stochastic model predictive control for Autonomous Bots in uncertain Environments using Reinforcement learning' or SABER.
\begin{figure}[!t]
\centering
\includegraphics[width=1\columnwidth]{Figures/Intro/intro_pic.pdf}
\caption{\textbf{SABER framework.} SABER combines controls (stochastic model predictive control), vision (simultaneous localization and mapping), and machine learning (RNN and DQN), to provide local and globally optimized solutions in unknown and uncertain environments.}
\label{intro_pic}
\end{figure}
\subsection*{Summary of Our Contributions}
(1) SABER is an end-to-end motion planning framework for a team of heterogeneous robots that unifies controls, vision, and machine learning approaches to plan paths that account for safety, optimality, and global solutions (our complete framework is shown on a UGV-UAV team).
(2) Cooperative localization algorithms are used for cross-communicating robots, which may include both non-Gaussian and Gaussian measurement noise, where uncertainty is modeled with recurrent neural networks (RNNs) for each agent's sensor configuration using outputs from simultaneous localization and mapping algorithms (SLAM).
(3) Instead of simple heuristics when sampling the map for target positions, we employ Deep Q-learning (DQN) for high-level path planning, which is easily modifiable for learning desired multi-agent behavior and finds global solutions (DQN scalability for more than two robots is also evaluated).
\begin{figure*}[!t]
\centering
\includegraphics[width=6.5in]{Figures/Methods/saber_methods.pdf}
\caption{\textbf{SABER Algorithm.} This figure demonstrates the overall SABER planning algorithm in the testing phase, which can plan paths for one or more robots simultaneously. At timestep k, the environment provides information to robots that either carry a LiDAR or RGB camera and IMU; for the LiDAR configuration, a particle-filter SLAM is implemented, while for the RGB configuration, Visual-Inertial Odometry SLAM (VIO SLAM) is implemented. The sensors provide either scans or distance to feature information to a recurrent neural network model (which serve as inputs), and outputs the propagation of state uncertainty for future timesteps. If two or more robots are within communication range, a distributed Kalman filter updates the current and future states and their uncertainties to a more accurate estimate. These updated states and uncertainties are used to update the chance constraints for obstacle avoidance. These constraints are then considered by a stochastic MPC controller, which follows a given target position, provided by a deep Q-learning (DQN) agent that aims to move the robot towards a global goal. DQN uses the relative distances between the robots and the respective obstacles as its states, provides a target position for all robots as its actions, and is trained on several different maps with obstacles randomly distributed in each. Note, that the SMPC, SLAM, and RNNs components run on each robot individually, however, the DQN is run on a centralized base (which may be on the robot itself).}
\label{fig_scram}
\end{figure*}
\section{Related Works}
\label{relatedWorks}
Several works that examine planning for heterogeneous robots (typically composed of a two robot UGV-UAV team) have focused mainly on fusing different sensor data to build a unified global map \cite{heter_3}, integrating several components such as path planning, sensor fusion, mapping, and motion control towards a single framework \cite{heter_1}, or strictly analyzing multi-agent localization (i.e., multi-SLAM) \cite{heter_2}. While we also consider a UGV-UAV team as done in the above works, here, we are more concerned with the feasibility (i.e., computation time) of such a complex system, and also in how uncertainty is not only estimated for robots with different sensor configurations, but how it's tightly coupled with a local stochastic model predictive controller (SMPC) towards coordinating multi-agent behavior.
We also seek to address the major challenge for multi-agent (or even single-agent) planners, which is to estimate a path that is both safe (e.g., considers uncertainty in agent/obstacle avoidance) and fast (e.g., finding the shortest path to the goal). This is significant because conservative approaches to safety would lead to over avoidance or non-optimal solutions, and high-risk behavior may cause undesired collisions. Currently, few motion planners fully investigate this problem. For example, in \cite{wurm_coordinating_2013}, the cost of reaching a target position for each robot in a heterogeneous team depends on its individual characteristics (e.g., varying sensors, travelling speed, and payloads). However, by not considering uncertainty in their planner, their cost-to-go function can be significantly affected by disturbances. Conversely, a multi-agent planner that does consider probabilistically-safe motion planning can be found in \cite{bajcsy_scalable_2019}. Still, their planner may lead to conservative solutions as they assume a worst-case behavior approach to safety. SABER addresses the problem of avoiding obstacles without over avoidance by using an RNN, which predicts and propagates future state uncertainty dynamically and does not make the assumption that uncertainty increases when no future measurements are received \cite{Schperberg_rnn}. The downside with an RNN (which is typical with learning-based approaches) is that the accuracy of the uncertainty estimates is directly correlated with the quality of the data collection.
The geometric representation of obstacles is also critical for planning. For example, FASTER \cite{tordesillas_faster_2019} is a decentralized and asynchronous planner where obstacles are represented as outer polyhedrals (estimated from convex decomposition) and applied as constraints into the optimization. In SABER, we also represent obstacles as polyhedral constraints for each timestep, however, we decompose them into a disjunction of linear chance constraints (thus, obstacle $\lq$size and location' are a function of exteroceptive and proprioceptive uncertainty). Using chance constraints in motion planning is not new, and has been shown with success in single-agent planners such as \cite{blackmore_chance-constrained_2011} and \cite{risk_aware}. In this work, our chance constraints are also influenced by the cross-communication uncertainty of a heterogeneous robot team (see \ref{methods:coop}). Additionally, while obstacle constraints can be explicitly used in the optimization, other works show a learning-based approach to avoid collisions by modeling the distribution of promising regions for travel \cite{region_learning} or predicting the separation distance between the robot and its surroundings \cite{chase_kew_neural_2021}. Our work is a hybrid approach, where we use the RNN to predict uncertainty of state estimates (which affect the $\lq$size' of polyhedral obstacles), but still use these obstacles as constraints in our SMPC optimization. This choice sacrifices computation time, but may be more generalizable to environments not observed in training and prevent collisions.
Finding a suitable path to the goal also has a wide array of different solutions. Most commonly, sampling-based methods which do not consider workspace topology (such as grid-sampling \cite{grid_sample} or rapidly-exploring random tree or RRT and its variants \cite{gammell_informed_2014}) can lead to very dense roadmaps and may not scale well when the shortest path to the goal is desired. This issue has been investigated in \cite{probabilistic_roadmaps}, which uses a self-supervised learning approach to build sparse probabilistic roadmaps (PRM) for bias sampling (sampling only regions the robot is likely to safely travel). Moreover, using a learning-based approach for path planning also has the potential for integrating semantic behavior that can be gained from multi-agent coordination, as evidenced in \cite{multi_agent_rendezvous} and \cite{Baker2020Emergent}. Motivated by these works, we use a DQN for high-level planning, where simple modifications of a reward function can yield desired multi-agent behavior (e.g., rewarding agents based on proximity or reaching the goal concurrently). Nevertheless, the tradeoff of using a DQN compared to sampling-based methods is that it cannot guarantee asymptotic optimality (e.g., RRT-star) or probabilistic completeness (e.g., PRM). However, for our DQN, we are primarily concerned with the feasibility relative to our application (i.e., finding a near-optimal path in a computationally efficient manner while satisfying multi-agent behavior).
\begin{algorithm}[!ht]
\SetAlgoLined
\DontPrintSemicolon
Initialize state $X_{k}^{i:n_{r}}$, goal $X_{goal}^{i:n_{r}}$, $dt$, horizon $N$, robot size $r_{i:n_{r}}$, timestep $k$, empty $Map^{i:n_{r}}$, uncertainty $\Sigma_{k:k+N+1}^{i:n_{r}}$, error to goal $\epsilon$, number of robots $n_r$ \\
\BlankLine
\While{$ \| X_{k}^{i:n_{r}} - X_{goal}^{i:n_{r}} \|_2 > \epsilon$}{
\textbf{if} LiDAR configuration: \\
\ \ $X_{k}^{i:n_{r}}, \Sigma_{k}^{i:n_{r}}, Map^{i:n_{r}} \leftarrow \text{Particle-filter SLAM}(\text{odom, scans, } Map^{i:n_{r}})$ \\
\ \ $\Sigma_{k+1:k+N+1}^{i:n_{r}} \leftarrow \text{RNN}^{i:n_{r}}(X_{k}^{i:n_{r}}, \text{scans})$ \\
\textbf{if} RGB camera configuration:
\ \ $X_{k}^{i:n_{r}}, \Sigma_{k}^{i:n_{r}}, Map^{i:n_{r}} \leftarrow \text{VIO SLAM}(\text{IMU, RGB, } Map^{i:n_{r}})$ \\
\ \ $\Sigma_{k+1:k+N+1}^{i:n_{r}} \leftarrow \text{RNN}^{i:n_{r}}(X_{k}^{i:n_{r}},\ \text{features})$ \\
\textbf{continue:}\\
$X_{ref}^{i:n_{r}} \leftarrow \text{Deep Q-Learning}(X_{k}^{i:n_{r}}, X_{goal}^{i:n_{r}}, Map^{i:n_{r}})$ \\
$\mathcal{O}_{j:n_{obs}} \leftarrow \text{checkObstacles}(X_{k}^{i:n_{r}}, r_{i:n_{r}}, Map^{i:n_{r}})$ \\
$X_{k+1:k+N+1}^{i:n_{r}}, U_{k+1}^{i:n_{r}} \leftarrow \text{SMPC}(X_{ref}^{i:n_{r}}, X_{k}^{i:n_{r}}, \mathcal{O}_{j:n_{obs}}, \Sigma_{k:k+N+1}^{i:n_{r}})$ \\
\textbf{if }$\forall robot \ i,j:n_{r} \text{ within communication range:}$ \\
\ \ $\Sigma_{k:k+N+1}^{i:n_r},X_{k:k+N+1} \leftarrow \text{CoopLocalization}(\Sigma_{k:k+N+1}^{i:n_{r}}, X_{k:k+N+1}) $\\
}
\caption{SABER}
\label{SABER}
\end{algorithm}
\section{Methods}
The SABER framework contains both learning (requiring data collection) and non-learning components (traditional control schemes). The non-learning components consist of an SMPC and a distributed Kalman filter, while the learning components consist of an RNN and DQN agent. The RNN and DQN components are trained separately and offline before being implemented into the overall system for online deployment (note, that the RNN is supervised by the SMPC controller on each robot, while the DQN is not integrated with any other component during training). Overall, the algorithm is structured as an SMPC problem, which moves a robot toward a target location as formulated in \ref{methods:smpc}. By using state uncertainties and obstacle locations, obstacles are represented as chance constraints within the SMPC cost function (\ref{methods:smpc:cc}, \ref{methods:smpc:cc2}). If two or more robots are in communication range, their state and uncertainty values are updated using a distributed Kalman filtering approach as described in \ref{methods:coop}. To quickly propagate state uncertainties for future timesteps, we use different RNN models based on the robot's sensor configuration, as explained in \ref{methods:rnn}. In \ref{methods:dqn}, we formulate a DQN approach, providing the SMPC with target locations which help generate trajectories that move the robots toward a global goal and prevent local minima solutions. See Algorithm \eqref{SABER}, Fig.~\ref{fig_scram}, or the attached video\footnote{\url{https://youtu.be/EKCCQtN5Z6A}} for an overview of the methods, and \ref{implementation} for implementation details.
\subsection{Stochastic Model Predictive Control Formulation}
\label{methods:smpc}
The goal of the cost function (equation \eqref{objfunction}) is to find the optimal control value $U_{k}$ that minimizes the distance between the current and predicted states ($X_{k\rightarrow N+1}$) with a reference state or trajectory ($X^{(ref)}$) while under equality and inequality constraints -- where $X_{k}^{i}$ is given by the results of localization for each robot $i$, and $k$ is the current timestep ($Q$ and $R$ are control matrices, and $P$ is described further below):
\mathchardef\mhyphen="2D
\begin{equation}
\label{objfunction}
\begin{split}
\min_{U_{k:k+N\mhyphen1}^{i}} \sum_{k}^{k+N-1} \lVert X_{k+1}^{i}-X_{k+1}^{(ref)i} \rVert^{2}_{Q^{i}}
+ U_{k}^{i\top}R^{i}U_{k}^{i} \\ + \lVert X_{k+N+1}^{i}-X_{k+N+1}^{(ref)i} \rVert^{2}_{P^{i}}
\end{split}
\end{equation}
\begin{alignat}{2}
& \text{s.t.} \quad
& & {X_{k+1}^{i}=f^{i}(X_{k}, U_{k})=A^{i} X_{k}^{i}+B^{i}U_{k}^{i}}+W_k^{i} \\
&&& X_{k}^{i} \sim \mathcal{N}(\Bar{X_{k}^{i}}, \Sigma_k^{i}), W_k^{i} \sim \mathcal{N}(0, \sigma^{2i}) \\
&&& X_{limit}^{i} \geq |X_{k}^{i}|, U_{limit}^{i} \geq |U_{k}^{i}|
\end{alignat}
\textit{Obstacle Constraints:}
\begin{gather}
\text{Pr}\left(\bigwedge_{j=1}^{N_{O}} \mathcal{O}_{j}\right) \geq 1-\Delta
\end{gather}
Constraint (2) represents multiple shooting constraints, which ensure that the next state is equivalent to a time-invariant linear discretized model, where $A$ and $B$ represent the robot's dynamic matrices. Uncertainty in state as well as the addition of a non-unit variance random Gaussian noise ($W$) is shown in (3). Note, that the propagation in state uncertainty ($\Sigma_{k+1\rightarrow N+1}^{i}$) is received from an RNN model (Section \ref{methods:rnn}), and can be affected by cooperative localization algorithms (Section \ref{methods:coop}) at timestep $k$, if multiple robots are in communication range at timestep $k$.
Limits on state and controls variables are imposed by constraint (4). For robustness \cite{rawlings_model_2017}, a terminal cost is included with a weighting matrix $P$ which can be obtained by solving the discrete-time Riccati equation \eqref{ricatti}:
\begin{equation}
\label{ricatti}
\begin{split}
A^{i\top}P^{i}A^{i} - P^{i} - A^{i\top}P^{i}B^{i}(B^{i\top}P^{i}B^{i}+R^{i})^{-1}B^{i\top}P^{i}A^{i}+\\Q^{i}=0
\end{split}
\end{equation}
\subsubsection{Chance Constraints for Obstacle Avoidance}
\label{methods:smpc:cc}
Constraint (5) represents chance constraints that enable obstacle avoidance subject to uncertainty in convex regions as done in \cite{blackmore_chance-constrained_2011}. This constraint can be rewritten as a disjunction of linear constraints for obstacle $\mathcal{O}_{j}$:
\begin{equation}\mathcal{O}_{j} \Longleftrightarrow \bigvee_{k \in T\left(\mathcal{O}_{j}\right)} \bigwedge_{i \in G\left(\mathcal{O}_{j}\right)} a_{i}^{{\top}} \Bar{X}_{k} - b_{i} \geq c_{i}\end{equation}
where $G\left(\mathcal{O}_{j}\right)$ is the set of linear constraints (indexed by $i$) for each obstacle (indexed by $j$), $T\left(\mathcal{O}_{j}\right)$ is the set of timesteps in the MPC prediction horizon (indexed by $k$), $a_{i}$ is the vector normal to each line constraint and directed toward state $\Bar{X}_{k}$, $r$ is the radius/size of the robot, and $c_{i}$ is given by:
\begin{equation}c_{i}=\sqrt{2 a_{i}^{\top} \Sigma_{k} a_{i}} \cdot \operatorname{erf}^{-1}(1-2 \delta_{j}), \ \delta_{j} \leq 0.5 \end{equation}
Important to consider is that the degree of $\lq$risk' can be controlled by changing the values of $\delta_{j}$ (related to $\Delta)$ for each obstacle $\mathcal{O}_{j}$. Lower values lead to more evasive behavior (robot moves further away from obstacles) while higher values lead to more risky behavior (robot moves closer to obstacles).
If obstacles are assumed circular (centered at $x_{o_j}$, $y_{o_j}$), we can use the following equation, where $a_{i}$ is equal to an identity vector, $x_{k}$ and $y_{k}$ is the center position of the robot, and only a single $c_{j}$ value needs to be calculated per obstacle:
\begin{equation}-\sqrt{\left(x_{k}-x_{o_j}\right)^{2}+\left(y_{k}-y_{o_j}\right)^{2}}+(r+ c_{j}) \leq 0 \end{equation}
\subsubsection{Mixed-Integer Nonlinear Programming}
\label{methods:smpc:cc2}
To more effectively consider the disjunctive convex program for polygon-shaped obstacles, as introduced by (7), we can change these constraints into a mixed integer format (we assume the line constraints are in the x-y plane, however, the same equations can be used for the x-z, and y-z planes respectively):
\begin{equation}\mathcal{O}_{j} \Longleftrightarrow \bigvee_{k \in T\left(\mathcal{O}_{j}\right)} \bigwedge_{i \in G\left(\mathcal{O}_{j}\right)} I_{i,j} \text{dist}(\Bar{X}_{k}, a_{i}, m_{i}, b_{i}) \geq I_{i,j}( r+c_{i})\end{equation}
\begin{equation}
x_{l} = a^{*}_{i}x_{k}-y_{k}+b_{i} / (a^{*}_{i} - m_{i})
\end{equation}
\begin{equation}
y_{l} = m_{i}x_{k}+b_{i}
\end{equation}
\begin{equation}\text{dist(*)}=\left|-m x_{k}+y_{k}-b_{i}\right| / \sqrt{m_{i}^{2}+1}\end{equation}
\begin{equation}\text {dist(*)}\left\{\begin{array}{ll}{\textbf{IF } \text{sign}(y_{l} - y_{k}) = \text{sign}(a_{y})} \bigvee \\ \ \ \ \ \text{sign}(x_{l}-x_{k})=\text{sign}(a_{x}), & {-\text{dist(*)}} \\ {\textbf{ELSE}} & {\text{dist(*)}}\end{array}\right.\end{equation}
\begin{equation}I_{i,j}=\{0,1\} \forall i,j\end{equation}
\begin{equation}\sum_{i=1}^{size(I_{j})} I_{i,j} \geq 1 \quad \forall j\end{equation}
where $m_{i}$ and $b_{i}$ are the slope and y-intercept of each line constraint ${i}$ belonging to obstacle $\mathcal{O}_{j}$, $a^{*}_{i}$ is $a_{y}/a_{x}$, $x_{k}$ and $y_{k}$ are the $x$ and $y$ position retrieved from robot state $\Bar{X}_{k}$, and the coordinates of the point on the line constraint closest to $\Bar{X}_{k}$ is represented by $x_{l}$ and $y_{l}$ (equations (11) and (12)). The dist(*) function approximates the distance between $\Bar{X}_{k}$ and one of the linear constraints of an obstacle as shown in equation (13). Equation (14) returns a positive distance if the robot is $\lq$outside' the obstacle boundary, and a negative distance if the robot is $\lq$inside' the obstacle boundary (a negative distance would cause the line constraint to fail). By definition of constraint (10), only one or more of the line constraints need to be satisfied per obstacle $\mathcal{O}_{j}$, which is ensured by using binary integer variables under constraints (15) and (16) (e.g., for line constraint $i$ belonging to $\mathcal{O}_{j}$, if $I_{i,j}=1$, the robot is outside this obstacle). For simplicity, we assume we have a $\lq$perfect' object detection system. If the robot is close enough to an obstacle, the obstacle is automatically $\lq$seen' and embedded into the SMPC cost function.
\subsection{Cooperative Multi-Agent Localization}
\label{methods:coop}
While the propagation of uncertainty for each robot is calculated using an RNN (see \ref{methods:rnn}), updating the uncertainty after information is exchanged with another robot is done using a distributed Kalman filtering approach \cite{coop_local2000}. Thus, when two or more robots are in communication range (as pre-specified by the user), their individual uncertainty estimates should be updated to correctly reflect the gain from additional sensor information.
Equations (17) - (25) describe how the pose for robot $i$ is updated while in communication range of another robot $j$. The same equations can be further extrapolated to consider additional robots as explained in \cite{coop_local2000}.\\
\textit{For $\forall i,j$ and $k \rightarrow k+N+1$:} \\
\textit{Propagation}:
\begin{equation}
\Sigma_{k+1}^{i}, \Sigma_{k+1}^{j} = RNN^{i}(*), RNN^{j}(*)
\end{equation}
\begin{equation}
X_{k+1}^{i}, X_{k+1}^{j} = f^{i}(X_{k}^{i}, U_{k}^{i}), f^{j}(X_{k}^{j}, U_{k}^{j})
\end{equation}
\textit{Update}:
\begin{equation}
\Bar{X}_{k+1}^{+i} = \Bar{X}_{k+1}^{i} + K_{k+1}^{i}(Z_{k+1}^{ij}-(\Bar{X}_{k+1}^{i}-\Bar{X}_{k+1}^{j}))
\end{equation}
\begin{equation}
S_{k+1}^{ij} = \Sigma_{k+1}^{i} + \Sigma_{k+1}^{j} + R_{k+1}^{ij}
\end{equation}
\begin{equation}
Z_{k+1}^{ij} = X_{k+1}^{i} - X_{k+1}^{j}
\end{equation}
\textit{Update A (first time robots meet)}:
\begin{equation}
\Sigma_{k+1}^{+ij} = \Sigma_{k+1}^{i}(S_{k+1}^{ij})^{-1}\Sigma_{k+1}^{j}
\end{equation}
\begin{equation}
K_{k+1}^{i} = \Sigma_{k+1}^{i}(S_{k+1}^{ij})^{-1}
\end{equation}
\textit{Update B (all other times robots meet)}:
\begin{equation}
\Sigma_{k+1}^{+ij} = \Sigma_{k+1}^{ij}-[\Sigma_{k+1}^{i}-\Sigma_{k+1}^{ij}](S_{k+1}^{ij})^{-1}\Sigma_{k+1}^{j}[\Sigma_{k+1}^{ij}-\Sigma_{k+1}^{j}]
\end{equation}
\begin{equation}
K_{k+1}^{i} = (\Sigma_{k+1}^{i}-\Sigma_{k+1}^{ij})(S_{k+1}^{ij})^{-1}
\end{equation}
where $Z_{k+1}^{ij}$ is the relative pose measurement between robot $i$ and robot $j$ ($X_{k}$ is received by current localization, and $X_{k+1\rightarrow k+N+1}$ can be retrieved by the SMPC solution), and $R^{ij}$ is the relative measurement noise between robot $i$ and robot $j$.
\subsection{Recurrent Neural Network for Uncertainty Propagation}
\label{methods:rnn}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{Figures/Methods/saber_methods_networks.pdf}
\caption{\textbf{Network structures.} We show the RNN structure used to model an EKF from a VIO SLAM algorithm in (\textit{A}), or a particle-filter SLAM algorithm in (\textit{B}) (\ref{methods:rnn}). The inputs are shown in orange, and correspond to either features/robot position (using VIO SLAM) or LiDAR scans/robot position (using particle-filter SLAM). The outputs are shown in red, and correspond to the x-y covariance matrix (which represents uncertainty in x-y position). The layer type is color coded below, where green represents a simple RNN layer, and purple a dense layer.}
\label{networks}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{Figures/Results/rnn_loss.pdf}
\caption{\textbf{Training loss.} Here we show the training loss for the RNNs, which were trained on uncertainty covariance outputs (in position) of a Visual-Inertial SLAM in (\textit{A}) and a particle-filter SLAM in (\textit{B}). The training was done using 500 epochs and on 4 different maps. Note, that the noise observed in (\textit{B}) may be due to the particle-filter estimations/simplifications done in \cite{Gmapping}.}
\label{rnn_loss}
\end{figure}
Because our SMPC calculates the optimal control ($U_{k}$) based on a prediction horizon ($X_{k+1\rightarrow k+N+1}$, $U_{k\rightarrow k+N}$), we must also provide as input the propagation of uncertainty ($\Sigma_{k+1 \rightarrow k+N+1}$) at each timestep. As described in more detail in \cite{Schperberg_rnn}, an RNN can provide a computationally fast prediction of future state uncertainties (as it only requires a small network) and can operate in continuous space, making it ideal for online replanning in complex environments. This is achieved as the RNN can model the behavior of a filter (e.g., particle filter or EKF) from SLAM algorithms. However, in this study, we have multiple robots with different sensor configurations, which requires training separate RNN models for each.
In this work, we estimate state uncertainty using two different SLAM algorithms, a Rao-Blackwellized particle-filter SLAM (LiDAR camera configuration) and a Visual-Inertial Odometry (VIO) SLAM (RGB camera configuration). In the particle-filter case, the following equation is used for factorization:
\begin{equation}\begin{array}{c}
p\left(x_{1: k}, m \mid z_{1: k}, u_{1: k-1}\right)= \\
p\left(m \mid x_{1: k}, z_{1: k}\right) \cdot p\left(x_{1: k} \mid z_{1: k}, u_{1: k-1}\right)
\end{array}\end{equation}
where $x_{1:k}$ = $x_{1},...,x_{k}$ is the robot's trajectory, $z_{1:k}$ = $z_{1},...,z_{t}$ are the given observations, $d_{1:k-1}$ = $d_{1},...,d_{k-1}$ are the odometry measurements, and $p(x_{1: k}, m \mid z_{1: k}, d_{1: k-1})$ is the joint posterior estimate about map $m$ (see \cite{AMCL}).
For training the RNN for the particle-filter SLAM case, we have 362 input units for each timestep $k$. The first 2 units is the center position of the robot ($x$ and $y$), and the 360 other units represent the range distances from LiDAR scans (e.g., for timestep k, we have $d_{k}^{1:360}$ relative scan distances). The output layers, which use a linear activation function, correspond to the robot's 2$\times$2 x-y covariance matrix which is flattened into a 4$\times$1 array or 4 output units. For the VIO SLAM configuration, we used the same methods for training the RNN as described in \cite{Schperberg_rnn}. See Fig.~\ref{networks} for an overview of the network structure, and \ref{implementation} for further implementation details.
\subsection{Deep Q-Learning (DQN) for Global Planning}
\begin{figure*}[!t]
\centering
\includegraphics[width=400px]{Figures/Methods/dqn_method.pdf}
\caption{\textbf{DQN Training and Testing Procedure.} In (\textit{A}), we show the neural network structure used in our DQN algorithm (\ref{methods:dqn}). The network maps the inputs (i.e., states or relative distances between robots and obstacles/goals) to the outputs (i.e., actions or next target positions for the robots). The states and actions are connected by a linear neural network model (blue). In (\textit{B}) we visually show the training process of the DQN for a 2 to 5 robot team, where all robots were trained to go to the goal location while avoiding obstacles (obstacles are randomized for each episode). The average rewards (calculated from 25 episodes at a time and divided by number of robots) are shown across the 35,000 episodes of training (training time was 5 hours). In (\textit{C}) we show an example of how the environment can be transcribed into a 2D plot and apply the DQN to traverse multiple robots toward their goals. We also allow the UAV to fly over the obstacles (and assume we know the height of the obstacle \textit{a priori}) while the UGV must avoid it. In (\textit{D}) we show another example of our DQN, but this time the 2-robot UAV and UGV team have separate goal locations, and we add a reward incentive when both robots are near each other at each timestep.}
\label{fig_dqn_methods}
\end{figure*}
\label{methods:dqn}
\subsubsection{DQN formulation}
To provide local target positions ($X_{ref}^{i:n_{r}}$) that direct the robots toward a global goal ($X_{goal}^{i:n_r}$), we implement a DQN agent and use the Bellman equation:
\begin{equation}\label{bellman}Q(s_{k}, a_{k})=r+\gamma \max _{a_{k+1}} Q\left(s_{k+1}, a_{k+1}\right)\end{equation}
where the state is represented by $s_{k} \in \mathbb{R}^{({N}_{j}\times n_{r})+n_{r}}$, action by $a_{k} \in \mathbb{R}^{9^{n_{r}}}$, learning rate by $\gamma$, $n_{r}$ is the number of robots, and reward function by $r$ (where ${N}_{j}$ is the number grid spaces required to represent each obstacle; described further below). The idea of DQN is to use the Bellman equation \eqref{bellman} and a function approximator (with neural networks) to reduce the loss function (we use the Adam optimizer) \cite{Q-learning}.
Ultimately, the goal of the DQN-agent is to generate target positions at each timestep k for multiple robots and to move them towards a global goal position. To accomplish this and generalize our methods to any map (or changing the location of obstacles after each episode), we use the relative distances between the robot and surrounding obstacles and the relative distance between the robot's position and the global goal as our states: $s_{k} = (d_{j}^{i:n_{r}}, d_{j+1}^{i:n_{r}}, d_{j+2}^{i:n_{r}},...d_{N_{O}}^{i:n_{r}}, d_{goal}^{i:n_{r}})$, where $d_{j}^{i:n_{r}}$ is the relative distance between the robot and obstacles and can be described by $\lVert X_{k}^{i:n_{r}}-\mathcal{O}_{j}^{i:n_{r}}\lVert$, and $d_{goal}^{i:n_{r}}$ by $\lVert X_{k}^{i:n_{r}}-X_{goal}^{i:n_{r}}\lVert$ ($X_{k}$ is assumed to be the $x$ and $y$ position of the robot). Our DQN agent is trained on a 2D-grid map, where the robots and obstacles are represented as squares (1m$^{2}$) within the grid. Thus, the actions permitted by the robot is an 8-directional x-y movement (or no movement) at each timestep. The reward function is simply formulated by providing a positive reward $(+1)$ if the robot gets to the goal and a negative reward $(-1)$ if the robot hits an obstacle, which results in termination of the episode (we allow the UAV to $\lq$fly' over some obstacles by modifying its reward function to not receive a penalty for hitting that obstacle). To test different behaviors, we also added a reward $(+0.1)$ at each timestep when both robots are within a 2 meter distance (see (D) in Fig.~\ref{fig_dqn_methods}).
We also assume the dimensions of each obstacle are known \textit{a priori} (to estimate this without this assumption may require an object detection pipeline). Thus, by knowing the height of each obstacle, a UAV can use this value in its chance constraint to fly over obstacles. Finally, mapping our states to our actions is done through a linear neural network. For an overview of the network structure, and the training/testing process, see Fig.~\ref{fig_dqn_methods}.
\section{Experimental Validation}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{Figures/Results/saber_results.pdf}
\caption{\textbf{SABER Algorithm Results.} This figure demonstrates the overall SABER planning algorithm. In (\textit{A}) and (\textit{B}) we first show the capability of the SMPC-RNN to navigate the UGV and UAV in a densely populated space. In (\textit{C}), we show that the SMPC-RNN of the UGV cannot get to the goal state, because of the occurrence of a local minima. However, with a DQN (which provides a global path illustrated by triangles), the UGV (orange) can correctly maneuver around the obstacle. The UAV (purple) can simply use it's z-axis to fly above the obstacle. A more complex example is shown in \textit{(D)}, where both robots are directed towards different goal locations simultaneously.}
\label{fig_saber}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.83\columnwidth]{Figures/Results/rnn_results.pdf}
\caption{\textbf{RNN Results.} In this figure, we show the true covariance value (where $\lq$xx' represents the covariance of the center of mass in the x position as an example) in blue and the predicted covariance in orange for about 430 seconds of data. This is done when modeling the uncertainty using VIO SLAM (\textit{A}) and particle-filter SLAM (\textit{B}) on a test map. Note, that the predicted and true values almost perfectly align, demonstrating the RNN's ability to make valid uncertainty predictions. Lastly, we show more explicitly in the graphs, that when more features are tracked (green arrow) the lower the estimated uncertainty, while fewer features corresponded to higher uncertainty (red arrow).}
\label{rnn_results}
\end{figure}
\subsection{Implementation details}
\label{implementation}
SABER is demonstrated on a UGV (Turtlebot3) equipped with a 360 degree LiDAR camera, and a UAV (Quadrotor) equipped with a RGB camera in a Gazebo simulation running in real-time with additive noise. The dynamic equation of motion for the UGV assumes states composed of center of mass position and heading angle ($X=[x, y, \theta]$), actions composed of linear and angular velocity ($U=[v, \omega]$) and the following matrices for the discretized equation:
\begin{equation*}A = \mathbb{I}\in \mathbb{R}^{3\times3}, \ B=\left[\begin{array}{cc}
\cos (\theta) \delta t & 0 \\
\sin (\theta) \delta t & 0 \\
0 & \delta t
\end{array}\right]\left[\begin{array}{l}
v \\
w
\end{array}\right]\end{equation*}
The UAV assumes similar dynamics as in \cite{quadrotor}, where the states are center of mass position, linear velocity ($x$, $y$, and $z$ components), angle, and angular velocity (we consider pitch and roll but keep yaw fixed) or $X=[x,v_{x},\theta_{1}, \omega_{1}, y, v_{y}, \theta_{2}, \omega_{2}, z, v_{z}]$, and actions composed of thrust $U \in \mathbb{R}^{3\times1}$. The $A$ and $B$ matrices and their parameters are fully described in \cite{quadrotor}, where ${A}\in \mathbb{R}^{10\times10}$ and ${B}\in \mathbb{R}^{10\times3}$.
To solve the cost function \eqref{objfunction}, we use the mixed-integer nonlinear programming (MINLP) solver $\lq$bonmin' \cite{bonami_algorithmic_2008}, as we need to consider both integer and continuous variables. Constraints and problem formulation were setup using CasADi \cite{Casadi} running on a laptop with an Intel Core i7‐8850H CPU, and NVIDIA Quadro P3200 GPU.
To collect data for training our RNN model for the UGV, we used the GMapping package \cite{Gmapping} to create a map of the environment, and then implemented the AMCL package \cite{AMCL} to track the robot's pose and receive the uncertainty covariances using this map with particle-filter SLAM (uncertainty outputs from the filter are considered as the $\lq$ground truth'). In the UAV case, we used the XIVO SLAM package \cite{xiao-semantic-mapping} to make localization and covariance estimations (XIVO uses an Extended-Kalman Filter). The Adamax optimizer was used for training and the Mean Squared Error (MSE) was implemented as the loss function for both RNN models (covariance matrices need to be converted to be positive semi-definite during usage, see \cite{Schperberg_rnn}). For training (for both models), we created 4 different maps with obstacles randomly distributed, where robots traverse about these maps via the SMPC. Note, that by definition of a particle-filter and an EKF, the former assumes non-Gaussian noise while the latter assumes Gaussian noise.
\subsection{Results}
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\columnwidth]{Figures/Results/system_perf.pdf}
\caption{\textbf{System performance.} We compare the results of SABER (SMPC-RNN-DQN) using the same map as \textit{(D)} in Fig. \ref{fig_saber}) against baselines for the UAV \textit{(A)} and UGV \textit{(B)} with distance to goal vs time as our metric. See \ref{validation} for details.}
\label{sys_perf}
\end{figure}
\subsubsection{Analysis of learning components and their significance}
The training loss for the RNN networks are shown in Fig.~\ref{rnn_loss}. The RNN networks for both SLAM algorithms were trained for 500 epochs (validation loss was observed to be close to the training loss), and shows a strong correlation between truth (output of SLAM in training) and predicted covariance. We also demonstrate that our RNN can model the behavior of covariance outputs generated by different sensor configurations (i.e., SLAM algorithms), as seen in (A) and (B) of Fig.~\ref{rnn_results} (e.g., we observed an increase in uncertainty for VIO SLAM when too close to obstacles, while seeing the opposite behavior for a particle-filter SLAM). Important to note, is that this result indicates that the RNN can model both non-Gaussian (particle-filter) and Gaussian (EKF) noise. By modeling the propagation of uncertainty of different measuring systems and integrating them into the SMPC prediction horizon through chance constraints, we ensure control values that avoid obstacle collision (without over avoidance) for each individual robot.
In (B) of Fig.~\ref{fig_dqn_methods}, we show that during training of our DQN, the rewards would increase over a majority of the 35,000 episodes, reaching a steady-state at approximately 27,000 (except for the 5-robot team). In Table \ref{table1}, we also compared our DQN to other 2D baseline algorithms (we chose resolution settings for RRT, RRT-star, and A-star so that their path lengths were similar to the DQN, then evaluated their computation time). The results indicate that on average, the DQN had the lowest computation time and was comparable to A-star in terms of its path length to the goal (considered optimal for our map resolution). Unlike the baseline algorithms, our DQN also has the additional functionality of learning multi-agent semantic behavior--it was successful at moving the robots simultaneously to their goal points as shown in (C), and, as expected, stayed closer to each other when rewarding them based on close proximity as shown in (D) of Fig.~\ref{fig_dqn_methods}. Although it would be possible to modify the baseline planners to consider multi-agent planning, the learning-based approach considers potential semantics between agents that would be difficult to quantify and define beforehand within a cost function.
\begin{table}[h!]
\caption{}
\label{table1}
\begin{tabular}{|P{1.13cm}|P{1.65cm}|P{1.22cm}|P{1.5cm}|P{1.14cm}|}
\hline
\multicolumn{5}{|c|}{\textit{High-level planner analysis (100 trials on map (D) of Fig.~\ref{fig_saber})}} \\
\hline
Planner& Path length(m)& std dev(m) & Solve time(s)& std dev(s) \\
\hline
RRT& 10.83& $\pm$1.28& 0.120& $\pm$0.091 \\
\hline
RRT-star& 10.40& $\pm$1.20& 0.084& $\pm$0.073 \\
\hline
A-star& 9.66& $\pm$0.00& 0.154& $\pm$0.002 \\
\hline
DQN& 9.68& $\pm$0.00& 0.051& $\pm$0.001 \\
\hline
\end{tabular}
\begin{tabular}{|P{1.3cm}|P{1.1cm}|P{0.9cm}|P{1.3cm}|P{0.8cm}|P{0.8cm}|}
\hline
\multicolumn{6}{|c|}{\textit{Average Computation time of SABER components (10 minutes of data)}} \\
\hline
Component& SMPC & RNN & VIO SLAM& PF SLAM & SABER \\
\hline
Solve time(s)& 0.0559& 0.0212 & 0.030& 0.073&0.193 \\
\hline
std dev(s)& $\pm$0.0187& $\pm$0.0077 & $\pm$0.003& $\pm$0.005 &$\pm$0.110 \\
\hline
\end{tabular}
\end{table}
The limitation of our DQN is that it's currently most capable in planning in 2D rather than 3D space, which helps lower training time and increase convergence (more complex DQN formulations may be necessary if the problem is either scaled to 3D, assumes a map bigger than 10$\times$10m, or uses more than 3 robots, see (B) of Fig.~\ref{fig_dqn_methods}). However, since we use an SMPC to avoid obstacles in 3D space using chance constraints, a 2D planner is sufficient for our application.
\subsubsection{Validation of the SABER algorithm}
\label{validation}
The complete planner is exemplified in Fig.~\ref{fig_saber} on a UGV-UAV team. We show that the SMPC of the UGV and UAV uses the state uncertainties estimated by an RNN to avoid colliding in obstacles in dense maps (see (A), and (B)). A special case is also shown in (C), where the SMPC of the UGV (without the DQN) reaches a local minima solution and is stuck behind the obstacle. However, when using the DQN's proposed path, the UGV can successfully reach its global goal. Note, that the SMPC and not the DQN considers both the dynamics of the robots and the uncertainty provided by their RNN models, thus, the actual path (shown in orange/purple) will differ slightly from the proposed DQN path (marked by triangles). We also evaluate the average computation time of the SABER algorithm (and its individual components) in Table \ref{table1}, showing a computation time of $\simeq{0.19}$ seconds/timestep. Lastly, in Fig.~\ref{sys_perf}, we verify that SABER performs best compared to several baselines (using distance to goal vs time as our metric on map (D) of Fig.~\ref{fig_saber}). The figure shows that MPC alone (no uncertainty is considered) causes an obstacle collision for the UGV and UAV (this likely occurred as the simulation contains noise, and the MPC is unaware of this noise during optimization). Both robots are able to get to their goals using a na\"ive stochastic MPC (uncertainty is considered by artificially inflating all obstacle boundaries), but due to over avoidance take longer to reach their goals. By adding our RNN to the SMPC allows both robots to reach their goals more quickly (uncertainty is now more accurately propagated within the SMPC prediction horizon). However, we observed that without a global planner, both robots run into local minima issues (i.e., robots would sometimes get stuck behind an obstacle for some time before reaching their goals). With a global planner (e.g., DQN, A-star), the robots reach their goals in the quickest manner (avoiding local minima issues). Although A-star and DQN provide near-optimal paths, the reason the DQN shows slightly better improvements (including better computation time) over A-star is because the DQN also accounts for multi-agent behavior, where robots were trained to stay close to each other when possible before reaching their respective goals (staying in close proximity decreases uncertainty via the Kalman filter).
\section{Conclusion}
In this work, we demonstrated that the SABER algorithm (which combines several fields of robotics including controls, vision, and machine learning into a single framework) is computationally feasible ($\simeq{0.19}$ seconds/timestep) and plans paths for heterogeneous robots to reach a global goal while satisfying diverse dynamics, constraints, and consideration of uncertainty. In future work, we plan to relax the assumption of a perfect object detection system and will focus on expanding our DQN to consider more complex behavior and tasks.
\bibliographystyle{IEEEtran}
|
2,869,038,154,860 | arxiv | \section{Introduction}
\label{sec:intro}
Recent advances in deep learning and more specifically in sequence-to-sequence modeling have led to dramatic improvements in ASR~\cite{chan2016listen, chiu2017state} \todo{more references?} and MT~\cite{sutskever2014sequence, cho2014properties, bahdanau2014neural, wu2016google, vaswani2017attention} tasks. These successes naturally led to attempts to construct end-to-end speech-to-text translation systems as a single neural network~\cite{berard2016listen, weiss2017sequence}. Such end-to-end systems have advantages over a traditional cascaded system that performs ASR and MT consecutively in that they
\begin{inparaenum}[1)]
\item naturally avoid compounding errors between the two systems;
\item can directly utilize prosodic cues from speech to improve translation;
\item have lower latency by avoiding inference with two models;
and
\item lower memory and computational resource usage.
\end{inparaenum}
However, training such an end-to-end ST model typically requires a large set of parallel speech-to-translation training data. Obtaining such a large dataset is significantly more expensive than acquiring data for ASR and MT tasks. This is often a limiting factor for the performance of such end-to-end systems. Recently explored techniques to mitigate this issue include multi-task learning \cite{weiss2017sequence, anastasopoulos2018tied} and pre-trained components \cite{berard2018end} in order to utilize weakly supervised data, i.e.\ speech-to-transcript or text-to-translation pairs, in contrast to fully supervised speech-to-translation pairs.
Although multi-task learning has been shown to bring significant quality improvements to end-to-end ST systems, it has two constraints which limit the performance of the trained ST model:
\begin{inparaenum}[1)]
\item the shared components have to compromise between multiple tasks, which can limit their performance on individual tasks;
\item for each training example,
the gradients are calculated for a single task,
parameters are therefore updated independently for each task,
which may lead to sub-optimal solution for the entire multi-task optimization problem.
\end{inparaenum}
In this paper, we train end-to-end ST models
on much larger datasets than previous work,
spanning up to 100 million training examples, including 1.3K hours of translated speech and 49K hours of transcribed speech. We confirm that multi-task learning and pre-training are still beneficial at such a large scale.
We demonstrate that performance of our end-to-end ST system can be significantly improved, even outperforming multi-task learning, by using a large amount of data synthesized from weakly supervised data such as typical ASR or MT training sets.
Similarly, we show that it is possible to train a high-quality end-to-end ST model without any fully supervised
training data by leveraging pre-trained components and data synthesized from weakly supervised datasets.
Finally, we demonstrate that data synthesized from fully unsupervised monolingual datasets can be used to improve end-to-end ST performance.
\section{Related Work}
\label{sec:related}
Early work on speech translation typically used a cascade of an ASR model and an MT model \cite{ney1999speech, matusov2005integration, post2013improved}, giving the MT model access to the predicted probabilities and uncertainties from the ASR. Recent work has focused on training end-to-end ST in a single model \cite{berard2016listen, weiss2017sequence}.
In order to utilize both
fully supervised data
and also
weakly supervised data,
\cite{weiss2017sequence, anastasopoulos2018tied} use multi-task learning to train the ST model jointly with the ASR and/or the MT model. By doing so, both of them achieved better performance with the end-to-end model than the cascaded model.
\cite{berard2018end} conducts experiments on a larger 236 hour English-to-French dataset and pre-trains the encoder and decoder prior to multi-task learning, which further improves performance. However, the end-to-end model performs worse than the cascaded model in that work. \cite{bansal2018pre} shows that pre-training a speech encoder on one language can improve ST quality on a different source language.
Using TTS synthetic data for training speech translation
was a requirement when no direct parallel training data is available, such as in \cite{berard2016listen, kano2018structured}.
In contrast, we show that even when a large fully supervised training set is available, using synthetic training data from a high quality multi-speaker TTS system can further improve the performance of an end-to-end ST model.
Synthetic data has also been used to improve ASR performance. \cite{tjandra2018machine} builds a cycle chain between TTS and ASR models, in which the output from one model is used to help the training of the other. Instead of using TTS, \cite{renduchintala2018multi} synthesizes repeated phoneme sequence from unlabeled text to mimic temporal duration in acoustic input. Similarly, \cite{Hayashi18} trains a pseudo-TTS model to synthesize the latent representation from a pre-trained ASR model from text, and uses it for data augmentation in ASR training.
The MT synthetic data in this work helps the system in a manner similar to knowledge distillation \cite{hinton2015distilling}, since the network is trained to predict outputs from a pretrained MT model. In contrast, synthesizing speech inputs using TTS is more similar to MT back-translation \cite{Sennrich16}.
\section{Models}
\label{sec:model}
\begin{figure}[t]
\centering
\scalebox{0.8}{
\begin{tikzpicture}[auto, font=\small\sffamily, node distance=2.2cm,auto,>=latex']
\pgfdeclarelayer{back}
\pgfdeclarelayer{middle}
\pgfsetlayers{back,middle,main}
\tikzstyle{cell} = [rectangle, draw, fill=green!20, text width=8em, text centered, minimum height=4.2ex]
\tikzstyle{component} = [draw, minimum height=4.5ex, minimum width=6em, fill=blue!20, rounded corners=6]
\tikzstyle{component_label} = [inner sep=2pt, align=left, font=\scriptsize\sffamily]
\tikzstyle{io_label} = [align=center, inner sep=0]
\node [io_label, name=input, align=center, font=\scriptsize\sffamily] {English speech};
\node[cell, above=0.7cm of input, fill=gray!30] (asr_enc_cell) {Bidi LSTM $\times 5$};
\node [component_label, above=0.09cm of asr_enc_cell] (asr_enc_label) {Pre-trained ASR encoder};
\begin{pgfonlayer}{middle}
\node[component, fit=(asr_enc_cell)(asr_enc_label)] (asr_enc) {};
\end{pgfonlayer}
\node[cell, above=0.3cm of asr_enc] (ast_enc_cell) {Bidi LSTM $\times 3$};
\node [component_label, above=0.09cm of ast_enc_cell] (ast_enc_label) {ST encoder};
\begin{pgfonlayer}{back}
\node[component, fit=(asr_enc)(ast_enc_cell)(ast_enc_label), fill=orange!30] (ast_enc) {};
\end{pgfonlayer}
\node[cell, above=0.3cm of ast_enc] (attn) {8-Head Attention};
\node[cell, above=0.5cm of attn] (MT_dec_cell) {LSTM $\times 8$};
\node [component_label, above=0.09cm of MT_dec_cell] (MT_dec_label) {Pre-trained MT decoder};
\begin{pgfonlayer}{middle}
\node[component, fit=(MT_dec_cell)(MT_dec_label)] (MT_dec) {};
\end{pgfonlayer}
\node [component_label, above=0.09cm of MT_dec] (ast_dec_label) {ST decoder};
\begin{pgfonlayer}{back}
\node[component, fit=(MT_dec)(ast_dec_label), fill=orange!30] (ast_dec) {};
\end{pgfonlayer}
\node [io_label, above=0.5cm of ast_dec, font=\scriptsize\sffamily] (output) {Spanish text};
\draw [->] (input) -- (ast_enc);
\draw [->] (asr_enc) -- (ast_enc_cell);
\draw [->] (ast_enc) -- (attn);
\draw [->] (attn) -- (ast_dec);
\draw [->] (ast_dec) -- (output);
\end{tikzpicture}
}
\caption{Overview of the end-to-end speech translation model. Blue blocks correspond to pre-trained components, grey components are frozen, and green components are fine-tuned on the ST task.}
\label{fig:model}
\end{figure}
Similar to \cite{weiss2017sequence}, we make use of three sequence-to-sequence models. Each one is composed of an encoder, a decoder, and an attention module. Besides the end-to-end ST model which is the major focus of this paper, we also build an ASR model and an MT model, which are used for building the baseline cascaded ST model, as well as for multi-task learning and encoder / decoder pre-training for ST.
All three models represent text using the same shared English/Spanish Word Piece Model (WPM) \cite{schuster2012japanese} containing 16K tokens.
\begin{itemize}[leftmargin=0em,label={}]
\item \textbf{ASR model:} Our ASR model follows the architecture of \cite{chiu2017state}. We use a 5 layer bidirectional LSTM encoder, with cell size 1024. The decoder is a 2 layer unidirectional LSTM with cell size 1024. The attention is 4-head additive attention. The model takes 80-channel log mel spectrogram features as input.
\item \textbf{MT model:} Our MT model follows the architecture of \cite{chen2018best}. We use a 6 layer bidirectional LSTM encoder, with cell size of 1024. The decoder is an 8 layer unidirectional LSTM with cell size 1024, with residual connection across layers. We use 8-head additive attention.
\item \textbf{ST model:} The encoder has similar architecture to the ASR encoder, and the decoder has similar architecture to the MT decoder. Throughout this work we experiment with varying the number of encoder layers. The model with the best performance is visualized in Figure \ref{fig:model}. It uses an 8 layer bidirectional LSTM for the encoder and an 8 layer unidirectional LSTM with residual connections for the decoder. The attention is 8-head additive attention, following the MT model.
\end{itemize}
\section{Synthetic training data}
\label{sec:synthetic}
Acquiring large-scale parallel speech-to-translation training data is extremely expensive.
The scarcity of such data is often a limiting factor on the quality of an end-to-end ST model.
To overcome this issue, we use two forms of weakly supervised data by:
synthesizing input speech corresponding to the input text in a parallel text MT training corpus, and synthesizing translated text targets from the output transcript in an ASR training corpus.
\subsection{Synthesis with TTS model}
\label{sec:synthetic-tts}
Recent TTS systems are able to synthesize speech with close to human naturalness \cite{shen2018natural}, in varied speakers' voices \cite{ping2017deep}, create novel voices by sampling from a continuous speaker embedding space \cite{jia2018transfer}.
In this work, we use the TTS model trained on LibriSpeech~\cite{Libri15} from \cite{jia2018transfer}, except that we use a Griffin-Lim \cite{griffin1984signal} vocoder as in \cite{yx2017tacotron} which has significantly lower cost, but results in reduced audio quality\footnote{Synthetic waveforms are needed only for audio data augmentation. Otherwise, the mel-spectrogram predicted by the TTS model can be directly fed as input to the ST or ASR models, bypassing the vocoder.}. We randomly sample from the continuous speaker embedding space for each synthesized
example, resulting in wide diversity in speaker voices in the synthetic data. This avoids unintentional bias toward a few synthetic speakers when using it to train an ST model and encourages generalization to speakers outside the training set.
\subsection{Synthesis with MT model}
\label{sec:synthetic-mt}
Another way to synthesize training data is to use an MT model to translate the transcripts in an ASR training set into the target language. In this work, we use the Google Translate service to obtain such translations. This procedure is similar to knowledge distillation \cite{hinton2015distilling}, except that it uses the final predictions as training targets rather than the predicted probability distributions. %
\section{Experiments}
\label{sec:experiments}
\subsection{Datasets and metrics}
We focus on an English speech to Spanish text conversational speech translation task. Our experiments make use of three proprietary datasets, all consisting of conversational language, including: \begin{inparaenum}[(1)]
\item an MT training set of 70M English-Spanish parallel sentences;
\item an ASR training set of 29M hand-transcribed English utterances,
collected from anonymized voice search log;
and
\item a substantially smaller English-to-Spanish speech-to-translation set of 1M utterances obtained by sampling a subset from the 70M MT set, and crowd-sourcing humans to read the English sentences.
\end{inparaenum}
The final dataset can be directly used to train the end-to-end ST model.
We use data augmentation on both speech corpora by adding varying degrees of background noise and reverberation in the same manner as \cite{chiu2017state}.
The WPM shared among all models is trained with the 70M MT set.
We use two datasets for evaluation: a held out subset of 10.8K examples from the 1M ST set, which contains read speech, and another 8.9K recordings of natural conversational speech in a domain different from both the 70M MT and 29M ASR sets. Both eval sets contain English speech, English transcript and Spanish translation triples, so they can be used for evaluating either ASR, MT, or ST. ASR performance is measured in terms of Word Error Rate (WER) and translation performance is measured in terms of BLEU~\cite{papineni-EtAl:2002:ACL} scores, both on case and punctuation sensitive reference text.
\subsection{Baseline cascaded model}
\label{sec:cascaded}
We build a baseline system by training an English ASR model and an English-to-Spanish MT model with the architectures described in Section~\ref{sec:model}, and cascading them together by feeding the predicted transcript from the ASR model as the input to the MT model. The ASR model is trained on a mixture of the 29M ASR set and the 1M ST set with 8:1 per-dataset sampling probabilities in order to better adapt to the domain of the ST set. The MT model is trained on the 70M MT set, which is a superset of the 1M ST set.
As shown in Table~\ref{tbl:cascaded}, the ST BLEU is significantly lower than the MT BLEU, which is the result of cascading errors from the ASR model.
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{clcc}
\toprule
Task & Metric & In-domain & Out-of-domain \\
\midrule
ASR & WER\footnote{} & 13.7\% & 30.7\% \\
MT & BLEU & 78.8 & 35.6 \\
ST & BLEU & 56.9 & 21.1 \\
\bottomrule
\end{tabular}
\caption{Performance of the baseline cascaded ST system and the underlying ASR and MT components on both test sets.}
\label{tbl:cascaded}
\end{small}
\end{table}
\footnotetext{We report WER based on references which are case and punctuation sensitive in order to be consistent with the way BLEU is evaluated. The same ASR model obtains a WER of 6.9\% (in-domain) and 14.1\% (out-of-domain) if trained and evaluated on lower-cased transcripts without punctuation.}
\subsection{Baseline end-to-end models}
\label{sec:baselines}
We train a vanilla end-to-end ST model with a 5-layer encoder and an 8-layer decoder directly on the 1M ST set. We then adopt pre-training and multi-task learning as proposed in previous literature \cite{weiss2017sequence, anastasopoulos2018tied, berard2018end, bansal2018pre} in order to improve its performance.
We pre-train the encoder on the ASR task, and the decoder on the MT task as described in
Section~\ref{sec:cascaded}. After initialization with pre-trained components (or random values for components not being pre-trained), the ST model is fine-tuned on the 1M ST set.
Finally, we make use of multi-task learning by jointly training a combined network on the ST, ASR, and MT tasks, using the 1M, 29M, 70M datasets, respectively. The ST sub-network shares the encoder with the ASR network, and shares the decoder with the MT network. For each training step, one task is sampled and trained with equal probability.
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{lcc}
\toprule
& In-domain & Out-of-domain \\
\midrule
Cascaded & 56.9 & 21.1 \\
\midrule
Vanilla & 49.1 & 12.1 \\
+ Pre-training & 54.6 & 18.2 \\
+ Pre-training + Multi-task & 57.1 & 21.3 \\
\bottomrule
\end{tabular}
\caption{BLEU scores of baseline end-to-end ST and cascaded models. All end-to-end models use 5 encoder and 8 decoder layers.}
\label{tbl:baselines}
\end{small}
\end{table}
Performance of these baseline models is shown in Table \ref{tbl:baselines}.
Consistent with previous literature, we find that pre-training and multi-task learning both significantly improve ST performance, because they increase the amount of data seen during training by two orders of magnitude.
When both pre-training and multi-task learning are applied, the end-to-end ST model slightly outperforms the cascaded model.
\subsection{Using synthetic training data}
\label{sec:exp-synthetic}
We explore the effect of using synthetic
training data as described in Section \ref{sec:synthetic}.
To avoid overfitting to TTS synthesized audio (especially since audio synthesized using the Griffin-Lim algorithm contains obvious and unnatural artifacts), we freeze the pre-trained encoder but stack a few additional layers on top of it. The impact of adding different number of additional layers when fine-tuning with the 1M ST set is shown in Table~\ref{tbl:additional-layers}.
Even with no additional layer, this approach outperforms the fully trainable model (Table \ref{tbl:baselines} row 3) on the out-of-domain eval set, which indicates that the frozen pre-trained encoder helps the ST model generalize better.
The benefit of adding extra
layers saturates after around 3 layers.
Following this result, we use 3 extra
layers for the following experiments, as visualized in Figure~\ref{fig:model}.
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{cccccc}
\toprule
& \multicolumn{5}{c}{\# additional encoder layers} \\
& 0 & 1 & 2 & 3 & 4 \\
\midrule
In-domain & 54.5 & 55.7 & 56.1 & 55.9 & 56.1 \\
Out-of-domain & 19.5 & 18.8 & 19.3 & 19.5 & 19.6 \\
\bottomrule
\end{tabular}
\caption{BLEU scores of the extended ST model, varying the number of additional encoder layers on top of the frozen pre-trained ASR encoder. The decoder is pre-trained but kept trainable. Multi-task learning is not used.}
\label{tbl:additional-layers}
\end{small}
\end{table}
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{lcc}
\toprule
Fine-tuning set & In-domain & Out-of-domain \\
\midrule
Real & 55.9 & 19.5 \\
\midrule
Real + TTS synthetic & 59.5 & 22.7 \\
Real + MT synthetic & 57.9 & 26.2 \\
\textbf{Real + both synthetic} & \textbf{59.5} & \textbf{26.7} \\
\midrule
Only TTS synthetic & 53.9 & 20.8 \\
Only MT synthetic & 42.7 & 26.9 \\
\textbf{Only both synthetic} & \textbf{55.6} & \textbf{27.0} \\
\bottomrule
\end{tabular}
\caption{BLEU scores of ST trained with synthetic data. All rows use the same model architecture as Figure \ref{fig:model}.}
\label{tbl:synthetic}
\end{small}
\end{table}
We analyze the effect of using different synthetic datasets in Table~\ref{tbl:synthetic} for fine-tuning.
The TTS synthetic data is sourced from the 70M MT training set, by synthesizing English speech as described in Section~\ref{sec:synthetic-tts}. The synthesized speech is augmented with noise and reverberation following the same procedure as is used for real speech.
The MT synthetic data is sourced from the 29M ASR training set, by synthesizing translation to Spanish as described in Section \ref{sec:synthetic-mt}.
The encoder and decoder are both pre-trained on ASR and MT tasks, respectively, but multi-task learning is not used.
The middle group in Table \ref{tbl:synthetic} presents the result of fine-tuning with both synthetic datasets and the 1M real dataset, sampled with equal probability.
As expected, adding a large amount of synthetic training data (increasing the total number of training examples by 1 -- 2 orders of magnitude), significantly improves performance on both in-domain and out-of-domain eval sets.
The MT synthetic data improves performance on the out-of-domain eval set more than it does on the in-domain set, partially because it contains natural speech instead of read speech, which is better matched to the out-of-domain eval set, and partially because it introduces more diversity to the training set and thus generalizes better.
Fine-tuning on the mixture of the three datasets results in dramatic gains on both eval sets, demonstrating that the two synthetic sources have complementary effects.
It also significantly outperforms the cascaded model.
The bottom group in Table~\ref{tbl:synthetic} fine-tunes using only synthetic data. Surprisingly, they achieve very good performance and even outperform training with both synthetic and real collected data on the out-of-domain eval set. This can be attributed to the increased sampling weight of training data
with natural speech (instead of read speech).
This result demonstrates the possibility of training a high-quality end-to-end ST system
with only weakly supervised data,
by using such data for components pre-training and generating synthetic parallel training data from them by leveraging on high quality TTS and MT models or services.
\subsection{Importance of frozen encoder and multi-speaker TTS}
To validate the importance of freezing the pre-trained encoder, we compare to a model where the encoder is fully fine-tuned on the ST task. As shown in Table~\ref{tbl:non-frozen}, full encoder fine-tuning hurts ST performance, by overfitting to the synthetic speech. The ASR encoder learns a high quality latent representation of the speech content when
pre-training on a large quantity of real speech with data augmentation. Additional fine-tuning on synthetic speech only hurts performance since the TTS data are not as realistic nor as diverse as real speech.
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{lcc}
\toprule
Fine-tuning set & In-domain & Out-of-domain \\
\midrule
Real + TTS synthetic & 58.7 & 21.4 \\
Only TTS synthetic & 35.1 & 9.8 \\
\bottomrule
\end{tabular}
\caption{BLEU scores using fully trainable encoder, which performs worse than freezing lower encoder layers as in Table \ref{tbl:synthetic}.}
\label{tbl:non-frozen}
\end{small}
\end{table}
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{lcc}
\toprule
Fine-tuning set & \hspace{-1.5ex}In-domain\hspace{-1.5ex} & Out-of-domain \\
\midrule
Real + one-speaker TTS synthetic & 59.5 & 19.5 \\
Only one-speaker TTS synthetic & 38.5 & 13.8 \\
\bottomrule
\end{tabular}
\caption{BLEU scores when fine-tuning with synthetic speech data synthesized using a single-speaker TTS system, which performs worse than using the multi-speaker TTS as in Table \ref{tbl:synthetic}.}
\label{tbl:single-speaker}
\end{small}
\end{table}
Similarly, to validate the importance of using a high quality multi-speaker TTS system to synthesize training data with wide speaker variation, we train models using data synthesized with the single speaker TTS model from \cite{shen2018natural}.
This model generates more natural speech than
the multi-speaker model used in Sec.~\ref{sec:exp-synthetic} \cite{jia2018transfer}.
To ensure a fair comparison, we use a Griffin-Lim vocoder and
a 16 kHz sampling rate.
We use the same data augmentation procedure described above.
Results are shown in Table~\ref{tbl:single-speaker}. Even though a frozen pre-trained encoder is used, fine-tuning on only single-speaker TTS synthetic data still performs much worse than fine-tuning with multi-speaker TTS data, especially on the out-of-domain eval set.
However, when trained on the combination of real and synthetic speech, performance
on the in-domain eval set is not affected.
We conjecture that this is because the in-domain eval set consists of read speech, which is better matched to the prosodic quality of the single-speaker TTS model.
The large performance degradation on the out-of-domain eval set again indicates worse generalization.
Incorporating recent advances in TTS to introduce more natural prosody and style variation \cite{skerry2018towards,wang2018style,hsu2018hierarchical} to the synthetic speech might further improve performance when training on synthetic speech. We leave such investigations as future work.
\subsection{Utilizing unlabelled monolingual text or speech}
In this section, we go further and show that unlabeled monolingual speech and text can be leveraged to improve performance of an end-to-end ST model, by using them to synthesize parallel speech-to-translation examples using available ASR, MT, and TTS systems. Even though such datasets are highly synthetic, they can still benefit an ST model trained with as many as 1M real training examples.
We take the English text from the 70M MT set as an unlabeled text set, synthesize English speech for it using a multi-speaker TTS model as in Section \ref{sec:exp-synthetic}, and translate it to Spanish using the Google Translate service.
Similarly, we take the English speech from the 29M ASR set as an unlabeled speech set, synthesize translation targets for it by using the cascaded model we build in Section \ref{sec:cascaded}. We use this cascaded model only to enable comparison to its own performance. Replacing it with a cascade of other ASR and MT models or services should not change the conclusion.
Since there is no parallel training data for ASR or MT in this case, pre-training does not apply. We use the vanilla model with a 5-layer encoder as in Section \ref{sec:baselines}.
\begin{table}[t]
\centering
\begin{small}
\begin{tabular}{lcc}
\toprule
Training set & In-domain & Out-of-domain \\
\midrule
Real & 49.1 & 12.1 \\
Real + Synthetic from text & 55.9 & 19.4 \\
Real + Synthetic from speech & 52.4 & 15.3 \\
Real + Synthetic from both & 55.8 & 16.9 \\
\bottomrule
\end{tabular}
\caption{BLEU scores for the vanilla model trained on synthetic data generated from unlabeled monolingual data, without pre-training.}
\label{tbl:unlabeled}
\end{small}
\end{table}
Results are presented in Table~\ref{tbl:unlabeled}.
Even though these datasets are highly synthetic, they still significantly improve performance over the vanilla model.
Because the unlabeled speech is processed with weaker models, it doesn't bring as much gain as the synthetic set from unlabeled text. Since it essentially distills knowledge from the cascaded model, it is also understandable that it does not outperform it.
Performance is far behind our best results in Section~\ref{sec:exp-synthetic} since pre-training is not used.
Nevertheless, this result demonstrates that with access to high-quality ASR, MT, and/or TTS systems, one can leverage large sets of unlabeled monolingual data to improve the quality of an end-to-end ST system, even if a small amount of direct parallel training data are available. \section{Conclusions}
\label{sec:conslusions}
We propose a weakly supervised learning procedure that leverages synthetic training data to fine-tune an end-to-end sequence-to-sequence
ST
model, whose encoder and decoder networks have been separately pre-trained on
ASR and MT
tasks, respectively.
We demonstrate that this approach outperforms multi-task learning in experiments on a large scale English speech to Spanish text translation task.
When utilizing synthetic speech inputs, we find that it is important to use a high quality multispeaker TTS model, and to freeze the pre-trained encoder to avoid overfitting to synthetic audio.
We explore even more impoverished data scenarios, and
show that it is possible to train a high quality end-to-end ST model by fine-tuning \emph{only} on synthetic data from readily available ASR or MT training sets.
Finally, we demonstrate that a large quantity of unlabeled speech or text can be leveraged to improve an end-to-end ST model when a small fully supervised
training corpus is available.
\section{Acknowledgments}
The authors thank Patrick Nguyen, Orhan Firat, the Google Brain team, the Google Translate team, and the Google Speech Research team for their helpful discussions and feedback, as well as Mengmeng Niu for her operational support on data collection.
\bibliographystyle{IEEEbib}
|
2,869,038,154,861 | arxiv | \section{Introduction}
In the last few decades it has become apparent that many problems in
Information Theory have analogies to certain problems in the
area of statistical physics of disordered systems.
Such analogies are useful because physical insights, as well as
statistical mechanical tools and analysis techniques
can be harnessed in order to advance the knowledge and the
understanding with regard to
the information--theoretic problem under discussion.
One important example of such an analogy is between the statistical physics
of disordered magnetic materials, a.k.a.\ spin glasses, and the behavior of
certain ensembles of random codes for source coding
(see, e.g., \cite{HK05},
\cite{Murayama02},
\cite{KH05},
\cite{Tadaki07})
and for channel coding
(see, e.g., \cite{MM06} and references therein,
\cite{KanS99},
\cite{PS99},
\cite{Sourlas89},
\cite{Sourlas94},
\cite{KSNS01},
\cite{KabS99}
\cite{SK06},
\cite{MR06},
\cite{Montanari05a},
\cite{FLMR02},
\cite{MU07},
\cite{Montanari01},
\cite{Rujan93},
\cite{Montanari05b},
\cite{DW99}).
Among the various models of interaction disorder
in spin glasses, one of the most fascinating models is the {\it random
energy model} (REM), invented by Derrida in
the early eighties \cite{Derrida80},
\cite{Derrida80b}, \cite{Derrida81} (see also, e.g.,
\cite{DW99},
\cite{CFP98},
\cite{Jana06},
for later developments). The REM
is on the one hand, extremely
simple and easy to analyze, and on the other hand, rich enough to exhibit
phase transitions. According to the REM, the different spin configurations
are distributed according to the Boltzmann distribution,
namely, their probabilities are proportional to an exponential function of
their negative energies, but the configuration energies themselves are i.i.d.\ random
variables, hence the name random energy model.\footnote{More details on this and other
terminology described in the remaining part of this Introduction, will be given in
the Section \ref{bg}.}
In \cite[Chap.\ 6]{MM06}, M\'ezard and Montanari draw an interesting analogy between
the REM and the statistical physics pertaining to
{\it finite temperature decoding} \cite{Rujan93} of ensembles of random block codes.
The relevance of the REM here is due to the fact that in this context, the
partition function that naturally arises has the log--likelihood function
(of the channel output given the input codeword) as its energy function (Hamiltonian),
and since the codewords are selected at random,
then the induced energy levels are random variables.
Consequently, the phase transitions of the REM are `inherited' by ensembles of random block codes,
as is shown in \cite{MM06}.
In \cite{Merhav07}, this subject was further studied and the free energies corresponding to
the various phases
were related to random coding exponents of the probability of error
at rates below capacity and to the probability of correct decoding at rates above capacity.
While the REM is a very simple and interesting model for capturing disorder, as described above,
it is not quite faithful for the description of
a real physical system. The reason is that according to
the REM, any two distinct spin configurations, no matter how similar and close to each other, have
independent, and hence unrelated, energies.
A more realistic model
must take into account the geometry and the structure of the physical system
and thus allow dependencies between energies associated with closely
related configurations.
This observation has motivated Derrida to develop the {\it generalized
random energy model} (GREM)
\cite{Derrida85}
(see also, e.g.,
\cite{DG86a},
\cite{DG86b},
\cite{DD01},
\cite{Saaskian97},
\cite{BK04},
\cite{BK02},
for later related work). The GREM extends the REM in that it introduces an
hierarchical structure in the form of a tree,
by grouping subsets of (neighboring) spin configurations
in several levels, where the leaves of this tree correspond to the various configurations.
According to the GREM, for every branch in this tree, there is an associated
independent randomly chosen
energy component. The total energy of each
configuration is then the sum of these energy components
along the branches that form the path from the root of the tree to the leaf corresponding
to this configuration. This way, the degree of dependency between the energies of two
different configurations depends on the `distance' between them on the tree: More
precisely, it depends on
the number of common branches shared by their paths from the root up to the node at which their
paths split. The GREM is somewhat more complicated to analyze than the REM, but not substantially so.
It turns out that the number of phase transitions in the GREM depends on the parameters of the
model. If the tree has $k$ levels, there can be up to $k$ phase transitions, but there can also
be a smaller number. For example, in the case $k=2$, under a certain condition, there is only one
phase transition and the behavior of the free energy
in both phases is
just like in the ordinary REM.
In analogy to the above described relationship between the REM and the statistical
physics of random block codes, the natural question
that now arises is whether the GREM and its phase transitions can give us some
insights about the behavior of code ensembles with some hierarchical structure
(e.g., tree--structured codes, successive refinement codes, etc.).
In particular, in what way do these phase transitions guide
us in the choice of the design parameters of these codes?
It is the purpose of this paper to explore these questions and to give at least some partial
answers.
We demonstrate that there is indeed an intimate relationship between the GREM
and certain ensembles of hierarchical codes. Consider, for example, a two--stage rate--distortion code
of block length $n=n_1+n_2$, where the first $n_1$
components of the reproduction vector, at rate $R_1$, depend
only on the first $n_1R_1$ bits of the compressed bitstream,
and the last $n_2$ symbols of the reproduction
codeword, at rate $R_2$, depend on the entire bitstream of length $n_1R_1+n_2R_2$.
The overall rate of this code is, of course, the weighted average of $R_1$ and $R_2$
with weights proportional to $n_1$ and $n_2$, respectively. An ensemble of codes with this
structure is defined as follows: First, we randomly draw a rate $R_1$ codebook
of block length $n_1$ according to some distribution.
Then, for each resulting codeword of length $n_1$, we randomly draw a rate $R_2$ codebook
of block length $n_2$.\footnote{Note
that this is different from using the same second--stage
codebook for all first--part codewords,
in which case, this is just a combination
two codebooks of length $n_1$ and $n_2$, operating
independently.}
Thus, the code has a tree structure with two
levels, like a two--level GREM. The overall distortion
of the code along the entire
$n$ symbols is the sum of partial
distortions along the two segments, in analogy to
the above described additivity of the partial energies along
the branches of the tree pertaining to the GREM, and since the codewords are
random, then so are the distortions they induce.
The motivation for this
class of codes, especially when the idea is generalized from two parts to a larger number
of $k$ parts, say, of equal length ($n_1=n_2=\ldots =n_k=n/k$), is that the delay, at least at the
decoder, is reduced from $n$ to $n/k$, because the decoder is causal in the level of segments
of length $n/k$. The following questions now arise:
Is there any inherent penalty, in terms of performance,
for this ensemble of reduced delay
decoding codes? If so, how can we minimize this penalty? If not, how should we choose the design
parameters (i.e., $n_i$ and $R_i$, $i=1,\ldots,k$, for a given overall average rate $R$)
such that this code will `behave' like a full block code of length $n$?
For simplicity, let us return to the case $k=2$. For a given $R$ and $n$, we have two degrees
of freedom: the choices of $R_1$ and $n_1$ (which will then dictate $R_2$ and
$n_2$). Is it better to choose $R_1 > R_2$ or $R_1\le R_2$, if at all it makes any difference?
A similar question can be asked concerning $n_1$ and $n_2$. The answer depends, of course,
on our figure of merit. Obviously, if one is interested only in the asymptotic distortion,
the question becomes uninteresting, because then by choosing two independent codes\footnote{c.f.\
footnote no.\ 2.} for the
two parts, both at rate $R$, the overall distortion will be given by the distortion--rate
function, $D(R)$, just like that of
the full unstructured code. For a given $n$, of course,
the redundancies will correspond to the shorter blocks $n_1$ and $n_2$, but this is a second
order effect. Here, we choose to examine performance in terms of the characteristic function of
the overall distortion, $\bE[\exp\{-s\cdot\mbox{distortion}\}]$.
This is, of course, a much more informative figure of merit than
the average distortion, because in principle, it gives information on the entire probability
distribution of the distortion. In particular, it generates all the moments of the distortion
by taking derivatives, and it is useful in deriving Chernoff bounds on probabilities of
large deviations events concerning the distortion. In the context of the analogy
with statistical physics
and the GREM, this characteristic function can easily be related to the partition function
whose Hamiltonian is given by the distortion.
It turns out that the characteristic
function of the distortion behaves in a rather
surprisingly interesting manner and
with a direct relation to the GREM. For $R_1 < R_2$,
when the corresponding GREM has $k=2$ phase transitions, the characteristic function of
the distortion behaves like that of two independent block codes of lengths $n_1$ and $n_2$
and rates $R_1$ and $R_2$, thus the dependency between the two parts of the code is not exploited in terms
of performance. For $R_1 > R_2$, which is the case where the analogous GREM has only one phase
transition (and behaves exactly like the ordinary REM, which
is parallel to an ordinary random block code
with no structure),
the characteristic function behaves like that of a full
unstructured optimum block code at rate $R$ across a certain interval
of small $s$,
but beyond a certain point,
it becomes inferior to that of a full code. For $R_1=R_2=R$, it behaves like
the unstructured code for the {\it entire} range of $s\ge 0$,
but then one might as well use two independent block codes (and reduce the search complexity
at the encoder from $e^{nR}$ to $2e^{nR/2}$). The choices of $n_1$
and $n_2$ are immaterial in that sense, as long as they both grow linearly with $n$.
Thus, the conclusion is that it is best to use $R_1=R_2$, but if communication protocol
constraints dictate different rates at different segments,\footnote{
For example, this can be the case if there are additional users in the system
and the bandwidth allocation for each user changes in a dynamical manner, or if different
parts of the encoded information are transmitted via separate links with different capacities.}
then performance is better when $R_1 > R_2$ than when $R_1 < R_2$.
These results can be extended to the
case of $k$ stages.
A parallel analysis can be applied to analogous ensembles
of (reduced delay) channel encoders
of block length $n=n_1+n_2$ (for the case $k=2$), which have a similar tree
structure: Here, the first $n_1$ channel letters of each block depend
only on the first $n_1R_1$ information bits, whereas the other
$n_2$ channel symbols depend on the entire information vector of
length $n_1R_1+n_2R_2$. The random codebook is again drawn
hierarchically in the same manner
as before. If the code performance is judged in terms
of the error exponent, then once again,
the choice $R_1\ge R_2$ is always better than
the choice $R_1< R_2$. Here, unlike the source coding problem,
there is an additional consideration: There are two types of incorrect
codewords that are competing with the correct one in the decoding process:
those for which the first $n_1$ channel inputs agree with those of the
correct codeword (the first segment is the same)
and those for which this is not the case. In this case, $R_2$ has to be
chosen sufficiently small so that the error term contributed by erroneous
codewords of the first kind would not dominate the probability of error.
Considering the case $n_1=n_2=n/2$,
if the overall average rate is not too small, it is possible to choose
$R_1$ and $R_2$ so that the error exponent of this ensemble of codes
is not worse than that of an ordinary
random code with no structure. This idea can be
extended to $k$ stages in a straightforward manner. In fact,
we propose a systematic procedure to allocate rates to the different stages in a way
that guarantees that the error exponent would be at least as good as
that of the classical random coding error exponent pertaining to an ordinary
random code at rate $R$.
The outline of this paper is as follows.
In Section 2, a few notation conventions are described.
In Section 3, we provide
some more detailed background in statistical physics, with emphasis on the REM
and the GREM.
Finally, in Section 4, we present our main results
on hierarchical code ensembles of the type described
above, along with their relationship to the GREM.
Readers who are not interested in the relationship
with statistical physics (although this is one of the main points in the
paper) may skip Section 3
and ignore, in Section 4, the comments on
the statistical mechanical aspects, all this without
essential loss of continuity.
\section{Notation Conventions}
Throughout this paper, scalar random
variables (RV's) will be denoted by capital
letters, like $X$ and $Y$, their sample values will be denoted by
the respective lower case letters, and their alphabets will be denoted
by the respective calligraphic letters.
A similar convention will apply to
random vectors and their sample values,
which will be denoted with the same symbols in the boldface font.
Thus, for example, $\bX$ will denote a random $n$-vector $(X_1,\ldots,X_n)$,
and $\bx=(x_1,...,x_n)$ is a specific vector value in ${\cal X}^n$,
the $n$-th Cartesian power of ${\cal X}$.
Sources and channels will be denoted generically by the letters $P$ and $Q$.
Specific letter probabilities corresponding to a source $Q$ will be
denoted by the corresponding lower case letters, e.g., $q(x)$ is the
probability of a letter $x\in{\cal X}$. A similar convention will be applied
to the channel $P$ and the
corresponding transition probabilities, $p(y|x)$,
$x\in{\cal X}$, $y\in{\cal Y}$. The expectation operator will be
denoted by $\bE\{\cdot\}$.
The cardinality of a finite set ${\cal A}$ will be denoted by $|{\cal A}|$.
For two positive sequences $\{a_n\}$ and $\{b_n\}$, the notation
$a_n\exe b_n$ means that $a_n$ and $b_n$ are asymptotically of the same
exponential order, that is, $\lim_{n\to\infty}\frac{1}{n}\ln\frac{a_n}{b_n}
=0$. Similarly, $a_n\lexe b_n$ means that $\limsup_{n\to\infty}\frac{1}{n}\ln\frac{a_n}{b_n}\le 0$,
etc. Information theoretic quantities like entropies and mutual
informations will be denoted following the usual conventions
of the Information Theory literature.
\section{Background}
\label{bg}
In this section, we provide some basic background in statistical physics,
focusing primarily on the REM, along with its relevance to ordinary ensembles of
source and channel block codes, and then we extend the scope to the GREM.
\subsection{General}
\label{general}
Consider a physical system with a large number $n$ of particles,
which can be in a variety of `microstates' pertaining to the various combinations
of the microscopic physical states (characterized by position, momentum, spin, etc.) that
these particles may have. For each such microstate of the system, which we shall
designate by a vector $\bx$, there is an
associated energy, given by an energy function (Hamiltonian)
${\cal E}(\bx)$.
One of the most
fundamental results in statistical physics (based on the law of energy conservation and
the basic postulate that all microstates of the same energy level are equiprobable)
is that when the system is in equilibrium, the probability of a microstate $\bx$ is
given by the Boltzmann distribution
\begin{equation}
\label{bd}
P(\bx)=\frac{e^{-\beta{\cal E}(\bx)}}{Z(\beta)}
\end{equation}
where $\beta$ is the inverse temperature, that is, $\beta=1/T$, $T$ being temperature,\footnote{
More precisely, $\beta=1/(kT)$, where $k$ is Boltzmann's constant, but following the common
abuse of the notation, we redefine $T\leftarrow kT$ as temperature (in units of energy).} and
$Z(\beta)$ is the normalization constant, called the {\it partition function}, which
is given by
$$Z(\beta)=\sum_{\bx} e^{-\beta{\cal E}(\bx)}$$
or
$$Z(\beta)=\int d\bx e^{-\beta{\cal E}(\bx)},$$
depending on whether $\bx$ is discrete or continuous. The role
of the partition function is by far deeper than just being a normalization factor, as
it is actually the key quantity from which many macroscopic physical quantities can be derived,
for example, the free energy is $F=-\frac{1}{\beta}\ln Z(\beta)$, the average internal
energy (i.e., the expectation of ${\cal E}(\bx)$ where $\bx$ drawn is according (\ref{bd}))
is given by the negative derivative of $\ln Z(\beta)$, the heat capacity is obtained from
the second derivative, etc.
One of the important examples of such a multi--particle physical system is that of a magnetic
material, in which each molecule has a magnetic moment, a three--dimensional vector which
tends to align with the magnetic field felt by that molecule. In addition to the
influence of a possible external magnetic
field, there is also an effect of mutual interactions between the magnetic moments of
various (neighboring) molecules. Quantum mechanical considerations dictate that the set of possible
configurations of each magnetic moment (spin) is discrete: in the simplest case, it has only
two possible values, which we shall designate by $+1$ (spin up) and $-1$ (spin down). Thus, a spin
configuration, i.e., the vector of spins of $n$ molecules,
is designated by a binary vector $\bx=(x_1,\ldots,x_n)$, where each
component $x_i$ takes values in $\{-1,+1\}$ according to the spin of the $i$--th molecule,
$i=1,2,\ldots,n$. When the spins of a certain magnetic material tend to align in the
same direction, the material is called {\it ferromagnetic}, and a customary model of the
Hamiltonian, the {\it Ising model}, is given by
\begin{equation}
\label{ham}
{\cal E}(\bx)=-J\sum_{i,j}x_ix_j-B\sum_{i=1}^nx_i
\end{equation}
where the in first term, pertaining to the interaction,
$J > 0$ describes the intensity of the interaction with the summation
being defined over pairs of neighboring spins (depending on the geometry of the problem),
and the second term is associated with an external magnetic field (proportional to) $B$.
When $J < 0$, the material is {\it antiferromagnetic}, namely, neighboring spins `prefer'
to be antiparallel. More general models allow interactions not only with immediate neighbors,
but also more distant ones, and then there are different strengths of interaction, depending
on the distance between the two spins. In this case, the first term is replaced, by the more
general form $-\sum_{i,j}J_{ij}x_ix_j$, where now the sum can be defined over all possible
pairs $\{(i,j)\}$.\footnote{Moreover, the interaction term may be generalized
to include also summations over
triples of spins, quadruples, etc., but we will limit the discussion to pairs.}
Here, in addition to the ferromagnetic case, where all $J_{ij} > 0$, and the
antiferromagnetic case, where all $J_{ij} < 0$, there is also a situation where
some $J_{ij}$ are positive and others are negative, which is the case if a {\it spin glass}.
Here, not all spin pairs can be in their preferred mutual position (parallel/antiparallel),
thus the system may be {\it frustrated.}
To model situations of disorder, it is common to model $J_{ij}$ as random variables (RV's) with,
say, equal probabilities of being positive or negative. For example,
in the Edwards--Anderson (EA) model \cite{EA75},
$J_{ij}$ are taken to be i.i.d.\ zero--mean Gaussian RV's when $i$ and $j$ are neighbors
and zero otherwise. In the Sherrington--Kirkpatrick (SK) model \cite{SK75}, all $\{J_{ij}\}$ are
i.i.d.\ zero--mean Gaussian RV's. Thus, the system has two levels of randomness: the
randomness of the interaction coefficients and the randomness of the spin configuration given
the interaction coefficients, according to the Boltzmann distribution. However, the two
sets of RV's are normally treated differently. The random coefficients are considered {\it
quenched} RV's in the terminology of physicists,
namely, they are considered fixed in the time scale at which the spin
configuration may vary. This is analogous to the situation
of coded communication in a random coding paradigm: A randomly
drawn code should normally be thought of as a quenched
entity, as opposed to the randomness of the source and/or the channel.
\subsection{The REM}
\label{rem}
In \cite{Derrida80},\cite{Derrida80b},\cite{Derrida81}, Derrida took the above described
idea of randomizing the (parameters of the) Hamiltonian to an extreme, and suggested a model of spin
glass with disorder under which the energy levels $\{{\cal E}(\bx)\}$ are simply i.i.d.\ RV's, without
any structure in the form of (\ref{ham}) or its above--described extensions. In particular,
in the absence of a magnetic field,
the $2^n$ RV's $\{{\cal E}(\bx)\}$ are taken to be zero--mean Gaussian RV's, all with variance
$nJ^2/2$, where $J$ is a parameter.\footnote{The variance scales linearly with $n$ to match
the behavior of the Hamiltonian (\ref{ham}) with
a limited number of interacting neighbors and random interaction parameters, which has a number of
independent terms that is linear in $n$.} The beauty of the REM is in that on the one hand, it
is very easy to analyze, and on the other hand, it consists of sufficient richness to exhibit
phase transitions.
The basic observation about the REM is that for a typical realization of the
configurational energies $\{{\cal E}(\bx)\}$, the number of configurations with energy
about $E$ (i.e., between $E$ and $E+dE$), $N(E)$, is proportional
(up to sub--exponential terms in $n$) to $2^n\cdot e^{-E^2/(nJ^2)}$, as long as
$|E|\le E_0\dfn nJ\sqrt{\ln 2}$, whereas energy levels outside this range are typically
not populated by spin configurations ($N(E)=0$), as the probability
of having at least one configuration with such an energy decays exponentially with $n$.
Thus, the asymptotic (thermodynamical) entropy per spin, which is defined by
$$S(E)=\lim_{n\to\infty}\frac{\ln N(E)}{n}$$
is given by
$$S(E)=\left\{\begin{array}{ll}
\ln 2 -\left(\frac{E}{nJ}\right)^2 & |E|< E_0\\
0 & |E|= E_0\\
-\infty & |E| > E_0 \end{array}\right.$$
The partition function of a typical realization of a REM spin glass
is then
\begin{eqnarray}
Z(\beta)&\exe&\int_{-E_0}^{E_0}dE\cdot N(E)\cdot e^{-\beta E}\nonumber\\
&\exe&\int_{-E_0}^{E_0}dE \cdot e^{nS(E)}\cdot e^{-\beta E}
\end{eqnarray}
whose exponential growth rate,
$$\phi(\beta)\dfn\lim_{n\to\infty}\frac{\ln Z(\beta)}{n},$$
behaves according to
\begin{eqnarray}
\phi(\beta)
&=&\max_{|E|\le E_0}\left[S(E)-\beta\cdot\frac{E}{n}\right]\nonumber\\
&=&\max_{|E|\le E_0}\left[\ln 2-
\left(\frac{E}{nJ}\right)^2-\beta J \cdot\left(\frac{E}{nJ}\right)\right].
\end{eqnarray}
Solving this simple optimization problem, we find that $\phi(\beta)$
is given by
$$\phi(\beta)=\left\{\begin{array}{ll}
\ln 2 +\frac{\beta^2J^2}{4} & \beta \le \frac{2}{J}\sqrt{\ln 2}\\
\beta J\sqrt{\ln 2} & \beta > \frac{2}{J}\sqrt{\ln 2}\end{array}\right.$$
which means that the asymptotic free energy per spin, a.k.a.\ the {\it free energy density},
which is obtained by
$$F(\beta)=-\frac{\phi(\beta)}{\beta},$$
is given by (cf.\ \cite[Proposition 5.2]{MM06}):
$$F(\beta)=
\left\{\begin{array}{ll}
-\frac{\ln 2}{\beta}-\frac{\beta J^2}{4} & \beta \le \frac{2}{J}\sqrt{\ln 2}\\
-J\sqrt{\ln 2} & \beta > \frac{2}{J}\sqrt{\ln 2}\end{array}\right.$$
Thus, the free energy density is subjected to a phase transition at the inverse
temperature $\beta_0\dfn \frac{2}{J}\sqrt{\ln 2}$. At high temperatures ($\beta < \beta_0$),
which is referred to as the {\it paramagnetic phase}, the partition function is dominated by
an exponential number of configurations with energy $E=-n\beta J^2/2$ and the entropy
grows linearly with $n$. When the system is cooled to $\beta=\beta_0$ and beyond,
which is the {\it glassy phase},
the system freezes but it is still in disorder -- the partition function
is dominated by a subexponential number of configurations of minimum energy $E=-E_0$.
The entropy, in this case, grows sublinearly with $n$, namely the entropy
per spin vanishes, and the free energy density no longer depends on $\beta$.
Further details about the REM can be found in \cite{MM06} and the references mentioned in
the Introduction.
\subsection{The REM and Random Code Ensembles}
\label{remc}
As described in \cite{MM06}, there is an interesting analogy between the REM
and the partition function pertaining to {\it finite temperature decoding} \cite{Rujan93}
of ensembles of channel block codes (see also \cite{Merhav07}).
In particular, consider a codebook ${\cal C}$ of $M=e^{nR}$ binary codewords
of length $n$, $\bx_1,\ldots,\bx_M$,
to be used across a binary symmetric channel (BSC)
with crossover probability $p$. Given a binary vector $\by$ at the channel output,
consider the generalized posterior parametrized by $\beta$:
\begin{eqnarray}
\label{pbeta}
P_\beta(\bx|\by)&=&\frac{P^\beta(\by|\bx)}{\sum_{\bx'\in{\cal C}}
P^\beta(\by|\bx')}\nonumber\\
&=&\frac{e^{-\beta Bd_H(\bx,\by)}}{\sum_{\bx'\in{\cal C}}
e^{-\beta Bd_H(\bx',\by)}}\nonumber\\
&\dfn&\frac{e^{-\beta B d_H(\bx,\by)}}{Z(\beta|\by)},
\end{eqnarray}
where $B\dfn\ln\frac{1-p}{p}$, $d_H(\bx,\by)$ is the Hamming distance between $\bx$ and $\by$, and
where the real posterior is obtained, of course, for $\beta=1$. This is identified as a Boltzmann
distribution whose energy function (which depends on the given $\by$) is
${\cal E}(\bx)=Bd_H(\bx,\by)$.
As described in \cite{MM06} and \cite{Merhav07},
there are a few motivations for introducing the temperature parameter $\beta$ here.
First, it allows a degree of freedom
in case there is some uncertainty regarding the channel
noise level (small $\beta$ corresponds to high noise level). Second, it is
inspired by the ideas behind simulated annealing techniques: by
sampling from $P_\beta$ while gradually increasing $\beta$
(cooling the system), the minima of the energy function
(ground states) can be found.
Third, by applying symbolwise MAP decoding, i.e., decoding the $\ell$--th
symbol of $\bx$ as
$\mbox{arg}\max_a P_\beta(x_\ell=a|\by)$, where
$$P_\beta(x_\ell=a|\by)=
\sum_{\bx\in{\cal C}:~x_\ell=a}P_\beta(\bx|\by),$$
we obtain
a family of
{\it finite--temperature decoders}
parametrized by $\beta$, where $\beta=1$ corresponds
to minimum symbol error probability (with respect to the true channel)
and $\beta\to\infty$ corresponds to minimum block error probability.
As in \cite{MM06}, we will distinguish between two contributions of $Z(\beta|\by)$:
One is $Z_c(\beta|\by)=e^{-\beta Bd_H(\bx_0,\by)}$, where $\bx_0$ is the actual codeword
transmitted, and the other is $Z_e(\beta|\by)=\sum_{\bx'\in{\cal C}\setminus{\bx_0}}e^{-\beta Bd_H(\bx',\by)}$,
pertaining to all incorrect codewords. The former is typically about $e^{-\beta Bnp}$
since $d_H(\bx_0,\by)$ concentrates about $np$. We next focus on the behavior of $Z_e(\beta|\by)$.
To this end, consider a random selection of the code ${\cal C}$, where every bit of every codeword
is drawn by an independent fair coin tossing.
For a given $\by$, the energy levels $\{Bd_H(\bx,\by)\}$
pertaining to all incorrect codewords are RV's
(exactly like in the REM) because of the random selection of these
codewords. Now, the total number of correct codewords is about $e^{nR}$,
and the probability that a randomly chosen $\bx$ would fall at distance $d=n\delta$ from $\by$
is exponentially $e^{n[h(\delta)-\ln 2]}$, where
$$h(\delta)=-\delta\ln \delta-(1-\delta)\ln(1-\delta),$$
then the typical number of codewords at normalized distance $\delta$ is about
$$N(\delta)=e^{n[R+h(\delta)-\ln 2]}$$
as long as $R+h(\delta)-\ln 2 \ge 0$ and $N(\delta)=0$ when $R+h(\delta)-\ln 2<0$.
Thus, letting $\delta(R)$ denote the small solution to the equation $R+h(\delta)-\ln 2=0$
(the Gilbert--Varshamov distance),
we find that, with a clear analogy to the REM,
the corresponding thermodynamical entropy is given by
\begin{equation}
\label{srem}
S(\delta)=\left\{\begin{array}{ll}
R+h(\delta)-\ln 2 & \delta(R)<\delta < 1-\delta(R)\\
0 & \delta=\delta(R)~~\mbox{or}~~\delta = 1-\delta(R)\\
-\infty & \delta<\delta(R)~~\mbox{or}~~\delta > 1-\delta(R)
\end{array}\right.
\end{equation}
Accordingly, the partition function $Z_e(\beta|\by)$ of a typical code is given by
\begin{equation}
\label{ze}
Z_e(\beta|\by)\exe \sum_{\delta=\delta(R)}^{1-\delta(R)} e^{n[R+h(\delta)-\ln 2]}\cdot
e^{-\beta Bn\delta}\exe \exp\{n[R-\ln 2+\max_{\delta(R)\le\delta\le 1-\delta(R)}(h(\delta)-
\beta B\delta)]\},
\end{equation}
and the free energy density pertaining to $Z_e$ behaves according to
\begin{equation}
F_e(\beta)= \left\{\begin{array}{ll}
\frac{\ln 2-R-h(p_\beta)}{\beta}+Bp_\beta & \beta \le \beta_0\\
B\delta(R) & \beta > \beta_0 \end{array}\right.
\end{equation}
where
$$p_\beta=\frac{p^\beta}{p^\beta+(1-p)^\beta}$$
and
$$\beta_0=\frac{\ln[(1-\delta(R))/\delta(R)]}{B},$$
and where, again, the first line of $F_e(\beta)$ corresponds to the paramagnetic phase
with exponentially many codewords at distance (energy) $np_\beta$ from $\by$,
and the second line is the glassy phase with subexponentially many codewords at distance
$n\delta(R)$. In \cite{Merhav07}, these free energies
are related to random coding exponents as mentioned in the Introduction.
By the same token, in rate--distortion source coding, if one defines
the partition function as
$$Z(\beta)=\sum_{\hat{\bx}\in{\cal C}} e^{-\beta d_H(\bx,\hat{\bx})}$$
with $\bx$ being the source vector, $\{\hat{\bx}\}$ being the reproduction codevectors,
and $d_H(\bx,\by)$ being the Hamming distortion measure, then the same analysis takes place.
In the sequel, we will motivate this definition of the partition function
of rate--distortion coding and use it.
\subsection{The GREM}
\label{grem}
As we have seen, the REM is an extremely simple model to analyze, but its
simplicity is also recognized as a drawback from the aspect of faithfully modeling a spin glass.
The reason for this is the lack of structure which is needed to allow dependencies between
energy levels of spin configurations that are closely related: For example, if $\bx$ and $\bx'$ differ
only in a single component, it is conceivable that the respective energies would be
close, as suggested by (\ref{ham}). To this end, as described in the Introduction, Derrida
proposed a generalized version of the REM -- the GREM, which introduces dependencies between
configurational energies in an hierarchical fashion. We next briefly review the GREM.
A GREM with $k$ levels can best be thought of as a tree
with $2^n$ leaves and depth $k$, where
each leaf represents one spin configuration. This tree
is defined by $k$ positive parameters, $\alpha_1,\ldots,\alpha_k$,
which are all in the interval $(1,2)$, and whose product, $\prod_{i=1}^k\alpha_i$, equals $2$.
The construction of this tree is as follows:
The root of the tree is connected to $\alpha_1^n$ distinct
nodes,\footnote{We are approximating $\alpha_1^n, \alpha_2^n,\ldots \alpha_k^n$ by integers.}
which will be referred to as first--level nodes. Each first--level
node is in turn connected to $\alpha_2^n$
distinct second--level nodes, thus a total of
$(\alpha_1\alpha_2)^n$ second--level nodes. In the case $k=2$, these second--level nodes are
the leaves of the tree and $\alpha_1\alpha_2=2$. If $k > 2$, the process continues, and
each second--level node is connected to $\alpha_3^n$ third--level nodes, and so on. At the
last step, each one of the $\prod_{i=1}^{k-1}\alpha_i^n$ nodes at level $k-1$ is connected
to $\alpha_k^n$ distinct leaves, thus a total of $\prod_{i=1}^k\alpha_i^n=2^n$ leaves.
The REM corresponds
to the degenerate special
case where $k=1$.
The random selection of energy levels for the GREM is defined by another
set of $k$ parameters, $a_1,a_2,\ldots,a_k$, which are all positive reals that sum to unity.
The random selection is carried out in the following manner:
For each one of the $\prod_{j=1}^i\alpha_j^n$ branches
emanating from $(i-1)$--th level nodes and connecting them to $i$--th level nodes ($i=1,2,\ldots,k$)
in the tree, we randomly choose an
independent RV, henceforth referred to as a {\it branch energy},
which is a zero mean, Gaussian RV with variance $nJ^2a_i/2$,
where $J$ is like in the REM and where $\{a_i\}_{i=1}^k$ are as described above.
Finally, the energy level of a given configuration
is given by the sum of branch energies along the path from the root to the leaf that
represents this configuration. Thus, the total energy, is the sum of $k$ independent
zero--mean Gaussian RV's with variances $nJ^2a_i/2$, and so, it is
zero--mean Gaussian RV with variance $nJ^2/2$, exactly like in the REM.
However, now the energy levels of different configurations may be clearly correlated if the
paths from the root to their corresponding leaves share some common branches before they split.
The degree of statistical dependence is according to their distance along the tree. For example,
if two configurations are first--degree siblings, i.e., they share the same parent node at level
$k-1$, then all their energy components are the same except
their last branch energies, which are independent. On the other extreme, if their paths are
completely distinct, then their energies are independent.
The GREM for $k=2$ is analyzed in \cite{Derrida85}.
We next present the derivation for this case (with a few more details than in \cite{Derrida85}).
Let $\alpha_1$ and $\alpha_2$ be positive numbers whose product equals $2$,
and let $a_1$ and $a_2$ be positive numbers whose sum equals $1$.
Now, every configuration with energy $E$ has some first--level branch energy $\epsilon$
and second--level branch energy $E-\epsilon$. For a typical realization of this GREM,
the number of first--level branches with energy about $\epsilon$ is exponentially
$$N_1(\epsilon)\exe\alpha_1^n\cdot\exp\left\{-\frac{\epsilon^2}{nJ^2a_1}\right\}=
\exp\left\{n\left[\ln\alpha_1-\frac{1}{a_1}\left(\frac{\epsilon}{nJ}\right)^2\right]\right\},$$
provided that the expression in the square brackets is non--negative, i.e.,
$|\epsilon|\le \epsilon_0\dfn nJ\sqrt{a_1\ln\alpha_1}$, and $N_1(\epsilon)=0$ otherwise.
Therefore, the number of configurations with total energy about $E$
is exponentially
$$N_2(E)\exe\int_{-\epsilon_0}^{\epsilon_0} d\epsilon\cdot N_1(\epsilon)\cdot
\exp\left\{n\left[\ln\alpha_2-\frac{1}{a_2}\left(\frac{E-\epsilon}{nJ}\right)^2\right]\right\},$$
whose exponential rate (the entropy per spin) is given by
$$S(E)=\lim_{n\to\infty}\frac{\ln N_2(E)}{n}=
\max_{|\epsilon|\le\epsilon_0}\left[\ln\alpha_1-\frac{1}{a_1}\left(\frac{\epsilon}{nJ}\right)^2+
\ln\alpha_2-\frac{1}{a_2}\left(\frac{E-\epsilon}{nJ}\right)^2\right].$$
Note that $S(E)$
is an even function, non--increasing in $|E|$, and it should be
kept in mind that beyond the value of $|E|$ at which $S(E)$ vanishes, denote it by $\hat{E}$,
we have $S(E)=-\infty$ since $N_2(E)$ is typically zero (as was the case with the REM).
We shall get back to this point
shortly, but for a moment, let us ignore it and solve
the maximization problem pertaining to the above expression of $S(E)$, as is.
Denoting the resulting maximum by $\tilde{S}(E)$ (to distinguish from $S(E)$, where $\hat{E}$
and the jump to $\infty$ are taken into account), we get:
\begin{equation}
\label{psibasic}
\tilde{S}(E)=\left\{\begin{array}{ll}
\ln 2 -\left(\frac{E}{nJ}\right)^2 & |E|\le E_1\\
\ln\alpha_2-\frac{1}{a_2}\left(\frac{E}{nJ}-\sqrt{a_1\ln\alpha_1}\right)^2 & |E|> E_1
\end{array}\right.
\end{equation}
where $E_1\dfn nJ\sqrt{(\ln\alpha_1)/a_1}$. Taking now into account the above mentioned
observation concerning the criticality of the point $|E|=\hat{E}$, we have to distinguish between
two cases. The first is the case where $\hat{E} < E_1$, namely, the first line of the
above expression of $\tilde{S}(E)$ vanishes for $|E|$ smaller than $E_1$. The first line
vanishes for $|E|=E_0=nJ\sqrt{\ln 2}$, so the condition for this case to hold is
$E_0\le E_1$, or equivalently, $(\ln\alpha_1)/a_1 \ge \ln 2$. In this case,
we then have:
$$S(E)=\left\{\begin{array}{ll}
\ln 2 -\left(\frac{E}{nJ}\right)^2 & |E|\le E_0\\
0 & |E|=E_0\\
-\infty & |E| > E_0 \end{array}\right.$$
which is exactly the same behavior as in the ordinary REM ($k=1$).
Consequently, the exponential rate of the partition function, which is given by
$$\phi(\beta)=\lim_{n\to\infty}\frac{\ln Z(\beta)}{n}=\max_E\left[S(E)-\beta\frac{E}{n}\right],$$
is also the same as in the REM, namely,
$$\phi(\beta)=\left\{\begin{array}{ll}
\ln 2+\frac{\beta^2J^2}{4} & \beta < \beta_0\\
\beta J\sqrt{\ln 2} & \beta\ge\beta_0\end{array}\right.$$
where $\beta_0$ is the above defined critical inverse temperature of the REM (see Subsection \ref{rem}).
We next consider the complementary case where $(\ln\alpha_1)/a_1 < \ln 2$.
In this case, the expression of $S(E)$ should take into account
the fact that it vanishes (and then becomes $-\infty$) according to the
second line of (\ref{psibasic}). This amounts to:
\begin{equation}
\label{2phasetransitions}
S(E)=\left\{\begin{array}{ll}
\ln 2 -\left(\frac{E}{nJ}\right)^2 & |E|\le E_1\\
\ln\alpha_2-\frac{1}{a_2}\left(\frac{E}{nJ}-\sqrt{a_1\ln\alpha_1}\right)^2 & E_1\le |E|< E_2\\
0 & |E|= E_2\\
-\infty & |E| > E_2
\end{array}\right.
\end{equation}
where $E_2\dfn nJ(\sqrt{a_1\ln\alpha_1}+\sqrt{a_2\ln\alpha_2})$.
Before we compute the corresponding partition function,
we make the following observation:
$$\ln 2=\frac{\ln\alpha_1+\ln\alpha_2}{a_1+a_2}\le \max_{i=1,2}\frac{\ln\alpha_i}{a_i},$$
where the inequality follows from the well--known inequality
$[\sum_{i=1}^m a_i]/[\sum_{i=1}^m b_i]\le\max_{1\le i \le m}a_i/b_i$ for positive $\{a_i\}$ and
$\{b_i\}$ \cite[Lemma 1]{CO96}. In the same manner, using the similar inequality
$[\sum_{i=1}^m a_i]/[\sum_{i=1}^m b_i]\ge\min_{1\le i \le m}a_i/b_i$, we get
$$\ln 2\ge \min_{i=1,2}\frac{\ln\alpha_i}{a_i}.$$
It follows then that the condition $(\ln\alpha_1)/a_1 < \ln 2$ is equivalent
to the condition $(\ln\alpha_1)/a_1 < \ln 2 < (\ln\alpha_2)/a_2$.
Defining
$$\beta_i=\frac{2}{J}\sqrt{\frac{\ln \alpha_i}{a_i}},~~i=1,2$$
we then have $\beta_1 < \beta_0 < \beta_2$.
Let us examine how $\phi(\beta)$
behaves as $\beta$ grows from zero to infinity.
For small enough $\beta$, the achiever of $\phi(\beta)$,
call it $E^*$, is still smaller in absolute value than $E_0$, and then it is
obtained from equating to zero the derivative of $[S(E)-\beta E/n]$, with $S(E)$
being according to first line of (\ref{2phasetransitions}), thus $E^*=-\frac{n}{2}\beta J^2$.
This remains true as long as $\frac{n}{2}\beta J^2\le E_1$, which means $\beta\le \beta_1$.
In this case, the partition function is dominated by
$\exp\{n[\ln\alpha_1-a_1\beta^2J^2/4]\}$ first--level branches with energy
$\epsilon^*=-\frac{a_1}{2}n\beta J^2$, each followed by
$\exp\{n[\ln\alpha_1-a_1\beta^2J^2/4]\}$ second--level branches with energy
$E^*-\epsilon^*=-\frac{a_2}{2}n\beta J^2$, and this is a pure paramagnetic phase.
As $\beta$ continues to grow beyond $\beta_1$, but is still below $\beta_2$, the partition
function is dominated by a subexponential number of first--level branches of energy
$-nJ\sqrt{a_1\ln\alpha_1}$ followed by $\exp\{n[\ln\alpha_1-a_1\beta^2J^2/4]\}$
second--level branches with energy $E^*+nJ\sqrt{a_1\ln\alpha_1}$. This is a ``semi--glassy''
phase, where the first--level branches are already glassy but the second--level ones are
still paramagnetic.
As $\beta$ exceeds $\beta_2$, this becomes a pure glassy phase where the partition function
is dominated by a subexponential number of first--level branches with energy
$-nJ\sqrt{a_1\ln\alpha_1}$ and
a subexponential number of second--level branches with energy
$-nJ\sqrt{a_2\ln\alpha_2}$. Accordingly, the function $\phi(\beta)$ exhibits two phase transitions
at inverse temperatures $\beta_1$ and $\beta_2$:
$$\phi(\beta)=\left\{\begin{array}{ll}
\ln 2+\frac{\beta^2J^2}{4} & \beta < \beta_1\\
\beta J\sqrt{a_1\ln\alpha_1}+\ln\alpha_2+\frac{a_2\beta^2J^2}{4} & \beta_1\le\beta<\beta_2\\
\beta J(\sqrt{a_1\ln\alpha_1}+\sqrt{a_2\ln\alpha_2}) & \beta\ge \beta_2 \end{array}\right.$$
Again, the free energy density is obtained by $F(\beta)=-\phi(\beta)/\beta$.
This different behavior of the GREM for the two different cases will
be pivotal to our later discussion on the parallel behavior of ensembles of codes.
When there is a general number $k$ of levels, the above analysis of the GREM
becomes, of course, more complicated and there are more cases to consider, but the
concepts remain the same. There can be up to $k$ phase transitions, but there can
be less, depending on the parameters of the model $\{a_i,\alpha_i\}_{i=1}^k$.
For details, the reader is referred to \cite{DG86a},\cite{DG86b}.
\section{Relations Between GREM and Hierarchical Code Ensembles}
\label{main}
In analogy to the relationship between the REM and ordinary
ensembles of block codes, as was described
in Subsection \ref{remc}, it is natural to wonder about the possibility of similar
relationships between the GREM and more general ensembles of block codes,
and to ask whether the fact that the GREM exhibits
different types of behavior (as we have seen in Subsection \ref{grem}), has
implications on the behavior of these ensembles of codes.
Since the GREM is defined by an hierarchical (tree) structure, it is plausible
to expect that if a relationship to coding exists, it will be in the context of
ensembles of codes which have hierarchical structures as well.
Hierarchically structured ensembles of codes are encountered in numerous applications
in Information Theory, including block--causal tree--structured
source codes and channel codes of the type described informally
in the Introduction, successive refinement
source codes \cite{EC91},\cite{Koshelev80},\cite{Rimoldi94},
codes for the broadcast channel \cite[Chap.\ 15.6]{CT06} and codes based on binning techniques
(see, e.g., \cite{WZ76},\cite{GP80},\cite{Wyner75}),
just to name a few. In this paper, we confine our attention to the first above--mentioned class of codes.
The fact that the GREM
behaves, in some situations, like the REM, and the REM is analogous to an ordinary
block code without any hierarchical structure (cf.\ \ref{remc}), may hint that
in the parallel situations in the realm of our coding problem, a typical code
from the hierarchical ensemble will perform essentially as well as a typical (good) code
without the hierarchical structure. In these situations then (which can be imposed by
a clever choice of certain design parameters),
it would be interesting to explore the question whether we may enjoy the benefit that
the hierarchical structure buys us (in our case, reduced delay) without
essentially paying in terms of performance. As we show in this section, the answer
to this question turns out to be affirmative to a large extent, both in the source coding
setting and in the channel coding setting.
Finally, in closing this introductory part of Section \ref{main}, a more technical comment is in order:
As in Subsection \ref{remc}, throughout the sequel,
we confine ourselves to the memoryless binary symmetric source (BSS)
with the Hamming distortion measure, in the context of source coding,
and to the binary symmetric channel (BSC) in the context of channel coding.
The random coding distribution in both problems will be i.i.d.\ and uniform,
i.e., each bit of each codeword will be drawn by independent fair coin tossing.
Also, we will focus mostly on the case $k=2$.
The reason for this is that
our purpose is this paper
is more to demonstrate certain concepts, and so,
we prefer to slightly sacrifice generality at the benefit of simplicity,
and so, better readability, and a smaller amount of space. Having said that,
all the derivations can be extended
to apply to more general memoryless sources, channels,
and random coding distributions (as was done in \cite{Merhav07}), as well
as to a general number $k$ of stages.
\subsection{Lossy Source Coding}
\label{lossysourcecoding}
Consider the BSS $X_1,X_2,\ldots$, $X_i\in\{0,1\}$ ($i$ -- positive integer)
and the Hamming distortion measure between two binary $n$--vectors $\bx$ and $\hat{\bx}$:
$$d_H(\bx,\hat{\bx})=\sum_{i=1}^n d_H(x_i,\hat{x}_i),$$
where $d_H(a,b)=1$ if $a\ne b$ and $d_H(a,b)=0$ if $a=b$, $a,b\in\{0,1\}$.
Before discussing ensembles of codes with hierarchical structures, let
us first confine attention to an ordinary ensemble with no structure.
Consider a random selection of a codebook of size $M=e^{nR}$ ($R$ being
the coding rate in nats per source bit), ${\cal C}=\{\hbx_1,\ldots,\hbx_M\}$,
$\hbx_i\in\{0,1\}^n$, $i=1,2,\ldots,M$,
where each component of each codeword is drawn randomly by an independent
fair coin tossing. For a given source vector $\bx$ and for a given such
randomly drawn codebook ${\cal C}$, let $\Delta(\bx)=\min_{\hbx\in{\cal C}}d_H(\bx,\hbx)$
denote the distortion associated with encoding $\bx$.
Instead of examining the expected distortion, $\bE\{\Delta(\bX)\}$, w.r.t.\ both
the source and the random codebook selection, as is traditionally done,
we will concern ourselves with a more refined and more informative objective function,
which is the characteristic function of $\Delta(\bX)$, namely,
$$\Psi_n(s,R)=\bE\{\exp[-s\Delta(\bX)]\},$$
or in particular, its exponential rate
$$\psi(s,R)=-\lim_{n\to\infty}\frac{\ln\Psi_n(s,R)}{n}$$
focusing on the range $s\ge 0$. As is well known, the characteristic function provides information
not only on the expected distortion, $\bE\{\Delta(\bX)\}$, but also on every moment of $\Delta(\bX)$
(by taking derivatives of $\Psi_n(s,R)$ at $s=0$). It is also intimately related to the tail
behavior (i.e., large deviations probabilities) of the distribution of $\Delta(\bX)$ via Chernoff
bounds.
In order to analyze $\Psi_n(s,R)$ and then $\psi(s,R)$, first, for an ordinary ensemble, and later for
an hierarchical structured ensemble, it is convenient to define,
for given $\bx$ and ${\cal C}$, the partition function\footnote{For a given $\bx$, the
partition function $Z(\beta|\bx)$ induced by a typical codebook is exactly the same as in
(\ref{ze}), with the minor modification that here $\beta$ is not scaled by $B$ as in (\ref{ze}).}
\begin{equation}
Z(\beta|\bx)=\sum_{\hbx\in{\cal C}} e^{-\beta d_H(\bx,\hbx)}.
\end{equation}
The function $\Psi_n(s,R)$ is obtained from the partition function by
$$\Psi_n(s,R)=
\bE\{\lim_{\theta\to\infty}Z^{1/\theta}(s\cdot\theta|\bX)\}=
\lim_{\theta\to\infty}\bE\{Z^{1/\theta}(s\cdot\theta|\bX)\}.$$
In the definition of the ensemble behavior of $\psi(s,R)$, there are now two options. The first
is to think of the above defined expectation of $Z^{1/\theta}(s\theta|\bX)$ as being
taken w.r.t.\ both the source $\bX$ and the code ensemble $\{{\cal C}\}$, and then to define
$\psi(s,R)$ as above. The second option is to define the above expectation of
$Z^{1/\theta}(s\theta|\bX)$ w.r.t.\ the source only, while keeping ${\cal C}$ fixed, and
then to define $\psi(s,R)$ as $-\lim_{n\to\infty}\bE\{\ln\Psi_n(s,R)\}/n$, where the latter expectation
is across the ensemble of codebooks $\{{\cal C}\}$. The difference between meanings of the two approaches
is in the point of view: In the former approach the randomness of both $\bX$ and ${\cal C}$
are treated on equal grounds, and this makes sense if $\bX$ and ${\cal C}$ vary on the same
time scale (e.g., when the codebook varies frequently according to
some secret key). In the parallel discussion on spin glasses (cf.\ Section 3.1),
this is analogous to the double randomness of both the spin configuration and the interaction
parameters, and in the language of statistical physicists, this is called {\it annealed}
averaging. The second approach, which physicists refer to as {\it quenched} averaging, fits
better the paradigm where the code ${\cal C}$ is held fixed over many realizations of the source $\bX$.
In the Information Theory literature, it is more customary to adopt an approach analogous to
annealed averaging\footnote{In particular, source and channel random coding exponents are
normally defined as exponential rates of ensemble--average error
probabilities, and not as ensemble--average
exponents of error probabilities.} and so, we shall do the same here.
\subsubsection{The Ordinary Ensemble}
\label{ordinary}
Let us begin the with the calculation of the annealed version of $\psi(s,R)$, first, for a
an ordinary non--hierarchical code:
\begin{eqnarray}
\bE\{Z^{1/\theta}(s\theta|\bX)\}&=&
\bE\left\{\left[\sum_{\hbx\in{\cal C}}\exp(-s\theta d_H(\bX,\hbx))\right]^{1/\theta}\right\}\nonumber\\
&=&\bE\left\{\left[\sum_{d=0}^nN(d)\cdot e^{-s\theta d}\right]^{1/\theta}\right\}\nonumber\\
&\exe&\bE\left\{\sum_{d=0}^nN^{1/\theta}(d)\cdot e^{-sd}\right\}\nonumber\\
&=&\sum_{d=0}^n\bE\{N^{1/\theta}(d)\}\cdot e^{-sd}
\end{eqnarray}
where $N(\delta)$ is the number of codewords whose normalized Hamming distance from $\bX$
is exactly $\delta$, and
where the third (exponential) equality
holds, even before taking the expectation,
because the summation over $d$ consists of a {\it subexponential} number of terms, and so, both
$[\sum_d N(d)e^{-s\theta d}]^{1/\theta}$ and
$\sum_d N^{1/\theta}(d)e^{-sd}$ are of the same exponential order as
$\max_d N^{1/\theta}(d)e^{-sd}=
[\max_d N(d)e^{-s\theta d}]^{1/\theta}$. This is different from the original summation over ${\cal C}$
which contains an {\it exponential} number of terms.
Now, as is shown in Subsection A.1 of the Appendix (see also \cite{Merhav07b}),
\begin{equation}
\label{moments}
\bE\{N^{1/\theta}(n\delta)\}\exe
\left\{\begin{array}{ll}
e^{n[R+h(\delta)-\ln 2]} & \delta < \delta(R)~~\mbox{or}~~ \delta > 1-\delta(R)\\
e^{n[R+h(\delta)-\ln 2]/\theta} & \delta(R)\le\delta\le 1-\delta(R)
\end{array}\right.
\end{equation}
where $\delta(R)$ is defined (cf.\ Subsection \ref{remc}) as the small solution to
the equation $R+h(\delta)-\ln 2=0$, which is also the distortion--rate function
of the BSS. This gives
\begin{eqnarray}
\bE\{Z^{1/\theta}(s\theta|\bX)\}&\exe&
\sum_{\delta<\delta(R)} e^{n[R+h(\delta)-\ln 2]}\cdot e^{-s\delta n}+
\sum_{\delta\ge\delta(R)} e^{n[R+h(\delta)-\ln 2]/\theta}\cdot e^{-s\delta n}\nonumber\\
&\dfn&A+B
\end{eqnarray}
Now, as $\theta\to\infty$, the term $B$ tends to $\sum_{\delta\ge\delta(R)}e^{-s\delta n}$,
which is of the exponential order of $e^{-ns\delta(R)}$. The term $A$, which is independent
of $\theta$, is of the exponential order of $e^{-nu(s,R)}$, where
$$u(s,R)\dfn\ln 2-R-\max_{\delta\le\delta(R)}[h(\delta)-s\delta]=
\left\{\begin{array}{ll}
s\delta(R) & s\le s_R\\
v(s,R) & s > s_R \end{array}\right.$$
where
$$s_R\dfn\ln\left[\frac{1-\delta(R)}{\delta(R)}\right].$$
and
$$v(s,R)\dfn \ln 2-R+s-\ln(1+e^s).$$
Since $v(s,R)$ never exceeds $s\delta(R)$ for $s> s_R$, the dominant term is $A$,
and therefore, for the ordinary block code ensemble, we have:
$$\psi(s,R)=u(s,R).$$
It is not difficult to show also, using sphere covering considerations,
that $u(s,R)$ is the best achievable performance in terms
of the exponential rate of the characteristic function of the distortion.
The function $u(s,R)$ is depicted qualitatively in Fig.\ \ref{gen}.
\begin{figure}[ht]
\hspace*{4cm}\input{usrn.pstex_t}
\caption{$u(s,R)$ as a function of $s$ for fixed $R$.}
\label{gen}
\end{figure}
\subsubsection{The Hierarchical Ensemble}
\label{hierarchical}
We proceed to define the ensemble of hierarchical codes and to analyze
its performance with relation to the GREM. Let $n=n_1+n_2$, where $n$, $n_1$
and $n_2$ are positive integers. For a given $R_1$,
consider a random selection of a codebook of size $M_1=e^{n_1R_1}$,
${\cal C}_1=\{\hbx_1,\ldots,\hbx_{M_1}\}$,
$\hbx_i\in\{0,1\}^{n_1}$, $i=1,2,\ldots,M_1$,
where each component of each codeword is drawn randomly by an independent
fair coin tossing. Next, given $R_2$, for each $i=1,2,\ldots,M_1$,
consider a similar random selection of a codebook of size $M_2=e^{n_2R_2}$,
${\cal C}_2(i)=\{\tbx_{i,1},\ldots,\tbx_{i,M_2}\}$,
$\tbx_{i,j}\in\{0,1\}^{n_2}$, $j=1,2,\ldots,M_2$.
The encoder works as follows: Given a source vector $\bx\in\{0,1\}^n$,
it finds a pair of indices $(i,j)$, $i=1,2,\ldots,M_1$, $j=1,2,\ldots,M_2$,
such that the distortion between $\bx$ and the concatenation of the codewords
$(\hbx_i,\tbx_{i,j})$ is minimum. The index $i$ is encoded by $n_1R_1$ nats
and the index $j$ (given $i$) is encoded by $n_2R_2$ nats, thus a total of
$nR\dfn n_1R_1+n_2R_2$ nats, where $R$ is the overall rate, given by
$$R=\lambda R_1+(1-\lambda)R_2, ~~~\lambda=\frac{n_1}{n}.$$
The decoder can, of course, generate the first--stage reproduction
$\hbx_i$ based on the first $n_1R_1$ nats received,
without having to wait for the $n_2R_2$ following ones. The extension of this hierarchical
structure to a larger number of stages $k$ should be obvious. In particular,
as mentioned in the Introduction, if $k$ divides $n$
and the $n$--block is divided to $k$ sub--blocks of length $n/k$ each, then the decoder
can generate chunks of the reproduction at a reduced delay of $n/k$ instead of $n$.
The analogy of this structure with the GREM should also be obvious. The code has a tree
structure and the configurational energies of the GREM play the same role as the distortion
here, as the overall distortion is the cumulative sum of the per--stage distortions.
Also, the coding rate $R_i$ here plays the same role as $\ln\alpha_i$ of the GREM ($i=1,2$).
Thus, it is natural to expect that the partition function $Z(\beta|\bx)$ of this code ensemble would
behave analogously to that of the GREM, as we shall see next.
For the sake of simplicity, we return to the case $k=2$, with the understanding that
our derivations can be extended without any essential difficulties to a general $k$.
Before analyzing the characteristic function of the distortion along with its exponential rate,
it is instructive to examine the partition function $Z(\beta|\bx)$ for a given $\bx$ and address
the analogy with that of the GREM.
For a given $\bx$ and a typical code in the ensemble,
there are $N_1(\delta_1)\exe e^{n_1[R_1+h(\delta_1)-\ln 2]}$
first-stage codewords $\{\hbx\}$ at distance $n_1\delta_1$ from
the vector formed by the first $n_1$ components of $\bx$, provided that $\delta_1\ge \delta(R_1)$
and $N_1(\delta_1)=0$ otherwise. For each one of these first--stage codewords, there are
$e^{n_2[R_2+h(\delta_2)-\ln 2]}$ second--stage codewords $\{\tbx\}$ at distance $n_2\delta_2$ from
the vector formed by the last $n_2$ components of $\bx$, provided that $\delta_2\ge \delta(R_2)$.
Thus, the total number of concatenated codewords $\{(\hbx,\tbx)\}$ at distance $n\delta=n_1\delta_1+n_2
\delta_2$ (that is, $\delta=\lambda\delta_1+(1-\lambda)\delta_2$) from $\bx$ is given by
\begin{eqnarray}
N_2(\delta)&\exe&\sum_{\delta_1=\delta(R_1)}^{1-\delta(R_1)} e^{n_1[R_1+h(\delta_1)-\ln 2]}\cdot
e^{n_2[R_2+h((\delta-\lambda\delta_1)/(1-\lambda))-\ln 2]}\nonumber\\
&\exe&\exp\left\{n\max_{\delta(R_1)\le\delta_1\le 1-\delta(R_1)}\left[R+\lambda h(\delta_1)
+(1-\lambda)h\left(\frac{\delta-\lambda\delta_1}{1-\lambda}\right)-\ln 2\right]\right\}.
\end{eqnarray}
Consequently, the exponential growth rate of $N_2(\delta)$ is given by
$$S(\delta)=\max_{\delta(R_1)\le\delta_1\le 1-\delta(R_1)}\left[R+\lambda h(\delta_1)
+(1-\lambda)h\left(\frac{\delta-\lambda\delta_1}{1-\lambda}\right)-\ln 2\right].$$
For large $\delta$, the constraint $\delta(R_1)\le\delta_1\le 1-\delta(R_1)$
is inactive and the achiever of $S(\delta)$ is $\delta_1=\delta$, and then
$$S(\delta)=
R+\lambda h(\delta)+(1-\lambda)h(\delta)-\ln 2=R+h(\delta)-\ln 2.$$
If we now gradually reduce $\delta$, the behavior depends on whether
we first encounter the value $\delta=\delta(R_1)$,
below which $\delta_1=\delta$ no longer satisfies the constraint, or the
the value $\delta=\delta(R)$, below which
$S(\delta)=R+h(\delta)-\ln 2$
vanishes. This in turn depends on whether $\delta(R_1)$ is larger or smaller than $\delta(R)$,
or equivalently, if $R_1< R< R_2$ or $R_1\ge R\ge R_2$.
Consider the case $R_1\ge R\ge R_2$ first. In this case, $\delta(R_1)\le\delta(R)\le \delta(R_2)$, and we have:
\begin{equation}
S(\delta)=\left\{\begin{array}{ll}
R+h(\delta)-\ln 2 & \delta(R)<\delta < 1-\delta(R)\\
0 & \delta=\delta(R)~~\mbox{or}~~\delta = 1-\delta(R)\\
-\infty & \delta<\delta(R)~~\mbox{or}~~\delta > 1-\delta(R)
\end{array}\right.
\end{equation}
exactly like in the ordinary, non--hierarchical ensemble (cf.\ eq.\ (\ref{srem})), and
then the corresponding exponential rate of the partition function is as in
Subsection \ref{remc}, except that here $\beta$ is not scaled by $B$, i.e.,
$\phi(\beta)=-u(\beta,R)$.
The other case is $R_1< R< R_2$, which is equivalent to $\delta(R_1)>\delta(R)> \delta(R_2)$.
Here, in analogy to the GREM with two phase transitions, we have:
$$\phi(\beta)=\left\{\begin{array}{ll}
-v(\beta,R) & \beta < \beta(R_1)\\
-\lambda\beta\delta(R_1)-(1-\lambda)v(\beta,R_2)
& \beta(R_1)\le \beta<\beta(R_2)\\
-\beta[\lambda\delta(R_1)+(1-\lambda)\delta(R_2)] & \beta > \beta(R_2)
\end{array}\right.$$
We now identify the first line as the purely paramagnetic phase,
the second line -- as the ``semi--glassy'' phase (where $\{\hbx\}$ are glassy but $\{\tbx\}$
are paramagnetic), and the third line -- as the purely glassy phase.
Note that the glassy phase here behaves as if the two parts of the code,
at rates $R_1$ and $R_2$, were operating independently, namely,
as if $\{{\cal C}_2(i)\}_{i=1}^{M_1}$ were all identical,
in which case, the distortion would have been minimized separately over the
two segments. We will get back to this point in the sequel.
We have seen then that the ensemble behaves substantially differently
depending on whether $R_1\ge R_2$ or $R_1 < R_2$. In the former case,
the above calculation may indicate that the ensemble performance is similar
to that of an ordinary block code of length $n$ without any structure.
We next carry out a detailed analysis of the characteristic function
and its exponential rate, which we shall denote by $\psi(s,R_1,R_2)$.
Similarly as before, we first compute $\bE\{Z^{1/\theta}(s\theta|\bX)\}$:
\begin{eqnarray}
\bE\{Z^{1/\theta}(s\theta|\bX)\}&=&\bE\left\{\left[\sum_{d_1=0}^{n_1}\sum_{d_2=0}^{n_2}
N(d_1,d_2)\cdot e^{-s\theta(d_1+d_2)}\right]^{1/\theta}\right\}\nonumber\\
&\exe&\sum_{d_1=0}^{n_1}\sum_{d_2=0}^{n_2}\bE\{N^{1/\theta}(d_1,d_2)\}\cdot e^{-s(d_1+d_2)},
\end{eqnarray}
where $N(d_1,d_2)$ is the number concatenated codewords $\{(\hbx,\tbx)\}$ for which the
first stage contributes distance $d_1$ and the second
stage contributes distance $d_2$. For the moments $\bE\{N^{1/\theta}(d_1,d_2)\}$,
or equivalently, $\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}$, the following is
proven in Section A.2 of the Appendix:
\begin{equation}
\label{nd1d2}
\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}\exe\left\{\begin{array}{ll}
\exp\{n[\lambda W_1+(1-\lambda)W_2]\} &
\delta_1 \in{\cal I}^c(R_1),~\delta_2\in{\cal I}^c(R_2)\\
\exp\{n[\lambda W_1+
(1-\lambda)W_2/\theta]\} & \delta_1\in{\cal I}^c(R_1),~\delta_2\in{\cal I}(R_2)\\
\exp\{n[\lambda W_1+(1-\lambda)W_2]/\theta\} &
\delta_1 \in{\cal I}(R_1),~\delta_2\in{\cal I}(R_2)\\
\exp\{n\eta[\lambda W_1+(1-\lambda)W_2]\} &
\delta_1 \in{\cal I}(R_1),~\delta_2\in{\cal I}^c(R_2)
\end{array}\right.
\end{equation}
where ${\cal I}(R)\dfn(\delta(R),1-\delta(R))$, ${\cal I}^c(R)=[0,1]\setminus{\cal I}(R)$,
$W_i=W(\delta_i,R_i)$, $i=1,2$, with $W(\delta,R)$ being defined as
$$W(\delta,R)\dfn R+h(\delta)-\ln 2$$
and
$$\eta=\eta(\theta,\delta_1,\delta_2,\lambda,R)=\left\{\begin{array}{ll}
1 & \lambda W_1+(1-\lambda)W_2< 0\\
\frac{1}{\theta} & \lambda W_1+(1-\lambda)W_2\ge 0
\end{array}\right.$$
Therefore,
\begin{eqnarray}
\bE\{Z^{1/\theta}(s\theta|\bX)\}&\exe&
\sum_{\delta_1\in{\cal I}^c(R_1)}\sum_{\delta_2\in{\cal I}^c(R_2)}
e^{n[R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)-\ln 2]}\times\nonumber\\
& &e^{-sn[\lambda\delta_1+(1-\lambda)\delta_2]}+\nonumber\\
& &\sum_{\delta_1\in{\cal I}^c(R_1)}\sum_{\delta_2\in{\cal I}(R_2)}
e^{n[\lambda(R_1+h(\delta_1)-\ln 2)+(1-\lambda)(R_2+h(\delta_2)-\ln 2)/\theta]}\times\nonumber\\
& &e^{-sn[\lambda\delta_1+(1-\lambda)\delta_2]}+\nonumber\\
& &\sum_{\delta_1\in{\cal I}(R_1)}\sum_{\delta_2\in{\cal I}(R_2)}
e^{n[\lambda(R_1+h(\delta_1)-\ln 2)+(1-\lambda)(R_2+h(\delta_2)-\ln 2)]/\theta}\times\nonumber\\
& &e^{-sn[\lambda\delta_1+(1-\lambda)\delta_2]}+\nonumber\\
& &\sum_{\delta_1\in{\cal I}(R_1)}\sum_{\delta_2\in{\cal I}^c(R_2)}
e^{n\eta[\lambda(R_1+h(\delta_1)-\ln 2)+(1-\lambda)(R_2+h(\delta_2)-\ln 2)]}\times\nonumber\\
& &e^{-sn[\lambda\delta_1+(1-\lambda)\delta_2]}\nonumber\\
&\dfn& A+B+C+D
\end{eqnarray}
Let us now handle each one of these four terms and take the limit $\theta\to\infty$.
This results in:
\begin{eqnarray}
A&\exe&\left[\sum_{\delta_1\in{\cal I}^c(R_1)}e^{n_1[R_1+h(\delta_1)-\ln 2-s\delta_1]}\right]\cdot
\left[\sum_{\delta_2\in{\cal I}^c(R_2)}e^{n_2[R_2+h(\delta_2)-\ln 2-s\delta_2]}\right]\nonumber\\
&\exe&e^{-n_1u(s,R_1)}\cdot e^{-n_2u(s,R_2)}\nonumber\\
&=& e^{-n[\lambda u(s,R_1)+(1-\lambda)u(s,R_2)]},
\end{eqnarray}
\begin{eqnarray}
B&\exe&\left[\sum_{\delta_1\in{\cal I}^c(R_1)}e^{n_1[R_1+h(\delta_1)-\ln 2-s\delta_1]}\right]\cdot
\left[\sum_{\delta_2\in{\cal I}(R_2)}e^{-n_2s\delta_2}\right]\nonumber\\
&\exe&e^{-n_1u(s,R_1)}\cdot e^{-n_2\delta(R_2)}\nonumber\\
&=& e^{-n[\lambda u(s,R_1)+(1-\lambda)\delta(R_2)]},
\end{eqnarray}
\begin{equation}
C\exe e^{-n[\lambda\delta(R_1)+(1-\lambda)\delta(R_2)]},
\end{equation}
and
\begin{equation}
D\exe e^{-nf(s,R_1,R_2)}
\end{equation}
where
$$f(s,R_1,R_2)=\min_{\delta_1\in{\cal I}(R_1),\delta_2\in{\cal I}^c(R_2)}
\{s[\lambda\delta_1+(1-\lambda)\delta_2]-
\mu(\delta_1,\delta_2)[R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)-\ln 2]\}$$
and where
$$\mu(\delta_1,\delta_2)=\left\{\begin{array}{ll}
1 & R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)<\ln 2\\
0 & R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)\ge\ln 2 \end{array}\right.$$
Among the terms $A$, $B$, and $C$, the term $A$ is exponentially the dominant one.
To check whether or not $A$ dominates also $D$, we will have to investigate the function
$f(s,R_1,R_2)$. This is done in Subsection A.3 of the Appendix, where it is
shown that
this function is as follows:
For $R_1 > R_2$:
\begin{equation}
\label{fr1gr2}
f(s,R_1,R_2)=\left\{\begin{array}{ll}
u(s,R) & 0\le s\le s_{R_1}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_1}
\end{array}\right.
\end{equation}
and for $R_1 < R_2$:
\begin{equation}
\label{fr1lr2}
f(s,R_1,R_2)=\left\{\begin{array}{ll}
s[\lambda\delta(R_1)+(1-\lambda)\delta(R_2)] & 0\le s\le s_{R_2}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_2}\end{array}\right.
\end{equation}
Finally, the overall exponential rate of the characteristic function, $\psi(s,R_1,R_2))$, we have to take
into account the contribution of $A$, as mentioned above. This gives:
$$\psi(s,R_1,R_2))=\min\{f(s,R_1,R_2),a(s,R_1,R_2)\}$$
where $a(s,R_1,R_2)\dfn \lambda u(s,R_1)+(1-\lambda)u(s,R_2)$.
Now, in the case $R_1> R_2$, for small $s$, the function $f$ is linear
with slope $\delta(R)$, whereas the function $a$
is linear with a slope of $\lambda\delta(R_1)+(1-\lambda)\delta(R_2)$ which is
larger. Thus, $f$ is smaller in some interval of small $s$. However, for
larger $s$, $f$ continues to have a linear term with slope $\lambda \delta(R_1)$ whereas
$a$ never exceeds the level of $\ln 2-R$.
Thus, there must be a (unique) point of intersection $s^*$. Consequently,
for $R_1> R_2$, we have
$$\psi(s,R_1,R_2)=\left\{\begin{array}{ll}
f(s,R_1,R_2) & s\le s^*\\
a(s,R_1,R_2) & s\ge s^*
\end{array}\right.$$
where $f(s,R_1,R_2)$ is as in (\ref{fr1gr2}).
Concerning the case $R_1 < R_2$, both $f$ (of eq.\ (\ref{fr1lr2})) and $a$
start as linear functions of the
same slope of $\lambda\delta(R_1)+(1-\lambda)\delta(R_2)$.
However, while the latter begins its
curvy part at $s=s_{R_1}$, the former continues
to be linear until the point $s=s_{R_2}> s_{R_1}$. In this case,
then it is easy to see that $\psi(s,R_1,R_2)$ is dominated by $a$ across
the entire range $s\ge 0$, i.e.,
$$\psi(s,R_1,R_2)=\lambda u(s,R_1)+(1-\lambda)u(s,R_2).$$
We see then that the ensemble performance
is substantially different in the two cases: For $R_1 < R_2$,
$\psi(s,R_1,R_2)$ is exactly the same as if we used two {\it independent}
block codes of lengths $n_1$ and $n_2$
at rates $R_1$ and $R_2$, respectively.
In particular, the corresponding average
distortion is $\lambda\delta(R_1)+(1-\lambda)\delta(R_2)$ which
is, of course, larger than $\delta(R)$.
In other words, we are
gaining nothing from the tree structure and the dependence between the two parts of the code.
For $R_1 > R_2$,
on the other hand, there is at least a considerable range of small $s$ for
which $\psi(s,R_1,R_2)=u(s,R)$, namely,
the ensemble performance is exactly like that of the
ordinary ensemble of full block code
of length $n$ and rate $R$, without any structure (which is also the best achievable exponential rate).
However, beyond a certain value of $s$, there is some loss
in comparison to the ordinary ensemble.
The case $R_1=R_2=R$ can be obtained as the limiting behavior of both
$R_1<R_2$ and $R_1> R_2$, by taking both rates to be arbitrarily close
to each other. In this case, we obtain $\psi(s,R_1,R_2)=u(s,R)$ throughout the
{\it entire} range $s\ge 0$ (cf.\ the discussion on this in the Introduction).
The conclusion then is that
if we use an hierarchical structure of the kind we consider in this paper,
it is best to assign equal rates at the two stages, but then we might as
well abandon the tree structure of the code altogether, and just encode the two parts
independently, both at rate $R$ (this will moreover save complexity at the encoder).
If, however, certain considerations
dictate different rates at different segments, then it is better to
encode at a larger rate in the first segment and at a smaller rate in the
second.
This derivation can be extended, in principle, to any finite
number $k$ of stages. The analysis is, of course, more complicated but
conceptually, the ideas are the same. We will not carry out this extension
in this paper.
\subsection{Channel Coding}
\label{channelcoding}
In complete duality to the source coding problem, one may consider
a channel code (for the BSC)
with a similar hierarchical structure: Given a binary
information vector of length $nR=n_1R_1+n_2R_2$ nats, we encode it in two
parts: The first segment, of length $n_1R_1$ nats, is encoded
to a binary channel input vector of length $n_1$, independently of the forthcoming
$n_2R_2$ nats (thus, the channel encoder is of reduced delay). Then, the
remaining $n_2R_2$ nats are mapped to another binary channel input vector of length
$n_2$ and it depends on the entire information vector of length $nR$.
The ensemble of codebooks is drawn similarly as before: first, a
randomly drawn first--stage codebook of size $e^{n_1R_1}$, and then, for each one of its
codewords, another codebook of size $e^{n_2R_2}$
is drawn independently. Once again, each bit of
each codeword is drawn by independent fair coin tossing.
The decoder applies maximum likelihood (ML) decoding based on the
entire channel output vector $\by$ of length $n=n_1+n_2$, pertaining to
the input $\bx$ of length $n$. The analogy with the GREM is that here,
the energy function is the log--likelihood, which is additive over
the two stages by the memorylessness of the channel.
In full analogy to the GREM and the
source coding problem of Subsection \ref{lossysourcecoding}, and as an extension
to the derivation in Subsection \ref{remc}, here too, the partition
function $Z_e(\beta|\by)$ has exactly the same two different
types of behavior, depending on whether $R_1\ge R_2$
or $R_1< R_2$. Therefore, we will not repeat this here.
Concerning the aspect of performance evaluation of this ensemble of codes,
and a comparison to the ordinary ensemble, here
the natural figure of merit is Gallager's random coding error exponent,
which can be analyzed using methods similar to those that we used
in Subsection \ref{lossysourcecoding}. We will not carry out a very refined analysis
as we did before, but we will make a few observations in this context,
although not quite directly related to the GREM.
Referring to the notation of Subsection \ref{remc}, let ${\cal C}=
\{\bx_1,\ldots,\bx_M\}$ be a given channel code of
size $M=e^{nR}$ and block length $n$, and let $\by$ designate
the output vector of the BSC, of length $n$.
Gallager's classical upper bound \cite[p.\ 65, eq.\ (2.4.8)]{VO79} on the probability
of error is well known to be given by
$$P_e \le \frac{1}{M}\sum_{m=1}^M\sum_{\by} P(\by|\bx_m)^{1/(1+\rho)}\cdot
\left[\sum_{m'\ne m} P(\by|\bx_{m'})^{1/(1+\rho)}\right]^\rho
~~~0\le\rho\le 1.$$
Consider first the ordinary ensemble, where all $M$ codewords are
chosen independently at random. In this case, taking the expectation
of both sides, the average
error probability is upper bounded by
$$\bar{P}_e \le \frac{1}{M}\sum_{m=1}^M\sum_{\by}
\bE\{P(\by|\bX_m)^{1/(1+\rho)}\}\cdot
\bE\left\{\left[\sum_{m'\ne m}
P(\by|\bX_{m'})^{1/(1+\rho)}\right]^\rho\right\}.$$
As is shown in \cite{Merhav07}, the second factor of the summand
is actually the expectation of the $\rho$--th moment of the
partition function $Z_e(\beta|\by)$ computed at the inverse temperature
$\beta=1/(1+\rho)$. Now, at least for the ordinary ensemble,
the traditional derivation, which is based on applying Jensen's inequality,
is good enough to yield an exponentially tight bound \cite{Gallager73} on the ensemble performance.
This amounts to inserting the expectation into the square brackets, i.e.,
$$\bar{P}_e \le \frac{1}{M}\sum_{m=1}^M\sum_{\by}
\bE\{P(\by|\bX_m)^{1/(1+\rho)}\}\cdot
\left[\sum_{m'\ne m}\bE\{
P(\by|\bX_{m'})^{1/(1+\rho)}\}\right]^\rho.$$
We shall not continue any further with the analysis of this expression.
Instead, we shall compare it as is, with a corresponding upper bound
for the hierarchical ensemble defined above.
In the hierarchical case with $k=2$ stages, the probability of error consists
of two contributions. The first pertains to all incorrect
codewords $\bx=(\bx',\bx'')$ whose first segment $\bx'$ agrees
with that of the correct codeword, and the second one is associated
with all other incorrect codewords. As for the former type of codewords,
the ML decoder actually compares the likelihood scores of the second segment
only (as those of the first segment are the same and hence cancel out),
and so, these incorrect codewords contribute a term of the order
of $e^{-n_2E_r(R_2)}$ to the average error probability, where $E_r(R)$
is the Gallager's random coding error exponent
function \cite[p.\ 139, eq.\ (5.6.16)]{Gallager68}. Concerning the second
set of incorrect codewords, we can apply an upper bound as above, except
that the expectations have to be taken w.r.t.\ the hierarchical ensemble.
However, it is easy to see that the expectation of
$\bE\{P(\by|\bX)^{1/(1+\rho)}\}$ is exactly the same as in the ordinary
ensemble, and thus, so is the upper bound
for this set of codewords, which is then
$e^{-nE_r(R)}$. The total average error probability is then
upper bounded by
$$\bar{P}_e\le e^{-nE_r(R)}+e^{-n_2E_r(R_2)}
= e^{-nE_r(R)}+e^{-n(1-\lambda)E_r(R_2)}.$$
This gives further motivation why $R_2$ should be chosen smaller than $R_1$:
If $R_2> R_1$, the second term definitely dominates the exponent, because
both $n_2 < n$ and $R_2> R$ and so $E_r(R_2) < E_r(R)$. For a given $R$ and $\lambda$, can we, and if so
how, assign the segmental rates $R_1$ and $R_2$ such that the second term
would not be dominant, i.e.,
$(1-\lambda)E_r(R_2)\ge E_r(R)$? If $R$ is large enough this is
possible. For example, one way to do this is to select $R_1=C$, where $C$ is the channel
capacity. In this case, we have, by the convexity of $E_r(\cdot)$:
$$E_r(R)=E_r(\lambda C+(1-\lambda)R_2)\le
\lambda E_r(C)+(1-\lambda)E_r(R_2)=(1-\lambda)E_r(R_2).$$
For this strategy to be applicable, $R$ must be at least as large as $\lambda C$.
How does this discussion extend to a general number of stages $k$
and is there a more systematic approach to allocate the segmental rates $R_1,\ldots,R_k$
for a given overall rate $R$?
For simplicity, let us suppose that the segment lengths
are all the same, i.e., $n_1=n_2=\ldots=n_k=n/k$.
The extension turns out to be quite straightforward: In the case of $k$ stages there
are $k$ types of incorrect codewords: Those that agree with the correct
codeword in all stages except the last stage, those that agree in all stages
except the last two stages, etc. Accordingly, using the same
considerations as above, it is easy to see then that the
upper bound on the average error
probability consists of $k$ contributions whose exponents are
$$\frac{k-i}{k}E_r\left(\frac{1}{k-i}\sum_{j=i+1}^k R_j\right), ~~i=0,1,\ldots,k-1.$$
For convenience, let us denote
$$\bar{R}_i=\frac{1}{k-i}\sum_{j=i+1}^k R_j.$$
Under what conditions and how can we assign the segmental rates such that
$$\frac{k-i}{k}E_r(\bar{R}_k)\ge E_r(R)$$
for all $i=1,2,\ldots,k-1$?
First, we must select $\bar{R}_1$ sufficiently small such that
$E_r(\bar{R}_1)\ge \frac{k}{k-1}E_r(R)$.
As $R$ is given, this will dictate the choice of
$R_1$ according to the identity
$$R=\bar{R}_0=\frac{1}{k}R_1+\frac{k-1}{k}\bar{R}_1.$$
Next, we choose $\bar{R}_2$ small enough such that
$$E_r(\bar{R}_2)\ge \frac{k}{k-2}E_r(R).$$
As $\bar{R}_1$ has already been
chosen, this will dictate the choice of $R_2$ according to the identity
$$\bar{R}_1=\frac{1}{k-1}R_2+\frac{k-2}{k-1}\bar{R}_2,$$
and so on. This procedure continues until in the last step
we choose
$R_k=\bar{R}_{k-1}$ such that $E_r(R_k)\ge kE_r(R)$, which
dictates the choice of $R_{k-1}$ via $\bar{R}_{k-2}=(R_k+R_{k-1})/2$,
where $\bar{R}_{k-2}$ was selected in preceeding step.
An obvious condition for this procedure to be applicable is that
$R$ would be
large enough such that $E_r(R)\le E_r(0)/k$. Note that if some of the
segmental rates exceed capacity (or even the log alphabet size), this
is not a problem, as long as the averages $\bar{R}_i$ are all small enough.
\section*{Appendix}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\subsection*{A.1 Proof of Eq.\ (\ref{moments})}
\label{moments1level}
We begin with a simple large deviations bound regarding
the distance enumerator, which appears also in \cite{Merhav07b},
but we present here too for the sake of
completeness.
For $a,b\in[0,1]$, consider the binary divergence
\begin{eqnarray}
D(a\|b)&\dfn&a\ln \frac{a}{b}+(1-a)\ln\frac{1-a}{1-b}\nonumber\\
&=&a\ln \frac{a}{b}+(1-a)\ln\left[1+\frac{b-a}{1-b}\right].
\end{eqnarray}
To derive a lower bound to $D(a\|b)$, let us use the inequality
\begin{equation}
\label{lnineq}
\ln(1+x)=-\ln\frac{1}{1+x}=-\ln\left(1-\frac{x}{1+x}\right)\ge \frac{x}{1+x},
\end{equation}
and then
\begin{eqnarray}
D(a\|b)&\ge&a\ln \frac{a}{b}+(1-a)\cdot\frac{(b-a)/(1-b)}
{1+(b-a)/(1-b)}\nonumber\\
&=&a\ln \frac{a}{b}+b-a\nonumber\\
&>&a\left(\ln\frac{a}{b}-1\right).
\end{eqnarray}
For every given $\by$, $N(d)$ is the sum of the $e^{nR}-1$
independent binary random variables, $\{1\{d(\bX_{m'},\by)=d\}\}_{m'\ne m}$,
where the probability that $d(\bX_{m'},\by)=n\delta$ is
exponentially $b\exe e^{-n[\ln 2- h(\delta)]}$.
The event $N(n\delta)\ge e^{nA}$, for $A\in[0,R)$,
means that the relative frequency of the event $1\{d(\bX_{m'},\by)=n\delta\}$
is at least $a=e^{-n(R-A)}$. Thus, by the Chernoff bound:
\begin{eqnarray}
\mbox{Pr}\{N(n\delta)\ge e^{nA}\}&\lexe&
\exp\left\{-(e^{nR}-1)D(e^{-n(R-A)}\|e^{-n[\ln 2-
h(\delta)]})\right\}\nonumber\\
&\lexe&
\exp\left\{-e^{nR}\cdot e^{-n(R-A)}(n[(\ln 2-R-
h(\delta)+A]-1)\right\}\nonumber\\
&\le&
\exp\left\{-e^{nA}(n[\ln 2-R-h(\delta)+A]-1)\right\}.
\end{eqnarray}
Denoting by ${\cal I}(R)$ the interval $(\delta(R),1-\delta(R))$ and by ${\cal I}^c(R)$,
the complementary range $[0,1]\setminus{\cal I}(R)$,
we have, for $\delta\in{\cal I}^c(R)$:
\begin{eqnarray}
\bE\{N^s(n\delta)\}&\le&e^{n\epsilon s}\cdot\mbox{Pr}\{1\le N(n\delta)\le e^{n\epsilon}\}+
e^{nRs}\cdot \mbox{Pr}\{ N(n\delta)\ge e^{n\epsilon}\}\nonumber\\
&\le&e^{n\epsilon s}\cdot\mbox{Pr}\{N(n\delta)\ge 1\}+
e^{nRs}\cdot \mbox{Pr}\{ N(n\delta)\ge e^{n\epsilon}\}\nonumber\\
&\le&e^{n\epsilon s}\cdot\bE\{N(n\delta)\}+
e^{nRs}\cdot e^{-(n\epsilon-1)e^{n\epsilon}}\nonumber\\
&\le&e^{n\epsilon s}\cdot e^{n[R+h(\delta)-\ln 2]}+
e^{nRs}\cdot e^{-(n\epsilon-1)e^{n\epsilon}}.
\end{eqnarray}
One can let $\epsilon$ vanish with $n$ sufficiently slowly that
the second term is still superexponentially small, e.g., $\epsilon=1/\sqrt{n}$.
Thus, for $\delta\in{\cal I}^c(R)$, $\bE\{N^s(n\delta)\}$ is exponentially bounded
by $e^{n[R+h(\delta)-\ln 2]}$ independently of $s$. For $\delta\in{\cal I}(R)$, we have:
\begin{eqnarray}
\bE\{N^s(n\delta)\}&\le&e^{ns[R+h(\delta)-\ln 2+\epsilon]}\cdot
\mbox{Pr}\{N(n\delta)\le e^{n[R+h(\delta)-\ln 2+\epsilon]}\}+\nonumber\\
& &e^{nRs}\cdot
\mbox{Pr}\{N(n\delta)\ge e^{n[R+h(\delta)-\ln 2+\epsilon]}\}\nonumber\\
&\le&e^{ns[R+h(\delta)-\ln 2+\epsilon]}
+e^{nRs}\cdot e^{-(n\epsilon-1)e^{n\epsilon}}
\end{eqnarray}
where again, the second term is exponentially negligible.
To see that both bounds are exponentially tight, consider
the following lower bounds. For $\delta\in{\cal I}^c(R)$,
\begin{eqnarray}
\bE\{N^s(n\delta)\}&\ge&1^s\cdot\mbox{Pr}\{N(n\delta)=1\}\nonumber\\
&=&e^{nR}\cdot\mbox{Pr}\{d_H(\bX,\by)=n\delta\}\cdot
\left[1-\mbox{Pr}\{d_H(\bX,\by)=n\delta\}\right]^{e^{nR}-1}\nonumber\\
&\exe&e^{nR}e^{-n[\ln 2-h(\delta)]}\cdot
\left[1-e^{-n[\ln 2-h(\delta)]}\right]^{e^{nR}}\nonumber\\
&=&e^{n[R+h(\delta)-\ln 2]}\cdot\exp\{e^{nR}\ln[1-e^{-n[\ln 2-h(\delta)]}]\}.
\end{eqnarray}
Using again the inequality in (\ref{lnineq}),
the second factor is lower bounded by
$$\exp\{-e^{nR}e^{-n[\ln 2-h(\delta)]}/(1-e^{-n[\ln 2-h(\delta)]})\}
=\exp\{-e^{-n[\ln 2-R-h(\delta)]}/(1-e^{-n[\ln 2-h(\delta)]})\}$$
which clearly tends to unity as $\ln 2-R-h(\delta) > 0$ for $\delta\in{\cal I}^c(R)$.
Thus, $\bE\{N^s(n\delta)\}$ is exponentially lower bounded by $e^{n[R+h(\delta)-\ln 2]}$.
For $\delta\in{\cal I}(R)$, and an arbitrarily small $\epsilon > 0$, we have:
\begin{eqnarray}
\bE\{N^s(n\delta)\}&\ge& e^{ns[R+h(\delta)-\ln 2-\epsilon]}\cdot
\mbox{Pr}\{N(n\delta)\ge e^{n[R+h(\delta)-\ln 2-\epsilon]}\}\nonumber\\
&=& e^{ns[R+h(\delta)-\ln 2-\epsilon]}\cdot\left(1-
\mbox{Pr}\{N(n\delta)< e^{n[R+h(\delta)-\ln 2-\epsilon]}\}\right)
\end{eqnarray}
where $\mbox{Pr}\{N(n\delta)< e^{n[R+h(\delta)-\ln 2-\epsilon]}\}$ is again
upper bounded, for an internal point in ${\cal I}(R)$,
by a double exponentially small quantity as above.
For $\delta$ near the boundary of ${\cal I}(R)$, namely, when $R+h(\delta)-\ln 2\approx 0$,
we can lower bound $\bE\{N^s(n\delta)\}$ by slightly reducing $R$ to $R'=R-\epsilon$
(where $\epsilon > 0$ is very small). This will make
$\delta$ an internal point of ${\cal I}^c(R')$ for which the previous bound applies,
and this bound
is of the exponential order of $e^{n[R'+h(\delta)-\ln 2]}$.
Since $R'+h(\delta)-\ln 2$ is still very close to zero, then $e^{n[R'+h(\delta)-\ln 2]}$
is of the same exponential order as $e^{ns[R+h(\delta)-\ln 2]}$
since both are about $e^{0\cdot n}$.
It should be noted that a similar double--exponential bound can be obtained for
the probability of the event $\{N(n\delta) \le e^{nA}\}$, where $A < R+h(\delta)-\ln 2$
and $R+h(\delta)-\ln 2> 0$.
Here we can proceed as above except that the in the lower bound on divergence $D(a\|b)$
we should take the second line of (A.3) (rather than the third),
which is of the exponential order of $b\exe e^{-n[\ln 2-h(\delta)]}$ (observe that
here $b$ is exponentially larger than $a$, as opposed to the earlier case). Thus,
we obtain $R+h(\delta)-\ln 2>0 $ at the second level exponent, and so the decay is double
exponential as before.
\subsection*{A.2. Proof of Eq.\ (\ref{nd1d2})}
\label{moments2levels}
First, let us write $N(n_1\delta_1,n_2\delta_2)$ as follows:
\begin{eqnarray}
N(n_1\delta_1,n_2\delta_2)&=&\sum_{i=1}^{M_1}1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\cdot
\sum_{j=1}^{M_2}1\{d_H(\bx'',\tbx_{i,j})=n_2\delta_2\}\nonumber\\
&\dfn&\sum_{i=1}^{M_1}1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\cdot N_i(n_2\delta_2)
\end{eqnarray}
where $\bx'$ and $\bx''$ designate $(x_1,\ldots,x_{n_1})$ and $(x_{n_1+1},\ldots,x_n)$,
respectively, and where $1\{\cdot\}$ denotes the indicator function of an event.
We now treat each one of the four cases pertaining to the combinations of both
$\delta_1$ and $\delta_2$ being or not being members of ${\cal I}(R_1)$ and ${\cal I}(R_2)$,
respectively.\\
\subsubsection*{Case 1: $\delta_1\in{\cal I}^c(R_1)$ and $\delta_2\in{\cal I}^c(R_2)$}
\label{case1}
For a given, arbitrarily small
$\epsilon > 0$, consider the event ${\cal E}=\{N(n_1\delta_1,n_2\delta_2)\ge e^{n\epsilon}\}$.
If both the
number of indices $i$ for which $d_H(\bx',\hbx_i)=n_1\delta_1$ is less than $e^{n_1\epsilon}$
and for each $i$,
$N_i(n_2\delta_2)\le e^{n_2\epsilon}$, then clearly, the event ${\cal E}$ does not occur.
Thus, for ${\cal E}$ to occur, at least one of these events must occur. In other words,
either the number of indices $i$ for which $d_H(\bx',\hbx_i)=n_1\delta_1$ is
larger than $e^{n_1\epsilon}$ or there exist $i$ for which
$N_i(n_2\delta_2)> e^{n_2\epsilon}$. The probability of the former
event is upper bounded by
$e^{-e^{n_1\epsilon}(n_1\epsilon-1)}$ (cf.\ Subsection A.1). Similarly,
the probability of the latter, for a given $i$, is bounded by $e^{-e^{n_2\epsilon}(n_2\epsilon-1)}$. Thus,
the probability of the union of events $\bigcup_i\{N_i(n_2\delta_2)>
e^{n_2\epsilon}\}$ is upper bounded by
$M_1e^{-e^{n_2\epsilon}(n_2\epsilon-1)}=e^{n_1R_1}\cdot e^{-e^{n_2\epsilon}(n_2\epsilon-1)}$, which
is still double exponential in $n$. Thus,
$$\mbox{Pr}\{{\cal E}\}\le e^{-e^{n_1\epsilon}(n_1\epsilon-1)}+e^{n_1R_1}\cdot
e^{-e^{n_2\epsilon}(n_2\epsilon-1)}.$$
Therefore,
\begin{eqnarray}
\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}&\le&
0^{1/\theta}\cdot\mbox{Pr}\{N(n_1\delta_1,n_2\delta_2)=0\}+
e^{n\epsilon/\theta}\cdot\mbox{Pr}\{1\le N(n_1\delta_1,n_2\delta_2)\le e^{n\epsilon}\}\nonumber\\
& &+e^{nR/\theta}\cdot\mbox{Pr}\{{\cal E}\}\nonumber\\
&\le&e^{n\epsilon/\theta}\cdot\mbox{Pr}\{N(n_1\delta_1,n_2\delta_2)\ge 1\}+
e^{nR/\theta}\cdot\mbox{Pr}\{{\cal E}\}\nonumber\\
&\le&e^{n\epsilon/\theta}\cdot\bE\{N(n_1\delta_1,n_2\delta_2)\}+
e^{nR/\theta}\cdot\mbox{Pr}\{{\cal E}\},
\end{eqnarray}
which is exponentially upper bounded by $e^{n[R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)-\ln 2]}$
since $\epsilon$ is arbitrarily small, $\bE\{N((n_1\delta_1,n_2\delta_2)\}\exe
e^{n[R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)-\ln 2]}$, and the last term is
double--exponential. To obtain the compatible lower bound, we use
\begin{eqnarray}
\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}&\ge&
1^{1/\theta}\cdot\mbox{Pr}\{N(n_1\delta_1,n_2\delta_2)=1\}\nonumber\\
&=&\mbox{Pr}\{N(n_1\delta_1,n_2\delta_2)=1\}.
\end{eqnarray}
Now, the event $\{N(n_1\delta_1,n_2\delta_2)=1\}$ is the event that there is exactly
one value of $i$ such that $d_H(\bx',\hbx)=n_1\delta_1$, and that for this $i$, there is
exactly one $j$ such that $d_H(\bx'',\tbx)=n_2\delta_2$. As shown in Subsection A.1,
the probability of the former is exponentially $e^{n_1[R_1+h(\delta_1)-\ln 2]}$ and
the probability of the latter is exponentially $e^{n_2[R_2+h(\delta_2)-\ln 2]}$. Thus,
by independence, $\mbox{Pr}\{N(n_1\delta_1,n_2\delta_2)=1\}$ is the product, which is
exponentially $e^{n[R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)-\ln 2]}$.
\subsubsection*{Cases 2 and 3: $\delta_2\in{\cal I}(R_2)$}
\label{cases23}
Define now the event ${\cal A}$ as
$${\cal A}=\bigcap_{i=1}^{M_1}\left\{N_i(n_2\delta_2)\le
\exp\{n_2[R_2+h(\delta_2)-\ln 2+\epsilon]\}\right\}.$$
As we have argued before, the probability of ${\cal A}$ is doubly exponentially
close to unity (since the probability of ${\cal A}^c$ is upper bounded by the sum of exponentially
many doubly-exponentially small probabilities). Now, clearly, if ${\cal A}$ occurs,
$$N(n_1\delta_1,n_2\delta_2)\le \exp\{n_2[R_2+h(\delta_2)-\ln 2+\epsilon]\}\cdot\sum_{i=1}^{M_1}
1\{d_H(\bx',\hbx_i)=n_1\delta_1\}.$$
Thus,
\begin{eqnarray}
\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}&\le&\mbox{Pr}\{{\cal A}\}\cdot\bE\left\{\left[
\exp\{n_2[R_2+h(\delta_2)-\ln 2+\epsilon]\}\times\right.\right.\nonumber\\
& &\left.\left.\sum_{i=1}^{M_1}
1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\right]^{1/\theta}\right\}\nonumber\\
& &+e^{nR/\theta}\cdot\mbox{Pr}\{{\cal A}^c\},
\end{eqnarray}
where the second term is again doubly--exponentially small. As for the first term,
we bound $\mbox{Pr}\{{\cal A}\}$ by unity and
\begin{eqnarray}
&&\bE\left\{\left[\exp\{n_2[R_2+h(\delta_2)-\ln 2+\epsilon]\}\sum_{i=1}^{M_1}
1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\right]^{1/\theta}\right\}\nonumber\\
&=&\exp\{n_2[R_2+h(\delta_2)-\ln 2+\epsilon]/\theta\}\cdot\bE\left\{\left[\sum_{i=1}^{M_1}
1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\right]^{1/\theta}\right\}
\end{eqnarray}
where the latter expectation (cf.\ Subsection A.1) is of the exponential order of
$e^{n_1[R_1+h(\delta_1)-\ln 2]}$ if $\delta_1\in{\cal I}^c(R_1)$ (Case 2) and
$e^{n_1[R_1+h(\delta_1)-\ln 2]/\theta}$ if $\delta_1\in{\cal I}(R_1)$ (Case 3).
Thus, in both cases, we obtain the desired exponential order as an upper bound.
For the lower bound, we argue similarly that the probability of the event
$${\cal A}'=\bigcap_{i=1}^{M_1}\left\{N_i(n_2\delta_2)\ge
\exp\{n_2[R_2+h(\delta_2)-\ln 2-\epsilon]\}\right\}$$
is doubly--exponentially close to unity, and so,
$$\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)\}\ge\mbox{Pr}\{{\cal A}\}\cdot\bE\left\{\left[
\exp\{n_2[R_2+h(\delta_2)-\ln 2-\epsilon]\}\sum_{i=1}^{M_1}
1\{d_H(\bx',\hbx_i)=n_1\delta_1\}\right]^{1/\theta}\right\},$$
and we again use the above result on the moments of
$\sum_{i=1}^{M_1}1\{d_H(\bx',\hbx_i)=n_1\delta_1\}$ in both cases
of $\delta_1$.
\subsubsection*{Case 4: $\delta_1\in{\cal I}(R_1)$ and $\delta_2\in{\cal I}^c(R_2)$}
\label{case4}
Since $\delta_1\in{\cal I}(R_1)$, then the event
$${\cal A}=\left\{e^{n_1[R_1+h(\delta_1)-\ln 2-\epsilon]}\le
\sum_{i=1}^{M_1}1\{d_H(\bx',\hbx_i)=n_1\delta_1\}
\le e^{n_1[R_1+h(\delta_1)-\ln 2+\epsilon]}\right\},$$
has a probability which is doubly--exponentially close to unity.
Thus, given that ${\cal A}$ occurs, there are
$$e^{n_1[R_1+h(\delta_1)-\ln 2+\epsilon]}\le L\le e^{n_1[R_1+h(\delta_1)-\ln 2+\epsilon]}$$
indices $i_1,i_2,\ldots,i_L$ for which $d_H(\bx',\hbx_i)=n_1\delta_1$. Given $L$ and given these
indices, $N(n_1\delta_1,n_2\delta_2)$ is the sum of
$LM_2\exe e^{n_1[R_1+h(\delta_1)-\ln 2+]+n_2R_2}$ i.i.d.\ Bernoulli trials,
$1\{d_H(\bx'',\tbx)=n_2\delta_2\}$, whose probability of success is exponentially $q\exe e^{n_2[h(\delta_2)
-\ln 2]}$. Thus, similarly as in the derivation in Subsection A.1,
$$\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)|{\cal A}\}\exe\left\{\begin{array}{ll}
LM_2q & q \gexe LM_2\\
(LM_2q)^{1/\theta} & q \lexe LM_2
\end{array}\right.$$
or, equivalently, in the notation of eq. (\ref{nd1d2}):
$$\bE\{N^{1/\theta}(n_1\delta_1,n_2\delta_2)|{\cal A}\}\exe\left\{\begin{array}{ll}
\exp\{n[\lambda W_1+(1-\lambda)W_2]\} & \lambda W_1+(1-\lambda)W_2 < 0\\
\exp\{n[\lambda W_1+(1-\lambda)W_2]/\theta\} & \lambda W_1+(1-\lambda)W_2 \ge 0\end{array}\right.$$
The total expectation should, of course, account for ${\cal A}^c$ as well,
but since the probability of this event is doubly exponentially small,
then the contribution of this term is negligible.
This completes the proof of eq.\ (\ref{nd1d2}).
\subsection*{A.3. The function $f(s,R_1,R_2)$}
\label{fsr1r2}
First, we observe that the constraints $\delta_1\in{\cal I}(R_1)$
and $\delta_2\in{\cal I}^c(R_2)$ can be replaced by their one-sided versions
$\delta_1\ge\delta(R_1)$ and $\delta_2\le\delta(R_2)$, respectively,
since values of $\delta_1$ and $\delta_2$ beyond $0.5$ cannot be better
than their corresponding reflections $1-\delta_1$ and $1-\delta_2$.
Next observe that $f(s,R_1,R_2)$ can be rewritten as follows:
$$f(s,R_1,R_2)=\min\{f_1(s,R_1,R_2)),f_2(s,R_1,R_2)\},$$
where
$$f_1(s,R_1,R_2)=s\min [\lambda\delta_1+(1-\lambda)\delta_2]$$
subject to the constraints $\delta_1\ge\delta(R_1)$, $\delta_2\le\delta(R_2)$,
and $R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)\ge \ln 2$, and
$$f_2(s,R_1,R_2)=\min\{\lambda[s\delta_1-R_1-h(\delta_1)+\ln 2]+
(1-\lambda)[s\delta_2-R_2-h(\delta_2)+\ln 2]\}$$
subject to the constraints $\delta_1\ge\delta(R_1)$, $\delta_2\le\delta(R_2)$,
and $R+\lambda h(\delta_1)+(1-\lambda)h(\delta_2)\le \ln 2$.
Note that the optimization problem associated with $f_1(s,R_1,R_2)$
is a convex problem, but the one pertaining to $f_2(s,R_1,R_2)$ is not,
because of its last constraint which is not convex.
At this point, we have to distinguish between two cases:
(i) $R_1>R_2$ and (ii) $R_2< R_1$ (the case $R_1=R_2$ will be
taken as a limit $R_1\to R_2$ of case (i)).
\subsubsection*{The Case $R_1> R_2$}
\label{r1gr2}
When $R_1> R_2$, we have $\delta(R_1)< \delta(R)<\delta(R_2)$.
As for $f_1$, it is easy to see that $\delta_1=\delta_2=\delta(R)$
is a solution that satisfies the necessary and sufficient Kuhn--Tucker conditions
for optimality of a convex problem, and so, $f_1(s,R_1,R_2)=s\delta(R)$.
Consider next the function $f_2(s,R_1,R_2)$.
Let us ignore, for a moment, the non--convex
constraint $R+\lambda h(\delta_1)+(1-\lambda) h(\delta_2)\le 2$,
and refer only to the constraints $\delta_1\ge \delta(R_1)$
and $\delta_2\le\delta(R_2)$. Denote by $\tilde{f}_2(s,R_1,R_2)$ the
corresponding maximum without the non--convex constraint.
The maximization problem associated with $\tilde{f}_2$
is now convex and it is to see that
$\delta_1^*=\max\{\delta(R_1),\nu_s\}$ and
$\delta_2^*=\min\{\delta(R_2),\nu_s\}$ satisfy
the necessary and sufficient conditions for optimality,
where $\nu_s\dfn 1/(1+e^s)$.
This is also a solution for $f_2$ if it satisfies
the non--convex constraint, namely, if
\begin{equation}
\label{cond}
\lambda h\left(\max\{\delta(R_1),\nu_s\}\right)+
(1-\lambda)h\left(\min\{\delta(R_2),\nu_s\}\right)+R \le \ln 2.
\end{equation}
Whether or not this condition is satisfied depends on $s$.
Since we are assuming $R_1> R_2$, we then have
$s_{R_1}> s_{R_2}$, where we remind that
$s_R\dfn\ln\frac{1-\delta(R)}{\delta(R)}$.
Consequently, there are three different ranges of $s$:
$s > s_{R_1}$, $s_{R_2}<s\le s_{R_1}$, and $s\le s_{R_2}$.
When $s> s_{R_1}> s_{R_2}$, this is equivalent to
$\nu_s < \delta(R_1)<\delta(R_2)$ in which case the
above necessary condition (\ref{cond}) becomes
$$\lambda h(\delta(R_1))+
(1-\lambda)h(\nu_s) < \ln 2-R.$$
To check whether this condition is satisfied,
observe that $h(\delta(R_1))\equiv\ln 2-R_1$, and so this is equivalent
to the condition $h(\nu_s)<\ln 2-R_2$, which is $\nu_s<\delta(R_2)$,
in agreement with the assumption on the range of $s$. Therefore,
the above solution is acceptable for $f_2$ and by substituting it back into the
objective function, we get:
\begin{eqnarray}
f_2(s,R_1,R_2)&=&\lambda[s\delta(R_1)-R_1-h(\delta(R_1))+\ln 2]+
(1-\lambda)[s\nu_s-R_2-h(\nu_s)+\ln 2]\nonumber\\
&=&\lambda s\delta(R_1)+(1-\lambda)v(s,R_2)
\end{eqnarray}
When $s_{R_1}\ge s > s_{R_2}$, this is equivalent to
$\delta(R_1)<\nu_s<\delta(R_2)$, in which case
the condition (\ref{cond}) becomes $h(\nu_s) < \ln 2-R$,
or equivalently, $\nu_s < \delta(R)$, which is $s> s_R$.
However, $s_R$ is between $s_{R_1}$ and $s_{R_2}$, and so,
the conclusion is that the
non--convex constraint is satisfied only in upper part of the interval $[s_{R_2},
s_{R_1}]$, i.e., $[s_R,s_{R_1}]$. In this range, $\delta_1^*=\delta_2^*=\nu_s$,
and this yields $f_2(s,R_1,R_2)=v(s,R)$.
For $s < s_{R}$, the condition (\ref{cond}) no longer holds.
In this case, the optimum solution should be sought on the boundary
of the non--convex constraint, namely, under the equality constraint
$R+\lambda h(\delta_1)+(1-\lambda) h(\delta_2)=\ln 2$, but this coincides
then with the solution to $f_1$ which was found on this boundary as well.
Thus, for $s\in[0,s_R]$, we have $f_2(s,R_1,R_2)=s\delta(R)$.
Summarizing our results for $f_2$ over the entire range of $s\ge 0$, we have
$$f_2(s,R_1,R_2)=\left\{\begin{array}{ll}
s\delta(R) & 0\le s\le s_R\\
v(s,R) & s_R<s\le s_{R_1}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_1}
\end{array}\right.$$
or, equivalently,
$$f_2(s,R_1,R_2)=\left\{\begin{array}{ll}
u(s,R) & 0\le s\le s_{R_1}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_1}
\end{array}\right.$$
Finally, $f$ should be taken as the minimum between $f_1$ and $f_2$.
Now, $f_1$ is linear and $f_2$ is concave (as it is the minimum of a linear
function in $s$), coinciding with $f_1$ along $[0,s_R]$. Thus $f_2$ cannot
exceed $f_1$ for any $s$, and so, $f=f_2$. Thus,
$$f(s,R_1,R_2)=\left\{\begin{array}{ll}
u(s,R) & 0\le s\le s_{R_1}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_1}
\end{array}\right.$$
\subsubsection*{The Case $R_1< R_2$}
\label{r1lr2}
In this case, $\delta(R_1)>\delta(R_2)$.
Once again, $f_1$ is associated with a convex program
whose conditions for optimality are easily seen to be satisfied by
the solution $\delta_1=\delta(R_1)$ and $\delta_2=\delta(R_2)$. Thus,
$$f_1(s,R_1,R_2)=s[\lambda\delta(R_1)+(1-\lambda)\delta(R_2)].$$
As for $f_2$, let us examine again the various ranges of $s$, where
this time, $s_{R_1}<s_R< s_{R_2}$. For $s > s_{R_2}$, we have $\nu_s<\delta(R_2)<\delta(R_1)$
and then the condition (\ref{cond}) is equivalent to $h(\nu_s)\le \ln 2-R_2$, which is
$\nu_s<\delta(R_2)$, in agreement with the assumption. This corresponds to $\delta_1=\delta(R_1)$
and $\delta_2=\nu_s$, which yields
$$f_2(s,R_1,R_2)=\lambda s\delta(R_1)+(1-\lambda)v(s,R_2).$$
For $s_{R_1}<s<s_{R_2}$, which means $\delta(R_2)<\nu_s<\delta(R_1)$, condition (\ref{cond})
is satisfied with equality, and the corresponding solution is $\delta_1=\delta(R_1)$
and $\delta_2=\delta(R_2)$, which yields
$$f_2(s,R_1,R_2)=s[\lambda \delta(R_1)+(1-\lambda)\delta(R_2)].$$
For $s < s_{R_1}$, eq.\ (\ref{cond}) is not satisfied, and we resort again to the
boundary solution, which, as mentioned earlier, is the same as $f_1$.
Summarizing our findings for the case $R_1 < R_2$, and applying
similar concavity considerations
as before (telling us that $f=f_2$), we have:
$$f(s,R_1,R_2)=\left\{\begin{array}{ll}
s[\lambda\delta(R_1)+(1-\lambda)\delta(R_2)] & 0\le s\le s_{R_2}\\
\lambda s\delta(R_1)+(1-\lambda)v(s,R_2) & s > s_{R_2}\end{array}\right.$$
|
2,869,038,154,862 | arxiv | \section{Introduction}
Primordial inflation is the most convincing paradigm for the early
Universe~\cite{infl}. The vacuum fluctuations created during inflation
also explain the observed temperature anisotropy of the cosmic
microwave background (CMB) radiation~\cite{WMAP}. However inflation
leaves the Universe cold and void of any thermal entropy. Entropy is
believed to be created from the decay of the coherent oscillations of
the inflaton which can happen perturbatively~\cite{Dolgov} and/or
non-perturbatively into bosons~\cite{Brandenberger,kls,chm} and
fermions~\cite{Greene}. It is necessary that the standard model (SM)
degrees of freedom are produced at this reheating stage, particularly
baryons which are required for the synthesis of light elements during
big bang nucleosynthesis (BBN) at a temperature ${\cal O}(\rm
MeV)$~\cite{bbn,subir,lowt}.
On the other hand, we do not know the full particle content of the
Universe beyond the electroweak scale, therefore we do not know what
degrees of freedom were excited right after inflation. In this respect,
supersymmetry (SUSY) acts as a building block which can explain the
hierarchy between the Planck and electroweak scales, if it is softly
broken in the observable sector at the TeV scale,
see~\cite{Nilles}. Besides its phenomenological implications, this
also has important cosmological consequences. The scalar potential of
the minimal supersymmetric standard model (MSSM) has nearly 300 flat
directions~\cite{gkm}. These flat directions can address cosmological
issues from reheating to density perturbations~\cite{Kari}. If
$R$-parity is conserved, the lightest supersymmetric particle (LSP)
will be stable and can act as a cold dark matter (CDM)
candidate. A neutralino LSP with a mass $m_{\chi} \sim 100$ GeV can
match the current observational limit when produced
thermally~\cite{Jungman}.
SUSY breaking at the TeV scale in the observable sector can be
achieved via gravity~\cite{Nilles}, gauge~\cite{gr} and
anomaly~\cite{anomaly} mediation, leading to different patterns of
supersymmetric particle masses. However, there is a priori no
fundamental reason why this scale should be favored by
nature~\footnote{String theory which is believed to be the most
fundamental theory does not provide us with a concrete answer. Rather
it provides us with a landscape with multiple vacua~\cite{Landscape},
where the SUSY breaking scale remains undetermined~\cite{Douglas},
this transcends into an uncertainty into the scale of inflation and the
required number of minimal e-foldings~\cite{Cliff0}.}.
Inspired by the string landscape~\cite{Landscape,Douglas}, there has
recently been an interesting proposal for SUSY breaking well above the
electroweak (but below the Planck) scale~\cite{split,adgr}. In this
new scheme, coined split SUSY, the masses of sfermions can be
arbitrarily larger than those of fermions. Although such a scheme does
not attempt to address the hierarchy problem, it removes fear from
flavor changing and $CP-$violating effects induced by the light
scalars at one-loop level. Successful gauge coupling unification
requires that the gauginos be kept lighter than $100$~TeV, while
spontaneous breaking of the electroweak symmetry requires that the mass of the
lightest Higgs be around ${\cal O}(100)$~GeV.
A priori there is no fundamental theory which fixes the scale of SUSY
breaking, but cosmological considerations can constrain it. For
example, the theory permits a very long-lived gluino. The
annihilation of gluinos alone may not efficiently reduce their
abundance below the experimental limits on the anomalous isotopes of
ordinary matter~\cite{split}. The decay of gluinos within the lifetime
of the Universe solves this potential problem, and requires
sfermions masses to be less than $10^{13}$~GeV~\cite{split,adgr}. Models
which give rise to such a {\it split} pattern of SUSY breaking masses
are typically more complicated than the conventional ones for TeV
scale SUSY breaking~\cite{split,adgr,models}. Of course, one might
also give upon supersymmetric gauge coupling unification and allow all
supersymmetric particles to be much heavier than TeV. In this case SUSY will
be irrelevant for physics at the electroweak scale. On the other hand,
even in the context of MSSM, some of the sfermions can have a mass $\gg
1$ TeV, as happens in the so-called inverted hierarchy
models~\cite{inverted}. Therefore, under general circumstances, it is
possible that at least some of the sfermions are much heavier
than TeV.
Local SUSY naturally embeds gravity, hence supergravity,
and implies the existence of
a new particle known as the gravitino which is the superpartner of the
graviton which is fairly long-lived. Massive gravitinos consist of
helicity $\pm 1/2$ (longitudinal) and helicity $\pm 3/2$ (transverse)
components. In the early Universe gravitinos can be produced from
thermal scatterings of gauge and gaugino quanta~\cite{thermal,gmsb},
and from the decay of sfermions~\cite{gmsb}. Gravitinos are also produced
non-thermally from the direct decay of the inflaton~\cite{nop,ajm,aem},
and from the vacuum fluctuations during the coherent oscillations of
the inflaton field~\cite{non-pert1,non-pert2,non-pert3,nps}. In the
minimal supergravity models, the gravitino mass $m_{3/2}$ is the same
as the soft breaking mass of scalars~\cite{Nilles}. However,
gravitinos can be much heavier once one goes beyond the minimal
supergravity, for example in no-scale models~\cite{no-scale}. It is
therefore possible that $m_{3/2} \gg 1$ TeV, even if SUSY is broken at
TeV scale in the observable sector.
Gravitinos with a mass below $50$~TeV decay during and/or after
BBN. Depending on the nature of the decay, there exist tight bounds
on the reheating temperature, see~\cite{cefo,kkm}. Obviously these bounds
do not apply to {\it supermassive} gravitinos, i.e. when $m_{3/2} \geq
50$~TeV. In such a case the reheating temperature could potentially be as
large as the inflaton mass, leading to many interesting
consequences. As an example, it opens up new regions of the parameter
space for thermal leptogenesis~\cite{plumacher}, a scenario which is
sensitive to the reheating temperature as it requires the excitation of the
lightest right-handed (RH) neutrinos and their supersymmetric partners
from the thermal bath.
However other considerations can constrain the abundance of
supermassive gravitinos. Every gravitino produces one LSP upon its
decay. If the gravitino decays after the thermal freeze-out of LSPs,
then it can alter the LSP abundance. In addition, gravitinos can
dominate the energy density of the Universe if they are produced
abundantly. Entropy release from gravitino decay in this case dilutes
any generated baryon asymmetry.
However, this may turn into a virtue, for example, in the case of
Affleck--Dine (AD) baryogenesis~\cite{ad}. Depending on the parameter
space, the AD mechanism (which utilizes supersymmetric flat
directions) can generate order one baryon asymmetry. This would then
be diluted to the observed value if the gravitino decay generates
enough entropy. Often the flat directions also fragment to form
non-topological solitons such as supersymmetric Q-balls. The Q-balls
decay slowly through their surface~\cite{cohen}, and can
themselves be a major source of late gravitino production.
In this article we consider various cosmological consequences of
models with superheavy gravitinos and/or sfermions, without delving
into model-building issues. Such particles will be inaccessible at
future colliders, and hence cosmology will be essentially the only
window to probe and/or constrain these models. The rest of this paper
is organized as follows. In Section II we briefly review various
sources of gravitino production and their individual contributions.
We then discuss gravitino decay and constraints from the LSP dark
matter on the abundance of supermassive gravitinos in Section III. We
briefly review thermal leptogenesis and the effects of the gravitino
in Section IV, and identify regions of the parameter space which allow
successful leptogenesis. Section V is devoted to supersymmetric flat
directions baryogenesis, including thermal effects and viability of
the AD mechanism in the presence of supermassive gravitinos. We also show
that long-lived Q-balls can be a
source for copious production of gravitinos. We summarize our results
and conclude the paper in the final Section VI.
\section{Gravitino Production}
The most important interaction terms of the gravitino field
$\psi_{\mu}$ come from its coupling to the supercurrent. In the
four-component notation, and in flat space-time, these terms are
written as (see for instance~\cite{moroi})
\begin{equation} \label{lagr}
{\cal L}_{\rm int} = -{1 \over \sqrt{2} M_{\rm P}} \partial_{\nu} X^{*} {\bar
{{\psi}_{\mu}}} \gamma^{\nu} \gamma^{\mu} \left({1 + \gamma_5 \over 2}\right)
\chi ~ - ~ {i \over 8 M_{\rm P}} {\bar {{\psi}_{\mu}}}
[\gamma^{\nu},\gamma^{\rho}] \gamma^{\mu} \lambda^{(a)} F^{(a)}_{\mu \nu}
~ + ~ {\rm h.c.}
\end{equation}
Here $X$ and $\chi$ denote the scalar and fermionic components of a
general chiral superfield, respectively, while $F^{(a)}_{\mu \nu}$ and
${\lambda}^{(a)}$ denote the gauge and gaugino field components of a
given vector superfield respectively.
In the limit of unbroken SUSY gravitinos are massless and the physical
degrees of freedom consist of the helicity $\pm 3/2$ (transverse)
components. After spontaneous SUSY breaking gravitino eats the
Goldstino and obtains a mass $m_{3/2}$ through the super Higgs
mechanism, and helicity $\pm 1/2$ (longitudinal) states appear as
physical degrees of freedom. When the value of $m_{3/2}$ is much smaller than
the
momentum of the gravitino, the wave-function of the helicity $\pm 1/2$
components of the gravitino can be written as
\begin{equation} \label{goldstino}
{\psi}_{\mu} \sim i \sqrt{2 \over 3} { 1 \over m_{3/2}} \partial_{\mu} \psi,
\end{equation}
with $\psi$ being the Goldstino. The helicity $\pm 1/2$ states of the
gravitino will in this case essentially interact like the Goldstino
and the relevant couplings are given by an effective Lagrangian
\begin{equation} \label{efflagr}
{\cal L}_{\rm eff} = {i \over \sqrt{3} m_{3/2} M_{\rm P}}
\left[(m^2_{X} - m^2_{\chi})
{X}^* {\bar \psi} \left({1 + \gamma_5 \over 2}\right) \chi ~ - ~
m_{\lambda}
{\bar \psi} [{\gamma}^{\mu} , {\gamma}^{\nu}] {\lambda}^{(a)}
F^{(a)}_{\mu \nu}\right] + {\rm h.c.}
\end{equation}
Here $m_{X}$ and $m_{\chi}$ denote the mass of $X$ and $\chi$ fields,
respectively, while $m_{\lambda}$ is the mass of gaugino field
$\lambda^{(a)}$. The interactions of helicity $\pm 1/2$ and helicity
$\pm 3/2$ states of the gravitino have essentially the same strength
when $\vert m_X - m_{\chi} \vert$ and $m_{\lambda}$ are smaller than
$m_{3/2}$. In the opposite limit, the rate for interactions of
helicity $\pm 1/2$ states with $X$ and $\chi$, respectively gauge and
gaugino fields, will be enhanced by a factor of $(m^2_X -
m^2_{\chi})^2/ E^2 m^2_{3/2}$ ($E$ being the typical energy involved
in the relevant process), respectively $m^2_{\lambda}/m^2_{3/2}$,
compared to that of helicity $\pm 3/2$ states.
Gravitinos are produced through various processes in the early
Universe, where the relevant couplings are given in~(\ref{lagr})
and~(\ref{efflagr}).
\begin{itemize}
\item {\it Thermal scatterings}:\\
Scatterings of gauge and gaugino quanta in the primordial thermal bath
is an important source of gravitino production, leading
to (up to logarithmic corrections)~\cite{thermal,gmsb}:
\begin{eqnarray} \label{scattering}
{\rm Helicity} \pm {1 \over 2}: \left({n_{3/2} \over s}\right)_{\rm sca}
&\simeq & \left(1 + {M^2_{\widetilde g} \over 12 m^2_{3/2}}\right)
\left({T_{\rm R} \over 10^{10}~{\rm GeV}}\right) \left[{228.75 \over
g_{*}(T_{\rm R})}\right]^{3/2} 10^{-12}\,, \nonumber \\
{\rm Helicity} \pm {3 \over 2}: \left({n_{3/2} \over s}\right)_{\rm sca}
&\simeq & \left({T_{\rm R} \over 10^{10}~{\rm GeV}}\right) \left[{228.75 \over
g_{*}(T_{\rm R})}\right]^{3/2} 10^{-12}\,;
\end{eqnarray}
where $T_{\rm R}$ denotes the reheating temperature of the Universe,
$M_{\widetilde g}$ is the gluino mass and $g_{*}(T_{\rm R})$ is the
number of relativistic degrees of freedom in the thermal bath at
temperature $T_{\rm R}$. Note that for $M_{\widetilde g} \leq m_{3/2}$
both states have essentially the same abundance, while for
$M_{\widetilde g} \gg m_{3/2}$ production of helicity $\pm 1/2$ states
is enhanced due to their Goldstino nature. The linear dependence of
the gravitino abundance on $T_{\rm R}$ can be understood
qualitatively. Due to the $M_{\rm P}$-suppressed couplings of the
gravitino, see~(\ref{lagr}) and~(\ref{efflagr}), the cross-section for
gravitino production is $\propto M^{-2}_{\rm P}$. The production rate
at temperature $T$ and the abundance of gravitinos produced within one
Hubble time will then be $\propto T^3$ and $\propto T$
respectively. We remind that the Hubble expansion rate at temperature
$T$ is given by $H = \sqrt{\left(g_{*} \pi^2/90 \right)} T^2/M_{\rm
P}$, where $g_*$ is the number of relativistic degrees of freedom in
the thermal bath at temperature $T$. This implies that gravitino
production from scatterings is most efficient at the highest
temperature of the radiation-dominated phase of the Universe,
i.e. when $T = T_{\rm R}$.
\item{\it Decaying sfermions}:\\
If $m_{3/2} < {\widetilde m}$, the decay channel $\mbox{\it sfermion}
\rightarrow
\mbox{\it fermion} + \mbox{\it gravitino}$ is kinematically open. When
${\widetilde m} > few
\times m_{3/2}$, the decay rate has a simple form
\begin{eqnarray} \label{decrate}
{\rm Helicity} \pm {1 \over 2}:~~~~~ \Gamma_{\mbox{\it sferm} \rightarrow
\mbox{\it ferm} +
\psi} &\simeq & {1 \over 48 \pi}{{\widetilde m}^5 \over m^2_{3/2}
M^2_{\rm P}}\,, \nonumber \\
{\rm Helicity} \pm {3 \over 2}:~~~~~ \Gamma_{\mbox{\it sferm} \rightarrow
\mbox{\it ferm} +
\psi} &\simeq & {1 \over 48 \pi}{{\widetilde m}^3 \over M^2_{\rm P}}\,.
\end{eqnarray}
Sfermions will reach thermal equilibrium abundances, provided that
$T_{\rm R} \geq {\widetilde m}$. They promptly decay through their gauge
interactions when the temperature drops below ${\widetilde m}$.
Gravitinos are produced from sfermion decays for the whole duration
sfermions exist in the thermal bath $t \sim M_{\rm P}/{\widetilde
m}^2$. The abundance of gravitinos thus produced will then
be~\cite{gmsb}
\begin{eqnarray} \label{decay}
{\rm Helicity} \pm {1 \over 2}: \left({n_{3/2} \over s}\right)_{\rm
dec} &\simeq & \left({{\widetilde m} \over m_{3/2}}\right)^2
\left({{\widetilde m} \over 1~{\rm TeV}}\right) \left[{228.75 \over
g_{*}({\widetilde m})}\right]^{3/2} \left({N \over 46}\right) 1.2
\times 10^{-19} \,, \nonumber \\
{\rm Helicity} \pm {3 \over 2}: \left({n_{3/2} \over s}\right)_{\rm
dec} &\simeq & \left({{\widetilde m} \over 1~{\rm TeV}}\right)
\left[{228.75 \over g_{*}({\widetilde m})}\right]^{3/2} \left({N \over
46}\right) 1.2 \times 10^{-19}\,,
\end{eqnarray}
where $g_{*}({\widetilde m})$ is the number of relativistic degrees of
freedom at $T = {\widetilde m}$, and $N$ is the number of all sfermions
such that $m_{3/2} < {\widetilde m} < T_{\rm R}$. This result is
independent of $T_{\rm R}$, so long as $T_{\rm R} > {\widetilde m}$. Note
that for ${\widetilde m} \roughly> m_{3/2}$ gravitinos of both helicities will
be produced with approximately the same abundance.
If ${\widetilde m} \gg m_{3/2}$, helicity $\pm 1/2$ states interact with
sfermions and fermions very efficiently and can actually reach thermal
equilibrium, thus leading to $\left(n_{3/2}/s \right)_{\rm dec} \simeq
10^{-2}$. This will happen when ${\widetilde m} \geq \left(10^{4}
m^2_{3/2} M_{\rm P}\right)^{1/3}$, for example, ${\widetilde m} \geq
10^{9}$~GeV if $m_{3/2} \simeq 50$ TeV.
\item{\it Inflaton decay}:\\
Reheating of the Universe also leads to gravitino
production~\cite{non-pert1,non-pert2,nop,ajm} (for related studies,
see~\cite{aem,kyy,ad4}). Here we consider the case where inflaton
decays perturbatively and a radiation-dominated Universe is
established immediately after the completion of its
decay.\footnote{Full thermal equilibrium is indeed achieved very
rapidly, provided that inflaton decay products have interactions of
moderate strength. For details on thermalization,
see~\cite{thermalization}.} This in general provides a valid
description of the last stage of inflaton decay, regardless of how
fast and explosive the first stage of reheating might be due to
various non-perturbative effects~\cite{Brandenberger,kls}.
Let us denote the SUSY-conserving mass of the inflaton multiplet by
$M_{\phi}$, and the mass difference between the inflaton $\phi$ and
inflatino ${\widetilde \phi}$ by $\Delta m_{\phi}$.\footnote{The mass
difference between the inflaton and inflatino $\Delta m_{\phi}$ is in
general different from that between the standard model fermions and
sfermions ${\widetilde m}$. As an example, consider the case where the
soft breaking $({\rm mass})^2$ of both the inflaton and sfermion fields
is ${\widetilde m}^2$. We will then have $\Delta m_{\phi}
\simeq \left({\widetilde m}^{2}/2 M_{\phi} \right) \ll {\widetilde m}$ if
${\widetilde m} \ll M_{\phi}$, while $\Delta m_{\phi} \simeq {\widetilde m}$
when ${\widetilde m} \geq M_{\phi}$.} If $\Delta m_{\phi} > m_{3/2}$, the
decay $\phi \rightarrow {\widetilde \phi} + \mbox{\it gravitino}$ is
kinematically
possible. For $\Delta m_{\phi} > few \times m_{3/2}$, the partial
decay width will be~\cite{nop,ad4}
\begin{eqnarray} \label{phidecrate}
{\rm Helicity} \pm {1 \over 2}: ~~~~~ \Gamma_{\phi \rightarrow
{\widetilde \phi} + \psi} &\simeq &{1 \over 48 \pi} {(m^2_{\phi}
- m^2_{\widetilde \phi})^4 \over
M_{\phi}^3 m^2_{3/2} M^2_{\rm P}}\, , \nonumber \\
{\rm Helicity} \pm {3 \over 2}:~~~~~ \Gamma_{\phi \rightarrow
{\widetilde \phi} + \psi} &\simeq &{1 \over 48 \pi} {(m^2_{\phi}
- m^2_{\widetilde \phi})^4 \over M_{\phi}^3 \Delta m^2_{\phi} M^2_{\rm P}}\, .
\end{eqnarray}
We can estimate the abundance of produced gravitinos with the help of
total inflaton decay rate $\Gamma_{\phi} = \sqrt{\left(g_{*} (T_{\rm
R}) \pi^2/90 \right)} T^2_{\rm R}/M_{\rm P}$, and the dilution factor
due to final entropy release which is given by $3 T_{\rm R}/4
M_{\phi}$.
If $\Delta m_{\phi} \ll M_{\phi}$, inflaton decay gives rise to
\begin{eqnarray} \label{phidecay}
{\rm Helicity} \pm {1 \over 2}: \left({n_{3/2} \over s}\right)_
{\rm reh} &\simeq &\left({\Delta m_{\phi}\over m_{3/2}}\right)^2
\left({\Delta m_{\phi}^2
\over T_{\rm R} M_{\rm P}}\right) \left[{228.75 \over
g_{*}(T_{\rm R})}\right]^{1/2} 1.6 \times 10^{-2}\, , \nonumber \\
{\rm Helicity} \pm {3 \over 2}: \left({n_{3/2} \over s}\right)_
{\rm reh} &\simeq & \left({\Delta m_{\phi}^2 \over T_{\rm R} M_{\rm P}}\right)
\left[{228.75 \over
g_{*}(T_{\rm R})}\right]^{1/2} 1.6 \times 10^{-2}\,.
\end{eqnarray}
An interesting point is that $M_{\phi}$ drops out of the calculation,
and hence the final results in Eq.~(\ref{phidecay}) have no explicit
dependence on $M_{\phi}$. If $M_{\phi} \leq \Delta m_{\phi}$, the
gravitino abundance will be smaller than that in~(\ref{phidecay}) by a
factor of $16$. Note again that for $\Delta m_{\phi} \roughly> m_{3/2}$
gravitinos of both helicities have approximately the same abundance.
Since from~(\ref{phidecay}) $n_{3/2} /s$
is inversely proportional to $T_{\rm R}$,
gravitino production from inflaton decay becomes more efficient at
lower reheating temperature. The reason is that a
smaller $T_{\rm R}$ means a smaller total decay rate $\Gamma_{\phi}$,
while the partial decay width~(\ref{phidecrate}) is independent from
$T_{\rm R}$. Therefore decreasing $T_{\rm R}$, while suppresses the
production from thermal scatterings~(\ref{scattering}) and sfermion
decays~(\ref{decay}), can actually enhance the overall production of
gravitinos. Obviously gravitino production from inflaton decay reaches
saturation when the partial decay width equals the total decay rate
$\Gamma_{\phi}$. In this case all inflatons decay to
inflatino-gravitino pairs, and the subsequent decay of inflatinos will
reheat the Universe. In consequence, one gravitino will be produced
per inflaton quanta, resulting in a gravitino abundance
$\left(n_{3/2~}/s \right)_{\rm reh} = 3 T_{\rm R}/4 M_{\phi}$.
Gravitino production
in two-body decays of the inflaton will be kinematically
forbidden if $m_{3/2} \geq \Delta m_{\phi}$. However, the
inflaton decay inevitably results in gravitino
production at higher orders of perturbation theory, provided that
$M_{\phi} > m_{3/2}$~\cite{ad1}. The leading order contributions come from the
diagrams describing the dominant mode of inflaton decay with gravitino
emission from the inflaton, its decay products and the decay
vertex. The partial width for inflaton decay to gravitinos is in this
case $\sim \left(M_{\phi}/M_{\rm P}\right)^2 \Gamma_{\phi}$ which,
after taking into account of the dilution factor, leads to
$\left(n_{3/2}/s \right)_{\rm reh} \sim \left(T_{\rm R}
M_{\phi}/M^2_{\rm P}\right)$.
If we impose the bound on the inflaton mass $M_{\phi} \leq
10^{13}$~GeV from the CMB for a simple chaotic type inflation model,
and if $m_{3/2} \geq \Delta m_{\phi}$, gravitino production from
inflaton decay is subdominant compared to that of thermal
scatterings~(\ref{scattering}), and hence can be neglected.\footnote{A
similar process, non-thermal production of gravitons from inflaton
decay, can become important in models with extra
dimensions~\cite{abgp}. The large multiplicity of the Kaluza-Klein
modes of the graviton can in this case easily overcome the suppression
factor $\left(M_{\phi}/M_{\rm P}\right)^2$.}
\item{\it Non-perturbative production:}\\
We note that besides various perturbative production mechanisms, both of
the helicity states can be excited non-perturbatively during the
coherent oscillations of the inflaton. This was first discussed
in~\cite{non-pert1} and then elaborated in \cite{non-pert2}. Right
after inflation the helicity $\pm 1/2$ component, i.e. the Goldstino,
is essentially the inflatino. For simple models with a single chiral
superfield, it was
shown that this component can be produced abundantly
$\left(n_{3/2}/s \right) \leq \left(T_{\rm
R}/M_{\phi}\right)$~\cite{non-pert2}. The reason is that its
couplings, given in Eqs.~(\ref{lagr}) and~(\ref{efflagr}), are not
necessarily $M_{\rm P}$-suppressed (contrary to the helicity $\pm 3/2$
states). However, as explicitly shown in~\cite{non-pert3}, it also
decays quickly along with the inflaton through derivative interactions, and
hence poses no danger. Realistic models include at least two chiral
superfields such
that the inflation sector is different from the sector responsible for
SUSY breaking in the vacuum. In these models also most of
the spin-$1/2$ fermions produced during inflaton oscillations decay in
form of inflatinos, provided that the scales of inflation and
present day SUSY breaking are sufficiently separated~\cite{nps}. The
helicity $\pm 3/2$ components of the gravitino have $M_{\rm P}$ suppressed
coupling all the time. In consequence, they are produced less abundantly
$\left(n_{3/2}/s \right) \leq
\left(M_{\phi}/M_{\rm P}\right) \left(T_{\rm R}/M_{\rm
P}\right)$~\cite{non-pert1}, compared to the direct decay of the
inflaton, thermal scatterings and sfermion decays. We will therefore
ignore the contribution from non-perturbative production of gravitinos
in the following.
\end{itemize}
\noindent
To summarize, the total gravitino abundance is given by
\begin{equation} \label{total1}
\left({n_{3/2} \over s}\right)=
\left({n_{3/2} \over s}\right)_{\rm sca} +
\left({n_{3/2} \over s}\right)_{\rm dec} +
\left({n_{3/2} \over s}\right)_{\rm reh}.
\end{equation}
As long as $m_{3/2} \leq T_{\rm R}$, gravitinos are
always produced in thermal scatterings of gauge and gaugino quanta.
In addition, sfermion and
inflaton decays also contribute to gravitino production if $m_{3/2} <
{\widetilde m} \leq T_{\rm R}$ and $m_{3/2} < \Delta m_{\phi}$ respectively.
Then it turns out
from~(\ref{scattering}),~(\ref{decay}) and~(\ref{phidecay}) that sfermion
decays will
be the dominant source of gravitino production unless $T_{\rm R} > 1.2
\times \left({\widetilde m}^3/m^2_{3/2}\right)$ and/or $\Delta
m_{\phi} > 0.37 {\widetilde m}$. Hence, for $m_{3/2} < {\widetilde m}
\leq T_{\rm R}$, the most important contribution in~(\ref{total1}) in
general comes from sfermion decays (see also footnote 3 on page 6).
\section{Gravitino Decay}
\begin{itemize}
\item{\it Stable gravitino}:\\
First, we briefly consider the case for stable gravitinos. If
the gravitino is the LSP, and $R$-parity is conserved, it will be
absolutely stable. Its total abundance (including both helicity $\pm
1/2$ and $\pm 3/2$ states) will in this case be constrained by the
dark matter limit $\Omega_{3/2} h^2 \leq 0.129$, leading to
\begin{equation}
\left(\frac{n_{3/2}}{s}\right) \leq
4.6 \times 10^{-10} \left({1~{\rm GeV} \over m_{\chi}}\right).
\end{equation}
This implies that the individual contributions from
Eqs.~(\ref{scattering}), ~(\ref{decay}) and~(\ref{phidecay}) should
respect this bound. As an example, consider the case with $m_{3/2} =
10$ GeV and $M_{\widetilde g} \simeq 1$ TeV. This results in the
constraints $T_{\rm R} \leq 5.5 \times 10^8$ GeV, ${\widetilde m} \leq
33$ TeV and $\Delta m_{\phi} \leq 140$ TeV.
\item{\it Unstable gravitino}:\\
An unstable gravitino decays to particle-sparticle pairs through the
couplings in~(\ref{lagr}), and the decay rate is given by~\cite{moroi}
\begin{equation}
\Gamma_{3/2} \simeq \left(N_g+\frac{N_{f}}{12}\right)
\frac{m^3_{3/2}}{32\pi M^2_{\rm P}}\,,
\end{equation}
where $N_g$ and $N_f$ are the number of available decay channels into
gauge-gaugino and fermion-sfermion pairs respectively. The gravitino
decay is completed when $H \simeq \Gamma_{3/2}$, when the
temperature of the Universe is given by
\begin{equation} \label{dectemp}
T_{3/2}
\simeq \left[\frac{10.75}{g_{\ast}(T_{3/2})}\right]^{1/4}
\left(\frac{m_{3/2}}{10^{5}~{\rm GeV}}\right)^{3/2}~6.8~{\rm MeV}\,.
\end{equation}
Here $g_*(T_{3/2})$ is the number of relativistic degrees of freedom
at $T_{3/2}$. If $m_{3/2} < 50$ TeV, gravitinos decay during or after
BBN~\cite{bbn} and can ruin its successful predictions for the
primordial abundance of light elements~\cite{subir}. If the gravitinos
decay radiatively, the most stringent bound $\left(n_{3/2}/s\right)
\leq 10^{-14}-10^{-12}$ arises for $m_{3/2} \simeq 100~{\rm GeV}-1$
TeV~\cite{cefo}. On the other hand, much stronger bounds are derived
if the gravitinos mainly decay through hadronic modes. In particular,
a branching ratio $\simeq 1$ requires that $\left(n_{3/2}/s\right)
\leq 10^{-16}-10^{-15}$ in the same gravitino mass range~\cite{kkm}.
To give a numerical example, consider the case when $m_{3/2} \simeq 1$
TeV. The abundance of a radiatively decaying gravitino is in this case
constrained to be $\left(n_{3/2}/s\right) \leq
10^{-12}$~\cite{cefo}. Then Eqs.~(\ref{scattering}),~(\ref{decay})
and~(\ref{phidecay}) result in the bounds $T_{\rm R} \leq 10^{10}$
GeV, ${\widetilde m} \leq 203$ TeV and $\Delta m_{\phi} < 1.1 \times
10^6$ GeV, respectively. If a TeV gravitino mainly decays into
gluon-gluino pairs, which will be the case if $m_{3/2} > M_{\widetilde
g}$, we must have $\left(n_{3/2}/s \right) \leq
10^{-16}$~\cite{kkm}. This leads to much tighter bounds $T_{\rm R}
\leq 10^6$ GeV, ${\widetilde m} \leq 9.4$ TeV and $\Delta m_{\phi}
\leq 11$ TeV.
\end{itemize}
\subsection{Decay of Supermassive Gravitinos and Dark Matter Abundance}
We now turn to {\it supermassive} gravitinos with a mass $m_{3/2} \geq
50$ TeV. If one insists on a successful supersymmetric gauge coupling
unification, the gaugino masses should be below $100$ TeV. This
implies that the gravitino will not be the LSP. The decay of
supermassive gravitinos happens sufficiently early in order not to
affect the BBN. Nevertheless, their abundance can still be constrained
due to different considerations. Gravitino decay produces one LSP per
gravitino. This non-thermal component may exceed the dark matter limit
if the decay happens below the LSP freeze-out temperature $T_{\rm
f}$. The freeze-out temperature is given by~\cite{Jungman}
\begin{equation} \label{freeze}
T_{\rm f} = {m_{\chi} \over x_{\rm f}}~,~ x_{\rm f} = 28 + {\rm ln}
\left\{{1~{\rm TeV} \over m_{\chi}} {c \over 10^{-2}} \left[{86.25 \over
g_*(T_{\rm f})}\right]^{1/2}\right\},
\end{equation}
where $m_{\chi}$ is the LSP mass and we have parameterized the
non-relativistic $\chi$ annihilation cross-section as
\begin{equation} \label{cross}
\langle \sigma_{\chi} v_{\rm rel} \rangle = {c \over m^2_{\chi}}.
\end{equation}
Note that neutralinos reach kinetic equilibrium with the thermal
bath, and hence become non-relativistic, very quickly at temperatures
above MeV~\cite{kamionkowski}. The exact value of $c$ depends on the
nature of $\chi$ and its interactions. For Bino-like LSP, $c$ can be
much smaller than for a Wino- or Higgsino-like one. When sfermions
are much heavier than the neutralinos, $c = 3 \times 10^{-3}$ for a
Higgsino LSP and $c = 10^{-2}$ for a Wino LSP (including the effects
of co-annihilation)~\cite{adgr}.
Gravitino decay occurs after the LSP freeze-out if $T_{3/2} < T_{\rm
f}$, which translates into an upper bound on the gravitino mass
\begin{equation} \label{after}
m_{3/2} < \left({m_{\chi} \over
1~{\rm TeV}}\right)^{2/3} \left[{g_{*}(T_{3/2}) \over 86.25}\right]^{1/6}~
4.3 \times 10^7~{\rm GeV}.
\end{equation}
This implies that for $m_{\chi} = 1$ TeV, gravitinos with a mass
$m_{3/2} < 4 \times 10^7$ GeV decay when thermal annihilation of LSPs
is already frozen. This decay produces one LSP per gravitino.
The dark matter limit $\Omega_{\chi} h^2 \leq 0.129$ constrains the
total LSP abundance to obey
\begin{equation} \label{dmlimit}
{n_{\chi} \over s} \leq 4.6 \times 10^{-10} \left({1~{\rm GeV} \over
m_{\chi}}\right).
\end{equation}
The final abundance of LSPs produced from gravitino decay depends on
their annihilation rate. The rate of annihilation of
non-relativistic LSPs is given by
\begin{equation} \label{ann}
\Gamma_{\chi} = \langle \sigma_{\chi} v_{\rm rel} \rangle ~ n_{\chi} =
c {n_{\chi} \over m^2_{\chi}}.
\end{equation}
If $\Gamma_{\chi} \geq \Gamma_{3/2}$, annihilation will be efficient and
reduce the LSP abundance to
\begin{equation} \label{lspeff}
{n_{\chi} \over s} \simeq 41.58 \left[{10^{-2} \over c}\right]
\left[{86.25 \over g_{*}(T_{3/2})}
\right]^{1/4} {m^2_{\chi} \over (m^3_{3/2} M_{\rm P})^{1/2}}.
\end{equation}
Otherwise, gravitino decay contributes an amount $n_{3/2}/s$ to the
LSP abundance. Having a large abundance of gravitinos, i.e.
$\left(n_{3/2}/s \right) > 4.6 \times 10^{-10} \left(1~{\rm
GeV}/m_{\chi}\right)$, is therefore potentially dangerous and requires
special attention.
The condition for efficient annihilation of LSPs whose abundance
$\left(n_{\chi}/s \right) \geq 4.6 \times 10^{-10} \left(1~{\rm
GeV}/m_{\chi}\right)$ at the time of gravitino decay translates into a
lower bound on the gravitino mass
\begin{equation} \label{eff}
m_{3/2} \geq \left[{10^{-2} \over c}\right]^{2/3}
\left[86.25 \over {g_{*}(T_{3/2})}\right]^{1/6}
\left({m_{\chi} \over 1~{\rm TeV}}\right)^{2}~ 2 \times 10^7~{\rm GeV}.
\end{equation}
If $m_{3/2}$ is in the window given by~(\ref{after}) and~(\ref{eff}),
the final abundance of non-thermal LSPs will be given
by~(\ref{lspeff}). It satisfies the dark matter limit~(\ref{dmlimit}),
and can account for the CDM for those values of $m_{\chi}$ and
$m_{3/2}$ which saturate the inequality in~(\ref{eff}).
Eqs.~(\ref{after}) and~(\ref{eff}) can be simultaneously satisfied
only if
\begin{equation} \label{effcond}
m_{\chi} \leq \left[{c \over 10^{-2}}\right]^{1/2}
\left[{g_{*}(T_{3/2}) \over 86.25}\right]^{1/4} ~ 1.8~{\rm TeV}.
\end{equation}
As a matter of fact, this is also the condition such that thermal
abundance of LSPs at freeze-out respects the dark matter limit. It is
not surprising as $T_{3/2} = T_{\rm f}$ when the inequality
in~(\ref{effcond}) is saturated. For the saturation value of
$m_{\chi}$ thermal LSP abundance gives rise to $\Omega_{\chi} h^2 =
0.129$. For smaller $m_{\chi}$ the thermal component is not
sufficient, while for larger $m_{\chi}$ thermal LSPs overclose the
Universe.
For values of $m_{\chi}$ respecting the bound in~(\ref{effcond}),
there always exists a window for $m_{3/2}$ such that gravitinos decay
after the freeze-out while at the same time non-thermal LSPs
efficiently annihilate and their abundance respects the dark matter
limit~(\ref{dmlimit}). In this mass window the abundance of thermal
LSPs is too low to account for dark matter. On the other hand,
non-thermal dark matter will be a viable scenario when the inequality
in Eq.~(\ref{eff}) is saturated (for non-thermal production of LSP dark
matter from gravitino decay, see also~\cite{adgr,ggw}). The gravitino mass
window becomes narrower as $m_{\chi}$ increases. It will eventually
disappear when $m_{\chi}$ reaches the upper bound
in~(\ref{effcond}).
For the canonical choice of $c = 10^{-2}$, the gravitino mass window
shrinks to a single point $m_{3/2} = 6.3 \times 10^7$ GeV at the
saturation value $m_{\chi} = 1.8$ TeV. For larger LSP masses it is
necessary that gravitinos which decay after the freeze-out are not
overproduced, i.e. that $\left(n_{3/2}/s \right) \leq 4.6 \times
10^{-10} \left(1~{\rm GeV}/m_{\chi}\right)$. Otherwise, gravitinos
must decay above the freeze-out temperature, i.e. the opposite
inequality as in~(\ref{after}) must be satisfied. Then gravitino decay
does not affect the final LSP abundance as $T_{3/2} > T_{\rm
f}$. However, for masses violating the bound in~(\ref{effcond})
thermal LSPs overclose the Universe. Therefore a viable scenario of
LSP dark matter in this case requires late entropy generation.
\subsection{Gravitino Non-domination}
We now consider the constraints from dark matter abundance on
gravitino decay in more detail. Assuming that there is no other stage
of entropy generation, the Universe will remain in the
radiation-dominated phase after reheating. During this period the
scale factor the Universe increases as $a \propto
H^{-1/2}$. Gravitinos become non-relativistic at
\begin{equation} \label{nonrel}
H_{\rm non} \simeq \left({m_{3/2} \over E_{\rm p}}\right)^2 H_{\rm p},
\end{equation}
where $H_{\rm p}$ denotes the expansion rate when (most of the)
gravitinos are produced and $E_{\rm p}$ is the energy of gravitinos
upon their production. If thermal scatterings are the main source of
gravitino production, $H_{\rm p} \sim T^2_{\rm R}/M_{\rm P}$ and
$E_{\rm p} \sim T_{\rm R}$. On the other hand, if sfermion decays
dominate gravitino production, $H_{\rm p} \sim {\widetilde m}^2/M_{\rm
P}$ and $E_{\rm p} = {\widetilde m}/2$. Finally, if most of the
gravitinos are produced in inflaton decay, $H_{\rm p} \sim T^2_{\rm
R}/M_{\rm P}$ and $E_{\rm p} \simeq \Delta m_{\phi}$.
For $H < H_{\rm p}$ the energy density of the gravitinos is redshifted
$\propto a^{-3}$, compared to $\propto a^{-4}$ for
radiation. Initially the gravitino energy density is $\rho_{3/2} = n_{3/2}
E_{\rm P}$, while the energy density in radiation is $\rho_{\rm rad} =
\left(\pi^2/30 \right) g_{*}(T_{\rm p}) T^4_{\rm p}$. Gravitinos will
dominate when $\rho_{3/2}/\rho_{\rm rad}$ is compensated by the slower
redshift of $\rho_{3/2}$. This happens at
\begin{equation} \label{domhubble}
H_{\rm dom} \simeq {16 \over 9} \left({n_{3/2} \over s}\right)^2
\left({E_{\rm p} \over
T_{\rm p}}\right)^2 H_{\rm non} = 8.9
\left[{g_{*}(T_{\rm p}) \over 228.75}\right]^{1/2} \left({n_{3/2}
\over s}\right)^2 {m^2_{3/2} \over M_{\rm P}},
\end{equation}
where we have used $s = \left(2 \pi^2 /45 \right) g_{*}(T_{\rm p})
T^3_{\rm p}$. Here $g_{*}(T_{\rm p})$ is the number of relativistic
degrees of freedom in the thermal bath at the temperature $T_{\rm p}$
when gravitinos are produced. Gravitino non-domination therefore
requires that $H_{\rm dom} < \Gamma_{3/2}$, i.e. that gravitinos
decay while their energy density is subdominant. This translates into
the bound
\begin{equation} \label{nodom}
{n_{3/2} \over s} < \left[{228.75 \over g_{*}(T_{\rm p})}\right]^{1/4}
\left({m_{3/2}
\over 10^5~{\rm GeV}}\right)^{1/2} ~ 2.4 \times 10^{-8}\,,
\end{equation}
on the gravitino abundance.
It is seen that for $m_{3/2} \geq 50$ TeV, gravitinos will dominate if
$\left(n_{3/2}/s \right) > 1.7 \times 10^{-8}$. Thermal scatterings
alone, see (\ref{scattering}), can yield such large abundances for
extremely large reheating temperatures $T_{\rm R} > 10^{14}$ GeV (we
consider $M_{\widetilde g} \leq 100$ TeV here). Therefore they do not
lead to gravitino domination in general. The sfermion and inflaton
decays, however, can produce a sufficiently large number of gravitinos
for much lower $T_{\rm R}$. As mentioned earlier, see the discussion
after Eq.~(\ref{total1}), sfermion decays are usually the dominant
source of gravitino production when $T_{\rm R} \geq {\widetilde
m}$. We therefore concentrate on the sfermions here.
Sfermion decays, see~(\ref{decay}), will not lead to gravitino
domination if
\begin{equation} \label{nosfermdom}
{\widetilde m} < \left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{5/6}
1.3 \times 10^8~{\rm GeV}\,.
\end{equation}
Here we have taken $g_{*}({\widetilde m}) = 228.75$ and $N = 46$
in~(\ref{decay}).
A successful scenario with gravitino non-domination should take the
constraints from LSP production into account. Fig. (1) depicts
different regions in the ${\widetilde m}-m_{3/2}$ plane for the choice
$c = 10^{-2}$ and $m_{\chi} = 100$ GeV. Above the solid line
gravitinos dominate, i.e. the opposite inequality as
in~(\ref{nosfermdom}) is satisfied, and hence excluded. The region
between the solid and dashed lines is defined by
\begin{equation} \label{nooversferm}
\left({1~{\rm GeV} \over m_{\chi}}\right)^{1/3} \left({m_{3/2} \over
10^5~{\rm GeV}}\right)^{2/3} 3.4 \times 10^7~{\rm GeV} < {\widetilde m}
< \left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{5/6}
1.3 \times 10^8~{\rm GeV}\,.
\end{equation}
In this region $\left(n_{3/2}/s \right) > 4.6 \times 10^{-10}
\left(1~{\rm GeV}/m_{\chi}\right)$, thus the density of LSPs produced
in gravitino decay exceeds the dark matter limit. The dotted and
dot-dashed vertical lines correspond to Eqs.~(\ref{after})
and~(\ref{eff}) respectively. The regions in black color are excluded since
either gravitinos dominate or gravitino decay produces
too many LSPs which do not sufficiently annihilate. In region $1$ gravitinos
decay after the
freeze-out but efficient annihilation reduces the abundance of
produced LSPs below the dark matter limit. Gravitino decay occurs
before the freeze-out in region 2 and does not affect the final LSP
abundance. Below the dashed
line sfermion decays do not overproduce gravitinos. In fact, below the
${\widetilde m} = m_{3/2}$ line such decays are kinematically
forbidden altogether. In this part of the ${\widetilde m}-m_{3/2}$
plane one has to worry about thermal scatterings though. If $T_{\rm R}
\geq 4.6 \times 10^{10} \left(1~{\rm GeV}/m_{\chi}\right)$ GeV,
scatterings will overproduce gravitinos. Therefore regions 3, 4 and 5 will not
be acceptable in this case, due to inefficient LSP annihilation.
For the values of $c$ and $m_{\chi}$ chosen in this plot, thermal
abundance of LSPs at freeze-out is too small to account for dark
matter. Non-thermal LSP dark matter from gravitino decay will in this
case be a viable scenario along the dot-dashed line.
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=9truecm\epsfbox{plot1.eps}
\vspace{0.0truecm}
\end{center}
\caption{Constraints from LSP production in gravitino non-domination
case for $c = 10^{-2}$ and $m_{\chi} = 100$ GeV. The solid and dashed
lines represent the upper and lower limits in Eq.~(\protect\ref{nooversferm})
respectively. The dotted and dot-dashed lines correspond to
Eqs.~(\protect\ref{after}) and~(\protect\ref{eff}) respectively. The regions
in black color are excluded. The solid red line is given by
$\widetilde m =m_{3/2}$.}
\end{figure}
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=9truecm\epsfbox{plot2.eps}
\vspace{0.0truecm}
\end{center}
\caption{Parameter constraints from LSP production in
gravitino-dominated case for $c = 10^{-2}$ and $m_{\chi} = 100$
GeV. The dotted and dot-dashed lines correspond to Eqs.~(\protect\ref{after})
and~(\protect\ref{eff}) respectively. The region in black color
is excluded.}
\end{figure}
\subsection{Gravitino-dominated Universe}
Gravitinos eventually dominate the energy density of the Universe if
$\Gamma_{3/2} \leq H_{\rm dom}$, which happens for
\begin{equation} \label{dom}
{n_{3/2} \over s} \geq \left[{228.75 \over g_{*}(T_{\rm p})}\right]^{1/4}
\left({m_{3/2}
\over 10^5~{\rm GeV}}\right)^{1/2} ~ 2.4 \times 10^{-8}.
\end{equation}
The scale factor of the Universe $a \propto H^{-2/3}$ in the interval
$\Gamma_{3/2} \leq H < H_{\rm dom}$. Gravitino decay will then
increase the entropy density by the factor $d = \left(g_*(T_{\rm
a})T^3_{\rm a}/g_*(T_{\rm b})T^3_{\rm b}\right)$.\footnote{To be more
precise, the dilution factor is given by $1+d$. The two definitions are
essentially equivalent when $d \ll 1$. Obviously there is no
gravitino dominaiton, and hence no dilution, when $d < 1$.} Here $T_{\rm
a},~T_{\rm b}$ denote the temperature of the thermal bath before and
after gravitino decay, respectively, while $g_*(T_{\rm a}),~g_*(T_{\rm
b})$ are the number of relativistic degrees of freedom at $T_{\rm a}$
and $T_{\rm b}$, respectively. Note that $d = \left(g_*(T_{\rm a})
\rho^3_{3/2}/g_*(T_{\rm b}) \rho^3_{\rm R}\right)^{1/4}$, with
$\rho_{3/2}$ and $\rho_{\rm R}$ being the energy density in the
gravitinos and radiation, respectively, at the time of gravitino
decay. The dilution factor $d$ is therefore given by
\begin{equation} \label{dilution}
d = \left({H_{\rm dom} \over \Gamma_{3/2}}\right)^{1/2}
\left({g_*(T_{\rm a}) \over g_*(T_{\rm b})}\right)^{1/4}
\simeq \left[{g_{*}(T_{\rm P}) \over 228.75}\right]^{1/4}
\left({10^5~{\rm GeV} \over m_{3/2}}\right)^{1/2}
\left[{(n_{3/2}/s) \over 2.4 \times 10^{-8}}\right]\,.
\end{equation}
Here we have taken $\left(g_*(T_{\rm a})/g_*(T_{\rm b}\right)^{1/4}
\simeq 1$, which is a good approximation since in a wide range $1~{\rm
Mev} \leq T \leq {\widetilde m}$ the number of relativistic degrees of
freedom $g_*$ changes between $10.75$ and $228.75$. To be more precise, the
dilution factor is given by $1+d$. Obviously for $d < 1$ there is no
gravitino domination, and hence no dilution. As mentioned
before, sfermions are usually the main source of gravitino production
for $T_{\rm R} \geq {\widetilde m}$, and hence we concentrate on them
here.
Gravitinos produced in sfermion decays dominate the Universe if
\begin{equation} \label{sfermdom}
\left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{5/6}
1.3 \times 10^8~{\rm GeV} \leq {\widetilde m} \leq T_{\rm R}\,,
\end{equation}
in which case the dilution factor is given by
\begin{equation} \label{sfermdilution}
d \simeq \left({10^5~{\rm GeV} \over
m_{3/2}}\right)^{5/2} \left({{\widetilde m} \over {1.3 \times
10^8~{\rm GeV}}}\right)^3\,.
\end{equation}
Gravitino decay dilutes the existing LSP abundance by a factor of $d$,
while at the same time producing one LSP per gravitino. Note that
$\left(n_{3/2}/s \right) \leq 10^{-2}$ as gravitinos at most reach
thermal equilibrium when ${\widetilde m} \gg
m_{3/2}$. Eq.~(\ref{dilution}) then implies that $d \leq 5.9 \times
10^5$.
It is evident from~(\ref{dom}) that the abundance of non-thermal LSPs
upon their production from the decay of supermassive gravitinos exceeds
the dark matter limit by several orders of magnitude. Therefore any
acceptable scenario with gravitino domination requires that the LSP
annihilation be efficient at the temperature $T_{3/2}$, i.e.
that~(\ref{eff}) be satisfied.
Fig.~(2) depicts different regions in the ${\widetilde m}-m_{3/2}$
plane for the choice $c = 10^{-2}$ and $m_{\chi} = 100$
GeV. Gravitinos dominate in the region above the solid line
corresponding to Eq.~(\ref{sfermdom}). Therefore regions below this line are
irrelevant. To the left of the dotted line,
which represents Eq.~(\ref{after}), gravitino decay occurs after the
freeze-out. To the right of the dot-dashed line, corresponding to
Eq.~(\ref{eff}), LSPs produced from gravitino decay efficiently
annihilate. The region in black color is excluded due to inefficient LSP
annihilation. Gravitinos decay after the freeze-out in region 1, but the final
abundance of LSPs respects the dark matter limit. On the other hand,
$T_{3/2} \geq T_{\rm f}$ in region 2, and hence gravitino decay does
not affect the LSP abundance. For the values of $c$ and $m_{\chi}$
chosen here, thermal abundance of LSPs at freeze-out is too low to
account for dark matter. However, non-thermal LSP dark matter from
gravitino decay is successful along the dot-dashed line.
In passing we note that the same discussions also apply when $T_{\rm
R} < {\widetilde m}$. In this case, however, the inflaton decay will
be the main source of gravitino production as sfermions are not
excited by the thermal bath. Constraints from efficient LSP annihilation
then lead to plots similar to those in Figs.~(1),~(2), with the
${\widetilde m} - m_{3/2}$ plane replaced by the $\Delta m_{\phi} -
m_{3/2}$ plane.
Entropy release from gravitino decay also dilutes any previously
generated baryon asymmetry. This implies that baryogenesis should
either take place after gravitino decay, or generates an asymmetry in
excess of the observed value by the dilution factor given in
(\ref{dilution}). It is seen from~(\ref{dectemp}) that $T_{3/2} \leq
100$ GeV unless $m_{3/2} > 10^8$ GeV. This implies that successful
baryogenesis after gravitino decay will be possible only if gravitinos
are extremely heavy. Therefore the more likely scenario in a
gravitino-dominated Universe is generating a sufficiently large baryon
asymmetry at early times which will be subsequently diluted by
gravitino decay. We will discuss leptogenesis and baryogenesis, and the
effect of gravitinos in detail in the next Sections.
One might invoke an intermediate stage of entropy release by the late
decay of some scalar condensate (beside inflaton) to prevent gravitino
domination. We shall notice, however, that any such decay will itself
produce gravitinos with an abundance which is inversely proportional
to the new (and lower) reheating temperature, see~(\ref{phidecay}).
\footnote{If the scalar field does not dominate the Universe, the
expression in~(\ref{phidecay}) should be multiplied by the fraction
$r$ of the total energy density $r$ which it carries.} This implies
that any stage of reheating, while diluting gravitinos which are
produced during the previous stage(s), can indeed produce more
gravitinos. Therefore entropy generation via scalar field decay is in general
not a helpful way to avoid a gravitino-dominated Universe.
One comment is in order before closing this subsection. In both of the
gravitino non-domination and domination
scenarios, having an LSP abundance in agreement with the dark matter
limit constrains its mass through Eq.~(\ref{effcond}). Heavier LSPs
overclose the Universe in one way or another. If $T_{3/2} \geq T_{\rm
f}$, gravitino decay will be irrelevant but thermal abundance of LSPs
will be too high. If $T_{3/2} < T_{\rm f}$, gravitino decay can in addition
make an unacceptably large non-thermal contribution. In case of gravitino
domination gravitino decay dilutes thermal LSPs. However, according
to~(\ref{dom}), the decay itself overproduces non-thermal LSPs
which will not sufficiently annihilate. Therefore gravitino
domination cannot rescue a scenario with thermally overproduced
LSPs. Indeed, for $m_{\chi} \gg 1$ gravitinos should never dominate the
Universe. The problem can be solved if $T_{\rm R} < T_{\rm f}$, or if another
stage of entropy release below $T_{\rm f}$ dilutes thermal LSPs. In both
case, however, reheating can overproduce gravitinos and the subsequent
gravitino decay may lead to non-thermal overproduction of LSPs. If $R$-parity
is broken, the LSP will be unstable and its abundance
will not be subject to the dark matter bound. Obviously its thermal and/or
non-thermal overproduction will not pose a danger in this case.
\subsection{Solving the Boltzmann Equation}
The entropy generated by gravitino decay can be estimated from
Eq.~(\ref{sfermdilution}). However, the evolution of gravitinos and
relativistic particles can be followed directly by solving the
Boltzmann equation. Assuming that gravitinos are non-relativistic at
decay and that $g_*$ is constant during the decay process, the
Boltzmann equations for gravitinos take the form
\begin{equation}
\dot{\rho}_{3/2} = -\Gamma_{3/2} \rho_{3/2} - 3 H \rho_{3/2}
\end{equation}
and
\begin{equation}
\dot{T} = -H T + \Gamma_{3/2} \frac{\rho_{3/2} T}{4 \rho_{\rm R}}.
\end{equation}
These should be solved together with the Friedmann equation
\begin{equation}
H^2 = \frac{1}{3 M^2_{\rm P}}\left[\rho_{3/2} + g_*(T) \frac{\pi^2}{30}
T^4\right].
\end{equation}
In Fig.~(\ref{figure1}) we show several examples of solving the
Boltzmann equation for different initial conditions, always assuming
the $g_* = 10.75$ during decay. The dilution factor $d$ can then be
found from
\begin{equation}
d(t) = \frac{(a(t)T(t))^3}{(a_{\rm i} T_{\rm i})^3},
\end{equation}
where $a_{\rm i}$ and $T_{\rm i}$ denote the initial values of the
scale factor and the radiation temperature respectively. The bottom
panel of Fig.~(\ref{figure1}) shows the dilution factor, and when
gravitinos dominate it agrees quite well with
Eq.~(\ref{sfermdilution}).
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=14truecm\epsfbox{figure1.eps}
\vspace{0.0truecm}
\end{center}
\caption{Plot of temperature, density and entropy increase as
functions of time for $m_{3/2}=100$ TeV. Curves are for
$(n_{3/2}/s)=10^{-10}, 10^{-9}, 10^{-8}, 10^{-7}$, and $10^{-6}$ in
increasing order.} \label{figure1}
\end{figure}
\section{Leptogenesis}
The baryon asymmetry of the Universe (BAU) parameterized as $\eta_{\rm
B}=(n_{\rm B}-n_{\bar{\rm B}})/s$, with $s$ being the entropy density,
is determined to be $0.9 \times 10^{-10}$ by recent analysis of WMAP
data~\cite{WMAP}. This number is also in good agreement with an
independent determination from primordial abundances produced during
BBN~\cite{cfo}. Three conditions are required for generating a baryon
asymmetry: $B-$ and/or $L-$violation, $C-$ and $CP-$violation, and
departure from thermal equilibrium~\cite{sakharov}. Since
$B+L$-violating sphalerons transitions are active at temperatures
$100~{\rm GeV} \leq T \leq 10^{12}$ GeV~\cite{krs}, any mechanism for
creating a baryon asymmetry at $T > 100$ GeV must create a $B-L$
asymmetry. The final asymmetry is then given by $B=a(B-L)$, where
$a=28/79$ in case of SM and $a=8/23$ for MSSM~\cite{khlebnikov}.
Leptogenesis postulates the existence of RH neutrinos, which are SM
singlets, with a lepton number violating Majorana mass
$M_N$. It can be naturally embedded in models
which explain the light neutrino masses
via the see-saw mechanism~\cite{seesaw}. A lepton asymmetry can then
be generated from the out-of-equilibrium decay of the RH neutrinos into
Higgs bosons and light leptons, provided $CP-$violating phases
exist in the neutrino Yukawa couplings~\cite{fy,luty,one-loop}.
The created lepton asymmetry will be
converted into a baryonic asymmetry via sphalerons
processes.
The on-shell RH neutrinos whose decay is responsible for the lepton
asymmetry can be produced thermally
via their Yukawa interactions with the standard model fields and their
superpartners~\cite{plumacher}, for which $T_{\rm R}\geq M_{1}\sim
10^{9}$~GeV,~\cite{buchmuller,sacha,gnrrs,strumia}, or non-thermally
for which $T_{\rm R}\leq M_{N}$,
see~\cite{infl,reheat,gprt}. Non-thermal leptogenesis can also
be achieved without exciting on-shell RH neutrinos~\cite{off-shell}.
In supersymmetric models one in addition has the RH sneutrinos which
serve as an additional source for leptogenesis~\cite{cdo}. The
sneutrinos are produced along with neutrinos in a thermal bath or
during reheating, and with much higher abundances in
preheating~\cite{bdps}. There are also additional possibilities for
leptogenesis from the RH sneutrinos some of which rely on soft SUSY
breaking effects~\cite{my,hmy,bmp,adm,ad3,soft}.
\subsection{Thermal Leptogenesis}
Let us concentrate on the supersymmetric standard model augmented with
three RH neutrino multiplets in order to accommodate neutrino masses
via the see-saw mechanism~\cite{seesaw}. The relevant part of the
superpotential is
\begin{equation} \label{superpot}
W \supset {1 \over 2} M_i {\bf N}_i {\bf N}_i + {\bf h}_{ij} {\bf H}_u
{\bf N}_i {\bf L}_j,
\end{equation}
where ${\bf N}$, ${\bf H}_u$, and ${\bf L}$ are multiplets containing
the RH neutrinos $N$ and sneutrinos $\tilde N$, the Higgs field giving
mass to, e.g., the top quark and its superpartner, and the left-handed
(s)lepton doublets, respectively. Here ${\bf h}_{ij}$ are the neutrino
Yukawa couplings and we work in the basis in which the Majorana mass
matrix is diagonal. The decay of a RH (s)neutrino with mass $M_i$ (we
choose $M_1 < M_2 < M_3$) results in a lepton asymmetry per
(s)neutrino quanta $\epsilon_i$, given by
\begin{equation} \label{asymmetry}
\epsilon_i = - {1 \over 8 \pi} {1 \over [{\bf h} {\bf h}^{\dagger}]_{ii}}
\sum_{j} {\rm Im}
\left( [{\bf h} {\bf h}^{\dagger}]_{ij}\right)^2 f \left({M^{2}_{j}
\over M^{2}_{i}} \right)\,,
\end{equation}
with~\cite{one-loop}
\begin{equation} \label{corrections}
f(x) = \sqrt{x} \left ({2 \over x-1} + {\ln} \left[{1 + x \over
x}\right ] \right ).
\end{equation}
The first and second terms on the right-hand side of
Eq.~(\ref{corrections}) correspond to the one-loop self-energy and
vertex corrections, respectively. Assuming strongly hierarchical RH
(s)neutrinos and an ${\cal O}(1)$ $CP-$violating phase in the Yukawa
couplings, it can be shown that~\cite{di}
\begin{equation} \label{hierarchical}
\vert \epsilon_1 \vert \roughly< {3 \over 8 \pi} {M_1 (m_3 - m_1) \over
{\langle H_u
\rangle}^{2}}\,,
\end{equation}
where $m_1 < m_2 < m_3$ are the masses of light, mostly left-handed
(LH) neutrinos. For a hierarchical spectrum of light neutrino masses
($m_1 \ll m_2 \ll m_3$),
we then have
\begin{equation} \label{epsilon1}
\vert \epsilon_1 \vert \roughly< 2 \times 10^{-7}
\left({m_3 - m_1 \over 0.05~{\rm eV}}\right) \left({M_1 \over
10^{9}~{\rm GeV}}\right)\,.
\end{equation}
To obtain this, we have used $m_3 - m_1 \simeq m_3\simeq 0.05$ eV (as suggested
by atmospheric neutrino oscillation data) and $\langle H_u \rangle \simeq
170$ GeV.
If the asymmetry is mainly produced from the decay of the lightest RH
states, after taking the conversion by sphalerons into account, we
arrive at
\begin{equation} \label{baryonthermal}
\eta_{\rm B}^{\mbox{}_{\rm MAX}} \simeq 3 \times 10^{-10}
\left({m_3 - m_1 \over 0.05~{\rm eV}}\right)
\left({M_1 \over 10^9~{\rm GeV}}\right) \kappa \, ,
\end{equation}
where we have assumed maximal $CP$-violation. Here $\kappa$ is the
efficiency factor accounting for the decay, inverse decay and
scattering processes involving the RH states~\cite{buchmuller,gnrrs}.
The decay parameter $K$ is defined as
\begin{equation} \label{decpar}
K \equiv {\Gamma_1 \over H(T = M_1)},
\end{equation}
where
\begin{equation} \label{Ndec}
\Gamma_1 = \sum_{i} {{\vert h_{1i} \vert}^2 \over 4 \pi} M_1,
\end{equation}
is the decay width of $N_1$ and ${\tilde N}_1$. One can also define
the effective neutrino mass
\begin{equation} \label{effmass}
{\tilde m}_1 \equiv \sum_{i} {{\vert h_{1i} \vert}^2 {\langle H_u \rangle}^2
\over M_1},
\end{equation}
which determines the strength of ${\tilde N}_1$ and $N_1$
interactions, with the model-independent bound $m_1 < {\tilde
m}_1$~\cite{fhy}.
\begin{itemize}
\item{{\it case(1)}:\\ If $K < 1$, corresponding to ${\tilde m}_1 <
10^{-3}$ eV, the decay of RH states will be out-of-equilibrium at all
times. In this case the abundance of RH states produced via Yukawa
interactions does not reach the thermal equilibrium value. The lepton
number violating scatterings can be safely neglected.
Hence this case is called
the weak washout regime. The efficiency factor is $\kappa \simeq 0.1$
when ${\tilde m}_1 = 10^{-3}$ eV. Generating sufficient asymmetry then
puts an absolute lower bound $few \times 10^9$ GeV on $M_1$, and
$T_{\rm R} \geq M_1$ will be required in this case~\cite{buchmuller}.}
\item{{\it case (2)}:\\ In the opposite limit $K > 1$, ${\tilde N}_1$
and $N_1$ will be in thermal equilibrium at temperatures $T > M_1$. In
particular, the efficiency of inverse decays erases any pre-existing
asymmetry (generated, for example, from the decay of heavier RH
states). This regime is called of strong washout. We note that this
regime includes the entire favored neutrino mass range $m_{\rm sol} ~
\roughly< {\tilde m}_1 ~ \roughly< m_{\rm atm}$. The inverse decays keep the
RH (s)neutrinos in equilibrium for sometime after $T$ drops below
$M_1$. The number density of quanta which undergo out-of-equilibrium
decay is therefore suppressed and reduces the efficiency factor.
Successful leptogenesis in the range $(m_{\rm sol},m_{\rm atm})$
requires that $10^{10}~{\rm GeV} < M_1 \roughly< 10^{11}$ GeV while, due
to the efficiency of inverse decays, $T_{\rm R}$ can be smaller than
$M_1$ by almost one order of magnitude~\cite{buchmuller}. The
efficiency factor $\kappa$ in this window varies between $few \times
10^{-3}$ and $few \times 10^{-2}$.}
\end{itemize}
It is possible to obtain (approximate) analytical expressions for the
efficiency factor $\kappa$. In the strong washout regime, the final
efficiency factor is maximal when $\Delta L = 2$ scatterings among the
LH (s)leptons can be neglected. So long as the scatterings can be
neglected, the efficiency factor $\kappa$ is independent from the lightest RH
(s)neutrino mass $M_1$ and
$\kappa(\tilde{m}_1)$
is given by~\cite{buchmuller,gnrrs,pilaftsis,strumia}
\begin{equation}
\kappa \simeq 10^{-2} \left( \frac{0.01 \ {\rm eV}}{\tilde{m}_1}
\right)^{1.1}\,.
\label{efficiency}
\end{equation}
Scatterings cannot be neglected for very large $M_1$ or large
light neutrino masses. In this case the efficiency factor can be
approximated as~\cite{buchmuller}:
\begin{equation}
\label{efficiencycomplete}
\kappa(\tilde{m}_1, M_1, \bar{m}^2) = \kappa (\tilde{m}_1)
e^{- \frac{\omega}{z_{\rm B}}
\left( \frac{M_1}{10^{10} \ {\rm GeV} }\right)\left({{\bar m} \over
1~{\rm eV}}\right)^2},
\end{equation}
where $\omega \simeq 0.186$, $z_{\rm B}= M_1/T_{\rm B}\sim $~few, with
$T_{\rm B}$ the temperature at which most of the lepton asymmetry is
produced, $\bar{m}^2$ is the sum over the squares of the light
neutrino masses. The efficiency factor $\kappa (\tilde{m}_1) $ is given in
Eq.~(\ref{efficiency}). For hierarchical light neutrinos, $\bar{m}^2 =
m_3^2 \simeq
\Delta m^2_{atm} \simeq 2.2 \times 10^{-3} \ {\rm eV}^2$,
where $\Delta m^2_{atm}$ is the mass squared difference
which controls the oscillations of atmospheric neutrinos. Washout
effects due to scatterings then become important for $M_1 \sim
10^{14}$~GeV. For quasi-degenerate neutrinos, $m_1 \simeq m_2 \simeq
m_3$, larger values of ${\bar m}$ are possible and $\kappa$ will be
exponentially
suppressed even for much smaller values of $M_1$.
For negligible $\Delta L = 2$ scatterings,
the baryon asymmetry is, in the case of maximal decay asymmetry,
given by
\begin{equation} \label{baryonlargeM1}
{\eta_{\rm B}^{\mbox{}_{\rm MAX}}} \simeq 0.9 \times 10^{-10} \ \left({m_3 -
m_1 \over 0.05~
{\rm eV}}\right) \left( \frac{0.01 \ {\rm eV}}{\tilde{m}_1} \right)^{1.1}
\left({M_1 \over 3.7 \times 10^{10}~{\rm GeV}}\right).
\end{equation}
The baryon asymmetry is proportional to $M_1$. The usual bound on the
reheating temperature $T_{\rm R} \leq 10^{10}$ GeV, given for $m_{3/2}
< 50$ TeV, does not apply to supermassive gravitinos. Hence, RH
(s)neutrinos with masses $M_1 \gg 10^{10}$~GeV can be thermally produced in the
early Universe. This can lead to larger amounts of baryon asymmetry
generated by thermal leptogenesis, with respect to the standard
scenario. The lowest reheating temperatures required for thermally
producing these RH (s)neutrinos is $T_R \sim M_1/{few} \ll 10^{14}
$~GeV. Note that the initial assumption of gravitino non-domination is
satisfied if gravitinos are dominantly produced by thermal
scatterings.
\subsection{Effects of the Gravitino on Thermal Leptogenesis}
Thermal leptogenesis completes when $T \sim M_1/
{few}$~\cite{buchmuller,gnrrs}. Eq.~(\ref{dectemp}) then implies that
gravitino decay takes place after leptogenesis unless they are
extremely heavy:
\begin{equation} \label{before}
m_{3/2} > \left( \frac{M_1}{10^9 \ {\rm GeV}}
\right)^{2/3} 10^{12} \ {\rm GeV}.
\end{equation}
On the other hand, for $m_{3/2} \geq 50$ TeV, gravitino decay occurs
at a temperature $T_{3/2} > 6.8 $ MeV which is compatible with a
successful BBN. Therefore both scenarios of gravitino domination and
non-domination are in agreement with the BBN constraints. Nevertheless
the effect of gravitino decay on the final baryon asymmetry need to be
taken into account. We consider both scenarios of gravitino domination
and non-domination.
\begin{itemize}
\item{{\it Gravitino non-domination}:\\
The condition for gravitino non-domination is given in
Eq.~(\ref{nodom}). There will be practically no dilution by gravitino
decay in this case, and thermal leptogenesis should generate
$\eta_{\rm B} \simeq 10^{-10}$ according to
Eq.~(\ref{baryonthermal}). Obtaining sufficient asymmetry in both of
the weak and strong washout regimes requires that $M_1 \roughly> T_{\rm R}
> 10^9$ GeV, and leptogenesis completes when $T \sim M_1 \geq 10^9$
GeV~\cite{buchmuller,gnrrs}.
Sfermions with a
mass ${\tilde m} \leq 10^9$ GeV certainly reach thermal equilibrium
and, for $m_{3/2} < {\tilde m}$, their decay will produce gravitinos
according to Eq.~(\ref{decay}).
If $m_{3/2} \geq {\tilde m}$, gravitino production from sfermion
decays is kinematically forbidden. Scatterings of gauge and gaugino
quanta in the thermal bath will nevertheless produce gravitinos so
long as $m_{3/2} \leq T_{\rm R}$. Late time domination of gravitinos
thus produced requires that the reheating temperature $T_{\rm R} \geq
10^{14}$ GeV, see Eq.~(\ref{scattering}). However, gravitinos can be
overproduced for much smaller $T_{\rm R}$. For $m_{\chi} = 100~{\rm
GeV}~(1$ TeV), gravitino decay results in LSP overproduction when
$T_{\rm R} \geq 3 \times 10^{10}~(3 \times 10^{9})$ GeV. This indeed
occurs for the bulk of the parameter space compatible with thermal
leptogenesis, particularly in the favored neutrino mass window
$m_{\rm solar} \leq {\tilde m}_1 \leq m_{\rm atm}$~\cite{buchmuller}.
The condition for sufficient annihilation of non-thermal LSPs in this
case sets a lower bound on the gravitino mass through Eq.~(\ref{eff}),
independently of whether $m_{3/2} < {\tilde m}$ or $m_{3/2} \geq
{\tilde m}$. Constraints from LSP annihilation, which determine acceptable
regions of the ${\widetilde m} - m_{3/2}$ plane, are summarized in Fig.~(1) and
the related discussion.
In the case of gravitino non-domination, the produced
lepton asymmetry is not subsequently
diluted by gravitino decays, even if taking place after leptogenesis.
As the bound on the reheating temperature $T_{\rm R} \leq 10^{10}$~GeV
does not apply for supermassive gravitinos,
RH neutrinos with masses $M_1 \gg 10^{10}$~GeV can be fully thermalized
for a sufficiently large reheating temperature.
From Eqs.~(\ref{baryonthermal}), (\ref{efficiency}) and
(\ref{efficiencycomplete}),
it follows that the baryon asymmetry is proportional
to the lightest RH neutrino mass, $M_1$,
up to $M_1 \sim 10^{14}$~GeV,
for $\bar{m}^2 \sim \Delta m^2_{atm}$.
For larger values of $M_1$,
the lepton asymmetry is washed out by
$\Delta L=2$ scatterings.
Then, the maximum baryon asymmetry is produced for
$M_1 \sim 5 \times 10^{12} \ z_{\rm B} /\omega$~GeV.
Here we have again taken $\bar{m}^2 \sim \Delta m^2_{atm}$.
From Eq.~(\ref{baryonlargeM1}), we notice that for large values of $M_1$
the generated $\eta_{\rm B}$ can be much larger
than the one required to explain the observations,
if the decay asymmetry is maximal.
Models which implement
the see-saw mechanism of neutrino mass generation
typically
assume the conservation of flavor symmetries
and/or special forms of the Yukawa couplings,
in order to explain the low energy
neutrino masses and mixing.
In many of these models,
the decay asymmetry is constrained to be non maximal
and $M_1$ larger than the typical values $10^9$--$10^{10}$~GeV
are needed to generate a sufficient baryon asymmetry.
Models with supermassive gravitinos allow to have reheating temperatures
high enough to thermally produce such heavy RH neutrinos.
In each specific model,
a detailed analysis is required
for establishing the feasibility of successfull leptogenesis
and, at the same time,
the possibility to explain the low energy neutrino mass matrix
(for a discussion of $CP$-violation in specific see-saw models and
leptogenesis, see, e.g., Ref.~\cite{CPVconn1}).
}
\item{{\it Gravitino-dominated Universe}:\\
If gravitinos are produced very abundantly, see Eq.~(\ref{dom}), the
Universe will become gravitino-dominated. Sfermion decays (which, as
mentioned earlier, usually dominate over thermal scatterings and
inflaton decay) produce such abundances of gravitinos
when Eq.~(\ref{sfermdom}) is satisfied.
Gravitino decay reheats the Universe to a temperature $T_{3/2}$,
see Eq.~(\ref{dectemp}), and increases the entropy density by a factor $d$
given in Eq.~(\ref{dilution}). Successful thermal leptogenesis {\it after}
gravitino decay will be only possible if gravitinos are extremely
heavy $m_{3/2} > 10^{12}$ GeV, see Eq.~(\ref{before}). According
to~Eq.~({\ref{dom}), gravitino domination in this case requires that
$\left(n_{3/2}/s \right) > 10^{-5}$. It follows
from Eq.~(\ref{sfermdom}) that sfermion decays yield this only if ${\tilde
m}$ (and $T_{\rm R}$) is $> 10^{14}$~GeV. In addition,
non-perturbative production of gravitinos with the necessary abundance
is also questionable.
Therefore, it is more realistic to consider the opposite situation
where leptogenesis occurs {\it before} gravitino decay. In this case,
due to the entropy release by gravitino decay, the generated asymmetry
must exceed the observed value of $\eta_{\rm B} \simeq 10^{-10}$ by a
factor of $d$. Eqs.~(\ref{sfermdilution}) and~(\ref{baryonthermal})
then imply that the final asymmetry is
\begin{equation} \label{domasym}
\eta_{\rm B} \simeq 3 \times 10^{-10}
\left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{5/2}
\left({1.3 \times 10^8~{\rm GeV} \over {\widetilde m}}\right)^3
\left({M_1 \over 10^9~{\rm GeV}}\right) \kappa.
\end{equation}
As a specific example, consider leptogenesis in the favored neutrino
mass window $m_{\rm sol} \leq {\tilde m}_1 \leq m_{\rm atm}$. In this
interval, which entirely lies within the strong washout regime, the
efficiency factor $\kappa \sim 10^{-2}$, and the reheating temperature
follows $T_{\rm R} \geq 0.1 M_1$~\cite{buchmuller}. Therefore
successful leptogenesis requires that
\begin{equation} \label{suclep}
M_1 \simeq \left({10^5~{\rm GeV} \over m_{3/2}}\right)^{5/2}
\left({{\widetilde m} \over 1.3 \times 10^8~{\rm GeV}}\right)^3
3 \times 10^{10}~{\rm GeV}.
\end{equation}
In Fig.~\ref{figure3} we show the value of $M_1$ needed to produce the
correct baryon asymmetry of $\eta_{\rm B} = 0.9 \times 10^{-10}$ as a
function of the lightest RH (s)neutrino mass $m_{3/2}$ and
${\widetilde m}$. The plot is produced assuming $\kappa = 10^{-2}$,
but since $\eta_{\rm B} \propto \kappa$ it can easily be rescaled for
other values. Without dilution, successful leptogenesis for $\kappa =
10^{-2}$ requires that $M_1 \simeq 3 \times 10^{10}$ GeV, see
Eq.~(\ref{baryonthermal}). Therefore having contours with $M_1 > 3
\times 10^{10}$ GeV indicates gravitino domination. As mentioned
earlier, thermal leptogenesis fails for $M_1 \geq 10^{14}$ GeV due to
the erasure of generated asymmetry by $\Delta L = 2$ scattering
processes. This happens in the light colored region, and hence excludes it.
The overlap between Figs.~(1) and~(3)
combines the constraints from leptogenesis and dark matter
considerations.
Thermal leptogenesis cannot generate a baryon asymmetry which exceeds
$10^{-2}$. The maximal value is obtained in the (hypothetical) case
when RH (s)neutrinos have thermal equilibrium abundance, and the
efficiency factor $\kappa$ and asymmetry parameter $\vert \epsilon_{1}
\vert$ are both $1$, see Eq.~(\ref{baryonthermal}). This implies that
successful leptogenesis in a gravitino-dominated Universe would be
impossible if the dilution factor $d$ was larger than $10^8$. However,
since gravitinos can at most reach thermal equilibrium, we always have
$d \leq 5.9 \times 10^5$, see the discussion after
Eq.~(\ref{sfermdilution}). This ensures that there will be no such
case where thermal leptogenesis is absolutely impossible in a
gravitino-dominated Universe.
}
}
\end{itemize}
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=9truecm\epsfbox{newfig.ps}
\vspace{0.0truecm}
\end{center}
\caption{The value of $\log(M_1)$ (in GeV) needed to produce
$\eta_{\rm B} = 0.9 \times 10^{-10}$ as a function of $m_{3/2}$ and
$\widetilde{m}$ in the gravitino-dominated scenario. We have chosen
the efficiency factor $\kappa = 10^{-2}$, which is the typical value in
the favored neutrino mass window. The dark color region does not have
gravitino domination, while in the light color region thermal
lepotgenesis fails due to washout by lepton number violating scatterings.}
\label{figure3}
\end{figure}
\section{Baryogenesis from Supersymmetric Flat Directions}
There are many gauge invariant combinations of the Higgs, squark and
slepton fields along which the scalar potential identically vanishes
in the limit of exact SUSY. Within the MSSM there are nearly $300$
flat directions which are both $F-$ and $D-$ flat and conserve
R-parity~\cite{gkm}. Soft terms, as well as non-renormalizable
superpotential terms, lift the flat directions when SUSY is broken.
A homogeneous condensate along can be formed along a flat direction in
the inflationary epoch, provided that the flat direction mass
${\widetilde m}$ is smaller than the Hubble expansion rate during
inflation. The condensate starts oscillating coherently when the
expansion rate $H \simeq {\widetilde m}$. During this epoch the
inflationary fluctuations of the condensate can be converted to
density perturbations~\cite{denspert}. The condensate can also help an
efficient reheating to the SM degrees of freedom~\cite{averdi}. In
addition, it can excite vector perturbations to explain the observed
large scale magnetic field~\cite{mag}. If the condensate carries a
non-zero baryon and/or lepton number, then the flat direction dynamics
can be responsible for baryogenesis via the Affleck-Dine (AD)
mechanism~\cite{drt} (for a review, see Ref.~\cite{Kari}). In the
following section we make a general discussion on the viability of AD
baryogenesis for arbitrarily heavy gravitinos and/or sfermions.
\subsection{Late Baryogenesis via the Affleck--Dine Mechanism}
The scalar potential for a flat direction $\phi$ (not to be confused
with the inflaton field in Section II) is given by~\cite{drt}
\begin{equation}
\label{adpot}
V = \left(\widetilde m^2 + c_H H^2 \right)|\phi|^2 +
\left(\frac{ A + a H}{n M^{n-3}}\lambda \phi^n + h.c. \right)
+\frac{\lambda^2}{M^{(2n-3)}}|\phi|^{2(n-1)}\,.
\end{equation}
Here ${\widetilde m}^2$ and $c_H H^2$ are the soft ${\rm mass}^2$ from
SUSY breaking in the vacuum and SUSY breaking by the non-zero energy
density of the inflaton respectively. Here $c_H$ can have either sign,
and its sign is also affected by radiative
corrections~\cite{Gaillard,manuel}. The last term on the right-hand
side of~(\ref{adpot}), which lifts the flat direction, arises from a
non-renormalizable superpotential term (i.e. $n \geq 4$) induced by
new physics at a high scale $M$. In general $M$ could be a string
scale, below which we can trust the effective field theory, or $M =
M_{\rm P}$. SUSY breaking in the vacuum and by the inflaton energy
density generate $A$-terms $A$ and $a H$, respectively, corresponding
to this non-renormalizable superpotential term. For minimal K\"ahler
terms and in case of gravity mediation $ {\widetilde m} \sim A \sim
m_{3/2}$ and $0 < c_H \sim 1$. Depending on the symmetries of the
inflaton sector, both $a \sim {\cal O}(1)$ and $a \ll 1$ are possible.
The equation of motion for the flat direction is given by
\begin{equation}
\label{adeqm}
\ddot\phi + 3 H \dot \phi + \frac{\partial V}{\partial\phi^*} = 0\,.
\end{equation}
The evolution is easiest to analyze by the field parameterization
\begin{equation}
\label{param}
\phi = \frac{1}{\sqrt{2}}\,\varphi e^{i\theta}\,,
\end{equation}
where $\varphi,\,\theta$ are real fields. Then the scalar potential
can be written in the form
\begin{equation}
\label{adpot2}
V(\varphi,\theta) = \frac{1}{2} \left( \widetilde m^2 + c_H H^2 \right)
\varphi^2 + \frac{|\lambda| f(\theta) }{2^{(n-2)/2} n M^{n-3}} \,
\varphi^n + \frac{|\lambda|^2}{2^{n-1} M^{2(n-3)}} \,
\varphi^{2(n-1)}\,.
\end{equation}
Here
\begin{equation}
\label{ftheta}
f(\theta) = |A| \cos(n\theta + \theta_A + \theta_{\lambda} ) + |a| H
\cos(n \theta + \theta_a + \theta_{\lambda})\,,
\end{equation}
with $\theta_A,~\theta_a,~\theta_{\lambda}$ being the angular
directions for $A,~a,~\lambda$ respectively. The baryon/lepton
number density is given by
\begin{equation}
\label{charge}
n_{\rm B,L} = \frac{\beta}{i} \left( \phi^* \dot\phi - \phi\dot\phi^*
\right) = \beta \dot\theta \, \varphi^2\,,
\end{equation}
where $\beta$ is the baryon/lepton number carried by the flat
direction. At the minima of the of potential
\begin{equation}
\label{extrep}
\varphi_{\rm min}^{n-2} = \frac{2^{n/n-2}\,M^{n-3}}{(n-1)\,|\lambda|}
\left\{ -f(\theta) \pm \left[f(\theta)^2 - 4(n-1)(\widetilde m^2 + c_H H^2)
\right]^{1/2} \right\}\,,
\end{equation}
and $n \theta_{\rm min} = (2p+1)\pi - \theta_a -\theta_{\lambda}$ if
$|a|H \gg |A|$, while $n \theta_{\rm min} = (2p+1)\pi - \theta_A -
\theta_{\lambda}$ if $|a|H \ll |A|$ (with $p=0,1,\ldots,n-1$).
The radial field $\varphi$ quickly settles at one of the minima given
in~(\ref{extrep}) during inflation. If $|a|H \gg |A|$, the phase field
$\varphi \theta$ (since $\theta$ is dimensionless) has a mass of order
$|a|H$ and it ends up in one of the discrete minima $n \theta_{\min} =
\pi -\theta_a -\theta_{\lambda}$. The phase field has a mass $\ll H$
if $|a| H \ll |A|$, and hence it freezes at a random value.
After inflation the Hubble rate decreases as the Universe expands, and
so does $\varphi_{\rm min}$. The $\varphi$ field tracks the
instantaneous minimum of the potential, so its evolution can be
qualitatively understood by looking at the evolution of the
minimum. Once ${\widetilde m}^2 \simeq |c_H| H^2$, the minimum of the
potential changes from $\varphi_{\rm min}$ to $\varphi = 0$ in a
non-adiabatic manner. At this time $\phi$ starts oscillating in the
radial direction with frequency ${\widetilde m}$. The motion of the
phase field, which is necessary for generating a baryon/lepton
asymmetry, requires the exertion of a torque. If $|a|H \sim |A|$, a
non-adiabatic change in the position of the minimum from $n
\theta_{\rm min} = \pi - \theta_a - \theta_{\lambda}$ to $n
\theta_{\rm min} = \pi - \theta_A -\theta_{\lambda}$ generates a
torque. If $|a| H \ll |A|$, the freezing of the phase field at a
random value generates the torque and leads to its motion towards the
minimum $n \theta_{\rm min} = \pi - \theta_A -\theta_{\lambda}$ if
$|a| H \ll |A|$. The potential along the angular direction will
quickly decrease due to the redshift of $\varphi$, see
Eq.~(\ref{adpot2}), once $\varphi$ starts its oscillations. In
consequence, $\phi$ starts freely rotating in the angular direction at
which time a net baryon/lepton asymmetry is generated.
Based on Eqs.~(\ref{adeqm}) and~(\ref{charge}), the baryon/lepton
asymmetry obeys the equation
\begin{equation}
\label{cheqm}
\dot n_{\rm B,L} + 3H n_{\rm B,L} = -\beta \frac{\partial V}{\partial\theta}\,.
\end{equation}
This can be integrated to give at late times $t \gg H^{-1}_{\rm osc}$
(see~\cite{Kari})
\begin{equation}
\label{chdens}
n_{\rm B,L} \simeq \beta \frac{2(n-2)}{3(n-3)} {\sin \delta \over
(H_{\rm osc} t)^2}|A| \varphi_{\rm osc}^2 \,,
\end{equation}
where $\varphi_{\rm osc},\, H_{\rm osc}$ denote the value of $\varphi,
\, H$ when the condensate starts oscillating. Here $\delta$ is a
measure of spontaneous $CP$-violation in the $\phi$ potential and
$\sin \delta \sim 1$. The baryon to entropy ratio will then be
\begin{equation}
\label{barentr}
\frac{n_{\rm B,L}}{s} = \frac{3 T_{\rm R} \, n_{\rm B,L}}{4\rho_{\rm R}} =
\frac{T_{\rm R} \, n_{\rm B,L}}{4 M_{\rm P}^2 H_{\rm osc}^2}\,.
\end{equation}
We parameterize the $A$-term as $|A| = \gamma \widetilde m$. For
gravity-mediated SUSY breaking typically ${\widetilde m} \sim |A| \sim
m_{3/2}$, while in gauge-mediated models $|A| \sim m_{3/2} \ll
\widetilde m$. In split SUSY $A \ll \widetilde m$, see the discussion
in Ref.~\cite{adgr}. Here we consider $\gamma$ to be arbitrary, and
hence $\widetilde m$ and $|A|$ be unrelated.
Since $H_{\rm osc} \simeq {\widetilde m}$ and $\varphi_{\rm
osc}^{n-2}\sim 2^{(n-2)/2} M^{n-3} \widetilde m / |\lambda|$, we find
\begin{equation}
\label{barentr2}
\frac{n_{\rm B,L}}{s} \simeq \frac{n-2}{6(n-3)} \,
\frac{|A|T_{\rm R}}{\widetilde m^2} \left( \frac{\widetilde m}{|\lambda|
M_{\rm P}} \right)^{2/(n-2)}\,,
\end{equation}
where $M = M_{\rm P}$ is taken.
For the lowest dimensional non-renormalizable term $n=4$, and
$\lambda\sim {\cal O}(1)$, the generated baryon asymmetry is $n_{\rm
B,L}/s\sim 0.1(|A|T_{\rm R}/\widetilde m M_{\rm P})$. If $|A| \sim
\widetilde m$ then $T_{\rm R} \sim 10^{9}$~GeV is adequate to generate
baryon asymmetry of order $10^{-10}$. If $|A| \ll \widetilde m$, one
would require even larger $T_{\rm R}$. However the most desirable
feature of AD baryogenesis lies in its flexibility to generate a
desirable baryon asymmetry even for very low reheating temperatures. For
instance, if $\widetilde m \sim 10^{7}$~GeV and $n=6$, the required
asymmetry can be generated for $T_{\rm R} \sim 10^{3}$~GeV when $|A|=
\widetilde m$.
At late times, i.e. $H \ll {\widetilde m}$, contributions from SUSY
breaking by the inflaton energy density are subdominant to soft terms
from SUSY breaking in the vacuum. If $|A|^2< 4(n-1) {\widetilde m}^2$,
the $\phi$ potential has only one minimum at $\varphi = 0$. However,
another minimum appears away from the origin when $4(n-1) {\widetilde
m}^2 \leq |A|^2$, see~(\ref{extrep}). In the AD scenario the $\phi$
field starts at large values of $\varphi$, and hence it gets trapped
in this secondary minimum in the course of its evolution in the early
Universe. If $4(n-1){\widetilde m}^2 \leq |A|^2 < n^2{\widetilde
m}^2$, the true minimum is still located at the origin. Tunneling
from the false vacuum could still save the situation in this case.
However, for $|A| \geq n {\widetilde m}$, the true minimum will be at
$\varphi \neq 0$. Since flat directions have non-zero charge and
color quantum numbers, this will lead to an unacceptable situation
with charge and color breaking in the vacuum. It is therefore
necessary to have $|A|^2 < 4 (n-1) {\widetilde m}^2$ in order to avoid
entrapment in such vacuum states. This is the case in
gravity-mediated and gauge-mediated models, as well as split
SUSY. Note that the same discussion applies to the soft terms induced
by non-zero energy density of the inflaton. However, these terms
disappear at late times and will be irrelevant in the present vacuum.
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=9truecm\epsfbox{barnograv.eps}
\vspace{0.0truecm}
\end{center}
\caption{The reheating temperature as a function of the scalar mass,
given by Eq.~(\protect\ref{barentr2}), for successful AD baryogenesis. The
cases $n=4$ and $n=6$ are represented by solid and dashed lines
respectively. The black and red lines are plotted for $|A|= \widetilde
m$ and $|A|=1$~TeV respectively.}
\end{figure}
\begin{itemize}
\item{{\it Thermal effects}:\\ According to the potential given in
Eqs.~(\ref{adpot}) and~(\ref{adpot2}), the flat direction condensate
starts oscillating when $H \simeq {\widetilde m}$. However thermal
effects from reheating may trigger an earlier oscillation and lead to
a larger value of $H_{\rm osc}$~\cite{ace,and}. The inflaton decay (in
the perturbative regime) is a gradual process which starts after the
end of inflation. Hence, even before the inflaton decay is completed,
the decay products constitute a thermal bath with instantaneous
temperature $T \simeq (H T^2_{\rm R} M_{\rm P})^{1/4} > T_{\rm
R}$~\cite{kt}. The flat direction has gauge and Yukawa couplings,
collectively denoted by $y$, to other fields. Its VEV gives a mass
$\sim y \varphi$ to these fields. If $y \varphi \leq T$, these fields
are excited in the thermal bath and reach thermal equilibrium. This,
in turn, results in a thermal correction $\sim y T$ to the $\phi$
mass. If $y T$ exceeds the Hubble parameter at early times, i.e. for
$H \gg {\widetilde m}$, the condensate starts early
oscillations~\cite{ace}. On the other hand, if $y \varphi > T$, the
fields coupled to $\phi$ will be too heavy to be excited. They will
decouple from the running of gauge coupling(s) at temperature $T$
instead, which induces a logarithmic correction to the free energy
$\sim T^4 \log \left(T/\varphi \right)$. This triggers early
oscillations of the condensate if $T^2/\varphi > H$ when $H \gg
{\widetilde m}$~\cite{and}. Note that according to Eq.~(\ref{barentr})
a larger value of $H_{\rm osc}$ results in a smaller baryon/lepton
asymmetry.
Refs.~\cite{ace,and} have studied thermal effects for the conventional
case with ${\widetilde m} \sim 1$ TeV. Thermal corrections of the
former type will become less important as ${\widetilde m}$
increases. The reason is that $\varphi^{n-2} \propto H$,
see~(\ref{extrep}), and hence $y \varphi > H$ will be more difficult
to satisfy for larger ${\widetilde m}$. Also, since $H \propto T^4$
at early times, $\left(T^2/\varphi \right)$ increases more slowly that
$H$. Therefore thermal corrections of the latter type will also become
less important when ${\widetilde m} \gg 1$ TeV.}
\end{itemize}
\subsection{Effects of Gravitino on Affleck-Dine Baryogenesis}
In Eq.~(\ref{barentr2}) no specific assumption is made about the
source which reheats the Universe. It can be either the inflaton
decay, as usually considered, or the decay of the flat direction
condensate. In case the inflaton decay reheats the Universe $T_{\rm R}
< {\widetilde m}$ and ${\widetilde m} \leq T_{\rm R}$ are both
possible. However, $T_{\rm R} \geq {\widetilde m}$ if the flat
direction is responsible for reheating the Universe. The energy
density in the condensate oscillations is ${\widetilde m}^2
{\varphi}^2$. The flat direction has gauge and Yukawa couplings to
other fields through which it induces a mass $\propto \varphi$ for the
decay products. The condensate will decay no later than the time when
$\varphi \roughly< {\widetilde m}$. Hence, since energy density in the
radiation is $\sim T^4$, we will have $T_{\rm R} \geq {\widetilde m}$
in this case. According to Eq.~(\ref{barentr2}), the yielded baryon
asymmetry is $\propto T_{\rm R}$. This implies a larger asymmetry for
larger values of $T_{\rm R}$. Having an unacceptably large baryon
asymmetry is indeed typical when the flat direction condensate has a
very large VEV such that it dominates the energy density and,
subsequently, reheats the Universe~\cite{ad}. Gravitino domination
can in this case help to dilute the excess of baryon asymmetry. It is
interesting that such a solution, invoked from the early days of AD
baryogenesis~\cite{ad}, can be naturally realized with supermassive
gravitinos.
When $T_{\rm R} \geq {\widetilde m}$ sfermions reach thermal
equilibrium after reheating and their decay will be the dominant
source of gravitino production. The condition for gravitino domination
and the dilution factor from gravitino decay are then given by
Eqs.~(\ref{sfermdom}) and~(\ref{sfermdilution}) respectively. The
final asymmetry generated via the AD mechanism in case of gravitino
domination will then be, see~(\ref{barentr2})
\begin{equation}
\label{barentr3}
\frac{n_{\rm B}}{s}\approx \frac{n-2}{6(n-3)}
\left(\frac{\widetilde m}{|\lambda| M_{\rm P}}\right)^{2/n-2}
\left(\frac{|A|}{\widetilde m}\right)
\left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{5/2}
\left({1.3 \times 10^8~{\rm GeV} \over {\widetilde m}}\right)^3\,.
\end{equation}
The results for the two extreme cases $|A| = {\widetilde m}$ and $|A|
= 1$ TeV are summarized in Fig.~(6) when $n = 4,6$. The combined
constraints from baryogenesis and dark matter considerations will be
included in the overlap of Figs.~(1) and~(6).
If $T_{\rm R} \ll {\widetilde m}$, sfermion quanta will not be excited
in the thermal bath. However, inflaton decay can in this case result
in efficient production of gravitinos according to
Eq.~(\ref{phidecay}). One can then repeat the same steps to find an
expression for the initial asymmetry similar to that
in~(\ref{barentr3}). Such an expression, and plots similar to those in
Fig.~(6), will however depend on $\Delta m_{\phi}$ as well as
${\widetilde m}$ and $m_{3/2}$. This leads to a more complicated and
model-dependent situation. Moreover, the scenario with gravitino
domination will be more constrained when $T_{\rm R} \ll {\widetilde
m}$ (specially if $|A| \ll {\widetilde m}$). The initial baryon
asymmetry is already suppressed in this case, see~(\ref{barentr2}),
and gravitino decay may dilute it to unacceptably small values.
\begin{figure}[htb]
\vspace*{-0.0cm}
\begin{center}
\epsfysize=9truecm\epsfbox{bargrav.eps}
\vspace{0.0truecm}
\end{center}
\caption{The scalar mass as a function of gravitino mass, given by
Eq.~(\protect\ref{barentr3}), for successful AD
baryogenesis in a gravitino-dominated Universe. The conventions are the same
as in Fig.~(5).}
\end{figure}
\subsection{Late Gravitino Production from Q-ball Decay}
So far we have assumed that gravitinos are mainly produced in sfermion
decays (if $T_{\rm R} \geq {\widetilde m}$), or in inflaton decay (if
$T_{\rm R} \ll {\widetilde m}$). The flat direction condensate
consists of zero-mode quanta of the sfermions, and hence its decay too
can lead to gravitino production. If $T_{\rm R} \geq {\widetilde m}$,
this will be subdominant to the contribution from the decay of thermal
sfermions. The reason is that the zero-mode quanta have at most an
abundance $\left(n/s \right)$ which is comparable to that of thermal
sfermions. If $T_{\rm R} \ll {\widetilde m}$, sfermions will not be
excited in the thermal bath. As mentioned before, the condensate
certainly decays no later than the time when $\varphi \roughly<
{\widetilde m}$. Even if $\varphi \sim M_{\rm P}$ initially, the
condition $\varphi \roughly< {\widetilde m}$ is satisfied at $H \roughly>
{\widetilde m}^2/M_{\rm P}$. Not that the scale factor of the Universe
$a \propto H^{-2/3}$ during reheating, in which phase the Universe is
dominated by inflaton oscillations. Also, the abundance of zero-mode
quanta $\left(n/s \right) < \left(3 T_{\rm R} /4 {\widetilde m}\right)
\ll 1$. This implies that the condensate contains a much smaller
number of quanta which survive (much) shorter than thermal
sfermions. Therefore gravitino production from the decay of the flat
direction condensate will not be as constraining as that in the decay
of thermal sfermion. In most case, it can be simply neglected.
The above discussion strictly applies to the oscillations of a
homogeneous condensate. However, it usually happens that the flat
direction oscillations fragment and forms non-topological solitons
known as Q-balls~\cite{coleman}. These Q-balls can decay much later
than a homogeneous condensate. Their late decay could then efficiently
produce gravitinos, even if Q-balls do not dominate the energy density
of the Universe.
To elucidate, let us consider the case when tree-level sfermion masses
at a high scale $M$ are given by ${\widetilde m}$. The potential for
the sfermions, after taking into account of one-loop corrections,
reads
\begin{equation}
\label{gravqball}
V = \widetilde m^2 |\phi|^2 \left[ 1 + K \textrm{ln}
\left( \frac{|\phi|^2}{M^2} \right) \right]\,,
\end{equation}
where $K$ is a coefficient determined by the renormalization group
equations, see~\cite{Nilles,enqvistetal}. In order to form Q-balls it
is necessary that the potential be flatter than $|\phi|^2$ at large
field values, i.e. that $K < 0$. Loops which contain gauginos make a
negative contribution $\propto - m^2_{1/2}$ to $K$, with $m_{1/2}$
being the gaugino mass. On the other hand, loops which contain
sfermions contribute $\propto + {\widetilde m}^2$. Then $K < 0$ can
be obtained, provided that $2 m_{1/2} \roughly> {\widetilde
m}$~\cite{enqvistetal}. In models of gravity-mediated SUSY breaking
$\widetilde m \sim m_{1/2}$, and hence $K<0$ is obtained for many flat
directions. When the spectrum is such that ${\widetilde m} \gg
m_{1/2}$, like in the case of split SUSY, there are no Q-balls as $K >
0$.
The potential can also be much flatter $\propto \textrm{ln} |\phi|$ at
large field values. This happens in models of gauge-mediated SUSY
breaking~\cite{gmsb}, and can arise from thermal
corrections~\cite{and}. The important point in any case is that the
scalar field profile within a Q-ball is such that the field value is
maximum at the center $\varphi_0$ and decreases towards the
surface. This implies that for $g \varphi_0 \geq {\widetilde m}$, with
$g$ being a typical coupling of the $\phi$ field to other fields, the
Q-ball decays through its surface as decay inside the Q-ball is not
energetically allowed~\cite{cohenetal}. Note that for a typical gauge
or Yukawa coupling $g \varphi_0 \geq {\widetilde m}$ if $\varphi_0 \gg
{\widetilde m}$. The decay rate of Q-ball which contains a
(baryonic/leptonic) charge $Q$ is in this case given
by~\cite{cohenetal}
\begin{equation} \label{qballrate}
{d Q \over d t} \leq {\omega^3 A \over 192 \pi^2},
\end{equation}
where $\omega \simeq {\widetilde m}$ and $A = 4 \pi R^2_{\rm Q}$ is the
surface area of the Q-ball. For example, consider Q-ball formation for
the potential given in~(\ref{gravqball}). A Q-ball with total charge
$Q$ then has a decay lifetime~\cite{enqvistetal}
\begin{equation} \label{qballife}
\tau_{\rm Q} \roughly> \left({|K| \over 0.03}\right) \left({1~{\rm TeV} \over
{\widetilde m}}\right) \left({Q \over 10^{20}}\right) \times
10^{-7}~{\rm sec},
\end{equation}
which corresponds to the decay temperature
\begin{equation} \label{qballtemp}
T_{\rm d} \roughly< \left({0.03 \over |K|}\right)^{1/2} \left({{\widetilde m}
\over 1~{\rm TeV}}\right)^{1/2} \left({10^{20} \over Q}\right)^{1/2}
\times 2~{\rm GeV}.
\end{equation}
Here we have used $\tau^{-1}_{\rm d} \sim \left(T^2_{\rm d}/M_{\rm P}
\right)$. The total baryonic/leptonic charge $Q$ of a Q-ball is given
by the multiplication of baryon/lepton number carried by the flat
direction and the total number of zero-mode quanta inside the Q-ball.
This, after using~(\ref{decrate}), leads to
\begin{eqnarray} \label{qballdecay}
{\rm Helicity} \pm {1 \over 2}: \left({n_{3/2} \over s}\right)_{\rm
Q-ball} &\sim & \left({{\widetilde m} \over m_{3/2}}\right)^2
\left({{\widetilde m} \over 1~{\rm TeV}}\right)^2
\left({Q \over 10^{20}}\right)
\left({n_{\rm B} \over s}\right) 2.5 \times 10^{-13} \,, \nonumber \\
{\rm Helicity} \pm {3 \over 2}: \left({n_{3/2} \over s}\right)_{\rm
Q-ball} &\sim & \left({{\widetilde m} \over 1~{\rm TeV}}\right)^2
\left({Q \over 10^{20}}\right)
\left({n_{\rm B} \over s}\right) 2.5 \times 10^{-13} \,,
\end{eqnarray}
Obviously $\left(n_{3/2}/s \right)_{\rm Q-ball} \leq \left(n_{\rm B}/s
\right)$, since the decay of each quanta inside the Q-ball can at most
produce one gravitino. This implies that gravitinos produced from the decay of
Q-balls will not dominate the Universe if the decay generates a baryon
asymmetry $\left(n_{\rm B}/s \right) \simeq 10^{-10}$, see
Eq.~(\ref{dom}). However, the situation will be different for larger
Q-balls which yield an asymmetry $\gg 10^{-10}$. If gravitinos from
Q-ball decay dominate the Universe, they will dilute the baryon
asymmetry. The final asymmetry will then have the correct size, see
Eqs.~(\ref{dilution}) and~(\ref{qballdecay}), if
\begin{equation} \label{qballbaryon}
Q \sim \left({m_{3/2} \over 10^5~{\rm GeV}}\right)^{1/2}
\left({1~{\rm TeV} \over {\widetilde m}}\right)^2 \left({m_{3/2}
\over {\widetilde m}}\right)^2 \times 10^{35}.
\end{equation}
Assuming that the Q-balls do not dominate the energy density of the
Universe, we must have $\left(n_{\rm B}/s \right) < \left(3 T_{\rm
d}/4 {\widetilde m}\right)$. The necessary condition for gravitino
domination, after using Eqs.~(\ref{dom}),~(\ref{qballtemp})
and Eq.~(\ref{qballdecay}), is then obtained to be
\begin{equation} \label{qballdom}
Q > \left({m_{3/2} \over 10^5~{\rm GeV}}\right)
\left({m_{3/2} \over {\widetilde m}}\right)^4
\left({1~{\rm TeV} \over {\widetilde m}}\right)^3 2.5 \times 10^{35}.
\end{equation}
For ${\widetilde m} \gg 1$ TeV and $m_{3/2} \ll {\widetilde m}$, this
lower bound on the Q-ball charge is compatible with the value in
Eq.~(\ref{qballbaryon}) required for successful baryogenesis. Note
that the Q-ball decay temperature $T_{\rm d}$ may be smaller than the
LSP freeze-out temperature $T_{\rm f}$. In this case Q-ball decay can
be dangerous as three LSP per baryon number will be produced. However,
this will not lead to problem so long as the gravitino decay
temperature $T_{3/2} < T_{\rm d}$ (which is typically the case) and
the condition for efficient LSP annihilation in Eq.~(\ref{eff}) is
satisfied. The Q-balls will dominate the Universe if they carry a
very large charge. The initial baryon asymmetry released by the
Q-balls then has a simple expression $\left(n_{\rm B}/s \right) \sim
\left(T_{\rm d}/{\widetilde m}\right)$.
The results in Eqs.~(\ref{qballife}),~(\ref{qballtemp}),
~(\ref{qballdecay}),~(\ref{qballbaryon}) and~(\ref{qballdom}) are
valid for the potential given in Eq.~(\ref{gravqball}). The same steps
(though more involved) can be followed to obtain similar results for
logarithmic potentials as in the case of gauge-mediated models. The
remarkable point in all cases is that the Q-ball decay lifetime
increases with its charge, implying a more efficient production of
gravitinos from the Q-ball decay. This leads to an attractive
solution that the large baryon asymmetry released by the Q-ball decay
can be naturally diluted by gravitinos produced in the same process.
Finally, we shall notice that in models of running mass inflation even
the inflaton condensate could fragment into non-topological
solitons~\cite{Kasuya}. In that case Q-balls would naturally dominate
the Universe. We do not discuss such possibility here.
\section{Summary and Conclusions}
In this paper we have investigated cosmological consequences of models
with superheavy gravitinos and/or sfermions. A priori there is no
fundamental reason which fixes the scale of SUSY breaking. Models with
weak-scale SUSY breaking in the observable sector have the promise to
solve the hierarchy problem. However this may not necessarily be the
case and the SUSY breaking scale can turn out to be very high. Under
general circumstances, arbitrarily heavy gravitino mass $m_{3/2}$
and/or sfermion masses ${\widetilde m}$ are quite
plausible. Therefore, inspired from the recent models of large scale
SUSY breaking, it becomes pertinent to re-examine the cosmological
and phenomenological consequences.
Gravitino are produced through various processes in the early
Universe. Scatterings of gauge and gaugino quanta in thermal bath,
sfermion decays and the inflaton decay are the main sources for
gravitino production. The main results are presented in
Eqs.~(\ref{scattering}),~(\ref{decay}) and~(\ref{phidecay}). Sfermion
decays usually dominate when the reheating temperature $T_{\rm R} \geq
{\widetilde m} > m_{3/2}$. On the other hand, the contribution from
the inflaton decay dominates when $T_{\rm R} \ll {\widetilde m}$.
Gravitinos which are heavier than $50$ TeV decay before primordial
nucleosynthesis, and hence are not subject to BBN bounds. However,
each gravitino produces one LSP upon its decay. Hence, in models with
conserved $R$-parity, the abundance of {\it supermassive} gravitinos
is constrained by the dark matter limit. Indeed, efficient
annihilation of LSPs produced in gravitino decay sets a lower bound on
$m_{3/2}$. When this lower bound is saturated, gravitino decay can
successfully produce non-thermal LSP dark matter.
This is also valid in a gravitino-dominated Universe, which happens
when gravitinos are produced very abundantly. However, for $m_{3/2}
\geq 50$ TeV, gravitino domination cannot rescue a scenario where
thermal LSP abundance at the freeze-out exceeds the dark matter
bound. For a Wino- or Higgsino-like LSP this is the case when the LSP
mass $m_{\chi} > 2$ TeV. The reason is that in this case gravitino
decay, while diluting the thermal abundance, leads to non-thermal
overproduction of LSPs. Therefore, if $R$-parity is conserved,
gravitinos should never dominate in models with such heavy LSPs. The
results for gravitino production in conjunction with the constraints
from the dark matter bound and LSP annihilation are summarized in
Eqs.~(\ref{after}),~(\ref{eff}),~(\ref{nosfermdom})
and~(\ref{sfermdom}). Figs.~(1) and~(2) depict the acceptable parts of
the ${\widetilde m}-m_{3/2}$ plane for successful scenarios of
gravitino non-domination and domination respectively.
We discussed some specific scenarios of baryogenesis in the presence
of supermassive gravitinos. The parameter space for thermal
leptogenesis is substantially relaxed when $m_{3/2} \geq 50$ TeV, as a
considerably larger reheating temperature $T_{\rm R}$ and/or right-handed
(s)neutrino mass $M_1$ will be allowed. This, however, implies that
gravitinos can also be efficiently produced for a wide range of
sfermion masses. Since gravitino decay takes place after the
completion of leptogenesis, unless they are extremely heavy, the
generated baryon asymmetry will be diluted in a gravitino-dominated
Universe. Successful leptogenesis then requires (much) larger
right-handed (s)neutrino masses than usual. However, it is known that
thermal leptogenesis fails for $M_1 \geq 10^{14}$ GeV since lepton
number violating scatterings in this case erase the generated
asymmetry. This leads to additional constraints on the ${\widetilde
m}-m_{3/2}$ parameter space in case of gravitino domination. Our
results are summarized in Eqs.~(\ref{domasym}),~(\ref{suclep}) and
Fig.~(4).
We also considered late time baryogenesis from supersymmetric flat
directions via the Affleck-Dine mechanism. Thermal effects which can
trigger early oscillations of the flat direction condensate, thus
suppressing the generate asymmetry, tend to be less important for
${\widetilde m} \gg 1$ TeV. A large expectation value for the
condensate at the onset of its oscillations usually leads to a baryon
asymmetry $\left(n_{\rm B}/s \right) \gg 10^{-10}$, as well as a large
reheating temperature $T_{\rm R} \geq {\widetilde m}$. Gravitinos
produced from sfermion decays can then dominate the Universe and
dilute the initially large asymmetry down to acceptable values. The
main results in Eqs.(\ref{barentr2}), ~(\ref{barentr3}) are depicted
in Figs~(5),~(6).
There is even a closer connection between large baryon asymmetry and
efficient gravitino production when oscillations of the flat direction
condensate fragment into Q-balls (as happen in many cases). Q-balls decay
(much) later than the homogeneous condensate, and the larger the
baryonic/leptonic charge they carry the longer their decay lifetime.
Hence the decay of large Q-balls is a natural source for
copious production of gravitinos which can even dominate the Universe and
dilute the large baryon asymmetry released by Q-balls. We explicitly
demonstrated this for a potential with logarithmic corrections,
Eqs.~(\ref{qballbaryon}) and~(\ref{qballdom}), but the same conclusions hold
for other types of flat potentials.
To conclude, models with superheavy gravitinos and/or sfermions have
very interesting cosmological consequences. These models can naturally
give rise to a large gravitino abundance in the early Universe. This,
contrary to models with a weak scale gravitino mass, can turn to a
virtue and lead to successful production of dark matter and baryon
asymmetry generation.
\section{Acknowledgments}
The work of R.A. is supported by the National Sciences and Engineering
Research Council. S.P thanks the kind hospitality of NORDITA and NBI during
part of the course of the present work.
|
2,869,038,154,863 | arxiv | \section{Introduction}
The high sensitivity of optically trapped Brownian particles, combined with long integration times available, makes optical traps outstanding metrological systems.
They have therefore been involved in many weak force experiments and have been recognized as outstanding systems for implementing and simulating many results and protocols that have been brought forward recently in the field of optomechanics and non-equilibrium statistical physics \cite{CilibertoPRX2017,Martinez2017,Bechhoefer_2020}.
Optical traps physically implement an Ornstein-Uhlenbeck process through the harmonic trapping force field. One of their interesting features is to give access to different diffusing dynamics for the trapped Brownian object, ranging from confined motion in the long timescales to free Brownian motion on the shortest ones, therefore probing the evolution of the Ornstein-Uhlenbeck process towards the Wiener process-like limit at short times \cite{Uhlenbeck,gardiner}. These two dynamic regimes have very different properties that make them more relevant for different experiments. In particular, the Ornstein-Uhlenbeck regime is well suited for force measurements \cite{Wu2009,MaiaNeto2015,Ricci2017,Li} while position detection benefits from the Wiener regime, allowing to achieve higher resolution \cite{Lukic2005,li2010,Schnoering2019}.
In this article, we address these differences from the viewpoints of noise stability and ergodicity for both regimes. We implement theoretical and experimental tools capable of characterizing motional noise (using an Allan-variance based analysis \cite{Oddershede,Saleh}) and ergodic signatures (developing a specific test of ergodicity \cite{Metzler2014,Metzler2015}) in an optical trap from the Ornstein-Uhlenbeck regime to its high frequency Wiener limit in a unified way.
This capacity is important in particular in the field of precision measurements involving optical traps. There indeed, the building of large motional statistical ensembles necessary to reach high resolution levels usually relies on strong assumptions related to the nature and stability of the driving noise. It also depends on the ergodicity of the corresponding Brownian motion. We show here precisely how these assumptions can be tested on overdamped harmonic optical traps, paving the way for reliable experiments at all measurement bandwidths. The tools we describe below are general: they can be used on underdamped and more complex systems and can thereby be exploited when colored noise and non-ergodic effects enrich the physics of Brownian motion, as in the realm of swimmers or active matter, for instance.
\section{Wiener vs. Ornstein-Uhlenbeck crossover in an optical trap}
Free Brownian motion driven only by the Gaussian white noise of thermal fluctuations is described by the Wiener process $W_t$. The displacement of the overdamped free Brownian object writes as:
\begin{equation}
dx_t = \sqrt{2D}dW_t ,
\label{wiener}
\end{equation}
working directly with the differential $dW_t$ with the following properties: $\langle dW_t \rangle = 0$, $\langle dW_t dW_{t'} \rangle = \delta(t, t') dt$.The diffusion coefficient $D = k_B T / \gamma$ involves the Boltzmann constant $k_B$, the temperature of the surrounding fluid $T$ and the Stokes drag coefficient $\gamma$.
Inside the trap, the harmonic optical potential modifies the stochastic process by exerting on the object a restoring force characterized by a constant stiffness $\kappa$. The same displacement now follows the Ornstein-Uhlenbeck process:
\begin{equation}
dx_t = -\frac{\kappa}{\gamma}x_tdt + \sqrt{2 D}dW_t .
\label{ou}
\end{equation}
\begin{figure}[htb]
\centering{
\includegraphics[width=0.8\linewidth]{fig/schema.png}}
\caption{{Schematic view of the optical trapping system: a $1~\mu$m polystyrene bead is trapped by a 785 nm laser beam, focused by a high numerical aperture (NA=$1.2$) water immersion objective. The instantaneous position of the bead trapped at the laser waist is recorder along the optical axis with an acquisition frequency of $2^{15}=32768$ Hz.}}
\label{schema}
\end{figure}
\vspace{3mm}
Our experiment, detailed in Appendix \ref{APPENDIX_samples}, consists in trapping a single Brownian object in the harmonic potential created at the waist of a focused laser beam, and recording the instantaneous overdamped position $x(t)$ of the trapped bead, as schematized in Fig. \ref{schema}. All the experimental results presented in this work are obtained from a 10 minutes long trajectory (i.e. $1.97\times 10^{7}$ successive position measurements acquired at a frequency of $2^{15}=32768$ Hz).
These data are compared, throughout this article, with numerical simulations obtained from an algorithm for the Wiener process:
\begin{equation}
x_{t+\Delta t} = x_t + \sqrt{2D\Delta t} \theta_t ,
\end{equation}
where $\theta$ is a dimensionless Gaussian white noise with $\langle \theta_t \rangle = 0$, $\langle \theta_t \theta_{t'} \rangle = \delta(t-t')$, according to the methods detailed in \cite{volpe}. By the same token, the algorithm for the Ornstein-Uhlenbeck process is:
\begin{equation}
x_{t+\Delta t} = x_t - \frac{\kappa}{\gamma}x_t \Delta t + \sqrt{2D\Delta t} \theta_t .
\end{equation}
This discretisation method, known as the Euler-Maruyama method, corresponds to an $ \mathcal{O}(\Delta t^{1/2})$ approximation of It\^o-Taylor expansions \cite{kloeden}. As discussed in details in Appendix \ref{APPENDIX_algorithm}, higher order terms lead to a more efficient algorithm known as the Mildstein algorithm, which our simulations are based on and which converges more quickly towards the analytical expression as $\Delta t$ decreases \cite{Higham2001,vanden-eijnden}.
From Eq. (\ref{ou}), the Brownian motion in the trap can be spectrally analyzed with the position's power spectral density (PSD):
\begin{equation}
S_{x}(f) = \frac{D}{2\pi^2(f_c^2 + f^2)} .
\label{psd_eq}
\end{equation}
As clearly seen on the experimental PSD displayed in Fig. \ref{psd}, the roll-off frequency $f_c = \kappa/ (2 \pi \gamma)$ separates the high frequency regime $S_x(f)\sim{D}/{(2\pi^2f^2)}$ of free Brownian motion -see Eq. (\ref{wiener})- from the low frequency trapping regime $S_{x}(f) \sim {D}/{( 2 \pi^2 f_c^2)} = 2 k_B T \gamma / (\kappa^2) $ -see Eq. (\ref{ou}). The PSD thus clearly reveals how a Wiener regime corresponds in the optical trap to the short time $\delta t \ll \gamma/\kappa$ limit of the Ornstein-Uhlenbeck process (in other words, when observed over such a short timescale, the Brownian object moves inside the trap as if it were freely diffusing without confinement).
\begin{figure}[htb]
\centering{
\includegraphics[width=1\linewidth]{fig/PSD_fit_units.png}}
\caption{{Experimental power spectrum density (PSD) evaluated for a trajectory $x(t)$ measured from $0.03$ \si{\hertz} to $100$ \si{\kilo \hertz}, displaying a large signal-to-noise ratio, spanning over 4 decades. We also note the transition, at the roll-off frequency ($53.6511$ Hz) between the high frequency almost-free regime and the low frequency trapped regime -vertical red dashed line. The thermal noise plateau $2 k_B T \gamma / \kappa $ (horizontal black dashed line) agrees well with the low frequency limit of the PSD, as expected. From the Lorentzian fit performed on the PSD, we can extract the stiffness $\kappa = 2.9614\cdot10^{-6} \pm 6.7339\cdot10^{-8}$ kg/s$^2$. The experiments are performed at room temperature, $T \approx 295$ \si{\kelvin} and the 1 \si{\micro\meter} bead experiences a drag coefficient $\gamma = 6 \pi \eta R $ kg/s where $\eta \approx 0.95 \cdot10^{-3}$ \si{\pascal\second}, hence $\gamma = 8.9837\cdot 10^{-9}$ kg/s. These parameters, with the stiffness extracted from the Lorentzian fit of the PSD, are used in all numerical and analytical results done throughout the paper.}}
\label{psd}
\end{figure}
\vspace{3mm}
\section{Noise stability: Allan variance and statistical tests}
In order to characterize the noise at play inside the optical trap, it is central to measure two of its properties: its nature (color, thermal weight, frequency contributions, etc), and its stability in time. Testing the nature of the noise can be done spectrally with the PSD that yields the different frequency contributions of the noise. Integrated PSD can also reveal the thermal nature of the noise through the fluctuation-dissipation theorem. However, the spectral approach turns out to be exposed to possible low frequency drifts that can modify noise properties \cite{Li,Oddershede,Saleh}. In order to avoid this stability issue, we work in the time-domain and perform an Allan-variance based test of the system, capable of revealing low frequency drifts within a stochastic signal \cite{Allan1966,Barnes1971}. This approach leads us to verify unambiguously the stationary and thermally limited properties of the noise at play in an experiment.
The Allan variance $\sigma^2(\tau)$ can be connected to the noise PSD $S(f)$ through the following relation \cite{Barnes1971}:
\begin{equation}
\sigma^2(\tau) = \frac{4}{\pi \tau^2} \int_{-\infty}^{+\infty} S(f)\sin^4(\pi f \tau) df
\end{equation}
It can therefore be explicitly evaluated analytically for the Ornstein-Uhlenbeck PSD $S_x(f)$ of Eq. (\ref{psd_eq}):
\begin{equation}
\sigma^2(\tau) = \frac{k_B T}{\kappa \tau^2} \left( 4\left[1 - e^{-\kappa\tau/\gamma}\right] - \left[1 - e^{-2\kappa\tau/\gamma}\right]\right) ,
\label{allanvar_eq}
\end{equation}
as detailed in Appendix \ref{APPENDIX_allan}.
The experimental Allan variance is shown in Fig. \ref{allan} following the same methodology presented in our earlier work \cite{Li}. This experimental Allan variance is compared with numerical simulations and with the analytical result of Eq. (\ref{allanvar_eq}). We note a remarkable experiment-theory agreement over more than 6 decades in time. These results show the very high level of noise stability up to $>250$ s that one can reach on a simple optical trap setup such as ours.
But they also reveal how the Ornstein-Uhlenbeck and the Wiener processes are characterized by different Allan variance signatures. Indeed, we identify here two clear asymptotic regimes. The short time regime ($\tau \ll \gamma/\kappa$) falls on the $\sigma_{\rm free} \sim t^{-1/2}$ slope, which is known to corresponds to the thermal white noise limit of free Brownian motion \cite{Oddershede,Li}. Interestingly, in the long time limit ($\tau \gg \gamma/\kappa$) of the Ornstein-Uhlenbeck process where the trapping action dominates the motional dynamics, the Allan variance shows a different slope with $\sigma_{trap} \sim t^{-1}$.
This change of signatures between the two regimes, accounting for the presence of the harmonic force field in the long time limit, is continuous. We observe a very good match between the experiments and theory in the transition between asymptotic regimes.
The slight differences at short time-lags between the theory and the experimental data will also be observed at the level of the mean squared displacement (MSD) Fig. \ref{MSD_fig} (a) and the ergodic parameter Fig. \ref{ergodicity}. As discussed in details in Appendix \ref{APPENDIX_err}, these deviations are due to tracking errors unavoidably induced experimentally by the photodiode and electronic system used for recording our Brownian trajectories.
\begin{figure}[htb]
\centering{
\includegraphics[width=1\linewidth]{fig/allan_dev_fit.png}}
\caption{{Allan standard deviation evaluated for the long trajectory experimentally recorded (blue open circles). We plot the simulated Allan standard deviation (orange continuous line) superimposed to the analytical result (black dashed line). We highlight the slopes in both free (purple continuous line) and trapped regimes (green continuous line). We observe that the whole time range from $\sim 10^{-4}s$ up to $\sim10^{2}s$ is perfectly captured by the theoretical expression built with experimental parameters --$\gamma, T, \kappa$, see Fig. \ref{psd}-- with a very good agreement. The small departure of the experimental data from the theoretical Allan variance is attributed to tracking errors discussed in Appendix \ref{APPENDIX_err_allan}.}}
\label{allan}
\end{figure}
\vspace{3mm}
We will now use an alternative method based on the autocorrelation and the MSD for identifying either a Wiener or an Ornstein-Uhlenbeck process. We however remind here that at thermal equilibrium, Wiener and Ornstein-Uhlenbeck processes generate trajectories $x(t)$ with different statistical properties. Indeed, the Ornstein-Uhlenbeck process of the trapped Brownian motion has a variance constant in time with the equipartition condition $\langle x_t^2 \rangle = k_B T / \kappa$.
In contrast, the Wiener process of free Brownian motion is non-stationary with a motional variance that grows linearly in time.
But looking at the statistical properties of successive displacements $dx_t$ whose dynamics is governed by Eqs. (\ref{wiener},\ref{ou}), it becomes possible to perform the same stationarity test for both processes. To do that, we will use the autocorrelation of displacements and the MSD, extracted from long trajectories.
We will verify stationarity --in the strong sense since the noise is Gaussian-- with $(i)$ a fixed mean (that can be removed without any loss of generality), $(ii)$ a finite variance $\overline{dx_t^2} $, and $(iii)$ a displacement covariance (autocorrelation) $\overline{dx_t dx_s}$ that depends only on the absolute time difference $\Delta = \mid t - s \mid$.
The covariance of displacements can be computed using Eq. (\ref{ou}) (details are given in Appendix \ref{APPENDIX_cov}, see Eq. (\ref{autocorrdim})) and yields:
\begin{equation}
\overline{dx_t dx_s} = -\frac{2\kappa k_B T}{\gamma^2} e^{-\kappa|t-s|/ \gamma} dt^2 + 2 D \delta(t-s) dt.
\label{cov}
\end{equation}
This theoretical expression is compared to the covariance evaluated experimentally as a time-average on successive displacements. The comparison, together with simulations, is shown on figure \ref{MSD_fig} (a). The convergence of the time averaging process for the covariance towards the theoretical expression, only function of $\Delta$, shows the absence of dependence on the absolute time $t$.
We can also evaluate the MSD directly from the measurement of successive positions separated by a given time-lag $\Delta$ (details are given in Appendix \ref{APPENDIX_MSD}, see Eq. (\ref{MSD_analytic})) as:
\begin{equation}
\overline{\delta x^2(\Delta) } = 2 \frac{k_B T}{\kappa}\left(1 - e^{-\kappa \Delta/\gamma}\right) .
\label{msd}
\end{equation}
Again, this theoretical result is compared to the experimental MSD which is given by evaluating the time average MSD of the entire trajectory. The comparison, also including simulations, shows a very good agreement displayed in Fig. \ref{MSD_fig} (b).
This agreement, together with the covariance, depending only on time-difference, confirms that our Brownian trap implements a strong stationary Ornstein-Uhlenbeck process. Clearly, our data demonstrate a smooth crossover between the linear MSD at short time lags associated with a Wiener regime and the constant MSD at longer time lags that reflects the confined nature of the diffusion for the Ornstein-Uhlenbeck process.
\begin{figure}[htb]
\centering{
\includegraphics[width=0.9\linewidth]{fig/Cov_fig_both.png}
\includegraphics[width=0.9\linewidth]{fig/MSD_fig.png}
}
\caption{{(a) Time average covariances of positions and displacements (inset). Experimental data are plotted (blue open circles) together with the simulation results (orange continuous line) and the analytical prediction (black dashed line). (b) Comparison between the measured mean square displacements (MSD) (blue open circles) and the analytical expression given in Eq. (\ref{msd}) obtained in the stationary regime (black dashed line). The comparison with simulation results is also displayed (orange continuous line). The very good agreement with both theory and simulations shows that the measured process can be considered as stationary. We note the same relaxation time of $ 3 \cdot 10^{-3} $s for all data, revealing the crossover between the free (Wiener) and trapped (Ornstein-Uhlenbeck) diffusion regimes. Again, the small departure of the experimental data with respect to the theoretical MSD is attributed to tracking errors discussed in Appendix \ref{APPENDIX_err_msd}.}}
\label{MSD_fig}
\end{figure}
\section{Test of ergodicity}
\label{SEC:ergo}
As reminded in the Introduction, the ergodic hypothesis is central for reaching high resolution levels in optical trapping experiments. Ergodicity \textit{per se} corresponds to the equality taken in the infinite time limit $\mathcal{T} \rightarrow \infty$, between the time average and the ensemble average for a given stochastic process. In order to test ergodicity, we first need to build an ensemble of trajectories $\{i\}$. To do this, we reshape our long trajectory into an ensemble of 600 trajectories $x_i(t)$ of 1 second duration each.
For such a trajectory $x_i(t)$ drawn from the ensemble, ergodicity is defined as:
\begin{equation}
\lim\limits_{\mathcal{T} \rightarrow \infty} \frac{1}{\mathcal{T}} \int_0^{\mathcal{T}} x_i(t)dt = \langle x_i(t) \rangle_{\{ i\}} .
\end{equation}
Although simple, this definition is however hardly operative in experiments that only yield ensembles of finite-time trajectories. Following the approach proposed in \cite{Metzler2014,Metzler2015}, we prefer resorting to an observable that can characterize the ergodic nature of an experiment performed over a finite integration time. This observable is grounded on the stationary nature of the MSD which is, as we shown above, independent of the choice from the initial time and only depends on the time-lag $\Delta$. In such conditions, ergodicity simply demands the time average MSD of any $i^{\rm th}$-trajectory, as defined above, to be equal, in the long $\mathcal{T}/\Delta$ limit, to the ensemble mean of individual time average taken over the ensemble $\{i\}$ of available trajectories:
\begin{equation}
\lim\limits_{\mathcal{T}/\Delta \rightarrow \infty} \overline{\delta x_i^2(\Delta)}= \biggl< \overline{\delta x_i^2(\Delta)} \biggl> .
\end{equation}
Formally, ergodicity demands that the $ \overline{\delta x_i^2(\Delta)}/ \biggl< \overline{\delta x_i^2(\Delta)} \biggl>$ ratio tends to a Dirac distribution as $\mathcal{T}/\Delta \rightarrow \infty$. A sufficient condition for ergodicity is therefore that the normalized variance of this ratio goes to zero in the limit $\mathcal{T}/\Delta \rightarrow \infty$:
\begin{equation}
\epsilon(\Delta) = \frac{\biggl< \overline{\delta x_i^2(\Delta)}^2 \biggl> - \biggl< \overline{\delta x_i^2(\Delta)} \biggl>^2}{\biggl< \overline{\delta x_i^2(\Delta)} \biggl>^2} .
\end{equation}
\begin{figure}[htb]
\centering{
\includegraphics[width=1\linewidth]{fig/ergodicity_with_sim_err_bar.png}}
\caption{{The normalized variance $\epsilon(\Delta)$ playing the role of an ergodic parameter is displayed (black dashed line) when calculated for the Ornstein-Uhlenbeck process at play in our optical trap. Experimental results (bleu open circles) for $\epsilon(\Delta)$ are compared to the theory within a $99.7\%$ confidence interval. We also show the results of a numerical simulation using $\mathcal{O}(3/2)$ algorithm (orange continuous line). The slight deviation at short times between the experiment and the theory comes again mainly from the position tracking errors whose impact on the ergodic parameter is discussed in Appendix \ref{APPENDIX_err_erg}.}}
\label{ergodicity}
\end{figure}
\vspace{3mm}
Handling therefore finite integration times, this normalized variance $\epsilon(\Delta)$ is the right observable needed to prove the ergodic nature of a stochastic process experimentally implemented. One very appealing aspect of $\epsilon(\Delta)$ is that it can be theoretically calculated for an Ornstein-Uhlenbeck process, as we do in Appendix \ref{APPENDIX_ergo}. This gives the capacity to characterize the ergodicity throughout the spectral range of the optical trap, therefore both in the long-time trapped and the short-time free diffusion regimes. These two regimes correspond to different time-lag evolutions of $\epsilon(\Delta)$, as clearly seen in Fig. \ref{ergodicity}. Here too, a smooth crossover between the long time-lag trapped (Ornstein-Uhlenbeck) regime and the short time-lag free (Wiener limit) regimes is revealed and measured, with the transition time-lag determined from the trap stiffness, as discussed in more details in Appendix \ref{APPENDIX_ergo}. The experimental evolution of $\epsilon(\Delta)$ corresponding to the recorded finite-time trajectories obtained for our trapping experiment is also shown. The excellent agreement with the theoretical $\epsilon(\Delta)$ in both the freely diffusing and in the trapped regimes confirms that our optical trapping process can be considered as ergodic with a high level of confidence. Because $\epsilon(\Delta)$ is formally a variance, the quality of its estimator on a finite-size ensemble can be quantified using a $\chi^2$-test. We perform in Fig. \ref{ergodicity} this test up to a $3\sigma$ level of confidence.
\section{Conclusion}
By implementing in a combined manner Allan variance-based, stationarity and ergodic tests, we have been able to fully characterize, through wide spectral ranges, the nature of the noise and the ergodicity of the stochastic regimes at play in our overdamped optical trap.
In particular, our observables have revealed distinctive features between the high and low frequency range of the trap.
There are clear differences from the viewpoint of noise stability and ergodicity between Wiener and Ornstein-Uhlenbeck processes notwithstanding that they are driven by the same Gaussian white thermal noise.
These differences appear in our results when comparing the different dynamical regimes.
In stochastic thermodynamics, ergodic processes are a very important subclass of stationary processes. When aiming at exploiting Brownian systems, it is therefore very important to be able to identify stationarity signatures. The simple and straightforward methodology proposed in our work is also relevant to many recent experiments involving Brownian systems coupled to non-thermal, colored, and more complex noise environments
\cite{VolpeRMP2016}.
\section{Acknowledgements}
Thanks are due to A. Canaguier-Durand and G. Schnoering for stimulating discussions. This work was supported in part by Agence Nationale de la Recherche (ANR), France, ANR Equipex Union (Grant No. ANR-10-EQPX-52-01), the Labex NIE projects (Grant No. ANR-11-LABX-0058-NIE), and USIAS within the Investissements d'Avenir program (Grant No. ANR-10-IDEX- 0002-02).
|
2,869,038,154,864 | arxiv | \section{Introduction}
It is well known that metric spaces are widely used in analysis. There are several common metric spaces, such as the numerical straight line $\mathbb{R}$, the $n$-dimensional Euclidean space $\mathbb{R}^n$, the continuous function space and the Hilbert space. Therefore, every metric space is an important kind of topological space. A function $d:X\times X \rightarrow \mathbb{R}^+$ is called a {\it metric} on a set $X$ if $d$ satisfies the following conditions for every $x, y,z\in X$:
(1) $d(x,y)=0$ if and only if $x=y$;
(2) $d(x,y)=d(y,x)$;
(3) $d(x,y)\leqslant d(x,z) + d(z,y)$.
Then set $X$ with $d$ is called a {\it metric space}, denoted by $(X,d)$.
We can obtain the generalizations of metric spaces when we weaken or modify the conditions of metric axiom. Pseudo-metrics are obtained when we change that condition `$\rho(x,y)=0$ if and only if $x=y$' into `$\rho(x,y)=0$ if $x=y$' \cite{GGS}. Quasi-metrics are defined by omitting the condition (2) \cite{14}. Symmetrics are defined by omitting the triangle inequality \cite{dc}. The ultrametric is a metric with the strong triangle inequality $d(x, y) \leqslant \max\{d(x, z), d(z, y)\}$, for $x, y,z\in X$ \cite{11}. There are many generalizations of metric spaces which have appeared in literatures (e.g. see \cite{6,4,3,9,5,CR})
Recently, Khatami and Mirzavaziri popularized the concept of metric. By extending the famous function which is called $t$-conorm, a new operation called $t$-definer is obtained. It is defined as:
\begin{definition} \label{def1} \cite[Definition 2.1]{AG} A {\it $t$-definer} is a function $\star$ : $[0,\infty)\times[0,\infty) \rightarrow [0,\infty)$ satisfying the following conditions for each $a, b,c \in [0,\infty)$:
\begin{enumerate}
\item[(T1)] $a \star b =b \star a$;
\item[(T2)] $a \star (b\star c) =(a \star b)\star c$;
\item[(T3)] if $a \leqslant b$, then $a \star c \leqslant b\star c $;
\item[(T4)] $a \star 0 = a$;
\item[(T5)] $\star$ is continuous in its first component with respect to the Euclidean topology.
\end{enumerate}
\end{definition}
When the condition (3) in the metric axiom is extended to the $\star$-triangle inequality, the following definition of $\star$-metrics can be obtained.
\begin{definition}\label{def2} \cite[Definition 2.2]{AG} Let $X$ be a nonempty set and $\star$ is a $t$-definer. If for every $x, y, z\in X$, a function $d^\star:X\times X \rightarrow [0,\infty)$ satisfies the following conditions:
\begin{enumerate}
\item[(M1)] $d^\star(x,y)=0$ if and only if $x=y$;
\item[(M2)] $d^\star(x,y)=d^\star(y,x)$;
\item[(M3)] $d^\star(x, y)\leqslant d^\star(x,z)\star d^\star(z,y)$,
\end{enumerate}
then $d^\star$ is called a {\it $\star$-metric} on $X$. The set $X$ with a $\star$-metric is called {\it $\star$-metric space}, denoted by $(X,d^\star)$.
\end{definition}
The following example shows that there are $\star$-metrics which are not metrics.
\begin{example}\cite[Example 2.4.]{AG} Clearly, $a \star b =(\sqrt{a}+\sqrt{b})^2$ is a t-definer. The function $d^\star(a,b) =(\sqrt{a}-\sqrt{b})^2$ forms an $\star$-metric which is not a metric. In fact,
\begin{equation}
\begin{aligned}
d^\star(a,b) &=(\sqrt{a}-\sqrt{b})^2=(\sqrt{a}-\sqrt{c}+ \sqrt{c}-\sqrt{b})^2\\
&\leqslant[\sqrt{(\sqrt{a}-\sqrt{c})^2}+\sqrt{(\sqrt{c}-\sqrt{b})^2}]^2\\
&=[\sqrt{d^\star(a,c)}+\sqrt{d^\star(c,b)}]^2\\
&=d^\star(a,c)\star d^\star(c,b)\nonumber
\end{aligned}
\end{equation}
while $d^\star(1,25)=16\nleqslant d^\star(1,16)+d^\star(16,25)=10.$
\end{example}
\begin{remark}
There are two most important $t$-definers:
$\bullet$ Lukasiewicz $t$-definer: $a \star_{L} b =a+b$;
$\bullet$ Maximum $t$-definer: $a \star_{m} b =\max{\{a,b\}}.$
Obviously, a $\star_L$-metric is actually a metric and a $\star_m$-metric is an ultrametric.
\end{remark}
Assume that $(M, d^\star)$ is a $\star$-metric space. For any $a\in M$ and $r > 0$, denote by
$$B_{d^\star}(a, r)=\{x\in M: d^\star(a,x)<r\}$$ and $$\mathscr{T}_{d^\star}=\{U\subseteq M: \text{~for each~}a\in U\text{~there is~}r>0 \text{~such that~}B_{d^\star}(a,r)\subseteq U\}.$$
Let $\mathscr{B}_{\frac{1}{n}}=\{B_{d^\star}(x,\frac{1}{n})\mid x\in X\}$ be a family open balls on a $\star$-metric space $(X, d^\star)$.
Khatami and Mirzavaziri proved the following result:
\begin{theorem}\cite[Theorems 3.2, 3.4, 3.5]{AG}
For every $\star$-metric space $(X, d^\star)$, the $\mathscr{T}_{d^\star}$ forms a Hausdorff topology on $X$ and the topological space $(X, \mathscr{T}_{d^\star})$ is first countable and satisfied the normal separation axiom.
\end{theorem}
The following question is posed naturally.
\begin{question}\label{q1}
Is the topological space $(X, \mathscr{T}_{d^\star})$ metrizable for every $\star$-metric space $(X, d^\star)$?
\end{question}
In this paper, we give a positive answer to Question \ref{q1}. Also, we extend the concepts of the totally boundedness and completeness in metric spaces into $\star$-metric spaces and discuss their properties.
This paper is organized as follows. Section 2 is given a positive answer to Question \ref{q1}. We obtain that let $(X, d^\star)$ be a $\star$-metric space; then the set $X$ with the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$ is metrizable (see Theorem \ref{Th1}). In Section 3 total boundedness of $\star$-metric spaces are studied. We prove that: (1) let $(X,d^\star)$ be a totally bounded $\star$-metric space; then for every subset $M$ of $X$ the $\star$-metric space $(M,d^\star)$ is totally bounded (see Theorem \ref{thm2}); (2) let $(X,d^\star)$ be a $\star$-metric space and for every subset $M$ of $X$ the space $(M,d^\star)$ is totally bounded if and only if $(\overline{M},d^\star)$ is totally bounded (see Theorem \ref{thm2.2}). We show that the Cartesian product and disjoint union of finite totally bounded $\star$-metric spaces are totally bounded under specific $\star$-metrics (see Theorems \ref{thm4} and \ref{thm5.2}). In Section 4, the completeness of $\star$-metric spaces are explored. We obtain that: (1) A $\star$-metric space $(X,d^\star)$ is compact if and only if $(X,d^\star)$ is complete and totally bounded (see Theorem \ref{thm13});
(2) A $\star$-metric space is complete if and only if for every decreasing sequence $F_{1}\supseteq F_{2}\supseteq F_{3}\supseteq\ldots$ of non-empty closed subsets of space $X$, such that $\lim_{n\rightarrow \infty }\delta(F_{n})=0$, the intersection $\bigcap_{n=1}^\infty F_{n}$ is a one-point set (see Theorem \ref{thm7});
(3) the completeness of the Cartesian product and disjoint union of complete $\star$-metric spaces under specific $\star$-metrics (see Theorems \ref{thm11} and \ref{thm11.2}).
(4) In a complete $\star$-metric space $(X,d^\star)$ the intersection $A=\bigcap_{n=1}^\infty A_{n}$ of a sequence $A_{1},A_{2},\dots$ of dense open subsets is a dense set (see Theorem \ref{thm12}).
\section{Metrizability of $\star$-metric spaces}
In this section, we shall give a positive answer to Question \ref{q1}. Let $(X,d^\star)$ be a $\star$-metric space, we shall prove that the set $X$ with the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$ is metrizable. We need to use some related symbols, terms, and preliminary facts.
Let $\mathscr{U}$ be a cover of a set $X$. For $x\in X$, denote by st($x,\mathscr{U}$)=$\bigcup\{U:U \in \mathscr{U},x\in U\}$ and st($A,\mathscr{U}$)=$\bigcup_{x\in A}\text{st}(x,\mathscr{U})$ for $A\subseteq X$. Let $\mathscr{U}$, $\mathscr{V}$ be two covers of a set $X$, we say that the cover $\mathscr{V}$ {\it refines} $\mathscr{U}$, if for every $V\in \mathscr{V}$, there exists an $U\in \mathscr{U}$ such that $V\subset U$, denoted by $\mathscr{U}< \mathscr{V}$; if $\{st(V,\mathscr{V}):V \in \mathscr{V}\} $ refines $\mathscr{U}$, then we said that $\mathscr{V}$ {\it star refines} $\mathscr{U}$, denoted by $\mathscr{U} \mathop{<}\limits_{}^*\mathscr{V}$.
\begin{definition} \cite[Definition 4.5.1]{GGS}\label{def} Let $X$ be a set and $\Phi=\{\mathscr{U}_\alpha:\alpha \in A\}$ a non-empty collection of covers of $X$ which satisfies:
\begin{enumerate}
\item[(U1)] if $\mathscr{U}$ is a cover of $X$ such that $\mathscr{U}_\alpha < \mathscr{U}$ for some $\alpha\in A$, then $\mathscr{U}\in\Phi$;
\item[(U2)] for any $\alpha, \beta\in A$, there exists an $\gamma\in A$ such that $\mathscr{U}_\gamma < \mathscr{U}_\alpha$, $\mathscr{U}_\gamma < \mathscr{U}_\beta$;
\item[(U3)] for every $\alpha\in A$, there exists an $\beta\in A$ such that $\mathscr{U}_\beta \mathop{<}\limits_{}^* \mathscr{U}_\alpha$;
\item[(U4)] if $x, y$ are different elements of $X$, then $x\notin \text{st}(y,\mathscr{U}_\alpha)$ for some $\alpha\in A$.
\end{enumerate}
Then $X$ is called a {\it uniform space} with the uniformity $\Phi$, denoted by $(X,\Phi)$. Let $X$ be a uniform space with the uniformity $\Phi=\{\mathscr{U}_\alpha:\alpha \in A\}$ and let $\{\mathscr{U}_\beta:\beta \in B\}$ be a subcollection of $\Phi$. If for every $\alpha\in A$, there exists a $\beta\in B$ such that $\mathscr{U}_\beta < \mathscr{U}_\alpha$, then the collection $\Phi'$ is called {\it a basis of the uniformity}.
Let $X$ be a uniform space with the uniformity $\Phi=\{\mathscr{U}_\alpha:\alpha \in A\}$
and
$$\mathscr{T}_{\Phi}=\{U:U\subseteq X, \text{~for each~}x\in U, \text{~there is~} \alpha \in A \text{~such that~} \text{st}(x,\mathscr{U}_{\alpha})\subset U\}.$$ Then $\mathscr{T}_{\Phi}$ is a topology on the $X$.
\end{definition}
Recalled that a topological space $X$ is said to be {\it metrizable} if there exists a metric $d$ on the set $X$ that induces the topology of $X$.
\begin{lemma}\cite[Theorem 4.5.9]{GGS}\label{lem2} Let $(X,\Phi)$ be a uniform space. Then the set with the topology $\mathscr{T}_{\Phi}$ induced by $\Phi$ is metrizable if and only if there is a base of the uniformity consisting of countably many covers.
\end{lemma}
The following theorem shows that $\star$ is continuous at the point $(0,0)$.
\begin{lemma}\label{lem3}
For $r>0$, there exists $r_{1}>0$ such that $[0,r_{1})\star[0,r_{1})\subseteq [0,r)$.
\end{lemma}
\begin{proof}
For $r>0$, we have $0 \star \frac{1}{2} r \in [0,r)$ by [Definition \ref{def1}, (T4)]. According to [Definition \ref{def1}, (T5)], there exists an $r_{0}>0$ such that $[0,r_{0})\star \frac{1}{2} r\subseteq [0,r)$. Without loss of generality, let $0\leqslant r_{0} \leqslant r$ by [Definition\ref{def1},(T3)]. Take $r_{1}=\frac{1}{2} r_{0}$. Then, we can claim that $[0,r_{1})\star[0,r_{1})\subseteq [0,r)$. For every $x, y\in [0,r_{1})$, we have that $$ x\star y\leqslant x\star \frac{1}{2} r.$$
Noting that $x\star \frac{1}{2} r \in [0,r_{1})\star \frac{1}{2} r \subseteq [0,r_{0}) \star \frac{1}{2} r \subseteq [0,r)$, we have that $x\star \frac{1}{2} r \in [0,r)$, which means $x\star \frac{1}{2} r < r$. Since $ x\star y\leqslant x\star \frac{1}{2} r $, $ x\star y < r $, therefore $x, y\in [0,r)$.
\end{proof}
The following theorem gives a positive answer to Question \ref{q1}.
\begin{theorem} \label{Th1}
Let $(X, d^\star)$ be a $\star$-metric space. Then the set $X$ with the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$ is metrizable.
\end{theorem}
\begin{proof}
First we shall show that $\mathscr{B}=\{\mathscr{B}_\frac{1}{n}:n \in \mathbb{N}\}$ is a base of a uniformity on set $X$, where $\mathscr{B}_\frac{1}{n}=\{B_{d^\star}(x,\frac{1}{n})\mid x\in X\}$. Indeed, it is enough to show that $\mathscr{B}$ satisfies (U2)$\sim$(U4) in Definition\ref{def1}.
(U2). For any $x \in X$, for any $n_{1},n_{2} \in \mathbb{N}$, take $n_{0} \in \mathbb{N}$, such that $n_{0}>n_{1}$, $n_{0}>n_{2}$. Take any $y\in B_{d^\star}(x,\frac{1}{n_{0}})$, then we have that $d^\star(x,y)< \frac{1}{n_{0}}< \frac{1}{n_{1}}$, thus $y\in B_{d^\star}(x,\frac{1}{n_{1}})$. This implies that $B_{d^\star}(x,\frac{1}{n_{0}})\subset B_{d^\star}(x,\frac{1}{n_{1}})$. Therefore $\mathscr{B}_\frac{1}{n_{0}}$ $< $ $\mathscr{B}_\frac{1}{n_{1}}$.
Similarly, we can prove that $\mathscr{B}_\frac{1}{n_{0}}< \mathscr{B}_\frac{1}{n_{2}}$. So, $\mathscr{B}$ satisfies (U2).
(U3). For any $n_{0}\in \mathbb{N}$, by Lemma \ref{lem3}, there exists an $r_{1}\in \mathbb{N}$ such that $r_{1}\star r_{1}\star r_{1}< \frac{1}{n_{0}}$. Take $n_{1}\in \mathbb{N}$ such that $\frac{1}{n_{1}}< r_{1}$. Now we shall prove that $\mathscr{B}_\frac{1}{n_{1}} \mathop{<}\limits_{}^* \mathscr{B}_\frac{1}{n_{0}}$. Hence the proof is completed once we show that st($B_{d^\star}(x,\frac{1}{n_{1}})$, $\mathscr{B}_\frac{1}{n_{1}}$)$\subseteq B_{d^\star}(x,\frac{1}{n_{0}})$, for any $x\in X$. Take any $y\in X$ such that $B_{d^\star}(y,\frac{1}{n_{1}}) \cap B_{d^\star}(x,\frac{1}{n_{1}})\neq \emptyset$. Then, there exists an $z_{1}\in B_{d^\star}(y,\frac{1}{n_{1}}) \cap B_{d^\star}(x,\frac{1}{n_{1}})$, for any $z_{2}\in B_{d^\star}(y,\frac{1}{n_{1}})$, we have
\begin{equation}
\begin{aligned}
d^\star(z_{2},x) &\leqslant d^\star(z_{2},y)\star d^\star(y,z_{1})\star d^\star(z_{1},x)\\
&<\frac{1}{n_{1}}\star \frac{1}{n_{1}}\star \frac{1}{n_{1}}<r_{1}\star r_{1}\star r_{1}\\
&< \frac{1}{n_{0}}.\nonumber
\end{aligned}
\end{equation}
Therefore, $z_{2}\in B_{d^\star}(y,\frac{1}{n_{0}})$, $\mathscr{B}$ satisfies (U3).
(U4). For $x,y \in X$ and $x\neq y $, let $d^\star(x,y)=r$ , then $r>0$. Take $n_{1} \in \mathbb{N}$, such that $\frac{1}{n_{1}}< r$, then we have $y \notin B_{d^\star}(x,\frac{1}{n_{1}})$. By Lemma \ref{lem3}, there exists an $n_{0} \in \mathbb{N}$ such that $\frac{1}{n_{0}}\star\frac{1}{n_{0}}< \frac{1}{n_{1}}$. Then, we claim that for any $B_{d^\star}(z_{0},\frac{1}{n_{0}})\in \mathscr{B}_\frac{1}{n_{0}}$ $(z_{0} \in X)$, $B_{d^\star}(z_{0},\frac{1}{n_{0}})$ can not contain both $x$ and $y$. Since, if $x,y\in B_{d^\star}(z_{0},\frac{1}{n_{0}})$, then $d^\star(x,y)\leqslant d^\star(x,z_{0}) \star d^\star(z_{0},y) < \frac{1}{n_{0}}\star\frac{1}{n_{0}} < \frac{1}{n_{1}} < r$, which is contradictory to $d^\star(x,y)=r$. So, $\mathscr{B}$ satisfies (U4).
Thus we have proved that $\mathscr{B}$ is a base of a uniformity on $X$, denote by $\Phi_{d^\star}$. According to Lemma \ref{lem2}, the set $X$ with the topology $\mathscr{T}_{\Phi_{d^\star}}$ induced by the uniformity $\Phi_{d^\star}$ is metrizable. Therefore, to complete the proof, it is enough to prove that the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$ is the same as the topology $\mathscr{T}_{\Phi_{d^\star}}$.
(1) For any $U\in \mathscr{T}_{d^\star}$, $x\in U$, there exists $n \in \mathbb{N}$ such that $B_{d^\star}(x,\frac{1}{n})\subset U$. By Lemma \ref{lem3}, there exists an $n_{0} \in \mathbb{N}$ such that $\frac{1}{n_{0}}\star\frac{1}{n_{0}}< \frac{1}{n}$. Then we shall prove that $\text{st}(x,\mathscr{B}_{\frac{1}{n_{0}}}) \subset B_{d^\star}(x,\frac{1}{n})\subset U$. Take any $z\in \text{st}(x,\mathscr{B}_{\frac{1}{n_{0}}})$, then there exists $y \in X$ such that $x\in \text{st}(y,\mathscr{B}_{\frac{1}{n_{0}}})$ and $z\in \text{st}(y,\mathscr{B}_{\frac{1}{n_{0}}})$. Since,
\begin{equation}
\begin{aligned}
d^\star(x,z) &\leqslant d^\star(x,y)\star d^\star(y,z)<\frac{1}{n_{0}}\star\frac{1}{n_{0}}<\frac{1}{n},\nonumber
\end{aligned}
\end{equation}
we have $z\in \text{st}(x,\mathscr{B}_{\frac{1}{n}})$, i.e. $\text{st}(x,\mathscr{B}_{\frac{1}{n_{0}}}) \subset U$, so $U\in \mathscr{T}_{\Phi_{d^\star}}$. Therefore $ \mathscr{T}_{d^\star} \subseteq \mathscr{T}_{\Phi_{d^\star}}$.
(2) For any $U\in \mathscr{T}_{\Phi_{d^\star}}$, $x\in U$, there exists $n \in \mathbb{N}$ such that $\text{st}(x,\mathscr{B}_{\frac{1}{n}})\subset U$. Since $B_{d^\star}(x,\frac{1}{n})\subset\text{st}(x,\mathscr{B}_{\frac{1}{n}}) \subset U$, we have $U\in \mathscr{T}_{d^\star}$, i.e. $ \mathscr{T}_{\Phi_{d^\star}}\subseteq \mathscr{T}_{d^\star}$. Then we get that $ \mathscr{T}_{\Phi_{d^\star}}= \mathscr{T}_{d^\star}$.
\end{proof}
Obviously, since $\star$-metric spaces are metrizable, we have the following two corollaries.
\begin{corollary} Let $(X,d^\star)$ be a $\star$-metric space and $X$ with the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$. Then the following statements equivalent:
\begin{enumerate}
\item[(1)] X has a countable basis;
\item[(2)] X is Lindel\"{o}f;
\item[(3)] every closed discrete subspace is a countable set;
\item[(4)] every discrete subspace is a countable set;
\item[(5)] every collection of disjoint open sets in X is countable;
\item[(6)] X has a countable dense subset.
\end{enumerate}
\end{corollary}
Recalled that a topological space $X$ is called {\it pseudo-compact} if every real valued continuous function on $X$ is bounded; $X$ is called {\it countably compact} if every countable open cover of $X$ has a finite subcover; $X$ is called {\it sequentially compact} if every sequence of points of $X$ has a convergent subsequence. A point $x$ is called {\it $\omega$-accumulation point} of $X$ if any neighborhood of point $x$ contains infinite points of $X$.
\begin{corollary} \label{col1} Let $(X,d^\star)$ be a $\star$-metric space and $X$ with the topology $\mathscr{T}_{d^\star}$ induced by $d^\star$. Then the following statements equivalent:
\begin{enumerate}
\item[(1)] X is pseudo-compact;
\item[(2)] every infinite subset of X has cluster point;
\item[(3)] every infinite subset of X has $\omega$-accumulation point;
\item[(4)] every sequence of X has cluster point;
\item[(5)] X is countably compact;
\item[(6)] X is sequentially compact;
\item[(7)] X is compact.
\end{enumerate}
\end{corollary}
\section{Total boundedness of $\star$-metric spaces}
Total boundedness is an important property in metric spaces. We generalize the concept of totally bounded into $\star$-metric spaces and study their properties. Now we need to give some related definitions.
\begin{definition}
A $\star$-metric space $(X,d^\star)$ is {\it totally bounded}, if for any $\epsilon >0$ there exists a finite set $F(\epsilon)\subseteq X$ such that $X=\bigcup_{x\in F(\epsilon)}B_{d^\star}(x,\epsilon)$. We also said that the finite set $F(\epsilon)$ is
{\it $\epsilon$-dense} in $X$.
\end{definition}
\begin{theorem}\label{thm3} Let $(X,d^\star)$ be a $\star$-metric space and every infinite subset of $X$ have an $\omega$-accumulation point in the topological space $X$ with the topology induced by $d^ \star$. Then $(X,d^\star)$ is totally bounded.
\end{theorem}
\begin{proof}
Suppose the contrary that there exists $\epsilon_{0}$, but there is no finite set $F(\epsilon_{0})$, so that $X=\bigcup_{x\in F(\epsilon_{0})}B_{d^\star}(x,\epsilon_{0})$ holds. Take any $x_{1}\in X$, then $X\neq B_{d^\star}(x_{1},\epsilon_{0})$; take any $x_{2}\in X-B_{d^\star}(x_{1},\epsilon_{0})$, since $X\neq \bigcup_{i=1}^2 B_{d^\star}(x_{i},\epsilon_{0})$; Repeat this procedure, we obtain an infinite set $\{x_{1},x_{2},\dots ,x_{n},\dots\}$. According to the above operation, we can get that $d^\star(x_{i},x_{j})\geqslant \epsilon_{0}$ $(i\neq j)$. By assuming, this infinite set has an $\omega$-accumulation point $x_{0}\in X$. Thus, by Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $\epsilon_{1}\star \epsilon_{1}<\epsilon_{0}$. Then the open-ball $B_{d^\star}(x_{0},\epsilon_{1})$ should contain infinite points of $\{x_{1},x_{2},\dots ,x_{n},\dots\}$. Let $x_{n},x_{m}\in B_{d^\star}(x_{0},\epsilon_{1})$, then $$d^\star(x_{m}, x_{n}) \leqslant d^\star(x_{m}, x_{0})\star d^\star(x_{0}, x_{n})< \epsilon_{1}\star \epsilon_{1}< \epsilon_{0}.$$ This contradicts $d^\star(x_{i},y_{j})\geqslant \epsilon_{0}$. Therefore $(X,d^\star),$ is totally bounded.
\end{proof}
By Corollary \ref{col1} and Theorem \ref{thm3}, we have the following corollary:
\begin{corollary} \label{col2}
Let $(X,d^\star)$ be a $\star$-metric space. If $X$ with the topology induced by $d^ \star$ is countably compact, then $(X,d^\star)$ is a totally bounded $\star$-metric space.
\end{corollary}
One can easily verify that for every subset $M\subseteq X$ of a $\star$-metric space $(X,d^\star)$, $(M, d^\star\mid_{M\times M})$ is a $\star$-metric space, where $d^\star\mid_{M\times M}$ is the restriction of the $\star$-metric $d^\star$ on $X$ to the subset $M$. The $\star$-metric space $(M, d^\star\mid_{M\times M})$ will also be denoted by $(M,d^\star)$.
\begin{theorem}\label{thm2} Let $(X,d^\star)$ be a totally bounded $\star$-metric space. Then for every subset $M$ of $X$ the $\star$-metric space $(M,d^\star)$ is totally bounded.
\end{theorem}
\begin{proof} Let $(X,d^\star)$ be a $\star$-metric space, $M\subset X$. For every $\epsilon> 0$, by Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $\epsilon_{1}\star \epsilon_{1}<\epsilon$. Take a finite set $F(\epsilon_{1})=\{x_{1},x_{2},\dots ,x_{k}\}$ which is $\epsilon_{1}$-dense in $X$. Let $\{x_{m_{1}},x_{m_{2}},\dots ,x_{m_{l}}\}$ be the subset of $F(\epsilon_{1})$ consisting of all points which satisfy that $d^\star(x,x_{i})< \epsilon_{1}$, for each $x\in M$ and $x_{i}\in F(\epsilon_{1})$.
Let $F'=\{x'_{1},x'_{2},\dots ,x'_{l}\}$ be a finite set satisfying $d^\star(x'_{j},x_{m_{j}})< \epsilon_{1}$ for $j\leqslant l$. We shall show that the set $F'$ is $\epsilon$-dense in $M$. Let $x\in M$, there exists $x_{i}\in F(\epsilon_{1})$ such that $d^\star(x,x_{i})< \epsilon_{1}$. Hence $x_{i}=x_{m_{j}}$ for some $j\leqslant l$, then we have $$d^\star(x, x'_{j}) \leqslant d^\star(x, x_{m_{j}})\star d^\star(x_{m_{j}}, x'_{j})< \epsilon_{1}\star \epsilon_{1}< \epsilon.$$
\end{proof}
\begin{theorem}\label{thm2.2} Let $(X,d^\star)$ be a $\star$-metric space and for every subset $M$ of $X$ the space $(M,d^\star)$ is totally bounded if and only if $(\overline{M},d^\star)$ is totally bounded.
\end{theorem}
\begin{proof}
Assume that $(M,d^\star)$ is totally bounded. For $\epsilon> 0$, Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $\epsilon_{1}\star \epsilon_{1}<\epsilon$, and take a finite set $F(\epsilon_{1})=\{x_{1},x_{2},\dots ,x_{k}\}$ which is $\epsilon_{1}$-dense in $M$. For each $x\in \overline{M}$ , we have $B_{d^\star}(x,\epsilon_{1})\cap M\neq \emptyset $. Take $y\in B_{d^\star}(x,\epsilon_{1})\cap M$ there exists $x_{i}\in F(\epsilon_{1})$ such that $d^\star(y,x_{i})< \epsilon_{1}$, then we have $d^\star(x, x_{i}) \leqslant d^\star(x, y)\star d^\star(y, x_{i})< \epsilon_{1}\star \epsilon_{1}< \epsilon.$
On the other hand, assume that $(\overline{M},d^\star)$ is totally bounded. One can easily get that $(M,d^\star)$ is totally bounded by Theorem \ref{thm2}, because $M$ is the subset of $\overline{M}$.
\end{proof}
\begin{corollary}
If the $\star$-metric space $(X,d^\star)$ has a dense totally bounded subspace, then $(X,d^\star)$ is totally bounded.
\end{corollary}
Let $\{(X_{i},d_{i}^\star)\}_{i=1}^n $ be a family of finite nonempty $\star$-metric spaces. Consider the Cartesian product $X=\prod_{i=1}^{n}X_{i}$ and for every pair $x=(x_i)_{i\leq 1\leq n}$, $y=(y_i)_{i\leq 1\leq n}$ of points of $X$ let
$$d_{T}^\star(x,y)=d_{1}^\star(x_{1},y_{1})\star d_{2}^\star(x_{2},y_{2})\star \dots \star d_{n}^\star(x_{n},y_{n})\quad\quad\quad\quad\quad\quad(3.1)$$
and
$$d_{\max}^\star(x,y)=\max_{1\leqslant i\leqslant n} d_{i}^\star(x_{i},y_{i})\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(3.2).$$
In \cite[Theorem 4.3]{AG}, Khatami and Mirzavaziri proved that the formulas (3.1) and (3.2) define two $\star$-metrics on the Cartesian product $X=\prod_{i=1}^{n}X_{i}$. Furthermore the induced topology of these two $\star$-metrics on $X$ is the same as the product topology on $X$.
\begin{theorem}\label{thm4} Let $\{(X_{i},d_{i}^\star)\}_{i=1}^n $ be a family of finite nonempty $\star$-metric spaces and $X=\prod_{i=1}^{n}X_{i}$ the Cartesian product. Then:
\begin{enumerate}
\item[(1)] $X$ with the $\star$-metric $d_{T}^\star$ defined by formula (3.1) is totally bounded if and only if all $\star$-metric spaces $(X_{i},d_{i}^\star)$ are totally bounded;
\item[(2)] $X$ with the $\star$-metric $d_{max}^\star$ defined by formula (3.2) is totally bounded if and only if all $\star$-metric spaces $(X_{i},d_{i}^\star)$ are totally bounded.
\end{enumerate}
\end{theorem}
\begin{proof}
(1). Necessity. Assume that the $\star$-metric space $(X,d_{T}^\star)$ is totally bounded. The subset $X_{m}^*=\prod_{i=1}^{n}A_{i}$ of $X$ , where $A_{m}=X_{m}$ and $A_{i}=\{x_{i}^*\}$ is a one-point subset of $X_{i}$ for $i \neq m$. Then the subspace $X_{m}^*$ is totally bounded by Theorem \ref{thm2}. One can easily verify that $p_{m}^*=p_{m}\mid _{X_{m}^*}:X_{m}^*\rightarrow X_{m}$ is a isometric isomorphism and according to the definition of $d_{T}^\star$, for $x^*,y^*\in X_{m}^*\subset X$, $d_{T}^\star(x^*,y^*)=d_{m}^\star(p_{m}(x^*),p_{m}(y^*))$. Therefore, if a finite set $F$ is $\epsilon$-dense in $(X_{m}^*,d_{T}^\star)$, then $p_{m}(F)$ is $\epsilon$-dense in $(X_{m},d_{T}^\star)$, and from this it follows further that $(X_{m},d_{T}^\star)$ is totally bounded.
Sufficiency. Let every $(X_{i},d_{i}^\star)$ is totally bounded. For $\epsilon> 0$, by Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $ \overbrace{\epsilon_{1}\star \epsilon_{1}\star \dots \star \epsilon_{1} }^{n \text{ times}}<\epsilon$. For every $i\leqslant n$ take a finite set $F_{i}$ which is $\epsilon_{1}$-dense in $X_{i}$. We define that $$F=\prod_{i=1}^{n}F_{i},$$ then $F$ is a finite set. To conclude the proof it suffices to show that $F$ is $\epsilon$-dense in the space $(X,d_{T}^\star).$ Let $x=(x_{1},x_{2},\dots ,x_{n})$ be an arbitrary point of $X$. For every $i\leqslant n$, since $F_{i}$ is $\epsilon$-dense in $X_{i}$, there exists a $y_{i} \in F_{i}$ such that $d_{i}^\star(x_{i},y_{i})< \epsilon_{1}$ and take a point $y=(y_{1},y_{2},\dots ,y_{n}) \in F$ we have $$d_{T}^\star(x,y)=d_{1}^\star(x_{1},y_{1})\star d_{2}^\star(x_{2},y_{2})\star \dots \star d_{n}^\star(x_{n},y_{n})<\overbrace{\epsilon_{1}\star \epsilon_{1}\star \dots \star \epsilon_{1} }^{n \text{ times}}<\epsilon.$$ By the foregoing, $F$ is $\epsilon$-dense in $(X,d_{T}^\star)$.
(2). Necessity. Assume that the $\star$-metric space $(X,d_{max}^\star)$ is totally bounded. Then the proof method is the same as that of necessity in (1).
Sufficiency. Let every $(X_{i},d_{i}^\star)$ is totally bounded. For $\epsilon> 0$, take a finite set $F_{i}$ which is $\epsilon$-dense in $X_{i}$, for every $i\leqslant n$. We define that $$F=\prod_{i=1}^{n}F_{i},$$ then $F$ is a finite set. To conclude the proof it suffices to show that $F$ is $\epsilon$-dense in the space $(X,d_{\max}^\star)$. Let $x=(x_{1},x_{2},\dots ,x_{n})$ be an arbitrary point of $X$. For every $i\leqslant n$, there exists a $y_{i}\in F_{i}$ such that $d_{i}^\star(x_{i},y_{i})< \epsilon$ and take a point $y=(y_{1},y_{2},\dots ,y_{n}) \in F$. Without loss of generality, choose $\max_{1\leqslant i\leqslant n} d_{i}^\star(x_{i},y_{i})=d_{k}^\star(x_{k},y_{k})$, then we have $$d_{\max}^\star(x,y)=\max_{1\leqslant i\leqslant n}d_{i}^\star(x_{i},y_{i})=d_{k}^\star(x_{k},y_{k})< \epsilon.$$ By the foregoing, $F$ is $\epsilon$-dense in $(X,d_{\max}^\star)$.
This completes the proof.
\end{proof}
Let $(X,d)$ be a metric space. Define $\tilde{d}(x,y)=\text{min}\{1,d(x,y)\}$ for each $x,y\in X$. It is well known that $\tilde{d}$ is a metric on $X$ such that the topology induced by $\tilde{d}$ is the same as induced by $d$. For $\star$-metric space, we have the following result.
\begin{proposition}\label{thm1} Let $(X,d^\star)$ be a $\star$-metric space and define $ \tilde{d^\star}(x,y)=\text{min}\{1,d^\star(x,y)\}$ for each $x,y\in X$. Then $(X,\tilde{d^\star})$ is also a $\star$-metric space on $X$. Furthermore the topology induced by $\tilde{d^{\star}}$ on $X$ is the same as induced by $d^{\star}$.
\end{proposition}
\begin{proof}
We shall verify that $\tilde{d^\star}$ is a $\star$-metric. Clearly, $\tilde{d^\star}$ satisfies (M1) and (M2) in the definition \ref{def2}. Suppose the contrary that exists $x,y,z\in X$ such that $$1\geqslant \tilde{d^\star} (x,z) > \tilde{d^\star} (x,y) \star \tilde{d^\star} (y,z),$$
then, $\tilde{d^\star} (x,y)<1$, $\tilde{d^\star} (y,z)<1$ (since, if $\tilde{d^\star}(x,y)\geqslant 1$, then $ \tilde{d^\star} (x,y)\star \tilde{d^\star} (y,z)\geqslant 1\star 0=1$, i.e $\tilde{d^\star} (x,z)>1$). Therefore $$\tilde{d^\star} (x,y)\star \tilde{d^\star}(y,z)=d^\star(x,y) \star d^\star(y,z)\geqslant d^\star(x,z).$$ This implies that $\tilde{d^\star} (x,z)>d^\star(x,z)$, which is a contradiction with $\tilde{d^\star} (x,z) \leqslant d^\star(x,z)$. Thus $(X,\tilde{d^\star})$ is a $\star$-metric space.
For any $\epsilon >0$, $x\in X$ we define $$B_{d^\star}(x,\epsilon)=\{y \in X:d^\star(x,y)< \epsilon\},$$ $$B_{\tilde{d^\star}}(x,\epsilon)=\{y \in X:d^\star(x,y)< \epsilon\}.$$
When $0< \epsilon <1$, $B_{d^\star}(x,\epsilon)=B_{\tilde{d^\star}}(x,\epsilon)$. Thus the topology induced by $\tilde{d^{\star}}$ on $X$ is the same as induced by $d^{\star}$. This completes the proof.
\end{proof}
Let $\{(X_{\alpha},d^\star_{\alpha})\}_{\alpha \in A}$ be a family of $\star$-metric spaces and $X=\bigoplus_{\alpha\in A}X_{\alpha}$ be the disjoint union of $\{X_{\alpha}\}_{\alpha \in A}$. By Proposition \ref{thm1}, one can suppose that $d^\star_{\alpha}(x,y)\leqslant 1$ for $x,y\in X_{\alpha}$ and $\alpha \in A$. For every $x,y\in X$, we define
$$d^\star_{q}(x,y)=
\begin{cases}
d^\star_{\alpha}(x,y),& \text{if $x,y\in X_{\alpha}$ for some $\alpha \in A$},\\
1,& \text{otherwise}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(3.3)
\end{cases}$$
Then $(X,d^\star_{q})$ is a $\star$-metric space.
Obviously, $d^\star_{q}$ satisfies conditions (M1) and (M2). It remains to show that condition (M3) $d^\star_{q}(x,z)\leqslant d^\star_{q}(x,y)\star d^\star_{q}(y,z)$ is also satisfied. Otherwise, if there exists $x,y,z\in X$, such that $1\geqslant d^\star_{q}(x,z)> d^\star_{q}(x,y)\star d^\star_{q}(y,z)$, then $d^\star_{q}(x,y)< 1$, $d^\star_{q}(y,z)<1$. Since, if $d^\star_{q}(x,y)\geqslant 1$, then $d^\star_{q}(x,y)\star d^\star_{q}(y,z)\geqslant 1\star 0=1$, i.e. $d^\star_{q}(x,z)>1$. This implies that $d^\star_{q} (x,z)>d^\star_{\alpha}(x,z)$, which is a contradiction with $d^\star_{q}(x,z) \leqslant d^\star_{\alpha}(x,z)$. Thus, there exists $\alpha \in A$ such that $x,y,z\in X_{\alpha}$, then we have $$d^\star_{q}(x,y)\star d^\star_{q}(y,z)=d^\star_{\alpha}(x,y)\star d^\star_{\alpha}(y,z)\geqslant d^\star_{\alpha}(x,z)=d^\star_{q}(x,z),$$ contradiction.
One can easily show that for every ${\alpha\in A}$, the set $X_{\alpha}$ is open in the space $X$ with the topology induced by $d^\star_{q}$. Since $d^\star_{\alpha}$ induces the topology on $X_{\alpha}$, then $d^\star_{q}$ induces on $X$ the topology of the disjoint union of topological spaces $\{X_{\alpha}\}_{\alpha\in A}$.
\begin{theorem}\label{thm5.2}
Let $\{(X_{i},d_{i}^\star)\}_{i=1}^n$ be a family of $\star$-metric spaces such that the metric $d_{i}^\star$ is bounded by $1$ for $1\leqslant i\leqslant n$, and $X=\bigoplus_{1\leqslant i\leqslant n} X_{i}$ the disjoint union of $\{X_{i}\}_{i\leq n}$. Then $(X,d^\star_{q})$ is totally bounded if and only if all spaces $(X_{i},d_{i}^\star)$ are totally bounded, where $d^\star_{q}$ is defined as the formula $(3.3)$.
\end{theorem}
\begin{proof}
Necessity. Assume that $(X,d^\star_{q})$ is totally bounded. One can easily get that $(X_{i},d_{i}^\star)$ is a subspace of $(X,d^\star_{q})$. So, all spaces $(X_{i},d_{i}^\star)$ are totally bounded by Theorem \ref{thm2}.
Sufficiency. Assume that all spaces $(X_{i},d_{i}^\star)$ are totally bounded. Then, for $\epsilon>0$, for any $x\in X_{i}$, there exists $y_{0}\in F_{i}(\epsilon)$ such that $d_{i}^\star(x,y_{0})<\epsilon$. Put $F(\epsilon)=\bigcup_{i=1}^n F_{i}(\epsilon)$, let $x$ be an arbitrary point of $X$, obviously, $x$ is also a point on some $X_{i}$. Thus, one can easily find a $y_{0}\in F_{i}(\epsilon)$, such that $d^\star_{q}(x,y_{0})=d_{i}^\star(x,y_{0})<\epsilon$. So, $(X,d^\star_{q})$ is totally bounded.
\end{proof}
\section{the completeness of $\star$-metric spaces}
Completeness is an important property in metric spaces. The completeness of metric spaces depend on the convergence of Cauchy sequences. Therefore, we extend the definition of Cauchy sequences and completeness in metric spaces to $\star$-metric spaces. Further, we study completeness properties of $\star$-metric spaces and give their characterization.
\begin{definition} Let $\{x_{n}\}_{n \in \mathbb{N}}$ be a sequence of a $\star$-metric space $(X,d^\star)$, and $x\in X$. If for every $\epsilon >0$, there exists ${k \in \mathbb{N}}$ such that $d^\star(x,x_{n})< \epsilon$ whenever $n\geqslant k$, then the sequence $\{x_{n}\}_{n \in \mathbb{N}}$ is said to {\it converge to $x$ under $d^\star$}, and we write $x_{n} \stackrel{d^\star}{\longrightarrow}x$.
\end{definition}
\begin{proposition} Let $(X,d^\star)$ be a $\star$-metric space. Then the following statements equivalent:
\begin{enumerate}
\item[(1)] $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $ \mathscr{T}_{d^\star}$;
\item[(2)] $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $ d^\star$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2). For every $\epsilon >0$, clearly, $B_{d^\star}(x_{0},\epsilon)$ is a neighborhood of $x_{0}$. Since $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $ \mathscr{T}_{d^\star}$, there exists ${k \in \mathbb{N}}$ such that $x_{n} \in B_{d^\star}(x_{0},\epsilon)$ whenever $n\geqslant k$, i.e. $d^\star(x_{n},x_{0})< \epsilon$. Therefore $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $ d^\star$.
(2) $\Rightarrow$ (1). For any neighborhood $U$ of the point $x_{0}$, there exists $\epsilon >0$ such that $B_{d^\star}(x_{0},\epsilon)\subset U$. Since $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $ d^\star$, there exists $k \in \mathbb{N}$ such that $d^\star(x_{n},x_{0})< \epsilon$ whenever $n\geqslant k$, i.e. $x_{n} \in B_{d^\star}(x_{0},\epsilon)$. Thus $x_{n}\in U$ whenever $n\geqslant k$. Therefore $\{x_{n}\}_{n \in \mathbb{N}}$ converges to $x_{0}$ under $\mathscr{T}_{d^\star}$.
\end{proof}
\begin{definition}
Let $(X,d^\star)$ be a $\star$-metric space, the sequence $\{x_{n}\}$ is called {\it Cauchy sequence} in $(X,d^\star)$ if for every $\epsilon>0$ there exists $k \in \mathbb{N}$ such that $d^\star(x_{n},x_{m})< \epsilon$ whenever $m, n\geqslant k$.
\end{definition}
\begin{proposition}\label{thm6} Let $\{x_{n}\}$ a Cauchy sequence in $\star$-metric space $(X,d^\star)$. If $\{x_{n}\}$ has a accumulation point $x_{0}$, then $x_{n} \stackrel{d^\star}{\longrightarrow}x$.
\end{proposition}
\begin{proof}
For every $\epsilon> 0$, by Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $\epsilon_{1}\star \epsilon_{1}<\epsilon$. Since $\{x_{n}\}$ is Cauchy sequence, there exists $k_{1} \in \mathbb{N}$ such that $d^\star(x_{n},x_{m})< \epsilon_{1}$ whenever $m, n\geqslant k_{1}$. Noting that $x_{0}$ is the accumulation point of $\{x_{n}\}$, there exists $k_{2}\in \mathbb{N}$ such that $d^\star(x_{0},x_{n})< \epsilon_{1}$ whenever $n \geqslant k_{2}$. Therefore, choose $k=\max\{k_{1},k_{2}\}$, while $m \geqslant k$, we have $$d^\star(x_{0}, x_{m}) \leqslant d^\star(x_{0}, x_{n})\star d^\star(x_{n}, x_{m})< \epsilon_{1}\star \epsilon_{1}< \epsilon.$$ This shows that $\{x_{n}\}$ converges to $x_{0}$.
\end{proof}
\begin{definition}
A $\star$-metric space $(X,d^\star)$ is {\it complete} if every Cauchy sequence in $(X,d^\star)$ is convergent to a point of $X$.
\end{definition}
\begin{theorem}\label{thm13}
A $\star$-metric space $(X,d^\star)$ is compact if and only if $(X,d^\star)$ is complete and totally bounded.
\end{theorem}
\begin{proof}
Necessity. Let $(X,d^\star)$ be a compact $\star$-metric space. According to Corollary \ref{col2}, $(X,d^\star)$ is totally bounded. According to Proposition \ref{thm6}, if a Cauchy sequence in space $(X,d^\star)$ has convergent subsequences, then this Cauchy sequence converges. Since compact $\star$-metric space is sequentially compact, which means that every sequence of points of $X$ has a convergent subsequence. By Corollary \ref{col1}, every Cauchy sequence in $(X,d^\star)$ is convergent to a point of $X$. This implies that $(X,d^\star)$ is complete.
Sufficiency. Let $(X,d^\star)$ be a complete and totally bounded $\star$-metric space. To conclude the proof it suffices to show that $X$ is sequentially compact which implies that $X$ is compact, by Corollary \ref{col1}.
Let $\{x_{n}\}$ be any sequence in $\star$-metric space $(X,d^\star)$. From the total boundedness of space $X$, there exists finite open-balls cover $X$ with radius $1$. At least one of the finite open-ball $B_{d^\star}^1$ contains infinite points $x_{n}$ in sequence $\{x_{n}\}$. Let the set formed by the subscript $n$ of $x_{n}$ contained in $B_{d^\star}^1$ be $N_{1}$, then $N_{1}$ is an infinite set, such that $x_{n}\in B_{d^\star}^1$ whenever $n \in N_{1}$. Then use finite open-balls to cover $X$ with radius $1/2$. Among these finite open-balls, there must be at least one open-ball $B_{d^\star}^2$ and an infinite subset $N_{2}$ of $N_{1}$, such that $x_{n}\in B_{d^\star}^2$ whenever $n \in N_{2}$. Generally speaking, taking the infinite subset $N_{k}$ of the positive integer set, we can select the open-ball $B_{d^\star}^{k+1}$ with radius of 1/(k+1) and the infinite set $N_{k+1}\subset N_{k}$, such that $x_{n}\in B_{d^\star}^{k+1}$ whenever $n \in N_{k+1}$.
Take $n_{1}\in N_{1}$, $n_{2}\in N_{2}$, which $n_{2}> n_{1}$. Generally speaking, $n_{k}$ has been taken, we can choose $n_{k+1}\in N_{k+1}$ so that $n_{k+1}> n_{k}$. Since for every $N_{k}$ is infinite set, the above method can be completed. For $i,j \geqslant k$, we have $n_{i},n_{j}\in N_{k}$ such that $x_{n_{i}},x_{n_{j}}\in B_{d^\star}^k$. This implies that $\{x_{n_{k}}\}$ is a Cauchy sequence, and it is convergent by completeness of $(X,d^\star)$.
\end{proof}
The {\it distance $D(x, A)$ from a point to a set $A$} in a $\star$-metric space $(X,d^\star)$ is defined by letting $$D(A,x)=D(x, A)=\inf_{y \in A}\{d^\star(x,y)\}, ~if~ A\neq\emptyset, ~and~ D(x,\emptyset)=D(\emptyset, x)=1.$$
\begin{proposition}\label{thm8} Let $(X,d^\star)$ be a $\star$-metric space, and $A\subset X$. Then $\overline{A}=\{x:D(A,x)=0\}$.
\end{proposition}
\begin{proof}
For any $x_{0}\in \overline{A}$ there exists $\{x_{n}\}_{n \in \mathbb{N}}\subset A$ such that $x_{n} \stackrel{d^\star}{\longrightarrow} x_{0}$. This implies that $d^\star(x_{n},x_{0})\rightarrow 0$. Since $0\leqslant d^\star(x_{0},A)\leqslant d^\star(x_{n},x_{0})\rightarrow 0$, we have that $d^\star(x_{0},A)=0$, which implies that $x_{0}\in \{x:D(A,x)=0\}$. Therefore $\overline{A}\subseteq \{x:D(A,x)=0\}$.
Suppose the contrary that take $y\in \{x:D(A,x)=0\}$ which satisfies $d^\star(y,A)=0$ and $y\notin \overline{A}$. Then there exists $\epsilon_{0} >0$ such that $B_{d^\star}(y,\epsilon_{0})\cap A=\emptyset$, which implies that $d^\star(y,A)\geqslant \epsilon_{0}$. This is a contradiction with $d^\star(y,A)=0$. Thus $\overline{A}\supseteq \{x:D(A,x)=0\}$.
This shows that $\overline{A}= \{x:D(A,x)=0\}$.
\end{proof}
\begin{definition}
Let $A$ be a subset of $\star$-metric space $(X,d^\star)$. We define $\delta(A)=\sup_{x,y\in A}\{d^\star(x,y)\}$ as the {\it diameter} of the set $A$; it can be finite or equal to $\infty$. We also define $\delta(\emptyset)=0$.
\end{definition}
Cantor theorem is an important characterization of complete metric spaces. Similarly, we extend the Cantor theorem in metric spaces into $\star$-metric spaces.
\begin{theorem}\label{thm7}
A $\star$-metric space is complete if and only if for every decreasing sequence $F_{1}\supseteq F_{2}\supseteq F_{3}\supseteq\ldots$ of non-empty closed subsets of space $X$, such that $\lim_{n\rightarrow \infty }\delta(F_{n})=0$, the intersection $\bigcap_{n=1}^\infty F_{n}$ is a one-point set.
\end{theorem}
\begin{proof}
Necessity. Let $(X,d^\star)$ be a complete $\star$-metric space, and $F_{1}, F_{2},\dots$ a sequence of non-empty closed subsets of $X$ such that $$\lim_{n\rightarrow \infty }\delta(F_{n})=0 ~and~ F_{n+1}\subset F_{n}~ for~ n=1,2,\dots$$ Choose $x_{n}\in F_{n}$, for every $n\in \mathbb{N}$. Now we shall prove that $\{x_{n}\}$ is a Cauchy sequence. According to $\lim_{n\rightarrow \infty }\delta(F_{n})=0$, for $\epsilon >0$, there exists a $k \in \mathbb{N}$ such that $\delta(F_{n})< \epsilon$ where $ n> k$. When $n\geqslant m>k$, we have $x_{n}\in F_{n}\subset F_{m}$ since $\{F_{n}\}$ is a decreasing sequence. Furthermore $x_{m}\in F_{m}$, so that $$d^\star(x_{n},x_{m})< \delta(F_{m})<\epsilon .$$ So, $\{x_{n}\}$ is a Cauchy sequence and thus is convergent to a point $x_{0}\in X$. Thus, any neighborhood of $x_{0}$ intersects $F_{n}$ $(n=1,2,\dots)$. The sets $F_{n}$ being closed, we have $x_{0}\in \bigcap_{n=1}^\infty F_{n}$
Now, we need proof $\bigcap_{n=1}^\infty F_{n}=\{x_{0}\}$. If there is an $y\in \bigcap_{n=1}^\infty F_{n}$, by $lim_{n\rightarrow \infty }\delta(F_{n})=0$, choose a $k\in \mathbb{N}$ such that $\delta(F_{n})< \epsilon$ where $ n> k$. Then we have $$d^\star(x_{0},y)< \delta(F_{n})<\epsilon, ~where~ x_{0}, y\in F_{n}.$$ Arbitrariness by $\epsilon$, $d^\star(x_{0},y)=0$, so $x_{0}=y$.
Sufficiency. Let $\{x_{n}\}$ be a Cauchy sequence of $(X,d^\star)$. For every $k\in \mathbb{N}$, there exists $l_{k}, ~r_{k}$ such that $d^\star (x_{l_{k}},x_{n})\leqslant \frac{1}{r_{k+1}}$ where $ n> l_{k}$. Let $x_{l_{k}}$ be the smallest positive integer with the above properties, so that $l_{k}\leqslant l_{k+1}, r_{k}\leqslant r_{k+1}(k=1,2,\dots)$. Construct the following sequence of closed subset $\{F_{n}\}$ which defined by letting $$F_{k}=\overline{B_{d^\star}(x_{l_{k}},\frac{1}{r_{k}})},~ (k=1,2,\dots),$$ here $B_{d^\star}(x_{l_{k}},\frac{1}{r_{k}})=\{y \in X:d^\star(x_{l_{k}},y)\leqslant \frac{1}{r_{k}}\}$. By Proposition \ref{thm8}, we can get that $d^\star (F_{k})\leqslant \frac{2}{r_{k}}$, this implies that $\lim_{n\rightarrow \infty }\delta(F_{n})=0$.
Now take subset $\{H_{n}\}$ of $\{F_{n}\}$. We define $H_{1}=F_{k_{1}}, k_{1}=1$. Then take $H_{2}=F_{k_{2}}$, by Lemma \ref{lem3}, we can set that $k_{2}=\min \{j\geqslant 2: \frac{1}{r_{j}}\star \frac{1}{r_{j}}<\frac{1}{r_{k_{1}}}\}$. Generally speaking, if we take the positive integer $k_{n}$, we can take $k_{n+1}=\min \{j\geqslant k_{n}+1: \frac{1}{r_{j}}\star \frac{1}{r_{j}}<\frac{1}{r_{k_{n}}}\}$, such that $H_{n+1}=F_{k_{n+1}}$. Now we shall show that $\{H_{n}\}$ satisfies the conditions in our theorem.
Let $y\in H_{n+1}$, then according to the selection method of $l_{k}$, we have $$d^\star(y,x_{l_{k_{n+1}}})< \frac{1}{r_{k_{n+1}}}, ~d^\star(x_{l_{k_{n+1}}},x_{l_{k_{n}}})< \frac{1}{r_{k_{n}+1}},$$
thus $$d^\star(y,x_{l_{k_{n}}})\leqslant d^\star(y,x_{l_{k_{n+1}}})\star d^\star(x_{l_{k_{n+1}}},x_{l_{k_{n}}})< \frac{1}{r_{k_{n+1}}}\star \frac{1}{r_{k_{n}+1}}< \frac{1}{r_{k_{n+1}}} \star \frac{1}{r_{k_{n+1}}}< \frac{1}{r_{k_{n}}}.$$ This implies that $y\in H_{n}$, i.e. $H_{n+1}\subset H_{n}$.
According to the assumption, there should be $\bigcap_{n=1}^\infty F_{n}=\{x_{0}\}$. Now we shall show that a Cauchy sequence $\{x_{n}\}$ is convergent to a point $x_{0}$. For every $\epsilon> 0$, by Lemma \ref{lem3}, take a $r_{k}\in \mathbb{N}$ such that $\frac{1}{r_{k}}\star \frac{1}{r_{k}}<\epsilon$, and $d^\star(x_{l_{k}},x_{n})\leqslant \frac{1}{r_{k+1}}$. In addition, $x_{0}\in F_{k}$, $d^\star(x_{l_{k}},x_{0})\leqslant \frac{1}{r_{k}}$, then we have $$d^\star(x_{0},x_{n})\leqslant d^\star(x_{0},x_{l_{k}})\star d^\star(x_{l_{k}},x_{n})< \frac{1}{r_{k}}\star \frac{1}{r_{k+1}}< \frac{1}{r_{k}}\star \frac{1}{r_{k}}< \epsilon.$$ Thus, $\lim_{n\rightarrow \infty }x_{n}=x_{0}$, $(X,d^\star)$ is a complete $\star$-metric space.
\end{proof}
\begin{theorem}
A $\star$-metric space is complete if and only if for every family of closed subsets of $X$ which has the finite intersection property and for every $\epsilon>0$ contains a set of diameter less than $\epsilon$ has non-empty intersection.
\end{theorem}
\begin{proof}
Sufficiency of the condition in our theorem for completeness of a $\star$-metric space follows from the Theorem \ref{thm7}.
We shall show that the condition holds in every complete $\star$-metric space $(X,d^\star)$. Consider a family $\{F_{s}\}_{s\in S}$ of closed subsets of $X$ which has the finite intersection property and which for every $j\in \mathbb{N}$ contains a set $F_{s_{j}}$, such that $\delta(F_{s_{j}})< \frac{1}{j}$. Let $F_{i}=\bigcap_{j\leqslant i}F_{s_{j}}$.
One easily sees that the sequence $F_{1}, F_{2},\dots$ satisfies the condition of the Cantor theorem, since $F_{n+1}\subset F_{n}$ and $\delta(F_{i})\leqslant \delta(F_{s_{i}})< \frac{1}{i}$ which means $\lim_{n\rightarrow \infty }\delta(F_{n})=0$.
So that there exists an $x\in \bigcap_{i=1}^\infty F_{i}$. Clearly, we have $\bigcap_{i=1}^\infty F_{i}=\{x\}$. Now, let us take an arbitrary $s_{0}\in S$; letting $F_{i}'=F_{s_{0}}\cap F_{i}$ for $i=1,2,\dots$ we obtain again a sequence $F_{1}',F_{2}',\dots$ satisfying the conditions of the Theorem \ref{thm7}. Since $$\emptyset \neq \bigcap_{i=1}^\infty F_{i}'=F_{s_{0}}\cap \bigcap_{i=1}^\infty F_{i}=F_{s_{0}}\cap \{x\},$$ we have $x\in F_{s_{0}}$. Hence $x\in \bigcap_{s\in S}F_{s}$.
\end{proof}
\begin{theorem} \label{thm10}
A subspace $(M,d^\star)$ of a complete $\star$-metric space $(X,d^\star)$ is complete if and only if $M$ is closed in $X$.
\end{theorem}
\begin{proof}
Necessity. Let $x\in \overline{M}$, and we define $F_{k}=M\cap \overline{B_{d^\star}(x,\frac{1}{k})}(k=1,2,\dots)$, then sequence $\{F_{k}\}$ is non-empty closed subsets in subspace $M$, so one can easily check that $\{F_{k}\}$ satisfies the conditions $(1),(2)$ in the Theorem \ref{thm7}. Since subspace $(M,d^\star)$ is complete, by Theorem \ref{thm7}, obviously $\bigcap_{k=1}^\infty F_{k}=\{x\}$, it follows that $x\in M$. Therefore $M= \overline{M}$.
Sufficiency. Let $M$ is a closed set, every Cauchy sequence of $\star$-metric space $(M,d^\star)$ is also a Cauchy sequence of complete $\star$-metric space $(X,d^\star)$, so it converges to a certain point $x\in X$. Since $M$ is closed in $X$, $x\in M$. This completes the proof.
\end{proof}
The following theorem shows that in a class of $\star$-metric spaces, the completeness is preserved by finite products.
\begin{theorem}\label{thm11} Let $\{(X_{i},d_{i}^\star)\}_{i=1}^n $ be a family of finite nonempty $\star$-metric spaces and $X=\prod_{i=1}^{n}X_{i}$ the Cartesian product. Then
\begin{enumerate}
\item[(1)]$X$ with the $\star$-metric $d_{T}^\star$ defined by formula (3.1) is complete if and only if all $\star$-metric spaces $(X_{i},d_{i}^\star)$ are complete;
\item[(2)]$X$ with the $\star$-metric $d_{max}^\star$ defined by formula (3.2) is complete if and only if all $\star$-metric spaces $(X_{i},d_{i}^\star)$ are complete.
\end{enumerate}
\end{theorem}
\begin{proof}
(1). Assume that the space $(X,d_{T}^\star)$ is complete. For every subspace $X_{m}^*=\prod_{i=1}^{n}A_{i}$ of $X$, where $A_{m}=X_{m}$ and $A_{i}=\{x_{i}^*\}$ is a one-point subset of $X_{i}$ for $i \neq m$, is closed in $(X,d_{T}^\star)$. Then the subspace $X_{m}^*$ is complete by Theorem \ref{thm10}. One can easily verify that $p_{m}^*=p_{m}\mid _{X_{m}^*}:X_{m}^*\rightarrow X_{m}$ is a isometric isomorphism, since $d_{T}^\star|_{X_{m}^*}(p_{m}^*(x),p_{m}^*(y))= d_{2}^\star(x,y)$. Therefore, for every Cauchy sequence $\{x_{n}\}$ in $(X_{m},d_{m}^\star)$, the sequence $\{{p_{m}^*}^{-1}(x_{n})\}$ is a Cauchy sequence in $X_{m}^*$. Then $$p_{m}^*(\lim_{n\rightarrow \infty} {p_{m}^*}^{-1}(x_{n}))=\lim_{n\rightarrow \infty}x_{n},$$ so that the space $(X_{m},d_{m}^\star)$ is complete.
Assume that all spaces $(X_{i},d_{i}^\star)$ are complete. Take any Cauchy sequence $\{y_{k}\}_{k \in \mathbb{N}}$ in $(X,d_{T}^\star)$, where $y_{k}=(x_{i}^k)$, for $1 \leqslant i\leqslant n$. Then the sequence $\{x_{i}^k\}_{k \in \mathbb{N}}$ is a Cauchy sequence in $(X_{i},d_{i}^\star)$ and thus converges to a point $x_{i}^0 \in X_{i}$. Now, we shall show that $\{y_{k}\}_{k \in \mathbb{N}}$ converges to a point $x^0 = (x_{i}^0)$. For $\epsilon> 0$, by Lemma \ref{lem3}, take an $\epsilon_{1}> 0$ such that $ \overbrace{\epsilon_{1}\star \epsilon_{1}\star \dots \star \epsilon_{1}}^{n \text{ times}}<\epsilon$. Since $\{x_{i}^k\}_{k \in \mathbb{N}}$ converges to a point $x_{i}^0$, there exists $m_{i} \in \mathbb{N}$, such that $d_{i}^\star(x_{i}^k,x_{i}^0)< \epsilon_{1}$, where $k\geqslant m_{i}$. Thus choose $m=\max_{1 \leqslant i\leqslant n}\{m_{i}\}$, such that $$d_{T}^\star(y_{k},x^0)=d_{1}^\star(x_{1}^k,x_{1}^0)\star d_{2}^\star(x_{2}^k,x_{2}^0)\star \dots \star d_{n}^\star(x_{n}^k,x_{n}^0)< \overbrace{\epsilon_{1}\star \epsilon_{1}\star \dots \star \epsilon_{1}}^{n \text{ times}}<\epsilon,$$ whenever $k\geqslant m$. We have shown that $(X,d_{T}^\star)$ is complete.
(2). Assume that the space $(X,d_{T}^\star)$ is complete. The method of proof is the same as (1).
Assume that all spaces $(X_{i},d_{i}^\star)$ are complete. Take any Cauchy sequence $\{y_{k}\}_{k \in \mathbb{N}}$ in $(X,d_{\max}^\star)$, where $y_{k}=(x_{i}^k)$, for $1 \leqslant i\leqslant n$. Then the sequence $\{x_{i}^k\}_{k \in \mathbb{N}}$ is a Cauchy sequence in $(X_{i},d_{i}^\star)$ and thus converges to a point $x_{i}^0 \in X_{i}$. Now, we shall show that $\{y_{k}\}_{k \in \mathbb{N}}$ converges to a point $x^0 = (x_{i}^0)$. Since $\{x_{i}^k\}_{k \in \mathbb{N}}$ converges to a point $x_{i}^0$. For every $\epsilon > 0$ there exists $m_{i} \in \mathbb{N}$, such that $d_{i}^\star(x_{i}^k,x_{i}^0)< \epsilon$, where $k\geqslant m_{i}$. Without loss of generality, let $\max_{1\leqslant i\leqslant n} d_{i}^\star(x_{i}^k,x_{i}^0)=d_{j}^\star(x_{j}^k,x_{j}^0)$, then while $k\geqslant m=m_{j}$, such that $$d_{\max}^\star(y_{k},x^0)=\max_{1\leqslant i\leqslant n}d_{i}^\star(x_{i}^k,x_{i}^0)=d_{j}^\star(x_{j}^k,x_{j}^0)< \epsilon.$$ We have shown that $(X,d_{\max}^\star)$ is complete.
\end{proof}
\begin{theorem} \label{thm11.2}
If $\{(X_{\alpha},d_{\alpha}^\star)\}_{\alpha\in A}$ be a family of $\star$-metric spaces such that the metric $d_{i}^\star$ is bounded for each $\alpha\in A$, and $X=\bigoplus_{\alpha\in A}X_{\alpha}$ be the disjoint union of $\{X_{\alpha}\}$. Then $(X,d^\star_{q})$ defined by formula (3.3) is complete if and only if all spaces $(X_{\alpha},d_{\alpha}^\star)$ are complete.
\end{theorem}
\begin{proof}
Necessity. Assume that $(X,d^\star_{q})$ is complete. Then it is easy to see that all sets $X_{\alpha}$ are open-and-closed in $X$. So, all spaces $(X_{\alpha},d_{\alpha}^\star)$ are complete by Theorem \ref{thm10}.
Sufficiency. Assume that all spaces $(X_{\alpha},d_{\alpha}^\star)$ are complete. Then we have $(X,d^\star_{q})$ is complete, because every Cauchy sequence of $\star$-metric space $(X_{\alpha},d_{\alpha}^\star)$ is also a Cauchy sequence of $(X,d^\star_{q})$ and it converges to a certain point $x\in X_{\alpha}\subseteq X$.
\end{proof}
Baire theorem is a very important result in complete metric spaces. We shall extend this theorem to complete
$\star$-metric spaces.
\begin{theorem}\label{thm12}
In a complete $\star$-metric space $(X,d^\star)$ the intersection $A=\bigcap_{n=1}^\infty A_{n}$ of a sequence $A_{1},A_{2},\dots$ of dense open subsets is a dense set.
\end{theorem}
\begin{proof}
Let $A=\bigcap_{n=1}^\infty A_{n}$, for every $A_{n}$ is an open dense subset of complete $\star$-metric space $(X,d^\star)$. Now, construct the sequence of closed subset $\{F_{n}\}$ which satisfies conditions in Theorem \ref{thm7}. Since $A_{1}$ is dense in $X$, and $U$ is a non-empty open set, then $A_{1}\cap U\neq \emptyset$. Take $x_{1}\in A_{1}\cap U$, since $ A_{1}\cap U$ is an open set, there exists $\epsilon _{1}$ which satisfies $0< \epsilon _{1}< 1/2^2$, such that $\overline{B_{d^\star}(x_{1},\epsilon _{1})} \subset A_{1}\cap U$. Since $A_{2}$ is dense in $X$, and $B_{d^\star}(x_{1},\epsilon _{1})$ is an open set, then $A_{2}\cap B_{d^\star}(x_{1},\epsilon _{1})\neq \emptyset$. Take $x_{2}\in A_{2}\cap B_{d^\star}(x_{1},\epsilon _{1})$, since $ A_{2}\cap B_{d^\star}(x_{1},\epsilon _{1})$ is an open set, there exists $\epsilon _{2}$ which satisfies $0< \epsilon _{2}< \epsilon _{1}/2$, such that $\overline{B_{d^\star}(x_{2},\epsilon _{2})} \subset A_{2}\cap B_{d^\star}(x_{1},\epsilon _{1})$. Obviously, $\overline{B_{d^\star}(x_{2},\epsilon _{2})} \subset \overline{B_{d^\star}(x_{1},\epsilon _{1})}$ and $\overline{B_{d^\star}(x_{2},\epsilon _{2})} \subset A_{2}\cap U$. Going on, one can easily obtain the sequence of closed subset $\{F_{n}\}=\{\overline{B_{d^\star}(x_{n},\epsilon _{n})}\}$ which satisfies $F_{n+1}\subset F_{n}$ and $\delta (F_{n}) \leqslant 1/2^n$ (n=1,2,\dots). This implies that $\{F_{n}\}$ satisfies conditions in Theorem \ref{thm7}. Noting that $F_{n}\subset A_{n}\cap U$, by Theorem \ref{thm7}, $\bigcap_{n=1}^\infty F_{n}\neq \emptyset$, then we have $$A\cap U=(\bigcap_{n=1}^\infty A_{n}) \cap U=\bigcap_{n=1}^\infty (A_{n} \cap U)\supset \bigcap_{n=1}^\infty F_{n} \neq \emptyset,$$ this implies that $A$ is dense in $X$.
\end{proof}
Every metric space is isometric to a subspace of a complete metric space. It would be interesting to find out whether this result remain valid in the class of $\star$-metric spaces:
\begin{question}
Is every $\star$-metric space isometric to a subspace of a complete $\star$-metric space?
\end{question}
|
2,869,038,154,865 | arxiv | \section{Introduction}
The vorticity-velocity formulation of the 3D Navier-Stokes equations
(NSE) reads
\begin{equation}\label{vorticity}
\partial_t \omega + (u \cdot \nabla)\omega = (\omega \cdot \nabla)u
+
\triangle \omega
\end{equation}
where $u$ is the velocity and $\omega = \, \mbox{curl} \, u$ is the
vorticity of the fluid (the viscosity is set to 1). A comprehensive
introduction to the mathematical study of the vorticity can be found
in \cite{MB02}.
\medskip
In 2D, the vortex-stretching term, $(\omega \cdot \nabla)u$, is
identically zero, and the nonlinearity is simply a part of the
transport of the vorticity by the flow.
\medskip
This allowed the authors to adopt to 2D a general mathematical
setting for the study of turbulent cascades and locality in
\emph{physical scales} of 3D incompressible viscous and inviscid
flows introduced in \cite{DaGr11-1} and \cite{DaGr11-2},
respectively, to establish existence of the enstrophy cascade and
locality in 2D \cite{DaGr11-3}.
\medskip
Since the vortex-stretching term is not a flux-type term, the only
way to establish existence of the enstrophy cascade in 3D in this
setting is to show that its contribution to the ensemble averaging
process can be suitably interpolated between integral-scale averages
of the enstrophy and the enstrophy dissipation rate.
\medskip
This is where \emph{coherent vortex structures} come into play. A
role of the coherent vortex structures in turbulent flows was
recognized as early as the 1500's in Leonardo da Vinci's ``deluge''
drawings. On the other hand, Kolmogorov's K41 phenomenology
\cite{Ko41-1, Ko41-2} does not discern geometric structures; the K41
eddies are essentially amorphous. As stated by Frisch in his book
\emph{Turbulence}, The Legacy of A.N. Kolmogorov \cite{Fr95}, ``Half
a century after Kolmogorov's work on the statistical theory of fully
developed turbulence, we still wonder how his work can be reconciled
with Leonardo's half a millennium old drawings of eddy motion in the
study for the elimination of rapids in the river Arno.'' This remark
was followed by a discussion on dynamical role, as well as
statistical signature of vortex filaments in turbulent flows.
Several (by now classical) directions in the study of the vortex
dynamics of turbulence -- both in 2D and 3D -- are presented in
Chorin's book \emph{Vorticity and Turbulence} (cf. \cite{Ch94} and
the references therein). The approach exposed in \cite{Ch94} is
essentially discrete (probabilistic lattice models); on the other
hand, the first rigorous continuous statistical theory of vortex
filaments was given by P.-L. Lions and Majda in \cite{LM00}.
\medskip
Local anisotropic behavior of the enstrophy, i.e., self-organization
of the regions of high vorticity in coherent vortex structures --
most notably vortex filaments/tubes -- is ubiquitous. In particular,
local alignment or anti-alignment of the vorticity direction, i.e.
\emph{local coherence}, is prominently featured in turbulent flows.
A strong numerical evidence, as well as several theoretical
arguments explaining the physical mechanism behind the formation of
coherent structures -- including rigorous estimates on the flow
directly from the 3D NSE -- can be found, e.g., in \cite{Co90,
SJO91, CPS95, GGH97, GFD99, Oh09, GM11}. One way to look at the
phenomenon of local coherence of the vorticity direction is to
interpret it as a manifestation of the general observation that the
regions of high fluid intensity are -- in the vorticity formulation
-- \emph{locally} `quasi 2D-like' (in 2D, the vorticity direction is
\emph{globally} parallel or antiparallel).
\medskip
The rigorous study of \emph{geometric depletion of the nonlinearity}
in the 3D NSE was pioneered by Constantin when he derived a singular
integral representation for the stretching factor in the evolution
of the vorticity magnitude featuring a geometric kernel depleted by
coherence of the vorticity direction (cf. \cite{Co94}). This was
followed by the paper \cite{CoFe93} where Constantin and Fefferman
showed that as long as the vorticity direction in the regions of
intense vorticity is Lipschitz-coherent, no finite-time blow up can
occur, and later by the paper \cite{daVeigaBe02} where Beirao da
Veiga and Berselli scaled the coherence strength needed to deplete
the nonlinearity down to $\frac{1}{2}$-H\"older.
\medskip
A full \emph{spatiotemporal localization} of the vorticity
formulation of the 3D NSE was developed by one of the authors and
the collaborators in \cite{GrZh06, Gr09, GrGu10-1, GrGu10-2}. The
main obstacle to the full localization of the evolution of the
enstrophy was the spatial localization of the vortex-stretching
term, $(\omega \cdot \nabla) u$. An explicit local representation
for the vortex-stretching term was given in \cite{Gr09}, and the
leading order term reads
\begin{equation}\label{vstloc}
P.V. \int_{B(x_0,2R)} \epsilon_{jkl} \frac{\partial^2}{\partial x_i
\partial y_k} \frac{1}{|x-y|} \phi(y,t) \omega_l(y,t) \, dy \
\phi(x,t) \omega_i(x,t) \omega_j(x,t)
\end{equation}
where $\epsilon_{jkl}$ is the Levi-Civita symbol and $\phi$ is a
spatiotemporal cut-off associated with the ball $B(x_0,R)$. The key
feature of (\ref{vstloc}) is that it displays both \emph{analytic}
(via a local non-homogeneous Div-Curl Lemma \cite{GrGu10-2}) and
\emph{geometric} (via coherence of the vorticity direction
\cite{Gr09, GrGu10-1}) \emph{cancelations}, inducing analytic and
geometric \emph{local depletion of the nonlinearity} in the 3D
vorticity model. (For a different approach to localization of the
vorticity-velocity formulation see \cite{ChKaLe07}.)
\medskip
The present work is envisioned as a contribution to the effort of
understanding the role that the geometry of the flow and in
particular, coherent vortex structures, plays in the theory of
turbulent cascades. More precisely, we show -- utilizing the
aforementioned localization within the general mathematical
framework for the study of turbulent cascades in physical scales of
incompressible flows introduced in \cite{DaGr11-1} (in this case,
via suitable ensemble-averaging of the local enstrophy equality) --
that \emph{coherence of the vorticity direction}, coupled with a
suitable condition on a modified Kraichnan scale, and under a
certain modulation assumption on evolution of the vorticity, leads
to existence of 3D enstrophy cascade in physical scales of the flow.
This furnishes a mathematical evidence that, in contrast to 3D
energy cascade, 3D enstrophy cascade is locally \emph{anisotropic},
providing a form of a reconciliation between Leonardo's and
Kolmogorov's views on turbulence on the \emph{enstrophy level}.
\medskip
It is worth pointing out that our theory of turbulent cascades in
physical scales of 3D incompressible flows is on the \emph{energy
level} \cite{DaGr11-1, DaGr11-2, DaGr12-1} consistent both with the
K41 theory of turbulence and the Onsager's predictions on existence
of the \emph{inviscid} energy cascade (and consequently, with the
phenomena of \emph{anomalous dissipation} and \emph{dissipation
anomaly}), as well as with the previous rigorous mathematical work
on existence of the energy cascade in the wavenumbers \cite{FMRT01}.
In particular, on the energy level, it does not `see' geometric
structures. It is only here, i.e., on the enstrophy level, that the
role of coherent vortex structures is revealed. A distinctive
feature of our theory that makes incorporating the geometry of the
flow and in particular, local coherence possible is the fact that
the cascade takes place in the actual physical scales of the flow.
To the best of our knowledge, there is nothing in the K41 theory
that would contradict existence of the 3D enstrophy cascade; on the
other hand, given that K41 takes place primarily in the Fourier
space, i.e., in the wavenumbers, formulating conditions (within K41)
faithfully reflecting various geometric properties of the flow is a
much more challenging enterprise.
\medskip
The paper is organized as follows. Section 2 recalls the
ensemble-averaging process introduced in \cite{DaGr11-1}, and
Section 3 the spatiotemporal localization of the evolution of the
enstrophy developed in \cite{GrZh06, Gr09, GrGu10-1, GrGu10-2}.
Existence and locality of anisotropic 3D enstrophy cascade is
presented in Section 4.
\section{Ensemble averages}
In studying a PDE model, a natural way of actualizing a concept of
scale is to measure distributional derivatives of a quantity with
respect to the scale. Let $x_0$ be in $B(0,R_0)$ ($R_0$ being the
\emph{integral scale}, $B(0,2R_0)$ contained in $\Omega$ where
$\Omega$ is the global spatial domain) and $0< R \le R_0$.
Considering a locally integrable physical density of interest $f$ on
a ball of radius $2R$, $B(x_0, 2R)$, a \emph{local physical scale
$R$} -- associated to the point $x_0$ -- is realized via bounds on
distributional derivatives of $f$ where a test function $\psi$ is a
refined -- smooth, non-negative, equal to 1 on $B(x_0, R)$ and
featuring optimal bounds on the derivatives over the outer $R$-layer
-- cut-off function on $B(x_0, 2R)$. More explicitly,
\begin{equation}\label{ps}
|(D^\alpha f, \psi)| \le \int_{B(x_0, 2R)} |f| |D^\alpha \psi| \le
\Bigl(c(\alpha) \frac{1}{R^{|\alpha|}} |f| ,
\psi^{\delta(\alpha)}\Bigr)
\end{equation}
for some $c(\alpha)>0$ and $\delta(\alpha)$ in $(0,1)$. (This is
reminiscent of Bernstein inequalities in the Littlewood-Paley
decomposition of a tempered distribution.)
\medskip
Henceforth, we utilize {\em refined} spatiotemporal cut-off
functions $\phi=\phi_{x_0,R,T}=\psi\,\eta$, where $\eta=\eta_T(t)\in
C^\infty (0,T)$ and $\psi=\psi_{x_0,R}(x)\in\mathcal{D}(B(x_0,2R))$
satisfying
\begin{equation}\label{eta_def}
0\le\eta\le1,\quad\eta=0\ \mbox{on}\ (0,T/3),\quad\eta=1\ \mbox{on}\
(2T/3,T),\quad\frac{|\eta'|}{\eta^{\rho_1}}\le\frac{C_0}{T}\;
\end{equation}
and
\begin{equation}\label{psi_def}
0\le\psi\le 1,\quad\psi=1\ \mbox{on}\ B(x_0,R),
\quad\frac{|\nabla\psi|}{\psi^{\rho_2}}\le\frac{C_0}{R}, \quad
\frac{|\triangle\psi|}{\psi^{2\rho_2-1}}\le\frac{C_0}{R^2}\;,
\end{equation}
for some $\frac{1}{2} <\rho_1,\rho_2 < 1$.
In particular, $\phi_0=\psi_0 \eta$ where $\psi_0$ is the spatial
cut-off (as above) corresponding to $x_0=0$ and $R=R_0$.
For $x_0$ near the boundary of the integral domain, $S(0,R_0)$, we
assume additional conditions,
\begin{equation}\label{psi_bd}
0\le\psi\le\psi_0
\end{equation}
and, if $B(x_0,R)\not\subset B(0,R_0)$, then
$\psi\in\mathcal{D}(B(0,2R_0))$ with $\psi=1\ \mbox{on}\ B(x_0,R)
\cap B(0,R_0)$ satisfying, in addition to (\ref{psi_def}), the
following:
\begin{equation}\label{psi_def_add1}
\begin{aligned}
&
\psi=\psi_0\ \mbox{on the part of the cone centered at zero and passing through}\\
& S(0,R_0)\cap B(x_0,R)\ \mbox{between}\ S(0,R_0)\
\mbox{and}\ S(0,2R_0)
\end{aligned}
\end{equation}
and
\begin{equation}\label{psi_def_add2}
\begin{aligned}
&
\psi=0\ \mbox{on}\ B(0,R_0)\setminus B(x_0,2R)\ \mbox{and outside the part of the cone}\\
&
\mbox{centered at zero and passing through}\ S(0,R_0)\cap B(x_0,2R)\\
&
\mbox{between}\ S(0,R_0)\ \mbox{and}\ S(0,2R_0).
\end{aligned}
\end{equation}
\medskip
A \emph{physical scale $R$} -- associated to the integral domain
$B(0,R_0)$ -- is realized via suitable ensemble-averaging of the
localized quantities with respect to `$(K_1,K_2)$-covers' at scale
$R$.
\medskip
Let $K_1$ and $K_2$ be two positive integers, and $0 < R \le R_0$. A
cover $\{B(x_i,R)\}_{i=1}^n$ of the integral domain $B(0,R_0)$ is a
\emph{$(K_1,K_2)$-cover at scale $R$} if
\[
\biggl(\frac{R_0}{R}\biggr)^3 \le n \le K_1
\biggr(\frac{R_0}{R}\biggr)^3,
\]
and any point $x$ in $B(0,R_0)$ is covered by at most $K_2$ balls
$B(x_i,2R)$. The parameters $K_1$ and $K_2$ represent the maximal
\emph{global} and \emph{local multiplicities}, respectively.
\medskip
For a physical density of interest $f$, consider time-averaged, per
unit mass -- spatially localized to the cover elements $B(x_i, R)$
-- local quantities $\hat{f}_{x_i,R}$,
\[
\hat{f}_{x_i,R} = \frac{1}{T} \int_0^T \frac{1}{R^3}
\int_{B(x_i,2R)} f(x,t) \phi^\delta_{x_i,R,T} (x,t) \, dx \, dt
\]
for some $0 < \delta \le 1$, and denote by $\langle F\rangle_R$ the
\emph{ensemble average} given by
\[
\langle F\rangle_R = \frac{1}{n} \sum_{i=1}^n
\hat{f}_{x_i,R}\,.
\]
\medskip
The key feature of the ensemble averages $\{\langle
F\rangle_R\}_{0<R\le R_0}$ is that $\langle F\rangle_R$ being
\emph{stable}, i.e., nearly-independent on a particular choice of
the cover (with the fixed parameters $K_1$ and $K_2$), indicates
there are no significant fluctuations of the sign of the density $f$
at scales comparable or greater than $R$. On the other hand, if $f$
does exhibit significant sign-fluctuations at scales comparable or
greater than $R$, suitable rearrangements of the cover elements up
to the maximal multiplicities -- emphasizing first the positive and
then the negative parts of the function -- will result in $\langle
F\rangle_R$ experiencing a wide range of values, from positive
through zero to negative, respectively.
\medskip
Consequently, for an \emph{a priori} sign-varying density, the
ensemble averaging process acts as a \emph{coarse detector of the
sign-fluctuations at scale $R$}. (The larger the maximal
multiplicities $K_1$ and $K_2$, the finer detection.)
\medskip
As expected, for a non-negative density $f$, all the averages are
comparable to each other throughout the full range of scales $R$, $0
< R \le R_0$; in particular, they are all comparable to the simple
average over the integral domain. More precisely,
\begin{equation}\label{k*}
\frac{1}{K_*} F_0 \le \langle F \rangle_R \le K_* F_0
\end{equation}
for all $0 < R \le R_0$, where
\[
F_0=\frac{1}{T}\int \frac{1}{R_0^3} \int f(x,t)
\phi_0^\delta (x,t) \, dx \, dt,
\]
and $K_* = K_*(K_1,K_2) > 1$.
\medskip
There are several properties of the ensemble averaging process that
although being plausible, deserve a precise analytic
description/quantification. Besides the above statement on
non-negative densities, perhaps the most elemental one is that if
the averages at a certain scale are nearly independent of a
particular choice of a $(K_1,K_2)$-cover ($K_1$ and $K_2$ fixed),
then essentially the same \emph{universality property} should
propagate to larger scales. In order to obtain a precise
quantitative propagation result, the universality is assumed on an
initial interval, rather than at a single scale (this is due to the
presence of smooth cut-offs with prescribed rates of change). Two
types of results are presently available.
\medskip
Let $f$ be a locally integrable function (a density), and $K_1$ and
$K_2$ two positive integers.
\medskip
\noindent TYPE I. \ Assume that there exists $R_* > 0$ such that for
any $R$ in $[R^*, 2R_*)$ and any $(27K_1,16K_2)$-cover at scale $R$,
the averages $\langle F\rangle_R$ are all comparable to some value
$F_*$; more precisely, $\displaystyle{\frac{1}{C_1} F_* \le \langle
F\rangle_R \le C_1 F_*}$. Then, for all $R \ge 2R_*$ and all
$(K_1,K_2)$-covers at scale $R$, the averages $\langle F\rangle_R$
satisfy $\displaystyle{\frac{1}{C_2} F_* \le \langle F\rangle_R \le
C_2 F_*}$.
\medskip
\noindent TYPE II. \ Assume that there exists $R_* > 0$ such that
for any $R$ in $(\frac{1}{2}R^*, \frac{5}{2}R_*)$ and any
$(K_1,K_2)$-cover at scale $R$, $\displaystyle{\frac{1}{C_1} F_* \le
\langle F\rangle_R \le C_1 F_*}$. Then, for all $R \ge \frac{5}{2}
R_*$ and all $(K_1,K_2)$-covers at scale $R$, the averages $\langle
F\rangle_R$ satisfy
\[
\frac{1}{C_3} \Bigl(\frac{R_*}{R}\Bigr)^{C_4} F_* \le \langle
F\rangle_R \le C_3 \Bigl(\frac{R}{R^*}\Bigr)^{C_4} F_*.
\]
\medskip
Shortly, in a Type I result, the universality is assumed with
respect to more refined covers, while in a Type II result, the
non-exactness of the propagation caused by the smooth cut-offs is
reflected in a correction to the universal value $F_*$ by the ratio
of the scales $R$ and $R^*$.
\medskip
The proofs are quite long and technical, and will be provided in a
separate publication together with computational results describing
the general statistics of the variation of the ensemble averages of
multi-scale sign-fluctuating densities across the scales. Here, we
provide a sample computation of the ensemble averages of a
time-independent (to emphasize the spatial behavior) 1D density
$f(x)=\cos^2 (x+5) \sin \Bigl(\frac{1}{2} (x-1)^2\Bigr)$ (Figure 1),
for $R_0=10$ and $K_1=K_2=3$; its global average is approximately
$-0.003880$. The $y$-values in Figure 2 represent the ensemble
averages with respect to the $(K_1,K_2)$-covers exhibiting maximal
positive and negative bias, across the range of scales from
$10^{-2}$ to $10^1=R_0$ (the $x$-values represent the powers of
$10$); the red line slightly below the $x$-axis corresponds to the
value of the global average. The response of the ensemble averages
to sign-fluctuations at several different scales is clearly visible.
\medskip
\begin{figure}
\centerline{\includegraphics[scale=.7]{f_graph.eps}}
\caption{$f(x)=\cos^2 (x+5) \sin \Bigl(\frac{1}{2} (x-1)^2\Bigr)$}
\label{g_f}
\end{figure}
\medskip
\begin{figure}
\centerline{\includegraphics[scale=.7]{k_ave.eps}}
\caption{Ensemble averages with positive and negative bias.}
\label{av_pos}
\end{figure}
\section{Spatiotemporal localization of evolution of the enstrophy}
The localization of the vorticity formulation introduced in
\cite{GrZh06, Gr09} was performed on an arbitrarily small parabolic
cylinder below a point $(x_0,t_0)$ contained in the spatiotemporal
domain $\Omega \times (0,T)$, with an eye on formulating the
conditions preventing singularity formation at $(x_0,t_0)$. Here --
in order to coordinate the notation with Section 2 -- the
localization will be performed on a spatiotemporal cylinder
$B(x_0,2R) \times (0,T)$ where $x_0$ belongs to the integral domain
$B(0,R_0)$ and $0<R\le R_0$.
\medskip
Suppose that the solution is smooth in $B(x_0,2R) \times (0,T)$.
Multiplying the equations by $\phi \, \omega$ ($\phi=\psi\eta$ being
the cut-off introduced in Section 2) and integrating over $B(x_0,2R)
\times (0,t)$ for some $2T/3<t<T$ yields
\begin{align}\label{locx0}
\int \frac{1}{2}|\omega(x,t)|^2\psi(x) \; dx &+ \int_0^t \int |\nabla\omega|^2\phi
\; dx \; ds\notag\\
&= \int_0^t \int \frac{1}{2}|\omega|^2 (\phi_t+\triangle\phi) \; dx
\; ds\notag\\
&+ \int_0^t \int \frac{1}{2}|\omega|^2 (u \cdot \nabla\phi) \; dx
\; ds +\int_0^t \int (\omega \cdot \nabla)u \cdot \phi \omega \; dx
\; ds.
\end{align}
\medskip
Suppressing the time variable, the localized vortex-stretching term
can be written as (cf. \cite{Gr09})
\begin{align}\label{locvst}
(\omega \cdot \nabla)u \cdot \phi \omega \, (x) & =
\phi^{\frac{1}{2}}(x) \, \frac{\partial}{\partial x_i} u_j(x) \, \phi^{\frac{1}{2}}(x) \,
\omega_i(x)\, \omega_j(x)\notag\\
& = -c \, P.V. \int_{B(x_0, 2r)} \epsilon_{jkl} \, \frac{\partial^2}{\partial x_i \partial y_k}
\frac{1}{|x-y|} \, \phi^{\frac{1}{2}} \, \omega_l \, dy \ \phi^{\frac{1}{2}}(x) \,
\omega_i(x) \, \omega_j(x) + \ \mbox{LOT}\notag\\
& = - c \, P.V. \int_{B_(x_0, 2r)} \bigl(\omega(x) \times
\omega(y) \bigr) \cdot G_\omega (x,y) \, \phi^{\frac{1}{2}}(y) \, \phi^{\frac{1}{2}}(x) \, dy +
\ \mbox{LOT}\notag\\
& = \ \mbox{VST} \ + \ \mbox{LOT}
\end{align}
where $\epsilon_{jkl}$ is the Levi-Civita symbol,
\[
\bigl( G_\omega (x,y) \bigr)_k = \frac{\partial^2}{\partial x_i
\partial y_k} \frac{1}{|x-y|} \, \omega_i(x)
\]
and LOT denotes the lower order terms.
\medskip
The above representation formula for the leading order
vortex-stretching term VST features both analytic and geometric
cancelations.
\medskip
The geometric cancelations were utilized in \cite{Gr09} to obtain a
full localization of $\frac{1}{2}$-H\"older coherence of the
vorticity direction regularity criterion, and then in
\cite{GrGu10-1} to introduce a family of \emph{scaling-invariant}
regularity classes featuring a precise balance between coherence of
the vorticity direction and spatiotemporal integrability of the
vorticity magnitude. Denote by $\xi$ the vorticity direction, and
let $(x,t)$ be a spatiotemporal point, $r>0$ and $0 < \gamma < 1$. A
$\gamma$-H\"older measure of coherence of the vorticity direction at
$(x,t)$ is then given by
\[
\rho_{\gamma, r}(x,t)=\sup_{y \in B(x, r), y \neq x}
\frac{|\sin \varphi \bigl(\xi(x,t), \xi(y,t)\bigr)|}{|x-y|^\gamma}.
\]
The following regularity class -- a scaling-invariant improvement
of $\frac{1}{2}$-H\"older coherence -- is included,
\begin{equation}\label{hybrid0}
\int_{t_0-(2R)^2}^{t_0} \int_{B(x_0, 2R)} |\omega(x,t)|^2 \
\rho^2_{\frac{1}{2}, 2R}(x,t) dx \, dt < \infty.
\end{equation}
\medskip
On the other hand, the analytic cancelations were utilized in
\cite{GrGu10-2} via a local non-homogeneous Div-Curl Lemma to obtain
a full spatiotemporal localization of Kozono-Taniuchi generalization
of Beale-Kato-Majda regularity criterion; namely, the
time-integrability of the $BMO$ norm of the vorticity.
\section{3D enstrophy cascade}
Let $\mathcal{R}$ be a region contained in the global spatial domain
$\Omega$. The inward enstrophy flux through the boundary of the
region is given by
\[
- \int_{\partial\mathcal{R}} \frac{1}{2}|\omega|^2 (u\cdot n) \, d\sigma
= - \int_\mathcal{R} (u\cdot\nabla)\omega \cdot \omega \, dx
\]
where $n$ denotes the outward normal (taking into account
incompressibility of the flow). Localization of the evolution of the
enstrophy to cylinder $B(x_0,2R) \times (0,T)$ (cf. Section 3) leads
to the following version of the enstrophy flux,
\begin{equation}\label{locflux}
\int \frac{1}{2}|\omega|^2 (u\cdot \nabla\phi) \, dx
= - \int (u\cdot\nabla)\omega \cdot \phi \omega \, dx
\end{equation}
(again, taking into account the incompressibility; here,
$\phi=\phi_{x_0,R,T}$ defined in Section 2). Since $\nabla\phi =
(\nabla\psi) \eta$, and $\psi$ can be constructed such that
$\nabla\psi$ points inward -- toward $x_0$ -- (\ref{locflux})
represents \emph{local inward enstrophy flux, at scale $R$} (more
precisely, through the layer $S(x_0,R,2R)$) \emph{around the point
$x_0$}. In the case the point $x_0$ is close to the boundary of the
integral domain $B(0,R_0)$, $\nabla\psi$ is not exactly radial, but
still points inward.
\medskip
Consider a $(K_1,K_2)$ cover $\{B(x_i,R)\}_{i=1}^n$ at scale $R$,
for some $0<R\le R_0$. Local inward enstrophy fluxes, at scale $R$,
associated to the cover elements $B(x_i,R)$, are then given by
\begin{equation}\label{locfluxi}
\int \frac{1}{2}|\omega|^2 (u\cdot \nabla\phi_i) \, dx,
\end{equation}
for $1 \le i \le n$ ($\phi_i = \phi_{x_i,R,T}$). Assuming
smoothness, the identity (\ref{locx0}) written for $B(x_i,R)$ yields
the following expression for time-integrated local fluxes,
\begin{align}\label{loc}
\int_0^t \int \frac{1}{2}|\omega|^2 (u \cdot \nabla\phi_i) \; dx
\; ds
&=
\int \frac{1}{2}|\omega(x,t)|^2\psi_i(x) \; dx + \int_0^t \int
|\nabla\omega|^2\phi_i
\; dx \; ds\notag\\
&- \int_0^t \int \frac{1}{2}|\omega|^2 \bigl((\phi_i)_s+\triangle\phi_i\bigr) \; dx
\; ds\notag\\
&-\int_0^t \int (\omega \cdot \nabla)u \cdot \phi_i \, \omega \; dx
\; ds,
\end{align}
for any $t$ in $(2T/3,T)$ and $1 \le i \le n$.
\medskip
Denoting the time-averaged local fluxes per unit mass associated to
the cover element $B(x_i,R)$ by $\hat{\Phi}_{x_i,R}$,
\begin{equation}\label{locfluxiav}
\hat{\Phi}_{x_i,R} = \frac{1}{t} \int_0^t \frac{1}{R^3} \int
\frac{1}{2}|\omega|^2 (u\cdot \nabla\phi_i) \, dx,
\end{equation}
the main quantity of interest is the ensemble average of
$\{\hat{\Phi}_{x_i,R}\}_{i=1}^n$ in the sense of Section 2; namely,
\begin{equation}\label{PhiR}
\langle\Phi\rangle_R = \frac{1}{n}\sum_{i=1}^n \hat{\Phi}_{x_i,R}.
\end{equation}
\medskip
Since the flux density, $-(u \cdot \nabla)\omega \cdot \omega$, is
an \emph{a priori} sign-varying density, the stability, i.e., the
near-constancy of $\langle\Phi\rangle_R$ -- while the ensemble
averages are being run over all $(K_1,K_2)$ covers at scale $R$ --
will indicate there are no significant sign-fluctuations of the flux
density at the scales comparable or greater than $R$.
\medskip
The main goal of this section is to formulate a set of physically
reasonable conditions on the flow in $B(0,2R_0) \times (0,T)$
implying the positivity and near-constancy of $\langle\Phi\rangle_R$
across a suitable range of scales -- \emph{existence of the
enstrophy cascade}.
\medskip
We will consider the case $\Omega = \mathbb{R}^3$. The main reason
is that in the case of a domain with the boundary, it is necessary
to utilize the full spatial localization (joint in $x$ and $y$) of
the vortex-stretching term given by (\ref{locvst}); this introduces
a number of the lower order terms which -- in turn -- lead to terms
of the form
\[
c_\gamma \frac{1}{R^\gamma} \iint |\omega|^2 \phi_i^{2\rho - 1} \,
dx \, ds
\]
for some $\gamma > 2$ ($c_\gamma$ is a suitable dimensional constant
-- scaling like $R^{\gamma-2}$). Such terms introduce a correction
to the dissipation cut-off in the enstrophy cascade; namely, the
modified Kraichnan scale $\sigma_0$ (see (A2) below) would have to
be replaced by $\sigma_0^\frac{2}{\gamma}$ (times a dimensional
constant). On the other hand, in $\mathbb{R}^3$, the full (spatial)
localization of the vortex-stretching term can be replaced by the
spatial localization in $x$ only,
\begin{equation}\label{vsti}
(\omega \cdot \nabla)u \cdot \phi_i \omega \, (x) =
- c \, P.V. \int \bigl(\omega(x) \times
\omega(y) \bigr) \cdot G_\omega (x,y) \, \phi_i^{\frac{1}{2}}(x) \,
\phi_i^{\frac{1}{2}}(x) \;
dy ;
\end{equation}
the $y$ integral is then split in suitable small and large scales
(similarly to \cite{GrZh06}) without introducing correction terms
(cf. the proof of the main result in this section).
\bigskip
\noindent \textbf{(A1) \, Coherence Assumption}
\medskip
\noindent Denote the vorticity direction field by $\xi$, and let
$M>0$ (large). Assume that there exists a positive constant $C_1$
such that
\[
|\sin\varphi\bigl(\xi(x,t),\xi(y,t)\bigr)| \le C_1
|x-y|^\frac{1}{2}
\]
for any $(x,y,t)$ in $\bigl(B(0,2R_0) \times
B(0,2R_0+R_0^\frac{2}{3}) \times (0,T)\bigr) \cap \{|\nabla u| >
M\}$ ($\varphi(z_1,z_2)$ denotes the angle between the vectors $z_1$
and $z_2$). Shortly, $\frac{1}{2}$-H\"older coherence in the region
of intense fluid activity (large gradients).
\medskip
Note that the previous local regularity results \cite{GrZh06, Gr09}
imply that -- under (A1) -- the \emph{a priori} weak solution in
view is in fact smooth inside $B(0,2R_0) \times (0,T)$, and can,
moreover, be smoothly continued (locally-in-space) past $t=T$; in
particular, we can write (\ref{loc}) with $t=T$.
\medskip
Let us briefly remark that in the aforementioned works on
regularity, the region of intense fluid activity is usually defined
as $\{|\omega| > M\}$ rather that $\{|\nabla u| > M\}$. Cutting-off
at $|\omega| = M$ here would eventually lead to replacing $E_0$ in
the definition of the modified Kraichnan scale $\sigma_0$ (see the
next paragraph) by
\[
E_0=\frac{1}{T}\int \frac{1}{R_0^3} \int \frac{1}{2}|\nabla u|^2
\phi_0^{2\rho-1} \, dx \, dt,
\]
and we prefer keeping $\sigma_0$ solely in terms of $\omega$.
\bigskip
\noindent \textbf{(A2) \, Modified Kraichnan Scale}
\medskip
\noindent Denote by $E_0$ time-averaged enstrophy per unit mass
associated with the integral domain $B(0,2R_0) \times (0,T)$,
\[
E_0=\frac{1}{T}\int \frac{1}{R_0^3} \int \frac{1}{2}|\omega|^2
\phi_0^{2\rho-1} \, dx \, dt,
\]
by $P_0$ a modified time-averaged palinstrophy per unit mass,
\[
P_0= \frac{1}{T}\int \frac{1}{R_0^3} \int |\nabla\omega|^2
\phi_0 \, dx \, dt
+ \frac{1}{T}\frac{1}{R_0^3} \int \frac{1}{2}|\omega(x,T)|^2
\psi_0(x) \, dx
\]
(the modification is due to the shape of the temporal cut-off
$\eta$), and by $\sigma_0$ a corresponding modified Kraichnan scale,
\[
\sigma_0=\biggl(\frac{E_0}{P_0}\biggr)^\frac{1}{2}.
\]
\medskip
Then, the assumption (A2) is simply a requirement that the modified
Kraichnan scale associated with the integral domain $B(0,2R_0)
\times (0,T)$ be dominated by the integral scale,
\[
\sigma_0 < \beta R_0,
\]
for a constant $\beta$, $0 < \beta < 1$, $\beta = \beta
(\rho,K_1,K_2,M,B_T)$, where $\displaystyle{B_T = \sup_{t \in (0,T)}
\|\omega(t)\|_{L^1}}$; this is finite provided the initial vorticity
is a finite Radon measure \cite{Co90}.
\bigskip
\noindent \textbf{(A3) \, Localization and Modulation}
\medskip
\noindent The general set up considered is one of the weak Leray solutions
satisfying (A1). As already mentioned, (A1) implies smoothness; however,
the control on regularity-type norms is only local. On the other hand,
the energy inequality on the global spatiotemporal domain
$\mathbb{R}^3 \times (0,T)$ implies
\[
\int_0^T \int_{\mathbb{R}^3} |\omega|^2 \, dx \, dt < \infty;
\]
consequently, for a given constant $C_2 > 0$, there exists $R_0^* >
0$ ($R_0^* \le \min \{\sqrt{T},1\}$; this is mainly for convenience)
such that
\begin{equation}\label{a3.1}
\int_0^T \int_{B(0,2R_0+R_0^\frac{2}{3})} |\omega|^2 \, dx \, dt
\le \frac{1}{C_2}
\end{equation}
for any $0 < R_0 \le R_0^*$. This is the localization assumption on
$R_0$; the precise value of the constant $C_2$ is given in the proof
of the theorem -- right after the inequality (\ref{4}).
\medskip
The modulation assumption on the evolution of local enstrophy on
$(0,T)$ -- consistent with the choice of the temporal cut-off $\eta$
-- reads
\[
\int |\omega(x,T)|^2 \psi_0(x) \, dx \ge \frac{1}{2} \sup_{t \in
(0,T)} \, \int |\omega(x,t)|^2 \psi_0(x) \, dx.
\]
\medskip
\begin{obs}
\emph{Assumption (A1) is simply a quantification of the degree of
local coherence of the vorticity direction -- a manifestation of the
local `quasi 2D-like' behavior of turbulent flows -- needed to
sufficiently deplete the nonlinearity. Assumption (A2) is a slight
modification of the condition implying existence of the enstrophy
cascade in 2D (cf. \cite{DaGr11-2}); namely, a requirement that the
modified Kraichnan scale be dominated by the integral scale. Once
unraveled, it postulates that the region of interest exhibits large
vorticity gradients (relative to the vorticity magnitude, in the
spatiotemporal average) -- indicating high spatial complexity of the
flow -- and is the enstrophy analogue of the condition implying
existence of the energy cascade in 3D (\cite{DaGr11-1}), i.e., large
velocity gradients (relative to the velocity magnitude, in the
spatiotemporal average). The last assumption is somewhat technical;
however, its purpose within our theory is physical -- to prevent
uncontrolled temporal fluctuations of $R_0$-scale enstrophy. Such
fluctuations would inevitably prevent the cascade over $B(0,R_0)$
(the region of interest).}
\end{obs}
\medskip
\begin{thm}\label{cascade}
Let $u$ be a Leray solution on $\mathbb{R}^3 \times (0,T)$ with the
initial vorticity $\omega_0$ being a finite Radon measure. Suppose
that $u$ satisfies (A1)-(A3) on the spatiotemporal integral domain
$B(0, 2R_0 + R_0^\frac{2}{3}) \times (0,T)$. Then,
\[
\frac{1}{4K_*} P_0 \le \langle\Phi\rangle_R \le 4K_* \ P_0
\]
for all $R, \, \frac{1}{\beta}\sigma_0 \le R \le R_0$ ($K_* > 1$ is
the constant in (\ref{k*})).
\end{thm}
\begin{proof}
Throughout the proof, $(K_1,K_2)$ cover parameters $\rho, K_1$ and
$K_2$, as well as the gradient cut-off $M$ will be fixed.
Henceforth, the quantities depending only on $\rho,K_1,K_2,M,B_T$
will be considered constants and denoted by a generic $K$ (that may
change from line to line).
\medskip
Let $0 < R \le R_0$. As already noted, we can write the expression
for a time-integrated local flux corresponding to the cover element
$B(x_i,R)$, (\ref{loc}), for $t=T$,
\begin{align}\label{locT}
\int_0^T \int \frac{1}{2}|\omega|^2 (u \cdot \nabla\phi_i) \; dx
\; ds
&=
\int \frac{1}{2}|\omega(x,T)|^2\psi_i(x) \; dx + \int_0^T \int
|\nabla\omega|^2\phi_i
\; dx \; ds\notag\\
&- \int_0^T \int \frac{1}{2}|\omega|^2 \bigl((\phi_i)_s+\triangle\phi_i\bigr) \; dx
\; ds\notag\\
&-\int_0^T \int (\omega \cdot \nabla)u \cdot \phi_i \, \omega \; dx
\; ds.
\end{align}
\medskip
The last two terms on the right-hand side need to be estimated.
\medskip
For the first term, the properties of the cut-off $\phi_i$ together
with the condition $T \ge R_0^2 \ge R^2$ yield
\begin{equation}\label{1}
\int_0^T \int \frac{1}{2}|\omega|^2
\bigl((\phi_i)_s+\triangle\phi_i\bigr) \; dx \; ds \le K
\frac{1}{R^2} \int_0^T \int |\omega|^2 \phi_i^{2\rho-1} \; dx \; ds.
\end{equation}
\medskip
For the second term, the vortex-stretching term
\[
\int_0^T \int (\omega \cdot \nabla)u \cdot \phi_i \, \omega \; dx
\; ds,
\]
the integration is first split into the regions in which $|\nabla u|
\le M$ and $|\nabla u| > M$. In the first region, the integral is
simply dominated by
\begin{equation}\label{2}
K \frac{1}{R^2} \int_0^T \int |\omega|^2 \phi_i^{2\rho-1} \; dx \;
ds
\end{equation}
($R \le R_0 \le 1$). In the second region, and for a fixed $x$, we
divide the domain of integration in the representation formula
(\ref{vsti}),
\[
(\omega \cdot \nabla)u \cdot \phi_i \omega \, (x) =
- c \, P.V. \int \bigl(\omega(x) \times
\omega(y) \bigr) \cdot G_\omega (x,y) \, \phi_i^{\frac{1}{2}}(x) \,
\phi_i^{\frac{1}{2}}(x) \,
dy,
\]
into the regions outside and inside the sphere $\{y: \,
|x-y|=R^\frac{2}{3}\}$. In the first case, the integral is bounded
by
\begin{align}\label{3}
\iint_{\{|\nabla u|>M\}} & \int_{\{y: \, |x-y| > R^\frac{2}{3}\}}
\frac{1}{|x-y|^3} |\omega| \, dy \ \phi_i |\omega|^2 \, dx \,
ds\notag\\
& \le K \frac{1}{R^2} \, \sup_t \, \|\omega(t)\|_{L^1} \int_0^T \int
|\omega|^2 \phi_i^{2\rho-1} \; dx \; ds\notag\\
& \le K \frac{1}{R^2} \int_0^T \int |\omega|^2 \phi_i^{2\rho-1} \;
dx \; ds.
\end{align}
In the second case -- utilizing (A1), the Hardy-Littlewood-Sobolev
and the Gagliardo-Nirenberg interpolation inequalities, and the
localization part of (A3) -- the following string of bounds
transpires.
\begin{align}\label{4}
\iint_{\{|\nabla u|>M\}} & \int_{\{y: \, |x-y| < R^\frac{2}{3}\}}
\frac{1}{|x-y|^\frac{5}{2}} |\omega| \, dy \ \phi_i |\omega|^2 \, dx
\,
ds\notag\\
& \le K \int_0^T \|\omega\|_{L^2\bigl(B(x_0,
2R_0+R_0^\frac{2}{3})\bigr)}
\| \, |\phi_i^\frac{1}{2} \omega|^2 \, \|_\frac{3}{2} \, ds \notag\\
& \le K \int_0^T \|\omega\|_{L^2\bigl(B(x_0,
2R_0+R_0^\frac{2}{3})\bigr)} \|\phi_i^\frac{1}{2} \omega\|_2 \,
\|\nabla(\phi_i^\frac{1}{2}
\omega)\|_2 \, ds\notag\\
& \le K \ \biggl(\int_0^T \|\omega\|_{L^2\bigl(B(x_0,
2R_0+R_0^\frac{2}{3})\bigr)}^2 \, ds\biggr)^\frac{1}{2} \,
\biggl(\frac{1}{2} \sup_{t \in (0,T)} \|\psi_i^\frac{1}{2}
\omega\|_2^2 + \int_0^T \|\nabla(\phi_i^\frac{1}{2} \omega)\|_2^2 \,
ds\biggr)\notag\\
& \le \frac{1}{8K_*^2} \ \biggl(\frac{1}{2} \sup_{t \in (0,T)}
\|\psi_i^\frac{1}{2} \omega\|_2^2 + \int_0^T
\|\nabla(\phi_i^\frac{1}{2}
\omega)\|_2^2 \, ds\biggr),\notag\\
\end{align}
where $K_*$ is the constant in (\ref{k*}) (in the last line, we used
the localization assumption (\ref{a3.1}) with $C_2 = 64 K^2 K_*^4$).
Incorporating the estimate
\begin{align}\label{5}
\int |\nabla(\phi_i^\frac{1}{2} & \omega)|^2 \, dx\notag\\
& \le 2 \int |\nabla\omega|^2 \phi_i \, dx + c \int
\biggl(\frac{|\nabla\phi_i|}{\phi_i^\frac{1}{2}}\biggr)^2 \,
|\omega|^2 \, dx\notag\\
& \le 2 \int |\nabla\omega|^2 \phi_i \, dx + K \frac{1}{R^2}
\int |\omega|^2 \phi_i^{2\rho-1} \, dx\notag\\
\end{align} in
(\ref{4}), we arrive at the final bound,
\begin{align}\label{6}
\iint_{\{|\nabla u|>M\}} & \int_{\{y: \, |x-y| < R^\frac{2}{3}\}}
\frac{1}{|x-y|^\frac{5}{2}} |\omega| \, dy \ \phi_i |\omega|^2 \, dx
\, ds\notag\\
& \le \frac{1}{4K_*^2} \ \biggl(\frac{1}{2} \sup_{t \in (0,T)}
\|\psi_i^\frac{1}{2} \omega\|_2^2 + \int_0^T \| \nabla\omega \,
\phi_i^\frac{1}{2}\|_2^2 \, ds\biggr) + K \frac{1}{R^2} \int
|\omega|^2 \phi_i^{2\rho-1} \, dx.\notag\\\notag\\
\end{align}
Collecting the estimates (\ref{1})-(\ref{6}) and using the
modulation part of (A3), the relation (\ref{locT})
yields
\begin{align}\label{locTT}
\int_0^T \int \frac{1}{2}|\omega|^2 (u \cdot \nabla\phi_i) \; dx
\; ds
&=
\int \frac{1}{2}|\omega(x,T)|^2\psi_i(x) \; dx + \int_0^T \int
|\nabla\omega|^2\phi_i
\; dx \; ds \, + \, Z\notag\\
\end{align}
where
\[
Z \le \frac{1}{2K_*^2} \ \biggl(\frac{1}{2}
\|\psi_i^\frac{1}{2}(\cdot) \omega(\cdot,T)\|_2^2 + \int_0^T \| \nabla\omega \,
\phi_i^\frac{1}{2}\|_2^2 \, ds\biggr) + K \frac{1}{R^2} \int
|\omega|^2 \phi_i^{2\rho-1} \, dx.
\]
Taking the ensemble averages and exploiting (\ref{k*}) multiple
times, we arrive at
\[
\frac{1}{4K_*} P_0 \le \langle \Phi \rangle_R \le 4K_* P_0
\]
for all $\frac{1}{\beta} \sigma_0 \le R \le R_0$, and a suitable
$\beta=\beta(\rho,K_1,K_2,M,B_T)$.
\end{proof}
\begin{obs}
\emph{The first mathematical result on existence of 2D enstrophy
cascade is in the paper by Foias, Jolly, Manley and Rosa
\cite{FJMR02}; the general setting is the one of infinite-time
averages in the space-periodic case, and the cascade is in the
Fourier space, i.e., in the wavenumbers. A recent work
\cite{DaGr11-3} provides existence of 2D enstrophy cascade in the
physical space utilizing the general mathematical setting for the
study of turbulent cascade in physical scales introduced in
\cite{DaGr11-1}. To the best of our knowledge, the present paper is
the first rigorous result concerning existence of the enstrophy
cascade in 3D.}
\end{obs}
\medskip
The second theorem concerns \emph{locality} of the flux. According
to turbulence phenomenology, the average flux at scale $R$ --
throughout the inertial range -- is supposed to be well-correlated
only with the average fluxes at nearby scales. In particular, the
locality along the \emph{dyadic scale} is expected to propagate
\emph{exponentially}.
\medskip
Denoting the time-averaged local fluxes associated to the cover
element $B(x_i,R)$ by $\hat{\Psi}_{x_i,R}$,
\begin{equation}\label{locfluxiavv}
\hat{\Psi}_{x_i,R} = \frac{1}{T} \int_0^T \int
\frac{1}{2}|\omega|^2 (u\cdot \nabla\phi_i) \, dx,
\end{equation}
the (time and ensemble) averaged flux is given by
\begin{equation}\label{PsiR}
\langle\Psi\rangle_R = \frac{1}{n}\sum_{i=1}^n \hat{\Psi}_{x_i,R} =
R^3 \, \langle\Phi\rangle_R.
\end{equation}
\medskip
The following locality result is a simple consequence of the
universality of the cascade of the time and ensemble-averaged local
fluxes \emph{per unit mass} $\langle\Phi\rangle_R$ obtained in
Theorem 4.1.
\begin{thm}\label{locality}
Let $u$ be a Leray solution on $\mathbb{R}^3 \times (0,T)$ with the
initial vorticity $\omega_0$ being a finite Radon measure. Suppose
that $u$ satisfies (A1)-(A3) on the spatiotemporal integral domain
$B(0, 2R_0 + R_0^\frac{2}{3}) \times (0,T)$, and let $R$ and $r$ be
two scales within the inertial range delineated in Theorem 4.1. Then
\[
\frac{1}{16K_*^2} \biggl(\frac{r}{R}\biggr)^3 \le \frac{\langle \Psi
\rangle_r}{\langle \Psi \rangle_R} \le 16K_*^2
\biggl(\frac{r}{R}\biggr)^3.
\]
In particular, if $r=2^k R$ for some integer $k$, i.e., through the
dyadic scale,
\[
\frac{1}{16K_*^2} \ 2^{3k} \le \frac{\langle \Psi \rangle_{2^k
R}}{\langle \Psi \rangle_R} \le 16K_*^2 \ 2^{3k}.
\]
\end{thm}
\begin{obs}
\emph{Previous locality results include locality of the flux via a
smooth filtering approach presented in \cite{E05} (see also
\cite{EA09}), and locality of the flux in the Littlewood-Paley
setting obtained in \cite{CCFS08}. The aforementioned results are
derived independently of existence of the inertial range, and are
essentially \emph{kinematic upper bounds} on the localized
flux in terms of a
suitable physical quantity localized to the nearby scales; the
corresponding lower bounds hold assuming saturation of certain
inequalities consistent with the turbulent behavior. In
contrast, our result is derived \emph{dynamically} as a direct
consequence of existence of the turbulent cascade in view, and
features \emph{comparable upper and lower bounds} throughout the
inertial range.}
\end{obs}
\bigskip
\bigskip
\noindent ACKNOWLEDGEMENTS \ The authors thank an anonymous referee
for a number of suggestions that lead to the present version of the
paper. Z.G. acknowledges the support of the \emph{Research Council of
Norway} via the grant number 213473 - FRINATEK.
\bigskip
\bigskip
|
2,869,038,154,866 | arxiv | \section{Acknowledgements}
\bibliographystyle{IEEEtran}
\section{Introduction}
Singing voice synthesis (SVS) uses music score and lyrics to generate natural singing voices of a target singer. Recently, SVS has attracted much research attention in the speech and music processing communities. However, due to high data annotation costs and strict musical copyright policies, there are limited public databases available for SVS. The shortage of high-quality data further leads to limited benchmarks in SVS, compared to various open-source activities in other generative tasks such as text-to-speech (TTS) and voice conversion (VC) \cite{schroder2011open, hayashi2021espnet2, tts_mozilla, tts_coqui}.
This paper introduces a new open-source toolkit called \textit{\textbf{Muskits}} that aims to provide a neural network-based end-to-end platform for music processing, specifically SVS. Unlike existing open-source SVS repositories \cite{oura2010recent, nnsvs, chandna2019wgansing, tae2021mlp, choi2020korean, shi2021sequence, wang2022opencpop}, Muskits provides different architectures for SVS in an end-to-end manner and offers a wide-range comparison between different architectures for singing voice synthesis. Muskits inherits the base framework from ESPnet \cite{hayashi2021espnet2, watanabe2018espnet} and follows ESPnet styles in data processing scripts and training recipes, demonstrating a complete setup for SVS. The contributions of this paper include: (1) a new platform for reproducible SVS modeling; (2) benchmarks on several public databases in both single- and multi-singer scenarios; (3) exploration on multilingual training for SVS; (4) studies on transfer learning
on a multi-style low-resource database.
\section{Related Works}
The initial effort on SVS starts with unit-selection models, where pre-recorded singing segments are concatenated to form the singing voice \cite{kenmochi2007vocaloid, bonada2016expressive}. However, because of the need for large corpora, this method is not flexible to achieve reasonable sound in low-resource scenarios. Later, statistical parametric methods (e.g., the hidden Markov model (HMM)), were proposed for singing synthesis \cite{saino2006hmm, oura2010recent}. They were shown to flexibly synthesize the singing voice, but also led to loss of naturalness. Deep neural networks (DNNs) have become dominant across several generation tasks in the recent decade. SVS is not an exception. Initially, several works applied basic structures, including fully-connected layers, convolutional neural networks (CNNs), and recurrent neural networks (RNNs) \cite{nishimura2016singing, kim2018korean, nakamura2019singing, hono2018recent}, which demonstrated reasonable quality and naturalness for singing generation. Some more advanced architectures, including generative adversarial networks (GANs), encoder-decoder, and diffusion-based models, also showed some quality improvements over specific databases \cite{blaauw2020sequence, gu2021bytesing, lu2020xiaoicesing, hono2019singing, lee2019adversarially, liu2021diffsinger, zhang2021visinger}.
As SVS is receiving more and more attention, there has been an increasing need for open-source platforms so as to allow fair comparison between the proposed models. Compared to similar tasks such as TTS, SVS has faced several issues in sharing benchmarks and their platforms. Different from speech, singing voice databases usually have more strict usage guidelines and copyright concerns. Meanwhile, singing voices require a professional high-quality recording environment and generally take longer to record than that for regular speech. Resources are further limited due to the need for detailed annotation (e.g., music score information).
Even with those limitations, there are still some relevant efforts in releasing databases and code bases for some SVS models. Sinsy \cite{oura2010recent, hono2018recent}, known as one of the initial open-source works in SVS, was released for research purposes. The toolkit applies the HMM-based architecture and can reach reasonable performances with low-resource singing voice corpus (e.g., 1 hour). It also demonstrates its extendability to multiple languages, including Japanese, Chinese, and English. NN-SVS \cite{nnsvs} is a DNN-based singing voice synthesis toolkit, which follows the cascaded neural networks in \cite{hono2018recent}. Seven Japanese recipes are supported in the toolkit. We also observe several open-source efforts related to specific models, including \cite{chandna2019wgansing, tae2021mlp, choi2020korean, shi2021sequence, wang2022opencpop}. They mostly include the whole training procedure and some also release the related databases \cite{tae2021mlp, wang2022opencpop}.
The aforementioned open-source efforts have greatly benefited the whole research community. However, it is difficult to provide a fair comparison of different models due to the differences in datasets, feature representations, and vocoders. In this work, our goal is to fill the gap with a new open-source toolkit, Muskits. With 12 recipes on 10 SVS databases, the toolkit has supported several choices of end-to-end SVS models with various ESPnet compatible vocoders\footnote{\scriptsize {\url{https://github.com/kan-bayashi/ParallelWaveGAN}}}. Furthermore, we explore multilingual training, pre-training, and transfer learning on a multi-style databases using Muskits, which have not been well investigated in SVS before.
\section{Functionality}
Muskits consists of two major parts: a Python library of several neural network models and recipes for running complete experiments on specific databases. The library uses PyTorch for neural networks, with standard music representation, various model architectures, and a uniform training/decoding framework. The recipes provide detailed examples in all-in-one style scripts, following the style in Kaldi and ESPnet \cite{hayashi2021espnet2, watanabe2018espnet, povey2011kaldi}.
\subsection{Music score representation}
\label{ssec: music representation}
Muskits performs all pre-processing steps on the fly, including acoustic feature extraction and two standard interfaces for music score information. The two interfaces are at the frame-level and syllable-level. The frame-level features need pre-aligned linguistic units (e.g., phoneme or syllable) and music note sequences. Based on timestamps, both sequences are first expanded to the sampling rate of a given audio (e.g., 24k). Then, they are aggregated with a sliding window that has the same window size and hop length to the target acoustic features (e.g., log-Mel filter-bank features). The same or similar features are used in \cite{shi2021sequence, kim2018korean, nakamura2019singing, hono2018recent, tae2021mlp}. The syllable-level features are an extended version of XiaoiceSing feature \cite{lu2020xiaoicesing}, including phoneme ids, music note ids\footnote{The music note ids are the same as MIDI 0-128 note ids.}, tempo/beats information, and corresponding duration information. Depending on the models and application scenarios, either frame-level or syllable-level features could be selected. Note that we always use syllable-level features for models that consider duration modeling (e.g., various sequence-to-sequence models) \cite{chen2020hifisinger, blaauw2020sequence, lu2020xiaoicesing, shi2021sequence}.
\subsection{Models}
Muskits supports three models, including a RNN-based model \cite{kim2018korean, shi2021sequence}, transformer-based model \cite{blaauw2020sequence}, and XiaoiceSing \cite{lu2020xiaoicesing}. All models are based on the encoder-decoder architecture and support both frame and syllable-level representations discussed in Sec~\ref{ssec: music representation}. The output of models is a sequence of acoustic features (e.g., Mel-filter bank features).
Similar to \cite{lu2020xiaoicesing, chen2020hifisinger, shi2021sequence}, in the RNN-based model, music notes and phoneme information are first passed to individual RNN encoders. Then, the combination of hidden states (either by concatenating or adding) is fed into another RNN decoder. If syllable-level representations are used for the RNN-based model, an additional duration predictor is introduced to expand the syllable-level hidden states to the frame level. The duration predictor is trained separately to predict the explicit duration of each hidden state. In training, the encoder-decoder model adopts ground truth duration. While in inference, the encoder output will be expanded by the predicted duration, and then fed into the decoder. The transformer-based model follows the architecture of \cite{blaauw2020sequence}. It first encodes the phoneme sequence by stacked gated linear unit (GLU) layers. Then, it expands the hidden representation of phonemes into frame-level with a rule-based duration model. After adding the pitch embedding and positional encoding, the hidden states are then fed into a transformer-based decoder for final outputs. The duration of phoneme sequences is defined using rules \cite{blaauw2020sequence}. XiaoiceSing adopts a FastSpeech-like architecture \cite{lu2020xiaoicesing, ren2020fastspeech}. The model sums over the syllable feature representations and adopts an encoder-decoder framework for singing voice generation.
\subsection{Training, inference, and evaluation}
\label{ssec: train, inference, eval}
The training procedures in Muskits are handled with a unified task processor adapted from ESPnet, which supports multi-GPU training and dynamic batch making. On-the-fly data augmentation is supported, including pitch augmentation and mixup-augmentation \cite{guo2022singaug}.
In addition to the default Griffin-Lim vocoder, Muskits also supports various neural vocoders, including ParallelWaveGAN, MelGAN, and HiFiGAN \cite{yamamoto2020parallel, kumar2019melgan, yang2021multi, mustafa2021stylemelgan, kong2020hifi}. The vocoder can be applied to both training and inference stages. In training, the vocoder could automatically generate audio from selected samples of development set for monitoring the training progress.
During inference, compatible vocoders can be used directly without additionally dumping intermediate features.
We implement three objective evaluation metrics for SVS in Muskits: Mel-cepstral distortion (MCD), voiced/unvoiced error rate (VUV\_E), and logarithmic rooted mean square error of the fundamental frequency (F$_0$RMSE). All the metrics are calculated by dynamic time-warping to consider the length mismatch between synthesized singing and ground truth singing.
\subsection{Recipe flow}
We provide 12 all-in-one recipes for 10 singing voice databases. All recipes follow a unified pipeline with explicit stages of preparation, training, inference, and evaluation. The stages are defined in the template \texttt{svs.sh}, as follows:
\noindent
\textit{\textbf{Stage 1}: Database-dependent data preparation}. The expected data generally follows Kaldi style but with additional \texttt{midi.scp} and \texttt{label} for music score and phoneme alignment information. For databases without explicit midi or MusicXML, we also provide rule-based automatic music transcription to extract related music information. Relevant functions can be found in KiSing recipe \cite{shi2020kising}.
\noindent
\textit{\textbf{Stage 2}: Standard audio and midi formatting}. This stage formats audio into a standard format by resampling, segmentation, and dumping from pipe-style input. MIDI is also normalized and segmented at this stage. Singer ID and Language ID are also defined if needed.
\noindent
\textit{\textbf{Stage 3}: Filtering}. Phrases in the training and development sets are filtered by a pre-defined length threshold.
\noindent
\textit{\textbf{Stage 4}: Token list generation}. Muskits collects tokens from the training set and generates a corresponding token list for training.
\noindent
\textit{\textbf{Stage 5}: Statistics collection}. The input information of training and development sets are collected for efficient batching.
\noindent
\textit{\textbf{Stage 6}: Model training}. The model is optimized based on the training objectives.
\noindent
\textit{\textbf{Stage 7}: Model inference}. If no vocoder is provided, Muskits adopts the Griffin-Lim vocoder. Otherwise, compatible vocoders can be loaded for inference. The predicted acoustic features are also stored for external vocoders.
\noindent
\textit{\textbf{Stage 8}: Objective evaluation}. Objective evaluation is conducted between paired reference data and inference outputs.
\noindent
\textit{\textbf{Stage 9}: Model packing}. The model is packed with all the necessary elements for inferences. The packed models of our recipes are also shared with the recipes.
Most public SVS databases only have a single singer, which limits the scalability of the SVS systems built upon them. By supporting several recipes with a standardized training interface, Muskits could easily extend to different genders, multi-singer, multilingual scenarios. To be specific, we already support two recipes on multi-singer and multilingual cases.
\section{Experiments}
\subsection{Experimental datasets}
\label{ssec: dataset}
To demonstrate the performance of Muskits, we conduct four sets of experiments under single-singer, multi-singer, multilingual, and transfer learning scenarios. For single-singer experiments, we use the Ofuton-P database \cite{futon2021ofuton} that contains 56 Japanese songs (61 minutes) of a male singer. For multi-singer experiments, we use a combination of four databases, including Ofuton-P, Oniku, Natsume, and Kiritan \cite{futon2021ofuton, kurumi2021oniku, amanokei2020natsume, moris2020kitian}. For multilingual training, we adopt the four databases mentioned above, CSD databases \cite{choi2020children}, and Opencpop \cite{wang2022opencpop}, which results in a ten-hour corpus in Japanese, English, Korean, and Mandarin. For transfer learning, we adopt the KiSing corpus, a one-hour female Mandarin corpus with multiple styles \cite{shi2020kising}.
Except for Opencpop and KiSing, other databases do not have official splits of train, development, and test sets, so we apply our own split in our reproducible recipes. All the songs are segmented based on the silence between lyrics. The data sizes for each experiment are listed in Table~\ref{tab: dataset}. For CSD, only syllables are provided for the alignment, but the training set is not enough to cover all the syllable combinations. Therefore, we split the syllable into phonemes and evenly assign duration information to each phoneme given the duration of syllables.
\begin{table}
\centering
\caption{\label{tab: dataset} Details of experimental data: \textbf{N$_{\text{src}}$}, \textbf{N$_{\text{lang}}$}, \textbf{N$_{\text{singer}}$} stands for number of databases, languages, and singers, respectively.}
\vspace{-6pt}
\begin{tabular}{l|c|ccc}
\toprule \multirow{2}{*}{\textbf{Task}} & \multirow{2}{*}{\textbf{N$_{\text{src}}$, N$_{\text{lang}}$, N$_{\text{singer}}$}} & \multicolumn{3}{|c}{\textbf{Duration(h)}} \\
& & \textbf{Train} & \textbf{Dev} & \textbf{Test} \\
\midrule
Single-singer & 1,1,1 & 0.63 & 0.08 & 0.07 \\
Multi-singer & 4,1,4 & 3.01 & 0.34 & 0.38 \\
Multilingual & 6,4,6 & 11.72 & 0.49 & 1.01\\
Transfer & 1,1,1 & 0.71 & 0.02 & 0.05\\
\bottomrule
\end{tabular}
\vspace{-15pt}
\end{table}
\vspace{-4pt}
\subsection{Experimental setups}
\label{ssec: exp_setup}
\vspace{-4pt}
The experiments are in two folds based on whether the model uses ground truth (G.T.) duration in order to factorize the evaluation of duration prediction and acoustic modeling.
The RNN-based model adopts three-layer 512-dimension bi-directional long short time memory (LSTM) encoder and a five-layer 1024-dimension decoder. The transformer-based model follows the original setting in \cite{blaauw2020sequence}. It consists of three 256-channel GLU layers with a kernel size of three as the encoder and a six-layer four-head-256-dimension transformer decoder. Local Gaussian constraints are injected into the self-attention in its decoder. The XiaoiceSing adopts the same network hyper-parameters as in \cite{lu2020xiaoicesing}, including an encoder and a decoder with the same setting (i.e., six transformer blocks, while each block has four-head-384-dimension self-attention block and 1536 dimensional feed-forward layer).
The decoder outputs of all models are projected to the dimension of acoustic features (i.e., 80) with a postnet inherited from Tacotron2 \cite{shen2018natural}. For settings with duration prediction, the transformer-based model adopts a rule-based duration modeling as defined in \cite{blaauw2020sequence}, while the others adopt two 384-channel convolutional layers with a kernel size of three to model duration information.
For multi-singer and multilingual training, we provide the option to apply additional embeddings from one-hot singer IDs and language IDs. The embedding size is the same as the dimension of the encoder outputs for all models. In the forward process, embeddings are added directly to the hidden states of encoders. As discussed in Sec~\ref{ssec: train, inference, eval}, the acoustic models can be easily used with compatible vocoders. We used the HiFi-GAN vocoder for all experiments. The vocoders are trained on the same training set from the Muskits' split introduced in Sec~\ref{ssec: dataset}.
During training, the RNN-based model uses Adam optimizer with a 0.001 learning rate without learning rate schedulers to keep the same as its original paper \cite{shi2021sequence}. All the other models employ Adam optimizer with a NoamLR scheduler. The warm-up steps are set to 4,000, and the peak learning rate is set to 1.0. We employ gradient clipping at the threshold of 1.0. All experiments are conducted on a single GPU and the batch size is set to 8. We apply a 0.1 dropout rate to encoder/decoder and a 0.5 dropout rate to postnets. We set the training epochs as 500 for models trained on single-singer databases and 250 for models trained on multi-singer or multilingual data.
As in Sec~\ref{ssec: train, inference, eval}, we adopt MCD, V/UV Error, and F$_0$RMSE for objective evaluation. For each subjective evaluation conducted online, we invite 25 subjects, including both musicians and non-professionals. The same set of 15 phrases are randomly selected for each system. Subjects are asked to score the phrases in one (unintelligible) to five (excellent quality). We report our results with 95\% confidence intervals. We also conduct the A/B test in the transfer learning scenario. The same 25 subjects are asked to compare two samples of the same phrase from a pair of models. 20 phrases are selected from the KiSing test set.
\subsection{Results and discussion}
\noindent\textbf{{Single singer:}}
Table~\ref{tab: models exp} presents the results of the single-singer setting. Among models using G.T. duration, the RNN-based model achieves the best performance over the subjective MOS metric; the Transformer-based model achieves the best MCD; XiaoiceSing achieves the best VUV\_E and F$_0$RMSE. For models without G.T. duration, the RNN-based model achieves the best performance on both objective and subjective metrics. We observe that there is a huge gap between models with and without G.T. duration. The primary reason could be the limited size of the Ofuton-P database (37.8 minutes of singing for training), which causes the difficulty of learning reasonable duration predictors. Moreover, the results in the subjective evaluation suggest that the RNN-based model is better than the other two, and the Transformer-based model is better than XiaoiceSing, in our single-singer scenario. This may indicate that simpler structures are more suitable for the task with low data-resource.
\begin{table*}
\centering
\caption{\label{tab: models exp}: Single- and multi-singer experiments: The experimental databases are Ofuton-P database for single-singer experiments and a combination of four databases as described in Sec~\ref{ssec: dataset}; the models are discussed in Sec~\ref{ssec: exp_setup}; the metrics are defined in Sec~\ref{ssec: train, inference, eval}. }
\vspace{-6pt}
\begin{tabular}{l|c|cccc|cccc}
\toprule \multirow{2}{*}{\textbf{Model}} & \textbf{G.T.} & \multicolumn{4}{c|}{\textbf{Single-Singer}} & \multicolumn{4}{c}{\textbf{Multi-Singer}} \\
& \textbf{Dur.} & \textbf{MCD$\downarrow$} & \textbf{VUV\_E$\downarrow$} & \textbf{F$_0$RMSE$\downarrow$} & \textbf{MOS$\uparrow$} & \textbf{MCD$\downarrow$} & \textbf{VUV\_E$\downarrow$} & \textbf{F$_0$RMSE$\downarrow$} & \textbf{MOS$\uparrow$}\\
\midrule
RNN \cite{shi2021sequence} & \cmark & 6.52 & 2.30 & 0.105 & \textbf{3.47 $\pm$ 0.11} & 7.15 & 3.87 & 0.195 & 3.25 $\pm$ 0.13 \\
Transformer \cite{blaauw2020sequence} & \cmark & \textbf{6.46} & 2.67 & 0.118 & 3.23 $\pm$ 0.11 & \textbf{6.68} & 3.90 & \textbf{0.184} & \textbf{3.40 $\pm$ 0.12} \\
XiaoiceSing \cite{lu2020xiaoicesing} &\cmark & 6.88 & \textbf{2.22} & \textbf{0.103} & 3.11 $\pm$ 0.11 & 6.81 & \textbf{3.86} & 0.190 & 3.23 $\pm$ 0.12\\
\midrule
RNN \cite{shi2021sequence} & / & \textbf{6.19} & \textbf{2.18} & \textbf{0.109} & \textbf{3.43 $\pm$ 0.11} & 7.05 & \textbf{3.55} & 0.226 & 2.84 $\pm$ 0.12 \\
Transformer \cite{blaauw2020sequence} & / & 6.95 & 5.61 & 0.172 & 2.39 $\pm$ 0.11 & 6.87 & 3.70 & 0.186 & \textbf{3.33 $\pm$ 0.12} \\
XiaoiceSing \cite{lu2020xiaoicesing} & / & 7.19 & 2.68 & 0.120 & 1.96 $\pm$ 0.08 & \textbf{6.79} & 3.85 & \textbf{0.154} & 2.83 $\pm$ 0.11\\
\midrule
G.T. & / & / & / & / & 4.89 $\pm$ 0.05 & / & / & / & 4.65 $\pm$ 0.10\\
\bottomrule
\end{tabular}
\vspace{-10pt}
\end{table*}
\vspace{3pt}
\noindent\textbf{{Multiple singer:}}
Table~\ref{tab: models exp} presents the experiment results of the multi-singer experiments setting. Even though multi-singer corpus has larger data size compared to single-singer, it may encounter additional issues like multiple styles of singing and singer variation. For example, we find singing voices in Oniku database mostly use falsetto, while the other databases use mostly modal voice. As in Table~\ref{tab: models exp}, evaluation metrics on multi-singer may not be necessarily better than single-singer's. However, as we compare the MOS between single and multi-singer scenarios\footnote{Other metrics are not suitable for comparison, given that the test data is different.}, we observe that the RNN-based model has some degradation on MOS, while the other two have some improvements. It might indicate that the transformer-based model and XiaoiceSing have a better modeling capacity and need a certain amount of data to converge. The RNN-based model, on the other hand, can efficiently converge on small data, but has the capacity issue when extending to multi-singer scenarios.
\vspace{3pt}
\noindent\textbf{{Multilingual singing:}}
We conduct five experiments on multilingual SVS and evaluate the performances on Ofuton-P database. The baseline is the XiaoiceSing without duration prediction presented in Table~\ref{tab: models exp}. The pre-train models are trained on the multilingual data introduced in Sec~\ref{ssec: dataset} with the same XiaoiceSing architecture. As in Sec~\ref{ssec: exp_setup}, we examine models with or without language IDs. Based on models trained on multilingual data, we also fine-tune the multilingual models on Ofuton-P database for another 500 epochs. As in Table~\ref{tab: multilingual}, the multilingual models clearly demonstrate a large gain over the MCD objective scores, but do not improve the V/UV error rate and F$_0$RMSE.
In subjective evaluation, directly using multilingual models does not bring improvements on MOS. However, pre-trained models could be useful when applying fine-tuning on the same database. Meanwhile, given that the model with language IDs always getting worse objective and subjective scores, it might indicate that additional language IDs could not improve the singing quality. One potential reason is that language IDs may restrict the model to transfer knowledge from multilingual data especially when the data size is in low-resource.
\begin{table}
\centering
\caption{\label{tab: multilingual} Multilingual experiments on Ofuton-P. All the models use the same XiaoiceSing architecture.
The pre-trained models are conducted over the multilingual data described in Sec.~\ref{ssec: dataset}.
}
\vspace{-10pt}
\begin{tabular}{l|cccc}
\toprule \textbf{Model} & \textbf{MCD$\downarrow$} & \textbf{VUV\_E$\downarrow$} & \textbf{F$_0$RMSE$\downarrow$} & \textbf{MOS$\uparrow$} \\
\midrule
Baseline & 6.88 & \textbf{2.22} & \textbf{0.103} & 3.05 $\pm$ 0.12 \\
Pre-train & \textbf{6.17} & 2.51 & 0.113 & 2.99 $\pm$ 0.12\\
\ + LangID & 6.32 & 2.94 & 0.123 & 2.90 $\pm$ 0.12\\
Fine-tune & 6.23 & 2.75 & 0.106 & \textbf{3.49 $\pm$ 0.12} \\
\ + LangID & 6.34 &2.55 & 0.110 & 3.09 $\pm$ 0.12\\
\midrule
G.T. & / & / & / & 4.77 $\pm$ 0.07 \\
\bottomrule
\end{tabular}
\vspace{-15pt}
\end{table}
\vspace{3pt}
\noindent\textbf{{Transfer learning:}}
In this experiment, we explore to use transfer learning on KiSing, a low-resource multi-style database. Although KiSing has 42.6 minutes of singing for training, it has content duplication in most songs, resulting in limited variances in terms of phoneme sequences and melodies. Furthermore, the singer has adopted both a Chinese folk style (a variant of the Peking opera) and a pop style in songs, which has more frequent changes in rhythm, harmony, and the usage of falsetto. We conducted three experiments on this database: Baseline is trained only on KiSing. Two pre-trained models Multilingual$*$ and Opencpop$*$ are first pre-trained on multilingual data and Opencpop data, respectively. And then, they are fine-tuned on KiSing for another 500 epochs. As discussed in Sec.~\ref{ssec: dataset}, Opencpop is 5.2-hour Mandarin pop singing voice database, which has the same language coverage with KiSing. As shown in Table~\ref{tab: transfer}, Multilingual$*$ achieves the best objective scores of MCD and F$_0$RMSE, while Opencpop$*$ achieves the best VUV\_E. For the subjective evaluation, we observe that both pre-trained models outperform the Baseline, while Multilingual$*$ is better than Opencpop$*$. The result indicates that larger data size and a larger coverage of phonemes could benefit the low-resource multi-style SVS.
\begin{table}
\centering
\caption{\label{tab: transfer} Objective evaluation for transfer learning experiments on KiSing (a low-resource multi-style singing database). All the models use the same XiaoiceSing architecture.
Models with $*$ are first pre-trained on the corresponding database and then fine-tuned on KiSing.
}
\vspace{-10pt}
\begin{tabular}{l|cccc}
\toprule \textbf{Model} &\textbf{MCD$\downarrow$} & \textbf{VUV\_E$\downarrow$} & \textbf{F$_0$RMSE$\downarrow$} \\
\midrule
Baseline & 8.74 & 3.74 & 0.214 \\
Opencpop$*$ & 8.52 & \textbf{3.54} & 0.190\\
Multilingual$*$ & \textbf{8.25} & 3.66 & \textbf{0.177} \\
\bottomrule
\end{tabular}
\vspace{-10pt}
\end{table}
\begin{figure}[tbp]
\centering
\centerline{\includegraphics[width=8.5cm]{sections/Muskit-abtest.pdf}}
\vspace{-10pt}
\caption{A/B test for transfer learning experiments on KiSing.}
\label{fig:abtest}
\vspace{-20pt}
\end{figure}
\section{Conclusions}
In this work, we introduce a new E2E-SVS toolkit, called Muskits. This toolkit is developed as an open platform which can support fair comparison of different SVS models, and we expect it to facilitate the research field of SVS. The toolkit is based on the ESPnet framework with 12 recipes covering over 10 public databases in 4 languages. We conduct experiments under four scenarios: single-singer, multi-singer, multilingual SVS, and transfer learning on a multi-style database. In single- and multi-singer scenarios, we compare three published works on SVS with the same music score representation and vocoder. In addition, we explore multilingual SVS by combining six databases in four languages. From multilingual pre-training, we demonstrate subjective and objective gain on a single-singer database, which could alleviate the low-resource issue in SVS. Further experiments on transfer learning also show the effectiveness of our framework on a multi-style low-resource database. Full documentation and real-time demos can be found at \url{https://github.com/SJTMusicTeam/Muskits}.
\vspace{-5pt}
\section{Acknowledgement}
This work was supported by the National Natural Science Foundation of China (No. 62072462) and the National Key R\&D Program of China (No. 2020AAA0108600). |
2,869,038,154,867 | arxiv | \section{Introduction. Description of main results}\label{S1}
Three-dimensional Zakharov--Kuznetsov equation (ZK)
\begin{equation}\label{1.1}
u_t+bu_x+ u_{xxx}+u_{xyy}+u_{xzz}+uu_x=f(t,x,y,z)
\end{equation}
($u=u(t,x,y,z)$, $b$ -- real constant) for the first time was derived in \cite{ZK} for description of ion-acoustic waves in plasma put in the magnetic field. Further, this equation became to be considered as a model equation for non-linear waves propagating in dispersive media in the preassigned direction $(x)$ with deformations in the transverse directions. A rigorous derivation of the ZK model can be found, for example, in \cite{LLS}. Zakharov--Kuznetsov equation generalizes Korteweg--de~Vries equation (KdV) $u_t+bu_x+u_{xxx}+uu_x=0$ in the multidimensional case.
In the present paper we consider an initial-boundary value problem on a layer $\Sigma =\mathbb R\times \Omega$, where $\Omega$ is a certain bounded domain in $\mathbb R^2$, with initial and boundary conditions
\begin{equation}\label{1.2}
u\big|_{t=0} =u_0(x,y,z),
\end{equation}
\begin{equation}\label{1.3}
u\big|_{(0,T)\times \partial\Sigma} =0,
\end{equation}
where $T>0$ is arbitrary. We establish results on global existence, uniqueness and large-time decay of weak solutions to this problem. Existence and uniqueness of weak solutions are also obtained for the initial value problem.
The theory of ZK equation is more or less developed in the two-dimensional case, that is for an equation
$$
u_t+bu_x+ u_{xxx}+u_{xyy}+uu_x=f(t,x,y),
$$
especially for the initial-value problem. In particular, classes of global well-posedness were constructed in \cite{F95} for initial data from the spaces $H^k(\mathbb R^2)$, $k\in\mathbb N$. Other results can be found in \cite{S1, F89, LP, LPS}.
Initial-boundary value problems on domains of the type $I\times \mathbb R$, where $I$ is a certain interval (bounded or unbounded), are studied in \cite{F02, F07, FB, F08, ST, F12, DL} and others. Initial-boundary value problems for $y$ varying in a bounded interval turned out to be the most complicated ones (\cite{LPS, LT, L13, BF, STW, DL}), although such problems seems to be more natural from the physical point of view. In particular, there are no results on global well-posedness in classes of regular solutions for the strips $\mathbb R\times I$.
The theory of equation \eqref{1.1} is on the initial level. Certain results on global existence of weak solutions (without uniqueness) for the initial value problem follow from \cite{S1, F89}. Local well-posedness for initial data from
$H^s(\mathbb R^3)$, $s>1$, is established in \cite{LS, RS}. Results similar to \cite{S1, F89}, that is global existence without uniqueness of weak solutions for initial-boundary value problems on domains of the type $I\times \mathbb R^2$ can be found in \cite{F02, FB, ST, F12} and on a bounded rectangle in \cite{STW, W2}. Regular solutions to one initial-boundary value problem on a bounded rectangle are considered in \cite{W1, L14} and in the last paper global regular solutions are constructed for small initial data.
Homogeneous equation \eqref{1.1} possesses two conservation laws for solutions to the considered problem:
\begin{equation}\label{1.4}
\iint_{\Sigma}u^2\,dxdydz=\text{const}, \mbox{ }\iint_{\Sigma}\left(u_x^2+u^2_y+u^2_z-
\frac 13 u^3\right)\,dxdydz=\text{const}.
\end{equation}
Of course, similar conservation laws exist for the initial value problem for homogeneous KdV equation. It is well-known that the number of conservation laws for KdV is infinite, while other ones for ZK are not found. The last circumstance, for example, did not allow to apply in \cite{LS, RS} their profound investigations of the linearized equation to establish global well-posedness. In the present paper we supplement these two conservation laws with some decay of solutions when $x\to +\infty$ and construct classes of global existence and uniqueness without any assumptions on the size of the initial data. According to our best knowledge it is the first result of such a type for equation \eqref{1.1}. For KdV such method was for the first time used in \cite{KF, K, F88}. In the two-dimensional case similar results for ZK were obtained in \cite{BF}.
In all the consequent results the domain $\Omega$ is bounded and satisfy the following assumption:
\noindent either 1) $\partial\Omega \in C^3$ (in the conventional sense, see, for example, \cite{Mikh}),
\noindent or 2) $\Omega$ is a rectangle $(0,L_1)\times (0,L_2)$ for certain positive $L_1, L_2$
\noindent (one can introduce certain more complicated assumptions on $\Omega$ such that the aforementioned domains are particular cases of more general ones and all the results of the paper hold but for simplicity we choose this variant).
The symbol $|\Omega|$ denotes the measure of $\Omega$.
Let $\Pi_T =(0,T)\times \Sigma$, $x_+=\max(x,0)$, $\mathbb R_+=(0,+\infty)$, $\Sigma_+=\mathbb R_+ \times \Omega$.
For an integer $k\geq 0$ let
$$
|D^k\varphi|=\Bigl(\sum_{k_1+k_2+k_3=k}(\partial^{k_1}_x\partial_y^{k_2}\partial_z^{k_3}\varphi)^2\Bigr)^{1/2}, \qquad |D\varphi|=|D^1\varphi|.
$$
Let $L_p=L_p(\Sigma)$, $L_{p,+}=L_p(\Sigma_+)$, $H^k=H^k(\Sigma)$, $H_0^1=H_0^1(\Sigma)=\{\varphi\in H^1:
\varphi|_{\partial\Sigma}=0\}$ (note that under the aforementioned assumptions on $\Omega$ the space $H_0^1$ coincides with the closure of the space $C_0^\infty(\Sigma)$ in the $H^1$-norm).
For a measurable non-negative on $\mathbb R$ function $\psi(x)\not\equiv \text{const}$, let
\begin{equation*}
L_2^{\psi(x)} =\{\varphi(x,y,z): \varphi\psi^{1/2}(x)\in L_2\}
\end{equation*}
with a natural norm. In particularly important cases we use the special notation
\begin{equation*}
L_2^\alpha=L_2^{(1+x_+)^{2\alpha}}\quad \forall\ \alpha\ne 0,\quad
L_2^0=L_2,\qquad
L_2^{\alpha,exp}=L_2^{1+e^{2\alpha x}}\quad \forall\ \alpha>0.
\end{equation*}
Restrictions of these spaces on $\Sigma_+$ are denoted by $L_{2,+}^{\psi(x)}$, $L_{2,+}^\alpha$,
$L_{2,+}^{\alpha,exp}$.
Let for an integer $k\geq 0$
\begin{equation*}
H^{k,\psi(x)}=\{\varphi: |D^j\varphi|\in L_2^{\psi(x)}, \ j=0,\dots,k\}
\end{equation*}
with a natural norm,
\begin{equation*}
H^{k,\alpha}=H^{k,(1+x_+)^{2\alpha}}\quad \forall\ \alpha\ne 0,\quad H^{k,0}=H^k,
\qquad H^{k,\alpha,exp}=H^{k,1+e^{2\alpha x}} \quad \forall\ \alpha>0.
\end{equation*}
Let $H^{1,\psi(x)}_0=\{\varphi\in H^{1,\psi(x)}: \varphi|_{\partial\Sigma}=0\}$ with similar notation for
$H_0^{1,\alpha}$ and $H_0^{1,\alpha,exp}$. Let $H^{-1,\psi(x)}= \{\varphi: \varphi\psi^{1/2}(x)\in H^{-1}\}$.
We say that $\psi(x)$ is an admissible weight function if $\psi$ is an infinitely smooth positive function on $\mathbb R$ such that $|\psi^{(j)}(x)|\leq c(j)\psi(x)$ for each natural $j$ and all $x\in\mathbb R$. Note that such a function has not more than exponential growth and not more than exponential decrease at $\pm\infty$. It was shown in \cite{F12} that $\psi^s(x)$ for any $s\in\mathbb R$ is also an admissible weight function.
As an important example of such functions, we introduce for $\alpha\geq 0$ special infinitely smooth functions $\rho_\alpha(x)$ in the following way: $\rho_\alpha(x)=1+e^{2x}$ when $x\leq -1$, $\rho_\alpha(x)=1+(1+x)^{2\alpha}$ for $\alpha>0$ and $\rho_0(x)=3-(1+x)^{-1/2}$ when $x\geq0$, $\rho'_\alpha(x)>0$ when $x\in (-1,0)$.
Note that both $\rho_\alpha$ and $\rho'_\alpha$ are admissible weight functions and $\rho'_\alpha(x) \leq c(\alpha)\rho_\alpha(x)$ for all $x\in\mathbb R$.
Moreover, for $\alpha\geq 0$
\begin{equation*}
L_2^{\rho_\alpha(x)}=L_2^\alpha, \qquad H^{k,\rho_\alpha(x)}=H^{k,\alpha}.
\end{equation*}
We construct solutions to the considered problem in spaces $X^{k,\psi(x)}(\Pi_T)$,\linebreak $k=0 \mbox{ or }1$, for admissible non-decreasing weight functions $\psi(x)\geq 1\ \forall x\in\mathbb R$, consisting of functions $u(t,x,y,z)$ such that
\begin{equation}\label{1.5}
u\in C_w([0,T]; H^{k,\psi(x)}), \qquad
|D^{k+1}u|\in L_2(0,T;L_2^{\psi'(x)})
\end{equation}
(the symbol $C_w$ denotes the space of weakly continuous mappings),
\begin{equation}\label{1.6}
\lambda(|D^{k+1} u|;T) =
\sup_{x_0\in\mathbb R}\int_0^T\!\! \int_{x_0}^{x_0+1}\!\! \iint_\Omega |D^{k+1}u|^2\,dydzdxdt<\infty, \quad
u\big|_{(0,T)\times\partial\Sigma}=0
\end{equation}
(let $X^{\psi(x)}(\Pi_T)=X^{0,\psi(x)}(\Pi_T)$).
In particularly important cases we use the special notation
\begin{equation*}
X^{k,\alpha}(\Pi_T)=X^{k,\rho_\alpha(x)}(\Pi_T),\quad X^\alpha(\Pi_T)=X^{0,\alpha}(\Pi_T)
\end{equation*}
and for $\alpha>0$
\begin{equation*}
X^{k,\alpha,exp}(\Pi_T)=X^{k,1+e^{2\alpha x}}(\Pi_T),\quad X^{\alpha,\exp}(\Pi_T)=X^{0,\alpha,\exp}(\Pi_T).
\end{equation*}
It is easy to see that $X^{k,0}(\Pi_T)$ coincides with a space of functions $u\in C_w([0,T]; H^k)$ for which \eqref{1.6} holds, $X^{k,\alpha}(\Pi_T)$, $\alpha>0$, -- with a space of functions $u\in C_w([0,T]; H^{k,\alpha})$ for which \eqref{1.6} holds and, in addition, $|D^{k+1} u|\in L_2(0,T;L_{2,+}^{\alpha-1/2})$; $X^{k,\alpha,exp}(\Pi_T)$ -- with a space of functions $u\in C_w([0,T]; H^{k,\alpha,exp})$ for which \eqref{1.6} holds and, in addition,
$|D^{k+1} u|\in L_2(0,T;L_{2,+}^{\alpha,exp})$.
\begin{theorem}\label{T1.1}
Let $u_0\in L_2^{\psi(x)}$, $f\in L_1(0,T; L_2^{\psi(x)})$ for certain $T>0$ and an admissible weight function $\psi(x)\geq 1\ \forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function. Then there exists a weak solution to problem \eqref{1.1}--\eqref{1.3} $u \in X^{\psi(x)}(\Pi_T)$.
\end{theorem}
\begin{theorem}\label{T1.2}
Let $u_0\in H_0^{1,\psi(x)}$, $f\in L_1(0,T; H_0^{1,\psi(x)})$ for certain $T>0$ and an admissible weight function $\psi(x)\geq 1\ \forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function. Then there exists a weak solution to problem \eqref{1.1}--\eqref{1.3} $u\in X^{1,\psi(x)}(\Pi_T)$ and it is unique in this space if $\psi(x)\geq \rho_{3/4}(x)$ $\forall x\in \mathbb R$.
\end{theorem}
\begin{remark}\label{R1.1}
It follows from Theorem~\ref{T1.2} that weak solutions to problem \eqref{1.1}--\eqref{1.3} are unique in the spaces
$X^{1,3/4}(\Pi_T)$ and in the spaces $X^{1,\alpha,exp}(\Pi_T)$ for any $\alpha>0$ (and exist under corresponding assumptions on $u_0$ and $f$).
\end{remark}
For small solutions to the considered problem the following large-time decay result holds.
\begin{theorem}\label{T1.3}
Let $\Omega_0=+\infty$ if $b\leq 0$, and if $b>0$ there exists $\Omega_0>0$ such that in both cases if
$|\Omega| <\Omega_0$ there exist $\alpha_0>0$, $\epsilon_0>0$ and $\beta>0$ such that if $u_0\in L_2^{\alpha,exp}$ for $\alpha\in (0,\alpha_0]$, $\|u_0\|_{L_2}\leq\epsilon_0$, $f\equiv 0$, then there exists a weak solution $u(t,x,y,z)$ to problem \eqref{1.1}--\eqref{1.3} from the space $X^{\alpha,exp}(\Pi_T)$ $\forall T>0$ satisfying an inequality
\begin{equation}\label{1.7}
\|e^{\alpha x}u(t,\cdot,\cdot,\cdot)\|_{L_2}\leq e^{-\alpha\beta t}\|e^{\alpha x}u_0\|_{L_2}\qquad \forall t\geq 0.
\end{equation}
\end{theorem}
The proof of this result, in particular, is based on Friedrichs inequality and, therefore, homogeneous Dirichlet conditions are essential. The idea that under such conditions Zakahrov--Kuznetsov equation possesses certain internal dissipation, which provides decay of such a type, was found out in \cite{LT}. Stabilization of solutions to three-dimensional linearized ZK equation is studied in \cite{DL15}.
Further we use the following auxiliary functions. Let $\eta(x)$ denote a cut-off function, namely, $\eta$ is an infinitely smooth non-decreasing on $\mathbb R$ function such that $\eta(x)=0$ when $x\leq 0$, $\eta(x)=1$ when $x\geq 1$, $\eta(x)+\eta(1-x)\equiv 1$.
For each $\alpha\geq 0$ and $\beta>0$ we introduce an infinitely smooth increasing on $\mathbb R$ function $\varkappa_{\alpha,\beta}(x)$ as follows:
$\varkappa_{\alpha,\beta}(x)=e^{2\beta x}$ when $x\leq -1$, $\varkappa_{\alpha,\beta}(x)=(1+x)^{2\alpha}$ for $\alpha>0$ and $\varkappa_{0,\beta}(x)=2-(1+x)^{-1/2}$ when $x\geq 0$, $\varkappa'_{\alpha,\beta}(x)>0$ when $x\in (-1,0)$.
Note that both $\varkappa_{\alpha,\beta}$ and $\varkappa'_{\alpha,\beta}$ are admissible weight functions, and $\varkappa'_{\alpha,\beta}(x)\leq c(\alpha,\beta)\varkappa_{\alpha,\beta}(x)$ for all $x\in \mathbb R$. It is obvious that one can take $\rho_\alpha(x) \equiv 1+\varkappa_{\alpha,1}(x)$.
Note also that if $u\in X^{k,\alpha}(\Pi_T)$ for $\alpha\geq 1/2$, then $|D^{k+1}u|\varkappa_{\alpha-1/2,\beta}^{1/2}(x)\in L_2(\Pi_T)$ for any $\beta>0$.
Further we omit limits of integration in integrals over the whole strip $\Sigma$. We need the following interpolating inequality.
\begin{lemma}\label{L1.1}
Let $\psi_1(x)$, $\psi_2(x)$ be two admissible weight functions such that $\psi_1(x)\leq c_0\psi_2(x)$
$\forall x\in\mathbb R$ for some constant $c_0>0$. Let $k=1$ or $2$, $m\in [0,k)$ -- integer, $q\in [2,6]$ if $k-m=1$ and $q\in [2,+\infty)$ if $k-m=2$. Then there exists a constant $c>0$ such that for every function $\varphi(x,y,z)$ satisfying
$|D^k\varphi|\psi_1^{1/2}(x)\in L_2$, $\varphi\psi_2^{1/2}(x)\in L_2$, the following inequality holds
\begin{equation}\label{1.8}
\bigl\| |D^m\varphi|\psi_1^s(x)\psi_2^{1/2-s}(x)\bigr\|_{L_q} \leq c
\bigl\| |D^k\varphi|\psi_1^{1/2}(x)\bigr\|^{2s}_{L_2}
\bigl\| \varphi\psi_2^{1/2}(x)\bigr\|^{1-2s}_{L_2}
+\bigl\| \varphi\psi_2^{1/2}(x)\bigr\|_{L_2},
\end{equation}
where $\displaystyle{s=s(k,m,q)=\frac{2m+3}{4k}-\frac{3}{2kq}}$. If $\varphi\big|_{\partial\Sigma}=0$ and either $k=1$ or $k=2,$ $m=0,$ $q\leq 6$ or $k=2,$ $ m=1,$ $q=2$ then the constant $c$ in \eqref{1.8} does not depend on $\Omega$.
\end{lemma}
\begin{proof}
Let first $k=1$. The proof is based on the well-known inequality (of course, it is valid for more general domains): for
$\varphi\in W^1_p$, $p\in [1,3)$, $p^*=3p/(3-p)$
\begin{equation}\label{1.9}
\|\varphi\|_{L_{p^*}} \leq c \bigl\||D\varphi|+|\varphi|\bigr\|_{L_p},
\end{equation}
where the constant $c$ does not depend on $\Omega$ in the case $\varphi\in H^1_0$ (see, for example, \cite{BIN,LSU}).
Then for $q\in [2,6]$ H\"older inequality yields that
$$
\|\varphi \psi_1^s \psi_2^{1/2-s}\|_{L_q} \leq \|\varphi\psi_1^{1/2}\|_{L_6}^{2s}
\|\varphi \psi_2^{1/2}\|_{L_2}^{1-2s},
$$
whence with the use of \eqref{1.9} for $p=2$ and the properties of the functions $\psi_1$ and $\psi_2$ the desired estimate follows.
Next, let $k=2,$ $m=1,$ $q=2$. Integration by parts yields an equality
\begin{multline*}
\iiint |D\varphi|^2\psi_1^{1/2}\psi_2^{1/2}\,dxdydz =
-\iiint \Delta\varphi \psi_1^{1/2} \cdot \varphi\psi_2^{1/2}\,dxdydz \\
-\iiint \varphi\varphi_x (\psi_1^{1/2}\psi_2^{1/2})'\,dxdydz +
\iint_{\partial\Sigma} \varphi (\varphi_y n_y +\varphi_z n_z)\psi_1^{1/2}\psi_2^{1/2} dS,
\end{multline*}
where $(n_y,n_z)$ is the exterior normal vector to $\Omega$. If $\varphi\big|_{\partial\Sigma}=0$ this equality immediately provides \eqref{1.8}, while in the general case one must also use for functions $\Phi\equiv \varphi^2$ and $\Phi\equiv\varphi_y^2$ or $\Phi\equiv \varphi_z^2$ the following well-known estimate on the trace (see, for example, \cite{BIN}):
$$
\|\Phi\|_{L_1(\partial\Omega)} \leq c \bigl\| |\Phi_y|+|\Phi_z|+|\Phi| \bigr\|_{L_1(\Omega)}.
$$
If $k=2,$ $m=1,$ $q\in (2,6]$ let $\sigma= \displaystyle \frac 32 -\frac 3q$, then
$$
\bigl\| |D\varphi|\psi_1^s\psi_2^{1/2-s}\bigr\|_{L_q} \leq
\bigl\| |D\varphi|\psi_1^{1/2}\bigr\|_{L_6}^\sigma
\bigl\||D\varphi|\psi_1^{1/4}\psi_2^{1/4} \bigr\|_{L_2}^{1-\sigma}
$$
and with the use of the already obtained estimates \eqref{1.8} for $k=1$ applied to $|D\varphi|$ and for $k=2,$ $m=1,$ $q=2$ derive \eqref{1.8} in this case.
Finally, let $k=2,$ $m=0$. If $q\leq 6$ then with the use of \eqref{1.8} for $k=1$ (where $\psi_1$ is substituted by
$\psi_1^{1/2}\psi_2^{1/2}$) and for $k=2,$ $m=1,$ $q=2$ we derive that
\begin{multline*}
\|\varphi \psi_1^s\psi_2^{1/2-s}\|_{L_q} =
\| \varphi(\psi_1^{1/2}\psi_2^{1/2})^{s(1,0,q)} \psi_2^{1/2-s(1.0,q)}\|_{L_q} \\ \leq
c\bigl\| |D\varphi|\psi_1^{1/4}\psi_2^{1/4}\bigr\|_{L_2}^{2s(1,0,q)}
\|\varphi\psi_2^{1/2}\|_{L_2}^{1-2s(1,0,q)} +c\|\varphi\psi_2^{1/2}\|_{L_2} \\ \leq
c_1\bigl\| |D^2\varphi|\psi_1^{1/2}\bigr\|_{L_2}^{2s}
\|\varphi\psi_2^{1/2}\|_{L_2}^{1-2s} +c_1\|\varphi\psi_2^{1/2}\|_{L_2}.
\end{multline*}
If $q\in (6,+\infty)$ choose $p\in (2,3)$ satisfying $q=p^*$, $\varkappa=1+(6-p)/(3p)$ and $\theta\in (5/6,1)$ satisfying
$\displaystyle \frac 1q = \frac \theta{\varkappa q} +\frac{1-\theta}2$ (of course, all these parameters can be expressed explicitly) and define
$$
\widetilde\varphi \equiv |\varphi|^\varkappa \cdot \sgn\varphi\cdot \psi_1^{1/2}\psi_2^{(\varkappa-1)/2}.
$$
It is easy to see that $s=\displaystyle\frac \theta{2\varkappa}$ and thus
$$
\|\varphi\psi_1^s\psi_2^{1/2-s}\|_{L_q} =
\|\widetilde\varphi^{\theta/\varkappa}\cdot (\varphi\psi_2^{1/2})^{1-\theta}\|_{L_q}
\leq \|\widetilde\varphi\|_{L_q}^{\theta/\varkappa}
\|\varphi\psi_2^{1/2}\|_{L_2}^{1-\theta}.
$$
Applying inequality \eqref{1.9} to the function $\widetilde\varphi$ we find that
\begin{multline*}
\|\widetilde\varphi\|_{L_q} \leq c\bigl\||D\widetilde\varphi|+|\widetilde\varphi|\bigr\|_{L_p} \leq
c_1\bigl\|(|D\varphi|+|\varphi|)\psi_1^{1/2}\cdot (|\varphi|\psi_2^{1/2})^{\varkappa-1}\bigr\|_{L_p} \\ \leq
c_2\bigl\|(|D\varphi|+|\varphi|)\psi_1^{1/2}\bigr\|_{L_6}
\|\varphi\psi_2^{1/2}\|_{L_2}^{\varkappa-1}.
\end{multline*}
Applying inequality \eqref{1.8} in the case $k=1$ to the function $|D\varphi|+|\varphi|$ we finish the proof.
\end{proof}
\begin{remark}\label{R1.2}
In the case $\psi_1=\psi_2\equiv 1$ inequality \eqref{1.8} is well-known (see, for example, \cite{BIN, LSU}. For the weighted spaces in the case $\Sigma=\mathbb R^3$ it was proved in \cite{F89} (in fact, in that paper the spatial dimension and natural $k$ were arbitrary).
\end{remark}
\begin{remark}\label{R1.3}
The constant $c$ in the right side of \eqref{1.8} depends on the corresponding constants evaluating the derivatives of the functions $\psi_1$ and $\psi_2$ by these functions themselves and the constant $c_0$ evaluating $\psi_1$ by $\psi_2$.
\end{remark}
For certain multi-index $\nu=(\nu_1,\nu_2)$ let $\partial^\nu_{y,z} =
\partial^{\nu_1}_y\partial^{\nu_2}_z$, $|\nu|=\nu_1+\nu_2$. Let $\Delta^{\bot} =\partial^2_y+\partial^2_z$,
$\Delta = \partial^2_x +\Delta^{\bot}$.
The paper is organized as follows. An auxiliary linear problem is considered in Section~\ref{S2}. Section~\ref{S3} is dedicated to problems on existence of solutions to the original problem. Results on continuous dependence of solutions on $u_0$ and $f$ are proved in Section~\ref{S4}. In particular, they imply uniqueness of the solution. Section~\ref{S5} is devoted to the large-time decay of solutions. The initial value problem is considered in Section~\ref{S6}.
\section{An auxiliary linear equation}\label{S2}
Consider a linear equation
\begin{equation}\label{2.1}
u_t+bu_x+\Delta u_x-\delta \Delta u=f(t,x,y,z)
\end{equation}
for a certain constant $\delta\in [0,1]$.
\begin{lemma}\label{L2.1}
Let $(1+|x|)^n \partial^j_x\partial^\nu_{y,z} u_0\in L_2$ for any integer non-negative $n$, $j$ and $|\nu|\leq 3$, $u_0\big|_{\partial\Sigma}=\Delta^{\bot}u_0\big|_{\partial\Sigma}=0$,
$(1+|x|)^n \partial^m_t \partial^j_x\partial^\nu_{y,z}f\in L_1(0,T; L_2)$ for any integer $n$, $j$ and
$2m+|\nu|\leq 3$, $f\big|_{(0,T)\times\partial\Sigma}=\Delta^{\bot} f\big|_{(0,T)\times\partial\Sigma}=0$.
Then there exists a solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} $u(t,x,y,z)$ such that $(1+|x|)^n \partial^m_t \partial^j_x\partial^\nu_{y,z}u\in C([0,T]; L_2)$ for any integer $n$, $j$ and
$2m+|\nu|\leq 3$, $\Delta^{\bot} u\big|_{(0,T)\times\partial\Sigma}=0$.
\end{lemma}
\begin{proof}
Let $\{\psi_l(y,z)$, $l=1,2\dots\}$ be an orthonormal in $L_2(\Omega)$ system of eigenfunctions for the operator $-\Delta^{\bot}$ on $\Omega$ with boundary conditions $\psi_l\big|_{\partial\Omega}=0$, $\lambda_l$ -- the corresponding eigenvalues. It is known (see, for example \cite{Mikh}) that such a system exists and satisfy the following properties: $\lambda_l>0$ $\forall l$, $\lambda_l\to +\infty$ when $l\to +\infty$, $\psi_l\in H^3(\Omega)$,
$\psi_l\big|_{\partial\Omega}=\Delta^{\bot}\psi_l\big|_{\partial\Omega}=0$ $\forall l$ and these functions are real-valued. If $(\varphi,\psi_l)$ denotes the scalar product in $L_2(\Omega)$ then for any
$\varphi\in L_2(\Omega)$
\begin{equation}\label{2.2}
\varphi =\sum\limits_{l=1}^{+\infty} (\varphi,\psi_l)\psi_l.
\end{equation}
If $\varphi \in H_0^1(\Omega)$ or $\varphi \in H^2(\Omega)\cap H_0^1(\Omega)$ then this series converges in these spaces respectively. Moreover,
\begin{gather}\label{2.3}
\varphi \in H_0^1(\Omega) \Longleftrightarrow \sum\limits_{l=1}^{+\infty} \lambda_l (\varphi,\psi_l)^2 <+\infty,
\quad \sum\limits_{l=1}^{+\infty} \lambda_l (\varphi,\psi_l)^2 =\|\varphi\|^2_{H_0^1(\Omega)}, \\
\label{2.4}
\varphi \in H^2(\Omega)\cap H_0^1(\Omega) \Longleftrightarrow
\sum\limits_{l=1}^{+\infty} \lambda_l^2 (\varphi,\psi_l)^2 <+\infty, \quad
\sum\limits_{l=1}^{+\infty} \lambda_l^2 (\varphi,\psi_l)^2 \sim \|\varphi\|^2_{H^2(\Omega)}.
\end{gather}
The last inequality is the particular case of an inequality valid for any function $\varphi \in H^2(\Omega)\cap H_0^1(\Omega)$:
\begin{equation}\label{2.5}
\|\varphi\|_{H^2(\Omega)} \leq c(\Omega)\|\Delta^{\bot}\varphi\|_{L_2(\Omega)},
\end{equation}
where the constant $c$ depends on the domain $\Omega$. Moreover, for any function $\varphi\in H^3(\Omega)$, such that $\varphi\big|_{\partial\Omega}=\Delta^{\bot} \varphi\bigl|_{\partial\Omega}=0$,
series \eqref{2.2} converges in this space and similarly to \eqref{2.3}, \eqref{2.4}
\begin{multline}\label{2.6}
\varphi\in H^3(\Omega), \varphi\big|_{\partial\Omega}=\Delta^{\bot} \varphi\bigl|_{\partial\Omega}=0
\Longleftrightarrow \sum\limits_{l=1}^{+\infty} \lambda_l^3 (\varphi,\psi_l)^2 <+\infty, \\
\sum\limits_{l=1}^{+\infty} \lambda_l^3 (\varphi,\psi_l)^2 \sim \|\varphi\|^2_{H^3(\Omega)}.
\end{multline}
For example, in the case $\Omega=(0,L_1)\times (0,L_2)$ these eigenfunctions are written in a simple form
$\displaystyle \left\{\frac 2{\sqrt{l_1l_2}} \sin\frac{\pi l_1 y}{L_1} \sin\frac{\pi l_2 z}{L_2}, l_1,l_2=1,2,\dots\right\}$.
Then with the use of Fourier transform for the variable $x$ and Fourier series for the variables $y, z$ a solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} can be written as follows:
\begin{equation}\label{2.7}
u(t,x,y,z)=\frac{1}{2\pi}\int\limits_{\mathbb R}\sum_{l=1}^{+\infty}e^{i\xi x}\psi_l(y,z)\widehat{u}(t,\xi,l)\,d\xi,
\end{equation}
where
\begin{multline*}
\widehat{u}(t,\xi,l)=\widehat{u_0}(\xi,l)e^{\bigl(i(\xi^3-b\xi+\xi\lambda_l)
-\delta(\xi^2+\lambda_l)\bigr)t} \\+
\int_0^t\widehat{f}(\tau,\xi,l)e^{\bigl(i(\xi^3-b\xi+\xi\lambda_l)-\delta(\xi^2+\lambda_l)\bigr)(t-\tau)}\,d\tau,
\end{multline*}
$$
\widehat{u_0}(\xi,l)\equiv\iiint e^{-i\xi x}\psi_l(y,z)u_0(x,y,z)\,dxdydz,
$$
$$
\widehat f(t,\xi,l)\equiv\iiint e^{-i\xi x}\psi_l(y,z) f(t,x,y,z)\,dxdydz.
$$
According to \eqref{2.3}--\eqref{2.6} and the properties of the functions $u_0$ and $f$ the function $u$ is the desired solution.
\end{proof}
\begin{lemma}\label{L2.2}
Let the hypothesis of Lemma~\ref{L2.1} be satisfied and, in addition, $\partial_x^j \partial_{y,z}^\nu u_0e^{\alpha x} \in L_{2,+}$, $\partial_x^j \partial_{y,z}^\nu fe^{\alpha x}\in L_2(0,T; L_{2,+})$ for any $\alpha>0$,
$j\geq 0$ and $|\nu|\leq 1$. Then $\partial_x^j \partial_{y,z}^\nu u e^{\alpha x} \in C([0,T]; L_{2,+})$ if
$|\nu|\leq1$, $\partial^m_t\partial_x^j \partial_{y,z}^\nu u e^{\alpha x} \in L_2(0,T;L_{2,+})$ if $2m+|\nu|= 2$ also for any $\alpha>0$ and $j\geq 0$, where $u$ is the solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} constructed in Lemma~\ref{L2.1}.
\end{lemma}
\begin{proof}
Let $v\equiv \partial^j_x u$, then the function $v$ satisfies an equation of \eqref{2.1} type, where $f$ is replaced by $\partial^j_x f$. Let $m\geq 3$. Multiplying this equation by $2x^m v$ and integrating over $\Sigma_+$, we derive an equality
\begin{multline}\label{2.8}
\frac{d}{dt}\iiint_{\Sigma_+}x^mv^2\, dxdydz +
m\iiint x^{m-1} (3v_x^2+v_y^2+v_z^2-bv^2)\,dxdydz \\ -
m(m-1)(m-2)\iiint_{\Sigma_+}x^{m-3}v^2 \,dxdydz +
2\delta \iiint_{\Sigma_+} x^m (v_x^2+v_y^2+v_z^2)\,dxdydz \\ -
\delta m(m-1)\iiint_{\Sigma_+}x^{m-2}v^2\,dxdyz =2\iiint_{\Sigma_+}x^m\partial^j_xf v\,dxdydz.
\end{multline}
Let $\alpha>0$, $n\geq3$. For any $m\in [3,n]$ multiplying the corresponding inequality by $\alpha^m/(m!)$ and summing by $m$ we obtain that for
\begin{gather*}
P_n(t)\equiv \iiint_{\Sigma_+}\sum_{m=0}^n\frac{(\alpha x)^m}{m!}v^2(t,x,y,z)\,dxdydz, \\
Q_n(t)\equiv \iiint_{\Sigma_+}\sum_{m=0}^n\frac{(\alpha x)^m}{m!}|Dv|^2(t,x,y,z)\,dxdydz
\end{gather*}
inequalities
$$
P_n'(t)+\alpha Q_{n-1}(t)+ 2\delta Q_n(t)\leq \gamma(t) P_n(t)+c, \quad P_n(0)\leq c, \quad
\|\gamma\|_{L_1(0,T)}\leq c
$$
hold uniformly with respect to $n$,
whence it follows that
\begin{equation}\label{2.9}
\sup_{t\in[0,T]}\iiint_{\Sigma_+}e^{\alpha x} v^2\,dxdydz
+\int_0^T\!\! \iiint_{\Sigma_+}e^{\alpha x} |Dv|^2 \,dxdydzdt<\infty.
\end{equation}
Multiplying the aforementioned equality for the function $v$ by $2e^{\alpha x} v$ and integrating over $\Sigma$ we derive similarly to \eqref{2.8} that for $R(t)\equiv \iiint e^{\alpha x} v^2\,dxdydz$ the following equality holds:
$$
R'(t)-(\alpha^3+\delta\alpha^2)R(t)= g(t)\in L_1(0,T),
$$
therefore, $R \in C[0,T]$.
Next, multiplying the corresponding equation by
$-2x^m\Delta^{\bot} v$ and integrating over $\Sigma_+$ we derive similarly to \eqref{2.8} that
\begin{multline*}
\frac{d}{dt}\iiint_{\Sigma_+}x^m(v_y^2+v_z^2)\, dxdydz \\+
m\iiint x^{m-1} (3v_{xy}^2+3v_{xz}^2+(\Delta^{\bot}v)^2-bv_y^2-bv_z^2)\,dxdydz \\ -
m(m-1)(m-2)\iiint_{\Sigma_+}x^{m-3}(v_y^2+v_z^2) \,dxdydz \\+
2\delta \iiint_{\Sigma_+} x^m (v_{xy}^2+v_{xz}^2+(\Delta^{\bot}v)^2)\,dxdydz \\ -
\delta m(m-1)\iiint_{\Sigma_+}x^{m-2}(v_y^2+v_z^2)\,dxdyz =
2\iiint_{\Sigma_+}x^m(\partial^j_xf_y v_y+\partial^j_xf_z v_z)\,dxdydz.
\end{multline*}
Taking into account inequality \eqref{2.5} similarly to \eqref{2.9} one obtains the rest properties for
$\partial^j_x\partial_{y,z}^\nu u$, $|\nu|\geq 1$. The properties of $\partial_t\partial_x^j u$ are derived with the use of equation \eqref{2.1} itself.
\end{proof}
We now pass to weak solutions.
\begin{definition}\label{D2.1}
Let $u_0\in L_2^{\psi(x)}$ for an admissible weight function $\psi$, $f\equiv f_0+f_1$, $f\in L_1(0,T; L_2^{\psi(x)})$,
$f_1\in L_2(0,T;H^{-1,\psi(x)})$.
A function $u \in L_2(0,T;L_2^{\psi(x)})$ is called a weak solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3}, if for any function $\varphi$, such that
$\partial^j_x\varphi \in C([0,T]; L_2^{1/\psi(x)})$,
$\partial_t^m\partial_x^j\partial_{y,z}^\nu \varphi \in L_2(0,T;L_2^{1/\psi(x)})$ for $j\geq 0$, $2m+|\nu|\leq 2$
and $\varphi\big|_{t=T}\equiv 0$, $\varphi\big|_{(0,T)\times \partial\Sigma}\equiv 0$, there holds the following equality:
\begin{multline}\label{2.10}
\int_0^T\!\! \iiint \big[u(\varphi_t+b\varphi_x+\Delta\varphi_x+\delta\Delta\varphi) +
f\varphi\big]\,dxdydzdt \\+
\iiint u_0\varphi\big|_{t=0}\,dxdydz=0.
\end{multline}
\end{definition}
\begin{lemma}\label{L2.3}
If there exists $\beta>0$ such that $\psi(x)\geq \varkappa_{0,\beta}(x)$ $\forall x\in\mathbb R$, then
a weak solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} is unique.
\end{lemma}
\begin{proof}
The proof is carried out by standard H\"olmgren's argument on the basis of Lemmas~\ref{L2.1} and~\ref{L2.2}. Let $F$ be an arbitrary function from the space $C_0^\infty(\Pi_T)$. Consider an auxiliary linear problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} for $u_0\equiv 0$ and $f(t,x,y,z)\equiv -F(T-t,-x,y,z)$. According to the aforementioned lemmas there exists a solution $\widetilde\varphi$ to this problem such that
$\partial_x^j \widetilde\varphi \in C([0,T];L_2^{1/\widetilde\psi(x)})$, where
$\widetilde\psi(x)\equiv \psi(-x)$, $\partial^m_t\partial_x^j \widetilde\varphi, \partial_x^j\partial_{y,z}^\nu \widetilde\varphi\in L_2(0,T;L_2^{1/\widetilde\psi(x)})$ if $2m+|\nu|\leq 2$ (note that $1/\widetilde\psi(x) \leq c e^{2\beta x}$ when $x\to +\infty$, $1/\widetilde\psi(x) \leq c$ when $x\to -\infty$).
Define $\varphi(t,x,y,z)\equiv \widetilde\varphi(T-t,-x,y,z)$. It is easy to see that this function satisfies the hypothesis of Definition~\ref{D2.1} and $\varphi_t+b\varphi_x+\Delta\varphi_x+\delta\Delta\varphi=F$ in the space
$L_2(0,T;L_2^{1/\psi(x)})$.
Therefore, if $u$ is a weak solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} for $u_0\equiv 0$ and $f\equiv 0$, it follows from \eqref{2.9} that $\langle u,F\rangle=0$ and so $u\equiv 0$.
\end{proof}
Now we present a number of auxiliary lemmas on solubility of the linear problem in non-smooth case.
\begin{lemma}\label{L2.4}
Let $u_0\in L_2^{\psi(x)}$ for a certain admissible weight function $\psi(x)$ such that $\psi'(x)$ is also an admissible weight function, $f\equiv f_0+\delta^{1/2}f_{1x}+f_{2x}$, where $f_0\in L_1(0,T; L_2^{\psi(x)})$, $f_1\in L_2(0,T;L_2^{\psi(x)})$,
$f_2\in L_2(0,T;L_2^{\psi^2(x)/\psi'(x)})$. Then there exists a weak solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} $u(t,x,y,z)$ from the space $C([0,T];L_2^{\psi(x)})\cap L_2(0,T;H_0^{1,\psi'(x)})$ and $\delta|Du|\in L_2(0,T;L_2^{\psi(x)})$. Moreover, for any $t\in (0,T]$ uniformly with respect to $\delta$
\begin{multline}\label{2.11}
\|u\|_{C([0,t];L_2^{\psi(x)})}+\|u\|_{L_2(0,t;H^{1,\psi'(x)})}+\delta^{1/2}\bigl\||Du|\bigr\|_{L_2(0,t;L_2^{\psi(x)})} \\ \leq c(T) \left[\|u_0\|_{L_2^{\psi(x)}}+\|f_0\|_{L_1(0,t;L_2^{\psi(x)})}+\|f_1\|_{L_2(0,t;L_2^{\psi(x)})}
+\|f_2\|_{L_2(0,T;L_2^{\psi^2(x)/\psi'(x)})}\right],
\end{multline}
\begin{multline}\label{2.12}
\iiint u^2(t,x,y,z)\psi(x)\,dxdydz+\int_0^t\!\! \iiint (3u_x^2+u_y^2+u_z^2)\psi'\,dxdydzd\tau \\
+2\delta \int_0^t \!\! \iiint |Du|^2\psi\,dxdydzd\tau -
\int_0^t \!\! \iiint u^2\cdot (b\psi'+\psi'''+\delta\psi'')\,dxdydzd\tau \\ =
\iiint u_0^2\psi\,dxdydz+2\int_0^t\!\iiint f_0u\psi \,dxdydzd\tau \\
-2\int_0^t\!\! \iiint (\delta^{1/2}f_1+f_2)(u\psi)_x \,dxdydzd\tau.
\end{multline}
\end{lemma}
\begin{proof}
Let at first $u_0\in C_0^\infty(\Sigma)$, $f_0,f_1,f_2 \in C_0^\infty(\Pi_T)$. Consider the corresponding solution from the class described in Lemmas~\ref{L2.1} and~\ref{L2.2}. Note that $\psi$ is non-decreasing and has not more than exponential growth at $+\infty$. Then
$\partial_x^j u\in C([0,T];H_0^{1,\psi(x)})$,
$\partial_x^j u_t, \partial_x^j\partial_{y,z}^\nu u \in L_2(0,T;L_2^{\psi(x)})$ if $|\nu|=2$ for any $j\geq 0$. Therefore, one can multiply equation \eqref{2.1} by $2u(t,x,y,z)\psi(x)$, integrate and as a result obtain equality \eqref{2.12}. Note that this equality provides estimate \eqref{2.11}, which in turn justifies the assertion of the lemma in the general case.
\end{proof}
\begin{corollary}\label{C2.1}
Let the hypothesis of Lemma~\ref{L2.4} be satisfied for $\psi(x)\geq 1$ $\forall x\in\mathbb R$ and $f_2\equiv 0$. Then for the (unique) weak solution $u\in C([0,T];L_2)$ and any $t\in (0,T]$
\begin{multline}\label{2.13}
\iiint u^2(t,x,y,z)\,dxdydz
+2\delta \int_0^t \!\! \iiint |Du|^2\,dxdydzd\tau =
\iiint u_0^2\,dxdydz \\+2\int_0^t\!\iiint f_0u \,dxdydzd\tau
-2\delta^{1/2}\int_0^t\! \iiint f_1u_x \,dxdydzd\tau.
\end{multline}
\end{corollary}
\begin{proof}
In the smooth case this equality is obvious and in the general case is obtained on the basis of estimate \eqref{2.11} via closure.
\end{proof}
\begin{lemma}\label{L2.5}
Let $u_0\in H_0^{1,\psi(x)}$ for a certain admissible weight function $\psi(x)$ such that $\psi'(x)$ is also an admissible weight function, $f\equiv f_0+\delta^{1/2}f_1$, where $f_0\in L_1(0,T;H_0^{1,\psi(x)})$, $f_1\in L_2(0,T;L_2^{\psi(x)})$. Then there exists a weak solution to problem \eqref{2.1}, \eqref{1.2}, \eqref{1.3} $u(t,x,y,z)$ from the space $C([0,T];H_0^{1,\psi(x)})\cap L_2(0,T;H^{2,\psi'(x)})$ and $\delta|D^2u|\in L_2(0,T;L_2^{\psi(x)})$ . Moreover, for any $t\in (0,T]$ uniformly with respect to $\delta$
\begin{multline}\label{2.14}
\|u\|_{C([0,t];H^{1,\psi(x)})}+\|u\|_{L_2(0,t;H^{2,\psi'(x)})}+
\delta^{1/2}\bigl\||D^2u|\bigr\|_{L_2(0,t;L_2^{\psi(x)})} \\ \leq c(T) \left[\|u_0\|_{H^{1,\psi(x)}}+
\|f_0\|_{L_1(0,t;H^{1,\psi(x)})}+\|f_1\|_{L_2(0,t;L_2^{\psi(x)})}\right],
\end{multline}
\begin{multline}\label{2.15}
\iiint|Du(t,x,y,z)|^2\psi(x)\,dxdydz
+c_0\int_0^t\!\iiint|D^2u|^2\cdot(\psi'+\delta\psi)\,dxdydzd\tau \\
\leq \iiint|Du_0|^2\psi \,dxdydz+
c\int_0^t\!\iiint |Du|^2\psi\,dxdydzd\tau \\
+2\int_0^t\!\iint(f_{0x}u_x+f_{0y}u_y+f_{0z}u_z)\psi \,dxdydzd\tau \\
-2\delta^{1/2}\int_0^t\iiint f_1[(u_x\psi)_x+u_{yy}\psi+u_{zz}\psi]\,dxdydzd\tau,
\end{multline}
where the constants $c_0$, $c$ depend on $b$ and the properties of the function $\psi$ and the domain $\Omega$.
\end{lemma}
\begin{proof}
In the smooth case $u_0\in C_0^\infty(\Sigma)$, $f_0,f_1 \in C_0^\infty(\Pi_T)$ multiplying \eqref{2.1} by $-2\bigl(u_x(t,x,y,z)\psi(x)\eta_n(x)\bigr)_x -2\Delta^{\bot}u(t,x,y,z)\psi(x)\eta_n(x)$, where $u$ is the solution constructed in Lemmas~\ref{L2.1} and~\ref{L2.2}, $\eta_n(x)\equiv \eta(n-|x|)$, and integrating we obtain an equality
\begin{multline}\label{2.16}
\iiint |Du(t,x,y,z)|^2\psi\eta_n \,dxdydz - \iiint |Du_0|^2\psi\eta_n dxdydz \\+
\int_0^t\!\!\iiint(3u_{xx}^2+4u^2_{xy}+4u^2_{xz} +(\Delta^{\bot}u)^2)(\psi\eta_n)' \,dxdydzd\tau \\
+2\delta\int_0^t\!\!\iiint(u^2_{xx}+2u^2_{xy}+2u^2_{xz}+(\Delta^{\bot}u)^2)\psi\eta_n \,dxdydzd\tau \\
-\int_0^t\!\!\iiint |Du|^2(b(\psi\eta_n)' +(\psi\eta_n)'''+\delta(\psi\eta_n)'')\,dxdydzd\tau \\
=2\int_0^t\!\!\iiint(f_{0x}u_x+f_{0y}u_y+f_{0z}u_z)\psi\eta_n \,dxdydxd\tau \\
-2\delta^{1/2}\int_0^t\!\!\iiint f_1\bigl((u_x\psi\eta_n)_x+\Delta^{\bot}u\psi\eta_n\bigr)\,dxdydzd\tau.
\end{multline}
Passing to the limit when $n\to +\infty$ and using the properties of the function $\psi$ and inequality \eqref{2.5} we obtain \eqref{2.15} in the smooth case. This inequality provides estimate \eqref{2.14}. The general case is handled via closure.
\end{proof}
\begin{corollary}\label{C2.2}
Let the hypothesis of Lemma~\ref{L2.5} be satisfied for $\psi(x)\geq 1$ $\forall x\in\mathbb R$. Then for the (unique) weak solution $u\in C([0,T];H_0^1)$ and any $t\in (0,T]$
\begin{multline}\label{2.17}
\iiint |Du(t,x,y,z)|^2\,dxdydz
+2\delta\int_0^t\!\!\iiint(u^2_{xx}+2u^2_{xy}+2u^2_{xz}+(\Delta^{\bot}u)^2)\,dxdydzd\tau \\ =
\iiint |Du_0|^2\,dxdydz +2\int_0^t\!\iiint(f_{0x}u_x+f_{0y}u_y+f_{0z}u_z) \,dxdydzd\tau \\
-2\delta^{1/2}\int_0^t\!\!\iiint f_1\Delta u\,dxdydzd\tau.
\end{multline}
\end{corollary}
\begin{proof}
In the smooth case this equality is derived from \eqref{2.16}, where formally one must set $\psi\equiv 1$, and the consequent passage to the limit when $n\to +\infty$ and in the general case is obtained on the basis of estimate \eqref{2.14} via closure.
\end{proof}
\begin{lemma}\label{L2.6}
Let the hypothesis of Lemma~\ref{L2.5} be satisfied for some $\delta>0$ and $\psi(x)\geq 1$ $\forall x\in \mathbb R$. Consider the (unique) weak solution $u\in C([0,T];H_0^{1,\psi(x)})\cap L_2(0,T;H^{2,\psi(x)})$. Then for any $t\in (0,T]$ the following equality holds:
\begin{multline}\label{2.18}
-\frac 13 \iiint u^3(t,x,y,z)\widetilde\psi(x)\,dxdydz
-b\int_0^t\!\!\iiint u^2u_x\widetilde\psi\,dxdydzd\tau \\
+2\int_0^t\!\!\iiint uu_x \Delta u\widetilde\psi \,dxdydzd\tau
+\int_0^t\!\!\iiint u^2 \Delta u\widetilde\psi' \,dxdydzd\tau \\
-2\delta\int_0^t\!\!\iiint u |Du|^2\widetilde\psi\,dxdydzd\tau
-\delta\int_0^t\!\!\iiint u^2u_x\widetilde\psi'\,dxdydzd\tau \\
=-\frac 13 \iiint u_0^3\widetilde\psi \,dxdydz
-\int_0^t\!\!\iiint f u^2\widetilde\psi \,dxdydzd\tau,
\end{multline}
where either $\widetilde\psi \equiv \psi$ or $\widetilde\psi\equiv 1$.
\end{lemma}
\begin{proof}
In the smooth case multiplying \eqref{2.1} by $-u^2(t,x,y,z)\widetilde\psi(x)$ and integrating one instantly obtains equality \eqref{2.18}.
In the general case we obtain this equality via closure. Note that by virtue of \eqref{1.8}
(for $q=4$, $\psi_1=\psi_2=\psi$) if
$u\in C([0,T];H_0^{1,\psi(x)})\cap L_2(0,T;H^{2,\psi(x)})$ then
$$
u\in C([0,T];L_4^{\psi(x)}),\qquad |Du|\in L_2(0,T; L_4^{\psi(x)})
$$
and this passage to the limit is easily justified.
\end{proof}
\section{Existence of weak solutions}\label{S3}
Consider the following equation:
\begin{equation}\label{3.1}
u_t+bu_x+\Delta u_x-\delta\Delta u+(g(u))_x=f(t,x,y,z), \quad \delta\in [0,1].
\end{equation}
\begin{definition}\label{D3.1}
Let $u_0\in L_2^{\psi(x)}$ for a certain admissible weight function $\psi(x)\geq 1$
$\forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function, $f\in L_1(0,T; L_2^{\psi(x)})$.
A function $u \in L_2(0,T;L_2^{\psi(x)})$ is called a weak solution to problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3}, if for any function $\varphi$, such that $\partial^j_x\varphi \in C([0,T]; L_2^{1/\psi'(x)})$,
$\partial_t^m\partial_x^j\partial_{y,z}^\nu \varphi \in L_2(0,T;L_2^{1/\psi'(x)})$ for $j\geq 0$, $2m+|\nu|\leq 2$
and $\varphi\big|_{t=T}\equiv 0$, $\varphi\big|_{(0,T)\times \partial\Sigma}=0$, the function
$g(u(t,x,y,z))\varphi_x\in L_1(\Pi_T)$ and there holds the following equality:
\begin{multline}\label{3.2}
\int_0^T\!\! \iiint \big[ u(\varphi_t+b\varphi_x+\Delta\varphi_x+\delta\Delta\varphi) +
g(u)\varphi_x + f\varphi\big]\,dxdydzdt \\
+\iiint u_0\varphi\big|_{t=0}\,dxdydz=0.
\end{multline}
\end{definition}
\begin{remark}\label{R3.1}
It is easy to see that if $g\equiv 0$ and a function $u$ is a weak solution to problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3}
in the sense of Definition~\ref{D3.1} then it is a weak solution to this problem in the sense of Definition~\ref{D2.1}.
\end{remark}
\begin{remark}\label{R3.2}
Let $g\in C(\mathbb R)$ and $|g(u)|\leq c(|u|+u^2)$ $\forall u\in \mathbb R$ for a certain constant $c$. Then it easy to see that for any function $u \in L_\infty(0,T;L_2^{\psi(x)})\cap L_2(0,T;H^{1,\psi'(x)})$, where $\psi\geq 1$ is an admissible weight function such that $\psi'$ is also an admissible weight function, and for any function $\varphi$ satisfying the hypothesis of Definition~\ref{D3.1} the function $g(u)\varphi_x\in L_1(\Pi_T)$. In fact, it follows from \eqref{1.8} that
\begin{multline}\label{3.3}
\iiint u^2|\varphi_x|\,dxdydz \leq
\bigl\|u(\psi')^{1/4}\psi^{1/4}\bigr\|_{L_3}^2 \bigl\|\varphi_x (\psi')^{-1/2}\bigr\|_{L_3} \\ \leq
c\bigl\|(|Du|+|u|)(\psi')^{1/2}\bigr\|_{L_2}\|u\psi^{1/2}\|_{L_2}
\bigl\|(|D^2\varphi|+|D\varphi|)(1/\psi')^{1/2}\bigr\|_{L_2}.
\end{multline}
\end{remark}
First of all, we prove a lemma on solubility of problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3} for spaces $L_2^{\psi(x)}$ in the "regularized" case.
\begin{lemma}\label{L3.1}
Let $\delta>0$, $g\in C^1(\mathbb R)$, $g(0)=0$ and $|g'(u)|\leq c\ \forall u\in\mathbb R$. Assume that $u_0\in L_2^{\psi(x)}$ for an admissible weight function $\psi(x)\geq 1$ $\forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function, $f\in L_1(0,T;L_2^{\psi(x)})$. Then problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3} has a unique weak solution
$u\in C([0,T];L_2^{\psi(x)})\cap L_2(0,T;H_0^{1,\psi(x)})$.
\end{lemma}
\begin{proof}
We apply the contraction principle. For $t_0\in(0,T]$ define a mapping $\Lambda$ on a set $Y(\Pi_{t_0})=C([0,t_0];L_2^{\psi(x)})\cap L_2(0,t_0;H_0^{1,\psi(x)})$ as follows: $u=\Lambda v\in Y(\Pi_{t_0})$ is a solution to a linear problem
\begin{equation}\label{3.4}
u_t+bu_x+\Delta u_x-\delta\Delta u=f-(g(v))_x
\end{equation}
in $\Pi_{t_0}$ with boundary conditions \eqref{1.2}, \eqref{1.3}.
Note that $|g(v)|\leq c|v|$ and, therefore,
\begin{equation}\label{3.5}
\|g(v)\|_{L_2(0,t_0;L_2^{\psi(x)})}\leq c||v||_{L_2(0,t_0;L_2^{\psi(x)})}<\infty.
\end{equation}
Thus, according to Lemma~\ref{L2.4} (where $f_1\equiv\delta^{-1/2}g(v)$) the mapping $\Lambda$ exists. Moreover, for functions $v,\widetilde{v}\in Y(\Pi_{t_0})$
$$
\|g(v)-g(\widetilde{v})\|_{L_2(0,t_0;L_2^{\psi(x)})}\leq c\|v-\widetilde{v}\|_{L_2(0,t_0;L_2^{\psi(x)})} \leq ct_0^{1/2}\|v-\widetilde{v}\|_{C([0,t_0];L_2^{\psi(x)})}.
$$
As a result, according to inequality \eqref{2.11}
$$
\|\Lambda v-\Lambda\widetilde{v}\|_{Y(\Pi_{t_0})}\leq
c(T,\delta)t_0^{1/2}\|v-\widetilde{v}\|_{Y(\Pi_{t_0})}.
$$
Since the constant in the right side of this equality is uniform with respect to $u_0$, one can construct the solution on the whole time segment $[0,T]$ by the standard argument.
\end{proof}
Now we pass to the proof of Theorem~\ref{T1.1}.
\begin{proof}[Proof of Theorem~\ref{T1.1}]
For $h\in (0,1]$ consider a set of initial-boundary value problems in $\Pi_T$
\begin{equation}\label{3.6}
u_t+bu_x+\Delta u_x-h\Delta u+(g_h(u))_x=f(t,x,y,z)
\end{equation}
with boundary conditions \eqref{1.2}, \eqref{1.3}, where
\begin{equation}\label{3.7}
g_h(u)\equiv\int_0^u\Bigl[\theta\eta(2-h|\theta|)+\frac{2\sgn\theta}{h}\eta(h|\theta|-1)\Bigr]\,d\theta.
\end{equation}
Note that $g_h(u)\equiv u^2/2$ if $|u|\leq 1/h$, $|g'_h(u)|\leq 2/h\ \forall u\in\mathbb R$ and $|g'_h(u)|\leq 2|u|$ uniformly with respect to $h$.
According to Lemma~\ref{L3.1} there exists a unique solution to each of these problems
$u_h\in C([0,T];L_2^{\psi(x)})\cap L_2(0,T;H_0^{1,\psi(x)})$. Note that similarly to \eqref{3.5}
$g_h(u_h) \in L_2(0,T;L_2^{\psi(x)})$.
Next, establish estimates for functions $u_h$ uniform with respect to $h$.
Write down corresponding equality \eqref{2.13} for functions $u_h$ (we omit the index $h$ in intermediate steps for simplicity):
\begin{multline}\label{3.8}
\iiint u^2 \,dxdydz
+2h\int_0^t\!\iiint |Du|^2\,dxdydzd\tau = \iiint u_0^2 \,dxdydz \\
+2\int_0^t\!\iiint f u \,dxdydzd\tau
-2\int_0^t\!\iiint g'(u)u_x u \,dxdydzd\tau.
\end{multline}
Since
\begin{equation}\label{3.9}
g'(u)u_xu=\Bigl(\int_0^u g'(\theta)\theta \,d\theta\Bigr)_x \equiv \bigl(g'(u)u\bigr)^*_x,
\end{equation}
where $g^*(u)\equiv \displaystyle \int_0^u g(\theta)\,d\theta$ denotes the primitive for $g$ such that $g^*(0)=0$,
we have that $\iiint g'(u)u_x u\,dxdydz=0$ and equality \eqref{3.8} yields that
\begin{equation}\label{3.10}
\|u_h\|_{C([0,T];L_2)}+h^{1/2}\|u_h\|_{L_2(0,T;H^1)}\leq c
\end{equation}
uniformly with respect to $h$ (and also uniformly with respect to $\Omega$).
Next, write down corresponding equality \eqref{2.12}, then with the use of \eqref{3.9}
\begin{multline}\label{3.11}
\iiint u^2(t,x,y,z)\psi(x)\,dxdydz+\int_0^t\! \iiint (3u_x^2+u_y^2+u_z^2)\psi'\,dxdydzd\tau \\
+2h \int_0^t \!\! \iiint |Du|^2\psi\,dxdydzd\tau -
\int_0^t \!\! \iiint u^2\cdot (b\psi'+\psi'''+h\psi'')\,dxdydzd\tau \\ =
\iiint u_0^2\psi\,dxdydz +2\int_0^t\!\iiint f u\psi \,dxdydzd\tau \\
+2\int_0^t\! \iiint (g'(u)u)^*\psi' \,dxdydzd\tau.
\end{multline}
Apply interpolating inequality \eqref{1.8} for $k=1$, $m=0$, $q=4$, $\psi_1=\psi_2\equiv\psi'$:
\begin{multline}\label{3.12}
\Bigl|\iiint (g'(u)u )^*\psi'\,dxdydz\Bigr|
\leq \iiint |u|^3\psi'\,dxdydz \\
\leq \Bigl(\iiint u^2\,dxdydz\Bigr)^{1/2} \Bigl(\iiint|u|^4(\psi')^2\,dxdydz\Bigr)^{1/2}
\leq c\Bigl(\iiint u^2\,dxdydz\Bigr)^{1/2} \\ \times \Bigl[\Bigl(\iiint |Du|^2\psi'\,dxdydz\Bigr)^{3/4}
\Bigl(\iiint u^2\psi' \,dxdydz\Bigr)^{1/4}
+\iiint u^2\psi' \,dxdydz\Bigr]
\end{multline}
(note that here the constant $c$ is also uniform with respect to $\Omega$).
Since the norm of the solution in the space $L_2$ is already estimated in \eqref{3.10}, it follows from \eqref{3.11} and \eqref{3.12} that
\begin{equation}\label{3.13}
\|u_h\|_{C([0,T];L_2^{\psi(x)})}
+\bigl\| |Du_h| \bigr\|_{L_2(0,T;L_2^{\psi'(x)})}
+h^{1/2}\|u_h\|_{L_2(0,T;H^{1,\psi(x)})}\leq c.
\end{equation}
Finally, write down the analogue of \eqref{3.11}, where $\psi(x)$ is substituted by \linebreak $\rho_0(x-x_0)$ for any $x_0\in\mathbb R$. Then it easily follows that (see \eqref{1.6})
\begin{equation}\label{3.14}
\lambda (|Du_h|;T)\leq c.
\end{equation}
In particular, $\|u_h\|_{L_2(0,T;H^1(Q_n))}\leq c(n)$ for any bounded domain $Q_n=(-n,n)\times \Omega$.
Since $|g_h(u)|\leq u^2$ we have that $\|g_h(u_h)\|_{L_\infty(0,T;L_1(Q_n))}\leq c(n)$. Using the well-known embedding $L_1(Q_n)\subset H^{-2}(Q_n)$ we first derive that $\|g_h(u_h)\|_{L_\infty(0,T;H^{-2}(Q_n))}\leq c(n)$, and then according to equation \eqref{3.1} itself that uniformly with respect to $h$
$$
\|u_{ht}\|_{L_1(0,T;H^{-3}(Q_n))}\leq c(n).
$$
Applying the compactness embedding theorem of evolutionary spaces from \cite{S2} we obtain that the set $\{u_h\}$ is precompact in $L_2((0,T)\times Q_n)$ for all $n$.
Now show that if $u_h\to u$ in $L_2((0,T)\times Q_n)$ for some sequence $h\to 0$, then $g_h(u_h)\to u^2/2$ in $L_1((0,T)\times Q_n)$.
Indeed,
\begin{multline*}
|g_h(u_h)-u^2/2|\leq |g_h(u_h)-g_h(u)|+|g_h(u)-u^2/2| \\
\leq 2(|u_h|+|u|)|u_h-u|+|g_h(u)-u^2/2|,
\end{multline*}
where $|g_h(u)-u^2/2|\leq 2u^2\in L_1((0,T)\times Q_n)$ and$g_h(u)\to u^2/2$ pointwise.
As a result, the required solution is constructed in a standard way as the limit of the solutions $u_h$ when $h\to 0$
(equality \eqref{3.2} is first derived for the functions $\varphi\eta(n-|x|)$ with consequent passage to the limit when $n\to +\infty$).
\end{proof}
\begin{remark}\label{R3.3}
Theorem~\ref{T1.1} remains valid if $\partial\Omega\in C^2$ but for simplicity we do not present here the corresponding argument.
\end{remark}
We now proceed to solutions in spaces $H_0^{1,\psi(x)}$ and first estimate a lemma analogous to Lemma~\ref{L3.1}.
\begin{lemma}\label{L3.2}
Let $\delta>0$, $g(u)\equiv u^2/2$. Assume that $u_0\in H_0^{1,\psi(x)}$ for an admissible weight function
$\psi(x)\geq 1$ $\forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function,
$f\in L_1(0,T;H_0^{1,\psi(x)})$. Then problem \eqref{3.1}, \eqref{1.2}, \eqref{1.3} has a unique weak solution $u\in C([0,T];H_0^{1,\psi(x)})\cap L_2(0,T;H^{2,\psi(x)})$.
\end{lemma}
\begin{proof}
Introduce for $t_0\in (0,T]$ a space $Y_1(\Pi_{t_0})=C([0,t_0];H_0^{1,\psi(x)})\cap L_2(0,t_0;H^{2,\psi(x)})$ and define a mapping $\Lambda$ on it in the same way as in the proof of Lemma~\ref{L3.1} with the substitution of $Y(\Pi_{t_0})$ by $Y_1(\Pi_{t_0})$ and equation \eqref{3.4} by an equation
$$
u_t+bu_x+\Delta u_x -\delta\Delta u = f - vv_x.
$$
By virtue of \eqref{1.8} (for $\psi_1=\psi_2\equiv \psi\geq 1$)
\begin{multline}\label{3.15}
\|vv_x\|_{L_2(0,t_0;L_2^{\psi(x)})} \leq
\Bigl[\int_0^{t_0}\|v_x\psi^{1/2}\|_{L_4}^2 \|v\psi^{1/2}\|_{L_4}^2\,dt\Bigr]^{1/2} \\ \leq
c\Bigl[\int_0^{t_0} \Bigl(\bigl\||D v_x|\bigr\|_{L_2^{\psi(x)}}^{3/2}\|v_x\|_{L_2^{\psi(x)}}^{1/2}+
\|v_x\|_{L_2^{\psi(x)}}^{2}\Bigr) \bigl\| |Dv|+|v|\bigr\|_{L_2^{\psi(x)}}^{2}\,dt\Bigr]^{1/2} \\ \leq
c_1 t_0^{1/8} \|v\|_{L_2(0,t_0;H^{2,\psi(x)})}^{3/4}\|v\|_{C([0,t_0];H^{1,\psi(x)})}^{5/4} \leq
c_1 t_0^{1/8} \|v\|_{Y_1(\Pi_{t_0})}^2
\end{multline}
and similarly
\begin{equation}\label{3.16}
\|vv_x-\widetilde v\widetilde v_x\|_{L_2(0,t_0;L_2^{\psi(x)})} \leq
c t_0^{1/8} \bigl(\|v\|_{Y_1(\Pi_{t_0})}+\|\widetilde v\|_{Y_1(\Pi_{t_0})}\bigr)
\|v-\widetilde v\|_{Y_1(\Pi_{t_0})}.
\end{equation}
In particular, the hypothesis of Lemma~\ref{L2.5} is satisfied (for $f_1\equiv -vv_x$) and, therefore, the mapping $\Lambda$ exists. Moreover, inequalities \eqref{2.14}, \eqref{3.15}, \eqref{3.16} provide that
\begin{equation}\label{3.17}
\|\Lambda v\|_{Y_1(\Pi_{t_0})}\leq c(T,\delta)\left(\|u_0\|_{H^{1,\psi(x)}}+
\|f\|_{L_1(0,T;H^{1,\psi(x)})} +
t_0^{1/8}\|v\|_{Y_1(\Pi_{t_0})}^2\right),
\end{equation}
\begin{equation}\label{3.18}
\|\Lambda v -\Lambda\widetilde v\|_{Y_1(\Pi_{t_0})}\leq c(T,\delta)
t_0^{1/8}\left(\|v\|_{Y_1(\Pi_{t_0})}+\|\widetilde v\|_{Y_1(\Pi_{t_0})}\right)
\|v-\widetilde v\|_{Y_1(\Pi_{t_0})}.
\end{equation}
Existence of unique weak solution to the considered problem in the space $Y_1(\Pi_{t_0})$ on the time interval $[0,t_0]$ depending on $\|u_0\|_{H^{1,\psi(x)}}$ follows from \eqref{3.17}, \eqref{3.18} by the standard argument.
Now we estimate the following a priori estimate: if $u\in Y_1(\Pi_{T'})$ is a solution to the considered problem for some $T'\in (0,T]$ then uniformly with respect to $\delta$
\begin{equation}\label{3.19}
\|u\|_{C([0,T'];H^{1,\psi(x)})}\leq c(T,\|u_0\|_{H^{1,\psi(x)}},
\|f\|_{L_1(0,T;H^{1,\psi(x)})}).
\end{equation}
Note that similarly to \eqref{3.13}
\begin{equation}\label{3.20}
\|u\|_{C([0,T'];L_2^{\psi(x)})} \leq c(T,\|u_0\|_{L_2^{\psi(x)}},\|f\|_{L_1(0,T;L_2^{\psi(x)})}).
\end{equation}
Let either $\widetilde\psi\equiv \psi$ or $\widetilde\psi\equiv 1$
Apply Corollary~\ref{C2.2} for $\widetilde\psi\equiv 1$ or Lemma~\ref{L2.5} for $\widetilde\psi\equiv \psi$, where $f_1\equiv -\delta^{-1/2}uu_x$, then it follows from \eqref{2.17} or \eqref{2.15} that
\begin{multline}\label{3.21}
\iiint|Du(t,x,y,z)|^2\widetilde\psi(x)\,dxdydz
+c_0\int_0^t\!\iiint|D^2u|^2\cdot(\widetilde\psi'+\delta\widetilde\psi)\,dxdydzd\tau \\
\leq \iiint|Du_0|^2\widetilde\psi \,dxdydz+
c\int_0^t\!\iiint |Du|^2\widetilde\psi\,dxdydzd\tau \\
+2\int_0^t\!\iint(f_{x}u_x+f_{y}u_y+f_{z}u_z)\widetilde\psi \,dxdydzd\tau \\
+2\int_0^t\iiint uu_x[(u_x\widetilde\psi)_x+u_{yy}\widetilde\psi+u_{zz}\widetilde\psi]\,dxdydzd\tau.
\end{multline}
Apply Lemma~\ref{L2.6} then it follows from \eqref{2.18} that
\begin{multline}\label{3.22}
-\frac 13 \iiint u^3(t,x,y,z)\widetilde\psi(x)\,dxdydz
-b\int_0^t\!\!\iiint u^2u_x\widetilde\psi\,dxdydzd\tau \\
+2\int_0^t\!\!\iiint uu_x \Delta u\widetilde\psi \,dxdydzd\tau
+\int_0^t\!\!\iiint u^2 \Delta u\widetilde\psi' \,dxdydzd\tau \\
-2\delta\int_0^t\!\!\iiint u |Du|^2\widetilde\psi\,dxdydzd\tau
-\delta\int_0^t\!\!\iiint u^2u_x\widetilde\psi'\,dxdydzd\tau \\
=-\frac 13 \iiint u_0^3\widetilde\psi \,dxdydz
-\int_0^t\!\!\iiint (f-uu_x) u^2\widetilde\psi \,dxdydzd\tau.
\end{multline}
Summing \eqref{3.21} and \eqref{3.22} provides an inequality
\begin{multline}\label{3.23}
\iiint\Bigl(|Du|^2-\frac {u^3}3\Bigr)\widetilde\psi \,dxdydz
+c _0\int_0^t\!\!\iiint|D^2u|^2\cdot(\widetilde\psi'+\delta\widetilde\psi) \,dxdydzd\tau \\
\leq \iint\Bigl(|Du_0|^2-\frac{u_0^3}3\Bigr)\widetilde\psi \,dxdydz
+c\int_0^t\!\!\iint|Du|^2\widetilde\psi \,dxdydzd\tau \\
+2\int_0^t\!\!\iiint (f_xu_x+f_yu_y+f_zu_z)\widetilde\psi \,dxdydzd\tau
-\int_0^t\!\!\iiint fu^2\widetilde\psi \,dxdydzd\tau \\
+2\int_0^t\!\!\iiint u u^2_x\widetilde\psi' \,dxdydzd\tau
-\int_0^t\!\!\iiint u^2\Delta u\widetilde\psi' \,dxdydzd\tau \\
+2\delta\int_0^t\!\!\iiint u|Du|^2\widetilde\psi \,dxdydzd\tau
-\frac 13\int_0^t\!\!\iiint u^3\cdot(\delta\widetilde\psi''+b\widetilde\psi') \,dxdydzd\tau \\
-\frac 14\int_0^t\!\!\iiint u^4\widetilde\psi' \,dxdydzd\tau.
\end{multline}
By virtue of \eqref{1.8} and \eqref{3.20} similarly to \eqref{3.12}
\begin{multline*}
\iiint |u|^3\widetilde\psi\,dxdydz
\leq \Bigl(\iiint u^2\,dxdydz\Bigr)^{1/2} \Bigl(\iiint |u|^4 \widetilde\psi^2\,dxdydz\Bigr)^{1/2} \\
\leq c\Bigl[\Bigl(\iiint |Du|^2\widetilde\psi\,dxdydz\Bigr)^{3/4}+1\Bigr].
\end{multline*}
Next,
\begin{multline*}
\iiint |f|u^2\widetilde\psi \,dxdydz\leq c\Bigl(\iiint f^2 \,dxdydz\Bigr)^{1/2}
\Bigl(\iiint u^4\widetilde\psi^2dxdydz\Bigr)^{1/2} \\
\leq c_1\Bigl(\iint f^2 \,dxdydz\Bigr)^{1/2}
\Bigl[\Bigl(\iiint |Du|^2\widetilde\psi dxdydz\Bigr)^{3/4}+1\Bigr],
\end{multline*}
\begin{multline*}
\iiint |u|\cdot |Du|^2\widetilde\psi'\,dxdydz \leq \Bigl(\iiint u^2\,dxdydz\Bigr)^{1/2}
\Bigl(\iiint |Du|^4(\widetilde\psi')^2\,dxdydz\Bigr)^{1/2} \\
\leq c\Bigl[\Bigl(\iiint |D^2 u|^2\widetilde\psi' dxdydz\Bigr)^{3/4}\Bigl(\iiint |Du|^2\widetilde\psi dxdydz\Bigr)^{1/4}+
\iiint |Du|^2\widetilde\psi dxdydz\Bigr],
\end{multline*}
\begin{multline*}
\delta\iiint |u|\cdot|Du|^2\widetilde\psi \,dxdydz \leq \delta\Bigl(\iiint u^2\,dxdydz\Bigr)^{1/2}
\Bigl(\iiint |Du|^4 \widetilde\psi^2\,dxdydz\Bigr)^{1/2} \\
\leq c\delta\Bigl[\Bigl(\iiint |D^2 u|^2\widetilde\psi dxdydz\Bigr)^{3/4}
\Bigl(\iiint |Du|^2\widetilde\psi dxdydz\Bigr)^{1/4}+
\iiint |Du|^2\widetilde\psi dxdydz\Bigr].
\end{multline*}
Choosing $\widetilde\psi\equiv 1$ we derive from \eqref{3.23} with the use of these estimates that
\begin{equation}\label{3.24}
\|u\|_{C([0,T'];H^1)} \leq c(T,\|u_0\|_{H^1},\|f\|_{L_1(0,T;H^1)}).
\end{equation}
Finally, since
\begin{multline*}
\iiint u^4\psi'\,dxdydz \leq c\Bigl(\iiint u^4\,dxdydz\Bigr)^{1/2}\Bigl(\iiint u^4\psi^2\,dxdydz\Bigr)^{1/2} \\
\leq c_1 \iint \bigl(|Du|^2+u^2\bigr)\,dxdydz \iint \bigl(|Du|^2+u^2\bigr)\psi\,dxdydz,
\end{multline*}
choosing in \eqref{3.23} $\widetilde\psi\equiv\psi$ with the use of \eqref{3.24} we obtain estimate \eqref{3.19}.
Inequalities \eqref{3.17} and \eqref{3.18} allow us to construct a solution to the considered problem locally in time by the contraction while estimate \eqref{3.19} enables us to extend it for the whole time segment $[0,T]$.
\end{proof}
At the end of this section we present the proof of the part of Theorem~\ref{T1.2} concerning existence of solutions.
\begin{proof}[Proof of Theorem~\ref{T1.2}, existence]
For $h\in (0,1]$ consider a set of initial-boundary value problems in $\Pi_T$
\begin{equation}\label{3.25}
u_t+bu_x+\Delta u_x -h\Delta u +uu_x =f(t,x,y,z)
\end{equation}
with boundary conditions \eqref{1.2}, \eqref{1.3}. It follows from Lemma~\ref{L3.2} that unique solutions to these problems $u_h\in C([0,T]; H_0^{1,\psi(x)})\cap L_2(0,T;H^{2,\psi(x)})$ exist. Moreover, according to \eqref{3.19} uniformly with respect to $h$
\begin{equation}\label{3.26}
\|u_h\|_{C([0,T];H^{1,\psi(x)})} \leq c.
\end{equation}
Next, inequality \eqref{3.23} in the case $\widetilde\psi\equiv \psi$ applied to the functions $u_h$ provides that uniformly with respect to $h$
\begin{equation}\label{3.27}
\bigl\| |D^2 u_h|\bigr\|_{L_2(0,T;L_2^{\psi'(x)})} \leq c.
\end{equation}
Finally, note that inequality \eqref{3.23} obviously holds for $\widetilde\psi\equiv \rho_0(x-x_0)$ for any
$x_0\in\mathbb R$, therefore, similarly to \eqref{3.14}
\begin{equation}\label{3.28}
\lambda(|D^2 u_h|;T)\leq c.
\end{equation}
The end of the proof is exactly the same as for Theorem~\ref{T1.1}.
\end{proof}
\section{Continuous dependence of weak solutions}\label{S4}
Present a theorem from which the result of Theorem~\ref{T1.2} on uniqueness of weak solutions follows.
\begin{theorem}\label{T4.1}
Let $u_0,\widetilde{u}_0\in H_0^{1,\alpha}$, $f,\widetilde{f}\in L_1(0,T;H_0^{1,\alpha})$ for some $\alpha\geq 3/4$, $u,\widetilde{u}$ be weak solutions to corresponding problems \eqref{1.1}--\eqref{1.3} from the class $X^{1,\alpha}(\Pi_T)$. Then for any $\beta>0$
\begin{multline}\label{4.1}
\|u-\widetilde{u}||_{L_\infty(0,T;L_2^{\varkappa_{\alpha,\beta}(x)})}+
\bigl\| |D(u-\widetilde{u})|\bigr\|_{L_2(0,T;L_2^{\varkappa_{\alpha-1/2,\beta}(x)})} \\
\leq c\Bigl(\|(u_0-\widetilde{u}_0)\|_{L_2^{\varkappa_{\alpha,\beta}(x)}} +
\|(f-\widetilde{f})\|_{L_1(0,T;L_2^{\varkappa_{\alpha,\beta}(x)})}\Bigr),
\end{multline}
where the constant $c$ depends on the norms of the functions $u,\widetilde{u}$ in the space $L_\infty(0,T;H^{1,3/4})$.
\end{theorem}
\begin{proof}
Let $\psi(x)\equiv \varkappa_{\alpha,\beta}(x)$, then $\psi'(x)\sim \varkappa_{\alpha-1/2,\beta}(x)$ and
$\psi^2(x)/\psi'(x)\sim \varkappa_{\alpha+1/2,\beta}(x)$. Since $\alpha\geq 1/2$
\begin{multline}\label{4.2}
\iiint u^4\varkappa_{\alpha+1/2,\beta}\,dxdydz
\leq c\iiint u^4 \rho^2_\alpha\,dxdydz \\
\leq c_1\Bigl(\iiint(u^2+|Du|^2)\rho_\alpha\,dxdydz\Bigr)^2,
\end{multline}
we have that $u^2\in L_\infty(0,T;L_2^{\psi^2(x)/\psi'(x)})$.
Denote $v\equiv u-\tilde{u}$, then the function $v$ is a solution to a linear problem
\begin{equation}\label{4.3}
v_t+bv_x+\Delta v_x=(f-\widetilde{f})-\frac 12 \bigl(u^2-\widetilde{u}^2\bigr)_x\equiv f_0+f_{2x},
\end{equation}
\begin{equation}\label{4.4}
v\big|_{t=0}=u_0-\widetilde{u}_0\equiv v_0,\qquad v\big|_{(0,T)\times\partial\Sigma}=0.
\end{equation}
The hypotheses of Lemmas~\ref{L2.3} and~\ref{L2.4} ($\delta = 0$) hold for this problem, therefore, by virtue of \eqref{2.12}
\begin{multline}\label{4.5}
\iiint v^2\psi\,dxdydz
+\int_0^t\!\!\iiint|Dv|^2\psi'\,dxdydzd\tau \leq \iiint v_0^2\psi\,dxdydz \\
+c\int_0^t\!\!\iiint v^2\psi\,dxdydzd\tau
+2\int_0^t\!\!\iiint f_0 v\psi\,dxdydzd\tau \\
-2\int_0^t\!\!\iiint f_2(v\psi)_x\,dxdydzd\tau.
\end{multline}
It is easy to see that
$$
\Bigl|\iiint f_2(v\psi)_x\,dxdydz \Bigr|
\leq c\iiint \bigl(|u_x|+|\widetilde{u}_x|+|u|+|\widetilde{u}|\bigr)v^2\psi\,dxdydz.
$$
With the use of \eqref{1.8} we derive that
\begin{multline}\label{4.6}
\iiint |u_x|v^2\psi\,dxdydz \\ \leq
\Bigl(\iiint u^2_x \Bigl(\frac{\psi}{\psi'}\Bigr)^{3/2}\,dxdydz\Bigr)^{1/2}
\Bigl(\iiint v^4(\psi')^{3/2}\psi^{1/2}\,dxdydz\Bigr)^{1/2} \\
\leq c\Bigl(\iiint u_x^2\rho_{3/4}\,dxdydz\Bigr)^{1/2}
\Bigl[\Bigl(\iiint|Dv|^2\psi'\,dxdydz\Bigr)^{3/4}
\Bigl(\iiint v^2\psi\,dxdydz\Bigr)^{1/4} \\
+\iiint v^2\psi\,dxdydz\Bigr].
\end{multline}
Other terms are estimated in a similar way and inequality \eqref{4.5} yields the desired result.
\end{proof}
\begin{remark}\label{R4.1}
It easy to see that continuous dependence of solutions can be also established by the same argument in spaces with exponential weights at $+\infty$. More precisely, if $u_0,\widetilde{u}_0\in H_0^{1,\alpha,\exp}$,
$f,\widetilde{f}\in L_1(0,T;H_0^{1,\alpha,\exp})$ for some $\alpha>0$ then for corresponding weak solutions from the space $X^{1,\alpha,\exp}(\Pi_T)$ the analogue of inequality \eqref{4.1} holds, where the functions $\varkappa_{\alpha,\beta}$, $\varkappa_{\alpha-1/2,\beta}$ are substituted by $e^{2\alpha x}$. Moreover, the constant $c$ in the right side depends here on the norms of functions $u$, $\widetilde u$ in the space $L_\infty(0,T;H^1)$ (since in this case $\psi\sim \psi'$, see \eqref{4.6}). Unfortunately, the applied technique does not allow to avoid exponentially decreasing weight at $-\infty$.
\end{remark}
\section{Large-time decay of solutions}\label{S5}
\begin{proof}[Proof of Theorem~\ref{T1.3}]
Let $\psi(x)\equiv e^{2\alpha x}$ for some $\alpha\in (0,1]$. As in the proof of Theorem~\ref{T1.1} consider the set of solutions $u_h\in C([0,T];L_2^{\alpha,\exp})\cap L_2(0,T;H_0^{1,\alpha\exp})$ to problems \eqref{3.6}, \eqref{1.2}, \eqref{1.3}. Of course, these solutions exist for all positive $T$.
First of all, note that equality \eqref{3.8} yields here that
\begin{equation}\label{5.1}
\|u_h(t,\cdot,\cdot,\cdot)\|_{L_2} \leq \|u_0\|_{L_2}.
\end{equation}
Now write down equality \eqref{3.11} (for $\psi\equiv e^{2\alpha x}$, we again temporarily omit the index $h$):
\begin{multline}\label{5.2}
\iiint u^2\psi\,dxdydz+2\alpha\int_0^t\!\! \iiint (3u_x^2+u_y^2+u_z^2)\psi\,dxdydzd\tau \\
+2h \int_0^t \!\! \iiint |Du|^2\psi\,dxdydzd\tau-2\alpha(b+4\alpha^2+2h\alpha)\int_0^t \!\! \iiint u^2 \psi\,dxdydzd\tau \\= \iiint u_0^2\psi\,dxdydz
+4\int_0^t\!\! \iiint \bigl(g'(u)u\bigr)^*\psi'\,dxdydzd\tau.
\end{multline}
Since $\bigl|\bigl(g'(u)u\bigr)^*\bigr|\leq u^2/h$ it is obvious that $\bigl(g'(u)u\bigr)^*\psi'\in L_\infty(0,T;L_1)$ and from equation \eqref{5.2} follows such an inequality in a differential form: for a.e. $t>0$
\begin{multline}\label{5.3}
\frac d{dt}\iiint u^2\psi\,dxdydz+2\alpha \iiint (3u_x^2+u_y^2+u_z^2)\psi\,dxdydz \\ \leq
2\alpha(b+4\alpha^2+2\alpha) \iiint u^2 \psi\,dxdydz+
2 \iiint \bigl(g'(u)u\bigr)^*\psi'\,dxdydz.
\end{multline}
Continuing inequality \eqref{3.12}, we find with the use of \eqref{5.1} that uniformly with respect to $\Omega$, $h$ and $\alpha$ (see Remark~\ref{R1.3})
\begin{multline}\label{5.4}
\Bigl|\iiint \bigl(g'(u)u\bigr)^*\psi'\,dxdydz\Bigr| \leq
\alpha \iiint |Du|^2\psi\,dxdydz \\+
c\alpha \bigl(\|u_0\|_{L_2}+\|u_0\|_{L_2}^4\bigr) \iiint u^2\psi\,dxdydz.
\end{multline}
Apply Friedrichs inequality (see, for example, \cite{LSU}): for $\varphi\in H_0^1(\Omega)$
\begin{equation}\label{5.5}
\|\varphi\|_{L_2(\Omega)} \leq c|\Omega|^{1/2}\bigl(\|\varphi_y\|_{L_2(\Omega)}+\|\varphi_z\|_{L_2(\Omega)}\bigr).
\end{equation}
Therefore, for certain constant $c_0$
\begin{equation}\label{5.6}
\iiint (u_y^2+u_z^2)\psi\,dxdydz \geq \frac{c_0}{|\Omega|}\iiint u^2\psi\,dxdydz.
\end{equation}
Combining \eqref{5.3}, \eqref{5.4}, \eqref{5.6} provides that uniformly with respect to $\Omega$, $h$ and $\alpha$
\begin{multline}\label{5.7}
\frac d{dt} \iiint u^2\psi\,dxdydz + \frac{c_0\alpha}{|\Omega|}\iiint u^2\psi\,dxdydz \\ \leq
c\alpha\bigl(b+6\alpha+\|u_0\|_{L_2}+\|u_0\|_{L_2}^4\bigr) \iiint u^2\psi\,dxdydz.
\end{multline}
Choose $\Omega_0 = \displaystyle \frac {c_0}{2cb}$ if $b>0$, $\alpha_0\leq 1$ and $\epsilon_0$ satisfying an inequality
$\displaystyle c(6\alpha_0+\epsilon_0+\epsilon_0^4) \leq \frac {c_0}{4|\Omega|}$,
$\beta =\displaystyle \frac{c_0}{8|\Omega|}$, then it follows from \eqref{5.7} that uniformly with respect to $h$
\begin{equation}\label{5.8}
\|u_h(t.\cdot,\cdot,\cdot)\|_{L_2^{\psi(x)}} \leq e^{-\alpha\beta t} \|u_0\|_{L_2^{\psi(x)}} \quad \forall t\geq 0.
\end{equation}
Passing to the limit when $h\to+0$ we derive \eqref{1.7}.
\end{proof}
\begin{remark}\label{R5.1}
Besides \eqref{5.5} Friedrichs inequality can be written in another form: if $\Omega\subset (0,L_1)\times (0,L_2)$ then
for $\varphi\in H_0^1(\Omega)$
$$
\|\varphi\|_{L_2(\Omega)} \leq \frac{\min(L_1,L_2)}{\pi}
\bigl(\|\varphi_y\|_{L_2(\Omega)}+\|\varphi_z\|_{L_2(\Omega)}\bigr),
$$
with the corresponding modification of the theorem.
\end{remark}
\section{The initial value problem}\label{S6}
Consider the initial value problem for equation \eqref{1.1} in $\mathbb R^3$ with initial condition \eqref{1.2}. Then all the aforementioned results except Theorem~\ref{T1.3} have analogues for this problem.
In fact, in the smooth case $u_0\in \EuScript S(\mathbb R^3)$, $f\in C^\infty([0,T];\EuScript S(\mathbb R^3))$ a solution to the initial value linear problem \eqref{2.1}, \eqref{1.2} from the space $C^\infty([0,T];\EuScript S(\mathbb R^3))$ can be constructed via Fourier transform. Exponential decay when $x\to+\infty$ similarly to Lemma~\ref{L2.2} can be established for any derivative of this solution. The consequent argument of Sections~\ref{S2}--\ref{S4} can be extended to the case of the initial value problem without any essential modifications. As a result the following theorems hold.
For a measurable non-negative on $\mathbb R$ function $\psi(x)\not\equiv \text{const}$, let
\begin{equation*}
L_2^{\psi(x)}(\mathbb R^3) =\{\varphi(x,y,z): \varphi\psi^{1/2}(x)\in L_2(\mathbb R^3)\},
\end{equation*}
\begin{equation*}
H^{k,\psi(x)}(\mathbb R^3)=\{\varphi: |D^j\varphi|\in L_2^{\psi(x)}(\mathbb R^3), \ j=0,\dots,k\}
\end{equation*}
endowed with natural norms. Introduce spaces $X^{k,\psi(x)}((0,T\times \mathbb R^3))$, $k=0 \mbox{ or }1$, for admissible non-decreasing weight functions $\psi(x)\geq 1\ \forall x\in\mathbb R$, consisting of functions $u(t,x,y,z)$ such that
$$
u\in C_w([0,T]; H^{k,\psi(x)}(\mathbb R^3)), \qquad
|D^{k+1}u|\in L_2(0,T;L_2^{\psi'(x)}(\mathbb R^3)),
$$
$$
\sup_{x_0\in\mathbb R}\int_0^T\!\! \int_{x_0}^{x_0+1}\!\!\iint_{\mathbb R^2} |D^{k+1}u|^2\,dydzdxdt<\infty
$$
(let $X^{\psi(x)}((0,T)\times \mathbb R^3)=X^{0,\psi(x)}((0,T)\times\mathbb R^3)$).
\begin{theorem}\label{T6.1}
Let $u_0\in L_2^{\psi(x)}(\mathbb R^3)$, $f\in L_1(0,T; L_2^{\psi(x)}(\mathbb R^3))$ for certain $T>0$ and an admissible weight function $\psi(x)\geq 1\ \forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function. Then there exists a weak solution to problem \eqref{1.1}, \eqref{1.2} $u \in X^{\psi(x)}((0,T)\times \mathbb R^3))$.
\end{theorem}
\begin{theorem}\label{T6.2}
Let $u_0\in H^{1,\psi(x)}(\mathbb R^3)$, $f\in L_1(0,T; H^{1,\psi(x)}(\mathbb R^3))$ for certain $T>0$ and an admissible weight function $\psi(x)\geq 1\ \forall x\in\mathbb R$ such that $\psi'(x)$ is also an admissible weight function. Then there exists a weak solution to problem \eqref{1.1}, \eqref{1.2} $u\in X^{1,\psi(x)}((0,T)\times\mathbb R^3))$ and it is unique in this space if $\psi(x)\geq \rho_{3/4}(x)$ $\forall x\in \mathbb R$.
\end{theorem}
The results on large-time decay can not be established by the same method because of absence of an analogue of Friedrichs inequality in the whole space.
Of course, one can extend the theory to the intermediate cases of domains, for example,
$\Omega= (0,L)\times \mathbb R$ (here the analogue of Theorem~\ref{T1.3} is also valid, see Remark~\ref{R5.1}).
|
2,869,038,154,868 | arxiv | \section{Window Sort}\label{sec-algorithm}
\textsc{Window Sort}\xspace\ consists of multiple iterations of the same procedure:
Starting with a permutation $\sigma$ and a \textit{window size} $w$,
we compare each element $x$ in $\sigma$ with its left $2w$ and right $2w$ adjacent elements (if they exist) and count its \emph{wins}, i.e., the number of times a comparison outputs $x$ as the larger element. Then, we obtain the \emph{computed rank} for each element based on its \textit{original position} in $\sigma$ and its wins:
if $\sigma(x)$ denotes the original position of $x$ in $\sigma$, the computed rank of $x$ equals $\max\{0,\sigma(x)-2w\}$ plus the number of its wins.
And we get a new permutation $\sigma'$ by placing the elements ordered by their computed ranks.
Finally, we set $w' = w/2$ and start a new iteration on $\sigma'$ with window size $w'$. In the very first iteration, $w=n/2$.
We formalize \textsc{Window Sort}\xspace\ in Algorithm \ref{alg:iter}.
In the following, w.l.o.g. we assume to sort elements $\{1,\dots,n\}$, i.e., we refer to both an element $x$ and its rank by $x$.
Let $\sigma$ denote the permutation of the elements at the beginning of the current iteration of \textsc{Window Sort}\xspace\ and let $\sigma'$ denote the permutation obtained after this iteration (i.e., the permutation on which the next iteration performs). Similarly, let $w$ and $w'=w/2$ denote the window size of the current and the next iteration.
Furthermore, let $\pi$ denote the sorted permutation. We define four important terms for an element $x$ in $\sigma$:
\begin{itemize}
\item \emph{Current/Original position}: The position of $x$ in $\sigma$: $\sigma(x)$
\item \emph{Computed rank}: The current position of $x$ minus $2w$ (zero if negative) plus its number of wins: $computed\_rank(x)$
\item \emph{Computed position}: The position of $x$ in $\sigma'$: $\sigma'(x)$
\end{itemize}
\begin{theorem}
\textsc{Window Sort}\xspace~takes $O(n^2)$ time. \end{theorem}
\begin{proof}
Consider the three steps in Algorithm \ref{alg:iter}.
The number of comparisons in an iteration in the outer loop is $4w$, for $w$ the current size of the window. Therefore, the first step needs $O(nw)$ time. For the second step we could apply for instance Counting Sort (see e.g. \cite{Cormen}), which takes $O(n)$ time, since all computed ranks lie between zero and $n$. Thus, the total running time is upper bounded by
$\sum_{i=0}^\infty O(\frac{4n^2}{2^i}) = O(8 n^2)$.
\end{proof}
\begin{algorithm2e}[t]
\vspace{2mm}
\textbf{Initialization:} The initial window size is $w=n/2$. Each element $x$ has two variables $wins(x)$ and $computed\_rank(x)$ which are set to zero.
\Repeat{$w<1$}{
\begin{enumerate}
\item \ForEach{\emph{{$x$ at position}} $l=1,2,3,\ldots,n$\emph{ in }$\sigma$}{
\ForEach{\emph{{$y$ whose position in} $\sigma$ is in} $[l-2w,l-1]$ or \emph{in} $[l+1,l+2w]$}{
\If{$x>y$}{
$wins(x) = wins(x) + 1$}}
$computed\_rank(x) = \max\{l-2w, 0\} + wins(x)$
} \label{alg:computed_rank}
\item {Place} the elements into $\sigma'$ ordered by non-decreasing $computed\_rank$,\\
break ties arbitrarily.
\item Set all $wins$ to zero, $\sigma=\sigma'$, and $w=w/2$.
\end{enumerate} }
\caption{\textsc{Window Sort}\xspace~ (on a permutation $\sigma$ on $n$ elements)}
\label{alg:iter}
\end{algorithm2e}
\subsection{Preliminaries}
We first introduce a condition on the errors in comparisons between an element $x$ and a fixed subset of elements which depends only on the window size $w$.
\begin{definition}
We define $\err{x}{w}$ as the set of errors among the comparisons between $x$ and every $y\in [x-4w,x+4w]$.
\end{definition}
\begin{theorem}\label{th:window-sort:error-analysis}
\textsc{Window Sort}\xspace returns a sequence of maximum dislocation at most $9w^{\star}$ whenever the initial comparisons are such that
\begin{equation}
|\err{x}{w}| \leq w/4 \label{eq:error-condition}
\end{equation}
hold for all elements $x$ and for all $w=n/2,n/4,\ldots,2w^\star$.
\end{theorem}
The proof of this theorem follows in the end of this section.
In the analysis, we shall prove the following:
\begin{itemize}
\item If the computed rank of each element is close to its (true) rank, then the dislocation of each elements is small (Lemma~\ref{lem-diff}).
\item The computed rank of each element is indeed close to its (true) rank if the number of errors involving the element under consideration is small (Lemma~\ref{lem-analyzable}).
\item The number of positions an element can move in further iterations is small (Lemma~\ref{lem-offset}).
\end{itemize}
We now introduce a condition that implies \Cref{th:window-sort:error-analysis}: Throughout the execution of \textsc{Window Sort}\xspace we would like every element $x$ to satisfy the following condition:
\begin{quote}\begin{itemize}
\item[{\Large\textbf{($\ast$)}}] For window size $w$, the dislocation of $x$ is at most $w$.
\end{itemize}
\end{quote}
We also introduce two further conditions, which essentially relax the requirement that all elements satisfy \textbf{($\ast$)}.
The first condition justifies the first step of our algorithm, while the second condition restricts the range of elements that get compared with $x$ in some iteration:
\begin{quote}\begin{itemize}
\item[{\Large\textbf{($\bullet$)}}] For window size $w$, element $x$ is larger (smaller) than all the elements lying apart by more than $2w$ positions left (right) of $x$'s original position.
\end{itemize}
\end{quote}
\begin{quote}\begin{itemize}
\item[{\Large\textbf{($\circ$)}}] For window size $w$, $x$ and its left $2w$ and right $2w$ adjacent elements satisfy condition~\textbf{($\ast$)}.
\end{itemize}
\end{quote}
Note that if \textbf{($\ast$)}\ holds for all elements, then \textbf{($\bullet$)}\ and \textbf{($\circ$)}\ also hold for all elements.
For elements that satisfy both \textbf{($\bullet$)}\ and \textbf{($\circ$)} , the computed rank is close to the true rank if there are few errors in the comparisons:
\begin{lemma}\label{lem-analyzable}
For every window size $w$, if an element $x$ satisfies satisfy both \textbf{($\bullet$)}\ and \textbf{($\circ$)}, then the absolute difference between the computed rank and its true rank is bounded by
\[
|computed\_rank(x) - x| \leq |\err{x}{w}| \enspace .
\]
\end{lemma}
\begin{proof}
This follows immediately from condition \textbf{($\bullet$)}.
\end{proof}
We now consider the difference between the computed rank and the computed position of an element, which we define as the \emph{offset} of this element. Afterwards, we consider the difference between the original position and the computed position of an element.
\begin{fact}\label{fact}
Observe that by Step~\ref{alg:computed_rank} of the algorithm it holds that,
for every permutation $\sigma$, every window size $w$ and every element $x$, the difference between $\sigma(x)$ and the computed rank of $x$ is at most $2w$,
\[
|computed\_rank(x) - \sigma(x)| \leq 2w \enspace .
\]
\end{fact}
\begin{lemma}\label{lem-difference}
For any permutation of $n$ elements and for each element $x$, if the difference between the computed rank and $x$ is at most $m$ for every element,
then the difference between the computed position and $x$ is at most $2m$ for every element.
\end{lemma}
The proof of this lemma is analogue to the proof of Lemma~\ref{lem-diff} below.
\begin{lemma}\label{lem-diff}
For every permutation $\sigma$ and window size $w$, the offset of every element $x$ is at most $2w$,
\[
|computed\_rank(x) - \sigma'(x)| \leq 2w \enspace .
\]
\end{lemma}
\begin{proof}
Let the computed rank of $x$ be $k$. The computed position $\sigma'(x)$ is larger than the number of elements with computed rank smaller than $k$,
and at most the number of elements with computed ranks at most $k$.
By Fact \ref{fact},
every element $y$ with $\sigma(y)<k-2w$ has a computed rank smaller than $k$,
and every element $y$ with $\sigma(y)>k+2w$ has a computed rank larger than $k$.
\end{proof}
\begin{lemma}\label{lem-offset}
Consider a generic iteration of the algorithm with permutation $\sigma$ and a window size $w$.
In this iteration, the position of each element changes by at most $4w$. Moreover, the position of each element changes by at most $8w$ until the algorithm terminates.
\end{lemma}
\begin{proof}
By Fact \ref{fact}, Lemma~\ref{lem-diff} and triangle inequality,
$$\lvert \sigma(x) - \sigma'(x) \rvert \leq \lvert computed\_rank(x) - \sigma(x) \rvert + \lvert computed\_rank(x) - \sigma'(x) \rvert \leq 2w + 2w =4w.$$
Since $w$ is halved after every iteration, the final difference is at most $\sum_{i=0}^{\infty}\frac{4w}{2^i}=8w$.
\end{proof}
Finally, we conclude \Cref{th:window-sort:error-analysis} and show that the dislocation of an element is small if the number of errors is small:
\begin{proof}[Proof of \Cref{th:window-sort:error-analysis}]
Consider an iteration of the algorithm with current window size $w$.
We show that, if \textbf{($\ast$)}{} holds for all elements in the current iteration, then \eqref{eq:error-condition} implies that \textbf{($\ast$)}\ also holds for all elements in the next iteration, i.e., when the window size becomes $w/2$.
In order for \textbf{($\ast$)}{} to hold for the next iteration,
the computed \emph{position} of each element should differ from the true rank by at most ${w}/{2}$,
\[
|\sigma'(x) - x| \leq w/2 \enspace .
\]
By Lemma \ref{lem-difference},
it is sufficient to require that the computed \emph{rank} of each element differs from its true rank by at most ${w}/{4}$,
\[
|computed\_rank(x) - x| \leq w/4
\]
By Lemma~\ref{lem-analyzable}, the above inequality follows from the hypothesis $|\err{x}{w}| \leq w/4$.
We have thus shown that after the iteration with window size $2w^*$, all elements have dislocation at most $w^*$. By Lemma \ref{lem-offset}, the subsequent iterations will move each element by at most $8w^*$ positions.
\end{proof}
\begin{remark}
If we care only about the maximum dislocation, then we could obtain a better bound of $w$ by simply stopping the algorithm at the iteration where the window size is $w$ (for a $w$ which guarantees the condition above with high probability). In order to bound also the total dislocation, we let the algorithm continue all the way until window size $w=1$. This will allow us to show that the total dislocation is linear in expectation.
\end{remark}
\section{Omitted Proofs}\label{app:proofs}
\subsection*{Proof of Lemma~\ref{lemma:swap_probability}}
We prove that, for any running time $t>0$, no algorithm $A$ can compute, within $t$ steps, a sequence in which $x$ and $y$ are in the correct relative order with a probability larger than $1-\frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1}$.
First of all, let us focus a deterministic version of algorithm $A$ by fixing a sequence $\lambda \in \{0, 1\}^t$ of random bits that can be used be the algorithm. We call the resulting algorithm $A_\lambda$ and we let $p_\lambda$ denote the probability of generating the sequence $\lambda$ of random bits (i.e., $2^{-t}$ if the random bits come from a fair coin).\footnote{Notice that $A_\lambda$ can use less than $t$ random bits, but not more (due to the limit on its running time).}
Notice that $A$ might already be a deterministic algorithm, in this case $A = A_{\lambda}$ for every $\lambda \in \{0, 1\}^t$.
Consider an instance $I = \langle \pi, C \rangle$, and let $\phi_I$ be the permutation of the elements in $S$ returned by $A_\lambda(I)$, i.e., $\phi_I(i)=j$ if the element $i \in S$ is the $j$-th element of the sequence returned by $A_\lambda$.
We define a new instance $I_{\pi,C} = \langle \pi', C' \rangle$ by ``swapping'' elements $x$ and $y$ of $I$ along with the results of all their comparisons. More formally we define: $\pi'(i)=\pi(i)$ for all $i \in S \setminus \{x, y\}$, $\pi'(x)=\pi(y)$, and $\pi'(y)=\pi(x)$; We define the comparison matrix $C' = (c'_{i,j})_{i,j}$ accordingly, i.e.:%
\indent\begin{multicols}{2}
\begin{itemize}
\item $c'_{i,j} = c_{i,j}$ if $i,j \in S \setminus \{x, y\}$ and $j>i$,
\item $c'_{i,x} = c_{i,y}$ if $i < x$,
\item $c'_{x,j} = c_{y,j}$ if $j>x$ and $j \neq y$,
\item $c'_{i,y} = c_{i,x}$ if $i < y$ and $i \neq x$,
\item $c'_{y,j} = c_{x,j}$ if $j > y$,
\item $c'_{x,y} = c_{y, x}$.
\end{itemize}
\end{multicols}%
\noindent Letting $R = ( r_{i,j} )_{i,j}$ be a random variable over the space of all comparison matrices, we get:
\begin{multline*}
\Pr(R = C') = \prod_{i<j} \Pr(r_{i,j} = c'_{i,j}) \\
= \prod_{\substack{i<j \\ i,j \not\in \{x, y\} }} \Pr(r_{i,j} = c_{i,j}) \cdot
\prod_{i<x} \Pr(r_{i,x} = c_{i,y}) \cdot
\prod_{x<j<y} \Pr(r_{x,j} = c_{y,j}) \cdot
\prod_{j>y} \Pr(r_{x,j} = c_{y,j}) \cdot \\
\qquad \prod_{i<x} \Pr(r_{i,y} = c_{i,x}) \cdot
\prod_{x<i<y} \Pr(r_{i,y} = c_{i,x}) \cdot
\prod_{j>y} \Pr(r_{y,j} = c_{x,j}) \cdot
\Pr(r_{x,y} = c_{y,x} ) \\
= \prod_{\substack{i<j \\ i,j \not\in \{x, y\} }} \Pr(r_{i,j} = c_{i,j}) \cdot
\prod_{i<x} \Pr(r_{i,y} = c_{i,y}) \cdot
\prod_{x<j<y} (1-\Pr(r_{x,j} = c_{j,y})) \cdot
\prod_{j>y} \Pr(r_{y,j} = c_{y,j}) \cdot \\
\qquad \prod_{i<x} \Pr(r_{i,x} = c_{i,x}) \cdot
\prod_{x<i<y} (1-\Pr(r_{i,y} = c_{x,i})) \cdot
\prod_{j>y} \Pr(r_{x,j} = c_{x,j}) \cdot
(1- \Pr(r_{x,y} = c_{x,y} )) \\
= \prod_{i<j} \Pr(r_{i,j} = c_{i,j}) \bigg/ \left( \prod_{x<j<y} \Pr(r_{x,j} = c_{x,j}) \cdot \prod_{x<i<y} \Pr(r_{i,y} = c_{i,y}) \cdot \Pr(r_{x,y} = c_{x,y} ) \right) \cdot \\
\quad \prod_{x<j<y} (1-\Pr(r_{x,j} = c_{j,y})) \cdot
\prod_{x<i<y} (1-\Pr(r_{i,y} = c_{x,i})) \cdot
(1- \Pr(r_{x,y} = c_{x,y} )) \\
\ge P(R=C) \cdot \left(\frac{1}{1-p}\right)^{2(y-x)-1} \cdot p^{2(y-x)-1}
= \Pr(R=C) \cdot \left(\frac{p}{1-p}\right)^{2(y-x)-1}
\end{multline*}
\noindent where we used the fact that $r_{i,j}$ and $r_{i',j'}$ with $i < j$ and $i' < j'$ have the same probability distribution.
Let $\widetilde{\pi}$ be a random variable whose value is chosen u.a.r. over all the permutations of $S$.
Let $\Pr(I)$ be the probability that instance $I=\langle \pi, C \rangle$ is to be solved and notice that $\Pr(I) = \Pr(\widetilde{\pi} = \pi) \Pr(R = C)$.
It follows from the above discussion that $\Pr(I_{\pi,c}) \ge \Pr( I ) \cdot \left( \frac{p}{1-p} \right)^{2(y-x)-1}$.
Let $X_I$ be an indicator random variable that is $1$ iff algorithm $A_\lambda$ on instance $I$ either does not terminates within $t$ steps, or it terminates returning a sequence in which $x$ appears after $y$. Let $U$ be the set of all the possible instances and
$U'$ be the set of the instances $I \in U$ such that $X_I = 0$ (i.e., $x$ and $y$ appear in the correct order in the output of $A$ on $I$).
Notice that there is a bijection between the instances in $I = \langle \pi, C \rangle$ and their corresponding instances $I_{\pi, C}$ (i.e., our transformation is injective). Moreover, since $I$ and $I_{\pi, C}$ are indistinguishable by $A_\lambda$, we have that
either (i) $A_\lambda$ does not terminate within $t$ steps on both instances, or (ii) $\sigma_I(x) < \sigma_I(y) \iff \sigma_{I_{\pi, C}}(x) > \sigma_{I_{\pi, C}}(y)$. As a consequence, if $X_I = 0$ then $X_{I_{\pi, C}} = 1$.
Now, either $\sum_{I \in U} X_I \Pr(I) \ge \frac{1}{2}$ or we must have $\sum_{I \in U'} \Pr(I) \ge \frac{1}{2}$, which implies:
\begin{align*}
\sum_{I \in U} X_I \Pr(I) &\ge \sum_{\langle \pi, C \rangle \in U} X_{I_{\pi, C}} \Pr(I_{\pi, C})
\ge \sum_{\langle \pi, C \rangle \in U'} X_{I_{\pi, C}} \Pr(I_{\pi, C})
= \sum_{\langle \pi, C \rangle \in U'} \Pr(I_{\pi, C}) \\
& \ge \sum_{ I \in U'} \Pr( I ) \cdot \left( \frac{p}{1-p} \right)^{2(y-x)-1}
\ge \frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1}.
\end{align*}
We let $Y$ (resp. $Y_\lambda$) be an indicator random variable which is $1$ iff either the execution of $A$ (resp. $A_\lambda$) on a random instance does not terminate within $t$ steps, or it terminates returning a sequence in which $x$ and $y$ appear in the wrong relative order.
By the above calculations we know that $\Pr(Y_\lambda = 1) = \sum_{I \in U} X_I \Pr(I) \ge \frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1} \; \forall \lambda \in \{0, 1\}^t$. We are now ready to bound the probability that $Y=1$, indeed:
\[
\Pr(Y = 1) = \sum_{ \lambda \in \{0, 1\}^t} \Pr(Y_\lambda=1) p_\lambda \ge
\frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1} \sum_{ \lambda \in \{0, 1\}^t } p_\lambda
= \frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1},
\]
\noindent where the last equality follows from the fact that $p_\lambda$ is a probability distribution over the elements $\lambda \in \{0, 1\}^t$, and hence $\sum_{ \lambda \in \{0, 1\}^t } p_\lambda = 1$.
\qed
\subsection*{Proof of Theorem~\ref{thm:lb_tot_dislocation}}
Let $A$ be any algorithm and let $\langle \pi,C \rangle$ be a random instance on an even number of elements $n$.
We define $X_k$ for $0 \le k < n/2$ to be an indicator random variable that is $1$ iff $A(\langle \pi,C \rangle)$
returns a permutation of the elements in $S$ in which elements $2k+1$ and $2k+2$ appear in the wrong order.
By Lemma~\ref{lemma:swap_probability} we know that $\Pr(X_k=1) \ge \frac{1}{2}\frac{p}{1-p}$. We can hence obtain a lower bound on the expected total dislocation $\Delta$ achieved by $A$ as follows:
\[
\mathrm{E}[\Delta] \ge \sum_{k=0}^{n/2 - 1} \mathrm{E}[X_k] \ge \sum_{k=0}^{n/2 - 1} \frac{1}{2} \frac{p}{1-p} \ge \frac{n}{4} \frac{p}{1-p} = \Omega(n).
\]
\qed
\section{Extension}\label{sec-extension}
The reason why we require the error probability $p$ to be smaller than $1/32$
is to analyze the probability that at most $w/4$ errors occur in $8w$ comparisons, for $w\geq 1$. This bound on the number of errors appears since we halve the window size in every iteration.
If we let the window size shrink by another rate $1/2<\alpha<1$,
the limit of $p$ will also change:
First, the running time of the adapted \textsc{Window Sort}\xspace\ will become $O(\frac{1}{1-\alpha}n^2)$.
Second, for any permutation $\sigma$ and window size $w$, in order to maintain condition~\textbf{($\ast$)}\ for an element $x$,
its computed position should differ from $x$ by at most $\alpha w$,
and thus $computed\_rank(x)$ should differ from $x$ by at most $\alpha w/2$.
Our new issue is thus the probability that at most $\alpha w/2$ errors occur in $8w$ comparisons:
Since the expected number of errors is $8wp$, we have
\[\frac{\alpha w}{2} = \frac{\alpha}{16p}\cdot 8wp=(1+\frac{\alpha-16p}{16p}) \cdot 8wp\, ,\]
and by the reasoning of Lemma~\ref{lem-pr-fail},
we have $\frac{\alpha-16p}{16p}> 0$, thus $p<{\alpha}/{16}$.
(Note that $f(p)$ should change accordingly.)
Finally, the number of windows for the weak invariant should also change accordingly. Let $m$ be the number of windows that matter for the weak invariant ($m=12$ when $\alpha=1/2$).
According to the analysis in Section~\ref{sub-weak}, we have $m-6\geq \alpha m$, implying that $m\geq \frac{6}{1-\alpha}$.
Of course, the constant inside the linear expected total dislocation will also change accordingly.
\begin{theorem}
For an error probability $p<{\alpha}/{16}$, where $1/2< \alpha <1$,
modified \textsc{Window Sort}\xspace\ on $n$ elements takes $O(\frac{1}{1-\alpha}n^2)$ time, has maximum dislocation $9\, g(p,\alpha) \log n$ with probability $1-1/n$,
and expected total dislocation $n \cdot (9+\frac{2}{1-\alpha})\cdot 6 \, g(p,\alpha) \log g(p,\alpha)$,
where
\[ g(p,\alpha) =
\begin{cases}
\frac{100p}{(\alpha-16p)^2} & \quad \text{for} \quad \alpha /32 < p < \alpha/16\ ,\\
\frac{4}{(\ln (\alpha/16p))-(\alpha-16p)} & \quad \text{for} \quad \alpha /96 < p \leq \alpha /32\ ,\\
6 & \quad \text{for} \quad p \leq \alpha /96\ .
\end{cases}
\]
\end{theorem}
\section{Experimental Results}\label{sec-exp}
In this section, we discuss some experimental results on the performance of \textsc{Window Sort}\xspace for increasing error probability $p$ (see Tables~\ref{tb-average} and \ref{tb-maximum}).
Our results suggest that in practice, the probability of error can be much higher than in our theoretical analysis (i.e. $p<1/32$). In these experiments we measure the average dislocation (which gives us an estimate of the expected total dislocation) and the maximum dislocation among all elements.
Note that in all the experiments,
\textsc{Window Sort}\xspace performs significantly better than the theoretical guarantees (see Theorem~\ref{thm-maximum-dislocation} and \ref{thm:TD} for the corresponding values of $p$).
Also, the experiments suggest that the expected total dislocation is $O(n)$ for $p<1/5$, since the average dislocation seems to not increase with $n$ in these cases.
Analogously, the maximum dislocation seems to be $O(\log n)$ for $p<1/4$.
We consider five different values for the input-size $n$ and ten different values for the error probability $p$.
Each setting consists of one-hundred instances. We use the error probability to generate a comparison table among the $n$ elements for simulating recurrent comparison errors.
Moreover, we set $\alpha$ to be $1/2$ as in the standard version of \textsc{Window Sort}\xspace.
\begin{table}[h]
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\hline
\backslashbox{$n$}{$p$} & 1/3 & 1/4 & 1/5 & 1/8 & 1/12 & 1/16 & 1/20 & 1/24 & 1/28 & 1/32 \\
\hline
1024 & 14.160 & 4.873 & 2.870 & 1.377 & 0.881 & 0.670 & 0.536 & 0.454 & 0.390 & 0.346 \\
\hline
2048 & 15.993 & 4.984 & 2.884 & 1.397 & 0.895 & 0.674 & 0.541 & 0.464 & 0.394 & 0.348 \\
\hline
4096 & 17.494 & 5.075 & 2.904 & 1.390 & 0.893 & 0.673 & 0.545 & 0.460 & 0.398 & 0.351 \\
\hline
8192 & 19.030 & 5.105 & 2.898 & 1.395 & 0.894 & 0.675 & 0.545 & 0.460 & 0.397 & 0.351 \\
\hline
16384 & 20.377 & 5.123 & 2.902 & 1.390 & 0.892 & 0.673 & 0.545 & 0.460 & 0.398 & 0.349 \\
\hline
\end{tabular}
\caption{The average dislocation of one element.}\label{tb-average}
\end{table}
\begin{table}[h]
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\hline
\backslashbox{$n$}{$p$} & 1/3 & 1/4 & 1/5 & 1/8 & 1/12 & 1/16 & 1/20 & 1/24 & 1/28 & 1/32 \\
\hline
1024 & 15.600 & 5.400 & 3.900 & 2.100 & 1.200 & 0.900 & 0.900 & 0.600 & 0.600 & 0.600 \\
\hline
2048 & 17.000 & 4.909 & 2.818 & 1.545 & 1.091 & 0.818 & 0.818 & 0.636 & 0.636 & 0.636 \\
\hline
4096 & 17.917 & 5.333 & 2.833 & 1.500 & 1.083 & 0.833 & 0.917 & 0.667 & 0.583 & 0.667 \\
\hline
8192 & 18.923 & 6.462 & 3.077 & 1.769 & 1.154 & 0.769 & 0.692 & 0.769 & 0.538 & 0.538 \\
\hline
16384 & 22.714 & 5.071 & 3.929 & 1.786 & 1.071 & 0.857 & 0.643 & 0.643 & 0.571 & 0.571 \\
\hline
\end{tabular}
\caption{The maximum dislocation divided by $\log n$.}\label{tb-maximum}
\end{table}
\section{Introdution}
We study the problem of \emph{sorting} $n$ distinct elements under \emph{recurrent} random comparison \emph{errors}. In this classical model,
each comparison is wrong with some fixed probability $p$, and correct with probability $1-p$. The probability of errors are independent over all possible pairs of elements, but errors are recurrent: If the same comparison is repeated at any time during the computation, the result is always the same, i.e., always wrong or always correct.
In such a scenario not all sorting algorithms perform equally well in terms of the output, as some of them are more likely to produce ``nearly sorted'' sequences than others.
To measure the quality of an output permutation in terms of sortedness, a common way is to consider the \emph{dislocation} of an element, which is the difference between its position in the permutation and its true rank among all elements. Two criteria based on the dislocations of the elements are the \emph{total dislocation} of a permutation, i.e., the sum of the dislocations of all $n$ elements,
and the \emph{maximum dislocation} of any element in the permutation.
Naturally, the running \emph{time} remains an important criteria for evaluating sorting algorithms.
To the best of our knowledge, for recurrent random comparison errors, the best results with respect to \emph{running time}, \emph{maximum}, and \emph{total dislocation} are achieved by the following two different algorithms:
\begin{itemize}
\item Braverman and Mossel~\cite{Braverman2008} give an algorithm which guarantees maximum dislocation $O(\log n)$ and total dislocation $O(n)$, both with high probability. The main drawback of this algorithm seems to be its running time, as the constant exponent can be rather high;
\item Klein et al.~\cite{Klein2011} give a much faster $O(n^2)$-time algorithm which guarantees maximum dislocation $O(\log n)$, with high probability. They however do not provide any upper bound on the total dislocation, which by the previous result is obviously $O(n\log n)$.
\end{itemize}
\medskip
In this paper we investigate whether it is possible to guarantee \emph{all} of these bounds together, that is, if there is an algorithm with running time $O(n^2)$, maximum dislocation $O(\log n)$, and total dislocation $O(n)$.
\begin{table}[t]
\centering
\begin{tabular}{|l|c|c||c|}
\hline
& Braverman and Mossel \cite{Braverman2008} & Klein et al~\cite{Klein2011} & Ours\\
\hline
\# Steps & $O(n^{3+24c})$ & $O(n^2)$ & $O(n^2)$ \\
\hline
Maximal Dislocation & w.h.p. $O(\log n)$ & w.h.p. $O(\log n)$ & w.h.p. $O(\log n)$ \\
\hline
Total Dislocation & w.h.p. $O(n)$ & w.h.p. $O(n \log n)$ & in exp. $O(n)$ \\
\hline
\end{tabular}
\vspace{10pt}
\caption{Our and previous results.
The constant $c$ in \cite{Braverman2008} depends on both the error probability $p$, and the success probability of the algorithm. For example, for a success probability of $1-1/n$, the analysis in \cite{Braverman2008} yields $c=\frac{110525}{(1/2-p)^4}$. The total dislocation in \cite{Klein2011} follows trivially by the maximum dislocation, and no further analysis is given.}
\label{tb-recurrent}
\end{table}
\subsection{Our contribution}
We propose a new algorithm whose performance guarantees are essentially the best of the two previous algorithms (see Table~\ref{tb-recurrent}). Indeed, our algorithm \textsc{Window Sort}\xspace
takes $O(n^2)$ time and guarantees the maximum dislocation to be $O(\log n)$ with probability $1-1/n$
and the expected total dislocation to be $O(n)$.
The main idea is to iteratively sort $n$ elements by comparing each element with its neighbouring elements lying within a \emph{window}
and to halve the window size after every iteration.
In each iteration, each element is assigned a rank based on the local comparisons,
and then they are placed according to the computed ranks.
Our algorithm is inspired by Klein et al.'s algorithm \cite{Klein2011} which
distributes elements into \emph{buckets} according to their computed ranks, compares each element with elements in neighboring buckets to obtain a new rank, and halves the range of a bucket iteratively.
Note however that the two algorithms operate in a different way, essentially because of the following key difference between \emph{bucket} and \emph{window}.
The number of elements in a bucket is not fixed, since the computed rank of several elements could be the same. In a window, instead, the number of elements is \emph{fixed}. This property is essential in the analysis of the total dislocation of \textsc{Window Sort}\xspace, but introduces a potential \emph{offset} between the computed rank and the computed position of an element. Our analysis consists in showing that such an offset is sufficiently small, which we do by considering a number of ``delicate'' conditions that the algorithm should maintain throughout its execution with sufficiently high probability.
We first describe a standard version of our algorithm which achieves the afore mentioned bounds for any error probability $p <1/32$. We then improve this result to $p<1/16$ by using the idea of shrinking the window size at a different rate. Experimental results (see \Cref{sec-exp}) show that
the performance of the standard version is significantly better than the theoretical guarantees.
In particular, the experiments suggest that the expected total dislocation is $O(n)$ for $p<1/5$,
while the maximum dislocation is $O(\log n)$ for $p<1/4$.
In addition, we prove that no sorting algorithm can guarantee the maximum dislocation to be $o(\log n)$ with high probability,
and no sorting algorithm can guarantee the expected total dislocation to be $o(n)$.
\subsection{Further Related Work on Sorting with Comparison Errors}
Computing with errors is often considered in the framework of a two-person game called \emph{R\'{e}nyi-Ulam Game}:
The \emph{responder} thinks of an object in the search space, and the \emph{questioner} has to find it by asking questions to which the responder provides answers.
However, some of the answers are incorrect on purpose; the responder is an \emph{adversarial lier}.
These games have been extensively studied in the past on various kinds of search spaces, questions, and errors; see Pelc's survey \cite{Pelc02} and Cicalese's monograph \cite{Cicalese13}.
Feige et al.~\cite{Feige1994} studied several comparison based algorithms with independent random errors
where the error probability of a comparison is less than half, the repetitions of an comparison can obtain different outcomes, and all the comparisons are independent.
They required the reported solution to be correct with a probability $1-q$, where $0<q<1/2$,
and proved that for sorting, $O(n\log(n/q))$ comparisons suffice, which gives also the running time.
In the same model, sorting by random swaps represented as Markovian processes have been studied under the question of the number of inversions (reversed pairs) \cite{gecco17,barbarapaolo}, which is within a constant factor of the total dislocation \cite{diaconis}.
Karp and Kleinberg~\cite{KarpK07} studied a noisy version of the classic binary search problem, where elements cannot be compared directly.
Instead, each element is associated with a coin that has an unknown probability of observing heads when tossing it and these probabilities increase when going through the sorted order.
For recurring errors,
Coppersmith and Rurda \cite{Coppersmith} studied a simple algorithm that gives a 5-approximation on the weighted feedback arc set in tournaments (FAST) problem if the weights satisfy probability constraints. The algorithm consists of ordering the elements based on computed ranks, which for unweighted FAST are identical to our computed ranks.
Alonso et al.\ \cite{alonso} and Hadjicostas and Lakshamanan \cite{hadji} studied Quicksort and recursive Mergesort, respectively, with random comparison errors.
\paragraph*{Paper organization}
The rest of this paper is organized as follows. We present the \textsc{Window Sort}\xspace\ algorithm in Section~\ref{sec-algorithm} and analyze the maximum and total dislocation in
Section~\ref{sec-max} and Section~\ref{sec-total}, respectively.
Then, we explain how to modify \textsc{Window Sort}\xspace\ to allow larger error probabilities in \Cref{sec-extension} and show some experimental results in \Cref{sec-exp}. Additionally, we prove lower bounds on both the maximum and average dislocation for any sorting algorithm in \Cref{sec:lowerbound}.
\section{A lower bound on the maximum dislocation}\label{sec:lowerbound}
In this section we prove a lower bound on both the maximum and the average dislocation that can be achieved w.h.p. by any sorting algorithm.
Let $S = \{1, 2, \dots, n\}$ be the set of elements to be sorted.
We can think of an instance of our sorting problem as a pair $\langle \pi, C \rangle$ where $\pi$ is a permutation of $S$ that represents the order of the element in the input collection and $C = (c_{i,j} )_{i,j}$ is a $n \times n$ matrix that encodes the result of the comparisons as seen by the algorithm. More precisely, $c_{i,j}$ is \text{``$<$''} if $i$ is reported to be smaller than $j$ when comparing $i$ and $j$ and $c_{i,j}=\text{``$<$''}$ otherwise.
Notice that $c_{i,j}=\text{``$<$''}$ iff $c_{j,i}=\text{``$>$''}$ and hence in what follows we will only define $c_{i,j}$ for $i < j$.
The following lemma -- whose proof is moved to \Cref{app:proofs} -- is a key ingredient in our lower bounds:
\begin{lemma}
\label{lemma:swap_probability}
Let $x,y \in S$ with $x < y$. Let $A$ be any (possibly randomized) algorithm. On a random instance, the probability that $A$ returns a permutation in which
elements $x$ and $y$ appear the wrong relative order is at least $\frac{1}{2} \left( \frac{p}{1-p} \right)^{2(y-x)-1}$.
\end{lemma}
As a first consequence of the previous lemma, we obtain the following:
\begin{theorem}
\label{thm:lb_max_dislocation}
No (possibly randomized) algorithm can achieve maximum dislocation $o(\log n)$ with high probability.
\end{theorem}
\begin{proof}
By Lemma~\ref{lemma:swap_probability}, any algorithm, when invoked on a random instance, must return a permutation $\rho$ in which elements $1$ and $h= \big\lfloor \frac{\log n }{2 \log {1-p}/{p}} \big\rfloor$ appear in the wrong order with a probability larger than $\frac{1}{n}$.
When this happens, at least one of the following two conditions holds: (i) the position of element $1$ in $\rho $ is at least $\lceil \frac{h}{2} \rceil$; or (ii) the position of element $h$ in $\rho$ is at most $\lfloor \frac{h}{2} \rfloor$.
In any case, the maximum dislocation must be at least $\frac{h}{2} - 1 = \Omega(\log n)$.
\end{proof}
Finally, we are also able to prove a lower bound to the total dislocation (whose proof is given in \Cref{app:proofs}).
\begin{theorem}
\label{thm:lb_tot_dislocation}
No (possibly randomized) algorithm can achieve expected total dislocation $o(n)$.
\end{theorem}
\section{Maximum Dislocation}\label{sec-max}
In this section we give a bound on the maximum dislocation of an element after running \textsc{Window Sort}\xspace\ on $n$ elements. We prove that it is a function of $n$ {and} of the probability $p$ that a single comparison fails. Our main result is the following:
\begin{theorem}\label{thm-maximum-dislocation}
For a set of $n$ elements, with probability $1-{1}/{n}$, the maximum dislocation after running \textsc{Window Sort}\xspace\
is $9\cdot f(p)\cdot \log n$ where
\[ f(p) =
\begin{cases}~
\frac{400p}{(1-32p)^2} & \quad \text{for} \quad 1 /64 < p < 1/32,\\~
\frac{4}{\ln \left(\frac{1}{32p}\right)-\left(1-32p\right)} & \quad \text{for} \quad 1/192 < p \leq 1/64,\\~
6 & \quad \text{for} \quad p \leq 1/192.
\end{cases}
\]
\end{theorem}
It is enough to prove that the condition in Theorem~\ref{th:window-sort:error-analysis} holds for all $w\geq 2f(p)\log n$ with probability at least $1 - 1/n$.
\begin{lemma}\label{le:error-condition}
For every fixed element $x$ and for every fixed window size $w\geq 2f(p)\log n$, the probability that
\begin{equation}
|\err{x}{w}| > w/4 \label{eq:error-condition-negated}
\end{equation}
is at most $1/n^3$.
\end{lemma}
By the union bound, the probability that \eqref{eq:error-condition-negated} holds for some $x$ and for some $w$ is at most $1/n$. That is, the condition of Theorem~\ref{th:window-sort:error-analysis} holds with probability at least $1 - 1/n$ for all $w\geq 2w^\star=2f(p)\log n$, which then implies Theorem~\ref{thm-maximum-dislocation}.
\subsection{Proof of Lemma~\ref{le:error-condition}}
Since each comparison fails with probability $p$ independently of the other comparisons, the probability that the event in \eqref{eq:error-condition-negated} happens is equal to the probability that at least ${w}/{4}$ errors occur in $8 w$ comparisons. We denote such probability as $\Pr(w)$, and show that
$
\Pr(w) \leq 1/n^3 \enspace.
$
We will make use of the following standard Chernoff Bounds (see for instance in \cite{MitzenmacherU05}):
\begin{theorem}[Chernoff Bounds]\label{thm-chernoff}
Let $X_1,\cdots,X_n$ be independent Poisson trials with $\Pr(X_i)=p_i$. Let $X=\sum_{i=1}^nX_i$ and $\mu=\mathrm{E}[X]$. Then the following bounds hold:
\begin{align}
(i)& \text{ For $0< \delta < 1$,}&&
\Pr(X\geq (1+\delta )\mu )\leq e^{-\frac{\mu\delta^2}{3}},\\
(ii)& \text{ For any $\delta >0$,}&&
\Pr(X\geq (1+\delta )\mu )< \bigg(\frac{e^\delta }{(1+\delta)^{(1+\delta )}}\bigg)^\mu,\\
(iii)& \text{ For $R\geq 6\mu$,}&& \Pr(X\geq R )\leq 2^{-R}.
\end{align}
\end{theorem}
\begin{lemma}\label{lem-pr-fail}
The probability $\Pr(w)$ (at least ${w}/{4}$ errors occur in $8 w$ comparisons) satisfies
\[ \Pr(w) \leq
\begin{cases}~
e^{-\frac{w\left(1-32p\right)^2}{384p}} & \quad \text{for} \quad1 /64 < p < 1/32,\\[6pt]
\bigg( \frac{e^{\frac{1-32p}{32p}}}{\left(\frac{1}{32p}\right)^{\frac{1}{32p}}}\bigg)^{8 w p}& \quad \text{for} \quad 1/192 < p \leq 1/64, \\[12pt]
2^{-\frac{w}{4}} & \quad \text{for} \quad p \leq 1/192.
\end{cases}
\]
\end{lemma}
\begin{proof}
Let the random variable $X$ denote the number of errors in the outcome of $8w$ comparisons. Clearly, $\mathrm{E}[X]=8w p$, and
\begin{align*}
\Pr (w)&=\Pr \left[ X\geq \frac{w}{4}\right]
=\Pr\left[X\geq \frac{\mathrm{E}[X]}{32p}\right]
= \Pr\left[X\geq \left(1+\frac{1-32p}{32p}\right) \mathrm{E}[X]\right].
\end{align*}
Let $\delta=\frac{1-32p}{32p}$.
If $1/64 < p <1/32$, then $0< \delta< 1$, and by Theorem~\ref{thm-chernoff}, case~($i$), we have
\[\Pr(w)\leq e^{-\frac{1}{3}\delta^2\mu}\leq e^{-\frac{w\cdot\left(1-32p\right)^2}{384p}}.\]
Similarly, if $p \leq 1/64$, then $\delta \geq1$, and by Theorem~\ref{thm-chernoff}, case ($ii$), we have
\[\Pr(w)< \left(\frac{e^\delta}{(1+\delta)^{(1+\delta)}}\right)^\mu \leq \bigg( \frac{{\scriptstyle e^{\frac{1-32p}{32p}}}}{{\scriptstyle \left(\frac{1}{32p}\right)}^{\frac{1}{32p}}}\bigg)^{8 w p}\ .\]
If $p\leq 1/192$, then ${w}/{4}\geq 48 w p =6\, \mathrm{E}[X]$, and by Theorem~\ref{thm-chernoff} case (iii),
$\Pr(w)\leq 2^{-\frac{w}{4}}.$
\end{proof}
\begin{lemma}\label{lemma-pr-cubic}
If $w\geq 2\, f(p)\log n$, with $n\geq 1$ and $f(p)$ as in Theorem~\ref{thm-maximum-dislocation}, then $\Pr(w)\leq 1/n^{3}$.
\end{lemma}
\begin{proof}
We show the first case, the other two are similar.
If $1/64 < p <1/32$, by Lemma~\ref{lem-pr-fail},
\[\Pr(w)\leq e^{-\frac{w\cdot\left(1-32p\right)^2}{384p}} \leq e^{-\frac{800}{384}\log n}\leq e^{-3\lg n}\leq 1/n^{3}.\qedhere\]
\end{proof}
\section{Total Dislocation}\label{sec-total}
In this section, we prove that \textsc{Window Sort}\xspace\ orders $n$ elements such that their total dislocation is linear in $n$ times a factor which depends only on $p$:
\begin{theorem}\label{thm:TD}
For a set of $n$ elements, the expected total dislocation after running \textsc{Window Sort}\xspace\ is at most $n \cdot 60 \, f(p) \log f(p)$.
\end{theorem}
The key idea is to show that for an element $x$, only $O(w)$ elements adjacent to its (true) rank matter in all upcoming iterations. If this holds, it is sufficient to keep the following \emph{weak invariant} for an element $x$ throughout all iterations:
\begin{quote}
\begin{itemize}
\item[{\Large\textbf{($\clubsuit$)}}] This invariant consists of three conditions that have to be satisfied:
\begin{itemize}
\item[~]
\begin{enumerate}[(a)]
\item $x$ satisfies condition \textbf{($\ast$)}.
\item All elements with original position in $[x-12w, x+12w]$ satisfy condition \textbf{($\ast$)}.
\item All elements with original position in $[x-10w, x+10w]$ satisfy condition \textbf{($\bullet$)}.
\end{enumerate}
\end{itemize}
\end{itemize}
\end{quote}
Note that if $x$ satisfies \textbf{($\clubsuit$)}, all elements lying in $[x-10w, x+10w]$ satisfy both \textbf{($\bullet$)}\ and \textbf{($\circ$)}.
The rest of this section is structured as follows: First we derive several properties of the weak invariant, then we prove an $n\log\log n$ bound on the expected total dislocation, and finally we extend the proof to achieve the claimed linear bound.
\subsection{Properties of the Weak Invariant}\label{sub-weak}
We start with the key property of the weak invariant \textbf{($\clubsuit$)}\ for some element $x$.
\begin{lemma}\label{lem-weak}
Let $\sigma$ be the permutation of $n$ elements and $w$ be the window size of some iteration in \textsc{Window Sort}\xspace. If the weak invariant \emph{\textbf{($\clubsuit$)}}\ holds for an element $x$ in $\sigma$ and the computed rank of every element $y$ with $\sigma(y)\in[x-10w, x+10w]$ differs from $y$ by at most ${w}/{4}$,
then \emph{\textbf{($\clubsuit$)}}\ still holds for $x$ in the permutation $\sigma'$ of next iteration with window size ${w}/{2}$.
\end{lemma}
\begin{proof}
Consider the set $X$ of all elements $y$ with $computed\_rank(y)\in[x-8w, x+8w]$.
Their computed ranks differ from their original positions by at most $2w$.
Thus, all these elements are in the set $Y\supseteq X$ of all elements whose original positions are in $[x-10w, x+10w]$.
By the assumption of the lemma, for each element $y\in Y$, $\lvert computed\_rank(y) - y \rvert \leq {w}/{4}$.
Using the same reasoning as in the proof of Lemma \ref{lem-diff}, we conclude that
\begin{aquote}{A}
for each element $y\in X$, $\lvert \sigma'(y)-y \rvert \leq{w}/{2}$.
\end{aquote}
Consider the set $Z$ of all elements $y$ with $\sigma'(y)\in[x-6w,x+6w]$.
By Lemma \ref{lem-diff}, their computed ranks lie in $[x-8w, x+8w]$, thus $Z\subseteq X$, and by (A), $\lvert \sigma'(y) - y \rvert\leq {w}/{2}$ for each $y\in Z$.
Thus, the second condition of \textbf{($\clubsuit$)}\ holds for the next iteration.
We continue with the third condition. Consider the set $T\subseteq Z$ of all elements $y$ with $\sigma'(y)\in [x-5w,x+5w]$. By the assumptions of the lemma, $y\in[x-5w-{w}/{2},x+5w+{w}/{2}]$ and $\sigma(y)\in[x-5w-{3w}/{2},x+5w+{3w}/{2}]$ for all $y\in T$.
It is sufficient to show that every element in $ T$ is larger (or smaller) than all elements whose computed positions are smaller than $x-6w$ (or larger than $x+6w$), the rest follows from the second condition. We show the former case, the latter is symmetric.
We distinguish three subcases: elements $y\in T$ with $\sigma'(y)<x-6w$ and with $\sigma(y)$ (i) smaller than $x-12$, (ii) between $x-12$ and $x-10w-1$, or (iii) between $x-10w$ and $x-4w-1$.
\begin{enumerate}[(i)]
\item This case follows immediately from the third condition of \textbf{($\clubsuit$)}.
\item This case follows immediately from the second condition of \textbf{($\clubsuit$)}.
\item By the assumption of our lemma, $\lvert computed\_rank(y) - y \rvert \leq {w}/{4}$.
Thus, if the computed rank $r$ of such an element $y$ is smaller than $x-6w$, then $y<x-6w+{w}/{4}$.
Otherwise, if $r\geq kx-6w$, then by (A), $\lvert y-\sigma'(y) \rvert \leq {w}/{2}$. Thus, $y<x-6w+{w}/{2}$.
\end{enumerate}
Since we assume \textbf{($\clubsuit$)}\ for $x$, $\sigma(x) \in [x-w, x+w]$ and $computed\_rank(x)\in[x-3w,x+3w]$. By Lemma \ref{lem-diff}, $\sigma'(x)\in [k-5w, k+5w]$, and thus $x\in Z$, which implies that the first condition of \textbf{($\clubsuit$)}\ will still be satisfied for $x$ for the next iteration. This concludes the proof.
\end{proof}
Next, we adopt Lemma~\ref{lemma-pr-cubic} to analyze the probability of keeping the weak invariant for an element $x$ and an arbitrary window size through several iteration of \textsc{Window Sort}\xspace.
\begin{lemma}\label{lem-weak-pr}
Consider an iteration of \textsc{Window Sort}\xspace\ on a permutation $\sigma$ on $n$ elements such that the window size is $w\geq 2\, f(p) \log w$, where $f(p)$ is defined as in Theorem \ref{thm-maximum-dislocation}.
If the weak invariant \emph{\textbf{($\clubsuit$)}}\ for an element $x$ holds,
then with probability at least $1-{42}/{w^2}$, \emph{\textbf{($\clubsuit$)}}\ still holds for $x$ when the window size is $f(p) \log w$ (after some iterations of \textsc{Window Sort}\xspace).
\end{lemma}
\begin{proof}
By Lemma~\ref{lem-weak}, the probability that \textbf{($\clubsuit$)}\ fails for $x$ before the next iteration is $(20w+1)\cdot \Pr(w)$.
Let $r=\log(\frac{w}{2\, f(p)\\log w})$, then the probability that \textbf{($\clubsuit$)}\ fails for $x$ during the iterations from window size $w$ to window size $f(p)\cdot \log w$ is
\begin{align*}
\sum_{i=0}^{r}\left(\frac{20w}{2^i}+1\right)\cdot\Pr\left(\frac{w}{2^i}\right) &\leq
\sum_{i=0}^{r}\left(\frac{21w}{2^i}\right)\cdot\Pr\left(2\, f(p) \log w\right) \leq 42w \cdot\Pr\left(2\, f(p) \log w\right),
\end{align*}
where the first inequality is by fact that $\Pr(w)$ increases when $w$ decreases.
By Lemma~\ref{lemma-pr-cubic}, $\Pr(2\, f(p) \log w)\leq {1}/{w^3}$, leading to the statement.
\end{proof}
\subsection{Double Logarithmic Factor (Main Idea)}\label{sub-d-log}
Given that \textsc{Window Sort}\xspace\ guarantees maximum dislocation at most $9\, f(p) \log n$ with probability at least $(1-{1}/{n})$ (Theorem~\ref{thm-maximum-dislocation}), this trivially implies that the expected total dislocation is at most $O(f(p)\log n)$. More precisely, the expected dislocation is at most
\begin{align*}
\left({1}/{n}\right)\cdot n \cdot n + \left(1-{1}/{n}\right)\cdot n \cdot 9 \, f(p) \log n\leq n\cdot (1+9\,f(p) \log n)\, ,
\end{align*}
since a fraction $1/n$ of the elements is dislocated by at most $n$, while the others are dislocated by at most $9\cdot f (p) \log n$.
We next describe how to improve this to $O(f(p)\log\log n)$ by considering in the analysis \emph{two phases} during the execution of the algorithm:
\begin{itemize}
\item \textbf{Phase 1:} The first phase consists of the iterations up to window size $w=f(p)\log n$. With probability at least $(1-1/n)$ all elements satisfy \textbf{($\ast$)}\ during this phase.
\item \textbf{Phase 2:} The second phase consists of the executions up to window size $w' = f(p)\log w$. If all elements satisfied \textbf{($\ast$)}\ at the end of the previous phase, then the probability that a fixed element violates \textbf{($\clubsuit$)}\ during this second phase is at most $42/w^2$.
\end{itemize}
More precisely, by \Cref{thm-maximum-dislocation} and the proof of \Cref{th:window-sort:error-analysis}, the
probability that \textbf{($\ast$)}\ holds for all elements when the window size is $f(p) \log n$ is at least $(1-{1}/{n})$. We thus restart our analysis with $w=f(p) \log n$ and the corresponding permutation $\sigma$.
Assume an element $x$ satisfies \textbf{($\clubsuit$)}.
By Lemma~\ref{lem-weak-pr},
the probability that \textbf{($\clubsuit$)}\ fails for $x$ before the window size is $f(p) \log w$ is at most ${42}/{w^2}$.
By Lemma~\ref{lem-offset},
an element moves by at most $8w$ positions from its original position, which is at most $w$ apart from its true rank. Therefore, the expected dislocation of an element $x$ is at most
\begin{equation*}
\left({1}/{n}\right)\cdot n+ {42}/{w^2}\cdot 9w + 9 \, f(p) \log w = O(1) + 9 \, f(p) \log (f(p) \log n)\ ,
\end{equation*}
where the equality holds for sufficiently large $n$ because $w=f(p)\log n$.
\subsection{Linear Dislocation (Proof of Theorem \ref{thm:TD})}\label{sub-logarithmic-dislocation}
In this section, we apply a simple idea to decrease the upper bound on the expected total dislocation after running \textsc{Window Sort}\xspace\ on $n$ elements to $60 \, f(p) \log f(p)$.
We recurse the analysis from the previous Section \ref{sub-d-log} for several \textit{phases}:
Roughly speaking, an \textit{iteration} in \textsc{Window Sort}\xspace\ halves the window size,
a \textit{phase} of iterations logarithmizes the window size.
\begin{itemize}
\item \textbf{Phase 1}: Iterations until the window size is $f(p)\log n$.
\item \textbf{Phase 2:} Subsequent iterations until the window size is $f(p)\cdot\log( f(p)\log n)$.
\item \textbf{Phase 3:} Subsequent iterations until the window size is $ f(p)\cdot\log(f(p)\cdot \log (f(p)\log n))$.
\item \dots
\end{itemize}
We bound the expected dislocation of an element $x$,
and let $w_i$ denote the window size after the $i$-th phase.
We have $w_0=n$, $w_1=f(p) \log n$, $w_2=f(p) \log ( f(p) \log n)$, and
\begin{equation}\label{eq:phase}
w_{i+1}= f(p) \log w_i,
\end{equation} if $i\geq 1$ and $w_i \geq 2\, f(p) \log w_i$.
Any further phase would just consist of a single iteration. In the remaining of this section, we only consider phases $i$ for which \Cref{eq:phase} is true, and we call them the \emph{valid phases}.
By Lemma~\ref{lem-weak-pr},
if the weak invariant \textbf{($\clubsuit$)}\ holds for $x$ and window size $w_{i-1}$,
the probability that it still holds for window size $w_{i}$
is at least $1-{42}/{w_{i-1}^2}$.
Similarly to the analysis in the Section~\ref{sub-d-log}, we get that a valid phase $i\geq 1$ contributes to the expected dislocation of $x$ by
\begin{eqnarray}\label{eq:disl}
{42}/{w_{i-1}^2}\cdot 9w_{i-1}={378}/{w_{i-1}}\, .
\end{eqnarray}
If we stop our analysis after $c$ valid phases,
then by \eqref{eq:disl} and Lemma \ref{lem-offset}, the expected dislocation of any element $x$ is at most
\begin{equation}\label{eq:wcup}
\sum_{i=0}^{c-1}{378}/{w_i} + 9w_c \leq {378}/{w_c} + 9w_c \, .
\end{equation}
The inequality holds since $\frac{w_{i-1}}{w_{i}}\leq 2$ for $1<i<c$.
We next define $c$ such that phase $c$ is valid and $w_c$ only depends on $f(p)$.
The term $\frac{w_{i-1}}{ f(p)\log w_{i-1}} \geq 2$ holds for every valid phase $i$ and decreases with increasing $i$.
For instance for $w=6f(p)\log f(p)$:
\begin{equation*}
\frac{w}{f(p)\log w} = \frac{6\,f(p)\log f(p)}{f(p)\log (6\,f(p)\log f(p))}
\geq \frac{6\,\log f(p)}{3\,\log f(p)}\geq 2\, .
\end{equation*}
Therefore, if we choose $c$ such that $w_{c-1}\geq 6\,f(p)\log f(p)> w_c$, we can use that $f(p)\geq 6$ and upper bound $w_c$
by
\begin{equation}\label{eq:wcdown}
w_c = f(p)\log w_{c-1} \geq f(p)\log (6\,f(p)\log f(p)) \geq 6 \log (36\log 6) \geq 39\ .
\end{equation}
\Cref{eq:wcup,eq:wcdown} and Lemma~\ref{lem-offset} imply the following:
\begin{lemma}
The expected dislocation of each element $x$ after running \textsc{Window Sort}\xspace\ is at most
\begin{equation*}
{378}/{w_c}+9w_c < 10+9w_c\leq 10 w_c \leq 60 \, f(p) \log f(p) \enspace.
\end{equation*}
\end{lemma}
This immediately implies Theorem~\ref{thm:TD}.
|
2,869,038,154,869 | arxiv | \section{Introduction}
Common envelopes (CEs) are events that often occur in binary systems when one component, the primary, evolves off the main-sequence \citep{Paczynski1976,Ivanova2013,Kochanek:2014aa}. Significant radial expansion of the primary during post-main-sequence evolution can lead to direct engulfment of the companion, Roche Lobe overflow, and/or orbital decay via tidal dissipation \citep{Nordhaus2006,Nordhaus:2010aa,Chen:2017aa}. The result is a binary system consisting of the primary's core and the companion embedded in a CE formed from the primary's envelope.
Once immersed inside a CE, the orbit of the system has been shown to decay rapidly ($\lesssim$$1-1000$ years) making direct detection difficult \citep{Ivanova:2013aa} and precursor emission signatures a promising means of identification \citep{MacLeod:2018aa}. However, in wide triple systems, the orbital decay timescale may be significantly longer \citep{Michaely2018}. Common envelope phases are thought to be the primary mechanism for producing short-period binaries in the universe \citep{Toonen2013,Kruckow:2018aa,Canals:2018aa}, though not the only method \citep{Fabrycky:2007aa,Thompson:2011aa,Shappee:2013aa, Michaely2016}.
During inspiral, energy and angular momentum are transferred from the orbit to the envelope \citep{Iben1993}. If sufficient to eject the CE, a tight binary emerges that contains at least one compact object. If the envelope cannot be ejected, the companion is destroyed leaving a ``single'' star that nevertheless underwent a binary interaction such that its evolution may be modified \citep{Nordhaus:2011aa}.
A method for parameterizing the outcomes of common envelopes, and hence predictions for their progeny populations, involves defining an ejection efficiency, $\bar{\alpha}_{\mathrm{eff}}$, based on energetic arguments. This ``$\alpha$-formalism'' is broadly used in population synthesis studies and often defined in the following way:
\begin{equation}
\bar{\alpha}_{\mathrm{eff}} = E_{\mrm{bind}}/\Delta E_{\mrm{orb}},
\label{eq:traditionalalpha}
\end{equation}
where $E_{\mrm{bind}}$ is the energy required to unbind the primary's envelope from its core and $\Delta E_{\mrm{orb}}$ is the orbital energy released during inspiral (for a detailed discussion see Sec~\ref{sec:energycalc}; \citealt{Tutukov1979, Iben1984, Webbink1984, Livio1988, DeMarco2011}). Given $\bar{\alpha}_{\mathrm{eff}}$, and knowledge of the primary's binding energy, the post-CE orbital separations can then be determined. Because the transfer of energy to the envelope is not perfectly efficient, and because predictions for the progeny populations are highly sensitive to adopted values \citep{Claeys2014}, better constraints on $\bar{\alpha}_{\mathrm{eff}}$, from either observations or theory, are topics of active research. In particular, an improved understanding of the ejection efficiency as a function of binary parameters or internal CE structure is needed as $\bar{\alpha}_{\mathrm{eff}}$ is often taken to be constant.
Observations of CE progenitors have allowed some estimates of $\bar{\alpha}_{\mathrm{eff}}$, albeit with significant uncertainties. \citet{Zorotovic2010} identified over 50 systems that are likely progeny of CE evolution, and determined that most are consistent with $\bar{\alpha}_{\mathrm{eff}}\simeq 0.2-0.3$. This is in general agreement with \cite{Cojocaru:2017aa}, who performed population synthesis studies of Galactic white dwarf main-sequence binaries using data from the Sloan Digital Sky Survey (SDSS) Data Release 12. Also utilizing population synthesis techniques, \citet{Davis2010} determined that $\bar{\alpha}_{\mathrm{eff}}>0.1$ reasonably describes all systems with late-type secondaries, but produced an overabundance of post-CE systems with orbital periods greater than a day. In a similar manner, \citet{Toonen2013} argue that the ejection efficiency must be low to explain the observed post-CE orbital period distribution present in the SDSS sample.
Additional studies that have tested the dependence of the ejection efficiencies on the mass ratio of the binary, $q=m_2/M_1$, have produced conflicting results. \citet{DeMarco2011} find that the mass ratio is in anti-correlation with $\bar{\alpha}_{\mathrm{eff}}$; an increased companion mass results in a decreased $\bar{\alpha}_{\mathrm{eff}}$. \citet{Zorotovic2011}, by way of orbital separation, find that the mass ratio is in fact in correlation with $\bar{\alpha}_{\mathrm{eff}}$, attributing the increased ejection efficiency to the increased initial orbital energy of the more massive companion. We discuss these contradictory findings in relation to our results in Section~\ref{sec:massratio}.
Three-dimensional hydrodynamic simulations of common envelopes have been carried out in recent years by multiple groups using diverse codes and numerical techniques \citep{Ricker2012,Passy2012,Ohlmann2015,Chamandy:2018aa}. While inspiral occurs rapidly, the envelope is pushed outward yet remains bound. This failure to eject the CE has resulted in proposed solutions that include additional energy sources (recombination/accretion/jets), processes that operate on longer timescales, or ejection through dust-driven winds \citep{Soker:2015aa,Ivanova:2015aa,Kuruwita:2016aa,Glanz:2018aa,Sabach2017,Grichener2018a,Kashi:2018aa,Ivanova:2018aa,Soker:2018aa}. While such effects may in fact prove necessary, it is first useful to consider the physical effects incorporated in simulations. The energy budget of the CE interaction is set by the initial orbital energy. As inspiral occurs, liberated orbital energy will be transferred to the CE unless it is lost via radiation. In this context, it is interesting to note that hydrodynamic simulations do not include radiation, and therefore provide incomplete analyses of the ejection efficiency. A full examination of the energy components in CE simulations and a robust discussion of why the envelope remains bound at large distances is needed, i.e.~\citet{Chamandy:2018ab}.
There have been several previous studies that investigate the effects of convection in conjunction with recombination energy. \citet{Grichener2018a} in particular consider a common envelope in which the inspiraling companion deposits energy and the envelope expands. The authors find that convection efficiently transports recombination energy to surface radiative regions where it is lost. The energy transport time is on the order of months and shorter than dynamical timescales. This is consistent with the results we present in this work. In an earlier study, \citet{Sabach2017} argue that when helium recombines, energy transport by convection cannot be neglected. While it may increase the luminosity of the event, it cannot be used to unbind the envelope. Whether recombination energy can be tapped to drive ejection remains a subject of vigorous debate \citep{Ivanova:2018aa,Soker:2018aa}.
In this paper, we focus on the general effects of convection, internal structure, and mass ratio on ejection efficiencies. Post-main sequence giants possess deep and vigorous convective envelopes which can carry energy to the surface where it can be lost via radiation, effectively lowering the ejection efficiency. In regions where radiative losses do not occur, convection can redistribute energy, carrying it to parcels of gas that are not in the direct vicinity of the binary, thus aiding ejection.
In Section 2, we describe our methodology, stellar models and the physics in which we ground our analysis. Our data are presented in Section 3. In Section 4 we discuss our results in the context of observational and theoretical work, and present our conclusions in Section 5.
\begin{figure*}
\begin{multicols}{2}
\centering
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{1_0M_timescales_121718}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{2_0M_timescales_121718}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{3_0M_timescales_121718}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{4_0M_timescales_121718}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{5_0M_timescales_121718}
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\includegraphics[width=\textwidth]{6_01M_timescales_121718}
\end{minipage}
\end{multicols}
\hfill
\caption{Comparative timescale plots for a sample of representative primary masses at their maximum radial extent and several test companion masses. The convective timescale profile of the primary star is shown in solid blue. The coloured, dashed lines show the inspiral timescale - the time it takes for the companion mass to spiral from its current radius to the centre of the primary star. The radius at which each companion mass shreds due to the gravity of the primary mass is marked with an X. The surface-contact convective regions (SCCRs) of the primary star that do not contribute to the unbinding of the envelope are shaded in yellow. Interior convective zones that do not extend to the primary's surface are shaded in pink.}
\label{fig:Timescales}
\end{figure*}
\section{Methodology}
\label{sec:methodology}
Our stellar models were computed using \textsc{mesa} (release 10108), an open-source stellar evolution code that allows users to produce spherically symmetric models of stellar interiors during all phases of a star's evolution \citep{Paxton:2011aa, Paxton:2018aa}\footnote{\textsc{mesa} is available at http://mesa.sourceforge.net}. Each star was evolved from the pre-main sequence to the white dwarf phase for zero-age-main-sequence (ZAMS) masses in the range of $0.8 M_{\odot} - 6.0 M_{\odot}$ in increments of $0.2M_\odot$ with finely-meshed time-stepping. Stars with initial masses below $0.8 M_\odot$ were not included as they have not evolved off the main sequence during the lifetime of the universe. Mass loss on the Red Giant Branch (RGB) followed a Reimer's prescription with $\eta_{\rm R}=0.7$ while mass loss on the Asymptotic Giant Branch (AGB) followed a Blocker prescription with $\eta_{\rm B}=0.7$ \citep{Reimers:1975aa, Bloecker:1995aa}. All models were assumed to have solar metallicity (z=0.02).
For each evolutionary model, the interior profile at maximum extent was chosen as this is a likely time for a CE to occur since the primary occupies its greatest possible volume for engulfing companions. This large volume and extended radius allow for strong tidal torques which shrink the companion's orbit \citep{Villaver:2009aa, Nordhaus:2010aa, Nordhaus:2013aa}. Each interior profile contains radial information about the mass, density, convective properties, and core and envelope boundaries. From these, we calculate the primary's binding energy, location of the convective zones, inspiral timescales, tidal disruption radii, and the energy released during orbital decay. These quantities are used to determine $\bar{\alpha}_{\mathrm{eff}}$ and the post-CE orbital separations for the companions that survive the CE interaction.
\subsection{Convective Regions}
\label{sec:convregions}
Post-main sequence stars host deep convective envelopes that can transport energy to optically thin surfaces where it is radiated away. As the companion inspirals, there may be interior regions where convection can effectively carry newly liberated orbital energy to the surface. The CE may then regulate itself with little-to-no orbital energy available for ejection until the companion reaches a region where the effects of convective transport no longer dominate.
To identify the convective regions of the primary star, we extract the calculated convective velocities ($v_{\mrm{conv}}$) from our interior profiles when each star is at the maximum radial extent in its evolution. The convective timescale can then be found:
\begin{equation}
t_{\mrm{conv}}[r] = \int_{r}^{R_{\star}}\frac{1}{v_{\mrm{conv}}} \mrm{d}r
\label{eq:tconv}
\end{equation}
\citep{Grichener2018a}.
Similarly for each radius in the primary, we can determine the time required for the orbit to fully decay. This inspiral timescale is given as,
\begin{equation}
t_{\mrm{inspiral}}[r] = \int_{r_{\mrm{i}}}^{r_{\mrm{shred}}}{\frac{\left(\frac{\mathrm{d}M}{\mathrm{d}r}-\frac{M[r]}{r}\right)\ \sqrt[] {v_r^2+\bar{v}_{\phi}^2}}{4 \xi \pi G m_2 r \rho[r]}\mrm{d}r}
\label{eq:tinsp}
\end{equation}
where $r_i$ is the initial radial position, $r_{\mrm{shred}}$ is the tidal shredding radius which can be estimated via $r_{\mrm{shred}}\sim R_2 \sqrt[3]{2M_{\mrm{core}}/m_2}$, and $\bar{v}_{\phi}=v_{\phi}-v_{\mrm{env}} \simeq v_{\phi}$ for slow rotators such as RGB/AGB stars \citep{Nordhaus:2007aa}. The parameter $\xi$ accounts for the geometry of the companion's wake, the gaseous drag of the medium, and the Mach number \citep{Park2017}. We assume a value of $\xi=4$, and note that the ejection efficiency is not sensitive to this value for the mass ratios considered in this work.
For regions in which the $t_{\mathrm{conv}} {\ll} t_{\mrm{inspiral}}$, convection will transport orbital energy radially outward. If the turbulent region reaches an optically thin area such as the surface of the star, this energy can be lost via radiation. We refer to the convection region that makes contact with the surface of the star as the surface-contact convective region (SCCR). For lower mass stars in our sample ($\lesssim 3.0 M_{\odot}$), there tends to be a single convective region at maximum extent. For stars more massive than $4.0 M_{\odot}$, a deeper, yet physically distinct, secondary convective layer is also present (see Figure~\ref{fig:Timescales}).
The inspiral and convective timescales are presented in Figure~\ref{fig:Timescales} for primary masses ranging from $1.0-6.0$ $M_\odot$ and companion masses ranging from $0.002-0.2$ $M_\odot$. SCCRs are shaded in yellow and secondary convective regions are shaded in pink. The tidal disruption locations are represented with X markers. The location and depth of the SCCR of the primary during inspiral is especially important and discussed in detail throughout the remainder of this paper.
\subsection{Energy and Luminosity Considerations}
\label{sec:energycalc}
The energy required to unbind the primary's envelope must be known to compute $\bar{\alpha}_{\mathrm{eff}}$. By carrying out calculations directly from our stellar evolution models, we avoid employing $\lambda$-formalisms, which approximate the primary's gravitational binding energy for situations in which the interior structure is not known \citep{DeMarco2011}.
The minimum energy required to strip the envelope's mass exterior to a radius, $r$, is then given by:
\begin{equation}
E_{\mathrm{bind}}[r]=-\int_M^{M_\mrm{o}}\frac{G M[r]}{r}\mrm{d}m[r],
\label{eq:Ebind}
\end{equation}
where $M_{\mrm{o}}$ is the total mass of the primary star. One necessary, but not exclusive, condition for CE ejection is that the orbital energy released during inspiral must exceed the binding energy of the envelope. Note that we focus exclusively on the gravitational component of the binding energy and do not include the internal energy of each shell in our calculations. While this could affect the ejection efficiency of the system, it has not been shown to make a significant contribution to the binding energy \citep{Han:1995aa,Ivanova2013}.
The energy released via inspiral is given by
\begin{equation}
\Delta E_{\mrm{orb}}[r]
=\frac{G m_2}{2} \left(\frac{M[r_\mrm{i}]}{r_\mrm{i}}-\frac{M[r]}{r}\right)
\label{eq:Eorb}
\end{equation}
where $m_2$ is the companion's mass and $r_\mrm{i}$ is the radius of the companion's orbit at the onset of energy transfer due to inspiral through the primary.
Equations~\ref{eq:Ebind} and~\ref{eq:Eorb} can then be combined with an efficiency to yield the following:
\begin{equation}
E_{\mathrm{bind}} = \bar{\alpha}_{\mathrm{eff}} \Delta E_{\mathrm{orb}},
\label{eq:alpha}
\end{equation}
where $\bar{\alpha}_{\mathrm{eff}}$ is the effective efficiency of energy transfer to the envelope from the decaying orbit. If $\bar{\alpha}_{\mathrm{eff}}=1$, then all transferred orbital energy remains in the system and can fully contribute toward ejection. If $\bar{\alpha}_{\mathrm{eff}}=0$, no orbital energy remains in the system and the CE would never be ejected. We discuss $\bar{\alpha}_{\mathrm{eff}}$ in the context of models and their convective zones in detail in Section~\ref{sec:transfer}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{ComboLum_010919.pdf}
\caption{The maximum luminosity carried by convection, $L_{\rm max,conv}$ for two primary masses in thick green curves, is shown with the drag luminosity of several companions for the two primary mass cases in dashed, coloured lines. The tidal disruption radii are marked with an X for the $1.0 M_{\odot}$ case and a triangle for the $6.0 M_{\odot}$ case. This demonstrates that convection can carry the liberated orbital energy to the surface where it is radiated away.}
\label{fig:lum}
\end{figure}
Within the primary's convective envelope, subsonic convection can accommodate additional power up to a maximum defined by:
\begin{equation}
L_{\rm max,conv} = \beta 4 \pi \rho[r] r^2 c_s^3[r],
\label{Lconv}
\end{equation}
where $\beta \simeq 5$ and $\rho[r]$ and $c_s[r]$ are the density and the sound speed of the envelope medium at radius $r$, respectively \citep{Grichener2018a}. If the additional luminosity generated from inspiral remains below this maximum, convection can transport the energy to other regions of the star. The drag luminosity, generated from inspiral, is given as:
\begin{equation}
L_{\rm drag} = \xi \pi r_{\rm acc}^2 \rho[r] v_{\phi}^3[r]
\label{Ldrag}
\end{equation}
where the accretion radius is $r_{\rm acc} = 2 G m_2/v_{\phi}^2[r]$ \citep{Nordhaus2006}.
We compare the maximum luminosity carried by convection to the drag luminosity at each radius. For all mass ratios in this study, the drag luminosity is less than the limit that convection can carry, as shown in Figure~\ref{fig:lum}. The thick, solid and dotted green curves show the maximum luminosity that can be carried by the convective envelope for the $1.0 M_{\odot}$ and $6.0 M_{\odot}$ models, respectively. These two curves bound the convective luminosity limits for all primary masses in this work. The drag luminosities due to companions of mass $0.002 M_{\odot} - 0.2 M_{\odot}$ inspiraling through the two limiting primaries can be seen in the coloured, dashed curves. The tidal disruption radii are shown with X symbols and triangles for companions orbiting within the $1.0 M_{\odot}$ and $6.0 M_{\odot}$ primaries, respectively.
The solid green curve denoting the maximum luminosity carried by a $1.0 M_{\odot}$ primary exceeds all of the drag luminosity curves which are marked by an X, just as the dotted green curve denoting the maximum luminosity carried by a $6.0 M_{\odot}$ primary exceeds all of the drag luminosity curves which are marked by a triangle. Therefore, the luminosity produced during inspiral can be carried by convection for the primary-companion mass pairs in this study, as the two cases presented here are representative of the extrema cases.
\color{black}
\subsection{Common Envelope Outcomes}
There are two possible outcomes for common envelope phases: (i.) the companion survives the interaction and emerges in a short-period, post-CE binary, or (ii.) it does not and is destroyed in the process.
The companion body's radius, $R_2$, is estimated according to its mass. For planet-mass objects ($m_2 \le 0.0026M_{\odot}$, \citealt{Zapolsky1969}) the radius is approximated as $R_2=R_{\mrm{Jupiter}}$. For brown dwarfs ($0.0026 M_{\odot} < m_2 < 0.077 M_{\odot}$, \citealt{Burrows1993}), the radius is calculated via:
\begin{equation}
R_2/R_{\odot}=0.117-0.054\log^2\left({\frac{m_2}{0.0026 M_\odot}}\right)+0.024\log^3\left({\frac{m_2}{0.0026 M_\odot}}\right) \nonumber
\label{eq:bdrad}
\end{equation} \citep{ReyesRuiz1999}.
For stellar main-sequence companions ($m_2 \ge 0.077M_{\odot}$), a power law is used:
\begin{equation}
R_2=\left(\frac{m_2}{M_{\odot}}\right)^{0.92} R_{\odot}
\label{eq:starrad}
\end{equation} \citep{ReyesRuiz1999}.
To determine whether a companion survives the CE and emerges in a short-period orbit, there must exist an orbital separation, $a$, such that: (i.) $\bar{\alpha}_{\mathrm{eff}} \Delta E_{\mathrm{orb}}[a] > E_{\mathrm{bind}}[a]$ and (ii.) $a > r_{\rm shred}$. The shredding radius of each companion mass is determined as described in Section~\ref{sec:convregions}. The shredding radii are presented as X symbols in Figures~\ref{fig:Timescales} and~\ref{fig:AlphaEorbs}. If the companion disrupts early in its descent through the envelope, the energy available to unbind the envelope will be minimized, whereas if the companion body remains intact through the majority of the envelope, the opposite will be true. This is discussed in more detail in Section~\ref{sec:transfer}.
\begin{figure*}
\begin{multicols}{2}
\centering
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{1_0M_alphaEorb_121718.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{2_0M_alphaEorb_121718.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{3_0M_alphaEorb_121718.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{4_0M_alphaEorb_121718.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{5_0M_alphaEorb_121718.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.45\textwidth}
\includegraphics[width=\textwidth]{6_01M_alphaEorb_121718.pdf}
\end{minipage}
\end{multicols}
\hfill
\caption{Comparative energy plots for a sample of representative primary masses at their maximum radial extent and several test companion masses. The binding energy for the primary star is shown in solid blue. The coloured, dashed lines show the change in orbital energy of the companion star as it inspirals, and the radius at which the companion shreds is marked with an X. (Several X's fall below $10^{44}$ erg.) For companion masses which the X falls below the binding energy curve, the companion will disrupt during inspiral before enough energy is transferred to unbind the envelope of the primary. These orbital energy curves take into account the convective zones of the primary, in that movement through the surface-contact convective regions (SCCRs) does not contribute energy to the ejecting of the envelope.
}
\label{fig:AlphaEorbs}
\end{figure*}
\section{Ejection Efficiency in Convective Regions}
\label{sec:transfer}
The efficiency with which orbital energy can be used to unbind the envelope is a function of position inside the CE. To determine a lower limit for $\bar{\alpha}_{\mathrm{eff}}$ in a star with convective regions, we proceed in the following manner. If $t_{\rm conv} < t_{\rm inspiral}$, and the companion is orbiting inside a surface-contact convective region (SCCR), then there is no contribution to $\bar{\alpha}_{\mathrm{eff}}$ as the orbital energy can be transported via convection to the surface and leave the system as photons. If the companion is orbiting in a region where $t_{\rm inspiral} < t_{\rm conv}$, then the orbital energy cannot escape the system and fully contributes, in some form, toward raising the negative binding energy of the primary. In this way, we construct an average ejection efficiency by determining individual binary $\alpha_{\mathrm{eff}}$ coefficients (0 or 1) at each position. We designate regions within the SCCR to have $\alpha_{\mathrm{eff}}=0$, as the energy can be carried to an optically thin layer and radiated away; we designate all other regions to have $\alpha_{\mathrm{eff}}=1$, as we assume that the energy transferred by the change in orbital energy of the companion gets evenly distributed throughout the mass in each radial shell. Then for each primary-companion pair, we determine $\bar{\alpha}_{\mathrm{eff}}$ by integrating from the surface to either the point of tidal disruption or the point of energy equivalence (i.e. where $E_{\mathrm{bind}} = \alpha_{\mathrm{eff}}[r] \Delta E_{\mathrm{orb}}$),
via:
\begin{equation}
\bar{\alpha}_{\mathrm{eff}} = \frac{\int_{r_{\mrm{i}}}^{r_{\mrm{f}}}\alpha_{\mathrm{eff}}[r] \mrm{d}E_{\mrm{orb}}[r]}{E_{\mrm{orb}}[r_{\mrm{f}}]-E_{\mrm{orb}}[r_{\mrm{i}}]},
\label{eq:abar}
\end{equation}
where $\mrm{d}E_{\mrm{orb}}[r]$ can be calculated discretely as in Equation~\ref{eq:Eorb} and $r_{\mrm{f}}$ is the final position of the companion. The $r_{\mrm{f}}$ limit will be the maximum of $r_{\mrm{shred}}$ and $r_{E_{\mathrm{bind}} = \alpha_{\mathrm{eff}}[r] \Delta E_{\mathrm{orb}}}$. This limit is set due to the ejection of the envelope (when $r_{\rm f} = r_{E_{\mathrm{bind}} = \alpha_{\mathrm{eff}}[r] \Delta E_{\mathrm{orb}}}$) or due to the tidal disruption of the companion (when $r_{\rm f} = r_{\rm shred}$). The maximum of these two values is taken as the integral's upper limit, as the inspiral advances from larger radii towards the core.
Note that we assume the internal structure of the primary is constant during inspiral. Since orbital energy is a function of enclosed mass of the primary and position, liberated orbital energy is distributed to the mass present in each location.
\begin{figure*}
\includegraphics[width=1.0\textwidth]{boxwhisker_011119_legend.pdf}
\caption{Box-and-whisker plot shows the range of convective depths during the final thermal pulse of the primary mass just prior to and including the time of maximum radius (does not include any time after the maximum radius). The orange line inside the box shows the median value of the boxplot, the vertical extent of the box marks the interquartile range (IQR: the middle 50\% of the range), and the upper and lower bounds of the whiskers mark the minimum and maximum convective depths, respectively. Note that for primary masses below $1.4 M_{\odot}$ the spanned range is very small, showing a stable convective region.
}
\label{fig:boxplot}
\end{figure*}
The curves showing $E_{\mrm{bind}}$ and $\Delta E_{\mrm{orb}}$ for primaries between $1.0~{-}~6.0 M_{\odot}$ and companions between $0.002-0.2 M_{\odot}$ can be seen in Figure~\ref{fig:AlphaEorbs}. Each subplot shows the binding energy of the primary (thick blue line), compared with the change in orbital energy of several companions, in dashed, coloured lines. The transfer of the released orbital energy is halted by the SCCR ($\alpha_{\mathrm{eff}}=0$), resulting in low, unchanging values of $E_{\mrm{orb}}$, and resumes once the companion has inspiraled deeper than the lower boundary of the SCCR ($\alpha_{\mathrm{eff}}=1$). The radius at which the companion tidally shreds is marked with an X symbol, halting energy transfer.
On the six subplots in Figure~\ref{fig:AlphaEorbs}, the location of the X-symbols show which companion masses unbind the envelope; if the X falls above the solid blue $E_{\mrm{bind}}$ curve, the companion survives the inspiral and emerges as a post-CE binary in a short-period orbit. We see that for a representative SCCR depth of $10^{11}$ cm, companions between $0.008-0.02 M_{\odot}$ (${\sim}8-20 M_\mrm{Jupiter}$) and greater will successfully unbind the envelope and survive the binary interaction.
\subsection{Variability of the SCCR}
\label{sec:boxplot}
As described in Section~\ref{sec:convregions}, we argue that mixing of the energy released by the inspiraling companion occurs within the convective regions of the primary star, with emphasis on the potential for the SCCR to carry energy that is then radiated away. Therefore, an understanding of the variability of the SCCR is imperative to a complete understanding of patterns in the ejection efficiency. An examination of the stability of the SCCR with mass is shown in Figure~\ref{fig:boxplot}. This box-and-whisker plot displays the range of convective depths of the SCCR during the final thermal pulse of all monitored primary masses (boxes and whiskers), the median SCCR depth (orange line), and the SCCR depth at the time of maximum radius (purple circle).
The variability of the SCCR becomes evident first in the $1.4M_{\odot}$ model and the range of SCCR depths remains large for all greater primary masses. The primary's time of maximum radius consistently corresponds with the minimum convective depth following and including the $3.2 M_{\odot}$ model, thus maximizing $\bar{\alpha}_{\mathrm{eff}}$. For the instances where the SCCR is at maximum depth and maximum radius concurrently, the companion tidally shreds within the deep SCCR or shortly thereafter, minimizing $\bar{\alpha}_{\mathrm{eff}}$.
The SCCR depth over time is of interest because of its inconsistency at the time of maximum radius, which is evident in Figure~\ref{fig:boxplot}. The SCCR depth of three representative models are plotted over time and can be seen in Figure~\ref{fig:convandradius}. The $1.0M_{\odot}$ model, $1.8M_{\odot}$ model, and $4.6M_{\odot}$ model are representative of a stable SCCR, a maximum-SCCR-depth-at-$\mrm{r_{max}}$ SCCR, and a \textit{minimum}-SCCR-depth-at-$\mrm{r_{max}}$ SCCR, respectively. The depth of the SCCR over time is plotted in the coloured, dashed lines. The respective radii of the primary over time are shown in blue, with dash patterns identical to those for the companion-mass SCCR depths. The lookback time is normalized to the maximum age of the star, and centred around the time of $\mrm{r_{max}}$, $t=0$. Each tick of normalized lookback time corresponds to a duration on the order of ${\sim}10^2$ years. For the $1.8M_{\odot}$ model, the time of peak radius corresponds with the time of deepest SCCR, thus minimizing the $\bar{\alpha}_{\mathrm{eff}}$ value. The $1.0M_{\odot}$ SCCR depth remains remarkably constant, and the depth of the $4.6M_{\odot}$ SCCR decreases with time, in stark contrast to that of the $1.8 M_{\odot}$ model. Companions will have been engulfed by $\mrm{r_{max}}$ or will never be engulfed. Therefore, though the SCCR depth of the $4.6M_{\odot}$ model continues to decrease during the time after $\mrm{r_{max}}$ (see Fig.~\ref{fig:convandradius}), the SCCR is considered at ``minimum depth'' in Figure~\ref{fig:boxplot} as the interior structure of the primary is only of interest prior to and including the time at maximum radial extent.
\begin{figure}
\includegraphics[width=\columnwidth]{convradius_0111119.pdf}
\caption{The SCCR depths over time are shown along with the primary's radius for three representative primary masses. The x-axis is a lookback time until maximum radius, described by: $\frac{t[r_{\mrm{max}}]-t[r]}{t[r_{\mrm{max}}]}$. The vertical black line marks the time of $r_{\mrm{max}}$, $t=0$. (Times after maximum radius are shown here for completeness but are not examined in this work.) The blue lines show the fraction of maximum radius in time. Two of the three blue lines overlap at unity. The coloured, dashed lines show the SCCR depth over time. Note the instability in the convective zone as it approaches the maximum radius for the $1.8 M_{\odot}$ model. (Convective depth is maximized with decreased interior convective radius.)
}
\label{fig:convandradius}
\end{figure}
\subsection{Ejection Efficiency, \boldmath{$\bar{\alpha}_{\mathrm{eff}}$}}
\label{sec:ejectioneff}
The ejection efficiency is unique for each primary-companion mass pair, since $\bar{\alpha}_{\mathrm{eff}}$ depends on the specific internal stellar structure (especially the properties of the SCCR) and the properties of the companion. We calculate $\bar{\alpha}_{\mathrm{eff}}$ values for a matrix of primary-companion pairs (see Equation~\ref{eq:abar}) and present the results in Figure~\ref{fig:colourmap}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{colormap_121618.pdf}
\caption{Colourmap of effective ejection efficiencies ($\bar{\alpha}_{\mathrm{eff}}$) based on surface-contact convective regions (SCCRs) of the primary star.}
\label{fig:colourmap}
\end{figure}
The distinct horizontal stripes between primary masses of $1.2M_{\odot}$ and $3.0M_{\odot}$ are of interest. This phenomenon can be attributed to variability of the SCCR during the final thermal pulse (${\sim}10^2$ years) of the primary star thus making the ejection efficiency sensitive to the time of the CE phase. In cases where $\bar{\alpha}_{\mathrm{eff}}{\sim}1$,
the SCCR is relatively deep with long convective timescales (see, e.g., $3.0 M_{\odot}$ panel of Figure~\ref{fig:Timescales}). Since the inspiral timescale is shorter than the convective timescale, energy is very efficiently distributed throughout the envelope and maximizes $\bar{\alpha}_{\mathrm{eff}}$. These cases also have a maximum SCCR depth at the time of maximum radius, unlike all other primary masses.
Low $\bar{\alpha}_{\mathrm{eff}}$ values are seen in the upper half of the colourmap, where primary masses $>3 M_{\odot}$. In these cases, the lower edge of each SCCR is at approximately $10^{11}$ cm from the center of the primary, and the inspiral timescales are greater with increasing primary mass. Because of these two factors, energy can be carried more readily by convective transport and lower $\bar{\alpha}_{\mathrm{eff}}$ in each case. A smoother spread of $\bar{\alpha}_{\mathrm{eff}}$ would be expected for lower masses as well if the size of the SCCR were stable at the time of maximum radius.
The distinct features in the colourmap show the ejection efficiency's strong dependence on SCCR depth, which varies during post-main-sequence evolution (see Figures~\ref{fig:boxplot}~and~\ref{fig:convandradius}). Therefore, the ejection efficiency of these systems is sensitive to the time of interaction, or the age of the primary when the CE phase occurs.
\section{Discussion of Results}
\label{sec:massratio}
In comparison to observational results presented by \citet{DeMarco2011} and \citet{Zorotovic2011}, we plot the $\bar{\alpha}_{\mathrm{eff}}$ vs. $q=m_2/M_1$ relation found from our simulated data in Figure~\ref{fig:mass_ratio}. The top panel is limited to the parameter space considered by \citeauthor{DeMarco2011} while the parameter space for our full suite of primary-companion mass pairs are shown in the bottom panel. Within the range of \citeauthor{DeMarco2011}, many of our ejection efficiencies do show an anti-correlation with mass ratio but with varying slopes. \citet{Zorotovic2011} are, however, in disagreement with \citeauthor{DeMarco2011}, finding larger final separations with lower companion masses. In the lower panel of Figure~\ref{fig:mass_ratio}, in which we present our full results, there are regions of parameter space that positively correlate with mass ratio.
It is worthwhile to note that \citeauthor{DeMarco2011} and \citeauthor{Zorotovic2011} estimate $\bar{\alpha}_{\mathrm{eff}}$ through observations of assumed CE progenitors, whereas we use stellar evolution models to probe the interior structure of primaries during the CE phase. For this reason, we note that a direct comparison is difficult as both studies estimate the CE binding energy via a $\lambda$ parameter. This can result in ejection efficiencies that are greater than unity making direct comparison challenging.
\citet{Politano2007} argue that $\bar{\alpha}_{\mathrm{eff}}$ is a function of the companion mass and the interior structure of the evolved primary, a statement with which we agree. Through this work, we find that the ejection efficiency is, in fact, highly sensitive to properties of the convective regions of the primary during the CE phase. To simulate CE systems and find the $\bar{\alpha}_{\mathrm{eff}}$ value, one must consider the effect of mixing within convective regions.
Convective mixing can affect the system in different ways depending on where it occurs. As the companion inspirals, convection can transport the released orbital energy away from the companion and distribute it throughout other regions. If convection occurs in the SCCR, then the convective eddies can carry the energy to the surface where it can be radiated away. This work assumes that all orbital energy released within the SCCR is radiated away and thus provides a lower limit on $\bar{\alpha}_{\mathrm{eff}}$ under the assumption that the liberated orbital energy is distributed evenly among the mass in a given layer.
\begin{figure}
\begin{minipage}{\textwidth}
\includegraphics[width=0.55\columnwidth]{massratio_demarco_121618.pdf}
\end{minipage}
\hfill
\begin{minipage}{\textwidth}
\includegraphics[width=0.55\columnwidth]{massratio_121618.pdf}
\end{minipage}
\hfill
\caption{Top: Axis-constrained natural logs of mass ratio and corresponding $\bar{\alpha}_{\mathrm{eff}}$. These axis limits are comparable to the parameter space examined by \citet{DeMarco2011}, who also found a negative slope in this range. Bottom: Mass ratio and $\bar{\alpha}_{\mathrm{eff}}$ for all primary-companion mass pairs in this study. }
\label{fig:mass_ratio}
\end{figure}
In some cases, the $\bar{\alpha}_{\mathrm{eff}}$ value of a specific system may deviate a bit from the values presented here, due to internal structure changes during inspiral (the depth of the SCCR can vary on ${\sim}10^2$ year timescales, as in Figure~\ref{fig:convandradius}, and the estimated duration of the entire CE phase is of comparable length). For this work, we assumed the internal structure to remain constant once the companion was engulfed. In any case, $\bar{\alpha}_{\mathrm{eff}}$ is sensitive to the time of the companion's inspiral through the envelope of primary, since the SCCR varies so rapidly during the evolution of a star. For this work, we used the profile of a primary at its maximum radius, and assumed that the secondary began to skim the primary's surface at that time. We chose this because at the maximum extent of the primary, the largest spatial volume in which tidal dissipation may lead to CE phases is the greatest. Given our findings of the dependence of the quickly-changing SCCR, the ejection efficiency values shown in Figure~\ref{fig:colourmap} cannot be generalized to those companion-mass pairs. Instead, the SCCR of the primary at the specific time of companion's engulfment must be known to calculate ejection efficiencies for unique configurations.
\subsection{Implications of Convection}
Convection allows the binary to naturally shrink to short orbital periods before the liberated orbital energy can be tapped to drive ejection. In many cases, these short periods are less than a day. This is, at least at initial glance, consistent with the steep drop off in observed post-CE systems that have orbital periods greater than a day \citep{Davis2010}. Investigating the predicted post-CE population distributions when $\bar{\alpha}_{\mathrm{eff}}$ from Equation~\ref{eq:abar} is adopted is an interesting future direction but requires determining when common envelope evolution starts.
The final state of the system also depends largely on the dominant timescale for the companions. If the inspiral timescale is shorter than the convective timescale, like those seen in $1.0-3.0 M_{\odot}$ panels of Figure~\ref{fig:Timescales}, the final orbital separations of the more massive companion bodies result in ${\sim}3$-day periods and the final separations of the less massive companions that exceed the energy required to unbind the envelope result in $\lesssim 1$-day periods. The intersection of the energy curves in Figure~\ref{fig:AlphaEorbs} shows where the envelope will be ejected, and thus the final orbital separation.
In some numerical simulations, the rate of orbital decay is slowed due to the gas reaching co-rotation \citep{Ricker2012,Ohlmann2015,Chamandy:2018aa}. However, co-rotation cannot be perfectly maintained in a turbulent medium and thus may lead to faster decay than is currently seen in such simulations. If the orbital decay timescale remains above the convective timescale in the SCCR, then the effect will be minimal as energy transport to the surface is the dominant process.
Note that there are several effects which we have neglected that may significantly increase the orbital decay timescale, and thus further increase the importance of convective effects. During inspiral, we have assumed that the gas is stationary and thus does not spin-up and reach near co-rotation with the orbit as is seen in some numerical simulations. If the gas is indeed near co-rotation with the orbit, then the orbital decay timescale is significantly larger than the values presented in this work. If that is the case, then even for our lowest-mass primaries, convection may dominate and carry the liberated orbital energy to the surface where it can be radiated away. In a similar vein, we have also assumed that the inspiral has no effect on the convection itself. However, the transfer of orbital energy to the gas during inspiral may result in larger convective velocities. Such an effect would shorten the convection transport timescales. The results presented here are conservative and would be improved by including both effects in future studies.
In regions where the orbital decay timescale is shorter than the convective turnover times, $\alpha_{\mathrm{eff}}[r]=1$. In principle, this means that we are implicitly assuming that the orbital energy is equally distributed to the mass in that region. At the inner boundary of the SCCR where this condition is satisfied, turbulent mixing may distribute sufficient orbital energy to enough of the mass to eject the full envelope. Primaries with masses $>3.0 M_\odot$ have secondary convective zones that may also aid in mixing sufficient orbital energy for those companions that reach it before being disrupted. These assumptions warrant investigation via numerical simulations that include convection (see \citealt{Chamandy:2018ab} for further discussion).
\section{Conclusions and future directions}
We have studied the effects of convection on the ejection efficiencies of common envelope interactions. Using detailed stellar evolution models at the time of maximal radial extent, we calculate $\bar{\alpha}_{\mathrm{eff}}$ values for a matrix of primary-companion mass pairs. The ejection efficiencies are most sensitive to the properties of the surface-contact convective region (SCCR). In this region, the orbital decay timescales are longer than the convective timescales, thereby allowing the star to effectively radiate released orbital energy and thus lower $\bar{\alpha}_{\mathrm{eff}}$. The inclusion of convection in CEs may solve the ejection problem seen in numerical simulations without the need for additional energy sources as the orbit must decay substantially before orbital energy can be tapped to drive ejection. Our considerations of convection also allow for post-CE orbital periods of less than a day in higher primary masses, an observational result that has been infrequently reproduced in population synthesis models that use universal, or constant, ejection efficiencies.
The results described are conservative, as changes in the envelope's gas are not considered during inspiral. For this reason, the co-rotation seen in numerical simulations cannot play a role in increasing the inspiral timescale, which may allow systems with even our lowest primary masses to end in sub-day orbital periods, just as our higher mass primary masses do. We also assume that the inspiraling companion's energy transfer does not affect the convective velocities. The released orbital energy may increase the convective velocities, consequently shortening the convective timescale. This, too, would strengthen the effect of convection on the system.
We provide a simple method to calculate $\bar{\alpha}_{\mathrm{eff}}$ if the properties of the SCCR are known. Since the ejection efficiencies are sensitive to the depth of the SCCR, they are inherently sensitive to the time of engulfment. If the SCCR depth changes substantially over time for a given stellar evolution model, then the time of the CE onset will determine the ejection efficiency. However, since RGB/AGB stars possess deep convective envelopes, the effects of convection remain important, independent of when a CE commences.
Future work can be advanced on multiple fronts. A more comprehensive study of the effects of co-rotation on the inspiral timescale and thus the ejection efficiency should be carried out. Numerical work should include high-resolution simulations of convection in common envelopes. Since this may be challenging in global simulations without convective sub-grid models, high-resolution local simulations of convective energy transport in stratified wind tunnels may be a natural starting point \citep{MacLeod:2017fk}. Massive stars also host deep convective zones and the impact on the ejection efficiencies should be investigated in the context of formation channels for the progenitors of gravitational-wave driven, compact-object mergers \citep{Belczynski:2016sf}. Finally, coupling these calculations to dynamical calculations that determine the time of engulfment can result in improved ejection efficiencies, which could then be incorporated into studies of populations \citep{Belczynski:2002nr,Moe:2006rm}.
\section*{Acknowledgements}
ECW and JN acknowledge support from the following grants: NASA HST-AR-15044, NASA HST-AR-14563 and NTID SPDI-15992. The authors thank Gabriel Guidarelli, Jeff Cummings, Joel Kastner, Luke Chamandy, Eric Blackman, John Whelan and Adam Frank for stimulating discussions.
|
2,869,038,154,870 | arxiv | \section{Introduction}
Recently, under the basic characteristics of mMTC, i.e, a large number of user devices and sporadic user traffic, the grant-free access strategy has been considered to allow the active devices to access the wireless network without a grant \cite{GraFrA2017, SenelGrant-Free2018}, which reduces both the access latency and signal processing overhead. Accordingly, the base station (BS) should simultaneously identify all the active user devices under grant-free random access. Furthermore, the BS is required to accurately acquire the channel state information for decoding uplink signals and executing downlink precoding after user activity detection. Hence, both active user detection (AUD) and channel estimation (CE) are required at the BS based on pilot sequences sent by the user devices.
Typically, the number of user devices in mMTC is very large, and thus assigning orthogonal pilot sequences to all user devices is prohibitive in practice, motivating the application of non-orthogonal pilot sequences in such situations. A central problem in the mMTC scenario is to jointly perform AUD and CE based on non-orthogonal pilot sequences.
Different from the sparse detection problem \cite{7563406, 7885116}, the joint AUD and CE requires both the support set and the corresponding amplitudes of the sparse vector, and can be formulated as a compressed sensing (CS) problem. Several CS-based solutions have been reported \cite{DiRe2020, LiuMassive12018, MRAcoCH2020, EPJAUDCE2019,jiang2021}. In \cite{DiRe2020}, a dimension reduction
method to reduce the pilot sequence length and computational complexity for joint AUD and CE has been proposed, which projects the original device state matrix onto a low-dimensional space by exploiting its sparse and low-rank structure. To fully utilize the statistical information of wireless channels, some Bayesian-based CS methods have been designed to achieve joint AUD and CE with higher accuracy. Specifically, the authors in \cite{LiuMassive12018} proposed an approximate message passing (AMP)-based joint AUD and CE strategy based on the randomly generated non-orthogonal pilot sequences for a massive MIMO system, and the statistical knowledge of both the channel and user sparsity was modeled using a prior distribution to facilitate the performance of AMP. Along this line, the authors in \cite{MRAcoCH2020} design an AMP-based algorithm to adaptively detect the active devices by exploiting the virtual
angular domain sparsity of the channels in an orthogonal frequency division multiplexing (OFDM) broadband system. Furthermore, an expectation propagation (EP)-based joint AUD and CE algorithm was proposed in \cite{EPJAUDCE2019} for massive access with a single-antenna BS, where the computationally intractable posterior probability of the involved sparse signals was approximated by a multivariate Gaussian distribution, enhancing the performance of joint AUD and CE. In addition, we have claimed in \cite{jiang2021} that the inherent temporal correlations of both the user channels and the active indicators between adjacent time slots can be used to enhance both the detection performance and channel estimation performance.
One critical issue for joint AUD and CE for massive connectivity is to analyze the performance of user detection and channel estimation. The AMP algorithm provides a corresponding theoretical framework called state evolution to accurately track the performance of AMP in each iteration. By using state evolution analysis, the authors in \cite{LiuSparse2018, LiuMassive12018} provide analytical characterizations of the missed detection and false alarm probabilities for device detection and channel error covariance for channel estimation under a massive MIMO Raleigh channel assumption. However, although state evolution provides a theoretical framework for analyzing the performance metrics of joint AUD and CE, there are some critical issues that are left untouched. First, the performance analysis is based on the fixed point of the state evolution function, but the connection between the performance and the specific choices of system parameters such as pilot length, transmit power, fraction of number of active users and number of antennas was not established. Second, for specific choices of system parameters, the AMP iterations are blocked in a sub-optimal fixed point, so that the Bayes-optimal AUD and CE performance cannot be achieved via the AMP framework \cite{2012Probabilistic}. The region of the system parameters where the AMP framework is sub-optimal cannot be measured via the state evolution analysis in \cite{LiuSparse2018, LiuMassive12018}. Finally, some related works \cite{RepMMVzhu2018, RepMMVG2018} have shown that the performance of AMP exhibits a \emph{phase transition} phenomenon where the AMP algorithm will exhibit disconnect performance variations with variations of the system parameters, which has not been analyzed for massive connectivity.
To address the above problems, in this paper, we propose a theoretical framework to analyze the performance of joint AUD and CE based on the replica method \cite{RepTan2002, 1985Entropy}, which is a classical tool for analyzing large systems that comes primarily from physics. The proposed framework established the connection between the system parameters and the performance of joint AUD and CE. Based on that, both the Bayes-optimal and the AMP achievable mean square error (MSE) can be predicted, and the region of system parameters where the AMP algorithm is sub-optimal can be determined. In addition, the phase transition phenomenon of our system can be analyzed, which guides the system design for massively connected networks.
Note that, different from \cite{LiuMassive12018}, where the performance analysis is performed based on the isotropic Raleigh channel model, the analysis in this paper is considered based on a more general channel model following the approach of \cite{JSDM2013}. Such a channel model has strong flexibility and reduces to particular channel models in multiple typical MIMO communication scenarios, such as the isotropic Raleigh channel model \cite{LiuSparse2018}, and the spatially correlated channel model \cite{MRAcoCH2020, CEMoG2015, CoCE2013}, etc. In particular, our performance analysis produces some novel results for both the isotropic Raleigh channel and spatially correlated channel case, and provides verifications for the analytical results in \cite{LiuSparse2018, LiuMassive12018}.
Our main contributions can be summarized as follow.
\begin{itemize}
\item
First, we consider a general grouping channel model for the joint AUD and CE scenario. We design a theoretical framework carefully tailored to our considered scenario based on the general idea of replica method. Based on that, we establish relations between the joint AUD and CE performance metrics and the system parameters: pilot length, transmit power, fraction of number of active users and number of BS antennas, under our considered channel model.
\item
Second, we analyze the isotropic channel scenario based on our theoretical framework. Concretely, we prove that the Bayes-optimal/AMP-achievable joint AUD and CE performance can be evaluated by a scalar valued function, which also provides a phase transition diagram. We further provide the analysis in the asymptotic MIMO regime and prove that the phase transition phenomenon disappears and the AMP can always achieve Bayes-optimal performance when the number of BS antennas is very large.
\item
Third, we analyze the performance of joint AUD and CE in the spatially correlated channel case. We prove that the performance of each user group can be separately evaluated by a particular free entropy function when the user groups are in mutually orthogonal subspaces, implying the per-group processing (PGP) will never bring the performance loss. In addition, we show that the spatially correlated channel is more likely to promote a phase transition compared with the isotropic channel, due to the small number of subspace dimensions and corresponding antenna gains, leading that
there will be a performance gap between AMP and Bayes-optimal performance. However, the pilot length required to step over the phase transition can be significantly reduced in that case.
\end{itemize}
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{fig/sys.eps}
\caption{Our model of the massive device communication network.}
\label{sys}
\end{figure}
\section{System Model}
\subsection{Massive Random Access Scenario}
We consider a uplink massive random access scenario in a single-cell cellular network with $N$ user devices. Each user is equipped with a single antenna, and the BS is equipped with $M$ antennas. The user traffic is assumed to be sporadic, i.e., most of user devices are idle at any given time.
Following the approach of \cite{JSDM2013}, in this paper, we consider the following channel model. We suppose that $N$ users are divided into $G$ groups based on the similarity of their covariance matrixes, where each group has $K_g$ users and the number of users is assumed same in each group for concise, i.e., $K_g = K = N/G$. Our model of the massive device communication network is shown in Fig. \ref{sys}. We denote $\boldsymbol h_{gk}\in\mathbb C^{M\times 1}$ as the channel response of $k$th user in the group $g$. We further assume that users in the same group have an identical channel probability density function (PDF) denoted as $Q_g(\boldsymbol h_{kg}) = \mathcal{CN}(\boldsymbol h_{kg}; \boldsymbol 0, \mathcal C_g)$ with covariance matrix $\mathcal C_g$. For the $k$th user in the group $g$, the user state, i.e., active or idle, is characterized by an activity indicator, denoted as $a_{gk}$, i.e., $a_{gk}= 1$ if it is active and $a_{gk}= 0$ otherwise.
We denote $\boldsymbol f_{kg}\in\mathbb C^{T\times 1}$ as the non-orthogonal pilot sequence that is assigned to the $k$th user in the group $g$ with independent and identically distributed (i.i.d.) components generated from a complex Gaussian distribution with zero mean and variance $1/T$, so that the pilot sequence has a unit norm\footnote{In the asymptotic regime that we consider, the pilot sequences have an unit power.}. Accordingly, the overall channel input-output is
\begin{equation}
\boldsymbol Y = \boldsymbol F\boldsymbol S + \boldsymbol W = \sum\limits_{g,k} a_{gk}\boldsymbol f_{gk}\boldsymbol h_{gk}^{T}+ \boldsymbol W, \label{channel}
\end{equation}
where $\boldsymbol F\in\mathbb C^{T\times N} \triangleq [\boldsymbol f_1,\dots, \boldsymbol f_{GK}]$ is the pilot matrix, $\boldsymbol S\in\mathbb C^{N\times M} \triangleq [\boldsymbol s_{11},\dots, \boldsymbol s_{GK}]^{T}$ with $\boldsymbol s_{gk}\in\mathbb C^{N\times 1}\triangleq a_{gk}\boldsymbol h_{gk}$, and $\boldsymbol W$ is the additive white Gaussian noise (AWGN) matrix with i.i.d. elements distributed as $\mathcal{CN}(0,\sigma_w^2)$. Each user device has a constant transmit power $P_t$, which is absorbed into the large-scale fading of the channel coefficients in (\ref{channel}) for concise.
We assume that the user devices are synchronized and each user decides whether or not to access the channel with a
probability $\rho$ in an i.i.d. manner \cite{LiuSparse2018, LiuMassive12018, MRAcoCH2020, HannakJoint2015}, and we further assume the fraction of active users $\rho$ and the PDF $Q_g(\boldsymbol h_{kg})$ of the $k$th user channel in the group $g$ are known for the BS\footnote{Note that the parameters learning strategy under mMTC scenario has been considered in the related work \cite{MRAcoCH2020, MAEC2020} based on the expectation-maximization method, which can be easily extended under our model. Since we focus on the theoretical analysis under our framework, the parameters learning issue is beyond the scope of this paper}. As a consequence, the corresponding PDF of the matrix $\boldsymbol S$ can be formulated as
\begin{equation}
p(\boldsymbol S) = \prod_{g=1}^{G}\prod_{k=1}^{K}[(1-\rho)\delta(\boldsymbol s_{gk})+\rho Q_g(\boldsymbol s_{gk})].\label{prs}
\end{equation}
Based on our model (\ref{channel}) and the prior distribution of the transmitted signals (\ref{prs})\footnote{Note that we assume joint AUD and CE is performed within one coherence time so that the channel coefficients remain unchanged.}, our task is to identify the active users, i.e., determine the active indicator $a_{gk}$ for each user, as well as estimate the corresponding channel coefficients $\boldsymbol h_{gk}$ for the active users.
\emph{Remark}: Note that our channel model assumption has a strong flexibility and reduces to particular channel models in multiple typical MIMO communication scenarios, such as the isotropic Rayleigh fading channels \cite{LiuSparse2018, LiuMassive12018}, where the channel coefficients of each user device are i.i.d., and the spatially correlated channels \cite{MRAcoCH2020, CEMoG2015, CoCE2013}, where different user groups are sufficiently well separated in the angular domain, and the channel covariance matrixes exhibit a low rank structure. We also note that although we consider the case where each group has the same number of users and the fraction of active users for each group is also the same, our following proposed theoretical framework can be easily extended to the case where the active ratio and user number are different for each group.
Based on our general channel model (\ref{prs}), in the following, we propose a novel theoretical framework by adopting replica method \cite{RepTan2002, 1985Entropy} to provide phase transition analysis, Bayes-optimality analysis of our scenario, and our theoretical framework can also provide the performance prediction of the AMP-based joint AUD and CE, which is the state-of-art algorithm under the Bayesian framework.
\section{Replica Analysis on Joint AUD and CE}
\label{a_re}
Our theoretical framework is based on statistical physics. In the statistical physics literature, the \emph{free entropy} of a system reflects the macro performance that characterizes some thermodynamic properties of the system \cite{1985Entropy}. Some related works in the signal processing literature have also shown that evaluating the fixed point of free entropy function provides the minimum MSE (MMSE) prediction for signal recovery and the achievable MSE for the AMP framework \cite{RepMMVzhu2018, RepMMVG2018}. For the massive connectivity scenario, evaluating the free entropy function of our system (\ref{channel}) provides an analytic tool to measure the performances metrics of joint AUD and CE.
Towards this end, we adopt the replica method in this paper to evaluate the free entropy function of our system. The replica method \cite{RepTan2002, 1985Entropy}, which is a classical tool in the statistical physics literature, provides an efficiency way to reduce the calculation of a certain free entropy into an optimization problem over specific covariance matrices. Such a reduction is based on a set of typical replica assumptions that include the self-averaging property, the validity of ``replica trick'', the ability to exchange certain limits and the replica symmetry \cite{2012Probabilistic}. It has been shown in many related works that the replica method provides correct predictions on many typical applications, such as signal processing \cite{RepCS2012} and physical layer communications \cite{RepADChe2018, RepADCW2017}.
Our analysis based on the replica method framework \cite{2012Probabilistic} is considered under a certain \emph{asymptotic regime} with $K\rightarrow\infty$, $T\rightarrow\infty$ and a fixed ratio $T/K\rightarrow\alpha$. Next, we provide the background of the statistical physics literature and obtain the free entropy function with respect to the system parameters under our scenario via replica method.
Although these system parameters can not be infinity in the practice, we still utilize the asymptotic regime to facilitate analysis, since the system parameters are typically large in the massive connectivity. Authors in \cite{LiuMassive12018} have shown that the recovery performances in the practical settings match the theoretical results in the large system limit.
\setcounter{TempEqCnt}{\value{equation}}
\setcounter{equation}{6}
\begin{figure*}[tbp]
\begin{equation}
\begin{split}
\Phi(\mathcal E) = &-{\rm Tr}\left(\left(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha}\right)^{-1}\left(\sum\limits_g\rho\mathcal C_g-\mathcal EG\right)\right)-\alpha M-\alpha \log\left|\boldsymbol\Delta+\frac{\mathcal EG}{\alpha}\right|\\
&+\sum\limits_g\int{\rm d}\boldsymbol s_g[(1-\rho)\delta(\boldsymbol s_g)+\rho Q_g(\boldsymbol s_g)]\int{\rm D}\boldsymbol z\log\left\{\int{\rm d}\boldsymbol x_g[(1-\rho)\delta(\boldsymbol x_g)+\rho Q_g(\boldsymbol x_g)]\right.\\
&\left.\exp\left\{-\boldsymbol x_g^H(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{-1}\boldsymbol x_g+2\mathfrak R\left(\boldsymbol x_g^H[(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{-1}\boldsymbol s_g+(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{-\frac{1}{2}}\boldsymbol z]\right)\right\}\right\}.\label{general_FE}
\end{split}
\end{equation}
\hrulefill
\end{figure*}
\setcounter{equation}{\value{TempEqCnt}}
\subsection{Free Entropy Function Derivation via Replica Method}
Under the statistical physics literature, a probabilistic inference approach to reconstruct the transmitted signal $\boldsymbol S$ aims to sample $\boldsymbol X$ from the following posterior distribution based on (\ref{channel}), given by
\begin{equation}
p(\boldsymbol X|\boldsymbol Y) = \frac{p(\boldsymbol X)}{Z}\prod_{t=1}^{T}\frac{\exp\left(-(\boldsymbol y_t-\boldsymbol f_t\boldsymbol X)\boldsymbol\Delta^{-1}(\boldsymbol y_t-\boldsymbol f_t\boldsymbol X)^{H}\right)}{\pi^M|\boldsymbol\Delta|},\label{posterior}
\end{equation}
where $\boldsymbol\Delta\triangleq \sigma_w^2\boldsymbol I$, $Z$ is the partition function of this distribution, and $p(\boldsymbol X) =\prod_{g,k}[(1-\rho)\delta(\boldsymbol x_{gk})+\rho Q_g(\boldsymbol x_{gk})]$, which can be regarded as the prior distribution of $\boldsymbol X$. Under the Bayes-optimal assumption, the prior distribution over $\boldsymbol X$ matches the PDF of transmitted signal $p(\boldsymbol S)$. As a consequence, $Z$ can be formulated as
\begin{align}
Z = \int {\rm d} \boldsymbol X&\prod_{t=1}^{T}\frac{1}{\pi^M|\boldsymbol\Delta|}e^{-(\boldsymbol y_t-\boldsymbol f_t\boldsymbol X)\boldsymbol\Delta^{-1}(\boldsymbol y_t-\boldsymbol f_t\boldsymbol X)^{H}}\nonumber\\
&\times\prod_{g=1}^{G}\prod_{k=1}^{K}[(1-\rho)\delta(\boldsymbol x_{gk})+\rho Q_g(\boldsymbol x_{gk})],\label{patition}
\end{align}
where $\boldsymbol f_t$ is the $t$th row of the pilot matrix $\boldsymbol F$. We note that above integration is performed over each element of matrix $\boldsymbol X$ and such a integral expression will be used throughout this paper. According to the statistical physics literature, the free entropy is defined as $\Phi \triangleq \log Z$. Utilizing the self-averaging assumption \cite{RepTan2002, 1985Entropy}: $\lim\limits_{N\to \infty} \Pr\left[\left|\frac{Z}{N}-\frac{\mathbb E(Z)}{N}\right|\ge\theta\right] = 0$, for any tolerance $\theta>0$, the free entropy can be reformulated as $\Phi = \frac{1}{N}\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}(\log Z)$. Hence, to determine the free entropy function, one requires to compute the average of the log-partition function $\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}(\log Z)$. Based on the standard procedure of the replica method \cite{2012Probabilistic}, we can alternatively determine the average log-partition function by the following relation
$
\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}(\log Z) = \lim\limits_{n\to 0}\frac{1}{n}\log\left(\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}Z^n\right)
$.
According to the \emph{replica trick} in \cite{2012Probabilistic, RepTan2002, 1985Entropy}, the quantity $\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}Z^n$ is carried out as if $n$ were an integer, and we take the fact that $n$ is a real number into consideration after obtaining a manageable enough expression. As a consequence, the free entropy can be formulated as
\begin{equation}
\Phi = \lim\limits_{N\to \infty}\lim\limits_{n\to 0}\frac{1}{Nn}\log\left(\mathbb E_{\boldsymbol F,\boldsymbol S,\boldsymbol W}Z^n\right).\label{re_trick}
\end{equation}
Based on this, we have the following theorem.
\begin{theorem}
Define the reconstruction MSE matrix over the system (\ref{channel}) as $\mathcal E\triangleq \frac{1}{G}\sum_g \mathcal E_g$, where \begin{equation}\mathcal E_g \triangleq \frac{1}{K}\sum\limits_{k = 1}^K\left(\hat{\boldsymbol x}_{gk}(\boldsymbol Y)-\boldsymbol s_{gk}\right)\left(\hat{\boldsymbol x}_{gk}(\boldsymbol Y)-\boldsymbol s_{gk}\right)^H,\end{equation} where $[\hat{\boldsymbol x}_{11}(\boldsymbol Y),\dots, \hat{\boldsymbol x}_{GK}(\boldsymbol Y)] = \hat{\boldsymbol X}$ is the sample of the posterior distribution $p(\boldsymbol X|\boldsymbol Y)$ with given $\boldsymbol Y$ defined in (\ref{posterior}). Under our certain asymptotic regime defined in Sec. \ref{a_re}, and using the standard assumptions of the replica method, i.e., the self-averaging assumption, the replica-symmetry assumption, and the replica trick in (\ref{re_trick}), the free entropy (\ref{re_trick}) can be calculated by optimizing a function with respect to $\mathcal E$, which is given in the equation (\ref{general_FE}). We note that the term ${\rm D}\boldsymbol z$ is a complex Gaussian integration measure.
\end{theorem}
\begin{proof}
Please see the appendix \ref{A}.
\end{proof}
We call the function (\ref{general_FE}) as \emph{free entropy function}\footnote{Since the free entropy function is closely related to the free entropy, we also use $\Phi$ to represent it. }.
According to \cite{RepMMVzhu2018, RepMMVG2018}, optimizing $\mathcal E$ that maximizes the free entropy function (\ref{general_FE}) corresponds to the minimum MSE (MMSE) of specific choices of system parameters, i.e., $\rho$, $\alpha$, $\mathcal C_g$, $\forall g$, in the Bayes-optimal condition. Setting the first order derivative of $\Phi(\mathcal E)$ with respect to the matrix $\mathcal E$ into zero, we get the following equation.
\setcounter{equation}{7}
\begin{align}
\mathcal E = \frac{1}{G}\sum\limits_{g}&\mathbb E_{\boldsymbol s_g, \boldsymbol z}\left[\left(\eta_g\left(\boldsymbol s_g+(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{\frac{1}{2}}\boldsymbol z\right)-\boldsymbol s_g\right.\right)\nonumber\\
&\times\left(\left.\eta_g\left(\boldsymbol s_g+(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{\frac{1}{2}}\boldsymbol z\right)-\boldsymbol s_g\right)^H\right],\label{SE}
\end{align}
where $\boldsymbol z$ obeys $\mathcal{CN}(\boldsymbol z; \boldsymbol 0, \boldsymbol I)$ and $\eta_g(\cdot)$ is the Bayes-optimal denoiser of the noisy measurement $\hat{\boldsymbol s}_g\triangleq \boldsymbol s_g+(\boldsymbol\Delta+\frac{\mathcal EG}{\alpha})^{\frac{1}{2}}\boldsymbol z$ with the latent signal $\boldsymbol s_g$ distributed as $(1-\rho)\delta(\boldsymbol s_g)+\rho Q_g(\boldsymbol s_g)$. We note that $\eta_g(\cdot)$ is the corresponding MMSE denoiser of noisy measurement $\hat{\boldsymbol s}_g$.
We can also observe from (\ref{SE}) that there is a close match between the fixed point of the AMP state evolution function in \cite{BayatiDynamics2011} and the stationary point of the free energy function (\ref{general_FE}). In addition, according to \cite{RepMMVzhu2018, RepMMVG2018}, the AMP algorithm will be blocked by a particular local maximum point of (\ref{general_FE}). This means that seeking the stationary points of the free entropy function (\ref{general_FE}) also provides accurate prediction of the performance of the AMP algorithm. We remark that the recovery MSE of AMP is sometimes sub-optimal since only the global maximum point that maximizes the free entropy function (\ref{general_FE}) corresponds to the MMSE, and we call it AMP-achievable MSE.
In the following, we establish connections between the fixed point of the AMP state evolution function predicted by the free entropy function (\ref{general_FE}), and the performance metrics of joint AUD and CE. Typically, we adopt the likelihood ratio test (LRT) detection for AUD, and adopt MMSE criterion for CE. We note that the author in \cite{OnRangan} claims that, in the case of large i.i.d. zero-mean Gaussian sensing matrix, the AMP methods exhibit fast convergence. Since in the massive access literature, the system is large enough and the i.i.d. zero-mean Gaussian sensing matrix is adopted, guaranteeing the convergence of the AMP algorithm in the following sections.
\setcounter{TempEqCnt}{\value{equation}}
\setcounter{equation}{13}
\begin{figure*}[tbp]
\begin{equation}
\begin{split}
\Phi(\tau) = &-\alpha M\left(\frac{\sigma_w^2}{\frac{1}{\alpha}\tau+\sigma_w^2}+\log(\sigma_w^2+\frac{1}{\alpha}\tau)\right)
+M\sum\limits_g\frac{(1-\rho)\sigma_g^2}{\sigma_g^2+\sigma_w^2+\frac{1}{\alpha}\tau}\\
&+\sum\limits_{g}\int{\rm D}\boldsymbol z\rho\log\left[(1-\rho)\exp\left\{-\frac{||\boldsymbol z||^2\sigma_g^2}{\sigma_w^2+\frac{1}{\alpha}\tau}\right\}+\rho\left(\frac{\frac{1}{\alpha}\tau+\sigma_w^2}{\frac{1}{\alpha}\tau+\sigma_w^2+\sigma_g^2}\right)^M\right]\\
&+\sum\limits_{g}\int{\rm D}\boldsymbol z(1-\rho)\log\left[(1-\rho)\exp\left\{-\frac{||\boldsymbol z||^2\sigma_g^2}{\sigma_g^2+\sigma_w^2+\frac{1}{\alpha}\tau}\right\}+\rho\left(\frac{\frac{1}{\alpha}\tau+\sigma_w^2}{\frac{1}{\alpha}\tau+\sigma_w^2+\sigma_g^2}\right)^M\right].\label{FE_tau}
\end{split}
\end{equation}
\hrulefill
\end{figure*}
\setcounter{equation}{\value{TempEqCnt}}
\subsection{Prediction of AMP-based Joint AUD and CE}
\label{pr_audce}
After executing the AMP algorithm on the transmitted signal $\boldsymbol Y$ in (\ref{channel}), the overall estimation problem is decoupled as a sequence of vector-valued estimation problems. For the users in the group $g$, the decoupled signal model is $\hat{\boldsymbol s}_g\triangleq \boldsymbol s_g+\boldsymbol \Sigma^{\frac{1}{2}}\boldsymbol z$, where we define $\boldsymbol \Sigma\triangleq \boldsymbol\Delta+\frac{\mathcal E^{\star}G}{\alpha}$ as the equivalent noise covariance matrix. The $\mathcal E^{\star}$ denotes the fixed point of AMP state evolution function that fulfills equation (\ref{SE}), so that it is also a stationary point of the free entropy function (\ref{general_FE}). Note that we have dropped the subscript about the user indicator $k$, much as the following expressions, since all the users within a group share a common probabilistic model.
As a consequence, for the user devices in the group $g$, the AUD rule based on log-likelihood ratio (LLR) with a threshold $l_g$ can be formulated as
\begin{align}
\text{LLR}(\hat{\boldsymbol s}_g) =& \log\left(\frac{p(\hat{\boldsymbol s}_g|a_g = 1)}{p(\hat{\boldsymbol s}_g|a_g = 0)}\right)\nonumber\\
=&\hat{\boldsymbol s}_g^H\tilde{\boldsymbol\Sigma}\hat{\boldsymbol s}_g+\log\frac{\left|\boldsymbol\Sigma\right|}{\left|\mathcal C_g+\boldsymbol\Sigma\right|}>l_g,\label{llr}
\end{align}
where we define $\tilde{\boldsymbol\Sigma}\triangleq\boldsymbol\Sigma^{-1}-\left(\mathcal C_g+\boldsymbol\Sigma\right)^{-1}$. Accordingly, the missed detection and false alarm probabilities $P_{M}$ and $P_{F}$ can be obtained via
\begin{align}
P_{F} = \Pr\left\{\hat{\boldsymbol s}_g^H\tilde{\boldsymbol\Sigma}\hat{\boldsymbol s}_g>l_g'|a_g = 0\right\},\label{pf}\\
P_{M} = \Pr\left\{\hat{\boldsymbol s}_g^H\tilde{\boldsymbol\Sigma}\hat{\boldsymbol s}_g<l_g'|a_g = 1\right\}\label{pm},
\end{align}
where $l_g' \triangleq l_g-\log\left|\boldsymbol\Sigma\right|+\log\left|\mathcal C_g+\boldsymbol\Sigma\right|$.
We then consider the performance analysis for CE. Since the CE for the active users are performed after the AUD, we thus consider the mean of conditional posterior distribution $p(\boldsymbol s_g|\hat{\boldsymbol s}_g, a_g = 1)$ as the channel estimator for active user devices. Based on the definition of noisy measurement $\hat{\boldsymbol s}_g$,
after performing the AMP algorithm, the CE for the active user devices in the group $g$ is
\begin{equation}
\hat{\boldsymbol h}_g = \left(\boldsymbol\Sigma^{-1}+\mathcal C_g^{-1}\right)^{-1}\boldsymbol \Sigma^{-1}\hat{\boldsymbol s}_g.\label{chan_est}
\end{equation}
For the ease of analysis, we only consider the CE error in the case where the AUD is executed perfectly. Under such a circumstance, the CE error matrix can be formulated as
\begin{align}
\mathbb E\left[\left(\boldsymbol s_g-\hat{\boldsymbol h}_g\right)\left(\boldsymbol s_g-\hat{\boldsymbol h}_g\right)^H\right]
= \left(\mathcal C_g^{-1}+\boldsymbol\Sigma^{-1}\right)^{-1},\label{cha_err_ma}
\end{align}
where the expectation is taken over $p(\boldsymbol s_g,\hat{\boldsymbol s}_g| a_g = 1)$.
As a consequence, the performance of joint AUD and CE can be analyzed after obtaining the stationary point $\mathcal E^{\star}$ of the free entropy (\ref{general_FE}). We note that the equation (\ref{general_FE}) and the corresponding performance of joint AUD and CE in this section is based on the general channel model (\ref{prs}). In the following sections, we will show our theoretical framework can reduce to two typical scenarios in the massive random access literature, i.e., the isotropic channel scenario and the spatially correlated channel scenario. Concretely, in Sec \ref{iso}, we provide the performance analysis of the isotropic channel case. While in the Sec. \ref{spc}, we provide the performance analysis of the spatially correlated channel case.
\section{Isotropic Channel}\label{iso}
In this section, we consider a typical scenario in wireless communications, where there are many small reflectors around the BS, and the Rayleigh fading MIMO channel with isotropic channel is assumed \cite{LiuSparse2018, LiuMassive12018}. In this case, the Rayleigh channel components of each user group are assumed to be i.i.d. complex Gaussian with zero means and unit variances across all the BS antennas, and function $Q_g(\boldsymbol h_g)$ in the prior distribution over the channel in the $g$th user group is $Q_g(\boldsymbol h_g) = \mathcal {CN}(\boldsymbol h_g;0,\sigma_g^2\boldsymbol I)$, where $\sigma_g^2$ is the product of the large scale coefficient and the user transmitted power. In the following, we will analyze the performance of joint AUD and CE under this case by first deriving the corresponding free entropy function as well as its expression under an asymptotic massive MIMO regime. Then, the performance analysis of joint AUD and CE is given.
\subsection{Free Entropy Function}
Before deriving the concrete free entropy function, we note that it is proved in \cite{LiuMassive12018} that the equivalent noise covariance matrix in the AMP state evolution iterations always in the form of a diagonal matrix with identical diagonal entries in the isotropic Rayleigh channel case, so that the fixed point of the state evolution function (\ref{SE}) remains the same form. We therefore restrict the domain of $\mathcal E$ in the form of $\mathcal E = \tau G^{-1}\mathbf I$, and the newly derived free entropy function with respect to $\tau$ is in the following theorem.
\begin{theorem}
With the assumption of the isotropic Rayleigh channel, i.e., $Q_g(\boldsymbol s_g) = \mathcal {CN}(\boldsymbol s_g;0,\sigma_g^2\mathbf I)$ and the matrix $\mathcal E$ is diagonal with identical entries, i.e., $\mathcal E = \tau G^{-1}\mathbf I$,
the free entropy function can be reformulated as a scalar valued function, given by (\ref{FE_tau}).
\end{theorem}
\begin{proof}
Please see the appendix \ref{T2}.
\end{proof}
Note that the MMSE under the isotropic channel case can be derived by calculating the global maximum point of the free entropy function (\ref{FE_tau}), and the fixed point of the AMP state evolution can be predicted by the largest $\tau$ that
associated with a local maximum of (\ref{FE_tau}). As we shall see in the following Sec. \ref{Num_1}, there exists a region of system parameters, i.e., pilot length $L$, transmit power $P_t$, fraction of number of active user devices $\rho$ and number of BS antennas $M$, where the MSE of AMP is blocked by a local maximum point, so that the AMP algorithm is suboptimal. However, the following proposition indicates that when considering an asymptotic MIMO regime with the number of BS antennas go to infinity, the region where the AMP algorithm is suboptimal vanishes.
\begin{proposition}
In an asymptotic massive MIMO regime, i.e., $M\to \infty$, by neglecting some constants and the irrelevant factor, the free entropy function (\ref{FE_tau}) is reduced to
\setcounter{equation}{14}
\begin{align}
\Phi(\tau) = &-\frac{\alpha\sigma_w^2}{\frac{1}{\alpha}\tau+\sigma_w^2}-\alpha\log(\sigma_w^2+\frac{1}{\alpha}\tau)\nonumber\\
&-\rho\sum\limits_g\log(1+\frac{\sigma_g^2}{\sigma_w^2+\frac{1}{\alpha}\tau}),\label{FE_tau_limit}
\end{align}
which has and only has one local maximum point, and can be derived by calculating the equation
\begin{equation}
\tau = \rho\sum\limits_g\left(\sigma_g^{-2}+\left(\frac{1}{\alpha}\tau+\sigma_w^2\right)^{-1}\right)^{-1}.\label{t_opti}
\end{equation}
\end{proposition}
\begin{proof}
Please see the appendix \ref{P1}.
\end{proof}
Therefore, with an asymptotic number of BS antennas, since there is only one local maximum point, the AMP algorithm always achieves Bayes-optimal performance in the isotropic Rayleigh channel case, and the fixed point of AMP state evolution can be derived by solving (\ref{t_opti}).
\subsection{Prediction of AMP-based AUD and CE}
\begin{figure*}[t]
\begin{center}
\subfigure[$P_t = 18$dBm.]{
\begin{minipage}[t]{0.31\textwidth}
\includegraphics[width=2.4 in]{fig/pt18M2.eps}\label{18M2}
\end{minipage}
}
\hfill
\subfigure[$P_t = 23$dBm.]{
\begin{minipage}[t]{0.31\textwidth}
\includegraphics[width=2.4 in]{fig/pt23M2.eps}\label{23M2}
\end{minipage}
}
\subfigure[$P_t = 33$dBm.]{
\begin{minipage}[t]{0.31\textwidth}
\includegraphics[width=2.4 in]{fig/pt33M2.eps}\label{33M2}
\end{minipage}
}
\caption{Free entropy as a function of MSE with $M = 2$ BS antennas under different settings of $\alpha$ and $P_t$. (The ``red cross'' denotes the MMSE point, the ``red circle'' denotes the AMP-achievable MSE point, the ``red square'' denotes the unachievable local maxima MSE point, and such marks will also be used in the following figures.)}
\end{center}
\begin{center}
\subfigure[$M = 4$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.4in]{fig/pt33M4.eps}
\label{33M4}
\end{minipage
}
\subfigure[$M = 8$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.4in]{fig/pt33M8.eps}
\label{33M8}
\end{minipage}
}
\subfigure[$M\to \infty$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.4in]{fig/pt33Mlim.eps}
\label{33Mlim}
\end{minipage}
}
\caption{Free entropy as a function of MSE with $P_t = 33$dBm transmit power under different settings of $\alpha$ and $M$.}
\end{center}
\end{figure*}
Define $\tau^{\star}$ as the largest stationary point of (\ref{FE_tau}) that associated with the fixed point of the AMP state evolution. Together with (\ref{llr}), the LLR in the isotropic channel case is
\begin{align}
&\text{LLR}(\hat{\boldsymbol s_g}) = M\log\frac{\sigma_w^2+\frac{1}{\alpha}}{\sigma_g^2+\sigma_w^2+\frac{1}{\alpha}\tau^{\star}}\nonumber\\
&+\hat{\boldsymbol s_g}^H\left((\sigma_w^2+\frac{1}{\alpha}\tau^{\star})^{-1}-(\sigma_g^2+\sigma_w^2+\frac{1}{\alpha}\tau^{\star})^{-1}\right)\hat{\boldsymbol s_g}.
\end{align}
Obviously, the corresponding detection sufficient statistic is $||\hat{\boldsymbol s_g}||^2$. As a consequence, we have
\begin{align}
P_D(M) &= \gamma\left(M, \left(\sigma_g^2+\sigma_w^2+\frac{1}{\alpha\tau^{\star}}\right)^{-1}l_g'\right)\Gamma^{-1}(M),\label{pm1}\\
P_{F}(M) &= \gamma\left(M, \left(\sigma_w^2+\frac{1}{\alpha\tau^{\star}}\right)^{-1}l_g'\right)\Gamma^{-1}(M).\label{pf1}
\end{align}
where $\gamma(M, \cdot)\Gamma(M)^{-1}$ is the CDF of the Chi-square distribution with $2M$ degrees of freedom, and
\begin{align}
l_g'\triangleq\sigma_g^{-2}&\left(\sigma_w^2+\frac{1}{\alpha\tau^{\star}}\right)\left(\sigma_g^2+\sigma_w^2+\frac{1}{\alpha\tau^{\star}}\right)\nonumber\\
&\left(l_g-M\log\frac{\sigma_w^2+\frac{1}{\alpha}\tau^{\star}}{\sigma_g^2+\sigma_w^2+\frac{1}{\alpha}\tau^{\star}}\right)\nonumber,
\end{align}
with arbitrary threshold $l_g$. Further, in the asymptotic massive MIMO regime, as $M$ go to infinity, we have $\lim\nolimits_{M\to \infty} P_{F}(M) = P_{M}(M) = 0$.
As for the CE performance, according to (\ref{chan_est}) and $\mathcal E = \tau G^{-1}\mathbf I$, after performing the AMP algorithm, the CE error for an active user in the group $g$ is
\begin{equation}
\mathbb E\left[\left(\boldsymbol s_g-\hat{\boldsymbol h}_g\right)\left(\boldsymbol s_g-\hat{\boldsymbol h}_g\right)^H\right]= \epsilon \mathbf I,
\end{equation}
where $\epsilon = \left(\left(\sigma_w^2+\frac{1}{\alpha}\tau^{\star}\right)^{-1}+\sigma_g^{-2}\right)^{-1}$.
Therefore, in the isotropic Rayleigh channel, perfect AUD and the Bayes-optimal channel estimation can be achieved as long as the number of antennas is large enough. We also note the above results validate and extend the analytical results in \cite{LiuMassive12018}. Besides, studying the free entropy function (\ref{FE_tau}) provides a phase transition diagram and an optimality analysis of the isotropic channel case with a finite number of BS antennas, as shown in the following section.
\subsection{Verification in the Isotropic Channel Scenario}
\label{Num_1}
In this section, we provide numerical examples to verify our results based on the replica method in the isotropic channel scenario. Specifically, we consider that each user accesses the channel with a probability $\rho = 0.1$ and we assume the served user devices have been divided into $5$ user groups and the distance $d_g$ between each user group is randomly distributed in the regime $[0.1, 1]$km. The path loss model of the wireless channel for each user group $g$ is given as ${\rm PL}_g = -128.1-36.7\log_{10}(d_g)$ in dB. Since, we assume no power adaption, we denote the transmit power for each user as $P_t$, and we have $\sigma_g^2 = P_t\times{\rm PL}_g$. We assume the bandwidth of the wireless channel are $1$MHz and the power spectral density of the AWGN at the BS is $-169$dBm/Hz. The
\subsubsection{Phase Transition of the Free Entropy Function}
First, we examine the phase transition of the free entropy function and demonstrating the optimal recovery MSE and the AMP-achievable MSE under our scenario, showing that for different settings of the system parameters, i.e., $P_t$ and $\alpha$, the MSE performance can be divided into some performance regions.
Specifically, we can notice that in the Fig. \ref{33M2}, there exists phase transitions and the MSE performance can be divided into several regions. In the first region, with $\alpha = 0.525$, the free entropy function has only one local maximum, indicating the optimal MSE performance, and the AMP-achievable MSE coincide. In the second region, with $\alpha = 0.550$, the second local maximum point with a lower function value than the first local maximum of the free entropy function appears, indicating the unachievable MSE point. In the third region, with $\alpha = 0.575$, the smaller local maximum point leads to a larger value of the free entropy function, which indicates the MMSE under such a choice of parameters. However, a larger local maximum point blocks the MSE performance of AMP, since it always converges to the local maximum point associated with the largest MSE. Hence, in this region, there exists a gap between the AMP-achievable MSE and the MMSE. Finally, in the region with $\alpha = 0.600\sim0.675$, one local maximum point disappears, and the AMP-achievable MSE and the MMSE coincide again. We can intuitively infer that there exists a hard threshold between the latter two regions, i.e, when the pilot length exceeds the threshold, the local maximum point associated with the largest MSE $\tau$ discards, and the AMP algorithm converges to the local maximum associated with a lower MSE, leading that there is a phase transition of the MSE performance of AMP.
We remark that in order to reduce the gap between the AMP-achievable MSE and the MMSE in the case that the AMP algorithm is sub-optimal, one can consider the idea of \emph{seeding matrix} introduced in \cite{2012Probabilistic} in the design of pilot matrix $\boldsymbol F$, which is beyond the scope of this manuscript.
In addition, Figs. \ref{18M2}-\ref{33M2} demonstrate the free entropy as a function of MSE with $M = 2$ BS antennas, and Figs. \ref{18M2}-\ref{33M2} demonstrate the free entropy function with $P_t = 33$dBm transmit power. We can notice that the region where AMP is suboptimal persists, but becomes smaller and eventually disappears as the number of BS antennas increases and transmit power decreases. Interestingly, we find that in the isotropic channel scenario, $M = 8$ antennas is enough to avoid phase transition in the case with the transmit power is lower than $P_t = 33$dBm.
\subsubsection{Performance Analysis of AUD}\label{Per_aud1}
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{fig/det_pre.eps}
\caption{Detection performance prediction of AMP with different settings of pilot length.}
\label{dtpre1}
\end{figure}
We then provide numerical results to analyze the missed detection and false alarm probabilities in the isotropic channel scenario. The number of user devices is set as $N = 10000$. Fig. \ref{dtpre1} demonstrates the prediction of AMP-based AUD by equation (\ref{pf1}) and (\ref{pm1}) versus pilot length, with $\tau^{\star}$ predicted by the free entropy function. The simulation curves are depicted by the empirical AMP algorithm \cite{ChenSparse2018, RepMMVG2018}. Different from the standard AMP algorithm, the empirical AMP algorithm adopts an empirical state evolution function in the iterations, substituting the standard state evolution function in the AMP algorithm, in order to avoid the complex expectation calculations. The numerical results shows that our predicted performances are consistent with that of the empirical AMP algorithm for most of the settings. The empirical AMP and the proposed theoretical prediction have the similar tendency and there exists phase transition phenomenons in both of them. We notice that when phase transition occurs, the required length of pilot sequences for our prediction is slightly shorter than that for the empirical algorithm. Since the prediction error is about $0.04$ over the length of pilot sequence, we think the theoretical results can provide a prediction for the detection performance in the practice. We also note that our predicted performance corresponding to the AMP achievable MSE provides a bound of the performance of the empirical AMP algorithm. The performance loss of the empirical AMP algorithm is because that the state evolution function that fulfills the fixed point condition (\ref{SE}) of the free entropy function is substituted by an empirical one.
\setcounter{TempEqCnt}{\value{equation}}
\setcounter{equation}{20}
\begin{figure*}[tbp]
\begin{align}
&\Phi_g(\mathcal E_g) = -\alpha{\rm Tr}\left(\left(\boldsymbol\Delta+\frac{\mathcal E_g}{\alpha}\right)^{-1}\left(\frac{1}{\alpha}\rho\mathcal C_g+\boldsymbol\Delta\right)\right)-\alpha\log\left|\boldsymbol\Delta+\frac{\mathcal E_g}{\alpha}\right|+\int{\rm d}\boldsymbol s_g[(1-\rho)\delta(\boldsymbol s_g)+\rho Q_g(\boldsymbol s_g)]\int{\rm D}\boldsymbol z\nonumber\\
&\log\left(\int{\rm d}\boldsymbol x_g[(1-\rho)\delta(\boldsymbol x_g)+\rho Q_g(\boldsymbol x_g)]\right.\exp\left(-\boldsymbol x_g^H(\boldsymbol\Delta+\frac{\mathcal E_g}{\alpha})^{-1}\boldsymbol x_g+2\mathfrak R\left(\boldsymbol x_g^H[(\boldsymbol\Delta+\frac{\mathcal E_g}{\alpha})^{-1}\boldsymbol s_g+(\boldsymbol\Delta+\frac{\mathcal E_g}{\alpha})^{-\frac{1}{2}}\boldsymbol z]\right)\right).\label{d_free}
\end{align}
\begin{align}
{\Phi}_g(\boldsymbol\Xi_g) = &-\alpha\sum\limits_{m = 1}^{r_g}\left(\frac{\sigma_w^2}{\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2}+\log\left(\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2\right)\right)+\sum\limits_{m = 1}^{r_g}\frac{(1-\rho)\lambda_{g,m}}{\lambda_{g,m}+\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}}\nonumber\\
&+\int{\rm D}\boldsymbol z\rho\log\left[(1-\rho)\prod_{m = 1}^{r_g}\exp\left\{-\frac{|z_m|^2\lambda_{g,m}}{\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}}\right\}+\rho\prod_{m = 1}^{r_g}\frac{\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2}{\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2+\lambda_{g,m}}\right]\nonumber\\
&+\int{\rm D}\boldsymbol z(1-\rho)\log\left[(1-\rho)\prod_{m = 1}^{r_g}\exp\left\{-\frac{|z_m|^2\lambda_{g,m}}{\lambda_{g,m}+\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}}\right\}+\rho\prod_{m = 1}^{r_g}\left(\frac{\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2}{\frac{1}{\alpha}\xi_{g,m}+\sigma_w^2+\lambda_{g,m}^2}\right)\right].\label{fe_si}
\end{align}
\hrulefill
\end{figure*}
\setcounter{equation}{\value{TempEqCnt}}
\subsubsection{Performance Analysis of CE}
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{fig/ce_pre.eps}
\caption{Channel estimation performance prediction of AMP with different settings of pilot length.}
\label{cepre1}
\end{figure}
The results of the CE error prediction is demonstrated in Fig. \ref{cepre1}. We can observe that the CE error prediction performance is consistent compared with the detection error prediction. Our numerical results show that when the phase transition occurs, the performances of the joint AUD and CE are highly improved. The predicted performance corresponding to the AMP achievable MSE reveals the minimum length of pilot sequences to make the AMP algorithm accomplish the phase transition. We think this is a guide to how long pilot sequences are needed when designing the system. For the prediction error that may occur when using the empirical AMP algorithm in the practice, we can appropriately increase the number of pilots to ensure that the phase transition occurs.
\section{Spatially Correlated Channel}\label{spc}
In this section, we concern another typical channel scenario in the wireless communications literature, where the scattering is localized around the user devices and the BS is elevated and thus has no scatterers in its near filed \cite{MassiveMIMON2017}, resulting in the spatially correlated channel. Assuming no line-of-sight propagation, the channel of each user $k$ in the group $g$ is distributed as $Q_g(\mathbf h_{g}) = \mathcal{CN}(\boldsymbol 0, \mathcal C_g)$, where $\mathcal C_g = \boldsymbol U_g\boldsymbol\Lambda_g\boldsymbol U_g^H$ with a rank $r_g\ll M$. Considering the number of BS antennas is sufficiently large, the $\boldsymbol\Lambda_g$ is approximated diagonal \cite{CEMoG2015, CoCE2013}.
We further suppose that different user groups are sufficiently well separated in the angle of arrival (AoA) domain and the angular spread (AS) of each group is sufficiently small. Accordingly, we assume that channels coefficients in all the user groups are in different mutually orthogonal subspaces such that $\boldsymbol U_g^H\boldsymbol U_j = \boldsymbol 0$, for $j\neq g$. We note that although directly achieving the mutually orthogonal subspaces is too restrictive, some user scheduling strategies can be adopted to guarantee the user groups in mutually orthogonal subspaces are served simultaneously \cite{JSDM2013}. In the following of this section, we will see that the above spatially correlated channel assumption reduces the expression of the free entropy function (\ref{general_FE}) and provides some novel propositions for joint AUD and CE.
\subsection{Free Entropy Function}
Before deriving the free entropy function in the spatially correlated channel scenario, we recall that the matrix $\mathcal E$ is defined as $\mathcal EG= \sum_g \mathcal E_g$, where $\mathcal E_g $ is the corresponding recovery MSE matrix of each user group $g$. As a consequence, we have the following lemma.
\begin{lemma}
Under the spatially correlated channel assumption, i.e., $\boldsymbol U_g^H\boldsymbol U_j = \boldsymbol 0$, for $j\neq g$, all the stationary points of the free entropy function (\ref{general_FE}) fulfill the equation: $\mathcal E^{(\text{s})} =
\sum_g\boldsymbol U_g\boldsymbol\Xi_g^{(\text{s})}\boldsymbol U_g^H$ for some symmetric positive semidefinite matrix $\boldsymbol\Xi_g^{(\text{s})}$.
\end{lemma}
\begin{proof}
Please see the appendix \ref{L_1}.
\end{proof}
Since the statistical characteristics of our considered scenario are all reflected in the stationary points of the free entropy function, we restrict the form of the matrix $\mathcal E$ as $\mathcal E =
\sum_g\boldsymbol U_g\boldsymbol\Xi_g\boldsymbol U_g^H$. As a consequence, the following theorem holds.
\begin{theorem}
Under the spatially correlated channel assumption, the free entropy function (\ref{general_FE}) can be decoupled into a summation of $G$ independent free entropy functions, i.e., $\Phi(\mathcal E) = \sum_{g = 0}^G\Phi_g(\mathcal E_g)$, where $\Phi_0(\mathcal E_0)$ is a constant, and the specific expression of each $\Phi_g(\mathcal E_g)$ is formulated in the equation (\ref{d_free}).
\end{theorem}
\begin{proof}
Please see the appendix \ref{T_3}.
\end{proof}
Accordingly, the recovery MSE of each user group can be evaluated separately via the equation (\ref{d_free}), indicating that the transmitted signals from users outside one group will not affect the reconstruction within such a group. In the following, the performance analysis of joint AUD and CE in the spatially correlated channel scenario will be provided based on the \emph{Theorem 3}.
\begin{figure*}
\begin{center}
\subfigure[$\alpha = 0.11$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.3in]{fig/pt18al0.11.eps}
\label{al011}
\end{minipage
}
\subfigure[$\alpha = 0.13$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.3in]{fig/pt18al0.13.eps}
\label{al013}
\end{minipage}
}
\subfigure[$\alpha = 0.15$.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.3in]{fig/pt18al0.15.eps}
\label{al015}
\end{minipage}
}
\caption{Free entropy as a function of MSE with $P_t = 18$dBm transmit power under different settings of $\alpha$ in spatially correlated channel scenario. (The ``red cross'' denotes the MMSE point, the ``red circle'' denotes the predicted AMP-achievable MSE point, the ``red square'' denotes the unachievable local maxima MSE point, and the ``blue triangle'' denotes the simulated AMP MSE point.)}
\end{center}
\end{figure*}
\subsection{Prediction of AMP-Based joint AUD and CE}
According to the \emph{Lemma 1}, we define $\boldsymbol\Xi_g^{\star}, \forall g$ as the matrixs that associated with the fixed point $\mathcal E^{\star}$ of AMP state evolution. We further restrict the structure of the matrix $\boldsymbol \Xi_g^{\star}$ to be diagonal\footnote{Restricting the matrix $\boldsymbol\Xi_g^{\star}$ into diagonal is reasonable. As proved in \cite{RepMMVG2018}, the state of state evolution function always maintains the same structure with iterations. The initial state of the state evolution in the AMP framework is always set to be $\mathcal E_g^{(0)} = \mathbb E(\boldsymbol s_g\boldsymbol s_g^H)$, and in our situation, we have $\mathcal E_g^{(0)} = \rho \mathcal C_g = \rho\boldsymbol U_g\boldsymbol \Lambda_g\boldsymbol U_g^H$. Hence, $\boldsymbol\Xi_g^{\star}$ will also be diagonal since the matrix $\boldsymbol\Lambda_g$ is approximately diagonal in our scenario with a large number of antennas.}, with the $m$th diagonal element denoted as $\xi_{g,m}^{\star}$. By combining \emph{Theorem 3} with \emph{Lemma 1}, the matrix $\boldsymbol\Xi_g^{\star}$ can be obtained by seeking the local maximum point of the equation (\ref{fe_si}), where the quantities $\xi_{g,m}$, $\lambda_{g,m}$ are defined as the $m$th element of the diagonal of $\boldsymbol\Xi_g$ and $\boldsymbol\Lambda_g$.
Then, we consider the performance analysis of joint AUD and CE. For AUD, defining $\underline{\boldsymbol s_g}\triangleq \boldsymbol U_g^H\hat{\boldsymbol s}_g$ and together with (\ref{llr}), the detection sufficient statistic in such a scenario can be formulated as
\setcounter{equation}{22}
\begin{equation}
\text{T}(\underline{\boldsymbol s_g}) = \sum\limits_{m = 1}^{r_g}\frac{\lambda_{g,m}|\underline{s_{g,m}}|^2}{\left(\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}^{\star}\right)\left(\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}^{\star}+\lambda_{g,m}\right)}.\label{det2}
\end{equation}
and the involved conditional probabilities of $\underline{\boldsymbol s_g}$ are given as
\begin{align}
p(\underline{\boldsymbol s_g}|a_g = 1) &= \mathcal{CN}(\underline{\boldsymbol s_g}; \boldsymbol 0, \boldsymbol\Lambda_g+\boldsymbol\Delta+\frac{1}{\alpha}\boldsymbol\Xi_g^{\star}),\nonumber\\
p(\underline{\boldsymbol s_g}|a_g = 0) &= \mathcal{CN}(\underline{\boldsymbol s_g}; \boldsymbol 0, \boldsymbol\Delta+\frac{1}{\alpha}\boldsymbol\Xi_g^{\star}).\label{CP}
\end{align}
As a result, we can then derive the specific functional forms of the detection performance metrics in the following proposition.
\begin{proposition}
The missed detection probability with the definition $P_D = {\rm Pr}\{\text{T}(\underline{\boldsymbol s_g})>l_g|a_g = 1\}$ and the false alarm probability with the definition $P_F = {\rm Pr}\{\text{T}(\underline{\boldsymbol s_g})>l_g|a_g = 0\}$ based on the conditional probabilities (\ref{CP}) and the detection sufficient statistic (\ref{det2}) are given as
\begin{align}
P_D = \sum\limits_{m = 1}^{r_g}\prod_{j\neq m}\frac{\omega_{g,j}}{\omega_{g,j}-\omega_{g,m}}\exp\{-\omega_{g,m}l_g\},\nonumber\\
P_F = \sum\limits_{m = 1}^{r_g}\prod_{j\neq m}\frac{\tilde{\omega}_{g,j}}{{\tilde\omega}_{g,j}-{\tilde\omega}_{g,m}}\exp\{-{\tilde\omega}_{g,m}l_g\},\label{pro2}
\end{align}
where $$\omega_{g,m}\triangleq (\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}^{\star})\lambda_{g,m}^{-1},$$ and $$\tilde{\omega}_{g,m}\triangleq (\sigma_w^2+\frac{1}{\alpha}\xi_{g,m}^{\star}+\lambda_{g,m})\lambda_{g,m}^{-1}.$$
\end{proposition}
\begin{proof}
Please see the appendix \ref{dp}.
\end{proof}
As for the analysis for CE, according to (\ref{cha_err_ma}), the CE error of the user group $g$ associated with the $m$th eigenvalue $\lambda_{g,m}$ can be formulated as $(\sigma_w^2+\lambda_{g,m} +\frac{1}{\alpha}\xi_{g,m}^{\star})^{-1}$.
After that, we can see that the performance of joint AUD and CE can be predicted by the local maximum point of the free entropy function (\ref{fe_si}). The verification of such a result, a phase transition diagram and an optimality analysis of joint AUD and CE in the spatially correlated channel scenario will be shown in the following Sec. \ref{ve_sc}.
\subsection{Discussion}
Before proceeding, we provide some discussions about our theoretical results. Multiplying $\boldsymbol U_g^{*}$ for each group by the right of the received signal $\boldsymbol Y$ in the original model (\ref{channel}), we can obtain $G$ a sub-model expressed as
\begin{equation}
\underline{\boldsymbol Y}_g = \underline{\boldsymbol F}_g\underline{\boldsymbol X}_g+\underline{\boldsymbol W}_g,\label{red_mo}
\end{equation}
where $\underline{\boldsymbol Y}_g\triangleq \boldsymbol Y\boldsymbol U_g^{*}$, $\underline{\boldsymbol X}_g\triangleq \boldsymbol X\boldsymbol U_g^{*}$, $\underline{\boldsymbol W}_g\triangleq \boldsymbol W\boldsymbol U_g^{*}$ and $\underline{\boldsymbol F}_g$ is the corresponding pilot matrix for the user device for the group $g$. We note that the free entropy function of (\ref{red_mo}) reflecting its statistical performances corresponds to (\ref{fe_si}), so that separately using model (\ref{red_mo}) for each group is equivalent to jointly processing all the groups with (\ref{channel}), if the mutually orthogonal subspaces condition holds. Such a per-group processing (PGP) idea has been successfully adopted in the case of downlink transmission \cite{JSDM2013} and our theoretical results provide the theoretical foundations for the PGP strategy in the uplink joint AUD and CE problem.
Complementally, when the mutually orthogonal condition does not perfectly hold, by carefully designing the processing matrix for each group, the PGP strategy can also be applied by permitting the affordable inter-group inference. According to the approach in \cite{JSDM2013}, when the BS equipped with a uniform linear array, the inter-group inference can be absorbed into the additive noise matrix with independent elements. Hence, by slightly modifying the free entropy function (\ref{fe_si}), our framework can also be adopted to evaluate performance of joint AUD and CE, where the mutually orthogonal condition does not perfectly hold.
\subsection{Verification in the Spatially Correlated Channel Scenario}\label{ve_sc}
In this section, we provide numerical results to verify the analysis based on the replica method in spatially correlated channel scenario. We consider the channel coefficients of all user groups are in orthogonal subspaces. Without loss of generality, we investigate the performance analysis results for one certain user group. We consider the typical spatially correlated channel model in \cite{MassiveMIMON2017}. For visualization, we assume the number of BS antennas is $M = 64$. The path loss model of the wireless channel, the distribution of the distance between the BS station and the user devices, the bandwidth the power spectral density and the total number of user devices are considered same with that in the settings of the isotropic channel case.
\subsubsection{Phase Transition of the Free Entropy Function}
We first consider the center angle of user angle domain is set as $45^{\circ}$ distributed as a normal distribution with a $1^{\circ}$ AS, where most of the power of channel coefficients is concentrated on a two-dimensional subspace.
Figs. \ref{al011}-\ref{al015} shows the different MSE performance regions of the free entropy function (\ref{fe_si}) in different settings. We can notice that the performance regions in the spatially correlated channel case is similar with that in the isotropic channel case. Specifically, Fig. \ref{al013} demonstrates the performance gap between the AMP-achievable MSE and the MMSE. The phase transitions appear in both our prediction and the simulated AMP algorithm. Differently from the isotropic channel scenario, it is observed that although there exists a large number of antennas in the BS, the phase transition phenomenon still exists. Fig. 6 also shows that our analytical MSE results predicted in the large system limit and AMP recovery MSE in practical settings, i.e., $T = 220, 260, 300$ and $K = 2000$ are consistent. Such results are evidences that our analytical results are valid when the length of pilot sequences and the number of user devices are large but finite.
\begin{figure*}
\begin{center}
\subfigure[$P_t = 13$dBm.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.2in]{fig/pra_Pt13.eps}
\end{minipage
}
\subfigure[$P_t = 23$dBm.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.2in]{fig/pra_Pt23.eps}
\end{minipage}
}
\subfigure[$P_t = 33$dBm.]{
\begin{minipage}[t]{0.31\linewidth}
\centering
\includegraphics[width=2.2in]{fig/pra_Pt33.eps}
\end{minipage}
}
\caption{NMSE performance of AMP versus pilot length in different settings of transmit power and AS.}
\label{pra}
\end{center}
\end{figure*}
Then, we demonstrate the case where the AS is not very small, which is more in line with the practical scenario. The phase transition phenomenons in all the considering settings can be observed in Fig. \ref{pra}. With the transmit power increasing, the phase transition phenomenon is becoming more and more obvious. In addition, with the AS increases, the number of the efficient dimensions in the subspace of the user channel becomes larger and the total energy will be allocated to these effective dimensions, leading that the phase transition phenomenon is weakened. Such an observation matches our previous results that the spatially correlated channel is more likely to promote a phase transition compared with the isotropic channel. Another observation is that smaller AS is beneficial to NMSE performance when pilot resources are abundant, while the pilot resources are scarce, the opposite is true.
\subsubsection{Performance Analysis of AUD}
\begin{figure}
\centering
\includegraphics[width=3.5in]{fig/det_pre2.eps}
\caption{Detection performance prediction.}
\label{detpre2}
\end{figure}
Fig. \ref{detpre2} depicts the prediction of AMP-based AUD as a function of pilot length with $1^{\circ}$ AS. It is observed that the phase transition of the detection error probabilities appears in both $P_t = 13$dBm and $P_t = 18$dBm settings, which is different from the results in the isotropic channel case. This is because the energy of a large number of antennas is concentrated in a small amount of subspace dimensions, increasing the signal-to-noise ratio in considered dimensions.
In summary, we note that the spatially correlated channel promotes the appearance of phase transition compared with the isotropic channel. This is because the energy of a large number of antennas is concentrated in a small amount of subspace dimensions. This reduces the effective dimensions of the BS antennas, while increasing the power on each dimension. Despite that, the pilot length required to step over the phase transition is highly reduced in the spatially correlated channel scenario, i.e., about $\alpha = 0.14$ for the spatially correlated channel case and about $\alpha = 0.625$ for the isotropic channel case with $5$ user groups.
\section{Conclusion}
In this paper, we have provided an analysis of joint AUD and CE problem under massive connectivity based on the replica method. Particularly, we have established a novel theoretical framework under a general channel model that reduces to multiple typical MIMO channel models. Based on the general framework, we have analyzed two typical scenarios in massive connectivity, i.e., the isotropic channel case and the spatially correlated channel case. We have provided the analysis of the Bayes-optimility, phase transition, and the predictions of the performance of joint AUD and CE in both the two cases. In addition, we have shown that the spatially correlated channel is more likely to promote a phase transition compared with the isotropic channel, due to the small number of subspace dimensions and corresponding antenna gains. However, thanks to the spatially correlated channel, where all the user group can be perfectly partitioned, the pilot length required to step over the phase transition can be significantly reduced.
Some future directions of research are also implied by this paper. 1) Our theoretical framework was established in the Bayes-optimal condition, if the condition is not met, what changes will exist in the analytical results. 2) Exploiting the potential user grouping information may bring some advantages, and how to design an algorithm to achieve joint user grouping, AUD and CE. 3) How to design the processing matrix in the PGP-based strategy for joint AUD and CE.
|
2,869,038,154,871 | arxiv | \section*{Introduction}
\label{sec:introduction}
A \emph{grain boundary} (GB) is the interface between two grains or crystals
in a polycrystalline material, and has an
atomic configuration significantly different
from that of a single crystal.
Since this
results in
peculiar mechanical and electrical properties of materials, one of the most important issues in materials research is determining the atomic configuration of an interface.
Experimental observations, such as the atomic-resolution transmission electron microscope (TEM) observations\cite{Haider1998} and theoretical simulations, such as first-principles calculations based on the density functional theory and static lattice calculations with empirical potentials, have been extensively performed to investigate interface structures\cite{Alfthan2010,Ikuhara2011}.
The macroscopic GB geometry is defined using five degrees of freedom (DOF) that fully describe the crystallographic orientation of one grain relative to the other (3 DOF) and the orientation of the boundary relative to one of the grains, i.e., the GB plane (2 DOF).
Besides these five macroscopic DOF, three other microscopic parameters exist for relative rigid body translation (RBT) of one grain to the other parallel and perpendicular to the GB plane.
It has been indicated that the most important parameter in determining the GB energy is the excess boundary volume \cite{Wolf1989}, which is related to the RBT perpendicular to the boundary.
Closely packed boundaries that have a local atomic density similar to that in the bulk will have low energies.
Thus, it is important to determine both the RBT and the number of atomic columns at the boundary \cite{Muller1999}.
These microscopic parameters are established based on energetic considerations and cannot be selected arbitrarily, and atomistic simulations are widely used to obtain stable GB structures.
To understand the whole nature of GBs, the stable interface structures for each rotation angle and rotation axis need to be determined.
A straightforward manner is optimizing all possible candidates of GB models, thereby determining the lowest-energy configuration.
However, determining the stable structures of GBs needs large-space searching due to the huge geometric DOF.
Although some databases of GB structures are available \cite{Olmsted2009,Erwin2012,Banadaki2016}, they contain only a limited number of systems because of considerable computational costs of simulations.
Therefore, developing efficient approaches to determining the interface structure without searching for all possible candidates is strongly demanded.
In recent years, materials-informatics techniques based on \emph{machine learning} have been introduced as an efficient way for data-driven material discovery and analysis \cite{Rodgers2006}.
For the structure search, which is our main focus in this study, a machine learning technique called \emph{Bayesian optimization} \cite{Shahriari16} has proven to be useful mainly in the application to determine stable bulk structures \cite{Seko2015,Ueno2016}.
Bayesian optimization iteratively \emph{samples} a candidate structure predicted by a probabilistic model that is statistically constructed by using already sampled structures.
Bayesian-model-based methods are quite general, and thus, they are apt for a variety of material-discovery problems, such as identifying the low-energy region in a potential energy surface \cite{Toyoura2016}.
For the interface structure, some studies \cite{Kiyohara2016,Kikuchi2017} proposed to apply Bayesian optimization to the GB-structure search, and its efficiency was confirmed, for example, by using the fcc-Cu $\Sigma5$ [001](210) CSL GB.
However, their search method is a standard Bayesian optimization method, i.e., same as the method in the case of bulk structures.
To our knowledge, a search methodology specifically for GBs has not been introduced so far.
As a general problem setting in the GB-structure search, we consider the exploration of a variety of rotation angles for a fixed rotation axis.
Suppose that we have $T$ different angles to search, and candidate structures are created by RBTs for each of them.
A naive approach to this problem is to apply some search method, such as Bayesian optimization \cite{Kiyohara2016,Kikuchi2017}, $T$ times separately.
However, this approach is not efficient because it ignores the following two important characteristics of the GB structure:
\begin{enumerate}
\item Energy-surface similarity:
%
The energy surfaces at different angles are often quite similar.
%
This similarity is explained by the \emph{structural unit model} \cite{Sutton1983a, Sutton1983b, Sutton1983c, Sutton1990}, which has been widely accepted to describe GB structures in many materials.
%
This model suggests that different GBs can contain common structural units, and that they share similar local atomic environments.
Although structurally similar GBs can produce similar energy surfaces, the naive search does not utilize this similarity and restarts the structure search from scratch for each angle.
\item Cost imbalance:
%
%
GB supercells usually have various sizes because of the
variations in
the $\Sigma$ value, which is the inverse of the density of lattice sites.
%
This means that the computational cost for large $\Sigma$ GBs dramatically increases because the number of atoms in a supercell increases.
%
Thus, the structure search for large $\Sigma$ GBs is significantly more time-consuming than that for small $\Sigma$ GBs.
%
For example, the computational time scale is $O(M) \sim O(M^{3})$ for $M$ number of atoms in the supercells, depending on the computational scheme.
\end{enumerate}
\figurename~\ref{fig:example-energy-surface} shows an example of this situation.
The figure contains (a) an illustration of RBT and an atom removal from the boundary, (b) calculated stable GB energies, and (c) energy surfaces created by two-dimensional RBTs for the rotation angles $141^\circ$ (top), $134^\circ$ (bottom left) and $145^\circ$ (bottom right).
The entire landscape of the surfaces in \figurename~\ref{fig:example-energy-surface} (c) are similar, while their computational costs are significantly
different since the biggest supercell ($\Sigma89$) contains almost $10$ times larger number of atoms than the smallest supercell ($\Sigma9$).
In this paper, we propose a machine-learning-based stable structure search method that is particularly efficient for the GB-structure search.
Our proposed method, called cost-sensitive multi-task Bayesian optimization (CMB), takes the above two characteristics of GB structures into account.
For energy-surface similarity, we introduce a machine-learning concept called \emph{transfer learning} \cite{Pan2010}.
The basic idea of transfer learning is to transfer knowledge among different (but related) tasks to improve the efficiency of machine-learning methods.
In this study, a GB-structure search for a fixed angle is considered to be a ``task''.
When a set of tasks are similar to each other, information accumulated for one specific task can be useful for other tasks.
In our structure-search problem, a sampled GB model for an angle provides information for other angles because of the energy-surface similarity.
For the cost imbalance issue, we introduce a \emph{cost-sensitive} search.
Our method incorporates cost information into the sampling decision, which means that we evaluate each candidate based on both the possibility of an energy improvement and the cost of sampling.
By combining the cost-sensitive search with transfer learning, CMB accumulates information by sampling low cost surfaces in the initial stage of the search, and can identify the stable structures in high cost surfaces with a small number of sampling steps by using the transferred surface information.
\figurename~\ref{fig:shematic-illust} shows a schematic illustration of the entire procedure of CMB, which indicates that knowledge transfer, particularly from the low cost surfaces to the high cost surfaces, is beneficial for the structure search.
As a case study, we evaluate the cost-effectiveness of our method based on fcc-Al [110] tilt GBs:
our proposed method determines
stable structures with $5$ mJ/m$^2$ average accuracy with only
about $0.2$ \% of the computational cost of the exhaustive search.
\clearpage
\section*{Methods}
\label{sec:method}
\subsection*{Problem Setting}
\label{subsec:prob-setting}
GB energy is defined against the total energy of the bulk crystal as
\begin{align}
E_{\rm GB} = \frac{E_{\rm GB}^{\rm tot} - E_{\rm bulk}}{ 2 S },
\end{align}
where $E_{\rm GB}^{\rm tot}$ is the total energy of the GB supercell, $E_{\rm bulk}$ is the bulk energy with the same number of atoms as the GB supercell, and $S$ is the cross-section area of the GB model in the supercell.
In the denominator, the cross-section area $S$ is multiplied by $2$ since the supercell contains two GB planes as shown in \figurename~\ref{fig:example-energy-surface} (a) which is an example of a $\Sigma$9 GB model.
Note that the GB energy for each GB model is calculated through \emph{atomic relaxation}.
Suppose that we have $t = 1, \ldots, T$ different rotation angles $\theta_t$, for each of which we have $N_t$ candidate-GB models created by \emph{rigid body translations} (RBTs) with or without atom removal.
\figurename~\ref{fig:example-energy-surface} (a) also illustrates RBTs by which $N_t$ GB models are created.
The total number of the GB models is denoted as $N = \sum_{t = 1}^T N_t$.
We would like to search the stable GB structures with respect to \emph{all} of the given rotation angles.
A set of GB energies for all $N$ GB models is represented as a vector $\bE = (E_{\rm GB}^{(1)}, \ldots, E_{\rm GB}^{(N)})^\top$, where $E_{\rm GB}^{(i)}$ is the GB energy of the $i$-th GB model.
A stable structure search for some fixed angles can be mathematically formulated as a problem to find low energy structures with a smaller number of ``model sampling steps'' from candidates.
The number of candidate structures is often too large to exhaustively compute their energies, and we usually do not know the exact energy surface as a function in the search space.
This problem setting is thus called the \emph{black-box optimization problem} in the literature.
We call a stable structure search for each angle a ``task''.
Let $\tau_i \in \{ 1, \ldots, T \}$ be the task index that the $i$-th GB model is included, and $C_t$ be the cost to compute the GB energy in the $t$-th task.
We assume that the cost can be estimated based on the number of atoms $M$ in the supercell.
For example, \emph{embedded atom method} (EAM) \cite{Mishin1999} with the cutoff radius needs $O(M)$ computations.
Then, we can set $C_{t}$ as $M$.
Instead of counting the number of model samplings, we are interested in the sum of the cost $C_t$ of
the search process, for a
practical evaluation of the search efficiency.
Assuming that a set $\cS \subseteq \{ 1, \ldots, N \}$ is an index set of sampled GB models, the total cost of sampling is written as
\begin{align}
C = \sum_{i \in \cS} C_{\tau_i}.
\label{eq:total-cost}
\end{align}
\subsection*{Knowledge-Transfer based Cost-Effective Search for GB Structures}
Our method is based on \emph{Bayesian optimization} which is a machine-learning-based method for solving general black-box optimization.
The basic idea is to estimate a stable structure iteratively, based on a probabilistic model that is statistically constructed by using already sampled structures.
\emph{Gaussian process regression} (GP) \cite{Rasmussen2005} is a probabilistic model usually employed in Bayesian optimization.
GP represents \emph{uncertainty} of unobserved energies by using a Gaussian random variable.
Let $\bx_i \in \RR^p$ be a $p$ dimensional descriptor vector for the $i$-th GB model, and $\bE_\cS$ be a energy vector for a set of sampled GB models.
The prediction of the $i$-th GB model is given by
\begin{align}
f_i \mid \bE_{\cS} \sim \cN(\mu(\bx_i),\sigma(\bx_i)),
\label{eq:pred-dist-GP}
\end{align}
where $f_i \mid \bE_{\cS}$ is a random variable $f_i$ after observing $\bE_\cS$, and $\cN(\mu(\bx_i),\sigma(\bx_i))$ is a Gaussian distribution having $\mu(\bx_i)$ and $\sigma(\bx_i)$ as the mean and the standard deviation, respectively.
Bayesian optimization iteratively predicts the stable structure based on $\mu(\bx_i)$ and $\sigma(\bx_i)$.
See Supplementary Information 1 for details regarding the Bayesian optimization.
Although energy surfaces for different angles are often quite similar, simple Bayesian optimization cannot utilize such similarity.
In machine learning, it has been known that, for solving a set of similar tasks, transferring knowledge across the tasks can be effective.
This idea is called \emph{transfer learning} \cite{Pan2010}.
In particular, we introduce a concept called \emph{multi-task learning}, in which knowledge is transferred among multiple tasks, to accelerate convergence of multiple structure-search tasks of GB.
In addition to the structure descriptor $\bx$, we introduce a descriptor which represents a task.
Let $\bz_t \in \RR^q$ be a descriptor of the $t$-th task, called a task-specific descriptor, through which the similarity among tasks is measured.
For example, a rotation angle can be a task-specific descriptor because surfaces for similar angles are often similar.
Hereafter, we refer to a descriptor $\bx$ as a structure-specific descriptor.
Given these two types of descriptors, we estimate the energy surface in the joint space of $\bx$ and $\bz$:
\begin{align}
f_i \mid \bE_{\cS} \sim \cN(\mu^{\rm (MT)}(\bx_i, \bz_{\tau_i}), \sigma^{\rm (MT)}(\bx_i, \bz_{\tau_i})).
\end{align}
Here, the mean $\mu^{\rm (MT)}$ and standard deviation $\sigma^{\rm (MT)}$ are functions of both of the structure-specific descriptor $\bx$ and the task-specific descriptor $\bz$.
This model is called \emph{multi-task Gaussian process regression} (MGP)\cite{Bonilla2008}, and \figurename~\ref{fig:mtgp} shows a schematic illustration.
In the figure,
information regarding the GP model is
transferred among tasks through ``task axis'', and it improves the accuracy of the surface approximation.
For a task-specific descriptor, we employed the rotation angle and radial distribution function in the later case study (See section ``GB Model and Descriptor'' for details).
Supplementary Information 2 provides for further mathematical details on MGP.
We propose combining MGP with Bayesian optimization, meaning that we determine the next structure to be sampled based on the probabilistic estimation of MGP.
Since knowledge transfer improves accuracy of GP (particularly for tasks in which there exists only a small number of sampled GB models), the efficiency of the search is also improved as illustrated in \figurename~\ref{fig:mtgp}.
After estimating the energy surface, Bayesian optimization calculates the \emph{acquisition function} using which we determine the structure to be sampled next.
A standard formulation of acquisition function is \emph{expected improvement} (EI) defined as the expectation of the energy decrease estimated by GP, which is also applicable in our multi-task GP case.
However,
EI does not consider the cost discrepancy for the surfaces, which may necessitate a large number of sampling steps for high cost surfaces.
In other words, the total cost Eq.~(\ref{eq:total-cost}) is not taken into account by usual Bayesian optimization.
We further introduce a cost-sensitive acquisition function to solve this issue, and then the method is called \emph{cost-sensitive multi-task Bayesian optimization} (CMB).
To select the next candidate, each GB model is evaluated based not only on the possible decrease of the energy, but also on the computational cost of that GB model.
Our cost-sensitive acquisition function for the $i$-th GB model is defined by
\begin{align}
{\rm EI}^{\rm (CMB)}_i =
\frac{
{\rm EI}_i
}{C_{\tau_i}},
\label{eq:CSAF}
\end{align}
where ${\rm EI}_i$ is the usual expected improvement for the $i$-th GB model which purely evaluates the possible improvement.
This cost-sensitive acquisition function selects the best GB model to be sampled
by considering
EI \emph{per} computational cost, while usual EI selects a structure
by considering
the improvement in the energy decrease per sampling iteration.
\figurename~\ref{fig:CMTB-demo} shows an illustrative demonstration of CMB.
In the figure, the two surfaces need low sampling costs and the other two surfaces need high sampling costs.
CMB first selects the low cost surfaces and accumulates surface information, using which the minimum energies for the high cost surfaces can be efficiently identified.
This illustrates that CMB is effective for minimizing the GB energy with a small amount of the total cost Eq.~(\ref{eq:total-cost}).
\clearpage
\section*{Results (Case Study on fcc-Al)}
\label{sec:results}
\subsection*{GB Model and Descriptor}
\label{sec:GB-model-setting}
We first constructed fcc-Al [110] symmetric tilt (ST) GBs using the coincidence site lattice (CSL) model.
The CSL is usually characterized by the $\Sigma$ value, which is defined as the reciprocal of the density of the coincident sites.
\figurename~\ref{fig:example-energy-surface} (a) shows an example of a supercell of a $\Sigma9$ STGB model.
Two symmetric GBs are introduced to satisfy three-dimensional periodicity.
To avoid artificial interactions between GBs, we set the distances between GBs to more than 10 \AA.
For the energy calculations and atomic relaxations, we used the EAM potential for Al in Ref. \cite{Mishin1999}, and the computational time scales as $O(M)$ for the number of atoms $M$ with the linked-list cell algorithm.
\figurename~\ref{fig:example-energy-surface} (a) also shows the construction of a supercell by RBT from the STGB model.
{
GB models contain largely different numbers of atoms in the supercells from
$36$
to
$388$
which results in a strong cost imbalance in the search space.
The number of atoms $M$ for all
$38$
angles are shown in Supplementary Information 3.
For each angle, the {three-dimensional RBTs, denoted as $\Delta X$, $\Delta Y$ and $\Delta Z$} which are illustrated in
\figurename~\ref{fig:example-energy-surface} (a), were generated.
The grid space is
$0.1 {\rm \AA}$
for the direction $\Delta X$,
$0.2 {\rm \AA}$
for the direction $\Delta Y$,
and
$0.1 {\rm \AA}$
for the direction $\Delta Z$.
In atomic columns, if the two atoms in an atomic pair are closer to each other than the cut-off distance, one atom from the pair is removed
More precisely, an atomic pair within the cutoff distance is replaced with a single atom located at the center of the original pair.
In this study, the cut-off distance
is varied between 1.43 and 2.72 \AA, i.e. 0.5 and 0.95 times the equilibrium atomic distance, respectively.
For example, two models for $\Sigma 9 $, where an atomic pair is replaced or not replaced, can be considered as illustrated in \figurename~\ref{fig:example-energy-surface} (a).
In total, we created
$157680$
candidate GB models for which the exhaustive search is computationally quite expensive.
}
As the structure-specific descriptor for each GB model $\bx$, we employed the {three-dimensional axes of RBTs: $\Delta X$, $\Delta Y$, and $\Delta Z$}.
For the task-specific descriptor $\bz$, we used the rotation angle $\theta$ and radial distribution function (RDF) of the
{$(\Delta X,\Delta Y,\Delta Z) = (0,0,0)$}
GB model.
As an angle descriptor, we applied the following transformation to the rotation angles: $\tilde{\theta}_t = \theta_t$ if $\theta_t \leq 90$, otherwise $\tilde{\theta}_t = 180 - \theta_t$.
In the case of the fcc-Al [100] GBs, $\theta_t$ and $\tilde{\theta}_t$ are equivalent.
Although the complete equivalence does not hold for fcc-Al [110] GBs, we used this transformed angle as an approximated similarity measure.
For the RDF descriptor, we created a $100$-dimensional vector $\brho \in \RR^{100}$ by taking $100$ equally spaced grids from $0$ to $6$ \AA.
The task-specific descriptor is thus written as $\bz_t = (\tilde{\theta}_t, \brho_t^\top)^\top$.
In other words, two tasks which have similar angles and RDFs simultaneously are regarded as similar in MGP.
The cost parameter $C_t$ was set by the number of atoms in each supercell.
Detail of the parameter setting of Bayesian optimization is shown in Supplementary Information 4.
\subsection*{Performance Evaluation}
\label{subsec:performance}
To validate the effectiveness of our proposed method, we compared the following four methods (methods 3 and 4 are newly proposed in this paper.):
\begin{enumerate}
\item random sampling (Random): At each iteration, the next candidate was randomly selected with uniform sampling.
%
\item single task Bayesian optimization (SB): SB is the usual Bayesian optimization for a single task.
%
At each iteration, a GB model which had the maximum EI was selected across all the angles.
%
%
\item multi-task Bayesian optimization (MB): MB is Bayesian optimization with multi-task GP in which knowledge of the energy surfaces is transferred to different angles each other.
%
The acquisition function is the usual EI.
%
\item cost-sensitive multi-task Bayesian optimization (CMB): CMB is MB with the cost-sensitive acquisition function defined by Eq.~(\ref{eq:CSAF}).
\end{enumerate}
All methods start with one randomly selected structure for each angle.
\figurename~\ref{fig:error-transition} shows the results.
We refer to the difference between the lowest energy identified by each search and the true minimum as an energy gap.
The vertical axis of the figure is the average of the gaps for the
{$38$}
different angles, and the horizontal axis is the total cost (\ref{eq:total-cost}).
All values are averages of {$5$} trials with different initial structures.
{We first see that CMB has smaller energy than the other three methods.}
Focusing on the difference between the single-task method and the multi-task-based methods, we see that the convergence of SB is much slower than that of multi-task based methods (MB and CMB).
We also see that the cost-sensitive search improved the convergence
(Note that although the cost-sensitive search is applicable to SB, it is not essentially beneficial because SB does not transfer information accumulated for low cost surfaces to high cost surfaces.).
{
To validate the effectiveness of our approach in a more computationally expensive setting, we consider the case that $O(M^3)$ computations are necessary for the atomic relaxation.
By setting the cost parameter $C_t$ as the cube of the number of atoms (i.e., $M^3$), we virtually emulated this situation with the same dataset.
\figurename~\ref{fig:error-transition-c3} shows the energy gap.
Here, the horizontal axis is the sum of the cube of the number of atoms $M^3$ for the calculated GB models.
Same as \figurename~\ref{fig:error-transition}, MB and CMB show better performance than the naive SB.
In particular, CMB rapidly decreased the energy gap than the other methods.
Because of the larger sampling cost, the cost-sensitive strategy showed a greater effect on the search efficiency.
}
\section*{Discussion}
The acceleration of the structure search is essential for material discovery in which a huge number of candidate structures are needed to be investigated.
In our case study using the fcc-Al [110] tilt GBs, the sum of the computational cost $C_t$ for all candidate structures is $\sum_{i=1}^N C_{\tau_i} = 33458160$ when $C_t$ is set as per the $M$ (i.e., $O(M)$ setting).
The total computational cost that CMB needed to reach
the average energy gaps
$10$ mJ/m$^2$ and
$5$ mJ/m$^2$
were
$0.001 \approx 43891.8 / 33458160.0$
and
$0.002 \approx 76937.6 / 33458160.0$,
respectively.
In other words, with only about 0.2 \% of the computation steps of the exhaustive search, CMB achieved 5 mJ/m$^2$ accuracy.
Figure~\ref{fig:cusp} compares the energy between the true stable structure and the structure identified by CMB, which shows that our method accurately identified the dependency of energy on the angle, with a low computational cost.
To summarize, we have developed a cost-effective simultaneous search method for GB structures based on two machine-learning concepts: transfer learning and cost-sensitive search.
Since amount of data is a key factor for data-driven search algorithms, knowledge transfer, by which data is shared across different tasks, is an important technique to accelerate the structure search.
Although the concept of multi-task learning is widely accepted in the machine-learning community, our method is the first study which utilizes it for fast exploration of stable structures.
Our other contribution is to introduce the concept of the cost-sensitive evaluation into the structure search.
For efficient exploration, the diversity of computational cost should be considered, though this issue has not been addressed in the context of the structure search.
Although we used the EAM potential as an example, the cost-imbalance issue would be more severe for computationally more expensive calculations such as density functional theory (DFT) calculations.
\section*{Data availability}
The gain-boundary structure data and our Bayesian optimization code are available on request.
\section*{Acknowledgement}
We would like to thank R. Arakawa for helpful discussions on GB models.
This work was financially supported by grants from the Japanese Ministry of Education, Culture, Sports, Science and Technology awarded to I.T. (16H06538, 17H00758) and M.K. (16H06538, 17H04694); from Japan Science and Technology Agency (JST) PRESTO awarded to M.K. (Grant Number JPMJPR15N2); and from the ``Materials Research by Information Integration'' Initiative (MI$^2$I) project of the Support Program for Starting Up Innovation Hub from JST awarded to T.T., I.T., and M.K.
\section*{Author contributions}
T.Y. implemented all machine learning methods. T.T. constructed the grain-boundary database, and contributed to writing the manuscript. I.T. conceived the concept and contributed to writing the manuscript. M.K. conceived the concept, designed the research, and wrote the manuscript.
\bibliographystyle{apsrev}
|
2,869,038,154,872 | arxiv | \section{1\quad Introduction}
\noindent Recent years have witnessed the promising development of Visual Question Answering (VQA) task\cite{vqa}, which is required to infer the answers based on an image and its relevant question text. Promoted by the continued researches on multimodal inference, the previously proposed Textbook Question Answering (TQA) task \cite{tqa} leads a new trend. Similarly, the TQA task also requires the model to give the answers based on complex multimodal inputs. Figure 1 illustrates an example of TQA. Two questions are listed on the right part. Different from VQA in general domain, TQA dataset relates to the domain of textbook, containing a large number of span-level terminologies. The evidence spans like $continental$ $slope$ in $Q2$ are crucial to TQA inference, but they are uncommon in the general domain. Meanwhile, the inference of questions relies on the joint consideration of abundant context and diagrams. Take $Q1$ as an instance, the model is required to focus on the span $benthic$ $zone$ in the question and find the most related instructional diagram $ID\raisebox{0mm}{-}2$ in the context. After extracting the corresponding information of $QD\raisebox{0mm}{-}1$ and $ID\raisebox{0mm}{-}2$, the answer can be worked out. From these perspectives, TQA task raises new challenges to multimodal inference.
Firstly, general language model (LM) is insufficient for the specific domain knowledge. Pretraining-based Transformer structures \cite{attention}, such as BERT \cite{bert} and GPT \cite{gpt}, show excellent performance on the general domain. However, TQA context ranges from astrophysics to life science, containing a large number of terminologies in the field of textbook (e.g., $benthic$ $zone$ and $continental$ $slope$ in Figure 1). There exists an obvious gap between specific domain and general domain. Some previous works \cite{dont,selectivemask} have been aware of this situation and adapted LM to some domains (e.g., News, Economics, and Reviews). However, they focus on the token-level information for inference, which fails to concern about span-level terminologies and evidences. To the best of our knowledge, so far there is no attempt to enhance LM for TQA by adding external knowledge or attending to span-level information.
\begin{figure}[t]
\large
\centering
\includegraphics[scale=0.47]{Figures/Dataset.jpg}
\caption{An example of TQA. $ID$ is short for instructional diagram while $QD$ is short for question diagram.}
\label{fig_dataset}
\end{figure}
Secondly, it is difficult to fully utilize abundant multimodal inputs. Different from VQA with only single natural image, TQA contains various instructional diagrams and one question diagram as visual input. The two types of diagrams are similar in structure and all have complementary information with text (e.g., $QD\raisebox{0mm}{-}1$ and $ID\raisebox{0mm}{-}2$ are closely connected under the guidance of text $benthic$ $zone$ in Figure 1). Although most popular VQA models \cite{butd,mcan,ban} are capable of fusing multimodal features, they lack the ability to update fine-grained features interactively between two diagrams. Several previous works in TQA \cite{isaaq,xtqa,igmn} have noticed the multiple types of diagrams, but they simply process two types in the same way, ignoring their independent effects.
In light of the above challenges, we propose a novel model MoCA, which incorporates multi-stage domain pretraining and cross-guided multimodal attention for the TQA task. We introduce multi-stage pretraining to conduct post-pretraining with the span mask strategy and pre-finetune sequentially between general pretraining-finetune paradigm. Especially for the employment of the terminology corpus, we propose a heuristic generation algorithm. Meanwhile, patch-level diagram representations are obtained through the Vision Transformer \cite{vit}. Then, we construct token-patch pair interaction and obtain attended features based on the multi-head guided attention. Inspired by the attention flow of human inference \cite{DBLP:conf/eccv/ChenJYZ20}, the features of text, question diagram and instructional diagram are updated in a progressive and interactive way. After the final fusion of all the features, the answer is predicted with a dual gating mechanism. The main contributions are shown as follows:
\begin{itemize}
\item A unified model MoCA is proposed to address both the representation for terminologies and the feature fusion of abundant multimodal inputs. We are the first to simultaneously focus on the two challenges in TQA.
\item We introduce a heuristic generation algorithm for terminology corpus. Based on the external knowledge, we are the first to design a span mask strategy in the pretraining stage for the TQA task.
\item To address the multimodal fusion challenges, a cross-guided attention mechanism is proposed to update the features of rich inputs. In a progressive manner, the interactive updates of the features are obtained.
\item Extensive experiments show that our model significantly improves the state-of-the-art (SOTA) results in the TQA task. Furthermore, ablation and comparison experiments prove the effectiveness of each module in our model.
\end{itemize}
\section{2\quad Related Work}
\textbf{Visual Question Answering.} VQA has aroused wide concerns \cite{DBLP:journals/pami/CaoLLL21, DBLP:conf/sigir/JainKKJRC21, DBLP:conf/acl/Khademi20, DBLP:journals/pr/YuZWZHT20}, as it is regarded as a typical multimodal task related to natural language processing and computer vision. Given an image as well as question text, the model is required to give the answer. Some early models attempted to jointly consider the multimodal inputs. \citet{vqa} encoded the image and text respectively and map them into a common space. \citet{mmresidual} proposed a residual structure to learn joint embeddings. These methods are limited to global and coarse information.
Therefore, more following works applied the attention mechanism to conduct fine-grained reasoning. Typically, \citet{ban} proposed a bilinear attention network to reduce computational cost. \citet{mutan} proposed a framework to efficiently parametrize bilinear multimodal interactions. \citet{mcan} included the self-attention and the question-guided attention within deep modular co-attention networks. \citet{butd} introduced the bottom-up and top-down attention to attend to object-level and salient information. However, these models are placed in an ideal scenario, which includes unitary input of an image and a short question. They lack the potential to process large context and abundant diagrams.
\noindent\textbf{Textbook Question Answering.} Much attention has been paid to the TQA task since it was proposed. For example, \citet{igmn} aimed to find contradictions between answers and context, and further employ memory network for inference. \citet{fgcn} built a multimodal context graph and introduced open-set learning based on self-supervised method. \citet{rafr} proposed fine-grained relation extraction to reason over the nodes of the constructed graphs. The above three methods focused more on inference process, but ignored the huge potential of multimodal encoding, causing relatively weak representation ability. Under this circumstance, \citet{isaaq} utilized the pretrained transformers for text encoding and bottom-up and top-down attention for multimodal fusion, significantly improving the performance. However, it neglected the span-level knowledge during pretraining and simply treated the question and instructional diagrams in the same way. \citet{xtqa} put the explainability at the first place and attempted to extract useful spans, but its text representation method is relatively weak for the textbook domain.
Considering the drawbacks of the above models, we apply external knowledge to enhance the span-level representation. Different from them, we treat two types of diagrams respectively to conduct fine-grained feature updates.
\section{3\quad Methods}
In this section, we will introduce our proposed model MoCA. The architecture of MoCA is shown in Figure 2. The left part is an input example of TQA. Then text and diagrams are encoded respectively, through \textbf{M}ulti-stage \textbf{P}retrain (MP) module and \textbf{P}atch-level \textbf{D}iagram \textbf{R}epresentation module (PDR). In PDR module, we employ Vision Transformer to obtain the patch-level representation of both types of the diagrams. Further, multimodal features are progressively updated based on \textbf{C}ross-\textbf{G}uided \textbf{M}ultimodal \textbf{A}ttention (CGMA). Finally, the answers are worked out by \textbf{G}ating \textbf{M}odel \textbf{E}nsemble (GME). The details of three main modules MP, CGMA and GME will be covered as follows.
\begin{figure*}[t]
\large
\centering
\includegraphics[scale=0.45]{Figures/ModelStructure.jpg}
\caption{The architecture of MoCA for the TQA task.}
\label{fig_model}
\end{figure*}
\subsection{3.1\quad Task Formulation}
Given the TQA dataset $\mathcal{D}$ with $M$ questions, the inference of $i^{th}$ question ($i \in [0,M-1]$) can be defined as follows:
\begin{equation}
\hat{a} =\underset{a_{i,j} \in A_{i}}{\arg \max } p\left(a_{i, j} \mid c_{i}, q_{i}, A_{i}, d_{i} ; \theta\right),
\end{equation}
where $c_{i}$, $q_{i}$, $A_{i}$ represent the text-only context, question sentence and candidate set respectively. $d_{i}$ includes question and instructional diagrams. The option number in $A_{i}$ is $n$, $j \in [0,n-1]$ and $a_{i,j}\in A_{i}$ represents $j^{th}$ option. $\hat{a}$ is the predicted option. $\theta$ is the trainable parameter.
\subsection{3.2\quad MP Module}
Although the two-stage general pretraining and finetune models has achieved great success in many tasks, there exists an obvious gap between general domain and textbook domain. We attribute the problem to the lack of data about both specific domain and task. Under this circumstance, we propose MP module for the enhancement of text representation.
MP includes four stages in total, which is constructed with two unsupervised pretraining stages and two supervised finetune stages. Stage \RNum{1} and Stage \RNum{4} inherit the traditional general paradigm, with pretraining on large general corpus and finetune on downstream task (TQA) respectively. We use RoBERTa \cite{roberta} as the base model for Stage \RNum{1}, which utilizes the dynamic random mask strategy to focus on token-level performance.
For Stage \RNum{2}, to adapt the language model to the textbook domain, we introduce a coarse-to-fine strategy to heuristically generate external domain corpus. In the beginning, we crawl the textbook-related websites, forming the large coarse-level domain corpus. To evaluate the similarity of the external knowledge with TQA, we employ vocabulary overlap of the Top 1k most frequent words, which excludes nearly 900 stopwords. We define the overlap operator as $\mathcal{O}(\mathbf{c}_{1}|\mathbf{c}_{2})$. It stands for the overlap vocabulary list of input corpus or text $\mathbf{c}_{2}$ with $\mathbf{c}_{1}$, while $L(\mathcal{O}(\mathbf{c}_{1}|\mathbf{c}_{2}))$ returns the number of overlap words. Based on the operator and coarse-level corpus, we design a heuristic method to generate fine-level domain corpus containing rich terminologies. For each line of the corpus, we retain the ones which contribute more to the vocabulary overlaps during every iteration (shown in Algorithm 1, more details are attached to Appendix). After obtaining fine-level corpus, we shuffle it with TQA text to prepare for the post-pretraining stage.
Considering that quite a lot of knowledge and question information consist of multiple words, we adopt span mask strategy to optimize domain-specific pretraining process. That is to say, given a sequence $S=\{t_{1},t_{2},...,t_{n}\}$ consisting of $n$ tokens, each time we randomly mask one to ten tokens based on geometric distribution. Until the mask percentage reaches 15\%, the process breaks automatically. Same as BERT, for 15\% tokens selected in sequence $S$, 80\% of them are replaced with $<$$mask$$>$ flag, 10\% of the tokens are substituted by random token and the others remain unchanged.
\begin{algorithm}[t]
\KwIn{Coarse-level Corpus $\mathbf{C}_{coarse}$, TQA Text $\mathbf{T}$, threshold $\delta$}
\KwOut{Fine-level Corpus $\mathbf{C}_{fine}$}
Calculate vocaburary overlap $\mathcal{O}(\mathbf{T}|\mathbf{C}_{coarse})$ of coarse corpus with TQA text.\\
\Repeat{$\mathbf{C}_{fine}$ is reduced to a certain specification}
{
Take $\mathbf{C}_{fine}$ as $\mathbf{C}_{coarse}$ if not the first iteration\\
\For{Line $l_{i} \in \mathbf{C}_{coarse}$}
{
$\mathbf{W}\leftarrow \mathcal{O}(\mathbf{T}|\mathbf{C}_{coarse})$\\
$Score(l_{i})\leftarrow \frac{L(\mathcal{O}(\mathbf{W}|l_{i}))}{L(\mathbf{W})}-\frac{L({l_{i}})}{L(\mathbf{C}_{coarse})}$\\
\If{$Score(l_{i})>\delta$}
{
Include $l_{i}$ into fine-level corpus $\mathbf{C}_{fine}$
}
}
}
\textbf{return} Fine-level Corpus $\mathbf{C}_{fine}$\\
\caption{Heuristic Terminology Corpus Generation}
\end{algorithm}
For Stage \RNum{3}, the model is trained to adapt to the specific task with the help of an external RACE dataset. Similar to TQA, RACE dataset is a multiple-choice task, which is collected from examinations. Through the process of pre-finetune, the performance of our model is enhanced.
The pretrained MP encoder is used to model the text-only part, including context, question text, and options. For each question, we concatenate the context $c_{i}$, question $q_{i}$ and option $a_{i,j}$ as the input sequence:
\begin{equation}
{\rm Input}(c_{i},q_{i},a_{i,j}) = [C L S] c_{i} [S E P] q_{i} [S E P] a_{i, j} [S E P].
\end{equation}
Feed it into our MP encoder and output the feature of the text $f_{T} \in \mathbb{R}^{N \times d}$, where $N$ represents the max length of the input sequence and $d$ represents the hidden size.
\subsection{3.3\quad CGMA Module}
Through PDR module, the original features of question diagram $f_{Q\raisebox{0mm}{-}D} \in \mathbb{R}^{P \times d}$ and instructional diagram $f_{I\raisebox{0mm}{-}D} \in \mathbb{R}^{P \times d}$ are obtained, where $P$ is patch number and $d$ is the dimension of hidden state. To align text and visual parts to the same space, we add a linear projection layer $\rm Linear(P,N)$. Thus, three features with the same dimension can function as the input of CGMA module.
\begin{figure}[t]
\large
\centering
\includegraphics[scale=0.43]{Figures/CGMA.jpg}
\caption{Progressive multimodal feature update in CGMA.}
\label{fig_model}
\end{figure}
Inspired by the human inference pattern, we propose the progressive attention flow for the TQA task, illustrated in Figure 3. Totally two progresses are included in CGMA. In Progress \RNum{1}, question diagram and text are considered as two inputs, while instructional diagram is added in Progress \RNum{2}. Three types of features are updated with three similar multi-head guided attention. Within each attention, we first build a fine-grained connection between each token-patch pair and focus on the important parts based on attention weights. Take question diagram feature update as an example. The text feature $f_{T}$ is utilized to instruct the feature update of question diagram $f_{Q\raisebox{0mm}{-}D}$. We first map the two features into three matrices $Q_{Q\raisebox{0mm}{-}D}$, $K_{T}$ and $V_{T}$.
\begin{equation}
\begin{aligned}
Q_{Q\raisebox{0mm}{-}D} &= f_{Q\raisebox{0mm}{-}D} \cdot{\rm W^{Q}},\\
K_{T} &= f_{T} \cdot {\rm W^{K}},\\
V_{T} &= f_{T} \cdot {\rm W^{V}},
\end{aligned}
\end{equation}
where ${\rm W^{Q}, W^{K}, W^{V}} \in \mathbb{R}^{d \times d}$ are projection matrices, and the obtained matrices $Q_{Q\raisebox{0mm}{-}D}, K_{T}, V_{T} \in \mathbb{R}^{N \times d}$. We compute the attention based on the query, key and value matrices.
\begin{equation}
Att(Q_{Q\raisebox{0mm}{-}D}, K_{T}, V_{T}) = {\rm softmax}(\frac{Q_{Q\raisebox{0mm}{-}D}K_{T}^{T}}{\sqrt{d}}) \cdot V_{T}.
\end{equation}
Multi-head guided attention is applied to improve the robustness and capacity of the attention module. The feature can be represented as follows, with the head number $H$.
\begin{equation}
Att_{MH}(Q_{Q\raisebox{0mm}{-}D}, K_{T}, V_{T}) = [Head_{1}, ..., Head_{H}] \cdot {\rm W^{H}},
\end{equation}
where ${\rm W^{H}}\in \mathbb{R}^{(H*d) \times d}$ is the projection matric, $Head_{i}=Att_{i}(Q_{Q\raisebox{0mm}{-}D}, K_{T}, V_{T})$, the input query, key and value matrix are obtained by linear projection of ${\rm W_{i}^{Q}, W_{i}^{K}, W_{i}^{V}} \in \mathbb{R}^{d \times d}$ respectively.
Then, we add the original question diagram features to the computed attention, followed with layer normalization.
\begin{equation}
f_{Q\raisebox{0mm}{-}D}^{\prime} = {\rm LayerNorm}(f_{Q\raisebox{0mm}{-}D}+Att_{MH}(Q_{Q\raisebox{0mm}{-}D}, K_{T}, V_{T})).
\end{equation}
Through the followed feedforward and layer normalization, the attended output feature of one multi-head guided attention layer is obtained. To further enhance the performance of the module, we concatenate $L$ paralleled layers and obtain the multi-layer updated feature $f^{\prime\prime}_{Q\raisebox{0mm}{-}D}$.
\begin{equation}
f^{\prime\prime}_{Q\raisebox{0mm}{-}D} = [Layer_{1}, ..., Layer_{L}] \cdot {\rm W^{L}},
\end{equation}
where ${\rm W^{L}} \in \mathbb{R}^{(L*d) \times d}$ is the projection matric and $Layer_{j}$ represents the output feature $f_{Q\raisebox{0mm}{-}D}^{\prime}\in \mathbb{R}^{N \times d}$ of $i\raisebox{0mm}{-}th$ layer.
Thus, the feature of question diagram is updated. The text feature can be updated with the similar process in Progress \RNum{1}. Considering that instructional diagram is similar to question diagram in structure and contains meaningful information related to text, we first integrate the multimodal features of text and question diagram and make it a guidance for the instructional diagram feature update in Progress \RNum{2}.
\subsection{3.4\quad GME Module}
To reduce the computational cost, we first shorten the large context into a fixed number of sentences with the help of information retrieval method. In consideration of single retrieval has limitations and may bring unnecessary noise, we perform background retrieval three times in different ways. Following ISAAQ \cite{isaaq}, we name them as $IR$, $NSP$ and $NN$. Since three retrieval methods are independent and contribute differently to the inference, we design a weighted ensemble method to fully exploit the advantages of each retrieval. It means that the final feature $f_{o}^{Ensemble}$ for the prediction consists of three parts with different weights.
\begin{equation}
f_{o}^{Ensemble} = \lambda_{1}f_{o}^{IR} + \lambda_{2}f_{o}^{NSP} + (1-\lambda_{1}-\lambda_{2})f_{o}^{NN},
\end{equation}
where $f_{o}^{Ensemble}, f_{o}^{IR}, f_{o}^{NSP}, f_{o}^{IR} \in \mathbb{R}^{N \times d}$, and weight parameters $\lambda_{1}, \lambda_{2}, 1-\lambda_{1}-\lambda_{2}\in[0,1)$.
Further, for diagram multiple-choice questions, diagrams are considered as an important part of the inference. However, in some cases, the help of diagram features can be negative, or the text-only feature can work out the correct answer. In order to reduce noise brought by diagram features, we integrate the text-only feature for inference in model ensemble. We set a gate parameter $\mu$ to model the importance of text-only features and multimodal features.
\begin{equation}
f_{o}^{GME} = (1-\mu) f_{t}^{Ensemble} + \mu f_{mm}^{Ensemble},
\end{equation}
where $f_{t}^{Ensemble}$ and $f_{mm}^{Ensemble}$ represents the weighted text-only feature and multimodal feature respectively. For each option, we obtain different final features. Through the training of the classifier, the answers can be obtained.
\section{4\quad Experiments}
In this section, extensive experiments are conducted to compare our model with SOTA methods in TQA. Comparison study and parameter analysis are followed to improve the comprehensiveness of the proposed model.
\subsection{4.1\quad Dataset and baselines}
We conduct the experiments on TQA dataset. Covered in the textbook domain, TQA includes 1,076 subjects, namely Life Science, Earth Physics, etc. It consists of 26,260 questions in the form of True/False (T/F), Text Multiple Choice (T-MC) and Diagram Multiple Choice (D-MC). The number of candidate answers and dataset split are shown in Table 1. Column $2$ to $5$ represent the number of questions while the last column represents the number of candidate answers.
\begin{table}[t]
\centering
\begin{tabular}{ccccc}
\hline
\textbf{Type} &\textbf{Train Set} &\textbf{Val Set} &\textbf{Test Set} &\textbf{Candidate}\\
\hline
T/F & 3,490 &998 &912 & 2\\
T-MC & 5,163 &1,530 &1,600 & 4-7\\
D-MC & 6,501 &2,781 &3,285 & 4\\
\hline
\end{tabular}
\caption{Dataset split and candidate answer number}
\label{tab:TQA}
\end{table}
To prove the superiority of our model, we employ the following baselines, including the SOTA model.
\begin{itemize}
\item \textbf{Random}: Results based on random prediction.
\item \textbf{MemN} \cite{tqa}: It employs the concept of memory network and diagram parse graph to construct the context graph for the TQA task.
\item \textbf{IGMN} \cite{igmn}: It grasps the contradiction between candidate answers and context for reasoning.
\item \textbf{FCC} \cite{fcc}: It jointly considers diagrams and image captions.
\item \textbf{f-GCN1} \cite{fgcn}: A new f-GCN module based on graph convolution network is proposed to address multimodal fusion challenges.
\item \textbf{XTQA} \cite{xtqa}: It designs a coarse-to-fine algorithm to generate span-level evidences.
\item \textbf{RAFR} \cite{rafr}: A fine-grained reasoning network is proposed to reason over the nodes of relation-based diagram graphs.
\item \textbf{ISAAQ} \cite{isaaq}: It improves the SOTA baseline by introducing the pretrained LM and bottom-up and top-down attention mechanism.
\end{itemize}
\begin{table}[t]
\centering
\begin{tabular}{cccc}
\hline
\textbf{Corpus} & \textbf{Domain} & \textbf{Size} & \textbf{Source}\\
\hline
TQA text & Textbook & 5MB & From TQA\\
External & Textbook & 350MB & Crawled\\
\hline
\textbf{Dataset} & \textbf{Domain} & \textbf{Size} & \textbf{Type}\\
\hline
TQA & Textbook &26,260 & Multimodality\\
RACE & Exams &97,687 & Text-only\\
VQA(Part) & General &90,000 & Multimodality\\
AI2D & Science &8,730 & Diagram-only\\
\hline
\end{tabular}
\caption{The detailed information of corpus and dataset employed in the training process.}
\label{tab:corpus}
\end{table}
\subsection{4.2\quad Implementation Details}
\begin{table}[t]
\centering
\begin{tabular}{m{1.75cm}<{\centering}ccccc}
\hline
Model & T/F & T-MC & T-All & D-MC & All\\
\hline
Random &50.10&22.88&33.62&24.96&29.08 \\
MemN &50.50&30.98&38.69&32.83&35.62 \\
IGMN &57.41&40.00&46.88&36.35&41.36 \\
FCC &-&36.56&-&35.30&-\\
fGCN &62.73&49.54&54.75&37.61&45.77\\
XTQA &58.24&30.33&41.32&32.05&36.46\\
RAFR &53.63&36.67&43.35&32.85&37.85\\
ISAAQ &\underline{81.36} &\underline{71.11} &\underline{75.16} &\underline{55.12} &\underline{64.66}\\
MoCA(Ours) &\textbf{81.56} &\textbf{76.14} &\textbf{78.28} &\textbf{56.49} &\textbf{66.87}\\
\hline
\end{tabular}
\caption{Experimental results on the validation split for TQA. The percentage signs (\%) of accuracy values are omitted. The optimal and suboptimal results are marked in bold and underline respectively (same for the following tables).}
\label{yab:performance_val}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{m{1.75cm}<{\centering}ccccc}
\hline
Model & T/F & T-MC & T-All & D-MC & All\\
\hline
XTQA &56.22&33.40&46.73&33.34&36.95\\
RAFR &52.75&34.38&41.03&30.47&35.04\\
ISAAQ &\underline{78.83} &\underline{72.06} &\underline{74.52} &\underline{51.81} &\underline{61.65}\\
MoCA(Ours) &\textbf{81.36} &\textbf{76.31} &\textbf{78.14} &\textbf{53.33} &\textbf{64.08}\\
\hline
\end{tabular}
\caption{Experimental results on test split for TQA.}
\label{tab:performance1}
\end{table}
All of the experiments are finished with a single GPU of Tesla V100. As for the encoder of text in MP module, we utilize the RoBERTa-large model for Stage \RNum{1} which has 1024-dimensional embeddings. For Stage \RNum{2}, we mix the external textbook corpus with TQA corpus and conduct the post-pretraining with 10 epochs based on the span mask strategy. For Stage \RNum{3}, we pre-finetune the model on RACE dataset with 4 epochs and select the best one based on the performance of test split. As for the background information retrieval, we follow the three methods introduced in ISAAQ \cite{isaaq} namely $IR$, $NSP$ and $NN$. Considering the length of retrieved context, we set the maximum input sequence length to 180, and the inputs shorter than 180 are padded to max. As for diagram representation, each diagram is cut into 14$\times$14 patches with the dimension of 1024. As for CGMA module, we empirically set the multi-head number to 8 and the number of paralleled layers is searched for the best in \{1,2,3,4\}. To enhance the classifier for D-MC questions, we further finetune the model on VQA \cite{vqa} and AI2D \cite{ai2d}. The detailed information of the corpus and dataset is shown in Table 2.
\subsection{4.3\quad Main Results}
MoCA model is evaluated on TQA dataset. The results on validation and test split are shown in Table 3 and Table 4 respectively. Since some previous baselines do not make the results of test split public, we only compare the validation split results for these models.
From the results, MoCA outperforms the SOTA results from all three types of questions. In general, we significantly improve the overall performance of SOTA by 2.21\% and 2.43\% on validation and test split respectively. It is worth mentioning that MoCA also shows better generalization capability. Especially for the text-only questions on the test split, MoCA outperforms the previous SOTA method by 3.62\%. Also, results on previous models illustrate the different distributions between validation and test split for D-MC questions, while MoCA narrows this obvious gap.
\subsection{4.4\quad Ablation and Comparison Experiments}
\begin{table}[t]
\centering
\begin{tabular}{p{3cm}ccc}
\hline
Model(Valid) & T/F & T-MC & T-All\\
\hline
MoCA &81.56 &76.14 &78.28\\
\quad w/o Stage \RNum{2} &79.86 &75.03 &76.94\\
\quad w/o Stage \RNum{3} &79.76 &75.88 &77.41 \\
\quad w/o Stage \RNum{2} \& \RNum{3} &79.26 &75.16 &76.78 \\
\quad w/o LSOs &81.16 &73.73 &76.66 \\
\hline
\hline
Model(Test) & T/F & T-MC & T-All\\
\hline
MoCA &81.36 &76.31 &78.14\\
\quad w/o Stage \RNum{2} &78.07 &75.19 &76.24\\
\quad w/o Stage \RNum{3} &78.51 &75.56 &76.63 \\
\quad w/o Stage \RNum{2} \& \RNum{3} &78.73 &74.00 &75.72 \\
\quad w/o LSOs &80.92 &72.88 &75.80 \\
\hline
\end{tabular}
\caption{Ablation experiments for MP module}
\label{tab:ablationMP}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{p{3.2cm}cccc}
\hline
Model(Valid) & IR & NSP &NN &T-MC\\
\hline
MoCA &73.33 &71.83 &68.17 &76.14\\
\quad - Random Mask &72.09 &71.11 &68.04 &75.36 \\
\quad - Whole Word Mask &72.09 &71.63 &68.69 &76.08 \\
\hline
\end{tabular}
\caption{Comparison results of mask strategy in MP module.}
\label{tab:mask}
\end{table}
\begin{table}[t]
\centering
\begin{tabular}{p{2.8cm}cccc}
\hline
Model (Valid) & IR & NSP & NN & D-MC\\
\hline
MoCA &54.15 &53.83 &51.60 &56.49\\
\quad w/o ID-Guided &53.72 &53.72 &50.74 &56.17\\
\quad w/o QD-Guided &53.94 &54.01 &51.06 &56.27\\
\quad w/o Text-Guided &53.65 & 52.68& 51.13 & 55.63\\
\quad w/o All Attention &53.15 &53.36 &50.99 &55.02\\
\quad w/o ID &53.94 &53.72 &51.31 &56.09\\
\hline
\hline
Model (Test) & IR & NSP & NN & D-MC\\
\hline
MoCA &52.12 &51.87 &51.26 &53.33\\
\quad w/o ID-Guided &51.69 &51.93 &51.29 &52.66\\
\quad w/o QD-Guided &51.57 &51.78 &50.75 &53.12\\
\quad w/o Text-Guided &51.63 &51.32 & 50.17&53.18 \\
\quad w/o All Attention &51.08 &51.32 &51.29 &53.15\\
\quad w/o ID &51.78 &51.11 &51.14 &52.94\\
\hline
\end{tabular}
\caption{Ablation experiments for CGMA module.}
\label{tab:ablationCGMA}
\end{table}
\begin{table}[h]
\centering
\begin{tabular}{p{4.8cm}cc}
\hline
Model & Valid & Test\\
\hline
MoCA &56.49 &53.33\\
\quad w/o $\mu$ effect (Three models) &55.45 &52.85\\
\quad w/o $\mu$ and $\lambda$ (Single model) &54.15 &52.12\\
\quad w/o $\mu$ and $\lambda$ (Six equal models) &55.88 &53.27\\
\hline
\end{tabular}
\caption{Ablation experiments for GME module.}
\label{tab:ablationGME}
\end{table}
As MoCA includes three main modules, we conduct the ablation experiments for each module to explore the effectiveness. Meanwhile, comparison experiments on the mask strategy in MP module are also presented.
\noindent\textbf{MP module.} We mainly experiment and analyze the effectiveness of multiple stages and the mask strategy in MP module. Firstly, we remove one or both of Stage \RNum{2} and Stage \RNum{3}. Without Stage \RNum{2} \& \RNum{3}, the text-only accuracy drops by 1.5\% and 2.42\% on validation and test split respectively. Within the removed two stages, Stage \RNum{2} contributes most to the model performance. It proves that the unsupervised training on external knowledge makes a difference.
Through exploratory data analysis, we also discover the characteristics of the options, which is the Latent Semantic Option(LSO). For example, the option `All of the above' has the further meaning that the answer includes the information of all options, which is hard to be reflected by LM. We divide them into two types: Postive and Negative. For the Positive type, like `All of these', `Both a and b', it means the final answer includes more than one option. We concatenate the options and replace the LSO with the spliced text. For the Negative one, like `None of these', `Neither a nor b', we set LSO to the empty string for simplicity. We also test the necessity of LSOs. The evaluation results are shown in Table 5. The consideration of LSOs brings 1.62\% and 2.34\% accuracy gain on validation and test split. From the results, the mapping of LSOs has equal contributions with MP. The former one improves the performance by manually designed rules while the latter one relies on the external knowledge.
Secondly, we select two popular mask strategies as comparison objects, that is random mask and whole word mask. Results on Table 6 show the superiority of the span mask strategy, especially on the retrieval of $IR$ with 1.24\% accuracy gain. Since pretraining on general domain (Stage \RNum{1}) utilizes the random mask strategy rather than span-level strategy, it still remains huge potentials for the improvement.
\noindent\textbf{CGMA module.} We remove one or all of the cross-guided attention. Since previous SOTA models seldom take ID into consideration or just simply confuse it with QD, we also remove the input of ID to show its effectiveness. The experimental results in Table 7 show that all the attention employed brings 1.47\% gain on the validation split. The consideration of ID improves the performance by 0.40\% and 0.49\% on the validation and test split respectively. Further, the update of text feature contributes most on the validation split and least on the test split. It shows the importance of text for D-MC in each split.
\noindent\textbf{GME module.} Our ensemble method has two gate parameters $\lambda$ ($\lambda_{1},\lambda_{2}$) and $\mu$, which has different type of roles. Therefore, we further compare three conditions. Firstly, utilize the multimodal features only for D-MC inference, which eliminates the effect of gate parameter $\mu$ ($\mu$=0.5). Secondly, simultaneously eliminate all the gate parameters, which means select the best single one as the final model ($\lambda,\mu$ not exist). Thirdly, keep both of the gate parameters while set all six models to equal contributions ($\lambda_{1}$=$\lambda_{2}$=1/3,$\mu$=0.5). Results are shown in Table 8. Generally, two gate parameters $\lambda$ and $\mu$ contribute 2.34\% and 1.21\% to the accuracy on the validation and test split respectively. On one hand, more models for ensemble bring better overall performance. On the other hand, single model of MoCA is still competitive to ensembled SOTA model.
\begin{figure}[t]
\large
\centering
\includegraphics[scale=0.50]{Figures/ParameterAnalysis_gate.jpg}
\caption{Analysis of gate parameter.}
\label{fig_gate}
\end{figure}
We select several $\mu$ with an interval of 0.1 for visualization. The results on validation and test split are shown in Figure 4. The trend illustrates that MoCA reaches the best performance when the gate parameter $\mu$ is 0.6. With $\mu$ increasing, accuracy on the validation and test split witnesses an obvious drop. More specifically, the inference of TQA task relies on text-only evidences. It also proves the effectiveness and necessity of GME module.
\subsection{4.5\quad Case Study}
\begin{figure}[t]
\large
\centering
\includegraphics[scale=0.35]{Figures/CaseStudy_main_small.jpg}
\caption{Case study.}
\label{fig_case}
\end{figure}
\begin{figure}[t]
\large
\centering
\includegraphics[scale=0.225]{Figures/CaseStudy_heatmap_small.jpg}
\caption{Visualization of attention maps. For better correspondence, we extend the sequence length of diagrams from 180 to 196, with the help of 2-D linear interpolation fitting.}
\label{fig_QA}
\end{figure}
As text-only questions can be regarded as a special case of diagram questions, we conduct the case study on the diagram questions only. Figure 5 shows a successful case and a failure case. For the successful one, MoCA makes good use of multimodal information for inference. For the failure one, text information is limited and it is required to figure out the number of volcano type directly from QD. It reflects that MoCA is weak in counting and numerical reasoning.
We select the successful case to visualize the effects of three cross-guided attention in MoCA, shown in Figure 6. For better visualization, we mark some important patches with blue boxes in QD and ID, and present the order number above. The left right presents the attention weight of QD-to-text guidance. Obviously from the attention map, Patch \#64 to Patch \#69 contribute most to the attention weight. These patches cover the crucial clues for the correct answer `Topsoil' in QD. The bottom left is the attention weight of text-to-QD guidance. We notice that tokens around \#100 are mainly focused on. These tokens consist of key sentence like `It is the layer with the most organic material' and key span like `rich organic upper layers (humus and topsoil)', which are extremely close to the answer. The bottom right part reflects the attention weight of multimodal guidance to ID. Its X-axis represents the multimodal sequence while Y-axis represents the ID sequence. From the correspondence, important patches of \#78 to \#80 and \#92 to \#94 obtain larger attention weights. The final softmax result after the classifier also proves the positive effect of the cross-guided attention.
\section{5\quad Conclusion}
We incorporate multi-stage domain pretraining and multimodal cross attention for the TQA task. Firstly, on the basis of general pretraining-finetune paradigm, we propose multi-stage domain pretraining module to bridge the gap between general domain and textbook domain. In the stage of domain post-pretraining, we propose a heuristic generation algorithm to employ terminology corpus. Span mask strategy is utilized to optimize the pretraining performance. Secondly, following the human inference pattern, we propose multimodal cross-guided attention to progressively update the features of text, question diagram and instructional diagram. Further, we adopt a dual gating mechanism to improve the ensembled model performance. Extensive experiments prove the superiority of our model and module effectiveness. In the future, we will pay more attention to the improvement of background text retrieval, as well as the employment of fine-grained diagram information.
|
2,869,038,154,873 | arxiv | \section{One Dimensional Linear Model} \label{sec:1d-linear}
In order to clarify our method for nonlinear hyperbolic systems
\eqref{equ:1d-sediment} and \eqref{equ:2d-sediment}, we first
present our basic idea by studying
the one dimensional linear hyperbolic system as follows
\begin{equation} \label{equ:linear}
\left\{
\begin{aligned}
&\bU_t + \mathbf{A} \bU_x = -\mathbf{g} B_x, \\
&B_t + \varepsilon \mathbf{c}^T\bU_x = 0.
\end{aligned}
\right.
\end{equation}
where $\bU=(u_1,\cdots,u_l)^T\in\mathbb R^l$ are fast variables, $B\in\mathbb R$ is
slow variable, $\mathbf{g},\mathbf{c}$ are two constant vectors in $\mathbb R^l$.
The constant matrix $\mathbf{A} \in\mathbb R^{l\times l}$ satisfies
\begin{equation} \label{equ:linear-A}
\mathbf{A} = \mathbf{X}^{-1}\mathbf{\Lambda}, \mathbf{X} \qquad \mathbf{\Lambda} = \text{diag}\{
\lambda_1, \cdots, \lambda_l
\}.
\end{equation}
With the purpose of scale separation on the characteristic speeds of
\eqref{equ:linear}, it requires $|\lambda_i|\gg\varepsilon$, thus
$\mathbf{A}$ is invertible. Moreover, we assume $\mathbf{A}$ has only discrete
eigenvalues (i.e. the algebraic multiplicity is one). We emphasize
that $t$ represents the fast time scale, and $\tau=\varepsilon t$
represents the slow time scale in the context of this paper.
Let
\begin{equation}\label{mat:C-Ceps0}
\mathbf{C} = \begin{pmatrix} \mathbf{A} & \mathbf{g} \\ 0 & 0 \\ \end{pmatrix}, \qquad
\mathbf{C}^\varepsilon = \begin{pmatrix} \mathbf{A} & \mathbf{g} \\ \varepsilon \mathbf{c}^T & 0 \\
\end{pmatrix}, \qquad \bV =
\begin{pmatrix}
\bU \\ B
\end{pmatrix}.
\end{equation}
Then, the original system \eqref{equ:linear} can be recast as:
\[
\bV_t+\mathbf{C}^\varepsilon \bV_x=0.
\]
It follows from \eqref{equ:linear-A} that $\mathbf{C}=\mathbf{K}^{-1}\mathbf{D}\mathbf{K}$, where
\[
\mathbf{K}=\begin{pmatrix} \mathbf{X} & \mathbf{\Lambda}^{-1}\mathbf{X} \mathbf{g} \\ 0 & 1
\end{pmatrix}, \qquad
\mathbf{D}=\begin{pmatrix} \mathbf{\Lambda} & 0\\0 & 0\end{pmatrix}.
\]
By the perturbation theory of discrete eigenvalues and eigenvectors in
\cite{wilkins1965eigenvalue},
\begin{equation}\label{decom-0}
\mathbf{C}^\varepsilon = (\mathbf{K}^\varepsilon)^{-1}
\mathbf{D}^\varepsilon \mathbf{K}^{\varepsilon},
\end{equation}
where
\[
\mathbf{K}^\varepsilon =
\begin{pmatrix} \mathbf{X}+\varepsilon\hat{\mathbf{X}} &
\mathbf{\Lambda}^{-1}\mathbf{X} \mathbf{g}+\varepsilon\hat{\vec{\alpha}} \\
\varepsilon\hat{\vec{\beta}}^T & 1+\varepsilon\hat{\theta} \\
\end{pmatrix}, \qquad
\mathbf{D}^\varepsilon=\begin{pmatrix}
\mathbf{\Lambda} + \varepsilon \hat{\mathbf{\Lambda}} & 0 \\ 0 & \varepsilon\mu\\
\end{pmatrix}.
\]
\subsection{Zeroth order model}
The first step of our method is to predict the slow variable $B$.
Eliminating the spatial derivative terms of fast variables in
\eqref{equ:linear}, we have
\[
B_t + \varepsilon\mathbf{c}^T\mathbf{A}^{-1}(-\mathbf{g} B_x - \bU_t) = 0,
\]
that is
\begin{equation} \label{equ:linear-0tau}
B_\tau - \mathbf{c}^T\mathbf{A}^{-1}\mathbf{g} B_x - \varepsilon \mathbf{c}^T\mathbf{A}^{-1} \bU_\tau =
0,
\end{equation}
The {\it zeroth order model} (or limiting equation) for the linear
hyperbolic system \eqref{equ:linear} is derived by $\varepsilon\to0$,
namely
\begin{equation} \label{equ:linear-0}
B^{(0)}_\tau + \lambda_B^{(0)} B^{(0)}_x = 0,
\end{equation}
where $\lambda_B^{(0)} = -\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}$. We note that the zeroth
order model \eqref{equ:linear-0} is established only on the hypothesis
that $\bU_\tau$ is bounded. In particular, this hypothesis holds when
the fast dynamics has a steady state (up to $\mathcal{O}(\varepsilon)$) with
fixed slow variable in the initial state. At this point, the zeroth
order model \eqref{equ:linear-0} provides a prediction for $B$ on the
interval $[0,\Delta\tau]$ as
\[
B^{(0)}(x,\tau) = B(x-\lambda_B^{(0)} \tau,0), \qquad
\tau \in[0, \Delta \tau].
\]
In light of the prediction, a modified fast dynamics $\bU^{(1)}$ can
be derived which is slightly different from that with fixed slow
variable. Our next step is to calculate the difference between the
steady variable $\bU^{(0)}(x,\tau)$ and $\bU^{(1)}(x,\tau)$ to get the
correction terms, and by this way the correction model for the
riverbed can be obtained. More precisely, we will show this process step by
step strictly for the linear system. First, we have the following
lemma for zeroth order model \eqref{equ:linear-0}:
\begin{lemma}\label{prop:1-conv}
Suppose that $\mathbf{A}\in \mathbb R^{l\times l}$ has only real and discrete
eigenvalues and all eigenvalues satisfy $|\lambda_i|>\delta \gg
\varepsilon$ for some positive $\delta$. The initial dynamics
satisfy
\begin{enumerate}
\item $\mathbf{A}\bU_x(x,0)+\mathbf{g} B_x(x,0) = \mathcal{O}(\varepsilon)$.
\item $B^{(0)}(x,0) = B(x,0) \in W^{1,\infty}(\mathbb R)$.
\end{enumerate}
Then
\[
\|B(x,t) - B^{(0)}(x,t)\|_{\infty} = \mathcal{O}(\varepsilon) \qquad
\text{for }~ t\sim \mathcal{O}(\varepsilon^{-1}).
\]
Moreover, if $B(x,0) \in W^{2,\infty}(\mathbb{R})$ and $\mathbf{A}\bU_{xx}(x,0)+\mathbf{g}
B_{xx}(x,0)=\mathcal{O}(\varepsilon)$, then
\[
\|B_x(x,t) - B^{(0)}_x(x,t)\|_{\infty} = \mathcal{O}(\varepsilon) \qquad
\text{for }~ t\sim \mathcal{O}(\varepsilon^{-1}).
\]
\end{lemma}
\begin{proof}
See Appendix \ref{proof:1-conv}.
\end{proof}
\subsection{First order model}
Now we consider the case in which $\varepsilon$ is small but does not
tend to zero. The basic idea here is to improve the accuracy of the
weakly
coupled term $-\varepsilon \mathbf{c}^T \mathbf{A}^{-1}\bU_\tau$ by zeroth order
model \eqref{equ:linear-0}. More specifically, we first substitute $B$
in \eqref{equ:linear} by the solution of the zeroth order model, i.e.
\begin{equation}\label{equ:U1}
\bU^{(1)}_t + \mathbf{A} \bU^{(1)}_x = -\mathbf{g} B^{(0)}_x(x,t).
\end{equation}
Then, the corrected equation of $B$ can be derived by
\eqref{equ:linear-0tau} and \eqref{equ:U1} that
\begin{equation}\label{equ:B1-hat}
\hat{B}^{(1)}_\tau -\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}
\hat{B}^{(1)}_x-\varepsilon\mathbf{c}^T\mathbf{A}^{-1}\bU^{(1)}_\tau=0.
\end{equation}
The following lemma indicates that the error between the solution of
\eqref{equ:B1-hat} and original equations \eqref{equ:linear} is of
$\mathcal{O}(\varepsilon^2)$ order.
\begin{lemma}\label{lemma:2-conv-1}
Under the same assumptions of Lemma \ref{prop:1-conv}, and assume the
initial dynamics satisfy
\begin{enumerate}
\item $(\mathbf{A}+\varepsilon \mathbf{X}^{-1}\mathbf{\Lambda}\hat{\mathbf{X}})\bU_x(x,0) +
(\mathbf{g}+\varepsilon\mathbf{X}^{-1}\mathbf{\Lambda}\hat{\alpha})B_x(x,0) =\mathcal{O}(\varepsilon^2)$.
\item $\mathbf{A}\bU_{xx}(x,0)+\mathbf{g} B_{xx}(x,0) =\mathcal{O}(\varepsilon)$.
\item $(\mathbf{A} + \varepsilon \mathbf{A})\bU_x^{(1)}(x,0) + (\mathbf{g} + \varepsilon
\mathbf{g} - \varepsilon\mathbf{A}^{-1}\mathbf{g}\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g})B_x^{(0)}(x,0) =
\mathcal{O}(\varepsilon^2)$.
\item $B^{(0)}(x,0) = \hat{B}^{(1)}(x,0) = B(x,0) \in
W^{2,\infty}(\mathbb R)$.
\end{enumerate}
Then
\[
\lVert B(x,t)-\hat{B}^{(1)}(x,t)\rVert_\infty=\mathcal{O}(\varepsilon^2) \qquad
\text{for }~ t\sim \mathcal{O}(\varepsilon^{-1}).
\]
\end{lemma}
\begin{proof}
See Appendix \ref{proof:2-conv-1}.
\end{proof}
\begin{remark}
It is expected that if we apply the above steps repeatedly (i.e.
substitute $B^{(0)}$ in \eqref{equ:U1} by the solution of
\eqref{equ:B1-hat}, and solve \eqref{equ:B1-hat} where $U^{(1)}$
substituted by the solution of the new equation), the results of
better accuracy will be achieved. In nonlinear cases, however, it may
be of some difficulties to get the correction terms (which will be
introduced later) of more than second order accuracy, so we only
consider the model up to second order accuracy, i.e. the first order
model.
\end{remark}
The equation \eqref{equ:B1-hat} does not provide a
convenient way to compute $\hat{B}^{(1)}$ due to the appearance of
$\bU^{(1)}_\tau$. Let us keep in mind that our aim is to deduce an
equation which only has the riverbed as the variable (just like
\eqref{equ:linear-0}). To this end, one needs to represent
$\bU^{(1)}_\tau$ by the riverbed up to the accuracy of order
$\mathcal{O}(\varepsilon^2)$.
Suppose $\bU^{(0)}$ is the steady state when $B$ is fixed to
$B^{(0)}(x,0)$, namely $\bU^{(0)}_\tau=0$. This implies that
$(\bU^{(1)}-\bU^{(0)})_\tau=\bU^{(1)}_\tau$, and thereafter we
consider $\vec{\varphi}=\bU^{(1)}-\bU^{(0)}$ other than $\bU^{(1)}$.
This trick is also used for the nonlinear systems: instead of solving
the coupled equations to get $\bU^{(1)}_\tau$, we use the steady
state $\bU^{(0)}$ and calculate the difference $\vec{\varphi}$ to
obtain $\bU^{(1)}$. Now, we derive the closed form of $\vec{\varphi}$.
First, we have the following equations:
\[
\begin{aligned}
\varepsilon \bU^{(0)}_{\tau} + \mathbf{A} \bU^{(0)}_x &= -\mathbf{g}
B^{(0)}_x(x, 0), \\
\varepsilon \bU^{(1)}_{\tau} + \mathbf{A} \bU^{(1)}_x &= -
\mathbf{g} B^{(0)}_x(x-\lambda_B^{(0)}\tau, 0).
\end{aligned}
\]
Taking the difference of the above dynamics and applying the
characteristic decomposition of $\mathbf{A}$, we have
\begin{equation}\label{equ:linear-varphi0}
\mathbf{X}\vec{\varphi}_\tau + \frac{\mathbf{\Lambda}}{\varepsilon}\mathbf{X}
\vec{\varphi}_x = -\frac{\mathbf{X}\mathbf{g}}{\varepsilon}
\big(B^{(0)}_x(x-\lambda^{(0)}_B\tau,0)- B^{(0)}_x(x,0)\big).
\end{equation}
The initial condition of $\vec{\varphi}$ can be derived from Lemma
\ref{lemma:2-conv-1} and steady state of $\bU^{(0)}$, i.e.
\begin{equation}\label{equ:varphi-init}
\mathbf{X}\vec{\varphi}_x(x,0)=
\varepsilon\mathbf{\Lambda}^{-2}\mathbf{X}\mathbf{g}\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}
B_x^{(0)}(x,0)+\mathcal{O}(\varepsilon^2).
\end{equation}
Then, \eqref{equ:linear-varphi0} can be solved analytically by the
method of characteristics as
\[
\begin{aligned}
& (\mathbf{X}\vec{\varphi})_k(x, \tau) \\
=&(\mathbf{X}\vec{\varphi})_k(x-\frac{\lambda_k}{\varepsilon}\tau,0)\\
&-\frac{(\mathbf{X}\mathbf{g})_k}{\varepsilon} \int_0^{\tau}
B^{(0)}_x\big( x-\frac{\lambda_k}{\varepsilon}\tau +
(\frac{\lambda_k}{\varepsilon}-\lambda_B^{(0)})s, 0\big) -
B^{(0)}_x\big( x-\frac{\lambda_k}{\varepsilon}\tau +
\frac{\lambda_k}{\varepsilon}s, 0 \big) \rd s \\
=& -\frac{(\mathbf{X}\mathbf{g})_k}{(\lambda_k-\varepsilon\lambda_B^{(0)})}
B^{(0)}(x-\lambda_B^{(0)}\tau,0) +
\frac{(\mathbf{X}\mathbf{g})_k}{\lambda_k}B^{(0)}(x, 0) \\
&+(\mathbf{X}\vec{\varphi})_k(x-\frac{\lambda_k}{\varepsilon}\tau,0)
+\frac{\varepsilon(\mathbf{X}\mathbf{g})_k\lambda_B^{(0)}}{\lambda_k(\lambda_k-\varepsilon \lambda_B^{(0)})}
B^{(0)}(x-\frac{\lambda_k}{\varepsilon}\tau, 0).
\end{aligned}
\]
That is,
\begin{flalign}
(\mathbf{X}\vec{\varphi})_k(x,\tau) = &-
\frac{\varepsilon(\mathbf{X}\mathbf{g})_k\lambda_B^{(0)}}{\lambda_k(\lambda_k-\varepsilon\lambda_B^{(0)})}
B^{(0)}(x, 0) & \cdots\cdots ~
\mathcal{O}(\varepsilon)~\text{term} \nonumber\\
&-\frac{(\mathbf{X}\mathbf{g})_k}{(\lambda_k-\varepsilon\lambda_B^{(0)})}
\big(B^{(0)}(x-\lambda_B^{(0)}\tau, 0) - B^{(0)}(x, 0)\big) &
\cdots\cdots ~ \mathcal{O}(\tau)~\text{term} \label{equ:linear-varphi}\\
&+(\mathbf{X}\vec{\varphi})_k(x-\frac{\lambda_k}{\varepsilon}\tau,0)
&\cdots\cdots ~ \text{high
order term}\nonumber\\
&+\frac{\varepsilon(\mathbf{X}\mathbf{g})_k\lambda_B^{(0)}}{\lambda_k(\lambda_k-\varepsilon \lambda_B^{(0)})}
B^{(0)}(x-\frac{\lambda_k}{\varepsilon}\tau, 0).
& \cdots\cdots ~ \text{high order term} \nonumber
\end{flalign}
We will show later that the last two terms provide
$\mathcal{O}(\varepsilon^2)$ term in the error estimation. This is
why they are called {\it high order terms}. In the nonlinear system,
the correction term will be shown to have a similar form.
Denote $\mathbf{e}_k (k=1,\cdots, l)$ the unit vectors of $\mathbb R^l$ and
${\text{diag}}(\bx)$ the diagonal matrix with vector $\bx$ in its diagonal
entries. By the decomposition \eqref{equ:linear-varphi} and initial
condition \eqref{equ:varphi-init}, we have
\[
\begin{aligned}
\vec{\varphi}_\tau(x,\tau) = &~ \lambda_B^{(0)}\mathbf{X}^{-1}(\mathbf{\Lambda}-
\varepsilon\lambda_B^{(0)}\mathbf{I})^{-1}\mathbf{X}\mathbf{g}
B_x^{(0)}(x-\lambda_B^{(0)}\tau,0)\\
&~-\frac{1}{\varepsilon}\mathbf{X}^{-1}\mathbf{\Lambda}
\left(\sum_{k=1}^l(\mathbf{X}\vec{\varphi}_x)_k(x-\frac{\lambda_k}{\varepsilon}\tau,0)\mathbf{e}_k\right)\\
&-\lambda_B^{(0)}\mathbf{X}^{-1}(\mathbf{\Lambda}-\varepsilon
\lambda^{(0)}_B\mathbf{I})^{-1}{\text{diag}}(\mathbf{X}\mathbf{g})
\left(\sum_{k=1}^l B^{(0)}_x(x-\frac{\lambda_k}{\varepsilon}\tau,0)
\mathbf{e}_k \right) \\
=&~ \lambda_B^{(0)}\mathbf{X}^{-1}\mathbf{\Lambda}^{-1}\mathbf{X}\mathbf{g}
B_x^{(0)}(x-\lambda_B^{(0)}\tau,0)\\
&~-\mathbf{X}^{-1}\mathbf{\Lambda}^{-1}{\text{diag}}(\mathbf{X}\mathbf{g}\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g})
\left(\sum_{k=1}^l B^{(0)}_x(x-\frac{\lambda_k}{\varepsilon}\tau,0)\mathbf{e}_k\right)\\
&~-\lambda_B^{(0)}\mathbf{X}^{-1}\mathbf{\Lambda}^{-1}
{\text{diag}}(\mathbf{X}\mathbf{g})\left(\sum_{k=1}^l B^{(0)}_x(x-\frac{\lambda_k}{\varepsilon}\tau,0)
\mathbf{e}_k \right) + \mathcal{O}(\varepsilon).
\end{aligned}
\]
Define
\[
\tilde{\mathbf{c}}^T = \mathbf{c}^T \mathbf{A}^{-1}\mathbf{X}^{-1}\mathbf{\Lambda}^{-1}
{\text{diag}}\left(\lambda_B^{(0)}\mathbf{X}\mathbf{g} +\mathbf{X}\mathbf{g}\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}
\right) \qquad \tilde{c}_k = \mathbf{c}^T \mathbf{e}_k.
\]
Note that $\bU^{(1)}_\tau=\vec{\varphi}_\tau$ , the model
\eqref{equ:B1-hat} can be rewritten as
\begin{equation}\label{equ:order1-approx1}
\begin{aligned}
\hat{B}^{(1)}_\tau - \mathbf{c}^T\mathbf{A}^{-1}\mathbf{g} \hat{B}^{(1)}_x = &~
\varepsilon\lambda_B^{(0)}\mathbf{c}^T\mathbf{A}^{-2}\mathbf{g}
B_x^{(0)}(x-\lambda_B^{(0)}\tau,0)\\
&-\varepsilon \sum_{k=1}^l\tilde{c}_k
B^{(0)}_x(x-\frac{\lambda_k}{\varepsilon}\tau,0) +
\mathcal{O}(\varepsilon^2).
\end{aligned}
\end{equation}
By discarding the last two terms in \eqref{equ:order1-approx1} and
denote its solution by $\tilde{B}^{(1)}$, we have
\begin{equation}\label{equ:order1-approx2}
\tilde{B}^{(1)}_\tau-\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}
\tilde{B}^{(1)}_x=\varepsilon\lambda_B^{(0)}\mathbf{c}^T\mathbf{A}^{-2}\mathbf{g}
B_x^{(0)}(x-\lambda_B^{(0)}\tau,0).
\end{equation}
Now, we show that $\lVert \hat{B}^{(1)}-\tilde{B}^{(1)}\rVert_\infty=
\mathcal{O}(\varepsilon^2)$ for $\tau\sim\mathcal{O}(1)$. Taking the difference of
\eqref{equ:order1-approx2} and \eqref{equ:B1-hat}. Let $\hat{E} =
\hat{B}^{(1)}-\tilde{B}^{(1)}$, since \eqref{equ:B1-hat} can be
rewritten as \eqref{equ:order1-approx1}, we have
\[
\hat{E}_\tau+\lambda_B^{(0)}\hat{E}_x
= \varepsilon \sum_{k=1}^l \tilde{c}_k
B_x^{(0)}(x-\frac{\lambda_k}{\varepsilon}\tau,0) +
\mathcal{O}(\varepsilon^2),
\]
with initial condition $\hat{E}(x,0)=0$. By the method of
characteristics again,
\[
\begin{aligned}
\hat{E}(x, \tau) &= \varepsilon \sum_{k=1}^l \tilde{c}_k
\int_0^\tau
B_x^{(0)}(x-\lambda_B^{(0)}\tau+(\lambda_B^{(0)} -
\frac{\lambda_k}{\varepsilon})s,0) \rd s + \mathcal{O}(\varepsilon^2)\\
& = \sum_{k=1}^l
\frac{\varepsilon^2}{\varepsilon\lambda_B^{(0)}-\lambda_k} \left(
B^{(0)}(x-\frac{\lambda_k}{\varepsilon}\tau,0)-B^{(0)}(x-\lambda_B^{(0)}\tau,0)
\right) + \mathcal{O}(\varepsilon^2) = \mathcal{O}(\varepsilon^2).
\end{aligned}
\]
Hence, $\tilde{B}^{(1)}$ is proven to be a $\mathcal{O}(\varepsilon^2)$ order
approximation to $\hat{B}^{(1)}$, and thus the $\mathcal{O}(\varepsilon^2)$
order approximation to original solution $B$ by Lemma
\ref{lemma:2-conv-1}. The final formula of {\it first order model}
can be acquired by modifying \eqref{equ:order1-approx2} as
\[
B^{(1)}_\tau-\mathbf{c}^T\mathbf{A}^{-1}\mathbf{g}
B^{(1)}_x=\varepsilon\lambda^{(0)}_B\mathbf{c}^T\mathbf{A}^{-2} \mathbf{g}
B^{(1)}_x(x,\tau).
\]
Or,
\begin{equation} \label{equ:linear-1}
B^{(1)}_\tau + \lambda_B^{(1)} B^{(1)}_x = 0,
\end{equation}
where $\lambda_B^{(1)} = \lambda_B^{(0)} - \varepsilon \lambda_B^{(0)}
\mathbf{c}^T\mathbf{A}^{-2}\mathbf{g}$.
It is straightforward to prove that $B^{(1)}$ is a
$\mathcal{O}(\varepsilon^2)$ order approximation to $\tilde{B}^{(1)}$ and thus
a $\mathcal{O}(\varepsilon^2)$ order approximation to original solution $B$,
which is precisely the following theorem:
\begin{theorem}\label{prop:2-conv}
Assume the initial dynamics satisfies
\begin{enumerate}
\item $(\mathbf{A}+\varepsilon \mathbf{X}^{-1}\mathbf{\Lambda}\hat{\mathbf{X}})\bU_x(x,0) +
(\mathbf{g}+\varepsilon\mathbf{X}^{-1}\mathbf{\Lambda}\hat{\alpha})B_x(x,0) =\mathcal{O}(\varepsilon^2)$.
\item $\mathbf{A}\bU_{xx}(x,0) + \mathbf{g} B_{xx}(x,0) = \mathcal{O}(\varepsilon)$.
\item $B^{(1)}(x,0) = B(x,0) \in W^{2,\infty}(\mathbb R)$.
\end{enumerate}
Then
\[
\|B(x,t) - B^{(1)}(x,t)\|_{\infty} = \mathcal{O}(\varepsilon^2) \qquad
\text{for}~ t\sim \mathcal{O}(\varepsilon^{-1}).
\]
\end{theorem}
In nonlinear cases, $\lambda_B^{(0)}$ and $\lambda_B^{(1)}$ are
functions of the steady state other than constants. Therefore, the
steady state as well as the correction term $\vec{\varphi}$ should be
computed at every step when solving zeroth order model or the first
order model.
Although we actually do not have to use $\vec{\varphi}$ in linear cases (we
only need to use $\bU^{(1)}_\tau$), the correction term $\vec{\varphi}$
is essential in nonlinear cases due to two reasons: (i) it is needed
to give the high order algorithm for solving first order model; (ii)
it will be used to give the prediction of fast variables at next step,
which could improve the efficiency of computing the steady state.
\section{One Dimensional Sediment Transport Model}\label{sec:1d}
Now we consider the one dimensional sediment transport model
\eqref{equ:1d-sediment}. The process here is quite similar with the
linear case. First, we derive the formulation of riverbed equation
through the original coupled system. Then, the zeroth order model is
obtained by taking $\varepsilon\rightarrow0$. After that, the
correction term is considered and first order model will then be
derived.
\subsection{Zeroth order model} \label{subsec:1d-0} First,
we reformulate the sediment transport systems
\eqref{equ:1d-sediment} with primitive variables as
\begin{equation} \label{equ:1d-equation}
\left\{
\begin{aligned}
&\begin{pmatrix}
h \\ u
\end{pmatrix}_t +
\begin{pmatrix}
u & h \\
g & u \\
\end{pmatrix}
\begin{pmatrix}
h \\ u
\end{pmatrix}_x = \begin{pmatrix}
0 \\ -gB_x
\end{pmatrix}, \\
&B_t + \varepsilon \big(\tilde{q}_b(|u|)+|u|\tilde{q}_b'(|u|)\big)
u_x = 0.
\end{aligned}
\right.
\end{equation}
When the flow is subcritical, namely $|u|<\sqrt{gh}$ everywhere, the
fast dynamics in \eqref{equ:1d-equation} deduces that
\[
\left\{
\begin{aligned}
h_x &= \frac{1}{u^2-gh} (-uh_t + hu_t + ghB_x), \\
u_x &= \frac{1}{u^2-gh} (gh_t - uu_t - guB_x).
\end{aligned}
\right.
\]
Eliminating the spatial derivative terms of fast variables in the
sediment transport equation, we have
\begin{equation} \label{equ:1d-0tau}
B_\tau - \frac{gu\tilde{\lambda}_b(|u|)}{u^2-gh}B_x + \varepsilon
\frac{\tilde{\lambda}_b(|u|)}{u^2-gh}(gh_\tau - uu_\tau) = 0,
\end{equation}
where $\tilde{\lambda}_b(|u|) = \tilde{q}_b(|u|) +
|u|\tilde{q}_b'(|u|)$. The {\it zeroth order model} (or the
{\it limiting equation}) for the 1D sediment transport systems
\eqref{equ:1d-sediment} is derived by taking $\varepsilon \to 0$ in
\eqref{equ:1d-0tau} as
\begin{equation} \label{equ:1d-0}
B^{(0)}_\tau + \lambda_B^{(0)}(h^{(0)},u^{(0)}) B^{(0)}_x = 0,
\end{equation}
where $u^{(0)},h^{(0)}$ are the steady states with fixed riverbed, and
$\lambda_B^{(0)}(h,u) = -\dfrac{gu\tilde{\lambda}_b(|u|)}{u^2-gh}$. It
should be noted that $h^{(0)}$ and $u^{(0)}$ are the functions of
riverbed $B^{(0)}$. Unlike the linear hyperbolic system, the
characteristic speed of \eqref{equ:1d-0} depends on the fast
variables. Similar to the discussion in section \ref{sec:1d-linear},
$h_\tau$ and $u_\tau$ are assumed to be bounded so that the last term
in \eqref{equ:1d-0tau} tends to zero when $\varepsilon\to0$. Usually,
the steady state of flow exists with fixed riverbed when appropriately
applying the boundary condition, which implies that $h_\tau$ and
$u_\tau$ are bounded for the sediment transport.
\begin{remark}
From the characteristic speed of riverbed in \eqref{equ:1d-0tau},
we know that $|\lambda_B^{(0)}|\rightarrow\infty$ if $|u|\rightarrow
\sqrt{gh}$, which means that our model can only handle the case
in which $|u|$ stays away from $\sqrt{gh}$ everywhere. This can be
guaranteed for the subcritical case $|u| < \sqrt{gh}$.
\end{remark}
\subsection{First order model} \label{subsec:1d-1}
As with linear case, the most essential step in deriving the
correction model is to compare the fast dynamics with fixed riverbed
to the modified one with riverbed moving according to zeroth order
model. Let $\tau_0$ be the base time with the riverbed $B^{(0)}(x,
\tau_0)$. At time $\tau = \tau_0 + \tilde{\tau}$,
\begin{equation} \label{equ:1d-diff}
\begin{aligned}
\begin{pmatrix}
h^{(0)} \\ u^{(0)}
\end{pmatrix}_\tau + \frac{1}{\varepsilon}
\begin{pmatrix}
u^{(0)} & h^{(0)} \\ g & u^{(0)}
\end{pmatrix}
\begin{pmatrix}
h^{(0)} \\ u^{(0)}
\end{pmatrix}_x &=
\begin{pmatrix}
0 \\ -\dfrac{g}{\varepsilon}B^{(0)}_x(x,\tau_0)
\end{pmatrix}, \\
\begin{pmatrix}
h^{(1)} \\ u^{(1)}
\end{pmatrix}_\tau + \frac{1}{\varepsilon}
\begin{pmatrix}
u^{(1)} & h^{(1)} \\ g & u^{(1)}
\end{pmatrix}
\begin{pmatrix}
h^{(1)} \\ u^{(1)}
\end{pmatrix}_x &=
\begin{pmatrix}
0 \\ -\dfrac{g}{\varepsilon}B^{(0)}_x(x,\tau_0 + \tilde{\tau})
\end{pmatrix}.
\end{aligned}
\end{equation}
Thereafter, we often consider the case in which $\tau_0 = 0$ to make
it more concise. Let
$$
\varphi_h(x,\tilde{\tau}) = h^{(1)}(x,\tilde{\tau})-h^{(0)}(x),
\qquad \varphi_u(x,\tilde{\tau}) = u^{(1)}(x,\tilde{\tau})-u^{(0)}(x)
$$ be
the correction of fast variables. Similar to
\eqref{equ:linear-varphi}, we intend to decompose $\varphi_h$ and
$\varphi_u$ into the sum of $\mathcal{O}(\varepsilon)$ term,
$\mathcal{O}(\tilde{\tau})$ term and high order term, namely
\begin{equation} \label{equ:1d-correction}
\begin{aligned}
\varphi_h(x, \tilde{\tau}) &= \varepsilon \varphi_h^{(0)}(x) +
\varphi_h^{(1)}(x,\tilde{\tau}) + \text{high order term}, \\
\varphi_u(x, \tilde{\tau}) &= \varepsilon \varphi_u^{(0)}(x) +
\varphi_u^{(1)}(x,\tilde{\tau}) + \text{high order term}.
\end{aligned}
\end{equation}
\subsubsection{$\mathcal{O}(\tilde{\tau})$ Term}
Let $\tilde{B}^{(0)} = B^{(0)}(x, \tilde{\tau})$. Taking the
difference of the two dynamics in \eqref{equ:1d-diff} to obtain
\[
\left\{
\begin{aligned}
&(\varphi_h)_\tau + \frac{1}{\varepsilon}
(h^{(1)}u^{(1)} - h^{(0)}u^{(0)})_x = 0, \\
&(\varphi_u)_\tau + \frac{1}{\varepsilon}
\left[ gh^{(1)} + \frac{1}{2}(u^{(1)})^2 + g\tilde{B}^{(0)}
- gh^{(0)}-\frac{1}{2}(u^{(0)})^2-gB^{(0)} \right]_x = 0.
\end{aligned}
\right.
\]
Then, eliminating $h^{(1)}, u^{(1)}$ and neglecting the high order term
to obtain the linearized equation as
\begin{equation} \label{equ:1d-linearization}
\left\{
\begin{aligned}
&(\varphi_h)_\tau + \frac{1}{\varepsilon}
(h^{(0)}\varphi_u + u^{(0)}\varphi_h)_x \approx 0, \\
&(\varphi_u)_\tau + \frac{1}{\varepsilon}
(g\varphi_h + u^{(0)}\varphi_u +g\tilde{B}^{(0)}-gB^{(0)})_x
\approx 0.
\end{aligned}
\right.
\end{equation}
Collecting the $\mathcal{O}(1/\varepsilon)$ terms and assuming that
$\varphi_h^{(1)},\varphi_u^{(1)}$ are zero when $|x|\to \infty$, we
have
\begin{equation}\label{equ:1d-varphi1-equ}
\left\{
\begin{aligned}
&h^{(0)}\varphi_u^{(1)} + u^{(0)}\varphi_h^{(1)}=0,\\
&g\varphi_h^{(1)}+u^{(0)}\varphi_u^{(1)}+g\tilde{B}^{(0)}-gB^{(0)}=0.
\end{aligned}
\right.
\end{equation}
Namely,
\begin{equation} \label{equ:1d-varphi1}
\varphi_h^{(1)} = \frac{gh^{(0)}}{(u^{(0)})^2-gh^{(0)}}(\tilde{B}^{(0)} -
B^{(0)}), \qquad
\varphi_u^{(1)} = -\frac{gu^{(0)}}{(u^{(0)})^2-gh^{(0)}}(\tilde{B}^{(0)} -
B^{0}).
\end{equation}
\subsubsection{$\mathcal{O}(\varepsilon)$ Term}
Consider the remaining terms in $\varphi_h,\varphi_u$ besides the
$\mathcal{O}(\tilde{\tau})$ terms.
Taking $\varphi_h = \varphi_h^{(1)} + \varepsilon \varphi_h^{(0)}$ and
$\varphi_u = \varphi_u^{(1)} + \varepsilon \varphi_u^{(0)}$ into
\eqref{equ:1d-linearization}, we have
\[
\left\{
\begin{aligned}
(u^{(0)}\varphi_h^{(0)} + h^{(0)}\varphi_u^{(0)})_x &=
\frac{gh^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x -
\varepsilon (\varphi_h^{(0)})_\tau, \\
(g\varphi_h^{(0)} + u^{(0)}\varphi_u^{(0)})_x &= -
\frac{gu^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x
-\varepsilon (\varphi_u^{(0)})_\tau.
\end{aligned}
\right.
\]
Collecting the $\mathcal{O}(1)$ term, we obtain
\begin{equation} \label{equ:1d-varphi0}
\left\{
\begin{aligned}
(u^{(0)}\varphi_h^{(0)} + h^{(0)}\varphi_u^{(0)})_x &=
\frac{gh^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x, \\
(g\varphi_h^{(0)} + u^{(0)}\varphi_u^{(0)})_x &= -
\frac{gu^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x.
\end{aligned}
\right.
\end{equation}
Notice that \eqref{equ:1d-varphi0} is a linear system for
$\varphi_h^{(0)}$ and $\varphi_u^{(0)}$, which is convenient to be
numerically solved, see Subsection \ref{sec:scheme-correction}.
\subsubsection{Slow variable correction}
Having the $\mathcal{O}(\tilde{\tau})$ and $\mathcal{O}(\varepsilon)$ terms,
\eqref{equ:1d-0tau} can be reformulated as
\[
\hat{B}^{(1)}_\tau -
\frac{gu^{(1)}\tilde{\lambda}_b(|u^{(1)}|)}{(u^{(1)})^2-gu^{(1)}}
\hat{B}^{(1)}_x + \varepsilon
\frac{g((u^{(1)})^2+gh^{(1)})\tilde{\lambda}_b(|u^{(1)}|)}
{((u^{(1)})^2-gh^{(1)})^2} \tilde{B}^{(0)}_\tau = 0,
\]
where $h^{(1)} = h^{(0)} + \varepsilon \varphi_h^{(0)} +
\varphi_h^{(1)}, u^{(1)} = u^{(0)} + \varepsilon \varphi_u^{(0)} +
\varphi_u^{(1)}$.
Similarly, the last term is regarded as a correction term on the
characteristic speed of slow variable. Let $\tilde{\tau} \rightarrow
0$, we derive the model with {\it first order correction} for the 1D
sediment transport systems \eqref{equ:1d-equation} as
\begin{equation} \label{equ:1d-1}
B^{(1)}_\tau + \lambda_B^{(1)}(h^{(0)}+\varepsilon \varphi_h^{(0)},
u^{(0)}+\varepsilon \varphi_u^{(0)})B^{(1)}_x = 0,
\end{equation}
where
\[
\lambda_B^{(1)}(h,u) = \lambda_B^{(0)}(h,u)-\varepsilon\lambda_B^{(0)}(h,u)
\frac{g(u^2+gh)\tilde{\lambda}_b(|u|)}{(u^2-gh)^2}.
\]
\section{Two Dimensional Sediment Transport Model} \label{sec:2d}
We move to the modelling of the two dimensional hyperbolic system
governing sediment transport. We will see later that there is an
essential difference between one dimensional model and two dimensional
model. In two dimensional case, the model cannot be obtained by
eliminating the spatial derivative terms of fast variables. Instead,
the two dimensional model is a convection equation with source term,
while is consistent with the one dimensional model.
\subsection{Zeroth order model}
Let
\[
\nabla\cdot\bu = u_x+v_y, \quad
\bu\cdot \nabla\bu = (uu_x+vu_y, uv_x+vv_y)^T,
\]
then
\[
\begin{aligned}
(u\tilde{q}_b(|\bu|))_x + (v\tilde{q}_b(|\bu|))_y &= \tilde{q}_b u_x
+ \frac{\tilde{q}_b'}{|\bu|}(u^2u_x+uvv_x)\\
&\quad + \tilde{q}_b v_y +\frac{\tilde{q}_b'}{|\bu|}(vuu_y+v^2v_y) \\
&= \tilde{q}_b(\nabla\cdot\bu) + \frac{\tilde{q}_b'}{|\bu|}\bu^T
(\bu\cdot\nabla\bu).
\end{aligned}
\]
We reformulate the 2D sediment transport system
\eqref{equ:2d-sediment} with primitive variables $\mathbf{W}=(h,\bu,B)^T$ as%
\begin{equation} \label{equ:2d-equation}
\left\{
\begin{aligned}
&h_t + h\nabla\cdot\bu + \bu^T\nabla h = 0, \\
&\bu_t + \bu\cdot\nabla\bu + g\nabla h = -g\nabla B, \\
&B_t + \varepsilon \big[ \tilde{q}_b(|\bu|)(\nabla\cdot\bu)
+ \frac{\tilde{q}_b'(|\bu|)}{|\bu|}\bu^T (\bu\cdot\nabla\bu)
\big] = 0.
\end{aligned}
\right.
\end{equation}
By the equations of mass and momentum conservation in
\eqref{equ:2d-equation}, we have
\[
\begin{aligned}
gh_t + gh\nabla\cdot\bu + g\bu^T\nabla h &=0, \\
\frac{1}{2}(|\bu|^2)_t + \bu^T(\bu\cdot\nabla\bu) + g\bu^T\nabla h
&= -g\bu^T\nabla B.
\end{aligned}
\]
Eliminating the spatial derivative of $h$ to obtain
\begin{equation} \label{equ:2d-du}
\bu^T(\bu\cdot\nabla\bu) - gh\nabla\cdot\bu = -\bu^T\bu_t + gh_t -
g\bu^T\nabla B.
\end{equation}
Notice that the spatial derivative of $\bu$ can not be solved from
\eqref{equ:2d-du}. Therefore, we introduce a rotational invariant
operator $\cL^{\mathbf{S}}$ as
\begin{equation} \label{equ:2d-L}
\cL^{\mathbf{S}} \bu \triangleq \bu^T(\bu\cdot\nabla\bu) - |\bu|^2
(\nabla\cdot\bu),
\end{equation}
which degenerates to null operator for 1D case $(v=0)$.
Then, the spatial derivative of $\bu$ from \eqref{equ:2d-du} and
\eqref{equ:2d-L} are represented as
\begin{equation} \label{equ:2d-duL}
\left\{
\begin{aligned}
&\nabla\cdot\bu = \frac{-\bu^T\bu_t + gh_t}{|\bu|^2-gh}
- \frac{g\bu^T\nabla B}{|\bu|^2-gh}
- \frac{\cL^\mathbf{S} \bu}{|\bu|^2-gh}, \\
&\bu^T(\bu\cdot\nabla\bu) =
\frac{-\bu^T\bu_t + gh_t}{|\bu|^2-gh}|\bu|^2
- \frac{g\bu^T\nabla B}{|\bu|^2-gh}|\bu|^2
- \frac{gh\cL^\mathbf{S} \bu}{|\bu|^2-gh}.
\end{aligned}
\right.
\end{equation}
Substituting \eqref{equ:2d-duL} into \eqref{equ:2d-equation}, we have
\begin{equation} \label{equ:2d-0tau}
B_\tau - \frac{g\tilde{\lambda}_b(|\bu|)\bu^T}{|\bu|^2-gh} \nabla B
+ \varepsilon
\frac{\tilde{\lambda}_b(|\bu|)}{|\bu|^2-gh} (gh_\tau
-\bu^T\bu_\tau) =
\frac{\tilde{q}_b+gh\frac{\tilde{q}_b'}{|\bu|}}{|\bu|^2-gh}\cL^\mathbf{S} \bu,
\end{equation}
where $\tilde{\lambda}_b(|\bu|) = \tilde{q}_b(|\bu|) +
|\bu|\tilde{q}_b'(|\bu|)$. Similar to 1D case, the {\it zeroth order
model} for the 2D sediment transport system \eqref{equ:2d-equation}
can be derived by taking $\varepsilon \to 0$,
\begin{equation} \label{equ:2d-0}
B^{(0)}_\tau + \vec{\lambda}_B^{(0)}(h^{(0)},\bu^{(0)}) \nabla B^{(0)} =
S_B^{(0)}(h^{(0)},\bu^{(0)}),
\end{equation}
where
\begin{equation}\label{equ:2d-lambda-S}
\vec{\lambda}_B^{(0)}(h^{(0)},\bu^{(0)}) =
-\frac{g\tilde{\lambda}_b(|\bu^{(0)}|)\bu^{(0)T}}{|\bu^{(0)}|^2-gh^{(0)}},
\qquad S_B^{(0)}(h^{(0)},\bu^{(0)}) =
\frac{\tilde{q}_b+gh^{(0)}\frac{\tilde{q}_b'}{|\bu^{(0)}|}}{|\bu^{(0)}|^2-gh^{(0)}}\cL^\mathbf{S}
\bu^{(0)},
\end{equation}
and $h^{(0)},\bu^{(0)}$ are steady states with respect to $B^{(0)}$.
Roughly speaking, \eqref{equ:2d-0} is not a convective equation with
source term, since $S_B^{(0)}$ in \eqref{equ:2d-0} involves the spatial
derivative of fast variables. From the comparison of \eqref{equ:1d-0}
and \eqref{equ:2d-0}, we find that the zeroth order model of 2D case
is consistent with the model of 1D case, namely, \eqref{equ:2d-0}
degenerates to \eqref{equ:1d-0} if $v=0$ and $B_y=0$. Therefore, one
may expect that \eqref{equ:2d-0} has captured the leading order part
of the characteristic speed for sediment transport.
\subsection{First order correction}
Similar to the 1D case, we first calculate the difference between the
fast dynamics with fixed slow variable and the modified one with
predicted slow variable. Using the similar notation with 1D case, we
have
\begin{equation} \label{equ:chap7-2d-diff}
\begin{aligned}
& \left\{
\begin{aligned}
h^{(0)}_\tau + \frac{1}{\varepsilon} [h^{(0)}\nabla\cdot\bu^{(0)} +
\bu^{(0),T}\nabla h^{(0)}]
&= 0, \\
\bu^{(0)}_\tau + \frac{1}{\varepsilon}
[\bu^{(0)}\cdot\nabla\bu^{(0)} + g\nabla h^{(0)}]
&= -\frac{1}{\varepsilon}g\nabla B^{(0)},
\end{aligned}
\right.\\
& \left\{
\begin{aligned}
h^{(1)}_\tau + \frac{1}{\varepsilon} [h^{(1)}\nabla
\cdot\bu^{(1)} + \bu^{(1),T} \nabla h^{(1)}] &= 0, \\
\bu^{(1)}_\tau + \frac{1}{\varepsilon} [\bu^{(1)}\cdot
\nabla\bu^{(1)} + g\nabla h^{(1)}]
&= -\frac{1}{\varepsilon}g\nabla \tilde{B}^{(0)}.
\end{aligned}
\right.
\end{aligned}
\end{equation}
Let
$$
\varphi_h = h^{(1)}(\bx,\tilde{\tau}) - h^{(0)}(\bx) \qquad
\vec{\varphi}_\bu = \bu^{(1)}(\bx,\tilde{\tau}) - \bu^{(0)}(\bx).
$$
The following linearized equation for $\varphi_h$ and $\varphi_\bu$
can be derived by dropping off the high order term
\begin{equation} \label{equ:2d-linearization}
\left\{
\begin{aligned}
&(\varphi_h)_\tau + \frac{1}{\varepsilon} \nabla\cdot
(h^{(0)}\vec{\varphi}_\bu + \bu^{(0)}\varphi_h) \approx 0, \\
&(\vec{\varphi}_\bu)_\tau + \frac{1}{\varepsilon} [
\vec{\varphi}_\bu\cdot\nabla\bu^{(0)} + \bu^{(0)}\cdot\nabla\vec{\varphi}_\bu
+ \nabla(g\varphi_h + g(\tilde{B}^{(0)}-B^{(0)}))] \approx \mathbf{0}.
\end{aligned}
\right.
\end{equation}
Similar to 1D case, we intend to decompose $\varphi_h$ and
$\vec{\varphi}_\bu$ into the sum of $\mathcal{O}(\varepsilon)$ term,
$\mathcal{O}(\tilde{\tau})$ term and high order term, i.e.
\begin{equation} \label{equ:2d-fast}
\begin{aligned}
\varphi_h &= \varepsilon \varphi_h^{(0)}(\bx) +
\varphi_h^{(1)}(\bx, \tilde{\tau}) + \text{high order term}, \\
\vec{\varphi}_\bu &= \varepsilon \vec{\varphi}_\bu^{(0)}(\bx) +
\vec{\varphi}_\bu^{(1)}(\bx, \tilde{\tau}) + \text{high order term}.
\end{aligned}
\end{equation}
\subsubsection{$\mathcal{O}(\tilde{\tau})$ Term}
We first try to find the $\mathcal{O}(\tilde{\tau})$ term analytically, which
is used to depict the change of the fast variables with the evolving
of slow variable. In comparison to the 1D case, the mass conservation
in \eqref{equ:2d-linearization} simply tells us that
$h^{(0)}\vec{\varphi}_\bu+\bu^{(0)}\varphi_h$ is divergence free when
neglecting the time derivative. However, the term
$\vec{\varphi}_\bu\cdot\nabla\bu^{(0)} + \bu^{(0)}\cdot\nabla\vec{\varphi}_\bu$
in the momentum conservative equation can not be formulated to a total
derivative . Therefore, we first define
\begin{equation} \label{equ:2d-Lu}
\cL^\bu(\vec{\varphi}_\bu) = \vec{\varphi}_\bu\cdot\nabla\bu +
\bu\cdot\nabla\vec{\varphi}_\bu - \nabla(\bu^T\vec{\varphi}_\bu).
\end{equation}
It is obvious that $\cL^\bu$ is a linear operator and degenerates to
null operator for 1D case ($\varphi_v=0$ or $v=0$). Hereafter, we
devote to analytically matching the flux and source term in
\eqref{equ:2d-linearization} with $\bar{\varphi}_h^{(1)}$ and
$\bar{\vec{\varphi}}_\bu^{(1)}$, namely
\[
h^{(0)}\bar{\vec{\varphi}}_\bu^{(1)} + \bu^{(0)} \bar{\varphi}_h^{(1)} = \mathbf{0},
\qquad \bu^{(0),T}\bar{\vec{\varphi}}_\bu^{(1)} + g\bar{\varphi}_h^{(1)} +
g(\tilde{B}^{(0)} - B^{(0)}) = 0,
\]
which yields
\begin{equation} \label{equ:2d-varphi1-1}
\bar{\varphi}_h^{(1)} = \frac{gh^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}(\tilde{B}^{(0)}-B^{(0)}), \quad
\bar{\vec{\varphi}}_\bu^{(1)} =
-\frac{g\bu^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}(\tilde{B}^{(0)}-B^{(0)}).
\end{equation}
Let $\hat{\varphi}_h^{(1)} = \varphi_h^{(1)} - \bar{\varphi}_h^{(1)}$
and $\hat{\vec{\varphi}}_\bu^{(1)} = \vec{\varphi}_\bu^{(1)} -
\bar{\vec{\varphi}}_\bu^{(1)}$ be the rest parts of $\mathcal{O}(\tilde{\tau})$
term. Notice that the $\mathcal{O}(\tilde{\tau})$ term is used to balance
\eqref{equ:2d-linearization} without time evolving terms, thus the
equation of $\hat{\varphi}_h^{(1)}$ and $\hat{\vec{\varphi}}_\bu^{(1)}$
can be proposed as
\begin{equation} \label{equ:2d-varphi1-2}
\left\{
\begin{aligned}
&\nabla \cdot (h^{(0)}\hat{\vec{\varphi}}_\bu^{(1)} +
\bu^{(0)}\hat{\varphi}_h^{(1)}) = 0, \\
&\cL^{\bu^{(0)}}(\hat{\vec{\varphi}}_\bu^{(1)}) + \nabla \cdot
(g\hat{\varphi}_h^{(1)} + \bu^{(0),T}\hat{\vec{\varphi}}_\bu^{(1)}) =
-\cL^{\bu^{(0)}}(\bar{\vec{\varphi}}_\bu^{(1)}).
\end{aligned}
\right.
\end{equation}
Therefore, the $\mathcal{O}(\tilde{\tau})$ term is composed of the
analytical terms from \eqref{equ:2d-varphi1-1} and the other terms
from \eqref{equ:2d-varphi1-2}. We also note that the
$\mathcal{O}(\tilde{\tau})$ term of 2D case is consistent with that of 1D case
by the degeneration of $\cL^\bu$ in 1D case.
\subsubsection{$\mathcal{O}(\varepsilon)$ Term}
The equations of $\varphi_h^{(0)}$ and $\vec{\varphi}_\bu^{(0)}$ are
deduced by taking the $\mathcal{O}(\tilde{\tau})$ term into linearized
equation \eqref{equ:2d-linearization} as well as neglecting the high
order term, namely
\begin{equation} \label{equ:2d-varphi0-equ}
\left\{
\begin{aligned}
&\nabla\cdot(h^{(0)}\vec{\varphi}_\bu^{(0)} + \bu^{(0)}\varphi_h^{(0)}) =
- (\bar{\varphi}_h^{(1)})_\tau - (\hat{\varphi}_h^{(1)})_\tau -
\varepsilon(\varphi_h^{(0)})_\tau, \\
&\cL^{\bu^{(0)}}(\vec{\varphi}_\bu^{(0)}) + \nabla
(\bu^{(0),T}\vec{\varphi}_\bu^{(0)} + g\varphi_h^{(0)}) =
-(\bar{\vec{\varphi}}_\bu^{(1)})_\tau - (\hat{\vec{\varphi}}_\bu^{(1)})_\tau
-\varepsilon(\vec{\varphi}_\bu^{(0)})_\tau.
\end{aligned}
\right.
\end{equation}
The time derivatives of $\bar{\varphi}_h^{(1)}$ and
$\bar{\vec{\varphi}}_\bu^{(1)}$ can be calculated by
\eqref{equ:2d-varphi1-1} as
\begin{equation} \label{equ:2d-varphi0-bar}
(\bar{\varphi}_h^{(1)})_\tau =
\frac{gh^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}\tilde{B}^{(0)}_\tau,
\qquad (\bar{\vec{\varphi}}_\bu^{(1)})_\tau =
-\frac{g\bu^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}
\tilde{B}^{(0)}_\tau.
\end{equation}
For the time derivatives of $\hat{\varphi}_{h}^{(1)}$ and
$\hat{\vec{\varphi}}_{\bu}^{(1)}$, we have
\begin{equation} \label{equ:2d-varphi0-hat}
\left\{
\begin{aligned}
&\nabla \cdot \big(h^{(0)} (\hat{\vec{\varphi}}_\bu^{(1)})_\tau +
\bu^{(0)} (\hat{\varphi}_h^{(1)})_\tau \big) = 0, \\
&\cL^{\bu^{(0)}} \left( (\hat{\vec{\varphi}}_\bu^{(1)})_\tau \right)
+ \nabla \big( g(\hat{\varphi}_h^{(1)})_\tau + \bu^{(0),T}
(\hat{\vec{\varphi}}_\bu^{(1)})_\tau \big) =
\cL^{\bu^{(0)}}\left(
\frac{g\bu^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}\tilde{B}^{(0)}_\tau
\right),
\end{aligned}
\right.
\end{equation}
by taking a time derivative on \eqref{equ:2d-varphi1-2}. Denote
$\hat{\varphi}_h^{(0)} = \left. (\hat{\varphi}_h^{(1)})_\tau
\right|_{\tilde{\tau}=0}$ and $\hat{\vec{\varphi}}_\bu^{(0)} = \left.
(\hat{\vec{\varphi}}_\bu)^{(1)})_\tau \right|_{\tilde{\tau}=0}$, and it is
easy to check that
\[
\left. \tilde{B}^{(0)}_\tau \right|_{\tilde{\tau}=0} = -\vec{\lambda}_B^{(0)}
\nabla B^{(0)} + S_B^{(0)}.
\]
Taking $\tilde{\tau}\rightarrow 0$ and collecting the $\mathcal{O}(1)$ term in
\eqref{equ:2d-varphi0-equ}, the $\mathcal{O}(\varepsilon)$ term satisfies
\begin{equation} \label{equ:2d-varphi0}
\left\{
\begin{aligned}
&\nabla\cdot(h^{(0)}\vec{\varphi}_\bu^{(0)} + \bu^{(0)}\varphi_h^{(0)}) =
\frac{gh^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}(\vec{\lambda}_B^{(0)} \nabla B^{(0)} - S_B^{(0)}) -
\hat{\varphi}_h^{(0)}, \\
&\cL^{\bu^{(0)}}(\vec{\varphi}_\bu^{(0)}) + \nabla
(\bu^{(0),T}\vec{\varphi}_\bu^{(0)} + g\varphi_h^{(0)}) =
\frac{g\bu^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}(-\vec{\lambda}_B^{(0)} \nabla B^{(0)} +
S_B^{(0)}) -
\hat{\vec{\varphi}}_\bu^{(0)},
\end{aligned}
\right.
\end{equation}
where $\hat{\varphi}_h^{(0)}$ and $\hat{\vec{\varphi}}_\bu^{(0)}$ satisfy
\begin{equation} \label{equ:2d-varphi0-hat0}
\left\{
\begin{aligned}
&\nabla \cdot (h^{(0)} \hat{\vec{\varphi}}_\bu^{(0)} + \bu^{(0)}
\hat{\varphi}_h^{(0)}) = 0, \\
&\cL^{\bu^{(0)}}(\hat{\vec{\varphi}}_\bu^{(0)}) + \nabla (g
\hat{\varphi}_h^{(0)} + \bu^{(0),T} \hat{\vec{\varphi}}_\bu^{(0)}) =
\cL^{\bu^{(0)}} \left(
\frac{g\bu^{(0)}}{|\bu^{(0)}|^2-gh^{(0)}}(-\vec{\lambda}_B^{(0)} \nabla B^{(0)} +
S_B^{(0)}) \right).
\end{aligned}
\right.
\end{equation}
We also note that the equations \eqref{equ:2d-varphi1-2},
\eqref{equ:2d-varphi0} and \eqref{equ:2d-varphi0-hat0} share the
similar form, which is able to be numerically solved, see Subsection
\ref{sec:scheme-correction}.
\subsubsection{Slow variable correction}
From \eqref{equ:2d-varphi1-1}, \eqref{equ:2d-varphi1-2} and
\eqref{equ:2d-varphi0}, the corrected fast variables in
\eqref{equ:2d-fast} can be obtained. Then, we substitute the fast
variables correction into \eqref{equ:2d-0tau} and omit the high order
term to have
\[
\begin{aligned}
\hat{B}^{(1)}_\tau & + \vec{\lambda}_B^{(0)}(h^{(1)}, \bu^{(1)})\nabla \hat{B}^{(1)} \\
& + \varepsilon \frac{g\tilde{\lambda}_b(|\bu^{(1)}|)
(|\bu^{(1)}|^2+gh^{(1)})} {(|\bu^{(1)}|^2-gh^{(1)})^2} \left[
-\vec{\lambda}_B^{(0)}(h^{(1)},\bu^{(1)}) \nabla \hat{B}^{(1)}
+S_B^{(0)}(h^{(1)},\bu^{(1)}) \right] \\
& + \varepsilon
\frac{\tilde{\lambda}_b(|\bu^{(1)}|)} {|\bu^{(1)}|^2-gh^{(1)}}
(h^{(1)}\hat{\varphi}_h^{(0)} - \bu^{(1),T}
\hat{\vec{\varphi}}_\bu^{(0)}) = S_B^{(0)}(h^{(1)}, \bu^{(1)}).
\end{aligned}
\]
Let $\tilde{\tau} \rightarrow0$, the {\it first order
correction model} for the 2D sediment transport system
\eqref{equ:2d-sediment} can be derived as
\begin{equation} \label{equ:2d-1}
\begin{aligned}
B^{(1)}_\tau &+ [\vec{\lambda}_B^{(1)}(h^{(0)}+\varepsilon
\varphi_h^{(0)}, \bu^{(0)}+\varepsilon \vec{\varphi}_\bu^{(0)})
\nabla B^{(1)} \\
&= S_B^{(1)} (h^{(0)}+\varepsilon \varphi_h^{(0)},
\bu^{(0)}+\varepsilon \vec{\varphi}_\bu^{(0)}),
\end{aligned}
\end{equation}
where
\begin{equation}\label{equ:2d-corr-lambda-S}
\begin{aligned}
\vec{\lambda}_B^{(1)}(h,\bu) &=\vec{\lambda}_B^{(0)}(h,\bu) -\varepsilon\vec{\lambda}_B^{(0)}(h,\bu)\frac{g(|\bu|^2+gh)
\tilde{\lambda}_b(|\bu|)}{(|\bu|^2-gh)^2}, \\
S_B^{(1)}(h,\bu) &= S_B^{(0)}(h,\bu)-\varepsilon\bigg[S_B^{(0)}(h,\bu)\frac{g(|\bu|^2+gh)
\tilde{\lambda}_b(|\bu|)}{(|\bu|^2-gh)^2} +
\frac{(h\hat{\varphi}_h^{(0)} - \bu^T\hat{\vec{\varphi}}_\bu^{(0)})
\tilde{\lambda}_b(|\bu|)}{|\bu|^2-gh}\bigg].
\end{aligned}
\end{equation}
\section{Numerical Scheme} \label{sec:scheme}
In this section, we develop the numerical scheme to solve the sediment
transport using the models introduced in the previous sections. Our
numerical scheme basically falls into the framework of HMM method
\cite{weinan2003heterogeneous}, which contains a micro-scale solver,
namely the steady state solver, and a macro-scale solver, namely the
riverbed solver. Meanwhile, our scheme also contains a fast variable
correction which differs from the traditional HMM method.
Briefly, the scheme contains three parts: solving the steady state of
flow, calculating the correction term, and solving the equation of the
riverbed. We only focus on last two parts in this section. After
introducing the algorithms to calculate the correction term and solve the
equation of the riverbed, we will give an implementation framework.
\subsection{Calculating the correction term}\label{sec:scheme-correction}
Correction term contains two parts: the $\mathcal{O}(\tilde{\tau})$ term and
$\mathcal{O}(\varepsilon)$ term. The former part needs $\tilde{B}^{(0)}$ which
is solved later in section \ref{subsec:advequ}. Here, we assume
$\tilde{B}^{(0)}$ can be acquired somehow.
\subsubsection{1D case}
The time correction term could be calculated according to
\eqref{equ:1d-varphi1} analytically after having $\tilde{B}^{(0)}$.
Hence, we only focus on the $\mathcal{O}(\varepsilon)$ term. For
\eqref{equ:1d-varphi0}, we will implement a method which provides an
inspiration on solving the correction term in 2D case.
We rewrite \eqref{equ:1d-varphi0} in the following formulation:
\begin{equation}\label{equ:1d-steadystate_varphi}
\bF(\vec{\varphi}^{(0)})_x = \mathbf{S}(x),
\end{equation}
where $\vec{\varphi}^{(0)}=(\varphi_h^{(0)}, \varphi_u^{(0)})^T$,
\[
\bF(\vec{\varphi}^{(0)}) = \begin{pmatrix} u^{(0)}\varphi_h^{(0)} +
h^{(0)}\varphi_u^{(0)}\\
g\varphi_h^{(0)} + u^{(0)}\varphi_u^{(0)}
\end{pmatrix} \qquad
\mathbf{S}(x) = \begin{pmatrix}
\dfrac{gh^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x\\
\dfrac{gu^{(0)}\lambda_B^{(0)}(h^{(0)},u^{(0)})}{(u^{(0)})^2-gh^{(0)}}B^{(0)}_x
\end{pmatrix}.
\]
Here, $\mathbf{S}$ is written as the function of $x$ since the steady states
can be computed with fixed $B^{(0)}$. Further, notice that
\[
\frac{\partial\bF}{\partial \vec{\varphi}^{(0)}} = \begin{pmatrix}
u^{(0)} & h^{(0)}\\
g^{(0)} & u^{(0)}\\
\end{pmatrix}
\]
shares the same eigenvalues with 1D shallow water equations.
Therefore, the {\it flux-based wave decomposition method}
\cite{bale2002fluxdecom} can be applied to solve $\vec{\varphi}^{(0)}$.
More precisely, let $s_{i-1/2}^p, \mathbf{r}_{i-1/2}^p (p=1,2)$ be the
eigenvalues and eigenvectors of Jacobi matrix $\left(\partial
\bF/\partial \vec{\varphi}^{(0)}\right)_{i-1/2}$ respectively. Here,
$\left(\partial \bF/\partial \vec{\varphi}^{(0)}\right)_{i-1/2}$ is
acquired by using the Roe averages $h_{i-1/2}^{(0)}, u_{i-1/2}^{(0)}$.
To make it more concise, we omit the superscript $^{(0)}$ below in
this subsection. The algorithm is described as follows:
\begin{enumerate}[leftmargin=*]
\item Decompose the fluxes as
\[
\bF_{i}-\bF_{i-1} =\sum_{p=1}^2 \alpha_{i-1/2}^p s_{i-1/2}^p
\mathbf{r}_{i-1/2}^p.
\]
where
\begin{gather*}
s_{i-1/2}^1=u_{i-1/2}+\sqrt{gh_{i-1/2}} \qquad
s_{i-1/2}^2=u_{i-1/2}-\sqrt{gh_{i-1/2}},\\
\mathbf{r}_{i-1/2}^1 = \begin{pmatrix}
\sqrt{\dfrac{h_{i-1/2}}{g}} \\ 1
\end{pmatrix} \qquad
\mathbf{r}_{i-1/2}^2 = \begin{pmatrix}
-\sqrt{\dfrac{h_{i-1/2}}{g}} \\ 1
\end{pmatrix},
\end{gather*}
and
\[
\begin{pmatrix}
\alpha_{i-1/2}^1s_{i-1/2}^1 \\ \alpha_{i-1/2}^2 s_{i-1/2}^2
\end{pmatrix}
=\begin{pmatrix}
\dfrac{1}{2}\sqrt{\dfrac{g}{h_{i-1/2}}} & \dfrac{1}{2} \\
-\dfrac{1}{2}\sqrt{\dfrac{g}{h_{i-1/2}}} & \dfrac{1}{2}
\end{pmatrix}
\begin{pmatrix}
u_i\varphi_{h,i}+h_i\varphi_{u,i}-u_{i-1}\varphi_{h,i-1}-h_{i-1}\varphi_{u,i-1}\\
g\varphi_{h,i}+u_{i}\varphi_{u,i}-g\varphi_{h,i-1}-u_{i-1}\varphi_{u,i-1}
\end{pmatrix}.
\]
\item Calculate wave fluctuations by
\[
\bF_{i-1/2}^{\pm} :=
\sum_{p=1}^2(s_{i-1/2}^p)^{\pm}\alpha_{i-1/2}^pr_{i-1/2}^p,
\]
where $(s)^+=\max(s,0), (s)^-=\min(s,0)$. By the subcritical
assumption,
\[
\bF_{i-1/2}^+ = \alpha_{i-1/2}^1
\begin{pmatrix}\sqrt{\dfrac{h_{i-1/2}}{g}}\\1 \end{pmatrix} \qquad
\bF_{i+1/2}^- = \alpha_{i+1/2}^2
\begin{pmatrix}-\sqrt{\dfrac{h_{i+1/2}}{g}}\\ 1\end{pmatrix}.
\]
\item Solve the algebraic linear system
\begin{equation} \label{equ:1d-solve}
\bF_{i-1/2}^+ + \bF_{i+1/2}^- = \Delta x \mathbf{S}_{i},
\end{equation}
where the central difference is used to discretize $\mathbf{S}_i$:
\[
\mathbf{S}_i=\begin{pmatrix}
S_{1,i}\\
S_{2,i}
\end{pmatrix} \quad
S_{1,i} = \frac{g h_i\lambda_B(h_i,u_i)}{u_i^2-gh_i}\cdot\frac{B_{i+1}-
B_{i-1}}{2\Delta x}\quad S_{2,i}=-\frac{u_i}{h_i}S_{1,i}.
\]
\end{enumerate}
Note that $\lambda_B(h,u) = 0$ if $u=0$ from \eqref{equ:1d-0}, then
the scheme is naturally well-balanced. Further, zero boundary
condition is enforced in a computational domain $[a,b]$, i.e.
$\varphi_{h}(a)=\varphi_{h}(b)=0$ and $\varphi_{u}(a) =
\varphi_{u}(b)=0$. For the linear system, the BiCGSTAB solver is
used whose parameters will be specified in the numerical test.
\subsubsection{2D case}\label{sec:2d-case-correction}
For 2D case, We need to combine \eqref{equ:2d-varphi1-2} with
\eqref{equ:2d-varphi1-1} to obtain the $\mathcal{O}(\tilde{\tau})$
term, and to solve \eqref{equ:2d-varphi0-hat} and
\eqref{equ:2d-varphi0-bar} to obtain the $\mathcal{O}(\varepsilon)$ term.
Notice that \eqref{equ:2d-varphi1-1}, \eqref{equ:2d-varphi0-bar}, and
\eqref{equ:2d-varphi0-hat} are of the same form as follow,
\begin{equation}\label{equ:varphi-old-form}
\left\{
\begin{aligned}
\nabla\cdot(h^{(0)}\vec{\phi}_\bu+\bu^{(0)}\phi_h) &= S_h, \\
\cL^{\bu^{(0)}}(\vec{\phi}_\bu)+\nabla(g\phi_h+\bu^{(0)T}\vec{\phi}_\bu)
&= \mathbf{S}_\bu.
\end{aligned}
\right.
\end{equation}
Thus, we only present the numerical scheme to solve
\eqref{equ:varphi-old-form}, which can be written as
\[
\begin{pmatrix}
h^{(0)}\phi_u + u^{(0)}\phi_h\\
u^{(0)}\phi_u + v^{(0)}\phi_v + g\phi_h\\
0
\end{pmatrix}_x
+
\begin{pmatrix}
h^{(0)}\phi_v + v^{(0)}\phi_h\\
0\\
u^{(0)}\phi_u + v^{(0)}\phi_v + g\phi_h
\end{pmatrix}_y
=\begin{pmatrix}
S_h\\
\mathbf{S}_\bu-\cL^{\bu^{(0)}}(\vec{\phi}_\bu)
\end{pmatrix}.
\]
This form is inappropriate to solve due to the degeneration of the
fluxes. To fix it, we add an additional term to both sides:
\[
\begin{pmatrix}
0\\
(v^{(0)}\phi_u)_y-(v^{(0)}\phi_v)_x\\
(u^{(0)}\phi_v)_x-(u^{(0)}\phi_u)_y
\end{pmatrix}.
\]
It is interesting to note that this fixing term degenerates to zero
for 1D case. Therefore, \eqref{equ:varphi-old-form} can be recast as
\begin{equation}\label{equ:varphi-new-form}
\bF(\vec{\phi},x,y)_x+\mathbf{G}(\vec{\phi},x,y)_y = \tilde{\mathbf{S}},
\end{equation}
where $\vec{\phi} = (h, \vec{\phi}_\bu)^T$,
\[
\bF(\vec{\phi},x,y)=\begin{pmatrix}
h^{(0)}\phi_u + u^{(0)}\phi_h\\
u^{(0)}\phi_u + g\phi_h\\
u^{(0)}\phi_v
\end{pmatrix}, \qquad
\mathbf{G}(\vec{\phi},x,y)=\begin{pmatrix}
h^{(0)}\phi_v + v^{(0)}\phi_h\\
v^{(0)}\phi_u\\
v^{(0)}\phi_v+g\phi_h
\end{pmatrix},
\]
and
\[
\tilde{\mathbf{S}}=\begin{pmatrix}
S_h\\
\mathbf{S}_\bu - \cL^{\bu^{(0)}}(\vec{\phi}_\bu) + \cL_f^{\bu^{(0)}}(\vec{\phi}_u)
\end{pmatrix}, \qquad
\cL_f^{\bu^{(0)}}(\vec{\phi}_u)=\begin{pmatrix}
v^{(0)}_y\phi_u-u^{(0)}_y\phi_v\\
u^{(0)}_x\phi_v-v^{(0)}_x\phi_u
\end{pmatrix}.
\]
We note again that $\partial\bF/\partial\vec{\phi},
\partial\mathbf{G}/\partial\vec{\phi}$ share the same eigenvalues with the 2D
shallow water equations. Denote $s_{i-1/2,j}^p, \mathbf{r}_{i-1/2,j}^p
(p=1,2,3)$ the eigenvalues and eigenvectors of Roe-averaged
$\left(\partial\bF/\partial \vec{\phi}\right)_{i-1/2,j}$, and
$s_{i,j-1/2}^p, \mathbf{r}_{i,j-1/2}^p (p=1,2,3)$ the eigenvalues and
eigenvectors of Roe-averaged $\left(\partial \mathbf{G}/\partial \vec{\phi}
\right)_{i,j-1/2}$. To be concise, we omit the
superscript $^{(0)}$ below in this subsection. The detailed algorithm
for \eqref{equ:varphi-old-form} is detailed as follows:
\begin{enumerate}[leftmargin=*]
\item Decompose fluxes as
\[
\begin{aligned}
\bF_{i,j}-\bF_{i-1,j} &=
\sum_{p=1}^3\alpha_{i-1/2,j}^ps_{i-1/2,j}^p \mathbf{r}_{i-1/2,j}^p,\\
\mathbf{G}_{i,j}-\mathbf{G}_{i,j-1} &=
\sum_{p=1}^3\alpha_{i,j-1/2}^ps_{i,j-1/2}^p \mathbf{r}_{i,j-1/2}^p,
\end{aligned}
\]
where
$$
\begin{aligned}
& s_{i-1/2,j}^1 = u_{i-1/2,j} + \sqrt{gh_{i-1/2,j}}, \quad s_{i-1/2,j}^2 =
u_{i-1/2,j}, \quad s_{i-1/2,j}^3 = u_{i-1/2,j} -
\sqrt{gh_{i-1/2,j}},\\
& \mathbf{r}_{i-1/2,j}^1 = \begin{pmatrix}
\sqrt{\frac{h_{i-1/2,j}}{g}} \\ 1 \\ 0
\end{pmatrix},
\qquad \quad \quad
\mathbf{r}_{i-1/2,j}^2 = \begin{pmatrix}
0 \\ 0 \\ 1
\end{pmatrix},
\qquad \quad
\mathbf{r}_{i-1/2,j}^3 = \begin{pmatrix}
-\sqrt{\frac{h_{i-1/2,j}}{g}} \\ 1 \\ 0
\end{pmatrix},
\end{aligned}
$$
and
$$
\begin{aligned}
& s_{i,j-1/2}^1 = v_{i,j-1/2} + \sqrt{gh_{i,j-1/2}}, \quad s_{i,j-1/2}^2 =
v_{i,j-1/2}, \quad s_{i,j-1/2}^3 = v_{i,j-1/2} -
\sqrt{gh_{i,j-1/2}},\\
& \mathbf{r}_{i,j-1/2}^1 = \begin{pmatrix}
\sqrt{\frac{h_{i,j-1/2}}{g}} \\ 0 \\ 1
\end{pmatrix},
\qquad \quad \quad
\mathbf{r}_{i,j-1/2}^2 = \begin{pmatrix}
0 \\ 1 \\ 0
\end{pmatrix},
\qquad \quad
\mathbf{r}_{i,j-1/2}^3 = \begin{pmatrix}
-\sqrt{\frac{h_{i,j-1/2}}{g}} \\ 0 \\ 1
\end{pmatrix}.
\end{aligned}
$$
\item Calculate wave fluctuations by
\[
\bF_{i-1/2,j}^{\pm} = \sum_{p=1}^3(s_{i-1/2,j})^{\pm}
\alpha_{i-1/2,j}^p \mathbf{r}_{i-1/2,j}^p, \qquad
\mathbf{G}_{i,j-1/2}^{\pm} = \sum_{p=1}^3(s_{i,j-1/2})^{\pm}
\alpha_{i,j-1/2}^p \mathbf{r}_{i,j-1/2}^p.
\]
\item Solve the algebraic linear system
\[
(\bF_{i-1/2,j}^+ + \bF_{i+1/2,j}^- )\Delta y
+(\mathbf{G}_{i,j-1/2}^+ + \mathbf{G}_{i,j+1/2})^- \Delta x = \Delta x\Delta y
\tilde{\mathbf{S}}_{i,j},
\]
where the central difference is used to discretize
$\tilde{\mathbf{S}}_{i,j}$. Similar to the 1D case, we need to write
$\alpha^p$ in terms of $\vec{\phi}_{i\pm1,j\pm1}$ to obtain a linear
system. We also note that $\tilde{\mathbf{S}}_{i,j}$ may contain $\vec{\phi}$,
the corresponding terms of which should be moved to left
hand side when building the linear system. Further, zero boundary
condition is enforced as the 1D case.
\end{enumerate}
The eigenvalues of $\partial \bF/\partial\vec{\phi}$ and $\partial
\mathbf{G}/\partial\vec{\phi}$ cannot guaranteed to be away from zero under the
subcritical assumption. Therefore, we make usage of the {\it Harten's
entropy fix} \cite{harten1983highresolution} to stablize algorithm.
The wave fluctuations after the entropy fix are as follows:
\[
\bF_{i-1/2,j}^{\pm}=\sum_{p=1}^3(s_{i-1/2,j})^{\pm}
\alpha_{i-1/2,j}^pr_{i-1/2,j}^p
\pm\frac{1}{2}\hat{\mathbf{M}}_{i-1/2,j}
(\vec{\phi}_{i,j}-\vec{\phi}_{i-1,j}),
\]
where
\[
\begin{aligned}
& \hat{\mathbf{M}}_{i-1/2,j} = \mathbf{R}_{i-1/2,j}\mathrm{diag}
\{\rho_{i-1/2,j}^p\} \mathbf{R}_{i-1/2,j}^{-1},\\
\rho_{i-1/2,j}^p &= \begin{cases}
0, & \text{if }|s_{i-1/2,j}^p|>\delta, \\
[(s_{i-1/2,j}^p)^2+\delta^2]/(2\delta) -|s_{i-1/2,j}^p|, & \text{otherwise,}
\end{cases} \qquad p = 1,2,3.
\end{aligned}
\]
Here, $\delta$ is a small positive constant, $\mathbf{R}_{i-1/2,j} =
[\mathbf{r}_{i-1/2,j}^1, \mathbf{r}_{i-1/2,j}^2, \mathbf{r}_{i-1/2,j}^3]$. We apply the entropy fix
in the $y$ direction in the same way.
\begin{remark}
In the subcritical assumption, the eigenvalues in 1D case are
guaranteed to be away from zero. Thus, entropy fix is not applied in
1D case.
\end{remark}
\begin{remark}
Here, we only use the first order scheme to solve the correction
terms in consideration of the accuracy. Specifically, the
$\mathcal{O}(\varepsilon)$ correction term $\varphi_h^{(0)}$ always
has the contribution as $\varepsilon\varphi_h^{(0)}$, whose error is
$\mathcal{O}(\varepsilon \Delta x)$ that consistent with the overall error.
\end{remark}
\begin{remark}
If $\bu^{(0)} = \mathbf{0}$, then $\bar{\vec{\varphi}}_{\bu}^{(1)} = \mathbf{0}$
from \eqref{equ:2d-varphi1-1}, which implies that $\hat{\varphi}_h^{(1)} =
0$ and $\hat{\vec{\varphi}}_h^{(1)} = 0$ in \eqref{equ:2d-varphi1-2}.
Further, $\bu^{(0)} = \mathbf{0}$ implies that $\vec{\lambda}_B^{(0)} = \mathbf{0}$
and $S_B^{(0)} = 0$ from \eqref{equ:2d-lambda-S}. Consequently, we
have $\hat{\varphi}_h^{(0)} = 0$ and $\hat{\vec{\varphi}}_\bu^{(0)} =
\mathbf{0}$ in \eqref{equ:2d-varphi0-hat0}, thus $\varphi_h^{(0)} = 0$ and
$\vec{\varphi}_\bu^{(0)} = \mathbf{0}$ in \eqref{equ:2d-varphi0}. Therefore,
the 2D scheme is well-balanced.
\end{remark}
\subsection{Solving the riverbed equation}\label{subsec:advequ}
The homogenized models of both zeroth order and first order can be
written in the following common form
\begin{equation} \label{equ:1d-adv}
B_{\tau}+\lambda B_x=0
\end{equation}
for 1D case and
\begin{equation}
B_{\tau}+\vec{\lambda} \cdot\nabla B=S
\label{equ:2d-adv}
\end{equation}
for 2D case, respectively. It suffices to describe the scheme for
\eqref{equ:2d-adv} since the numerical scheme for 1D case is a
simplification of that for 2D case. In light of \eqref{equ:2d-1} and
\eqref{equ:2d-corr-lambda-S}, we have
$\vec{\lambda}=\vec{\lambda}^{(1)}(h^{(0)}+\varepsilon\varphi_h^{(0)},\bu^{(0)}
+\varepsilon\vec{\varphi}_\bu^{(0)}),S=S^{(1)}(h^{(0)}+\varepsilon\varphi_h^{(0)},\bu^{(0)}+\varepsilon\vec{\varphi}_\bu^{(0)})$.
Here, $h^{(0)}(\bx,\tau),\bu^{(0)}(\bx,\tau)$ represent the steady
state of shallow water equations when $B$ is fixed to $B(x,\tau)$,
and $\varphi_h^{(0)},\vec{\varphi}_\bu^{(0)}$ are the functions of
$h^{(0)},\bu^{(0)}$ according to \eqref{equ:2d-varphi0} and
\eqref{equ:2d-varphi0-hat}.
First, assume $\vec{\lambda}$ and $S$ are known. We modify the second order
{\it TVD Runge-Kutta scheme} \cite{gottlieb1998tvdrungekutta} to solve
\eqref{equ:2d-adv} as \begin{equation}\label{equ:riverbed-1}
\begin{aligned}
\tilde{B}_{i,j}^{n+1} = B_{i,j}^n&-\frac{\Delta \tau}{\Delta x}
\lambda_{i,j}^{x,n}(B_{i,j}^{n,R}-B_{i,j}^{n,L}) \\
&-\frac{\Delta\tau}{\Delta y}
\lambda_{i,j}^{y,n}(B_{i,j}^{n,U}-B_{i,j}^{n,D})+
\Delta\tau S_{i,j}^n,
\end{aligned}
\end{equation}
\begin{equation}\label{equ:riverbed-2}
\begin{aligned}
B_{i,j}^{n+1} = \frac{1}{2}(B_{i,j}^n+\tilde{B}_{i,j}^{n+1})&-
\frac{\Delta \tau}{2 \Delta x}
\lambda_{i,j}^{x,n+1}
(\tilde{B}_{i,j}^{n+1,R}-\tilde{B}_{i,j}^{n+1,L}) \\
&-\frac{\Delta\tau}{2\Delta y}
\lambda_{i,j}^{y,n+1}(\tilde{B}_{i,j}^{n+1,U}-\tilde{B}_{i,j}^{n+1,D})+
\frac{\Delta\tau}{2} S_{i,j}^{n+1},
\end{aligned}
\end{equation}
where
$$
\begin{aligned}
B_{i,j}^{n,L} &=f^{\text{upwind}}(B_{i-1/2,j}^{n,L}, B_{i-1/2,j}^{n,R},
\lambda_{i,j}^{x,n}), \quad B_{i,j}^{n,R}
=f^{\text{upwind}}(B_{i+1/2,j}^{n,L},B_{i+1/2,j}^{n,R},
\lambda_{i,j}^{x,n}),\\
B_{i,j}^{n,D} &=f^{\text{upwind}}(B_{i,j-1/2}^{n,D},
B_{i,j-1/2}^{n,U}, \lambda_{i,j}^{y,n}), \quad
B_{i,j}^{n,U} =f^{\text{upwind}}(B_{i,j+1/2}^{n,D},
B_{i,j+1/2}^{n,U}, \lambda_{i,j}^{y,n}).
\end{aligned}
$$
Here, $f^{\text{upwind}}$ is an upwind flux function that
\[
f^{\text{upwind}}(a,b,\lambda)=\begin{cases}
a,\quad\text{if }\lambda>0,\\
b,\quad\text{if }\lambda<0.
\end{cases}
\]
To achieve the second order spatial discretization, we apply the {\it
MUSCL-type slope limiter} \cite{vanleer1979muscl} to obtain
\[
\begin{aligned}
B_{i-1/2,j}^{n,L}&=B_{i-1,j}^n+\frac{1}{2}\phi(r_{i-1,j}^{x,n})
(B_{i,j}^n-B_{i-1,j}^n),\\
B_{i-1/2,j}^{n,R}&=B_{i,j}^n-\frac{1}{2}\phi(r_{i,j}^{x,n})
(B_{i+1,j}^n-B_{i,j}^n),
\end{aligned}
\]
where $r_{i,j}^{x,n}=(B_{i,j}^n-B_{i-1,j})/(B_{i+1,j}^n-B_{i,j}^n)$
and $\phi(r)=\max(0,\min(1,r))$ is the minmod limiter. Discretization
on $y$ direction takes the same form. In multidimensional cases, this
slope limiter scheme may bring spurious oscillations in regions with
large gradients in conservation laws. When having a riverbed with
sharp shape, we can require the limiter to satisfy a limiting
condition in \cite{kim2005mlp} by setting the {\it MLP-type limiter}
as an upper bound.
It remains to show how to obtain $\vec{\lambda}^{n},\vec{\lambda}^{n+1}$ and
$S^n,S^{n+1}$. Based on $B^{n}$, the steady states $h^{(0),n}$ and
$\bu^{(0),n}$ can be computed, as well as $\varphi_h^{(0),n},
\vec{\varphi}_\bu^{(0),n}$ due to \eqref{equ:2d-1} and
\eqref{equ:2d-corr-lambda-S}. We note that slope limiters of the
$h^{(0)}$ and $\bu^{(0)}$ are applied to calculate the source term
$S^n$. For $\vec{\lambda}^{n+1}, S^{n+1}$, one option is to repeat the
above procedure when fixing $B$ to $\tilde{B}^{n+1}$. Another option
is to apply the $\mathcal{O}(\tilde{\tau})$ correction to approximate
the desired
terms
\begin{equation} \label{equ:runge-kutta-approximation}
\begin{aligned}
h^{(0),n+1}+\varepsilon\varphi_h^{(0),n+1} &\approx h^{(0),n}+
\varphi_h^{(1),n}(\Delta\tau) + \varepsilon\varphi_h^{(0),n},\\
\bu^{(0),n+1}+\varepsilon\varphi_\bu^{(0),n+1} &\approx
\bu^{(0),n}+ \varphi_\bu^{(1),n}(\Delta\tau) +
\varepsilon\varphi_\bu^{(0),n},\\
\end{aligned}
\end{equation}
where $\varphi_h^{(1),n},\vec{\varphi}_\bu^{(1),n}$ can be acquired by
$B^{n},\tilde{B}^{n+1},h^{(0),n},\bu^{(0),n}$ by
\eqref{equ:2d-varphi1-1} and \eqref{equ:2d-varphi1-2}.
Here, we use $\varepsilon\vec{\varphi}^{(0),n}$ to approximate
$\varepsilon\vec{\varphi}^{(0),n+1}$ with error
$\mathcal{O}(\varepsilon\Delta\tau)$, and use $h^{(0),n}+
\varphi_h^{(1),n}(\Delta\tau)$ to approximate $h^{(0),n+1}$ with error
$\mathcal{O}(\Delta \tau^2)$ (the same with $\bu$).
The error can be roughly estimated as below. First, the error of
$\vec{\lambda}^{n+1}$ and $S^{n+1}$ are of order
$\mathcal{O}(\varepsilon^2 + \varepsilon\Delta\tau + \Delta\tau^2)$.
Since the TVD Runge-Kutta scheme is applied up to time
$\mathcal{O}(1)$ in $\tau$ scale, the error in computing the riverbed
is as $\mathcal{O}(\Delta x^2+\Delta \tau ^2)$. Hence, the total error
is approximately of order $\mathcal{O}(\varepsilon^2 + \varepsilon
\Delta \tau + \Delta x^2 + \Delta \tau^2)$. Then, by the CFL condition
we have that $\Delta \tau \times (\text{speed of the riverbed
evolving}) \sim \Delta x$, the total error is of order
\begin{equation}\label{equ:error-nondim}
\mathcal{O}(\varepsilon^2+\varepsilon\Delta x+\Delta x^2).
\end{equation}
As shown above, we solve the steady state $h^{(0),n}$, and then use
$h^{(0),n}+ \varphi_h^{(1),n}(\Delta\tau)$ to approximate
$h^{(0),n+1}$. Actually, such approximation can be repeated for
several successive steps, i.e. using $h^{(0),n+1}+
\varphi_h^{(1),n+1}(\Delta\tau)$ to approximate $h^{(0),n+2}$. In a
practical simulation, in order to make the computation more efficient,
we will apply this approximation for fixed steps (denote as $K$
later on, $K$ is not big, say 2 or 3), i.e. we only solve steady state
for $h^{(0),n}$ and approximate $h^{(0),n+1},\cdots,h^{(0),n+K}$ through
time correction term. During these steps, we use the same
$\mathcal{O}(\varepsilon)$ correction $\varepsilon \varphi_h^{(0),n}$,
which does not affect the overall error.
\subsection{Sediment transport algorithm}\label{subsec:alg}
After all the preparations above, we are ready to give the
second order algorithm for sediment transport. We will only give the
algorithm in 2D case below for conciseness.
\begin{description}
\item[Step 1] {\bf Initialization}: Let $t=0, n=0$, and set the
initial data $B^0$. Give a positive integer $K$ and we will take $K$
macro steps forward for every sample.
\item[Step 2] {\bf Sampling and calculating the $\mathcal{O}(\varepsilon)$ term
correction.}
\begin{itemize}[leftmargin=*]
\item Sampling: Fix $B=B^n$, apply the steady state solver to
obtain $h^{(0),n}$, $\bu^{(0),n}$.
\item Solve \eqref{equ:2d-varphi0-hat0} to obtain
$\hat{\varphi}_h^{(0),n},\hat{\vec{\varphi}}_\bu^{(0),n}$:
\[
\left\{
\begin{aligned}
&\nabla \cdot (h^{(0),n} \hat{\vec{\varphi}}_\bu^{(0),n} + \bu^{(0),n}
\hat{\varphi}_h^{(0),n}) = 0, \\
&\cL^{\bu^{(0)},n}(\hat{\vec{\varphi}}_\bu^{(0),n}) + \nabla (g
\hat{\varphi}_h^{(0),n} + (\bu^{(0),n})^T \hat{\vec{\varphi}}_\bu^{(0),n}) =
\cL^{\bu^{(0)},n} \left(
\frac{g\bu^{(0),n}}{|\bu^{(0),n}|^2-gh^{(0),n}}(-\vec{\lambda}_B^{(0),n} \nabla B^{n} +
S_B^{(0),n}) \right).
\end{aligned}
\right.
\]
where $\vec{\lambda}_B^{(0),n}=\vec{\lambda}_B^{(0)}(h^{(0),n},\bu^{(0),n}),
S_B^{(0),n}=S_B^{(0)}(h^{(0),n},\bu^{(0),n})$ by \eqref{equ:2d-lambda-S}.
\item Solve \eqref{equ:2d-varphi0} to obtain $\varphi_h^{(0),n},\vec{\varphi}_\bu^{(0),n}$
\[
\left\{
\begin{aligned}
&\nabla\cdot(h^{(0),n}\vec{\varphi}_\bu^{(0),n} + \bu^{(0)}\varphi_h^{(0),n}) =
\frac{gh^{(0),n}}{|\bu^{(0),n}|^2-gh^{(0),n}}(\vec{\lambda}_B^{(0),n} \nabla B^n - S_B^{(0),n}) -
\hat{\varphi}_h^{(0),n}, \\
&\cL^{\bu^{(0),n}}(\vec{\varphi}_\bu^{(0),n}) + \nabla
(\bu^{(0),T}\vec{\varphi}_\bu^{(0),n} + g\varphi_h^{(0),n}) =
\frac{g\bu^{(0),n}}{|\bu^{(0),n}|^2-gh^{(0),n}}(-\vec{\lambda}_B^{(0),n} \nabla B^n +
S_B^{(0),n}) -
\hat{\vec{\varphi}}_\bu^{(0),n}.
\end{aligned}
\right.
\]
\item Apply the $\mathcal{O}(\varepsilon)$ correction: Let $m=0$, $t = t^n$
and
$$
B^{n,0}=B^{n}, \qquad h^{n,0}=h^{(0),n}+\varepsilon\varphi_h^{(0),n},
\qquad \bu^{n,0} = \bu^{(0),n}+\varepsilon\vec{\varphi}_\bu^{(0),n}.
$$
\end{itemize}
\item[Step 3] {\bf Riverbed prediction}
\begin{itemize}[leftmargin=*]
\item Use $h^{n,m},\bu^{n,m}$ to calculate characteristic speed
$\vec{\lambda}^{n,m}$ and source term $S^{n,m}$ according to
\eqref{equ:2d-corr-lambda-S}:
\[
\vec{\lambda}^{n,m} =\vec{\lambda}_B^{(1)}(h^{n,m},\bu^{n,m}),\quad
S^{n,m}=S_B^{(1)}(h^{n,m},\bu^{n,m}).
\]
\item Calculate $\tilde{B}^{n,m+1}$ using \eqref{equ:riverbed-1}:
\[
\begin{aligned}
\tilde{B}_{i,j}^{n,m+1} = B_{i,j}^{n,m}&-\frac{\Delta \tau}{\Delta x}
\lambda_{i,j}^{x,n,m}(B_{i,j}^{n,m,R}-B_{i,j}^{n,m,L}) \\
&-\frac{\Delta\tau}{\Delta y}
\lambda_{i,j}^{y,n,m}(B_{i,j}^{n,m,U}-B_{i,j}^{n,m,D})+
\Delta\tau S_{i,j}^{n,m}.
\end{aligned}
\]
Here, the time step $\Delta\tau^{n,m}$ is determined by the CFL
condition, namely
\[
\Delta\tau^{n,m} =
C_{\mathrm{cfl}}\cdot\frac{1}{\max\limits_{i,j}\{
|\lambda_{i,j}^{x,n,m}|/\Delta x+|\lambda_{i,j}^{y,n,m}|/\Delta y\}}.
\]
where $0<C_{\mathrm{cfl}}<1$.
\end{itemize}
\item[Step 4] {\bf Approximate the steady state by time correction}
\begin{itemize}[leftmargin=*]
\item Let $\bar{h}^{n,m}=h^{n,m}-\varepsilon\varphi_h^{(0),n},
\bar{\bu}^{n,m}=\bu^{n,m}-\varepsilon\vec{\varphi}_\bu^{(0),n}$.
In this step, we use $\bar{h}^{n,m},\bar{\bu}^{n,m}$ other
than $h^{n,m},\bu^{n,m}$ to approximate the steady states.
\item Solve $\bar{\varphi}_h^{(1),n,m},\bar{\vec{\varphi}}_\bu^{(1),n,m}$ by
\eqref{equ:2d-varphi1-1}
\begin{gather*}
\bar{\varphi}_h^{(1),n,m} = \frac{g\bar{h}^{n,m}}{|\bar{\bu}^{n,m}|^2-
g\bar{h}^{n,m}}(\tilde{B}^{n,m+1}-B^{n,m}), \\
\bar{\vec{\varphi}}_\bu^{(1),n,m} =
-\frac{g\bar{\bu}^{n,m}}{|\bar{\bu}^{n,m}|^2-g\bar{h}^{n,m}}(\tilde{B}^{n,m+1}-
B^{n,m}).
\end{gather*}
\item Solve $\hat{\varphi}_h^{(1),n,m},\hat{\vec{\varphi}}_\bu^{(1),n,m}$ by
\eqref{equ:2d-varphi1-2}:
\[
\left\{
\begin{aligned}
&\nabla \cdot (\bar{h}^{n,m}\hat{\vec{\varphi}}_\bu^{(1),n,m} +
\bar{\bu}^{n,m}\hat{\varphi}_h^{(1),n,m}) = 0,\\
&\cL^{\bar{\bu}^{n,m}}(\hat{\vec{\varphi}}_\bu^{(1),n,m}) + \nabla \cdot
(g\hat{\varphi}_h^{(1),n,m} + (\bu^{n,m})^T\hat{\vec{\varphi}}_\bu^{(1),n,m}) =
-\cL^{\bar{\bu}^{n,m}}(\bar{\vec{\varphi}}_\bu^{(1),n,m}).
\end{aligned}
\right.
\]
\item Update the steady state:
\[
h^{n,m+1}= h^{n,m}+\bar{\varphi}_h^{(1),n,m}+
\hat{\varphi}_h^{(1),n,m} + \varepsilon \varphi_h^{(0),n},
\qquad
\bu^{n,m+1} = \bar{\vec{\varphi}}_\bu^{(1),n,m}+
\hat{\vec{\varphi}}_{\bu}^{(1),n,m} + \varepsilon \varphi_\bu^{(0),n}.
\]
\end{itemize}
\item[Step 5] {\bf Riverbed correction}
\begin{itemize}[leftmargin=*]
\item Calculate $\vec{\lambda}^{n,m+1},S^{n,m+1}$ using $h^{n,m+1},\bu^{n,m+1}$
according to \eqref{equ:2d-corr-lambda-S}.
\item Update riverbed $B^{n,m+1}$ by \eqref{equ:riverbed-2}:
\[
\begin{aligned}
B_{i,j}^{n,m+1} = \frac{1}{2}(B_{i,j}^{n,m}+\tilde{B}_{i,j}^{n,m+1})&-
\frac{\Delta \tau^{n,m}}{2 \Delta x}
\lambda_{i,j}^{x,n,m+1}
(\tilde{B}_{i,j}^{n,m+1,R}-\tilde{B}_{i,j}^{n,m+1,L}) \\
&-\frac{\Delta\tau^{n,m}}{2\Delta y}
\lambda_{i,j}^{y,n,m+1}(\tilde{B}_{i,j}^{n,m+1,U}-
\tilde{B}_{i,j}^{n,m+1,D})+
\frac{\Delta\tau^{n,m}}{2} S_{i,j}^{n,m+1}.
\end{aligned}
\]
Update current time $t\rightarrow t+\Delta\tau^{n,m}/\varepsilon$
and set $m\rightarrow m+1$.
\end{itemize}
\item[Step 6] If $m\ge K$, set $B^{n+1}=B^{n,m}$, $n\rightarrow n+1$,
go to {\bf Step 2}, otherwise go to {\bf Step 3}.
\end{description}
If the $\mathcal{O}(\varepsilon)$ correction and {\bf Step 5} are omitted,
then the resulting scheme becomes a first order discretization, whose
overall error becomes $\mathcal{O}(\varepsilon + \Delta x)$ due to the CFL
condition. We call the scheme in such a simplified version the {\it
first order scheme}, and the scheme contains all the steps above will
be referred as the {\it second order scheme} later on.
\subsection{Nondimensionalization}
We often nondimensionalize the parameters in practice\cite{hudson2001numerical}.
Suppose the order of length, height, velocity, time and gravity
constant are $L, H, U, T, G$. We set
\[
\begin{aligned}
&x=Lx^*, \quad y=Ly^*,\quad h=Hh^*, \quad B=HB^*,\\
&u=Uu^*, \quad v=Uv^*,\quad t=Tt^*,\quad g=Gg^*,
\end{aligned}
\]
and the system can be reformulated in terms of $x^*,y^*,B^*,h^*,t^*$.
We may set $T=L/U,G=U^2/H$ thus the parameter $A_g$ (see
\eqref{equ:sed-Grass} or \eqref{equ:sed-MPM}) is set to be
$A_g\tilde{Q}_B/H$, where $\tilde{Q}_B=
\tilde{q}_b(U)$. Based on the error estimate in section
\ref{subsec:advequ}, the error after nondimensionalization becomes
\begin{equation}\label{error}
\mathrm{Error}\sim
\begin{cases}
\mathcal{O}(\dfrac{\tilde{Q}_B}{H}\varepsilon+
\dfrac{\Delta x}{L}), & \text{first order scheme,}\\
\mathcal{O}((\dfrac{\tilde{Q}_B}{H}\varepsilon)^2+
(\dfrac{\Delta x}{L})^2+\dfrac{\tilde{Q}_B}{LH}\varepsilon\Delta x),
&\text{second order scheme}.\\
\end{cases}
\end{equation}
\section{Numerical Results} \label{sec:num}
In this section, we present several numerical results to validate the
effectiveness of the second order time homogenized model for sediment
transport. For all sediment transport problems considered, the initial
setup of the flow are obtained by solving the steady state on the
initial riverbed. All the computations are carried out on a laptop
computer with core speed of 2.3 GHz and the algorithm is implemented
using C++ programming language.
\subsection{One dimensional case}
We consider the examples studied in \cite{hudson2001numerical,
benkhaldoun2009solution}. The channel is of length with $1000 \rmm$ and
the initial riverbed is given as
\begin{equation} \label{chap7-dune-initB}
B(x,0)=
\left\{
\begin{array}{ll}
\sin^2\left(\dfrac{(x-300)\pi}{200}\right), & 300\leq x\leq 500, \\
0, & \text{else where}. \\
\end{array}
\right.
\end{equation}
The initial water level is set to be $10\rmm$, and $Q$ is a constant
discharge taken case by case. Therefore, the nondimensionalized
parameters are
\[
L = 1000, \quad H = 10, \quad U = Q/10, \quad \tilde{Q}_B =
(Q/10)^{m-1}.
\]
The porosity constant $\gamma=0.4$, and the constant $A_g$
is set to $0.001$ representing the slow interaction of the riverbed
with water flow. Thus, the time scaling parameter in this case turns
to be $\varepsilon = 0.001/0.6$. The CFL number is set to 0.65 when
updating the riverbed.
In the Step 2 in the algorithm described in Section \ref{subsec:alg}, we
need to get the steady state, which can be acquired by the
standard flux-limited Roe scheme (see \cite{hudson2001numerical,
hudson2005numerical,deng2013robust}) for a long time so that
$\lVert h^{n+1}-h^{n}\rVert_1+\lVert h^{n+1}u^{n+1}-
h^{n}u^n\rVert_1<10^{-6}$ or iteration number is bigger than 20000.
For the initial condition of the flow, we use this iteration until the
steady state is reached. For the boundary condition used in the
steady solver, we fix the upstream discharge with $10m^2/s$ and use
the transmissive boundary condition for downstream.
Also in the Step 2 and Step 4 in the algorithm, correction terms are
need to compute. We use the zero boundary condition and use the
BiCGSTAB solver with SSOR preconditioner in deal-II\footnote{see the
webpage at \tt{http://www.dealii.com}.} to solve the linear system.
The tolerance of the BiCGSTAB solver is $10^{-6}$ and the relaxation
parameter of SSOR preconditioner is $0.955$.
\subsubsection{Basic results}\label{basic-results}
First, we present the results obtained when $Q=10 \rmm^2/\rs$ with
ending time $T=238079\rs$. The Grass model with $m=3$ (see
\eqref{equ:sed-Grass}) for the sediment transport flux is
considered at first, then numerical results of other models are given.
Later on, the convergence order of the first order and the second
order multi-scale algorithms will be computed. To make a comparison,
we have included a reference solution computed by the Roe's scheme
with the second order flux-limited method \cite{hudson2001numerical,
hudson2005numerical} using a fine mesh with 4096 grid points.
Figure \ref{fig:water} displays the sampling results (i.e.
the depth and velocity of water) at initial time and end time.
Figure \ref{fig:dune-schemes} displays the riverbed when applying the
first order scheme and the second order scheme on mesh with
$N=256$. We set $K = 2$ to accelerate the computing. We also plot the
solution of Roe's scheme for comparison. It is clear that the first
order scheme produces the diffusive riverbed. However, this numerical
diffusive has been reduced remarkably by the second order scheme.
\begin{figure}[!htb]
\centering
\subfigure[Depth of water at initial time]{
\includegraphics[width=0.45\textwidth]{initialDepth.pdf}
}
\subfigure[Velocity of water at initial time]{
\includegraphics[width=0.45\textwidth]{initialVelocity.pdf}
}
\subfigure[Depth of water at end time]{
\includegraphics[width=0.45\textwidth]{endDepth.pdf}
}
\subfigure[Velocity of water at end time]{
\includegraphics[width=0.45\textwidth]{endVelocity.pdf}
}
\caption{Sampling results.}
\label{fig:water}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\textwidth]{fig1_256.pdf}
\caption{Comparison of different methods when $N=256$, $T=238079\rs$. }
\label{fig:dune-schemes}
\end{figure}
\subsubsection{Meyer-Peter-M\"{u}ler Model}
Figure \ref{fig:other-models} shows the
comparison between Grass model and Meyer-Peter-M\"{u}ler model when
$u_{cr}=0.5$, $1.0$, and $1.04$. Here, all computations are carried
out using the second order multi-scale algorithms, with parameters the
same as above.
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\textwidth]{fig2_256_MPM.pdf}
\caption{Comparison between Grass model and Meyer-Peter-M\"{u}ler}
\label{fig:other-models}
\end{figure}
\subsubsection{Convergence results}
Let us examine the convergence order of the multi-scale schemes. The
test will be based on the Grass model. The Roe's scheme
\cite{hudson2001numerical, hudson2005numerical} on an extremely fine
mesh with $16384$ grid points to is applied to produce the reference
solution. Due to the limitation of our computing capacity, the
computing time is comparatively short, says $T=90000\rs$. Actually,
the time $T = 150/\varepsilon$ is enough for convergence order
study. Here, we set $K = 1$ and compute using both the first order and
the second order algorithms. Besides, to study the effect of the
$\mathcal{O}(\varepsilon)$ correction, we use the second order solver while
discarding the $\mathcal{O}(\varepsilon)$ correction in the third test.
The Table \ref{table:convergece-order} shows the convergence
order for each algorithm, $\hat{B}$ is the approximate solution and
$B^*$ is the reference solution. One can see our algorithm has
satisfactory convergence order, and the $\mathcal{O}(\varepsilon)$ correction
is essential to improve the accuracy.
\begin{table}[!htb]
\centering
\begin{tabular}{c|c|c|c|c|c|c}
\hline\hline
&\multicolumn{2}{|c|}{first order} & \multicolumn{2}{|c|}{second order}
&\multicolumn{2}{|c}{without $\varepsilon$ correction}\\
\hline
$N$ &$\lVert\hat{B}-B^*\rVert_1$ & order & $\lVert\hat{B}-B^*\rVert_1$ & order & $\lVert\hat{B}-B^*\rVert_1$ & order\\
\hline
128 & 7.05& &3.22 & &3.24 & \\
\hline
256 & 3.68& 0.94&1.03 & 1.65&1.05 &1.63 \\
\hline
512 & 1.88& 0.97&3.25e-1 & 1.66&3.57e-1& 1.55\\
\hline
1024& 9.60e-1& 0.97&9.01e-2& 1.85&1.47e-1 &1.27 \\
\hline
2048& 4.88e-1& 0.97&2.39e-2& 1.91&9.67e-2& 0.61\\
\hline\hline
\end{tabular}
\caption{Convergence order of different algorithms.}
\label{table:convergece-order}
\end{table}
\subsubsection{Computing time comparison}
We will show the computing times with different $A_g$'s and mesh sizes
in this subsection. The ending time $T = 150/\varepsilon$, the
porosity constant is 0.4, and $K=2$ in the computations.
For different cases that $A_g=0.01,0.005,0.001$ and $N=256,512$, Roe
scheme, the first order multi-scale scheme and the second order
multi-scale scheme are tested. From the computing times shown in the
Table \ref{Time}, we can see that for different $A_g$'s, the computing
times of first order and second order scheme do not change a lot.
It's because the main computational cost attributes to solving the
steady states, which does not change a lot for different $A_g$'s.
These results demonstrate the efficiency of our multi-scale schemes,
especially when $A_g$ is small enough.
\begin{table}[!htb]
\centering
\begin{tabular}{c|c|c|c|c}
\hline\hline
$A_g$ & $N$ & Roe scheme & first order& second order\\
\hline
0.01 & 256 & 4.05 & 0.10 & 0.22\\
\hline
0.01 & 512 & 15.41 & 0.65 & 0.97\\
\hline
0.005 & 256 & 8.15 & 0.10 & 0.20\\
\hline
0.005 & 512 & 30.29 & 0.66 & 0.98\\
\hline
0.001 & 256 & 39.15 & 0.10 & 0.20\\
\hline
0.001 & 512 & 152.01 & 0.65 & 0.99\\
\hline\hline
\end{tabular}
\caption{Computing times (seconds) for different cases.}
\label{Time}
\end{table}
\subsection{Two dimensional example}
This example has been studied in \cite{hudson2001numerical,
hudson2005numerical, delis2008relaxation}. We adopt the 2D case
where the sediment transport takes place in a $1000m \times 1000m$
channel, with the initial dune profile as
\begin{equation} \label{equ:dune2d-B}
B(x,y,0) =
\left\{
\begin{array}{ll}
\sin^2(\frac{(x-300)\pi}{200}) \sin^2(\frac{(y-400)\pi}{200}),
& \text{if}~ 300\leq x\leq 500, 400\leq y \leq 600, \\
0, & \text{else}.
\end{array}
\right.
\end{equation}
The initial water surface level is $10$m everywhere with the uniformly
horizontal discharge $Q=10m^2/s$, namely
\[
h(x,y,0) = 10 - B(x,y,0), \quad u(x,y,0) = \frac{Q}{h(x,y,0)}, \quad
v(x,y,0) = 0.
\]
In this test, the Grass model with $m=3$ is used. The porosity is
$0.4$ and time scaling parameter $\varepsilon = 0.001/(1-0.4)$ to
coincide with the model in \cite{hudson2001numerical}.
When solving the steady state, we fix the discharge of $x$-direction
to be $Q=10m^2/s$ at the upstream boundary, and the transmissive
boundary condition is applied to the downstream boundary. The
reflective boundary condition is adopted on the both sides of the
channel. We also use the flux-limited Roe scheme
\cite{hudson2001numerical,hudson2005numerical} to solve the steady
state. As with 1D case, we solve the shallow water
equations until the residual is less than $10^{-6}$ or the
iteration number is bigger than 20000, and then the result is
approximated to be the steady state.
We compute this
channel test problem using the second order multi-scale method until
$T=3.6\times10^5$s on a $128\times128$ mesh. The CFL number is set to
be $0.5$ and $K$ is set
to be $2$. When solving the correction terms, we use the reflective
boundary condition on the $y=0,1000m$, and use the zero boundaries
condition on $x=0,1000m$. As with 1D case, the
BiCGSTAB solver with SSOR preconditioner is used to solve the correction
terms. The tolerance of the BiCGSTAB solver is $10^{-6}$ and the
relaxation parameter of SSOR preconditioner is $0.955$.
\begin{figure}[!htb]
\centering
\subfigure[Riverbed at initial time]{
\includegraphics[width=0.45\textwidth]{2dinitialB.pdf}
}
\subfigure[Riverbed at end time]{
\includegraphics[width=0.45\textwidth]{2dendB.pdf}
}
\subfigure[Top view of riverbed at initial time]{
\includegraphics[width=0.45\textwidth]{2dinitialBTopView.pdf}
}
\subfigure[Top view of riverbed at end time]{
\includegraphics[width=0.45\textwidth]{2dendBTopView.pdf}
}
\caption{Numerical results of riverbed.}
\label{fig:2dbed}
\end{figure}
\begin{figure}[!htb]
\centering
\subfigure[Top view of $u$ at initial time]{
\includegraphics[width=0.45\textwidth]{2dinitialUTopView.pdf}
}
\subfigure[Top view of $u$ at end time]{
\includegraphics[width=0.45\textwidth]{2dendUTopView.pdf}
}
\subfigure[Top view of $v$ at initial time]{
\includegraphics[width=0.45\textwidth]{2dinitialVTopView.pdf}
}
\subfigure[Top view of $v$ at end time]{
\includegraphics[width=0.45\textwidth]{2dendVTopView.pdf}
}
\caption{Sampling results of water velocity.}
\label{fig:2dvelocity}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.7\textwidth]{2dspreadAngle.pdf}
\caption{Spread angle.}
\label{fig:spreadangle}
\end{figure}
Figure \ref{fig:2dbed} shows the riverbed at initial time and end
time. The steady state of velocities are shown in Figure
\ref{fig:2dvelocity}. As shown in \cite{hudson2001numerical,
hudson2005numerical, delis2008relaxation}, the initial dune will
gradually deform to a star-shaped pattern. From the Figure
\ref{fig:2dbed} one sees clearly that our method captures correctly
such behaviors.
More precisely, the spread angle of the riverbed is important to show whether
our model and scheme work \cite{de2009energetically,de19872dh}.
Assume that the interaction between the sediment layer and flow is
low, the following approximation of the spread angle is proposed by De
Vriend \cite{de19872dh}
\[
\tan \theta=\frac{3\sqrt{3}(m-1)}{9m-1}.
\]
For the case in which $m=3$, the angle is approximately
$21.78^{\circ}$. Figure \ref{fig:spreadangle} shows the contour of
the riverbed at the ending time and also the angle
$\theta=21.78^\circ$. From the figure, we observe that the spread
angle of our scheme is very approximately to the one derived by De
Vriend.
At last, the computing time of our multi-scale method is $492$s, which
is much less than that of the flux-limited Roe's method
\cite{hudson2001numerical,hudson2005numerical} ($16806$s).
\section{Conclusions} \label{sec:conclusion}
In this paper, a second order time homogenized model and the
corresponding numerical methods for the sediment transport are
proposed. Through the numerical experiments, the multi-scale method
shows significant effectiveness, especially for the long time
simulation of sediment transport while provides a considerably good
approximation to the coupled system.
\section*{Acknowledgment}
This is a succeeding research of a project supported by
ExxonMobile. The authors appreciate the financial supports provided by
the \emph{National Natural Science Foundation of China (NSFC)} (Grant
91330205 and 11325102).
|
2,869,038,154,874 | arxiv | \section{Introduction}
Being one of the first space geodetic techniques, lunar laser ranging (LLR) has routinely provided observations for more than 35 years. The LLR data are collected as normal points, i.e.\ the combination of lunar returns obtained over a short time span of 10 to 20 minutes. Out of $\approx 10^{19}$ photons sent per pulse by the transmitter, less than 1 is statistically detected at the receiver \cite{wil96}; this is because of the combination of several factors, namely energy loss (i.e. $1/R^4$ law), atmospherical extinction and geometric reasons (rather small telescope apertures and reflector areas). Moreover, the detection of real lunar returns is rather difficult as dedicated data filtering (spatially, temporally and spectrally) is required. These conditions are the main reason, why only a few observatories worldwide are capable of laser ranging to the Moon.
Observations began shortly after the first Apollo 11 manned mission to the Moon in 1969 which deployed a passive retro-reflector on its surface. Two American and two French-built reflector arrays (transported by Soviet spacecraft) followed until 1973.\footnote{One of the reflector-arrays (of the Soviet Luna 17 mission, see also Fig.\ \ref{refmoon}) has not been tracked operationally. The reason could be that the coordinates are not known well enough or that the reflectors are covered by dust or the transport cap.} Most observations are taken to the largest reflector array, that of the Apollo 15 mission. Over the years more than 16,000 LLR measurements have by now been made of the distance between Earth observatories and lunar reflectors. Most LLR data have been collected by the Observatoire de la C\^ote d'Azur (OCA, France), the McDonald observatory (Texas, USA) and - until 1990 - Haleakala (Hawaii, USA). The new data are still coming, but today only the first two stations operate regularly. Understanding unexpected and small effects is very difficult with only one or two operating stations, because possible instrumental systematics of the ranging system can not be separated from real scientific effects reliably. In order to further increase the impact of LLR in Relativity and Earth sciences more stations, with a wide geographical distribution, are needed. Therefore the Italian colleagues have set up a new site in Matera which has provided first LLR data quite recently. A new site with lunar capability is currently being built at the Apache Point Observatory, New Mexico, USA. This station, called APOLLO, is designed for a mm-level accuracy ranging \citep{Murphy_etal_2000,wil04b}. However, to fully exploit the available LLR potential, a few more sites capable of tracking the Moon are needed, especially at diverse locations including the Southern hemisphere.
\begin{figure*}[t!]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=obs_ann.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=synbin.eps}
\end{minipage}
\caption{(a) Left: Lunar observations per year, 1970 - 2005. (b) Data distribution as a function of the synodic angle $D$.
\label{ann_syn}}
\vskip -5pt
\end{figure*}
Fig.~\ref{ann_syn}a shows the number of LLR normal points per
year since 1970. As shown there and in Fig.~\ref{ann_syn}b, the range data have not been accumulated uniformly; substantial variations in data density exist as a function of synodic angle~$D$, these phase angles are represented by 36 bins of 10 degree width. In Fig.~\ref{ann_syn}b, data gaps are seen near new Moon (0 and 360 degrees) and full Moon (180 degrees) phases, and asymmetry about quarter Moon (90 and 270 degrees) phases also is exhibited. The former properties of this data distribution are a consequence of operational restrictions, such as difficulties to operate near the bright sun in daylight (i.e.\ new Moon) or of high background solar illumination noise (i.e.\ full Moon). Note also asymmetry about quarter Moon phases. The uneven distribution with respect to the lunar sidereal angle shown in Fig.~\ref{sid_rms}a represents the increased difficulty of making observations from northern hemisphere observatories to the Moon when it is located over the southern hemisphere. Here, the situation will only be improve if a southern observatory (e.g.\ in Australia) will start to track the Moon.
It might be possible that new missions to the Moon could be helpful in this respect; the deployment of active laser transponders could allow satellite laser ranging systems to participate in LLR.
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=sidbin.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=drho_rms2.eps}
\end{minipage}
\caption{(a) Data distribution as a function of the sidereal angle $S,$ where the Moon is south of equator from 0$^\circ$ to 180$^\circ$.
(b) Weighted residuals (observed-computed Earth-Moon distance) annually averaged. \label{sid_rms}}
\vskip -5pt
\end{figure*}
While measurement precision for all model parameters benefit from the ever-increasing improvement in precision of individual range measurements (which now is at the few cm level, see also Fig.\ \ref{sid_rms}b), some parameters of scientific interest, such as time variation of Newton's coupling parameter $\dot G/G$ or precession rate of lunar
perigee, particularly benefit from the long time period (35 years and growing) of range measurements.
In the 1970s LLR was an early space technique for determining Earth orientation parameters (EOP). Today LLR still competes with other space geodetic techniques, and because of large improvements in ranging precision (30 cm in 1969 to $\sim2$~cm today), it now serves as one of the strongest tools in the solar system for testing general relativity. Moreover, parameters such as the station coordinates and velocities
contributed to the International Terrestrial Reference Frame ITRF2000. EOP quantities were used in combined solutions of the International Earth Rotation and Reference Systems Service IERS ($\sigma$~=~0.5~mas).
\section{LLR Model and Relativity}
The existing LLR model at IfE has been developed to compute the LLR observables --- the roundtrip travel times of laser pulses between stations on the Earth and passive reflectors on the Moon (see e.g.\ M\"uller et al.\ 1996, M\"uller and Nordtvedt 1998 or M\"uller 2000, 2001, M\"uller and Tesmer 2002, Williams et al.\ 2005b and the references therein). The model is fully relativistic and is complete up to first post-Newtonian ($1/c^2$) level; it uses the Einstein's general theory of relativity -- the standard theory of gravity. The modeling of the relativistic parts is much more challenging than, e.g., in SLR, because the relativistic corrections increase the farther the distance becomes. The basic observation equation reads
\begin{equation}
d = c \frac{\tau}{2} = \left| \bf r_{\rm em} - r_{\rm station} + r_{\rm reflector}\right|+ c\; \Delta\tau
\label{eq:1}
\end{equation}
where $d$ is the station-reflector distance, $c$ the speed of light, $\tau$ the pulse travel time, $\bf r_{\rm em}$ the vector connecting the geocenter and the selenocenter, $\bf r_{\rm station}$ the geocentric position vector of the observatory, and $\bf r_{\rm reflector}$ the selenocentric position vector of the reflector arrays. $\Delta\tau$ describes corrections of the travel time caused by atmospheric effects, but also (relativistic) transformations into the right time system as well as the light time equation (Shapiro effect). In order to apply Eq.~(\ref{eq:1}), all vectors have to be transformed in one common reference frame (in our case the inertial frame) which requires consistent relativistic transformations, so-called pseudo-Lorentz transformations. The Earth-Moon vector $\bf r_{\rm em}$ can only be obtained by numerical integration of the corresponding equation of motion (here in simplified version):
\begin{eqnarray}
{\bf \ddot r_{\rm em}} = - \frac{GM_{\rm e+m}}{r_{\rm em}^3}{\bf r_{\rm em}} + GM_s \left( \frac{\bf r_{\rm s} - r_{\rm em}}{\left| {\bf r_{\rm s} - r_{\rm em}}\right|^3} - \frac{\bf r_{\rm s}}{r_{\rm s}^3} \right)
+ {\bf b_{\rm newtonian}} + {\bf b_{\rm relativistic}} .
\end{eqnarray}
$\bf \ddot r_{\rm em}$ is the relative acceleration vector between Earth and Moon, $GM_{\rm e+m}$ is the Earth-Moon mass times the gravitational constant, $\bf r_{\rm s}$ the geocentric position vector of the Sun, $r_{\rm s}$ the Earth-Sun distance, $M_s$ the solar mass, ${\bf b_{\rm newtonian}}$ comprises all further Newtonian terms like the effect of the other planets or the gravitational fields of Earth and Moon, and $\bf b_{\rm relativistic}$ indicates all 'relativistic' terms, i.e.\ those entering the Einstein-Infeld-Hofmann (EIH) equations. Corresponding relativistic equations are applied to describe the rotational motion of the Moon. The rotation angles are then applied to rotate the selenocentric reflector coordinates of Eq.\ (1) into the inertial frame. For the transformation of the geocentric station coordinates, the well known rotation matrices (precession, nutation, GAST, etc.) are used, see IERS 2003.
The modeling of the 'Newtonian' parts has been set up according to IERS Conventions (IERS 2003), but it is restricted to the 1 cm level. The weights are based upon the accuracy estimates for the normal points as provided by the observatories. Based upon this model, two groups of parameters (170 in total) are determined by a weighted least-squares fit of the observations. The first group includes the so-called 'Newtonian' parameters such as
\begin{itemize}
\item geocentric coordinates of three Earth-based LLR stations and their velocities;
\item a set of EOPs (luni-solar precession constant, nutation coefficients of the 18.6 years period, Earth's rotation
UT0 and variation of latitude by polar motion);
\item selenocentric coordinates of four retro-reflectors;
\item rotation of the Moon at one initial epoch (physical librations);
\item orbit (position and velocity) of the Moon at this epoch;
\item orbit of the Earth-Moon system about the Sun at one epoch;
\item mass of the Earth-Moon system times the gravitational constant;
\item the lowest mass multipole moments of the Moon;
\item lunar Love number and a rotational energy dissipation parameter;
\item lag angle indicating the lunar tidal acceleration responsible for the increase of the Earth-Moon distance
(about 3.8 cm/yr), the increase in the lunar orbit period and the slowdown of Earth's angular velocity.
\end{itemize}
The second group of parameters is used to perform LLR tests of viable modifications of the general theory of relativity. The general theory of relativity is not expected to be perfect, because Einstein's theory and quantum mechanics are fundamentally incompatible. Therefore, it is important in physics to find out at what level of accuracy it fails. Relativistic parameters to be determined by LLR analyses are (values for general relativity are given in parentheses):
\begin{enumerate}
\item Strong equivalence principle (EP) parameter $\eta$, which for metric theories is $\eta=4\beta-3-\gamma$ (= 0).
In LLR a violation of the equivalence principle would show up as a displacement of the lunar orbit along the direction to the Sun. LLR is the dominant test of the strong equivalence principle, i.e.\ for self-gravitating bodies.
\item Space-curvature parameter $\gamma$ (= 1) and non-linearity parameter $\beta$ (= 1). LLR also has the capability of determining the PPN parameters $\beta$ and $\gamma$ directly from the point-mass
orbit perturbations (i.e.\ as described by the EIH equations), but, e.g., $\beta$ may be derived much better by combining the EP parameter $\eta$ with an independent determination of $\gamma$ (see below).
\item Time variation of the gravitational coupling parameter $\dot G/G$ (= 0 $\rm yr^{-1}$). This is the second most important gravitational physics result that LLR provides. Einstein's theory does not predict a changing $G$, but some other theories do. So it is important to measure this as well as possible. The sensitivity improves like the square of the data span.
\item Geodetic de {Sitter} precession ${\Omega}_{\rm dS}$ of
the lunar orbit ($\simeq 1.92$~"/cy). LLR has also provided the only accurate determination of the geodetic precession. The dedicated space mission GP-B will provide an improved accuracy, if that mission is successfully completed.
\item Coupling constant $\alpha$~(=~0) of Yukawa potential for the Earth-Moon distance which corresponds to a test of Newton's inverse square law.
\item $\alpha_1$ (= 0) and $\alpha_2$ (= 0) which parametrize 'preferred frame' effects in metric gravity.
\item Combination of parameters $\zeta_1 - \zeta_0 - 1$ (= 0) derived in the Mansouri and Sexl (1977) formalism indicating a violation of special relativity (there: Lorentz contraction parameter $\zeta_1 = 1/2$, time dilation parameter $\zeta_0 = - 1/2$).
\item EP-violating coupling of normal matter to 'dark matter' at the galactic center. It is a similar test to item 1 above, but now different periods are affected (mainly the sidereal month).
\item A further application is the detection of the Sun's $J_2$ ($\approx 10^{-7}$) from LLR data (independent to other methods as solar seismology), which affects the anomalous perihelion shift of Mercury, one of the classical tests of relativity. The present uncertainty ($\approx 10^{-6}$) is larger than the expected value. The parameter $J_2$ is not further discussed in this paper and will be addressed in more detail in an upcoming study.
\end{enumerate}
The determination of the relativistic parameters indicated above can be accomplished either by modifying the equations of motion (i.e.\ parametrizing present terms or adding new ones) or by deriving analytical expressions for their effect on the Earth-Moon distance. In the first case the needed partial derivatives can be computed by
numerical differentiation, in the second case by direct derivation of the analytical terms.
Many relativistic effects produce a sequence of periodic perturbations of the Earth-Moon range
\begin{equation}
\Delta r_{EM} = \sum_{i=1}^n \Big[A_i \cos (\omega_i \Delta t + \Phi_i)+B_i \Delta t\sin (\omega_i \Delta t + \Phi_i)\Big].
\end{equation}
$A_i$ and $B_i,\; \omega_i,$ and $\phi_i$ are the amplitudes, frequencies, and phases, respectively, of the various perturbations. Some sample periods of perturbations important for the measurement of various parameters are given in Table~1.\footnote{Note: the designations should not be used as formulae for the computation of the corresponding periods, e.g.\ the period `sidereal-2$\cdot$annual' has to be calculated as $1/(1/27.32^d - 2/365.25^d) \approx 32.13^d$. In addition, `secular + emerging periodic' means the changing orbital frequencies induced by $\dot G/G$ are starting to become better signals than the secular rate of change of the Earth-Moon range in LLR.}
{}
\begin{table}[bht]
\caption{Typical periods of some relativistic quantities, taken from M\"uller et al.\ (1999).}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Parameter & Typical Periods \\ \hline\hline
$\eta$ & synodic (29$^d$12$^h$44$^m$2.9$^s$)\\ \hline
$\dot G/G$ & secular + emerging periodic \\ \hline
$\alpha_1$ & sidereal, annual, sidereal-2$\cdot$annual,\\
& anomalistic\ (27$^d$13$^h$18$^m$33.2$^s$) $\pm$ annual, synodic\\ \hline
$\alpha_2$ & 2$\cdot$sidereal, 2$\cdot$sidereal-anomalistic, nodal (6798$^d$) \\ \hline
$\zeta_1-\zeta_0-1$ & annual (365.25$^d$)\\ \hline
$\delta g_{\rm galactic}$ & sidereal (27$^d$7$^h$43$^m$11.5$^s$)\\ \hline
\end{tabular}
\end{center}
\vskip -10pt
\end{table}
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=gdotpartial.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=partials1.eps}
\end{minipage}
\caption{(a) Sensitivity of LLR with respect to $\dot G / G$ assuming $\Delta \dot G / G = 8\cdot 10^{-13}\ {\rm yr}^{-1}$.
(b) Sensitivity of LLR with respect to space curvature $\gamma$, non-linearity couplings $\beta$ and geodetic precession using their present LLR accuracy (see Table 2) as perturbation value.
\label{gpar_par1}}
\vskip -5pt
\end{figure*}
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=partials2.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=partials3.eps}
\end{minipage}
\caption{(a) Sensitivity of LLR with respect to Yukawa interaction parameter $\alpha$, equivalence principle violation $\eta$ and time-variable gravitational constant $\dot G / G$ using their present LLR accuracy (see Table 2) as perturbation value.
(b) Sensitivity of LLR with respect to preferred-frame effects $\alpha_1$ and $\alpha_2$ using their present LLR accuracy (see Table 2) as perturbation value.
\label{par2_par3}}
\vskip -5pt
\end{figure*}
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=etaspec.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=gdotspec.eps}
\end{minipage}
\caption{
(a) Power spectrum of a possible equivalence principle violation assuming $\Delta (m_G / m_I) \approx 10^{-13}$.
(b) Power spectrum of the effect of $\dot G / G$ in the Earth-Moon distance assuming $\Delta \dot G / G
= 8\cdot 10^{-13}\ {\rm yr}^{-1}$.
\label{gspec_espec}}
\vskip -5pt
\end{figure*}
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=gpmspec.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=yukawaspec.eps}
\end{minipage}
\caption{(a) Power spectrum of an additional (deviating from Einstein's value) geodetic precession assuming
$\Delta gpm = 10^{-2}$.
(b) Power spectrum of a possible Yukawa term using $\Delta \alpha
= 2\cdot 10^{-11}$.
\label{gpm_yukspec}}
\vskip -5pt
\end{figure*}
\begin{figure*}[ht]
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=a1spec.eps}
\end{minipage}
\hfill
\begin{minipage}[b]{.46\linewidth}
\centering \psfig{width=7.4cm, file=a2spec.eps}
\end{minipage}
\caption{(a) Power spectrum of a possible preferred-frame effect by $\alpha_1$ assuming $\Delta \alpha_1
= 9\cdot 10^{-5}$.
(b) Power spectrum of a possible preferred-frame effect by $\alpha_2$ assuming $\Delta \alpha_2
= 2.5\cdot 10^{-5}$.
\label{a1_a2spec}}
\vskip -5pt
\end{figure*}
\section{Sensitivity Study}
As indicated in Eqs.~(1)-(2), LLR is affected in various ways and at various levels by relativity. Relativity enters the equation of motion, i.e.\ the orbit and the rotation of the Moon. More precisely, the
Earth-Moon system behaves according to relativity. But also the light propagation and the transformations between the reference and time frames which are used have to be modeled in agreement with general relativity. The lunar measurements contain the summed signal of all effects in one, so that the separation of the individual effects is a big challenge. To better understand what the individual contributions of the different relativistic effects are, sensitivity studies have been carried out, as
\begin{equation}
\Delta r_{em}^p = \frac{\delta r_{em}}{\delta p}\Delta p.
\end{equation}
$\Delta r_{em}^p$ is the perturbation of the Earth-Moon distance caused by a parameter $p$ which is one of the relativistic parameters described in Section 2. $\delta r_{em}/\delta p$ is the partial derivative of the Earth-Moon distance with respect to $p$; it is obtained by numerical differentiation. $\Delta p$ is a small value indicating the variation in $p$, here we have used the present realistic error as derived from LLR analyses (see Table~2). As an example, Fig.\ \ref{gpar_par1}a represents the sensitivity of the Earth-Moon distance with respect to a possible temporal variation of the gravitational constant in the order of $8\cdot 10^{-13}$ yr$^{-1}$, the present accuracy of that parameter. It seems as if perturbations of up to 9 meters are still caused, but this range (compared to the ranging accuracy at the cm level) can not fully be exploited, because the lunar tidal acceleration perturbation is similar. Fig.\ \ref{gpar_par1}b, Fig.\ \ref{par2_par3}a and Fig.\ \ref{par2_par3}b show the results of corresponding investigations for all relativistic parameters which were investigated during the present study (i.e.\ without the parameters discussed in items 7--9 of the previous Section). The perturbations vary between 5 cm ($\alpha_1$) and 25 m ($\beta$) which indicates that the former parameter is determined quite well from LLR data, as the sensitivity values are close to the observation accuracy, whereas the latter is only poorly determined because of its high correlation with the Newtonian orbit perturbations. Nevertheless, the continuation of LLR over a longer time span will help to further de-correlate the various parameters.
To better understand those couplings, corresponding power spectra have been computed. The largest periods for the EP-parameter are shown in Fig.\ \ref{gspec_espec}a and for $\dot G / G$ in Fig.\ \ref{gspec_espec}b. Obviously many periods are affected simultaneously, because the perturbations, even if caused by a single beat period only (e.g.\ the synodic month for $\eta$ parameter\footnote{Note, here and throughout the paper, the relation $m_G / m_I = 1 + \eta (U/M c^2)$ has been used equivalently, where the second term describes the self energy of a body, cf.\ Williams et al.\ 2005b.}), change the whole lunar orbit (and rotation) and therefore excite further frequencies.
For comparisons also the spectra of the geodetic precession $gpm$ and the Yukawa coupling parameter $\alpha$ are indicated (Fig.\ \ref{gpm_yukspec}a and Fig.\ \ref{gpm_yukspec}b). Again a different combination of periods is visible. As before mainly the monthly (e.g.\ sidereal, synodic, anomalistic), half-monthly, etc., periods dominate, but longer periods (low frequencies) are also present, e.g.\ annual, 3 years or combinations of the monthly and annual frequencies.
Similar pictures are obtained when considering the preferred-frame parameters $\alpha_1$ and $\alpha_2$ (Fig.\ \ref{a1_a2spec}a and Fig.\ \ref{a1_a2spec}b). A huge variety of periods show up again, but they differ partly from each other (note, e.g., the very low-frequency contributions). The spectra of $\gamma$ and $\beta$ (not shown here) are very similar to the geodetic precession spectrum. Although there are big similarities between the various spectra, the typical properties of each can be used to identify and separate the different effects and to determine corresponding parameters.
\section{Results}
The global adjustment of the model by least-squares-fit procedures gives improved values for the estimated parameters and their formal standard errors, while consideration of parameter correlations obtained from the covariance analysis and of model limitations lead to more 'realistic' errors. Incompletely modeled solid Earth tides, ocean loading or geocenter motion, and uncertainties in values of fixed model parameters have to be considered in those estimations. See Table 2 for the determined values for the relativistic quantities and their realistic errors.
The EP-parameter $\eta$ $\left( =(6 \pm 7) \cdot 10^{-4}\right)$ benefits most from highest accuracy over a sufficient long time span (e.g.\ one year) and a good data coverage over the synodic month, as far as possible. Also any observations reducing the asymmetry about the quarter Moon phases (compare Fig.\ \ref{ann_syn}b) would improve the fit of $\eta$. The improvement of the EP parameter was not so big since 1999, as the LLR RMS residuals increased a little bit in the past years, see Fig.\ \ref{sid_rms}b. The reason for that increase is not completely understood and has to be investigated further.
In combination with the recent value of the space-curvature parameter $\gamma_{\rm Cassini}$ $\left(\gamma -1 = (2.1 \pm 2.3) \cdot 10^{-5}\right)$ derived from Doppler measurements to the Cassini spacecraft
(Bertotti et al.\ 2003), the non-linearity parameter $\beta$ can be determined by applying the relationship $\eta=4\beta-3-\gamma_{\rm Cassini}$. One obtains $\beta - 1 = (1.5 \pm 1.8) \cdot 10^{-4}$ (note that using the EP test to determine parameters $\eta$ and $\beta$ assumes that there is no composition-induced EP violation).
The LLR result for the space-curvature parameter $\gamma$ as determined from the EIH equations is less accurate than the results derived from other measurements, because its effect is very similar to the Newtonian orbit perturbations.
For the temporal variation of the gravitational constant, $\dot G / G = (6\pm 8)\cdot 10^{-13}$ has been obtained, where the formal standard deviation has been scaled by a factor 3 to yield the given value. This parameter benefits most from the long time span of LLR data and has experienced the biggest improvement over the past years (cf.\ M\"uller et al.\ 1999).
For the estimation of the de Sitter precession of the lunar orbit, a Coriolis-like term is added to the equation of motions, which adds the precession effect as predicted by Einstein for a second time. This term is scaled by a parameter called $gpm$ which has to give 0 if Einstein is correct. $gpm = 0$ is obtained with an accuracy of
about 1 percent.
The Yukawa coupling parameter $\alpha$ has been determined by adding corresponding perturbation terms to the equations of motion, where the partials were computed by numerical differentiation. The recently determined value shows small deviations from the expected value; it will be further investigated in future.
The preferred-frame parameters $\alpha_1$ and $\alpha_2$ can either be determined by extending the equations of motion or by adding analytical terms to the Earth-Moon distance. In both cases quite similar results are obtained (see M\"uller et al.\ 1996, 1999). Recent determinations are given in Table 2.
The Mansouri-Sexl parameters $\zeta_0$ and $\zeta_1$ as well as the quantity indicating a possible EP-violating coupling with dark matter were not investigated during our present studies; the values given in Table 2 are taken from M\"uller et al.\ (1999).
A further question to be considered in more detail in future is the right combination of the parameters. That means, shall all relativistic parameters be estimated together in one global adjustment, or each one alone (together with the parameters of the standard solution)? We carried out several tests considering the correlation of the relativistic quantities with each other, but also with the 'classical' ones, e.g.\ with the orbital parameters or site velocities (Koch 2005). It is too early to make a final decision. On the one hand 'over-estimation' of an effect has to be avoided, on the other hand over-constraining by fixing too many parameters should also be avoided.
Final results for all relativistic parameters obtained from the IfE analysis are shown in Table 2 (see also M\"uller et al. 2005). The realistic errors are comparable with those obtained in other recent investigations, e.g.\ at JPL (see Williams et al.\ 1996, 2004a, 2004b, 2005b).
\begin{table}[htb]
\caption{Determined values for the relativistic quantities and their realistic errors.}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Parameter & Results \\ \hline\hline
Equivalence\ Principle parameter\ $\eta$ & $(6 \pm 7) \cdot 10^{-4} $ \\ \hline
Metric parameter \ $\gamma - 1$ & $(4 \pm 5) \cdot 10^{-3} $ \\ \hline
Metric parameter \ $\beta - 1$: direct measurement & $(-2\pm 4) \cdot 10^{-3} $ \\
and from $\eta=4\beta-3-\gamma_{\rm Cassini}$ & $(1.5\pm 1.8) \cdot 10^{-4} $ \\ \hline
Time varying\ gravitational \ constant\ $\dot G / G$ [${\rm yr}^{-1}$] & $(6\pm 8)\cdot 10^{-13} $ \\ \hline
Differential\ geodetic\ precession\ $\Omega_{\rm GP}$ - $\Omega_{\rm deSit}$ [''/cy] & $(6\pm 10)\cdot 10^{-3} $ \\ \hline
Yukawa\ coupling\ constant\ $\alpha$ (for $\lambda = 4\cdot 10^5\, {\rm km})$
& $(3\pm 2) \cdot 10^{-11} $ \\ \hline
`Preferred frame' parameter $\alpha_1$ & $(-7\pm 9) \cdot 10^{-5} $
\\ \hline
`Preferred frame' parameter $\alpha_2$ & $(1.8 \pm 2.5) \cdot 10^{-5} $
\\ \hline
Special\ relativistic\ parameters $\zeta_1 - \zeta_0 - 1$ & $(-5\pm 12) \cdot 10^{-5} $
\\ \hline
Influence\ of dark matter $\delta g_{\rm galactic}$ [cm/s$^2$] & $(4\pm 4) \cdot 10^{-14} $
\\ \hline
\end{tabular}
\end{center}
\vskip -10pt
\end{table}
\section{Further Applications}
In addition to the relativistic phenomena discussed above, more effects related to lunar physics, geosciences, and geodesy can be investigated. The following items are of special interest (see also M\"uller et al. 2005), as they touch recent activities in the afore-mentioned disciplines:
\begin{enumerate}
\item {\it Celestial reference frame:} A dynamical realization of the International Celestial Reference System (ICRS) by the lunar orbit is obtained ($\sigma$~=~0.001") from LLR data. This can be compared and analyzed with respect to the kinematical ICRS from Very Long Baseline Interferometry (VLBI). Here, the very
good long-term stability of the lunar orbit is of great advantage.
\item {\it Terrestrial reference frame:} The results for the station coordinates and velocities, which are estimated simultaneously in the standard solution, contribute to the realization of the international terrestrial reference frame, e.g.\ to the last one, the ITRF2000.
\item {\it Earth rotation:} LLR contributes, among others, to the determination of long-term nutation parameters, where again the stable, highly accurate orbit and the lack of non-conservative forces from air-drag or solar radiation pressure (which affect satellite orbits substantially) is very convenient. Additionally UT0 and VOL values are computed, which stabilize the combined EOP series, especially in the 1970s when less data from other space geodetic techniques were available. The precession rate is another example in this respect.
\item {\it Relativity:} As discussed in the previous sections, the dedicated investigation of Einstein's theory of relativity is of major interest. With an improved accuracy of the LLR measurements and the modeling (see next Section) the investigation of further effects (e.g.\ the Lense-Thirring precession) or those of alternative theories might become possible.
\item {\it Lunar physics:} By the determination of the libration angles of the Moon, LLR gives access to underlying processes affecting lunar rotation (e.g.\ Moon's core, dissipation), cf. Williams et al.\ (2005a). A better distribution of the retro-reflectors on the Moon (see Fig.\ \ref{refmoon}) would be very helpful.
\item {\it Selenocentric reference frame:} The determination of a selenocentric reference frame, the combination with high-resolution images and the establishment of a better geodetic network on the Moon is a further big item, which then allows accurate lunar mapping.
\item {\it Earth-Moon dynamics:} The mass of the Earth-Moon system, the lunar tidal acceleration, possible geocenter variations and related processes as well as further effects can be investigated in detail.
\item {\it Time scales:} The lunar orbit can also be considered as a long-term stable clock so that LLR can be used for the independent realization of time scales, which can then be compared or combined with other determinations.
\end{enumerate}
Those features shall be addressed in the future, when more and better LLR data are available and the analysis models have been improved to the mm level, see next Section.
\begin{figure}[tbh]
\begin{center}
\epsfig{width=7.4cm, file=refl_moon.eps}
\end{center}\vskip -10pt
\caption{Distribution of retro-reflectors on the lunar surface.}
\label{refmoon}
\vskip -10pt
\end{figure}
\section{Model and Observation Refinement}
To exploit the full available potential of LLR, the theoretical models as well as the measurements require optimization. Using the 3.5 m telescope at the APOLLO site in New Mexico, USA, a mm-level ranging becomes possible. To allow an order of magnitude gain in determination of various quantities in the complete LLR solution, the current models have to be up-dated according to the IERS conventions 2003, and made compatible with the IAU 2000 resolutions.
This requires, e.g., to better model
\begin{itemize}
\item higher degrees of the gravity fields of Earth and Moon and their couplings;
\item the effect of the asteroids (up to 1000);
\item relativistically consistent torques in the rotational equations of the Moon;
\item relativistic spin-orbit couplings;
\item torques caused by other planets like Jupiter;
\item the lunar tidal acceleration with more periods (diurnal and semi-diurnal);
\item ocean and atmospheric loading by updating the corresponding subroutines;
\item nutation using the recommended IAU model;
\item the tidal deformation of Earth and Moon;
\item Moon's interior (e.g.\ solid inner core) and its coupling to the Earth-Moon dynamics.
\end{itemize}
Besides modeling, the overall LLR processing shall be optimized. The best strategy for the data fitting procedure needs to be explored for (highly) correlated parameters.
Finally LLR should be prepared for a renaissance of lunar missions where transponders or new retro-reflectors may be deployed on the surface of the Moon which would enable many pure SLR stations to observe the Moon. NASA is planning to return to the Moon by 2008 with Lunar Reconnaissance Orbiter (LRO), and later with robotic landers, and then with astronauts until 2018. The primary focus of these planned missions will be lunar exploration and preparation for trips to Mars, but they will also provide opportunities for science, particularly if new reflectors are placed at more widely separated locations than
the present configuration (see Fig.\ \ref{refmoon}). New installations on the Moon would give stronger determinations of lunar rotation and tides. New reflectors on the Moon would provide additional accurate surface positions for cartographic control (Williams et al.\ 2005b), would also aid navigation of surface vehicles or spacecraft at the Moon, and they also would contribute significantly to research in fundamental and gravitational physics, LLR-derived ephemeris and lunar rotation. Moreover in the case of co-location of microwave transponders, the connection to the VLBI system may become possible which will open a wide range of further activities such as frame ties.
\section{Conclusions}
LLR has become a technique for measuring a variety of relativistic gravity parameters with unsurpassed precision. Sensitivity studies have been performed to estimate the order of magnitude of relativistic effects on lunar ranges and the potential capability to better determine certain relativistic features. Spectral analyses showed the typical frequencies related to each effect, indicating as well, how (highly) correlated parameters might be separated. No violations of general relativity are found in our investigations. Both the weak and strong forms of the EP are verified, while strong empirical limitations on any inverse square law violation, time variation of $G$, and preferred frame effects are also obtained.
LLR continues as an active program, and it can remain as one of the most important tools for testing Einstein's general relativity theory of gravitation if appropriate observations strategies are adopted and if the basic LLR model is further extended and improved down to the millimeter level of accuracy. The deployment of transponders on the Moon would considerably improve the performance for lunar ranging applications. Lunar science, fundamental physics, control networks for surface mapping, and navigation would benefit. Demonstration of active devices would prepare the way for very accurate ranging to Mars and other solar system bodies.
\vspace{0.3cm}\noindent
{\bf\ Acknowledgments.}
Current LLR data is collected, archived and distributed under the auspices of the International Laser Ranging Service (ILRS). All former and current LLR data is electronically accessible through the European Data Center (EDC) in Munich, Germany and the Crustal Dynamics Data Information Service (CDDIS) in Greenbelt, Maryland. The following web-site can be queried for further information: {\tt http://ilrs.gsfc.nasa.gov}. We also acknowledge with thanks, that the more than 35 years of LLR data, used in these analyses, have been obtained under the efforts of personnel at the Observatoire de la C\^ote d'Azur, in France, the LURE Observatory in Maui, Hawaii, and the McDonald Observatory in Texas.
A portion of the research described in this paper was carried out at the Jet Propulsion Laboratory of the California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
|
2,869,038,154,875 | arxiv | \section{Introduction}
The MUNICS project is a wide area $K'$-band selected photometric
survey in the $VRIJK'$ passbands aiming at two main scientific goals,
namely
\begin{itemize}
\item the identification of galaxy clusters at redshifts around unity, and
\item the selection of a fair sample of field early-type galaxies at
similar redshifts for evolutionary studies.
\end{itemize}
Near-IR selection is an efficient tool for tracing the massive galaxy
population at redshifts around unity because of its high sensitivity
for evolved stellar populations even in the presence of moderate star
formation activity. Thus a $K$-band selected survey can provide a
very useful database for the investigation of the formation and
evolution of the cluster as well as the field population of massive
galaxies.
\begin{figure}[ht!]
\plottwo{models.eps}{cmd.eps}
\caption{{\em Left panel:} $K'$ vs. $J\!-\!K'$ color-magnitude diagram
for population synthesis models for a simple stellar population and 3
exponential star formation histories with timescales given in the
legend. The galaxies are assumed to form at $z_f = 4$ and to have a
luminosity of $L_*$ today. {\em Right panel:} the same diagram for
0.35 square degrees of the MUNICS data. The sequence at a color $\leq 0.8$
are late-type stars.}
\end{figure}
Clusters of galaxies are of prime interest in extragalactic astronomy
and cosmology. The $z$ evolution of their number density and
correlation function are sensitive tests for structure formation
theories and especially the density parameter $\Omega$. Models of
structure formation predict that if $\Omega_0 = 1$, the number density
of clusters of richness class 1 declines by a factor of $10^3$ between
$z=0$ and $z=1$. On the other hand, if $\Omega_0 = 0.3$, the number
density declines only by a factor of $\leq 10$ in the same redshift
range (Eke, Cole, \& Frenk 1996; Bahcall, Fan, \& Cen 1997; Fan,
Bahcall, \& Cen 1997). Furthermore, clusters of galaxies allow to
find large numbers of massive galaxies at higher redshift and thus represent
unique laboratories to study the evolution of galaxies in high density
regions as a function of redshift. While the number of clusters known
at redshifts $z>0.5$ is steadily increasing, {\em uniformly selected}
samples of clusters at high redshift are still deficient in the
optical and near-IR wavelength ranges.
The formation and evolution of the population of massive galaxies is
still a matter of lively and controversial debate. No general
agreement has been reached yet regarding the formation era of
spheroidals. While models of hierarchical galaxy formation (Cole et
al. 1993; Kauffman \& Charlot 1998) consistently predict a steep
decline in the number density of massive spheroidals, they have a
rather large number of free parameters, some of which involve
ill-understood processes. Observation has not yet been successful in
constraining the ranges of the involved model parameters tightly
enough, so that comparisons between theory and experiment are difficult
to interpret. Moreover, measuring the evolution of the number density
of early-type galaxies $\partial N(z)/\partial z$ to redshifts of
unity is by itself a difficult undertaking, suffering from too small
samples and strong selection effects, therefore yielding contradictory
results (e.g. Totani \& Yoshii 1998; Benitez et al. 1999; Shade et
al. 1999; Broadhurst \& Bouwens 1999; Barger et al. 1999).
\section{The MUNICS Project}
\begin{figure}[ht!]
\begin{center}
\plottwo{o651.eps}{zhist.eps}
\end{center}
\caption{{\em Left panel:} Magnitudes and best fit SED for an object
at $z=0.9$. Black dots denote the measured fluxes in $VRIJK'$, the
solid line is the best fit redshifted galaxy SED used to derive the
photometric redshift, a 5 Gyr old `elliptical' in this case. The dashed
line is the best fit local stellar SED, a M5 type star. Note that the
$J$-band flux is the only distinguishing feature in such a
case. {\em Right panel:} photometric redshift distributon for 371
galaxies selected out of a $13.2\arcmin\times 13.2\arcmin$ field.}
\end{figure}
The MUNICS project uniformly covers 1 square degree in the $J$ and
$K'$ near-IR bands. The survey area consists of 8 $13.2\arcmin \times
26.2\arcmin$ randomly selected fields at high galactic latitude, as
well as 13 $7\arcmin \times 7\arcmin$ fields targeted towards
$0.6<z<1.5$ QSOs. The $3\sigma$ detection limits for a point source
are $19.5$ in the $K'$-band and $21.5$ in the $J$-band. The data have
been acquired at the 3.5m telescope at Calar Alto Observatory using
the $\Omega\!-\!{\rm Prime}$ camera.
Optical photometry in the $V$, $R$, and $I$ bands was obtained
for a subsample of the survey fields covering 0.35 square degrees in
total. These data have been obtained at the 2.2m telescope at Calar
Alto Observatory and the 2.7m telescope at McDonald Observatory. These
data will enable us to determine photometric redshifts for the
galaxies and thus are of great importance in selecting and confirming
cluster candidates as well as individual galaxies for follow-up
spectroscopy.
Figure 1 shows $K'$ vs. $J\!-\!K'$ color magnitude diagrams for
population synthesis models based on the Bruzual \& Charlot 1995 code
(Bruzual \& Charlot 2000), together with a
subsample of the MUNICS data. Note that the $J\!-\!K'$ color of a late
M-type star is $\leq 0.8$, and therefore any object having redder
color must be redshifted, with the exception of extreme stellar
objects like brown dwarfs. Those are not expected to be very numerous
down to our sensitivity limits. Therefore the $J\!-\!K'$ color is a
very powerful selection tool for picking out the evolved populations
of massive galaxies at redshifts around unity and larger.
Figure 2 (right panel) shows the photometric redshift distribution of
371 galaxies which were detected in {\em all} 5 passbands in a small
subarea of $13.2\arcmin \times 13.2\arcmin$ and have a $J\!-\!K'$
color $\geq 1.3$, confirming the usefulness of the $J\!-\!K'$
selection. The left panel demonstrates that we can reliably
reconstruct the SED of early-type galaxies at high redshift from our
$VRIJK$ data.
Data reduction and calibration of the near-IR data is completed and
reduction of the optical data is almost completed. Here we present
first results regarding detection of clusters and the surface density
of EROs.
\section{The Surface Density of EROs}
The surface density of Extremely Red Objetcs (EROs) defined in terms
of $J\!-\!K'$ as determined from 0.35 square degrees of data is given
in Table 1. As soon as the optical data become fully available further
investigation of the nature of such objects will be possible, as well
as comparisons to previous studies which mostly use $R\!-\!K$ or
$I\!-\!K$ for defining EROs, and have surveyed much smaller areas at
comparable depth.
\begin{table}
\begin{center}
\begin{tabular}[ht!]{c|r|l}
\hline
$J\!-\!K'$ & $N_{total}$ & $\frac{N}{arcmin^2}$ \\
\hline
\hline
$> 1.50$ & 3030 & 2.41 $\pm$ 0.04 \\
$> 1.75$ & 1521 & 1.21 $\pm$ 0.03 \\
$> 2.00$ & 706 & 0.56 $\pm$ 0.021 \\
$> 2.25$ & 301 & 0.24 $\pm$ 0.014 \\
$> 2.50$ & 124 & 0.098 $\pm$ 0.009 \\
$> 2.75$ & 51 & 0.040 $\pm$ 0.006 \\
$> 3.00$ & 27 & 0.021 $\pm$ 0.004 \\
$> 3.25$ & 11 & 0.009 $\pm$ 0.003 \\
$> 3.50$ & 5 & 0.004 $\pm$ 0.001 \\
\hline
\end{tabular}
\caption{Surface densities as a function of $J\!-\!K'$ color as derived
from a total of 0.35 square degrees.}
\end{center}
\end{table}
Assuming that the $J\!-\!K'$ and $R\!-\!K'$ colors for LBDS 53W091
(Spinrad et al. 1997) are typical for EROs, all objects with $J\!-\!K'
> 1.75$ have to be considered as ERO candidates. The values of Table 1
then point to higher surface densities than the values obtained by
Thompson et al. (1999), which were based on an $R\!-\!K' > 6$ color
and a survey area of 0.04 square degrees.
\section{Detection of Galaxy Clusters}
In recent years it became clear that an early type population in
clusters was well in place at redshifts of at least 0.8
(e.g. Stanford, Eisenhardt, \& Dickinson 1998). Thus we may hope to
detect these clusters by looking for overdensities of red objects with
colors resembling the color sequence of cluster early-type
galaxies. The surface density field is divided into (overlapping)
slices in $J\!-\!K'$ color. These slices are then smoothed by a kernel
of the angular size of a cluster core at the apropriate redshift for
that particular color range. Cluster candidates are identified as
overdensities in this data cube.
Figure 3 shows the redshift distribution for objects in the vicinity
of a high redshift cluster candidate detected in this fashion as a
demonstration of the efficiency of the technique. The cluster
candidate is first detected as an overdensity of red objects in
$J\!-\!K'$. Then this detection is verified by looking at the
histogram of photometric redshifts in the vicinity of the cluster
candidate.
\begin{figure}[ht!]
\begin{center}
\begin{minipage}{0.4\textwidth}
\plotone{clusterz.eps}
\end{minipage}
\end{center}
\caption{Photometric redshift distribution of objects in the vicinity of a cluster candidate detected as an overdensity of red objects as described in the text. The cluster has an estimated redshift of 0.6.}
\end{figure}
|
2,869,038,154,876 | arxiv | \section{Introduction}\label{s:1}
For $n\geq 3$, let $r=r(n)$ and $\ell$ be integers such that $r=r(n)\geq 3$ and $2\leq\ell\leq r-1$.
A hypergraph ${H}$ on vertex set $[n]$ is an \textit{$r$-uniform hypergraph}
(\textit{$r$-graph} for short) if each edge is a set of $r$ vertices.
An $r$-graph is called a partial Steiner $(n,r,\ell)$-system
if every subset of size $\ell$ is contained in at most one edge of $H$.
In particular, $(n,r,2)$-systems are also called \textit{linear hypergraphs}, which implies that
any two edges intersect in at most one vertex.
Partial Steiner $(n,r,\ell)$-systems and the stronger version, Steiner $(n,r,\ell)$-systems,
where every $\ell$-set is contained in precisely one edge of ${H}$,
are widely studied combinatorial designs. Little is known about the number of %
distinct partial Steiner $(n,r,\ell)$-systems, denoted by $s(n,r,\ell)$.
Grable and Phelps~\cite{grable96}
used the R\"{o}dl nibble algorithm~\cite{rodl85} to obtain
an asymptotic formula for $\log s(n,r,\ell)$ as $\ell\leq r-1$
and $n\to\infty$.
Asratian and Kuzjurin gave another proof~\cite{asas00}.
An interesting problem is the enumeration of hypergraphs with
given number of edges. Let $\mathcal{H}_r(n,m)$ denote the set of $r$-graphs on
the vertex set $[n]$ with $m$ edges, and let $\mathcal{L}_r(n,m)$ denote
the set of all linear hypergraphs in $\mathcal{H}_r(n,m)$.
Dudek et al.~\cite{dudek13} used the switching method
to obtain the asymptotic number of $k$-regular $r$-graphs for fixed $r$
and $n\to\infty$ with $k=o(n^{1/2})$.
For $r=r(n)\geq 3$ an integer and a sequence of positive integers
$\boldsymbol{k}=\boldsymbol{k}(n)=(k_1,\ldots,k_n)$, define $M=M(n)=\sum_{i=1}^nk_i$.
Let $\mathcal{H}_r(\boldsymbol{k})$ denote the set of $r$-graphs on the
vertex set $[n]$ with degree sequence $\boldsymbol{k}$, and
$\mathcal{L}_r(\boldsymbol{k})$ denote the set of all linear hypergraphs
in $\mathcal{H}_r(\boldsymbol{k})$.
Blinovsky and Greenhill~\cite{vlaejoc,valelec} extended the asymptotic
enumeration result on the number of $k$-regular $r$-graphs
to the general $\mathcal{H}_r(\boldsymbol{k})$
when $r^4k_{\rm max}^3=o(M)$ and $n\to\infty$.
By relating the incidence matrix of a hypergraph to the biadjacency matrix of a
bipartite graph, they used switching arguments together with previous
enumeration results for bipartite graphs to obtain the asymptotic
enumeration formula for $\mathcal{L}_r(\boldsymbol{k})$
provided $r^4k_{\rm max}^4(k_{\rm max}+r)=o(M)$ and $n\to\infty$~\cite{valelec}.
Recently, Balogh and Li~\cite{balgoh17} obtained an upper bound on the
total number of linear $r$-graphs with given girth for fixed $r\geq 3$.
Apart from these few results, the literature on the enumeration of linear
hypergraphs is very sparse.
In particular, there seems to be no asymptotic enumeration of linear
hypergraphs by the number of edges, which is the subject of this paper.
The result of Blinovsky and Greenhill~\cite{valelec} could in principle
be summed over degree sequences to obtain $\mathcal{L}_r(n,m)$
for $m=o\bigl( \min\{r^{-2}n^{\frac54},r^{-\frac83}n^{\frac43}\}\bigr)$,
but we prefer a direct switching approach.
Note that $m=O(r^{-2}n^2)$ for all linear hypergraphs; we get as far
as $m=o(r^{-3}n^{\frac32})$.
Our
application of the switching method combines several different switching
operations into a single computation, which was previously used in~\cite{green06} to
count sparse $0$-$1$ matrices with irregular row and column sums, in~\cite{green08} to
count sparse nonnegative integer matrices with specified row and column sums, and
in~\cite{green13} to count sparse multigraphs with given degrees.
We will use the falling factorial $[x]_t=x(x-1)\cdots(x-t+1)$
and adopt $N$ as an abbreviation for $\binom nr$.
All asymptotics are with respect to $n\to\infty$.
Our main theorem is the following.
\begin{theorem}\label{t1.1}
Let $r=r(n)\geq 3$
and $m=m(n)$ be integers with $m=o(r^{-3}n^{ \frac32})$.
Then, as $n\to \infty$,
\begin{align}
|\mathcal{L}_r(n,m)|&=
\binom{N}{m}
\exp\biggl[- \frac{[r]_2^2[m]_2}{4n^2}- \frac{[r]_2^3(3r^2-15r+20)m^3}{24n^4}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr] \label{maineqn}\\
&={ \frac{N^m}{m!}}
\exp\biggl[- \frac{[r]_2^2[m]_2}{4n^2}- \frac{[r]_2^3(3r^2-15r+20)m^3}{24n^4}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr]. \notag
\end{align}
\end{theorem}
\begin{proof}[Proof of Theorem~\ref{t1.1}]
Note that the condition $m=o(r^{-3}n^{ \frac32})$ implies that
either $m=0$ or $r=o(n^{\frac12})$. In the former case the theorem
is trivially true, while in the latter we can apply
Remark~\ref{r4.5} and Lemma~\ref{l6.6} to obtain~\eqref{maineqn}
for $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
Equivalent expressions follow from Remark~\ref{r7.3} and
Lemma~\ref{l7.7} when $\log(r^{-2}n)\leq m=O(r^{-2}m)$
and from Remark~\ref{r8.2} and Lemma~\ref{l8.6}
when $1\leq m=O(\log(r^{-2}n))$.
\end{proof}
Let $\mathbb{P}_r(n,m)$ denote the probability that an
$r$-graph $H\in \mathcal{H}_r(n,m)$ chosen uniformly at random is linear. Then
\[
|\mathcal{L}_r(n,m)|= \binom{N}{m} \mathbb{P}_r(n,m).
\]
Hence, our task is reduced to computing $\mathbb{P}_r(n,m)$ and
it suffices to show that $\mathbb{P}_r(n,m)$ equals the exponential
factor in Theorem~\ref{t1.1}.
Recall that a random $r$-graph ${H}_r(n,p)=([n],E_{n,p})$ refers to an $r$-graph
on the vertex set $[n]$, where each $r$-set is an edge randomly and
independently with probability~$p$. Also it might be surmised that random
hypergraphs with edge probability~$p$ have about the same probability of
being linear as a random hypergraph with $Np$ edges, that is not the case
when $Np$ is moderately large.
Let $\mathcal{L}_r(n)$ be the set of all
linear $r$-graphs with $n$ vertices.
\begin{theorem}\label{t1.3}
Let $r=r(n)\geq 3$ and let
$pN=m_0$ with $m_0=o(r^{-3}n^{ \frac32})$.
Then, as $n\to\infty$,
\begin{align*}
\mathbb{P}&[H_r(n,p)\in \mathcal{L}_r(n)]\\
&=\begin{cases}
\exp\Bigl[- \frac{[r]_2^2m_0^2}{4n^2}+O\bigl( \frac{r^6m_0^2}{n^3}\bigr)\Bigr],
&\text{ if }m_0=O(r^{-2}n);\\[2ex]
\exp\Bigl[- \frac{[r]_2^2m_0^2}{4n^2}+ \frac{[r]_2^3(3r-5)m_0^3}{6n^4}+
O\bigl(\frac{\log^3(r^{-2}n)}{\sqrt{m_0}} + \frac{r^6m_0^2}{n^3}\bigr)\Bigr],
&\text{ if }r^{-2}n\leq m_0=o(r^{-3}n^{ \frac32}).
\end{cases}
\end{align*}
\end{theorem}
From the calculations in the proof of Theorem~\ref{t1.3}, we have a corollary about the distribution on the
number of edges of $H_r(n,p)$ conditioned on it being linear.
\begin{corollary}\label{c1.4}
Let $r=r(n)\geq 3$ and let $Np=m_0$
with $r^{-2}n\leq m_0=o(r^{-3}n^{ \frac32})$.
Suppose that $n\to \infty$.
Then the number of edges of $H_r(n,p)$ conditioned on being
linear converges in distribution to the normal distribution with mean $m_0- \frac{[r]_2^2m_0^2}{2n^2}$
and variance $m_0$.
\end{corollary}
Consider $H\in\mathcal{L}_r(n,m)$ chosen uniformly at random.
Using a similar switching method, we also obtain the
probability that $H$ contains a given hypergraph as a subhypergraph.
\begin{theorem}\label{t1.5}
Let $r=r(n)\geq 3$, $m=m(n)$ and $k=k(n)$ be integers with
$m=o(r^{-3}n^{ \frac32})$ and
$k=o\bigl(\frac{n^3}{r^6m^2}\bigr)$.
Let $K=K(n)$ be a linear $r$-graph on $n$ vertices with $k$ edges.
Let $H\in\mathcal{L}_r(n,m)$ be chosen uniformly at random. Then, as $n\to\infty$,
\[
\mathbb{P}[K\subseteq H]= \frac{[m]_k}{N^k}\exp\biggl[ \frac{[r]_2^2k^2}{4n^2}+
O\Bigl( \frac{r^4k}{n^2}+ \min\Bigl\{ \frac{r^6m^2k}{n^3}, \frac{r^5mk}{n^2}\Bigr\}\Bigr)\biggr].
\]
\end{theorem}
The remainder of the paper is structured as follows. Notation and auxiliary results
are presented in Section~\ref{s:2}. From Section~\ref{s:3} to Section~\ref{s:6},
we mainly consider the case $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
In Section~\ref{s:3}, we define subsets $\mathcal{H}^+_r(n,m)$ and
$\mathcal{H}^{++}_r(n,m)$ of $\mathcal{H}_r(n,m)$ and show that they
are almost all of $\mathcal{H}_r(n,m)$. In Section~\ref{s:4}, we
show that the same is true when $\mathcal{H}^+_r(n,m)$ and
$\mathcal{H}^{++}_r(n,m)$ are restricted by certain counts of clusters of
edges that overlap in more than one vertex.
We define four other kinds of
switchings on $r$-graphs in $\mathcal{H}_r^{+}(n,m)$
which are used to remove some hyperedges with two or more common vertices, and
analyze these switchings in Section~\ref{s:5}. In Section~\ref{s:6}, we complete
the enumeration for the case $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$
with the help of some calculations performed in~\cite{green06,green08,green13}.
In Sections~\ref{s:7}--\ref{s:8}, we consider the cases $\log (r^{-2}n)\leq m= O(r^{-2}n)$
and $1\le m=O(\log (r^{-2}n))$, respectively. In Section~\ref{s:9}, we prove Theorem~\ref{t1.3},
while in Section~\ref{s:10}, we prove Theorem~\ref{t1.5}.
\section{Notation and auxiliary results}\label{s:2}
To state our results precisely, we need some definitions.
Let $H$ be an $r$-graph in $\mathcal{H}_r(n,m)$.
For $U\subseteq [n]$, the \textit{codegree} of $U$ in $H$, denoted by $\codeg({U})$,
is the number of edges of $H$ containing~$U$. In particular, if $U=\{v\}$ for
$v\in [n]$ then $\codeg({U})$ is the degree of $v$ in $H$, denoted by $\deg (v)$.
Any $2$-set $\{x,y\}\subseteq [n]$ in an
edge $e$ of $H$ is called a \textit{link} of $e$ if $\codeg ({x,y})\geq 2$.
Two edges $e_i$ and $e_j$ in $H$ are called \textit{linked edges}
if $|e_i\cap e_j|=2$.
Let $G_H$ be a simple graph whose vertices are the edges of~$H$,
with two vertices of $G$ adjacent iff the corresponding edges of~$H$ are linked.
An edge induced subgraph of $H$ corresponding to a non-trivial
component of $G_H$ is called a \textit{cluster} of $H$.
The standard asymptotic notations $o$ and $O$ refer to $n\to\infty$.
The floor and ceiling signs are omitted whenever they are not crucial.
In order to identify several events which have low probabilities
in the uniform probability space $\mathcal{H}_r(n,m)$ as
$m=o(r^{-3}n^{ \frac32})$, the following lemmas will be useful.
\begin{lemma}\label{l2.1}
Let $r=r(n)\geq 3$, $t=t(n)\geq 1$ be integers.
Let $e_1,\ldots,e_{t}$ be distinct $r$-sets of $[n]$ and $H$ be an
$r$-graph that is chosen uniformly at random from $\mathcal{H}_r(n,m)$.
Then the probability that $e_1,\ldots,e_t$ are edges of $H$ is at most
$\bigl( \frac{m}{N}\bigr)^t$.
\end{lemma}
\begin{proof} Since $H$ is an $r$-graph
that is chosen uniformly at random from $\mathcal{H}_r(n,m)$, the probability
that $e_1,\ldots,e_t$ are edges of $H$ is
\begin{align*}
\frac{ \binom{N-t}{m-t}}{ \binom{N}{m}}&= \frac{[m]_t}{[N]_t}=
\prod_{i=0}^{t-1} \frac{m-i}{N-i}\leq\Bigl( \frac{m}{N}\Bigr)^t. \qedhere
\end{align*}
\end{proof}
\begin{lemma}\label{l2.2}
Let $r=r(n)\geq 3$ be an integer with $r=o(n^{ \frac12})$.
Let $t$ and $\alpha$ be integers such that $t=O(1)$ and $0\leq\alpha\leq rt$.
If a hypergraph $H$ is chosen uniformly at random from $\mathcal{H}_r(n,m)$,
then the expected number of sets of $t$ edges of $H$ whose union has
$rt-\alpha$ or fewer vertices is $O\bigl( t^\alpha r^{2\alpha}m^t n^{-\alpha})$.
\end{lemma}
\begin{proof}
Let $e_1,\ldots,e_t$ be distinct $r$-sets of $[n]$.
According to Lemma~\ref{l2.1}, the probability that $e_1,\ldots,e_t$
are edges of $H$ is at most $(m/N)^t$.
Here, we firstly count
how many $e_1,\ldots,e_t$ such that $|e_1\cup\cdots\cup e_t|=rt-\beta$
for some $\beta\geq \alpha$.
Suppose that there is a sequence among the edges $\{e_1,\ldots,e_t\}$ and
we have chosen the edges $\{e_1,\ldots,e_{i-1}\}$, where $2\leq i\leq t$.
Let $a_i=|(e_1\cup\cdots\cup e_{i-1})\cap e_i|$, thus we have
$\sum_{i=2}^ta_i=\beta$ and
\[
O\biggl(N\prod_{i=2}^{t}(tr)^{a_i} \binom{n}{r-a_i}\biggr)
\]
ways to choose $\{e_1,\ldots,e_t\}$. The expected number of these
$t$ edges is
\begin{align*}
O\biggl(\Bigl( \frac{m}{N}\Bigr)^tN\prod_{i=2}^{t}(tr)^{a_i} \binom{n}{r-a_i}\biggr)
&=O\biggl(m^tt^\beta r^\beta\prod_{i=2}^{t} \frac{[r]_{a_i}}{[n-r]_{a_i}}\biggr)\\
&=O\biggl(m^tt^\beta r^\beta\prod_{i=2}^{t}\Bigl( \frac{r}{n-r}\Bigr)^{a_i}\biggr)
=O\biggl( \frac{t^\beta m^t r^{2\beta}}{n^\beta}\biggr),
\end{align*}
where we use the fact that $\prod_{j=0}^{a_i-1} \frac{r-j}{n-r-j}\leq\bigl( \frac{r}{n-r}\bigr)^{a_i}$
is true as $r\leq \frac{n}{2}$ and $\bigl( \frac{r}{n-r}\bigr)^{a_i}=O(( \frac{r}{n})^{a_i})$ because $a_i< r$,
$r=o(n^{ \frac12})$ and $a_i r=o(n)$.
The expected number of sets of $t$ edges whose union has at most
$rt-\alpha$ vertices is
\[
O\biggl(\sum_{\beta\geq\alpha} \frac{t^\beta m^t r^{2\beta}}{n^\beta}\biggr)
=O\biggl( \frac{t^\alpha m^t r^{2\alpha}}{n^\alpha}\biggr),
\]
because $\beta=\alpha$ corresponds to the
largest term as $t=O(1)$ and $r=o(n^{ \frac12})$.\end{proof}
\begin{remark}\label{r2.3}
Throughout the following sections we assume that $n\to\infty$.
From Sections~\ref{s:3}--\ref{s:6},
we assume that $r^{-2}n \leq m=o(r^{-3}n^{ \frac32})$.
In Sections~\ref{s:7}--\ref{s:8} , we assume
that $\log (r^{-2}n)\leq m=O(r^{-2}n)$
and $1\le m=O(\log (r^{-2}n))$, respectively.
Recall that these conditions all imply that $r=o(n^{ \frac12})$.
\end{remark}
\section{Two important subsets of $\mathcal{H}_r(n,m)$}\label{s:3}
Define
\begin{equation}\label{e3.1}
\begin{split}
M_0^*&=\biggl\lceil\log (r^{-2}n)+ \frac{3^4r^2m}{n}\biggr\rceil, \\
M_0&=\biggl\lceil\log (r^{-2}n)+ \frac{3^4r^2m}{n}\biggr\rceil+3, \\
M_1&=\biggl\lceil\log(r^{-2}n)+ \frac{3^4r^8 m^3}{2n^4}\biggr\rceil, \\
M_2&=\biggl\lceil\log(r^{-2}n)+ \frac{3^4r^7m^3}{2n^4}\biggr\rceil, \\
M_3&=\biggl\lceil\log(r^{-2}n)+ \frac{3^4r^6m^3}{2n^4}\biggr\rceil, \\
M_4&=\biggl\lceil\log(r^{-2}n)+ \frac{3^4r^4m^2}{2n^2}\biggr\rceil.
\end{split}
\end{equation}
Now define $\mathcal{H}_r^+(n,m)\subseteq\mathcal{H}_r(n,m)$
to be the set of $r$-graphs $H$ which satisfy the following
properties $\bf(a)$ to $\bf(g)$.
$\bf(a)$\ The intersection of any two edges contains at most two vertices.
$\bf(b)$\ $H$ only contains the four types of clusters that are shown in Figure~\ref{fig:1}.
(This implies that any three edges of $H$ involve at least $3r-4$ vertices and
any four edges involve at least $4r-5$ vertices. Thus, if there are
three edges of $H$, for example $\{e_1,e_2,e_3\}$, such that $|e_1\cup e_2\cup e_3|=3r-4$,
then $|e\cap (e_1\cup e_2\cup e_3)|\leq 1$ for any
edge $e$ other than $\{e_1,e_2,e_3\}$ of $H$.)
\begin{figure}[!htb]
\centering
\includegraphics[width=1.0\textwidth]{component.pdf}
\caption{The four types of clusters allowed in $H\in\mathcal{H}_r^+(n,m)$.\label{fig:1}}
\end{figure}
$\bf(c)$\ The intersection of any two clusters contains at most one vertex.
$\bf(d)$\ Any three distinct Type-$1$, Type-$2$ or Type-$3$ clusters
involve at least $9r-13$ vertices. (Together with~$\bf(c)$,
this implies that if a pair of Type-$1$, Type-$2$ or Type-$3$ clusters
have exactly one common vertex, then any other
Type-$1$, Type-$2$ or Type-$3$ clusters
of $H$ must be vertex-disjoint from them.)
$\bf(e)$\ Any three distinct Type-$4$ clusters involve at least $6r-8$ vertices.
(Together with~$\bf(c)$,
this implies that if a pair of Type-$4$ clusters of $H$ have exactly one common vertex, then any other Type-$4$
cluster of $H$ shares at most one vertex with them.)
$\bf(f)$\ There are at most $M_i$ Type-$i$ clusters, for $1\leq i\leq 4$.
$\bf(g)$\ $\deg (v)\leq M_0$ for every vertex $v\in [n]$.
We further define $\mathcal{H}_r^{++}(n,m)\subseteq\mathcal{H}_r^+(n,m)$
to be the set of $r$-graphs $H$
by replacing the property $\bf(g)$ with a stronger constraint $\bf(g^*)$.
$\bf(g^*)$\ $\deg (v)\leq M_0^*$ for every vertex $v\in [n]$.
\begin{remark}\label{r3.1}
From property $\bf(g)$, it is natural to obtain
\[
\sum_{v\in [n]} \binom{\deg (v)}{2}=O\Bigl(M_0\sum_{v\in [n]}\deg (v)\Bigr)=O\bigl(rmM_0\bigr)
=O\Bigl(rm\log (r^{-2}n)+ \frac{r^3m^2}{n}\Bigr)
\]
for $H\in\mathcal{H}_r^+(n,m)$.
\end{remark}
We now show that the expected number of $r$-graphs
in $\mathcal{H}_r(n,m)$
not satisfying the properties of $\mathcal{H}^+_r(n,m)$ and $\mathcal{H}^{++}_r(n,m)$ is quite small,
which implies that these $r$-graphs make asymptotically insignificant contributions.
The removal of these $r$-graphs from our main proof will lead
to some welcome simplifications.
\begin{theorem}\label{t3.2}
Suppose that $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ and $n\to \infty$. Then
\[
\frac{|\mathcal{H}^+_r(n,m)|}{|\mathcal{H}_r(n,m)|}=1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr),\quad \frac{|\mathcal{H}^{++}_r(n,m)|}{|\mathcal{H}_r(n,m)|}=1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
\end{theorem}
\begin{proof} Consider $H\in \mathcal{H}_r(n, m)$ chosen uniformly at random. It's
enough for us to prove the second equation. We apply Lemma~\ref{l2.2} several times
to show that $H$ satisfies the properties $\bf(a)$-$\bf(g^*)$ with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(a)$\ Applying Lemma~\ref{l2.2} with $t=2$ and $\alpha=3$, the expected number of
two edges involving at most $2r-3$ vertices is $O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
Hence, the property $\bf(a)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(b)$\ Applying Lemma~\ref{l2.2} with $t=3$ and $\alpha=5$, the expected number of
three edges involving at most $3r-5$ vertices is
\[
O\Bigl( \frac{r^{10}m^3}{n^{5}}\Bigr)
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr),
\]
where the last equality is true because $m=o(r^{-3}n^{ \frac32})$.
Similarly, applying Lemma~\ref{l2.2} with $t=4$ and $\alpha=6$, the expected
number of four edges involving at most $4r-6$ vertices is
\[
O\Bigl( \frac{r^{12}m^4}{n^{6}}\Bigr)
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
If there is a cluster with four or more edges, then there must be
four edges involving at most $4r-6$ vertices.
Hence, every cluster contains at most three edges and the property
$\bf(b)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(c)$\ Applying Lemma~\ref{l2.2}, the expected number of two clusters such
that their intersection contains
two or more vertices is
\[
O\Bigl( \frac{r^{20}m^6}{n^{10}}+ \frac{r^{16}m^5}{n^8}+ \frac{r^{12}m^4}{n^6}
\Bigr)=O\Bigl( \frac{r^6m^2}{n^3}\Bigr),
\]
where the first term arises from the case between Type-$1$ (or Type-$2$
or Type-$3$) cluster and Type-$1$ (or Type-$2$ or Type-$3$)
cluster and there are at most $6r-10$ vertices if their intersection
contains at least two vertices, the second term arises from the case
between Type-$1$ (or Type-$2$ or Type-$3$) cluster and Type-$4$ cluster
and there are at most $5r-8$ vertices if their intersection contains
at least two vertices, while the last term arises from the case between
Type-$4$ cluster and Type-$4$ cluster and there are at most $4r-6$ vertices
if their intersection contains at least two vertices.
Hence, the property $\bf(c)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(d)$\ The expected number of three Type-$1$, Type-$2$ or Type-$3$ clusters
involving at most $9r-14$ vertices is
\[
O\Bigl( \frac{r^{28}m^9}{n^{14}}\Bigr)
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
Hence, the property $\bf(d)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(e)$\ The expected number of three Type-$4$
clusters involving at most $6r-9$ vertices is
\[
O\Bigl( \frac{r^{18}m^6}{n^{9}}\Bigr)
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
Hence, the property $\bf(e)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
$\bf(f)$\ Define four events $\mathcal{E}_i$ as
\[
\mathcal{E}_i=\bigl\{\text{There are at most $M_i$ Type-$i$ clusters in $H$}\bigr\},
\]
and $\mathcal{\overline{E}}_i$ as the complement of
the event $\mathcal{E}_i$, where $1\leq i\leq 4$.
We show that $\mathbb{P}[\mathcal{E}_i]=1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$
for $1\leq i\leq 4$.
Let
\[
Q_1=\biggl\lceil \log (r^{-2}n)+\frac{3^{4}r^8 m^3}{4n^4}\biggr\rceil
\]
and define $\ell_1=Q_1+1$. We first show that the expected number of
sets of $\ell_1$ vertex-disjoint Type-$1$ clusters in $H$ is $O\bigl( \frac{r^6}{n^3}\bigr)$.
These $\ell_1$ vertex-disjoint Type-$1$ clusters contain $3\ell_1$ edges and $(3r-4)\ell_1$
vertices. Note that Lemma~\ref{l2.2} is not appropriate here because $3\ell_1$ is not a constant.
By Lemma~\ref{l2.1}, it follows that the expected number of sets of $\ell_1$
vertex-disjoint Type-$1$ clusters in $H$ is at most
\begin{align*}
\binom{n}{(3r-4)\ell_1} &\binom{(3r-4)\ell_1}{3r-4,\ldots,3r-4} \frac{1}{\ell_1!}
\biggl[ \binom{3r-4}{r} \binom{r}{4} \binom{4}{2} \binom{2r-4}{r-2}\biggr]^{\ell_1}\biggl( \frac{m}{N}\biggr)^{3\ell_1}\\
&=O\biggl(\biggl( \frac{[r]_4[r]_2^2em^3}{4\ell_1n^4}\biggr)^{\ell_1}\biggr)
=O\Bigl(\Bigl( \frac{e}{3^4}\Bigr)^{\ell_1}\Bigr)
=O\Bigl( \frac{r^6}{n^3}\Bigr),
\end{align*}
where the first two equalities are true because $\ell_1!\geq \bigl( \frac{\ell_1}{e}\bigr)^{\ell_1}$
and $\ell_1> \frac{3^4r^8 m^3}{4n^4}$, and the
last equality is true because of the assumption $\ell_1>\log (r^{-2}n)$.
Assuming that property $\bf(d)$ holds, any Type-$1$ cluster is
either vertex-disjoint from all other Type-$1$ clusters of $H$, or
shares one vertex with precisely one
other Type-$1$ cluster of~$H$, then it follows that
$\mathbb{P}[\mathcal{\overline{E}}_1\mid\text{Property {\bf(d)} holds}]
=O\bigl( \frac{r^6}{n^3}\bigr)$.
By the total probability formula, we have
\[
\mathbb{P}[\mathcal{\overline{E}}_1]
\leq\mathbb{P}[\text{Property {\bf(d)} does not hold}]+
\mathbb{P}[\mathcal{\overline{E}}_1\mid\text{Property {\bf(d)} holds}]
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr)
\]
and $\mathbb{P}\left[\mathcal{E}_1\right]=1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
Similarly, we have $\mathbb{P}\left[\mathcal{E}_2\right]=1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$, $\mathbb{P}\left[\mathcal{E}_3\right]=1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
At last, we show that
$\mathbb{P}[\mathcal{E}_4]=1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$ is also true.
Let $\ell_4=M_4+1$. Let $\{x_i,y_i\}\in \binom{[n]}{2}$ be a set of $\ell_4$ links
with edges $e_i$ and $e_i'$, here $i\in\{1,\ldots,\ell_4\}$.
These $\ell_4$ links are called \textit{paired-distinct} if these $2\ell_4$ edges
are all distinct. Assuming the property $\bf(b)$ holds, note that the number of
Type-$4$ clusters is no greater than the number of paired-distinct links.
Define a event $\mathcal{E}_4'$ as
\[
\mathcal{E}_4'=\bigl\{\text{There are at most $M_4$ paired-distinct links in $H$}\bigr\},
\]
and $\mathcal{\overline{E}}_4'$ as the complement of the event $\mathcal{E}_4'$.
We firstly show that $\mathbb{P}[\mathcal{\overline{E}}_4']=O\bigl( \frac{r^6}{n^3}\bigr)$ by
\[
\mathbb{P}\bigl[\mathcal{\overline{E}}_4'\bigr]
=O\biggl( \binom{n}{r-2}^{2\ell_4} \binom{ \binom{n}{2}}{\ell_4}\biggl( \frac{m}{N}\biggr)^{2\ell_4}\biggr)
=O\biggl(\Bigl( \frac{r^4e m^2}{2\ell_4n^2}\Bigr)^{\ell_4}\biggr)
=O\Bigl( \frac{r^6}{n^3}\Bigr),
\]
where the last two equalities are true because $\ell_4> \frac{3^4r^4m^2}{2n^2}$
and $\ell_4>\log(r^{-2}n)$.
Then it follows that $\mathbb{P}[\mathcal{\overline{E}}_4\mid\text{Property {\bf(b)}
holds}]\leq\mathbb{P}[\mathcal{\overline{E}}_4']=O\bigl( \frac{r^6}{n^3}\bigr)$.
By the law of total probability,
\[
\mathbb{P}[\mathcal{\overline{E}}_4]
\leq\mathbb{P}[\text{Property {\bf(b)} does not hold}]+
\mathbb{P}[\mathcal{\overline{E}}_4\mid\text{Property {\bf(b)} holds}]
=O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
$\bf(g^*)$\ Define $d=M_0^*+1$.
The expected number of vertices $v$ such that $\deg (v)\geq d$ is
\[
n \binom{n-1 }{r-1}^d \frac{1}{d!}\Bigl( \frac{m}{N}\Bigr)^{d}
=O\Bigl(n\Bigl( \frac{emr}{dn}\Bigr)^{d}\Bigr)
=O\Bigl(n\Bigl( \frac{e}{3^4r}\Bigr)^{d}\Bigr)
=O\Bigl( \frac{r^6}{n^3}\Bigr),
\]
where the second equality is true because $d!\geq \bigl( \frac{d}{e}\bigr)^{d}$,
and the last equality is true because of the assumption $d>M_0^*$ made in~\eqref{e3.1}.
Thus, there are no vertices with degree at least $d$
in $H$ holds with probability $1-O\bigl( \frac{r^6}{n^3}\bigr)$.
This completes the proof of Theorem~\ref{t3.2}.
\end{proof}
\begin{remark}\label{r3.3}
For nonnegative integers $h_1$,
$h_2$, $h_3$ and $h_4$, define $\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$
to be the set of $r$-graphs in $H\in \mathcal{H}_r^+(n,m)$ with
exactly $h_i$ clusters of Type~$i$, for $1\leq i\leq 4$.
Define $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}$ similarly.
By the definitions of $H\in \mathcal{H}_r^+(n,m)$ and
$H\in \mathcal{H}_r^{++}(n,m)$ we have
\begin{align*}
|\mathcal{H}_r^+(n,m)|&=\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}
\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1}|\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}|,\\
|\mathcal{H}_r^{++}(n,m)|&=\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}
\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1}|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|.
\end{align*}
\end{remark}
We will estimate the relative sizes of these subsets by means of
switching operations. The following is an essential tool that we
will use repeatedly.
\begin{lemma}\label{l3.5}
Assume that $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
Let $H\in \mathcal{H}_r^{+}(n,m-\xi)$
and $N_t$ be the set of $t$-sets of $[n]$ of which no two vertices belong to
the same edge of $H$, where $r\leq t\leq 3r-4$ and $\xi=O(1)$.
Suppose that $n\to\infty$. Then
\[
|N_t|=\biggl[ \binom{n}{t}- \binom{r}{2}m \binom{n-2}{t-2}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
\end{lemma}
\begin{proof} We will use inclusion-exclusion. Let $A_{e(i,j)}$ be the
event that a $t$-set of $[n]$ contains two vertices $i$ and $j$ of the edge $e$.
Thus, we have
\[
\binom{n}{t}-\sum_{\{e,\{i,j\}\}}|A_{e(i,j)}|\leq|N_t|\leq \binom{n}{t}-\sum_{\{e,\{i,j\}\}}|A_{e(i,j)}|+\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}}|A_{e(i,j)}\cap A_{e'(i',j')}|.
\]
Clearly, $|A_{e(i,j)}|= \binom{n-2}{t-2}$ for each edge $e$
and $\{i,j\}\subset e$. We have $|N_t|\geq \binom{n}{t}- \binom{r}{2}(m-\xi) \binom{n-2}{t-2}$.
Now we consider the upper bound. For the case $e=e'$, we have
\begin{align}\label{e3.2}
\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}} & |A_{e(i,j)}\cap A_{e'(i',j')}| \notag \\
&=(m-\xi) \binom{r}{3} \binom{3}{2} \binom{n-3}{t-3}+ \frac12(m-\xi) \binom{r}{4} \binom{4}{2} \binom{n-4}{t-4} \notag \\
&=O\biggl((m-\xi)r^3 \binom{n-3}{t-3}\biggr).
\end{align}
For the case $e\neq e'$ and $\{i,j\}\cap \{i',j'\}=\emptyset$, we have
\begin{align}\label{e3.3}
\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}} & |A_{e(i,j)}\cap A_{e'(i',j')}| \notag\\
&=O\biggl( \binom{m-\xi}{2} \binom{r}{2}^2 \binom{n-4}{t-4}\biggr) \notag\\
&=O\biggl((m-\xi)^2r^4 \binom{n-4}{t-4}\biggr).
\end{align}
For the case $e\neq e'$ and $|\{i,j\}\cap \{i',j'\}|=1$, by Remark~\ref{r3.1}, we have
\begin{align}\label{e3.4}
\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}} & |A_{e(i,j)}\cap A_{e'(i',j')}| \notag\\
&=\sum_{v\in [n]} \binom{\deg(v)}{2}(r-1)^2 \binom{n-3}{t-3} \notag\\
&=O\biggl(\Bigl( \frac{r^3(m-\xi)n\log (r^{-2}n) +r^5(m-\xi)^2}{n}\Bigr) \binom{n-3}{t-3}\biggr).
\end{align}
For the case $e\neq e'$ and $\{i,j\}=\{i',j'\}$, since there are at most $M_s$ Type-$s$
clusters with $1\leq s\leq 4$ in $H$, as the equation shown in~\eqref{e3.1}, we have
\begin{align}\label{e3.5}
\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}} & |A_{e(i,j)}\cap A_{e'(i',j')}| \notag\\
&=O\biggl(\sum_{s=1}^4M_s \binom{n-2}{t-2}\biggr) \notag\\
&=O\biggl(\Bigl(\log (r^{-2}n)+ \frac{r^4(m-\xi)^2}{n^2}\Bigr) \binom{n-2}{t-2}\biggr).
\end{align}
By equations~\eqref{e3.2}--\eqref{e3.5} and the assumption $m\geq r^{-2}n$, we have
\[
\sum_{\{e,\{i,j\}\}\neq \{e',\{i',j'\}\}}|A_{e(i,j)}\cap A_{e'(i',j')}|
=O\biggl(\Bigl( \frac{r^3mn\log (r^{-2}n) +r^5m^2}{n}\Bigr) \binom{n-3}{t-3}\biggr).
\]
Thus, we complete the proof of Lemma~\ref{l3.5} by
\[
\frac{ \binom{n-3}{t-3}}{ \binom{n}{t}}= \frac{[t]_3}{[n]_3}=O\Bigl( \frac{r^3}{n^3}\Bigr),
\frac{ \binom{r}{2}\xi \binom{n-2}{t-2}}{ \binom{n}{t}}=O\Bigl( \frac{r^4}{n^2}\Bigr)
=O\Bigl( \frac{r^8m^2}{n^4}\Bigr),
\]
because $r\leq t\leq 3r-4$, $m\geq r^{-2}n$ and $n\to\infty$.
\end{proof}
\begin{lemma}\label{l3.6}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
Let $H\in\mathcal{H}_r^{+}(n,m-\xi)$ and $N_{2r-2}'$ be the set of $(2r-2)$-sets of $[n]$
of which exactly two vertices belong to the same edge of $H$, where $\xi=O(1)$.
Suppose that $n\to\infty$. Then
\[
|N_{2r-2}'|= \binom{r}{2}m \binom{n-2}{2r-4}
\biggl(1+O\Bigl( \frac{r^2n\log(r^{-2}n)+r^4m}{n^2}\Bigr)\biggr).
\]
\end{lemma}
\begin{proof} It is clear that
\[
\sum_{\{e,\{i,j\}\}}|A_{e(i,j)}|-\sum_{\{e,\{i,j\}\}
\neq \{e',\{i',j'\}\}}|A_{e(i,j)}\cap A_{e'(i',j')}|\leq |N_{2r-2}'|\leq \sum_{\{e,\{i,j\}\}}|A_{e(i,j)}|.
\]
Therefore, as shown in the proof of Lemma~\ref{l3.5}, we have
\begin{align*}
|N_{2r-2}'|&\leq \binom{r}{2}(m-\xi) \binom{n-2}{2r-4},\\
|N_{2r-2}'|&\geq \binom{r}{2}(m-\xi) \binom{n-2}{2r-4}-
O\biggl(\Bigl( \frac{r^3mn\log (r^{-2}n) +r^5m^2}{n}\Bigr) \binom{n-3}{2r-5}\biggr).
\end{align*}
We complete the proof by noting that
$\frac{ \binom{n-3}{2r-5}}{ \binom{r}{2}m \binom{n-2}{2r-4}}
=O\bigl( \frac{1}{rmn}\bigr)$ and $ \frac{\xi}{m}=O\bigl( \frac{r^4m}{n^2}\bigr)$.
\end{proof}
The switching method relies on the fact that the ratio of the sizes of the two
parts of a bipartite graph is reciprocal to the ratio of their average degrees.
For our purposes we need a generalization as given in the following lemma.
\begin{lemma}\label{l3.7}
Let $G$ be a bipartite graph with vertex sets $A$ and $B$, where $A=A_1\cup A_2$ with $A_1\cap A_2=\emptyset$ and
$B=B_1\cup B_2$ with $B_1\cap B_2=\emptyset$. Let
$d_{min}^{A_1}$ and $d_{min}^{B_1}$ be the minimum degrees of vertices in $A_1$ and $B_1$, respectively.
Let $d_{max}^A$ and $d_{max}^B$ be the maximum degrees of vertices in $A$ and $B$, respectively. Then
\[
\frac{d_{min}^{B_1}}{d_{max}^A}\biggl(1+ \frac{|A_2|}{|A_1|}\biggr)^{\!\!-1}
\leq \frac{|A_1|}{|B_1|}
\leq \frac{d_{max}^B}{d_{min}^{A_1}}\biggl(1+ \frac{|B_2|}{|B_1|}\biggr).
\]
\end{lemma}
\begin{proof} Let $E$ be the set of edges between $A_1$ and $B_1$ in $G$. We have
\begin{align*}
|A_1|d_{min}^{A_1}-|B_2|d_{max}^B \leq |E| &\leq |A_1|d_{max}^A,~~\text{and}\\
|B_1|d_{min}^{B_1}-|A_2|d_{max}^A \leq |E| &\leq |B_1|d_{max}^B.
\end{align*}
Combining these inequalities, we have
\[
\frac{|A_1|d_{min}^{A_1}-|B_2|d_{max}^B}{|B_1|d_{max}^B}\leq1
\]
which gives the upper bound on $ \frac{|A_1|}{|B_1|}$, and
\[
\frac{|B_1|d_{min}^{B_1}-|A_2|d_{max}^A}{|A_1|d_{max}^A}\leq1
\]
which gives the lower bound.
\end{proof}
\section{Partitions in $\mathcal{H}_r^{+}(n,m)$ and $\mathcal{H}_r^{++}(n,m)$}\label{s:4}
We firstly show that $|\mathcal{C}_{0,0,0,0}^{++}|$ and
$|\mathcal{L}_r(n,m)|$ are almost equal.
\begin{theorem}\label{t4.1}
Assume that $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$
and $n\to\infty$. Then
\[
|\mathcal{C}_{0,0,0,0}^{++}|=\Bigl(1-O\Bigl( \frac{r^6}{n^3}\Bigr)\Bigr)|\mathcal{L}_r(n,m)|.
\]
\end{theorem}
\begin{proof}
Consider $H\in \mathcal{L}_r(n,m)$ chosen uniformly at random.
We show that there are no vertices with degree greater
than $M_0^*$ with probability $1-O\bigl( \frac{r^6}{n^3}\bigr)$.
Fix $v\in[n]$.
Assume $\deg (v)=d$ for some integer $d\geq 1$. Define
the set $\mathcal{S}_v(d\to d-1)$ to be the set of switching operations
that consist of removing one edge containing $v$ and placing it somewhere
else such that it doesn't contain $v$ but the linearity property is preserved.
Applying Lemma~\ref{l3.5} to $H-\{v\}$, we have
$|\mathcal{S}_v(d\to d-1)|\geq d\bigl[ \binom{n-1}{r}- \binom{r}{2}m \binom{n-3}{r-2}\bigr]$.
In the other direction, assume $\deg (v)=d-1$ for some integer
$d\geq 1$. Define the set $\mathcal{S}_v(d-1\to d)$ as the set of
operations inverse to $\mathcal{S}_v(d\to d-1)$.
Clearly, $|\mathcal{S}_v(d-1\to d)|\leq m \binom{n-1}{r-1}$.
Thus, we have
\begin{align*}
\frac{\mathbb{P}[\deg (v)=d]}{\mathbb{P}[\deg (v)=d-1]}
&= \frac{|\mathcal{S}_v(d-1\to d)|}{|\mathcal{S}_v(d\to d-1)|}\\
&\leq \frac{m \binom{n-1}{r-1}}{d\bigl[ \binom{n-1}{r}- \binom{r}{2}m \binom{n-3}{r-2}\bigr]}\\
&= \frac{rm}{dn}\Bigl(1+O\Bigl( \frac{r^4m}{n^2}\Bigr)\Bigr)
< \frac{2rm}{dn},
\end{align*}
where the last equality is true because $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
If $d>M_0^*$, following the recursive relation as above, then we have
\[
\mathbb{P}[\deg (v)=d]\leq \prod_{i=4\lceil \frac{r^2m}{n}\rceil+1}^d\Bigl( \frac{2rm}{in}\Bigr)
\mathbb{P}\Bigl[\deg (v)=4\Bigl\lceil \frac{r^2m}{n}\Bigr\rceil\Bigr]
\leq\Bigl( \frac{1}{2r}\Bigr)^{d-4\bigl\lceil \frac{r^2m}{n}\bigr\rceil}.
\]
The probability that $\deg (v)>M_0^*$ is
\[
\mathbb{P}[\deg (v)>M_0^*]\leq \sum_{d>M_0^*}\Bigl( \frac{1}{2r}\Bigr)^{d-4
\bigl\lceil \frac{r^2m}{n}\bigr\rceil}
<2\Bigl( \frac{1}{2r}\Bigr)^{M_0^*-4\bigl\lceil \frac{r^2m}{n}\bigr\rceil}
=O\Bigl( \frac{r^6}{n^4}\Bigr).
\]
The probability that there is a vertex with degree greater than $M_0^*$ is
\[
\mathbb{P}\bigl[\exists_{v\in[n]}:\deg (v)>M_0^*\bigr]
\leq n\mathbb{P}\bigl[\deg (v)>M_0^*\bigr]
=O\Bigl(\frac{r^6}{n^3}\Bigr),
\]
to complete the proof.
\end{proof}
Next we show that
$|\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}|$
and $|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|$
are almost equal if $h_i\leq M_i$ for $1\leq i\leq 4$.
Though the proof is similar to that of Theorem~\ref{t4.1}, its switching
operations are different in order to keep the number of Type-$i$ clusters
intact for $1\leq i\leq 4$.
\begin{theorem}\label{t4.2}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ and that
$0\leq h_i\leq M_i$ for $1\leq i\leq 4$. Suppose that $n\to\infty$. Then
\[
|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|=
\Bigl(1-O\Bigl( \frac{r^6}{n^3}\Bigr)\Bigr)|\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}|.
\]
\end{theorem}
\begin{proof}
Consider $H\in \mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$ chosen uniformly at random.
We will show that there are no vertices with degree
greater than $M_0^*$ with probability $1-O\bigl( \frac{r^6}{n^3}\bigr)$.
Fix $v\in[n]$. Assume that $\deg (v)=d$ for some integer
$d\geq 1$. Let $d_i$ be the number of Type-$i$ clusters containing $v$.
By the property $\bf(c)$ for $\mathcal{H}_{r}^{+}(n,m)$, the
intersection of every two clusters here is only the vertex $v$. Let $d_0=d-d_1-d_2-d_3-d_4$.
Define the set $\mathcal{S}_v(d\to d-1)$
as a set of switching strategies to decrease the degree of $v$ from $d$
to $d-1$ while keeping the number of Type-$i$ clusters unchanged for
$1\leq i\leq 4$ in $H$. Each strategy involves moving one edge or one cluster.
If we choose an edge containing $v$ in a Type-$1$, Type-$2$ or Type-$3$
cluster, then we switch this cluster to a $(3r-4)$-set of $[n]-\{v\}$ with
no two vertices in the same edge of $H-\{v\}$; if we choose an
edge containing $v$ in a Type-$4$ cluster, then we switch this cluster to
a $(2r-2)$-set of $[n]-\{v\}$ with no two vertices in the same
edge of $H-\{v\}$; otherwise we switch the edge containing $v$ to
an $r$-set of $[n]-\{v\}$ with no two vertices in the same edge
of $H-\{v\}$. Applying Lemma~\ref{l3.5} to $H-\{v\}$, we have
\begin{align*}
|\mathcal{S}_v(d\to d-1)|&\geq d_0\biggl[ \binom{n-1}{r}- \binom{r}{2}m \binom{n-3}{r-2}\biggr]\\
&{\qquad}+(d_1+d_2+d_3)\biggl[ \binom{n-1}{3r-4}- \binom{r}{2}m \binom{n-3}{3r-6}\biggr]\\
&{\qquad}+d_4\biggl[ \binom{n-1}{2r-2}- \binom{r}{2}m \binom{n-3}{2r-4}\biggr]\\
&>d\biggl[ \binom{n-1}{r}- \binom{r}{2}m \binom{n-3}{r-2}\biggr].
\end{align*}
Assume $\deg (v)=d-1$ for some integer $d\geq 1$. Define the
set $\mathcal{S}_v(d-1\to d)$ as the switching
strategies inverse to the above to increase the degree of $v$ from $d-1$ to $d$.
Clearly, $|\mathcal{S}_v(d-1\to d)|\leq m \binom{n-1}{r-1}$.
Thus, we have
\[
\frac{\mathbb{P}[\deg (v)=d]}{\mathbb{P}[\deg (v)=d-1]}
= \frac{|\mathcal{S}_v(d-1\to d)|}{|\mathcal{S}_v(d\to d-1)|}
< \frac{m \binom{n-1}{r-1}}{d\bigl[ \binom{n-1}{r}- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}
= \frac{rm}{dn}\Bigl(1+O\Bigl( \frac{r^4m}{n^2}\Bigr)\Bigr),
\]
after which we complete the proof as in Theorem~\ref{t4.1}.
\end{proof}
\begin{remark}\label{r4.3}
From the proof of Theorem~\ref{t4.2}, for any given $h_1,\ldots,h_4$,
we have shown that $|\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}|=0$ iff
$|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|=0$.
Similarly, from the proof of Theorem~\ref{t4.1}, we have shown if $|\mathcal{L}_r(n,m)|\neq0$,
then $|\mathcal{C}_{0,0,0,0}^{++}|\neq0$.
\end{remark}
\begin{remark}\label{r4.4}
In fact, by Theorem~\ref{t3.2}, we have $|\mathcal{H}^+_r(n,m)|\neq0$.
There exist $h_i$ with $1\leq i\leq 4$ such that $|\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}|\neq0$.
By the switching operations in Section~\ref{s:5} below, we obtain $|\mathcal{L}_r(n,m)|\neq0$.
\end{remark}
\begin{remark}\label{r4.5}
By Theorem~\ref{t3.2}, Remark~\ref{r3.3}, Theorem~\ref{t4.1}, Remark~\ref{r4.3} and Remark~\ref{r4.4}, it follows that
\begin{align*}
\frac{1}{\mathbb{P}_r(n,m)}&= \biggl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr)
\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1}
\frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}
{|\mathcal{L}_r(n,m)|}\\
&= \biggl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr)
\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1} \frac{
|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}.
\end{align*}
We estimate the above sum using switching operations designed
to remove these Type-$i$ clusters for $1\leq i\leq 4$ in the next two sections.
\end{remark}
\section{Switchings on $r$-graphs in $\mathcal{H}_r^{+}(n,m)$}\label{s:5}
Now our task is reduced to calculating the ratio
$|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|/|\mathcal{C}_{0,0,0,0}^{++}|$
when $0\leq h_i\leq M_i$ for $1\leq i\leq 4$.
\subsection{Switchings of Type-$1$ clusters}\label{s:5.1}
Let $H\in \mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$. A \textit{Type-$1$ switching}
from $H$ is used to reduce the number of Type-$1$ clusters in $H$,
which is defined in the following four steps.
\noindent{\bf Step 0.}\ Take a Type-$1$ cluster
$\{e,f,g\}$ and remove it from~$H$.
Define $H_0$ with the same vertex set $[n]$ and the edge
set $E(H_0)=E(H)\backslash \{e,f,g\}$.
\noindent{\bf Step 1.}\ Take any $r$-set from $[n]$ of which
no two vertices belong to the same edge of $H_0$ and %
add it as a new edge. The new graph is denoted by $H'$.
\noindent{\bf Step 2.}\ Insert another new edge at an $r$-set
of $[n]$ of which no two vertices belong to the same edge of $H'$.
The resulting graph is denoted by~$H''$.
\noindent{\bf Step 3.}\ Insert an edge at an $r$-set of which no
two vertices belong to
the same edge of $H''$. The resulting graph is denoted by $H'''$.
A Type-$1$ switching operation is illustrated in Figure~\ref{fig:2} below.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.9\textwidth]{Switching111.pdf}
\caption{An example of a Type-$1$ switching between $H$ and $H'''$\label{fig:2}}
\end{figure}
Note that any two new edges may or may not have a vertex in common.
\begin{remark}\label{r5.1}
A Type-$1$ switching reduces the number of Type-$1$ clusters in $H$ by
one without changing the other types of clusters.
Moreover, conditions $\bf(a)$--$\bf(f)$ remain true.
Since a vertex might gain degree during Steps 1--3, a Type-1 switching
does not necessarily map $\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$ into
$\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}$ or map $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}$ into
$\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++}$.
However, it always maps $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}$ into
$\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}$.
\end{remark}
A \textit{reverse Type-$1$ switching} is the reverse of a Type-$1$ switching.
A reverse Type-$1$ switching from
$H'''\in \mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}$
is defined by sequentially removing three edges of $H'''$ not containing a link, then
choosing a $(3r-4)$-set $T$ from $[n]$
of which no two vertices belong to any remaining edges of $H'''$,
then inserting three edges into $T$ such that they
create a Type-$1$ cluster.
This operation is depicted in Figure~\ref{fig:2} by following the arrow in reverse.
\begin{remark}\label{r5.2}
If $H'''\in \mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++}$, then a reverse
Type-$1$ switching from $H'''$ may also violate the condition $\bf(g^*)$
for $\mathcal{H}_r^{++}(n,m)$, but the resulting graph~$H$ is in
$\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$
because $\deg (v)\leq M_0^*+2<M_0$ for every vertex $v\in[n]$ of $H$.
\end{remark}
Next we analyze Type-$1$ switchings to find a relationship between the sizes of
$\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}$ and
$\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++}$.
\begin{lemma}\label{l5.3}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ and
that $0\leq h_i\leq M_i$ for $1\leq i\leq 4$ with $h_1\geq 1$.\\
$(a)$\ Let $H\in \mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$.
Then the number of Type-$1$ switchings for $H$ is
\[
h_1\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^3
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
$(b)$\ Let $H'''\in \mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}$. The number of
reverse Type-$1$ switchings for $H'''$ is
\begin{align*}
&36 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2} \binom{m-3(h_1-1)-3h_2-3h_3-2h_4}{3}\\
&{\qquad}\times\biggl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{lemma}
\begin{proof} $(a)$\ Let $H\in \mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$.
Let $\mathcal{S}(H)$ be set the of all Type-$1$ switchings which can
be applied to $H$. There are exactly $h_1$ ways to choose a Type-$1$ cluster.
In each of the steps 2--4 of the switching, by Lemma~\ref{l3.5}, there are
\[
\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr)
\]
ways to choose the new edge.
Thus, we have
\[
\bigl|\mathcal{S}(H)\bigl|=h_1\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^3
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
$(b)$\ Conversely, suppose that $H'''\in \mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}$.
Similarly, let $\mathcal{S}'(H''')$ be the set of all reverse Type-$1$ switchings for $H'''$.
There are exactly $6 \binom{m-3(h_1-1)-3h_2-3h_3-2h_4}{3}$ ways to
delete three edges in sequence such that
none of them contain a link. By Lemma~\ref{l3.5}, there are
\[
\biggl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr)
\]
ways to choose a $(3r-4)$-set $T$ from $[n]$ of which
no two vertices belong to any remaining edges of $H'''$. For every $T$,
there are $ \binom{3r-4}{r} \binom{r}{4} \binom{4}{2} \binom{2r-4}{r-2}$
ways to create a Type-$1$ cluster in $T$. Thus, we have
\begin{align*}
|\mathcal{S}'(H''')|&=36 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2} \binom{m-3(h_1-1)-3h_2-3h_3-2h_4}{3}\\
&{\qquad}\times\biggl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\biggr]\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).\qedhere
\end{align*}
\end{proof}
\begin{corollary}\label{c5.4}
With notation as above,
for $0\leq h_i\leq M_i$ when $1\leq i\leq 4$, the following hold:\\
$(a)$\ If $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}=\emptyset$, then $\mathcal{C}_{h_1+1,h_2,h_3,h_4}^{++}=\emptyset$.
\\
$(b)$\
Let $h_1'=h_1'(h_2,h_3,h_4)$ be the first value of $h_1\leq M_1$
such that $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}=\emptyset$,
or $h_1'=M_1+1$ if no such value exists. Suppose that $n\to\infty$.
Then uniformly for $1\leq h_1< h_1'$,
\begin{align*}
\frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++}|}&=
\frac{36 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2} \binom{m-3(h_1-1)-3h_2-3h_3-2h_4}{3}
\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{h_1\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{corollary}
\begin{proof} $(a)$\ Suppose $\mathcal{C}_{h_1+1,h_2,h_3,h_4}^{++}\neq\emptyset$
and let $H\in\mathcal{C}_{h_1+1,h_2,h_3,h_4}^{++}$.
We apply a Type-$1$ switching from $H$ to obtain an $r$-graph
$H'''\in\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}$. By Remark~\ref{r4.3}, we have
$|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|\neq0$.
$(b)$\ By $(a)$, if $h_1'<M_1$ such that $\mathcal{C}_{h_1',h_2,h_3,h_4}^{++}=\emptyset$, then
$$\mathcal{C}_{h_1'+1,h_2,h_3,h_4}^{++},\ldots,\mathcal{C}_{M_1,h_2,h_3,h_4}^{++} =\emptyset.$$
By the definition of $h_1'$, the left hand ratio is well defined.
By Remark~\ref{r5.1} and Remark~\ref{r5.2}, we take
\begin{align*}
A_1&=\mathcal{C}_{h_1,h_2,h_3,h_4}^{++},&
A_2&=\mathcal{C}_{h_1,h_2,h_3,h_4}^{+}-\mathcal{C}_{h_1,h_2,h_3,h_4}^{++},\\
B_1&=\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++},&
B_2&=\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{+}-\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++},
\end{align*}
in Lemma~\ref{l3.7}.
By Theorem~\ref{t4.2}, we have $ \frac{|B_2|}{|B_1|}=O\bigl( \frac{r^6}{n^3})$
and $ \frac{|A_2|}{|A_1|}=O\bigl( \frac{r^6}{n^3}\bigr)$. By Lemma~\ref{l5.3}, we also have
\[
\frac{d_{min}^{B_1}}{d_{max}^A}= \frac{\min_{H'''\in B_1}|{\mathcal{S}}'(H''')|}{\max_{H\in A}|\mathcal{S}(H)|},\quad
\frac{d_{max}^B}{d_{min}^{A_1}}= \frac{\max_{H'''\in B}|{\mathcal{S}}'(H''')|}{\min_{H\in A_1}|\mathcal{S}(H)|}
\]
to complete the proof of $(b)$, where
$O\bigl( \frac{r^6}{n^3}\bigr)$ is absorbed into
$O\bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\bigr)$.
\end{proof}
\subsection{Switchings of Type-$2$ clusters}\label{s:5.2}
Let $H\in \mathcal{C}_{0,h_2,h_3,h_4}^{+}$. A \textit{Type-$2$ switching}
from $H$ is used to reduce the number of Type-$2$ clusters in $H$.
It is defined in the same manner as Steps $0$-$3$ in Section~\ref{s:5.1}.
Since we will only use this switching after all Type-$1$ clusters
have been removed, we can assume~$h_1=0$.
\begin{remark}\label{r5.5}
A Type-$2$ switching reduces the number of Type-$2$
clusters in $H$ by one without affecting other types of clusters.
If $H\in \mathcal{C}_{0,h_2,h_3,h_4}^{++}$, then a Type-$2$
switching from $H$ may violate property
$\bf(g^*)$ for $\mathcal{H}_r^{++}(n,m)$, but
the resulting graph $H'''$ is in $\mathcal{C}_{0,h_2-1,h_3,h_4}^{+}$
because $\deg (v)\leq M_0^*+3= M_0$
for every vertex $v\in[n]$ of $H'''$.
\end{remark}
A \textit{reverse Type-$2$ switching} is the reverse of a Type-$2$ switching.
A reverse Type-$2$ switching from $H'''\in \mathcal{C}_{0,h_2-1,h_3,h_4}^{++}$
is defined by sequentially removing three edges not
containing a link, then choosing a $(3r-4)$-set $T$ from $[n]$
such that no two vertices belong to any remaining edges of $H'''$, then inserting three edges into
$T$ such that they create a Type-$2$ cluster.
\begin{remark}\label{r5.6}
If $H'''\in \mathcal{C}_{0,h_2-1,h_3,h_4}^{++}$, then a reverse
Type-$2$ switching from $H'''$ may also
violate the property $\bf(g^*)$ for $\mathcal{H}_r^{++}(n,m)$, but
the resulting graph $H\in\mathcal{C}_{0,h_2,h_3,h_4}^{+}$ because $\deg (v)\leq M_0^*+3=M_0$
for every vertex $v\in[n]$ of $H$.
\end{remark}
Next we analyze Type-$2$ switchings to find a relationship
between the sizes of $\mathcal{C}_{0,h_2,h_3,h_4}^{++}$ and
$\mathcal{C}_{0,h_2-1,h_3,h_4}^{++}$.
\begin{lemma}\label{l5.7}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$
and that $1\leq h_2\leq M_2$, $0\leq h_3\leq M_3$
and $0\leq h_4\leq M_4$.\\
$(a)$\ Let $H\in \mathcal{C}_{0,h_2,h_3,h_4}^{+}$.
Then the number of Type-$2$ switchings for $H$ is
\[
h_2\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^3\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
$(b)$\ Let $H'''\in \mathcal{C}_{0,h_2-1,h_3,h_4}^{+}$.
Then the number of reverse Type-$2$ switchings for $H'''$ is
\begin{align*}
&18 \binom{3r-4}{r} \binom{r}{3} \binom{2r-4}{r-2} \binom{m-3(h_2-1)-3h_3-2h_4}{3}\\
&{\qquad}\times\biggl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{lemma}
\begin{proof}
The proof follows the same logic as the proof of Lemma~\ref{l5.3},
so we omit it.
\end{proof}
\begin{corollary}\label{c5.8}
With notation as above, for some $0\leq h_i\leq M_i$ with $2\leq i\leq 4$,\\
$(a)$\ If $\mathcal{C}_{0,h_2,h_3,h_4}^{++}=\emptyset$, then $\mathcal{C}_{0,h_2+1,h_3,h_4}^{++}=\emptyset$.
\\
$(b)$\
Let $h_2'=h_2'(h_3,h_4)$ be the first value of $h_2\leq M_2$
such that $\mathcal{C}_{0,h_2,h_3,h_4}^{++}=\emptyset$,
or $h_2'=M_2+1$ if no such value exists. Suppose that $n\to\infty$.
Then uniformly for $1\leq h_2< h_2'$,
\begin{align*}
\frac{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2-1,h_3,h_4}^{++}|}
&= \frac{18 \binom{3r-4}{r} \binom{r}{3} \binom{2r-4}{r-2} \binom{m-3(h_2-1)-3h_3-2h_4}{3}
\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{h_2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{corollary}
\begin{proof}
This is proved in the same way as Corollary~\ref{c5.4}.
\end{proof}
\subsection{Switchings of Type-$3$ clusters}\label{s:5.3}
Let $H\in \mathcal{C}_{0,0,h_3,h_4}^{+}$. A \textit{Type-$3$ switching}
from $H$ is used to reduce the number of Type-$3$ clusters in $H$,
after the numbers of Type-$1$ and Type-$2$ clusters have been
reduced to zero.
It is defined in the same manner as Steps $0$-$3$ in Section~5.1.
\begin{remark}\label{r5.9}
A Type-$3$ switching reduces the number of Type-$3$ clusters
in $H$ by one without affecting other types of clusters.
If $H\in \mathcal{C}_{0,0,h_3,h_4}^{++}$, then a Type-$3$ switching
from $H$ may violate the property
$\bf(g^*)$ for $\mathcal{H}_r^{++}(n,m)$, but
the resulting graph $H'''$ is in $\mathcal{C}_{0,0,h_3-1,h_4}^{+}$
because $\deg (v)\leq M_0^*+3= M_0$
for every vertex $v\in[n]$ of $H'''$.
\end{remark}
A \textit{reverse Type-$3$ switching} is the reverse of a Type-$3$ switching.
A reverse Type-$3$ switching from $H'''\in \mathcal{C}_{0,0,h_3-1,h_4}^{++}$
is defined by sequentially removing three edges not
containing a link, then choosing a $(3r-4)$-set $T$ from $[n]$
such that no two vertices belong to any remaining edges of $H'''$,
then inserting three edges into $T$ such that
they create a Type-$3$ cluster.
\begin{remark}\label{r5.10}
If $H'''\in \mathcal{C}_{0,0,h_3-1,h_4}^{++}$, then a reverse Type-$3$ switching from $H'''$ may also
violate the property $\bf(g^*)$ for $\mathcal{H}_r^{++}(n,m)$, but
the resulting graph $H$ is $\mathcal{C}_{0,0,h_3-1,h_4}^{+}$
because $\deg (v)\leq M_0^*+3=M_0$ for every vertex $v\in[n]$ of $H$.
\end{remark}
Next we analyze Type-$3$ switchings to find a relationship
between the sizes of $\mathcal{C}_{0,0,h_3,h_4}^{++}$ and
$\mathcal{C}_{0,0,h_3-1,h_4}^{++}$.
\begin{lemma}\label{l5.11}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$,
$1\leq h_3\leq M_3$ and $0\leq h_4\leq M_4$.\\
$(a)$\ Let $H\in \mathcal{C}_{0,0,h_3,h_4}^{+}$.
Then the number of Type-$3$ switchings for $H$ is
\begin{align*}
h_3\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^3
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
$(b)$\ Let $H'''\in \mathcal{C}_{0,0,h_3-1,h_4}^{+}$. Then the number
of reverse Type-$3$ switchings for $H'''$ is
\begin{align*}
& \binom{3r-4}{r} \binom{r}{2} \binom{2r-4}{r-2} \binom{m-3(h_3-1)-2h_4}{3}\\
&{\qquad}\times\biggl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{lemma}
\begin{proof}
The proof follows the same logic as the proof of Lemma~\ref{l5.3},
so we omit it.
\end{proof}
\begin{corollary}\label{c5.12}
With notation as above, for some $0\leq h_i\leq M_i$ with $3\leq i\leq 4$,\\
$(a)$\ If $\mathcal{C}_{0,0,h_3,h_4}^{++}=\emptyset$,
then $\mathcal{C}_{0,0,h_3+1,h_4}^{++}=\emptyset$.\\
$(b)$\
Let $h_3'=h_3'(h_4)$ be the first value of $h_3\leq M_3$
such that $\mathcal{C}_{0,0,h_3,h_4}^{++}=\emptyset$,
or $h_3'=M_3+1$ if no such value exists. Suppose that $n\to\infty$.
Then uniformly for $1\leq h_3< h_3'$,
\begin{align*}
\frac{|\mathcal{C}_{0,0,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,h_3-1,h_4}^{++}|}
&= \frac{ \binom{3r-4}{r} \binom{r}{2} \binom{2r-4}{r-2} \binom{m-3(h_3-1)-2h_4}{3}
\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{h_3\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{corollary}
\begin{proof}
This is proved in the same way as Corollary~\ref{c5.4}.
\end{proof}
\subsection{Switchings of Type-$4$ clusters}\label{s:5.5}
Let $H\in \mathcal{C}_{0,0,0,h_4}^{+}$.
A \textit{Type-$4$ switching} from $H$
is used to reduce the number of Type-$4$ clusters in $H$ after the
Type-$1$, Type-$2$ and Type-$3$ clusters have been removed.
\noindent{\bf Step 0.}\ Take any one of these $h_4$ Type-$4$ clusters in $H$, denoted by $\{e,f\}$,
and remove it from $H$.
Define $H_0$ with the same vertex set $[n]$ and the edge set $E(H_0)=E(H)\backslash \{e,f\}$.
\noindent{\bf Step 1.}\ Take any $r$-set from $[n]$ such that no two vertices
belong to the same edge of $H_0$ and insert one new edge
to the $r$-set. The new graph is denoted by $H'$.
\noindent{\bf Step 2.}\ Repeat the process of Step 1 in $H'$, that is, insert another new edge to
an $r$-set of $[n]$ such that %
no two vertices belong to the same edge of $H'$.
The resulting graph is denoted by $H''$.
Note that the two new edges may or may not have a vertex in common.
\begin{remark}\label{r5.13}
If $H\in \mathcal{C}_{0,0,0,h_4}^{++}$, then a Type-$4$ switching
from $H$ may violate the property $\bf(g^*)$ for $\mathcal{H}_r^{++}(n,m)$, but
the resulting graph $H''$ is in $\mathcal{C}_{0,0,0,h_4-1}^{++}$
because $\deg (v)\leq M_0^*+2< M_0$ for each vertex $v\in[n]$ of $H''$.
\end{remark}
A \textit{reverse Type-$4$ switching} is the reverse of a Type-$4$ switching.
A reverse Type-$4$ switching from $H''\in \mathcal{C}_{0,0,0,h_4-1}^{++}$
is defined by sequentially removing two edges not containing a
link in $H''$, then choosing a $(2r-2)$-set $T$ from $[n]$
such that at most two vertices belong to some remaining edge of $H''$,
then inserting two edges into $T$ such that
they create a Type-$4$ cluster.
\begin{remark}\label{r5.14}
If $H''\in \mathcal{C}_{0,0,0,h_4-1}^{++}$, then a reverse
Type-$4$ switching from $H''$ may also violate the property $\bf(g^*)$
for $\mathcal{H}_r^{++}(n,m)$, but the resulting graph $H$ is
in $\in\mathcal{C}_{0,0,0,h_4}^{+}$
because $\deg (v)\leq M_0^*+2<M_0$ for each vertex $v\in[n]$ of $H$.
\end{remark}
Next we analyze Type-$4$ switchings to find a relationship
between the sizes of $\mathcal{C}_{0,0,0,h_4}^{++}$ and
$\mathcal{C}_{0,0,0,h_4-1}^{++}$.
\begin{lemma}\label{l5.15}
Assume that $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ and $1\leq h_4\leq M_4$.\\
$(a)$\ Let $H\in \mathcal{C}_{0,0,0,h_4}^{+}$.
Then the number of Type-$4$ switchings for $H$ is \\
\[
h_4\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^2
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
$(b)$\ Let $H''\in \mathcal{C}_{0,0,0,h_4-1}^{+}$. Then the number of reverse
Type-$4$ switchings for $H''$ is
\begin{align*}
& \binom{2r-2}{2} \binom{2r-4}{r-2} \binom{m-2(h_4-1)}{2}
\biggl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\biggr]\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{lemma}
\begin{proof}
Since the likely number of Type-$4$ clusters in a random hypergraph is greater than
that of the other cluster types, our counting here must be more careful.
$(a)$\ Let $H\in \mathcal{C}_{0,0,0,h_4}^{+}$.
Define the set $\mathcal{S}(H)$ of all Type-$4$ switchings which can be applied to $H$.
By the same proof as given for Lemma~\ref{l5.3}, Lemma~\ref{l5.7} and Lemma~\ref{l5.11}, we have
\[
|\mathcal{S}(H)|=h_4\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]^2
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\]
$(b)$\ Conversely, suppose that $H''\in \mathcal{C}_{0,0,0,h_4-1}^{+}$.
Similarly, let $\mathcal{S}'(H'')$ be the set of all reverse Type-$4$ switchings for $H''$.
There are exactly $2 \binom{m-2(h_4-1)}{2}$ ways to delete two edges in sequence
such that neither of them contain a link in $H''$.
Unlike the reverse Type-$1$, Type-$2$ and Type-$3$ switchings, the chosen $(2r-2)$-set
may include two vertices belong to the same edge of $H''$, as shown by the
dashed lines in Figure~\ref{fig:3}.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.3\textwidth]{Switching4.pdf}
\caption{$(2r-2)$-subset in the reverse Type-$4$ switching\label{fig:3}}
\end{figure}
Firstly, by Lemma~\ref{l3.5}, there are
\begin{align*}
\biggl[ \binom{n}{2r-2}- \binom{r}{2}m \binom{n-2}{2r-4}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr)
\end{align*}
ways to choose a $(2r-2)$-set $T$ from $[n]$ such that no two vertices
belong to the same edge of~$H''$.
For every such~$T$,
there are $\frac12 \binom{2r-2}{2} \binom{2r-4}{r-2}$ ways to create a Type-$4$ cluster.
Secondly, by Lemma~\ref{l3.6}, there are
\begin{align*}
\binom{r}{2}m \binom{n-2}{2r-4}\biggl(1+O\Bigl( \frac{r^2n\log(r^{-2}n)+r^4m}{n^2}\Bigr)\biggr)
\end{align*}
ways to choose a $(2r-2)$-set $T$ from $[n]$ such that exactly two vertices
belong to the same edge of $H''$. For every such $T$,
there are $ \binom{2r-4}{2} \binom{2r-6}{r-3}$ ways to create a Type-$4$
cluster. Thus, we have
\begin{align*}
|\mathcal{S}'&(H'')|\\
&=2 \binom{m-2(h_4-1)}{2}\biggl[ \frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}}{2}
\biggl( \binom{n}{2r-2}- \binom{r}{2}m \binom{n-2}{2r-4}\biggr)\\
&{\qquad}+
\binom{2r-4}{2} \binom{2r-6}{r-3} \binom{r}{2}m \binom{n-2}{2r-4}\biggr]
\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr)\\
&= \binom{2r-2}{2} \binom{2r-4}{r-2} \binom{m-2(h_4-1)}{2}
\biggl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\biggr]\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).\qedhere
\end{align*}
\end{proof}
\begin{corollary}\label{c5.16}
With notation as above, for some $0\leq h_4\leq M_4$,\\
$(a)$\ If $\mathcal{C}_{0,0,0,h_4}^{++}=\emptyset$, then $\mathcal{C}_{0,0,0,h_4+1}^{++}=\emptyset$.
\\
$(b)$\
Let $h_4'$ be the first value of $h_4\leq M_4$
such that $\mathcal{C}_{0,0,0,h_4}^{++}=\emptyset$,
or $h_4'=M_4+1$ if no such value exists. Suppose that $n\to\infty$.
Then uniformly for $1\leq h_4< h_4'$,
\begin{align*}
\frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,h_4-1}^{++}|}
&=
\frac{ \binom{2r-2}{2} \binom{2r-4}{r-2} \binom{m-2(h_4-1)}{2}
\bigl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{h_4\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\Bigr)\biggr).
\end{align*}
\end{corollary}
\begin{proof}
This is proved in the same way as Corollary~\ref{c5.4}.
\end{proof}
\section{Analysis of switchings}\label{s:6}
In this section, we estimate the sum
\[
\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1}
\frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
\]
to finish the proof of the case
$r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ in accordance with Remark~\ref{r4.5}.
We will need the following summation lemmas from \cite{green06}, and
state them here for completeness.
\begin{lemma}[\cite{green06}, Corollary~4.3]\label{l6.1}
Let $K$, $N$ be integers with $N\geq 2$ and $0\leq K\leq N$. Suppose that
there are real numbers $\delta_i$ for $1\leq i\leq N$, and
$\gamma_i$ for $0\leq i\leq K$ such that
$\sum_{j=1}^{i}|\delta_j|\leq \sum_{j=0}^K \gamma_j [i]_j< \frac15$
for $1\leq i\leq N$. Let real numbers $A(i)$, $B(i)$ be given such that
$A(i)\geq 0$ and $1-(i-1)B(i)\geq 0$. Define $A_1=\min_{i=1}^NA(i)$, $A_2=\max_{i=1}^NA(i)$,
$B_1=\min_{i=1}^NB(i)$ and $B_2=\max_{i=1}^NB(i)$. Also suppose all $A\in [A_1,A_2]$,
$B\in [B_1,B_2]$ and $c$ be real numbers such that
$c>2e$, $0\leq Ac< N-K+1$ and $|BN|<1$. Define $n_0,n_1,\ldots, n_N$ by $n_0=1$ and
\[
\frac{n_i}{n_{i-1}}= \frac{A(i)}{i}\bigl(1-(i-1)B(i)\bigr)\bigl(1+\delta_i\bigr)
\]
for $1\leq i\leq N$, with the following interpretation: if $A(i)= 0$ or
$1-(i-1)B(i)=0$, then $n_j=0$ for $i\leq j\leq N$. Then
\[
\Sigma_1\leq \sum_{i=0}^{N}n_i\leq \Sigma_2,
\]
where
\begin{align*}
\Sigma_1&=\exp\Bigl[A_1- \dfrac12A_1^2B_2-4\sum_{j=0}^{K}\gamma_j
\bigl(3A_1\bigr)^j\Bigr]- \dfrac14\bigl(2e/c\bigr)^N,\\
\Sigma_2&=\exp\Bigl[A_2- \dfrac12A_2^2B_1+ \dfrac12A_2^3B_1^2+
4\sum_{j=0}^{K}\gamma_j\bigl(3A_2\bigr)^j\Bigr]+ \dfrac14\bigl(2e/c\bigr)^N.
\end{align*}
\end{lemma}
\begin{lemma}[\cite{green06}, Corollary~4.5]\label{l6.2}
Let $N\geq 2$ be an integer, and for $1\leq i\leq N$, let real numbers
$A(i)$, $B(i)$ be given such that $A(i)\geq 0$ and $1-(i-1)B(i)\geq 0$.
Define $A_1=\min_{i=1}^NA(i)$, $A_2=\max_{i=1}^NA(i)$, $C_1=\min_{i=1}^NA(i)B(i)$
and $C_2=\max_{i=1}^NA(i)B(i)$. Suppose that there exists a real number $\hat{c}$ with $0<\hat{c}< \frac{1}{3}$
such that $\max\{A/N,|C|\}\leq \hat{c}$ for all $A\in [A_1,A_2]$, $C\in[C_1,C_2]$.
Define $n_0$, $n_1$, $\ldots$, $n_N$ by $n_0=1$ and
\[
\frac{n_i}{n_{i-1}}= \frac{A(i)}{i}\left(1-(i-1)B(i)\right)
\]
for $1\leq i\leq N$, with the following interpretation: if $A(i)= 0$ or $1-(i-1)B(i)=0$, then $n_j=0$
for $i\leq j\leq N$. Then
\[
\Sigma_1\leq \sum_{i=0}^{N}n_i\leq \Sigma_2,
\]
where
\begin{align*}
\Sigma_1&=\exp\Bigl[A_1- \dfrac12A_1C_2\Bigr]-\bigl(2e\hat{c}\bigr)^N,\\
\Sigma_2&=\exp\Bigl[A_2- \dfrac12A_2C_1+ \dfrac12A_2C_1^2\Bigr]+\bigl(2e\hat{c}\bigr)^N.
\end{align*}
\end{lemma}
\begin{lemma}\label{l6.3}
Assume
$r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
If $\bigl|\mathcal{C}_{0,h_2,h_3,h_4}^{++}\bigl|\neq0$, then
\begin{align*}
\sum_{h_1=0}^{M_1} \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}
=\exp\biggl[ \frac{6 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}-
\binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].
\end{align*}
\end{lemma}
\begin{proof} Let $h_1'=h_1'(h_2,h_3,h_4)$ be the first value of $h_1\leq M_1$
such that $\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}=\emptyset$,
or $h_1'=M_1+1$ if no such value exists, which is defined in Corollary~\ref{c5.4} $(b)$.
Since $|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|\neq0$, we know $h_1'\geq1$.
Define $n_{1,0},\ldots,n_{1,M_1}$ by $n_{1,0}=1$,
\[
n_{1,h_1}= \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}
\]
for $1\leq h_1<h_1'$ and $n_{1,h_1}=0$ for $h_1'\leq h_1\leq M_1$.
Note that
\begin{align*}
& \binom{m-3(h_1-1)-3(h_2+h_3)-2h_4}{3}\\
&{\qquad}= \frac{m^3}{6}\biggl(1+O\Bigl(\frac{r^4m}{n^2}+
\frac{\log(r^{-2}n)}{m}\Bigr)\biggr)\Bigl(1-(h_1-1) \frac{9}{m}\Bigr).
\end{align*}
By Corollary~\ref{c5.4} $(b)$, we have for $1\leq h_1< h_1'$,
\begin{equation}\label{e6.1}
\begin{split}
\frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}
&= \frac{1}{h_1} \frac{|\mathcal{C}_{h_1-1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}
\frac{6 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}-
\binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3} \\
&{\qquad}\times\biggl(1+O\biggl(\frac{r^4m}{n^2}+ \frac{\log(r^{-2}n)}{m}
+ \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\biggr)\biggr) \\
&{\qquad}\times\Bigl(1-(h_1-1) \frac{9}{m}\Bigr).
\end{split}
\end{equation}
Define
\begin{align*}
A(1,h_1)&= \frac{6 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}
- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}\\
&{\qquad}\times\biggl(1+O\biggl(\frac{r^4m}{n^2}+\frac{\log(r^{-2}n)}{m}
+ \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\biggr)\biggr)
\end{align*}
for $1\leq h_1< h_1'$ and $A(1,h_1)=0$ for $h_1'\leq h_1\leq M_1$, and
$B(1,h_1)= \frac{9}{m}$ for $1\leq h_1\leq M_1$.
Let $A_{1,1}=\min_{h_1=1}^{M_1}A(1,h_1)$,
$A_{1,2}=\max_{h_1=1}^{M_1}A(1,h_1)$, $C_{1,1}=\min_{h_1=1}^{M_1}A(1,h_1)B(1,h_1)$
and $C_{1,2}=\max_{h_1=1}^{M_1}A(1,h_1)B(1,h_1)$.
By~\eqref{e6.1} we have
\[
\frac{n_{1,h_1}}{n_{1,h_1-1}}= \frac{A(1,h_1)} {h_1}\bigl(1-(h_1-1)B(1,h_1)\bigr).
\]
Let $\hat{c}= \frac{1}{3^4}$, then we have $\max\bigl\{A/M_1,|C|\bigr\}\leq \hat{c}< \frac{1}{3}$
for all $A\in [A_{1,1},A_{1,2}]$, $C\in[C_{1,1},C_{1,2}]$.
Therefore, Lemma~\ref{l6.2} applies and we obtain
\begin{align*}
\sum_{h_1=0}^{M_1}& \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,h_2,h_3,h_4}^{++}|}\\
&=\exp\biggl[ \frac{6 \binom{3r-4}{r} \binom{r}{4} \binom{2r-4}{r-2}m^3
\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}
{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}
\\&{\qquad}+O\biggl(\frac{r^{12}m^4}{n^6}+
\frac{r^{8}m^2\log(r^{-2}n)}{n^4}+
\frac{r^{14}m^4n\log(r^{-2}n)+r^{16}m^5}{n^8}\biggr)\biggr]+O\Bigl(\Bigl( \frac{2e}{3^4}\Bigr)^{M_1}\Bigr).
\end{align*}
Note that
$O\bigl( \frac{r^{12}m^4}{n^6}+ \frac{r^{8}m^2\log(r^{-2}n)}{n^4}+
\frac{r^{14}m^4n\log(r^{-2}n)+r^{16}m^5}{n^8}\bigr)
=O\bigl( \frac{r^6m^2}{n^3}\bigr)$ and $O\bigl(\bigl( \frac{2e}{3^4}\bigr)^{M_1}\bigr)=O\bigl( \frac{r^6}{n^3}\bigr)$ as
$r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$,
which gives the expression in the lemma.
\end{proof}
Note that the values of $h_2,h_3,h_4$ disappear into the error term
of Lemma~\ref{l6.3}. This means that Type-$2$ switchings can be
analysed in the same way using Corollary~\ref{c5.8}. The values
of $h_3$ and $h_4$ again disappear into the error term, so we
can analyse Type-$3$ switchings in the same way using Corollary~\ref{c5.12}.
As these two analyses are essentially the same as Lemma~\ref{l6.3}, we
will just state the results without proof.
\begin{lemma}\label{l6.4}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
If $|\mathcal{C}_{0,0,h_3,h_4}^{++}|\neq0$, then
\begin{align*}
\sum_{h_2=0}^{M_2}&\sum_{h_1=0}^{M_1} \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,h_3,h_4}^{++}|}\\
&{}=\exp\biggl[ \frac{\bigl[6 \binom{r}{4}+3 \binom{r}{3}\bigr] \binom{3r-4}{r}
\binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}
{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].\quad\qed
\end{align*}
\end{lemma}
\begin{lemma}\label{l6.5}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
If $|\mathcal{C}_{0,0,0,h_4}^{++}|\neq0$, then
\begin{align*}
\sum_{h_3=0}^{M_3}&\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1} \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}
{|\mathcal{C}_{0,0,0,h_4}^{++}|}\\
&=\exp\biggl[ \frac{\bigl[6 \binom{r}{4}+3 \binom{r}{3}+ \frac{1}{6} \binom{r}{2}\bigr]
\binom{3r-4}{r} \binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}
{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].\qed
\end{align*}
\end{lemma}
\begin{lemma}\label{l6.6}
Assume $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$.
Then
\begin{align*}
\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}&\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1} \frac{
|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}\\
&=\exp\biggl[ \frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}m^2\bigl[ \binom{n}{2r-2}-
\frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}- \frac{ \binom{2r-2}{2}^2 \binom{2r-4}{r-2}^2m^3\bigl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)(2r-3)}
\binom{r}{2}m \binom{n-2}{2r-4}\bigr]^2}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^4}\\
&{\qquad}+ \frac{\bigl[6 \binom{r}{4}+3 \binom{r}{3}+ \frac{1}{6} \binom{r}{2}\bigr] \binom{3r-4}{r}
\binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}
{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].
\end{align*}
\end{lemma}
\begin{proof}
Due to the larger number of Type-$4$ clusters, Lemma~\ref{l6.2} is not
accurate enough and we must use Lemma~\ref{l6.1}.
Let $h_4'$ be the first value of $h_4\leq M_4$
such that $\mathcal{C}_{0,0,0,h_4}^{++}=\emptyset$
or $h_4'=M_4+1$ if no such value exists, which is defined in Corollary~\ref{c5.16} $(b)$.
Note that
\[
\sum_{h_4=0}^{M_4}\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1} \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
=\sum_{h_4=0}^{h_4'-1}\biggl( \frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1} \frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,h_4}^{++}|}\biggr).
\]
Thus, by the definition of $h_4'$, $|\mathcal{C}_{0,0,0,h_4}^{++}|\neq0$ for $0\leq h_4<h_4'$.
By Lemma~\ref{l6.5}, we have
\begin{align*}
\sum_{h_4=0}^{M_4}&\sum_{h_3=0}^{M_3}\sum_{h_2=0}^{M_2}\sum_{h_1=0}^{M_1}
\frac{|\mathcal{C}_{h_1,h_2,h_3,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}\\
&=\exp\biggl[ \frac{\bigl[6 \binom{r}{4}+3 \binom{r}{3}+ \frac{1}{6} \binom{r}{2}\bigr]
\binom{3r-4}{r} \binom{2r-4}{r-2}m^3\bigl[ \binom{n}{3r-4}- \binom{r}{2}m \binom{n-2}{3r-6}\bigr]}
{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^3}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr] \\
&{\qquad}\times\sum_{h_4=0}^{h_4'-1}
\frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}.
\end{align*}
By Remark~\ref{r4.3} and Remark~\ref{r4.4}, we have $|\mathcal{C}_{0,0,0,0}^{++}|\neq0$.
Define $n_{4,0},\ldots,n_{4,M_4}$ by $n_{4,0}=1$,
\[
n_{4,h_4}= \frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
\]
for $1\leq h_4<h_4'$ and $n_{4,h_4}=0$ for $h_4'\leq h_4\leq M_4$, where
$M_4$ is shown in the equation~\eqref{e3.1}.
Note that as $n\to\infty$, $r^{-2}n\leq m=o(r^{-3}n^{ \frac32})$ and $h_4\leq M_4$,
\[
\binom{m-2(h_4-1)}{2}= \frac{m^2}{2}\biggl(1+O\Bigl( \frac{r^8m^2}{n^4}+
\frac{1}{m}\Bigr)\biggr)\Bigl(1-(h_4-1) \frac{4}{m}\Bigr).
\]
By Corollary~\ref{c5.16} $(b)$, we have for $1\leq h_4\leq h_4'$,
\begin{equation}\label{e6.2}
\begin{split}
\frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
&= \frac{1}{h_4} \frac{|\mathcal{C}_{0,0,0,h_4-1}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
\frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}m^2\bigl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)
(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2} \\
&{}\times\biggl(1+O\biggl(\frac{r^8m^2}{n^4}+ \frac{1}{m}
+ \frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\biggr)\biggr)\Bigl(1-(h_4-1) \frac{4}{m}\Bigr).
\end{split}
\end{equation}
Define
\begin{align*}
A(4,h_4)&= \frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}m^2\bigl[ \binom{n}{2r-2}-
\frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}\times
\biggl(1+O\biggl( \frac{r^8m^2}{n^4}+ \frac{1}{m}+
\frac{r^6mn\log (r^{-2}n)+r^8m^2}{n^{4}}\biggr)\biggr)
\end{align*}
for $1\leq h_4< h_4'$ and $A(4,h_4)=0$ for $h_4'\leq h_4\leq M_4$,
\[
B(4,h_4)= \frac{4}{m}\text{~~and~~}
\delta_{h_4}=0
\]
for $1\leq h_4\leq M_4$. Define $K=1$, $c=3^{4}$ and $\gamma_j=0$ for $0\leq j\leq K$.
By~\eqref{e6.2}, we further have
\[
\frac{n_{4,h_4}}{n_{4,h_4-1}}=
\frac{A(4,h_4)}{h_4}\bigl(1-(h_4-1)B(4,h_4)\bigr).
\]
Let $A_{4,1}=\min_{h_4=1}^{M_4}A(4,h_4)$, $A_{4,2}=\max_{h_4=1}^{M_4}A(4,h_4)$,
$B_{4,1}=\min_{h_4=1}^{M_4}B(4,h_4)= \frac{4}{m}$, $B_{4,2}=\max_{h_4=1}^{M_4}B(4,h_4)= \frac{4}{m}$.
Since $Ac\leq \frac{3^4r^4m^2}{4n^2}<M_4-K+1$ and $BM_4=o(1)<1$ for all $A\in[A_{4,1},A_{4,2}]$
and $B\in[B_{4,1},B_{4,2}]$, Lemma~\ref{l6.1} applies and we obtain
\begin{align*}
\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
&=\exp\biggl[ \frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}m^2\bigl[ \binom{n}{2r-2}-
\frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}- \frac{ \binom{2r-2}{2}^2 \binom{2r-4}{r-2}^2m^3\bigl[ \binom{n}{2r-2}-
\frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]^2}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^4}\\
&{\qquad}+O\biggl(\frac{r^{12}m^4}{n^6}+ \frac{r^4m}{n^2}+
\frac{r^{10}m^3n\log(r^{-2}n) +r^{12}m^4}{n^6}\biggr)\biggr]+O\Bigl(\Bigl( \frac{2e}{3^{4}}\Bigr)^{M_4}\Bigr).
\end{align*}
Note that $O\bigl( \frac{r^{12}m^4}{n^6}+ \frac{r^4m}{n^2}+
\frac{r^{10}m^3n\log(r^{-2}n) +r^{12}m^4}{n^6}\bigr)=O\bigl(
\frac{r^6m^2}{n^3}\bigr)$ and $O\bigl(\bigl( \frac{2e}{3^{4}}\bigr)^{M_3}\bigr)=O\bigl( \frac{r^6}{n^3}\bigr)$,
which further leads to
\begin{align*}
\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{0,0,0,h_4}^{++}|}{|\mathcal{C}_{0,0,0,0}^{++}|}
&=\exp\biggl[ \frac{ \binom{2r-2}{2} \binom{2r-4}{r-2}m^2\bigl[ \binom{n}{2r-2}-
\frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N-
\binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}- \frac{ \binom{2r-2}{2}^2 \binom{2r-4}{r-2}^2m^3
\bigl[ \binom{n}{2r-2}- \frac{(r^2-r-1)}{(r-1)(2r-3)} \binom{r}{2}m \binom{n-2}{2r-4}\bigr]^2}
{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^4}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].
\end{align*}
This gives the expression in the lemma statement.
\end{proof}
\section{The case $\log\, (r^{-2}n)\leq m=O(r^{-2}n)$}\label{s:7}
We consider the case $\log (r^{-2}n)\leq m=O(r^{-2}n)$
in Theorem~\ref{t1.1}.
Recall that these inequalities imply $r=o(n^{\frac12})$.
Define
\begin{equation}\label{e7.1}
\begin{split}
M_0^*&=\Bigl\lceil\log (r^{-2}n)\Bigr\rceil, \\
M_0&=\Bigl\lceil\log (r^{-2}n)\Bigr\rceil+2, \\
M_4&=\Bigl\lceil\log (r^{-2}n)\Bigr\rceil.
\end{split}
\end{equation}
Let $\mathcal{H}_r^+(n,m)\subseteq\mathcal{H}_r(n,m)$
be the set of $r$-graphs $H$ which satisfy
properties $\bf(a)$ to $\bf(e)$:
$\bf(a)$\ The intersection of any two edges contains at most two vertices.
$\bf(b)$\ $H$ only contains one type of cluster (Type-$4$ cluster).
(This implies that any three edges involve at least $3r-3$ vertices. Thus, if there are
two edges, for example $\{e_1,e_2\}$, such that $|e_1\cup e_2|= 2r-2$, then $|(e_1\cup e_2)\cap e|\leq 1$ for any
edge $e$ other than $\{e_1,e_2\}$ of $H$.)
$\bf(c)$\ Any two distinct Type-$4$ clusters in $H$ are vertex-disjoint.
(This implies that any four edges involve at least $4r-4$ vertices.)
$\bf(d)$\ There are at most $M_4$ Type-$4$ clusters in $H$.
$\bf(e)$\ $\deg (v)\leq M_0$ for every vertex $v\in [n]$.
Similarly, we further define $\mathcal{H}_r^{++}(n,m)\subseteq\mathcal{H}_r^+(n,m)$
to be the set of $r$-graphs $H$ obtained
by replacing the property $\bf(e)$
with a stronger constraint
$\bf(e^*)$\ $\deg (v)\leq M_0^*$ for every vertex $v\in [n]$.
\begin{remark}\label{r7.1}
From property $\bf(e)$, it easily follows that
\[
\sum_{v\in [n]} \binom{\deg (v)}{2}=O\Bigl(M_0\sum_{v\in [n]}\deg _0(v)\Bigr)
=O\Bigl(rm\log (r^{-2}n)\Bigr)
\]
for $H\in\mathcal{H}_r^+(n,m)$.
\end{remark}
Similarly to Section~\ref{s:3}, we find that the number of $r$-graphs
in $\mathcal{H}_r(n,m)$
not satisfying the properties of $\mathcal{H}^+_r(n,m)$ and
$\mathcal{H}^{++}_r(n,m)$ is quite small.
\begin{theorem}\label{t7.2}
Assume that $\log (r^{-2}n)\leq m=O(r^{-2}n)$ and
$n\to \infty$. Then
\[
\frac{|\mathcal{H}^+_r(n,m)|}{|\mathcal{H}_r(n,m)|}=1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr),\quad \frac{|\mathcal{H}^{++}_r(n,m)|}{|\mathcal{H}_r(n,m)|}=1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
\end{theorem}
\begin{proof}
The proof is much the same as that of Theorem~\ref{t3.2}, so we will
omit the proofs for $\bf(a)$--$\bf(d)$.
To prove property $\bf(e^*)$, define $d=M_0^*+1$.
The expected number of sets consisting of a vertex $v$
and $d$ edges that include $v$ is
\[
n \binom{n-1 }{r-1}^d \frac{1}{d!}\Bigl(\frac{m}{N}\Bigr)^{d}
=O\Bigl(n\Bigl( \frac{rem}{dn}\Bigr)^{d}\Bigr)
=O\Bigl(n\Bigl( \frac{e}{rd}\Bigr)^{d}\Bigr)
=O\Bigl( \frac{r^6}{n^3}\Bigr),
\]
where the second equality is true because $d!\geq \bigl( \frac{d}{e}\bigr)^{d}$ and $m=O(r^{-2}n)$,
and the last equality is true because of the choice $d>\log (r^{-2}n)$.
\end{proof}
\begin{remark}\label{r7.3}
Let
$\mathcal{C}_{h_4}^{+}$ {\rm(}resp. $\mathcal{C}_{h_4}^{++}${\rm)}
be the set of $r$-graphs $H\in \mathcal{H}_r^+(n,m)$ {\rm(}resp.
$H\in \mathcal{H}_r^{++}(n,m)${\rm)}
with exactly $h_4$ Type-$4$ clusters. Then, by Theorem~\ref{t7.2},
\[
|\mathcal{H}_r^+(n,m)|=\sum_{h_4=0}^{M_4}|\mathcal{C}_{h_4}^{+}|,\quad
|\mathcal{H}_r^{++}(n,m)|=\sum_{h_4=0}^{M_4}|\mathcal{C}_{h_4}^{++}|.
\]
By the same discussion as in Section~\ref{s:4}, we also have
\[
|\mathcal{C}_{0}^{++}|=\Bigl(1-O\Bigl( \frac{r^6}{n^3}\Bigr)\Bigr)|\mathcal{L}_r(n,m)|\quad
\text{and}\quad |\mathcal{C}_{h_4}^{++}|=\Bigl(1-O\Bigl( \frac{r^6}{n^3}\Bigr)\Bigr)|\mathcal{C}_{h_4}^{+}|.
\]
We also have $|\mathcal{L}_r(n,m)|\neq0$
and $|\mathcal{C}_{0}^{++}|\neq0$.
Thus,
\[
\frac{1}{\mathbb{P}_r(n,m)}=\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{L}_r(n,m)|}
\Bigl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\Bigr)
=\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}
\Bigl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\Bigr).
\]
\end{remark}
By Remark~\ref{r7.1} and the same arguments as used for Lemma~\ref{l3.5}
and Corollary~\ref{c5.4}, we also have the following two lemmas.
\begin{lemma}\label{l7.5}
Assume $\log(r^{-2}n)\leq m=O(r^{-2}n)$.
Let $H\in \mathcal{H}_r^{+}(n,m-\xi)$
and let $N_t$ be the set of $t$-sets of $[n]$ of which
no two vertices belong to the same edge of $H$, where
$r\leq t\leq 2r-2$ and $\xi=O(1)$. Then
\[
|N_t|=\biggl[ \binom{n}{t}- \binom{r}{2}m \binom{n-2}{t-2}\biggr]
\biggl(1+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m\log(r^{-2}n)}{n^3}\Bigr)\biggr).
\]
\end{lemma}
\begin{lemma}\label{l7.6}
Assume $\log(r^{-2}n)\leq m=O(r^{-2}n)$.
With notation as above, \\
$(a)$\ If $\mathcal{C}_{h_4}^{++}=\emptyset$, then $\mathcal{C}_{h_4+1}^{++}=\emptyset$.\\
$(b)$\ Let $h_4'$ be the first
value of $h_4\leq M_4$
such that $\bigl|\mathcal{C}_{h_4}^{++}\bigl|=0$, or $h_4'=M_4+1$ if no such value exists.
Suppose that $n\to\infty$, then uniformly for $1\leq h_4< h_4'$,
\begin{align*}
\frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{h_4-1}^{++}|}&=
\frac{ \binom{2r-2}{2} \binom{2r-4}{r-2} \binom{m-2(h_4-1)}{2}\bigl[ \binom{n}{2r-2}
- \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{h_4\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}\times\biggl(1+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m\log(r^{-2}n)}{n^3}\Bigr)\biggr).
\end{align*}
\end{lemma}
\begin{lemma}\label{l7.7}
Assume $\log(r^{-2}n)\leq m=O(r^{-2}n)$. Then
\[
\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}
=\exp\biggl[ \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}-
\binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}
+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].
\]
\end{lemma}
\begin{proof} Let $h_4'$ be the first
value of $h_4\leq M_4$
such that $|\mathcal{C}_{h_4}^{++}|=0$ or $h_4'=M_4+1$ if no such value exists,
which is defined in Lemma~\ref{l7.5}. By Remark~\ref{r7.3},
we have $|\mathcal{C}_{0}^{++}|\neq0$.
Define $n_{0},\ldots,n_{M_4}$ by $n_{0}=1$,
\[
n_{h_4}= \frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}
\]
for $1\leq h_4<h_4'$ and $n_{h_4}=0$ for $h_4'\leq h_4\leq M_4$, where
$M_4$ is shown in~\eqref{e7.1}.
Note that
\[
\binom{m-2(h_4-1)}{2}= \frac{[m]_2}{2}\Bigl(1-(h_4-1) \frac{4m-4h_4+2}{m(m-1)}\Bigr).
\]
By Lemma~\ref{l7.5}, for $1\leq h_4\leq h_4'$, we have
\begin{equation}\label{e7.2}
\begin{split}
\frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}&= \frac{1}{h_4} \frac{|\mathcal{C}_{h_4-1}^{++}|}
{|\mathcal{C}_{0}^{++}|}
\frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}- \binom{r}{2}
m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2} \\
&{\qquad}\times\biggl(1+O\Bigl(\frac{r^4}{n^2}+ \frac{r^6m\log(r^{-2}n)}
{n^3}\Bigr)\biggr)\biggl(1-(h_4-1) \frac{4m-4h_4+2}{m(m-1)}\biggr).
\end{split}
\end{equation}
Define
\[
A(h_4)= \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}-
\binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}
\biggl(1+O\Bigl(\frac{r^4}{n^2}+ \frac{r^6m\log(r^{-2}n)}{n^3}\Bigr)\biggr)
\]
for $1\leq h_4< h_4'$, $A(h_4)=0$ for $h_4'\leq h_4\leq M_4$ and
$B(h_4)= \frac{4m-4h_4+2}{m(m-1)}=O\bigl( \frac{1}{m}\bigr)$ as $m\geq \log (r^{-2}n)$. Let $A_{1}=\min_{h_4=1}^{M_4}A(h_4)$,
$A_{2}=\max_{h_4=1}^{M_4}A(h_4)$, $C_{1}=\min_{h_4=1}^{M_4}A(h_4)B(h_4)$
and $C_{2}=\max_{h_4=1}^{M_4}A(h_4)B(h_4)$.
We further have
\[
\frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}=
\frac{A(h_4)}{h_4} \frac{|\mathcal{C}_{h_4-1}^{++}|}{|\mathcal{C}_{0}^{++}|}
\bigl(1-(h_4-1)B(h_4)\bigr).
\]
Note that $\max\{A/M_4,|C|\}=o(1)$ for all $A\in [A_{1},A_{2}]$ and $C\in[C_{1},C_{2}]$
as $\log(r^{-2}n)\leq m=O(r^{-2}n)$ and $h_4\leq M_4=\lceil\log (r^{-2}n)\rceil$.
Let $\hat{c}= \frac{1}{2\cdot 3^4}$, then $\max\{A/M_4,|C|\}\leq \hat{c}< \frac{1}{3}$.
Lemma~\ref{l6.2} applies to obtain
\begin{align*}
\sum_{h_4=0}^{M_4} \frac{|\mathcal{C}_{h_4}^{++}|}{|\mathcal{C}_{0}^{++}|}
&=\exp\biggl[ \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}- \binom{r}{2}
m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}+O\Bigl( \frac{r^8m^2}{n^{4}}+ \frac{r^{10}
m^3\log(r^{-2}n)}{n^5}\Bigr)\biggr]+O\Bigl(\Bigl( \frac{e}{3^{4}}\Bigr)^{M_4}\Bigr)\\
&=\exp\biggl[ \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}-
\binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}
m \binom{n-2}{r-2}\bigr]^2}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr],
\end{align*}
where the last equality is true because $O\bigl( \frac{r^8m^2}{n^{4}}+
\frac{r^{10}m^3\log(r^{-2}n)}{n^5}\bigr)
=O\bigl( \frac{r^6m^2}{n^3}\bigr)$ as $m=O(r^{-2}n)$ and $O\bigl(\bigl( \frac{e}{3^{4}}\bigr)^{M_4}\bigr)=O\bigl( \frac{r^6}{n^3}\bigr)$.
\end{proof}
\section{The case $1\leq m= O(\log (r^{-2}n))$}\label{s:8}
Let $\mathcal{H}_r^+(n,m)\subset\mathcal{H}_r(n,m)$ be the set of
$r$-graphs $H$ which satisfy properties $\bf(a)$ to $\bf(d)$:
$\bf(a)$\ The intersection of any two edges contains at most two vertices.
$\bf(b)$\ $H$ only contains one type of cluster (Type-$4$ cluster).
(This implies that any three edges involve at least $3r-3$ vertices. Thus, if there are
two edges, for example $\{e_1,e_2\}$, such that $|e_1\cup e_2|= 2r-2$,
then $|(e_1\cup e_2)\cap e|\leq 1$ for any
edge $e$ other than $\{e_1,e_2\}$ of $H$.)
$\bf(c)$\ Any two distinct Type-$4$ clusters in $H$ are vertex-disjoint.
(This implies that any four edges involve at least $4r-4$ vertices.)
$\bf(d)$\ There are at most two Type-$4$ clusters in $H$. (This implies that
any six edges involve at least $6r-5$ vertices.)
\begin{theorem}\label{t8.1}
Assume $1\leq m= O(\log (r^{-2}n))$ and $n\to \infty$. Then
\[
\frac{|\mathcal{H}^+_r(n,m)|}{|\mathcal{H}_r(n,m)|}=1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\]
\end{theorem}
\begin{proof} Consider $H\in \mathcal{H}_r(n, m)$ chosen uniformly at random.
We can apply Lemma~\ref{l2.2} several times to show that $H$ satisfies properties
$\bf(a)$-$\bf(d)$ with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
We only prove the property $\bf(d)$ because the proof of other conditions are
exactly same with the proof in Theorem~\ref{t3.2}.
Applying Lemma~\ref{l2.2} with $t=6$ and $\alpha=6$,
the expected number of sets of six edges involving at most $6r-6$ vertices is
\[
O\Bigl( \frac{r^{12}m^6}{n^{6}}\Bigr)=O\Bigl( \frac{r^6m^2}{n^3}\Bigr)
\]
because $m=O(\log (r^{-2}n))$.
Hence, property $\bf(d)$ holds with probability $1-O\bigl( \frac{r^6m^2}{n^3}\bigr)$.
\end{proof}
\begin{remark}\label{r8.2}
By Theorem~\ref{t8.1}, for a nonnegative integer $h_4$, let $\mathcal{C}_{h_4}^{+}$
be the set of $r$-graphs $H\in \mathcal{H}_r^+(n,m)$
with exactly $h_4$
Type-$4$ clusters. Thus, we have $|\mathcal{H}_r^+(n,m)|=\sum_{h_4=0}^2|\mathcal{C}_{h_4}^{+}|$,
$|\mathcal{C}_{0}^{+}|=|\mathcal{L}_r(n,m)|\neq\emptyset$ and
\[
\frac{1}{\mathbb{P}_r(n,m)}=\sum_{h_4=0}^2 \frac{|\mathcal{C}_{h_4}^{+}|}
{|\mathcal{L}_r(n,m)|}\biggl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr)
=\sum_{h_4=0}^2 \frac{|\mathcal{C}_{h_4}^{+}|}{|\mathcal{C}_{0}^{+}|}
\biggl(1-O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr).
\]
\end{remark}
By arguments similar to Lemma~\ref{l3.5} and
Lemma~\ref{l7.5}, Corollary~\ref{c5.4} and Lemma~\ref{l7.6}, we have
the following two lemmas.
\begin{lemma}\label{l8.4}
Assume $1\leq m=O(\log (r^{-2}n))$.
Let $H\in \mathcal{H}_r^{+}(n,m-\xi)$
and $N_t$ be the set of $t$-sets of $[n]$ such that no two vertices belong to the same edge of $H$,
where $r\leq t\leq 2r-2$ and $\xi=O(1)$ are positive integers. Suppose that $n\to\infty$. Then
\[
|N_t|=\biggl[ \binom{n}{t}- \binom{r}{2}m \binom{n-2}{t-2}\biggr]\biggl(1+O\Bigl( \frac{r^4}{n^2}\Bigr)\biggr).
\]
\end{lemma}
\begin{lemma}\label{l8.5}
Assume $1\leq m=O(\log (r^{-2}n))$.
With notation as above, \\
$(a)$\ If $\mathcal{C}_{h_4}^{+}=\emptyset$, then $\mathcal{C}_{h_4+1}^{+}=\emptyset$.
\\
$(b)$\ Let $h_4'$ be the first
value of $h_4\leq 2$
such that $\bigl|\mathcal{C}_{h_4}^{+}\bigl|=0$ or $h_4'=3$ if no such value exists.
Suppose that $n\to\infty$, then uniformly for $1\leq h_4< h_4'$,
\[
\frac{|\mathcal{C}_{h_4}^{+}|}{|\mathcal{C}_{h_4-1}^{+}|}=
\frac{ \binom{2r-2}{2} \binom{2r-4}{r-2} \binom{m-2(h_4-1)}{2}\bigl[ \binom{n}{2r-2}
- \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}{h_4\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\biggl(1+O\Bigl( \frac{r^4}{n^2}\Bigr)\biggr).
\]
\end{lemma}
\begin{lemma}\label{l8.6}
Assume $1\leq m=O(\log (r^{-2}n))$. Then
\begin{align*}
\frac{|\mathcal{C}_{0}^{+}|+|\mathcal{C}_{1}^{+}|+|\mathcal{C}_{2}^{+}|}
{|\mathcal{C}_{0}^{+}|}
&=\exp\biggl[ \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}- \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr].\\
\end{align*}
\end{lemma}
\begin{proof} By Remark~\ref{r8.2}, we have $|\mathcal{C}_{0}^{+}|\neq0$.
By Lemma~\ref{l8.4}, we have
\[
\frac{|\mathcal{C}_{1}^{+}|}{|\mathcal{C}_{0}^{+}|}=
\frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}- \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\biggl(1+O\Bigl( \frac{r^4}{n^2}\Bigr)\biggr)
\]
and
\begin{align*}
\frac{|\mathcal{C}_{2}^{+}|}{|\mathcal{C}_{1}^{+}|}=
\frac{[m-1]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}
\bigl[ \binom{n}{2r-2}- \binom{r}{2}m \binom{n-2}{2r-4}\bigr]}
{4\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}
\biggl(1+O\Bigl( \frac{r^4}{n^2}\Bigr)\biggr).
\end{align*}
Thus, we have
\begin{align*}
\sum_{h_4=0}^2 \frac{|\mathcal{C}_{h_4}^{+}|}{|\mathcal{C}_{0}^{+}|}
&=\biggl(1+ \frac{[m]_2 \binom{2r-2}{2} \binom{2r-4}{r-2}\bigl[ \binom{n}{2r-2}- \binom{r}{2}
m \binom{n-2}{2r-4}\bigr]}{2\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}\\
&{\qquad}+ \frac{[m]_4 \binom{2r-2}{2}^2 \binom{2r-4}{r-2}^2\bigl[ \binom{n}{2r-2}- \binom{r}{2}
m \binom{n-2}{2r-4}\bigr]^2}{8\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^4}+O\Bigl( \frac{r^{8}m^2}{n^4}\Bigr)\biggr),
\end{align*}
which simplifies to the expression in the lemma statement.
\end{proof}
\section{Proof of Theorem~\ref{t1.3}}\label{s:9}
As in the statement of Theorem~\ref{t1.3}, we use $m_0=Np$.
The binomial distribution is $\Bin(n,p)$.
Recall that $\mathbb{P}_r(n,m)$ denotes the probability that
an $r$-graph $H\in \mathcal{H}_r(n,m)$ chosen uniformly
at random is linear. We found an expression for $\mathbb{P}_r(n,m)$
in Theorem~\ref{t1.1}.
By the law of total probability, we have
\begin{equation}\label{e9.2}
\mathbb{P}\bigl[H_r(n,p)\in \mathcal{L}_r\bigr]=\sum_{m=0}^{N}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}.
\end{equation}
In the proof of Theorem~\ref{t1.3},
we need the following lemmas. We firstly show $\mathbb{P}_r(n,m)$
is decreasing in $m$ (Lemma~\ref{l9.1}). Some approximations will make
use of the
Chernoff inequality (Lemma~\ref{l9.3}) and the normal approximation
of the binomial distribution (Lemmas~\ref{l9.4} and~\ref{l9.5}).
\begin{lemma}\label{l9.1}
$\mathbb{P}_r(n,m)$ is a non-increasing function of $m$.
\end{lemma}
\begin{proof}
Choosing $m$ distinct edges at random gives the same distribution as
choosing $m-1$ distinct edges at random and then an $m$-th edge at
random distinct from the first $m-1$. So $\mathbb{P}_r(n,m)\le\mathbb{P}_r(n,m-1)$.
\end{proof}
\begin{lemma}[\cite{cher52}]\label{l9.3}
For $X\sim \Bin(N, p)$ and any $0<t\leq Np$,
\[
\mathbb{P}\bigl[|X - Np|>t\bigr]<2\exp\left[-t^2/\left(3Np\right)\right].
\]
\end{lemma}
\begin{lemma}\label{l9.4}
Let $r=r(n)\geq 3$ be an integer with $r=o(n^{ \frac12})$ and
$X\sim \Bin(N,p)$, where $r^{-2}n\leq Np=o(r^{-3}n^{ \frac32})$.
Then
\[
\mathbb{P}\bigl[X=\lfloor Np\rfloor+t\bigr]= \frac{1}{\sqrt{2\pi Np}}
\exp\biggl[- \frac{t^2}{2Np}+O\biggl( \frac{\log^3 (r^{-2}n)}{\sqrt{Np}}
+ \frac{r^6(Np)^2}{n^3}\biggr)\biggr],
\]
where $t\in \mathbb{Z}$ and
\[
t\in\biggl[-\log(r^{-2}n)\sqrt{Npq}- \frac{[r]_2^2(Np)^2}{2n^2},\log (r^{-2}n)\sqrt{Npq}\biggr].
\]
\end{lemma}
\begin{proof} Note that
\[
\binom{N}{\lfloor Np\rfloor+t}p^{\lfloor Np\rfloor+t}q^{\lceil Nq\rceil -t}=
\binom{N}{\lfloor Np\rfloor}p^{\lfloor Np\rfloor}q^{\lceil Nq\rceil }
\frac{ \binom{N}{\lfloor Np\rfloor+t}p^tq^{-t}}{ \binom{N}{\lfloor Np\rfloor}}.
\]
By Stirling's formula for $\lfloor Np\rfloor!$ and $\lceil Nq\rceil !$,
we have
\[
\binom{N}{\lfloor Np\rfloor}p^{\lfloor Np\rfloor}q^{\lceil Nq\rceil}
= \frac{1}{\sqrt{2\pi Np}}\exp\Bigl(O\Bigl( \frac{1}{Np}\Bigr)\Bigr)
\]
and as $Np\to\infty$,
\begin{align}
\frac{ \binom{N}{\lfloor Np\rfloor+t}p^tq^{-t}}{ \binom{N}{\lceil Np\rceil}}
&=\prod_{i=0}^{t-1} \frac{\Bigl(1- \frac{i}{\lceil Nq\rceil}\Bigr)}
{\Bigl(1+ \frac{1+i}{\lfloor Np\rfloor}\Bigr)} \notag\\
&=\exp\biggl[\sum_{i=0}^{t-1}\ln\biggl( \frac{1- \frac{i}{\lceil Nq\rceil}}
{1+ \frac{1+i}{\lfloor Np\rfloor}}\biggr)\biggr]\notag\\
&=\exp\biggl[- \frac{t^2}{2Np}+O\biggl(\frac{t}{Np}+ \frac{t^3}{(Np)^2}\biggr)\biggr].\label{e9.3}
\end{align}
Since
\[
t\in\biggl[-\log (r^{-2}n)\sqrt{Npq}- \frac{[r]_2^2(Np)^2}{2n^2},\log (r^{-2}n)\sqrt{Npq}\biggr],
\]
then we have
\[
O\biggl(\frac{t}{Npq}+ \frac{t^3}{(Npq)^2}\biggr)
=O\biggl(\frac{\log^3 (r^{-2}n)}{\sqrt{Np}}+
\frac{r^6(Np)^2}{n^3}\biggr).\qedhere
\]
\end{proof}
By the proof of Lemma~\ref{l9.4}, we also have the following lemma.
\begin{lemma}\label{l9.5}
Let $r=r(n)\geq 3$ be an integer with $r=o(n^{ \frac12})$ and
$X\sim \Bin(N,p)$, where $r^{-2}n\leq Np=o(r^{-3}n^{ \frac32})$.
Then
\[
\mathbb{P}\bigl[X=\lfloor Np\rfloor+t\bigr]= \frac{1}{\sqrt{2\pi Np}}
\,O\biggl(\exp\biggl(- \frac{t^2}{2Np}\biggr)\biggr),
\]
where $t\in \mathbb{Z}$ and
\[
t\in\biggl[- Np+\log (r^{-2}n),- \frac{[r]_2^2(Np)^2}{2n^2}-\log (r^{-2}n)\sqrt{Npq}\biggr].
\]
\end{lemma}
\begin{proof} By the same proof as Lemma~\ref{l9.4}, we find that~\eqref{e9.3}
still holds, which implies the lemma.
\end{proof}
We prove Theorem~\ref{t1.3} separately for the two cases
$r^{-2}n\leq Np=o(r^{-3}n^{ \frac32})$ and $0<Np=O(r^{-2}n)$.
\begin{theorem}\label{t9.6}
Assume $Np=m_0$ with $r^{-2}n\leq m_0=o(r^{-3}n^{ \frac32})$.
Then
\[
\mathbb{P}\bigl[H_r(n,p)\in \mathcal{L}_r(n)\bigr]
=\exp\biggl[- \frac{[r]_2^2m_0^2}{4n^2}+
\frac{[r]_2^3(3r-5)m_0^3}{6n^4}+
O\biggl( \frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+ \frac{r^6m_0^2}{n^3}\biggr)\biggr].
\]
\end{theorem}
\begin{proof} Let
\[
m_0^*=m_0- \frac{[r]_2^2m_0^2}{2n^2}.
\]
Note that $m_0^*=m_0(1+o(1))$, enabling us to use
$m_0$ in place of $m_0^*$ in error terms.
We will divide the sum~\eqref{e9.2} into four domains:
\[
\mathbb{P}[H_r(n,p)\in \mathcal{L}_r]
=\sum_{m\in I_0\cup I_1\cup I_2\cup I_3}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m},
\]
where
\begin{align*}
I_0&=\bigl[0,\log (r^{-2}n)\bigr], \\
I_1&=\bigl[\log (r^{-2}n), m_0^*-\log (r^{-2}n)\sqrt{m_0q}\bigr], \\
I_2&=\bigl[m_0^*-\log (r^{-2}n)\sqrt{m_0q},m_0+\log (r^{-2}n)\sqrt{m_0q}\bigr], \\
I_3&=\bigl[m_0+\log (r^{-2}n)\sqrt{m_0q},N\bigr].
\end{align*}
The theorem follows from a sequence of claims which we show next.
\bigskip
{\bf Claim 1}.~~$\displaystyle\mathbb{P}_r(n,m_0^*)=
\exp\Bigl[- \frac{[r]_2^2m_0^2}{4n^2}+ \frac{[r]_2^3(3r^2+9r-20)m_0^3}{24n^4}+O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)\Bigr].$
\begin{proof}[Proof of Claim 1] From Theorem~\ref{t1.1} we have
\[
\mathbb{P}_r(n,m_0^*)
=\exp\biggl[- \frac{[r]_2^2\bigl[m_0- \frac{[r]_2^2m_0^2}{2n^2}\bigr]_2}{4n^2}- \frac{[r]_2^3(3r^2-15r+20)\bigl(m_0- \frac{[r]_2^2m_0^2}{2n^2}\bigr)^3}{24n^4}+
O\Bigl( \frac{r^6{m_0^*}^2}{n^3}\Bigr)\biggr],
\]
which simplifies to give the claim.
\end{proof}
\bigskip
{\bf Claim 2}.~~If
$m=m_0^*+s \in I_1\cup I_2$, then
\[
\mathbb{P}_r(n,m)=\mathbb{P}_r(n,m_0^*)\,
\exp\biggl[- \frac{[r]_2^2m_0^*s}{2n^2}- \frac{[r]_2^2s^2}{4n^2}
+O\Bigl( \frac{r^6{m_0^2}}{n^3}\Bigr)\biggr].
\]
\begin{proof}[Proof of Claim 2]
Since $m\in I_1\cup I_2$, we have
\[
s\in\Bigl[-m_0+ \frac{[r]_2^2m_0^2}{2n^2}+\log (r^{-2}n),\log (r^{-2}n)\sqrt{m_0q}+ \frac{[r]_2^2m_0^2}{2n^2}\Bigr].
\]
By Theorem~\ref{t1.1}, we have
\begin{align*}
\mathbb{P}_r(n,m)
&=\mathbb{P}_r(n,m_0^*+s)\\
&=\exp\biggl[- \frac{[r]_2^2[m_0^*+s]_2}{4n^2}
- \frac{[r]_2^3(3r^2-15r+20)(m_0^*+s)^3}{24n^4}
+O\Bigl( \frac{r^6(m_0^*+s)^2}{n^3}\Bigr)\biggr],
\end{align*}
which gives the claim because
$O\bigl( \frac{r^4(m_0^*+s)}{n^2}\bigr)=O\bigl( \frac{r^6{m_0^2}}{n^3}\bigr)$,
$O\bigl( \frac{r^8{m_0^*}^2s}{n^4}\bigr)=O\bigl( \frac{r^6{m_0^2}}{n^3}\bigr)$ and
$O\bigl( \frac{r^8{m_0^*}s^2}{n^4}\bigr)=O\bigl( \frac{r^6{m_0^2}}{n^3}\bigr)$.
\end{proof}
\bigskip
{\bf Claim 3}.~~$\displaystyle
\sum_{s=-\log (r^{-2}n)\sqrt{m_0q}}^{\log (r^{-2}n)
\sqrt{m_0q}+ \frac{[r]_2^2m_0^2}{2n^2}}\exp\biggl[- \frac{s^2}{2m_0}\biggr]=\sqrt{2\pi m_0}
\biggl(1+O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)\biggr)$.
\begin{proof}[Proof of Claim 3]
This is an elementary summation that is easily proved either using the
Euler-Maclaurin summation formula or the Poisson summation formula.
\end{proof}
\bigskip
{\bf Claim 4}.~~$\displaystyle
\sum_{m\in I_2} \mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m} \newline
{\kern10em}=\exp\biggl[- \frac{[r]_2^2m_0^2}{4n^2}+ \frac{[r]_2^3(3r-5)m_0^3}{6n^4}+O\Bigl(\frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+
\frac{r^6m_0^2}{n^3}\Bigr)\biggr]$.
\begin{proof}[Proof of Claim 4]
For $m=m_0- \frac{[r]_2^2m_0^2}{2n^2}+s\in I_2$, we have
\[
s\in\Bigl[-\log (r^{-2}n)\sqrt{m_0q},\log (r^{-2}n)\sqrt{m_0q}+ \frac{[r]_2^2m_0^2}{2n^2}\Bigr].
\]
By Lemma~\ref{l9.4}, we have
\[
\binom{N}{m}p^mq^{N-m}= \frac{1}{\sqrt{2\pi m_0}}
\exp\biggl[- \frac{{\bigl(s- \frac{[r]_2^2m_0^2}{2n^2}\bigr)}^2}{2m_0}+O\Bigl( \frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+ \frac{r^6m_0^2}{n^3}\Bigr)\biggr].
\]
By Claim 2, we further have
\begin{align*}
\sum_{m\in I_2} & \mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m} \\
&= \frac{\mathbb{P}_r(n,m_0^*)}{\sqrt{2\pi m_0}}\sum_{s=-\log (r^{-2}n)\sqrt{m_0q}}^{\log (r^{-2}n)\sqrt{m_0q}+ \frac{[r]_2^2m_0^2}{2n^2}}\exp\biggl[- \frac{{\bigl(s- \frac{[r]_2^2m_0^2}{2n^2}\bigr)}^2}{2m_0q}
- \frac{[r]_2^2s^2}{4n^2}- \frac{[r]_2^2\bigl(m_0- \frac{[r]_2^2m_0^2}{2n^2}\bigr)s}{2n^2} \\
&{\qquad}+O\Bigl( \frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+
\frac{r^6m_0^2}{n^3}\Bigr)\biggr] \\
&= \frac{\mathbb{P}_r(n,m_0^*)}{\sqrt{2\pi m_0}}\exp\biggl[- \frac{[r]_2^4m_0^3}{8n^4}+O\Bigl( \frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+
\frac{r^6m_0^2}{n^3}\Bigr)\biggr]\sum_{s=-\log (r^{-2}n)\sqrt{m_0q}}^{\log (r^{-2}n)\sqrt{m_0q}+ \frac{[r]_2^2m_0^2}{2n^2}}\!\!\!\exp\biggl[- \frac{s^2}{2m_0}\biggr],
\end{align*}
because $O\bigl( \frac{r^8{m_0^*}s^2}{n^4}\bigr)=O\bigl( \frac{r^6{m_0^2}}{n^3}\bigr)$ and $ \frac{[r]_2^2s^2}{4n^2}=O\bigl(\frac{\log^3 (r^{-2}n)}{\sqrt{m_0}}+
\frac{r^6m_0^2}{n^3}\bigr)$.
Now we apply the value of $\mathbb{P}_r(n,m_0^*)$ from Claim~1 and the
summation from Claim~3.
\end{proof}
Secondly, we show the value of
\[
\sum_{m\in I_1}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}
\]
in the following two claims.
\vskip 0.3cm
{\bf Claim 5}.~~$\displaystyle
\frac{1}{\sqrt{m_0}}\!\!\sum_{s=-m_0+ \frac{[r]_2^2m_0^2}{2n^2}
+\log (r^{-2}n)}^{-\log (r^{-2}n)\sqrt{m_0q}}\!\!
\exp\biggl[- \frac{[r]_2^2s^2}{4n^2}- \frac{s^2}{2m_0}
+ \frac{[r]_2^4m_0^2s}{4n^4}\biggr]=O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)$.
\begin{proof}[Proof of Claim 5]
Since the summand is increasing in the range of summation, it suffices to
take the number of terms times the last term.
\end{proof}
\bigskip
{\bf Claim 6}.~~$\displaystyle
\sum_{m\in I_1}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}=
O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)\sum_{m\in I_2}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}$.
\begin{proof}[Proof of Claim 6]
If $m=m_0^*+s\in I_1$, then we have
\[
s\in\Bigl[-m_0+ \frac{[r]_2^2m_0^2}{2n^2}+\log (r^{-2}n),-\log (r^{-2}n)\sqrt{m_0q}\Bigr].
\]
By Lemma~\ref{l9.5} and Claim 2, we have
\begin{align*}
&\sum_{m\in I_1}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}
= \frac{\mathbb{P}_r(n,m_0^*)}{\sqrt{2\pi m_0}} \\
&{\qquad}\times\sum_{s=-m_0+ \frac{[r]_2^2m_0^2}{2n^2}+\log (r^{-2}n)}^{-\log (r^{-2}n)\sqrt{m_0q}}
\exp\biggl[- \frac{[r]_2^2m_0^*s}{2n^2}- \frac{[r]_2^2s^2}{4n^2}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr]
O\biggl(\exp\biggl[- \frac{\bigl(s- \frac{[r]_2^2m_0^2}{2n^2}\bigr)^2}{2m_0}\biggr]\biggr)
\displaybreak[1]\\
&= \frac{\mathbb{P}_r(n,m_0^*)}{\sqrt{2\pi m_0}}
\kern-0.4em\sum_{s=-m_0+ \frac{[r]_2^2m_0^2}{2n^2}
+\log (r^{-2}n)}^{-\log (r^{-2}n)\sqrt{m_0q}}\kern-0.4em
O\biggl(\exp\biggl[- \frac{[r]_2^2s^2}{4n^2}- \frac{s^2}{2m_0}+ \frac{[r]_2^4m_0^2s}{4n^4}- \frac{[r]_2^4m_0^3}{8n^4}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\biggr]\biggr).
\end{align*}
By Claim 1 and Claim 4, we further have
\[
\mathbb{P}_r(n,m_0^*)\,O\biggl(\exp\Bigl[- \frac{[r]_2^4m_0^3}{8n^4}+O\Bigl( \frac{r^6m^2}{n^3}\Bigr)\Bigr]\biggr)=
O\biggl(\sum_{m\in I_2}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}\biggr),
\]
which completes the proof together with Claim~5.
\end{proof}
\bigskip
{\bf Claim 7}.~~$\displaystyle
\sum_{m\in I_0}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}=O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)\sum_{m\in I_2}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}$.
\begin{proof}[Proof of Claim 7]
Since $m=m_0+t\in I_0$, then we have
\[
t\in\Bigl[-m_0,-m_0+\log (r^{-2}n)\Bigr].
\]
Since $\mathbb{P}_r(n,m)\leq 1$ for $m\in I_0$,
Lemma~\ref{l9.3} gives
\[
\sum_{m\in I_0}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}
=O\bigl(\exp\bigl[- \dfrac13 m_0\bigr]\bigr).
\]
Together with Claim~4, this proves the required bound.
\end{proof}
\bigskip
{\bf Claim 8}.~~$\displaystyle
\sum_{m\in I_3}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}=
O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr)\sum_{m\in I_2}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}$.
\begin{proof}[Proof of Claim 8]
Since $m=m_0+t\in I_3$, then we have
\[
t\in\Bigl[\log (r^{-2}n)\sqrt{m_0q},N-m_0\Bigr].
\]
By Lemma~\ref{l9.1}, we have $\mathbb{P}_r(n,m)\leq \mathbb{P}_r(n,m_0)$
for $m_0\in I_3$. Therefore
\[
\sum_{m\in I_3}\mathbb{P}_r(n,m) \binom{N}{m}p^mq^{N-m}
\leq \mathbb{P}_r(n, m_0) \sum_{m\in I_3}
\binom{N}{m}p^mq^{N-m},
\]
from which the claim follows using Theorem~\ref{t1.1} and
Lemma~\ref{l9.3}.
\end{proof}
\bigskip
To complete the proof of Theorem~\ref{t9.6}, add together Claims 4, 6, 7 and 8.
\end{proof}
Note that in the process of proving Theorem~\ref{t9.6} we also proved
Corollary~\ref{c1.4}.
The second case of Theorem~\ref{t1.3} is $0<m_0=O(r^{-2}n)$.
\begin{theorem}\label{t9.8}
Assume that $0<m_0=O(r^{-2}n)$. Then
\[
\mathbb{P}\bigl[H_r(n,p)\in \mathcal{L}_r\bigr]=\exp\biggl[- \frac{[r]_2^2m_0^2}{4n^2}+O\biggl( \frac{r^6m_0^2}{n^3}\biggr)\biggr].
\]
\end{theorem}
\begin{proof} Let $X_{i}$
denote the number of pairs of edges with $i$ common vertices in $H_r(n,p)$,
where $2\leq i\leq r-1$. Let $X_{\rm{link}}=\sum_{i=2}^{r-1}X_i$.
Let $M_{i,1},\ldots,M_{i,t_i}$ be all unordered pairs of $r$-sets $\{e_1,e_{2}\}$ in
$[n]$ with $|e_1\cap e_2|=i$, where $2\leq i\leq r-1$ and
\[
t_i= \dfrac12N \binom{r}{i} \binom{n-r}{r-i}.
\]
Firstly, we have
\begin{align*}
\mathbb{E}\left[X_2\right]&=\sum_{j=1}^{t_2}\mathbb{P}\bigl[M_{2,j}
\text{ is in }H_r(n,p)\bigr]\\
&= \dfrac12N \binom{r}{2} \binom{n-r}{r-2}\biggl( \frac{m_0}{N}\biggr)^2
= \frac{[r]_2^2m_0^2}{4n^2}\Bigl(1+\Bigl( \frac{r}{n}\Bigr)\Bigr).
\end{align*}
We also have
\[
\mathbb{E}\biggl[\sum_{i\geq 3}X_i\biggr]
=O\biggl(\sum_{i\geq 3}N \binom{r}{i} \binom{n-r}{r-i}\biggl( \frac{m_0}{N}\biggr)^2\biggr)
=O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr).
\]
Thus, we have
\[
\mathbb{E}\bigl[X_{\rm{link}}\bigr]=\sum_{i=2}^{r-1}
\sum_{j=1}^{t_i}\mathbb{P}\bigl[M_{i,j}\text{ is in }H_r(n,p)\bigr]
= \frac{[r]_2^2m_0^2}{4n^2}+O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr).
\]
Similarly,
\[
\sum_{i_1,i_2=2}^{r-1}\sum_{j_1=1}^{t_{i_1}}\sum_{j_2=1}^{t_{i_2}}\mathbb{P}\bigl[M_{i_1,j_1},M_{i_2,j_2}\text{ are in }H_r(n,p)\bigr]=O\Bigl( \frac{r^8m_0^3}{n^4}\Bigr).
\]
By inclusion-exclusion, we conclude that
\[
\mathbb{P}\bigl[X_{{\rm link}}=0\bigr]= 1- \frac{[r]_2^2m_0^2}{4n^2}+O\Bigl( \frac{r^6m_0^2}{n^3}\Bigr),
\]
where $O\bigl( \frac{r^8m_0^3}{n^4}\bigr)=O\bigl( \frac{r^6m_0^2}{n^3}\bigr)$ because $0<m_0=O(r^{-2}n)$.
\end{proof}
Theorems~\ref{t9.6} and~\ref{t9.8} together complete the proof of Theorem~\ref{t1.3}.
\section{Proof of Theorem~\ref{t1.5}}\label{s:10}
As in the theorem statement, we will assume $m=o(r^{-3}n^{\frac32})$
and $k=o\bigl(\frac{n^3}{r^6m^2}\bigr)$.
The bound on $m$ implies that either $m=0$ (a trivial case we will ignore)
or $r=o(n^{\frac12})$.
We will also assume that $k\leq m$, since otherwise the theorem is trivially true
because $[m]_k=0$ if~$k>m$.
Let $K=K(n)$ be a given linear $r$-graph on $[n]$
vertices with edges $\{e_1,\ldots,e_k\}$.
Consider
$H\in\mathcal{L}_r(n,m)$ chosen uniformly at random. Let $\mathbb{P}[K\subseteq H]$
be the probability that
$H$ contains $K$ as a subhypergraph.
If $\mathbb{P}[K\subseteq H]\neq 0$, then we have
\begin{align}
\mathbb{P}[K\subseteq H]&=\mathbb{P}[e_1,\ldots,e_k\in H] \notag\\
&=\prod_{i=1}^{k}\, \frac{\mathbb{P}[e_1,\ldots,e_i\in H]}{\mathbb{P}[e_1,\ldots,e_i\in H]+\mathbb{P}[e_1,\ldots,e_{i-1}\in H,e_i\notin H]} \notag\\
&=\prod_{i=1}^{k}\, \biggl(1+ \frac{\mathbb{P}[e_1,\ldots,e_{i-1}\in H,e_i\notin H]}
{\mathbb{P}[e_1,\ldots,e_i\in H]}\biggr)^{\!\!-1}.\label{e10.1}
\end{align}
For $i=1,\ldots,k$, let $\mathcal{L}_r(n,m:\overline{e}_i)$ be the set of all linear
hypergraphs in $\mathcal{L}_r(n,m)$ which contain edges
$e_1,\ldots,e_{i-1}$ but not edge $e_i$.
Let $\mathcal{L}_r(n,m:e_i)$ be the set of all linear
hypergraphs in $\mathcal{L}_r(n,m)$ which contain edges
$e_1,\ldots,e_{i}$. We have the ratio
\begin{equation}\label{e10.2}
\frac{\mathbb{P}[e_1,\ldots,e_{i-1}\in H,e_i\notin H]}{\mathbb{P}[e_1,\ldots,e_i\in H]}
= \frac{|\mathcal{L}_r(n,m:\overline{e}_i)|}{|\mathcal{L}_r(n,m:e_i)|}.
\end{equation}
Note that $|\mathcal{L}_r(n,m)|\neq 0$
by Theorem~\ref{t1.1}.
We will show below that none of the denominators in~\eqref{e10.2} are zero.
\medskip
Let $H\in\mathcal{L}_r(n,m:e_i)$ with $1\leq i\leq k$.
An \textit{$e_i$-displacement} is defined in two steps:
\noindent{\bf Step 0.}\ Remove the edge $e_i$ from $H$.
Define $H_0$ with the same vertex set $[n]$
and the edge set $E(H_0)=E(H)\setminus\{e_i\}$.
\noindent{\bf Step 1.}\ Take any $r$-set distinct from $e_i$ of which no
two vertices belong to the same edge
of $H_0$ and add it as an edge to~$H_0$. The new graph is denoted by $H'$.
\begin{lemma}\label{l10.1}
Assume $m=o(r^{-3}n^{ \frac32})$ and $1\leq i\leq k$.
Let $H\in\mathcal{L}_r(n,m-1)$ and let $N_r$ be
the set of $r$-sets distinct from $e_i$ of which no two vertices belong to
the same edge of $H$. Then
\[
|N_r|=\biggl[N- \binom{r}{2}m\binom{n-2}{r-2}\biggr]
\biggl(1+O\Bigl( \frac{r^4}{n^2}+\frac{r^6m^2}{n^3}\Bigr)\biggr)
\]
\end{lemma}
\begin{proof} It is clear that $|N_r|\geq\bigl[N- 1- \binom{r}{2}(m-1) \binom{n-2}{r-2}\bigr]$.
The proof of Lemma~\ref{l3.5} applies if we replace the bound on
$\sum_{v\in[n]} \binom{\deg (v)}{2}$ in~\eqref{e3.4} by
$O(rm^2)$ and note that $ \frac{ \binom{r}{2} \binom{n-2}{r-2}}{ \binom{n}{r}}=O ( \frac{r^4}{n^2})$.
\end{proof}
An \textit{$e_i$-replacement} is the inverse of an $e_i$-displacement.
An $e_i$-replacement from $H'\in\mathcal{L}_r(n,m:\overline{e}_i)$
consists of removing any edge in $E(H')-\{e_1,\ldots,e_{i-1}\}$,
then inserting $e_i$. We say that the $e_i$-replacement is \textit{legal} if
$H\in\mathcal{L}_r(n,m:e_i)$, otherwise it is \textit{illegal}.
\begin{lemma}\label{l10.2}
Assume $m=o(r^{-3}n^{ \frac32})$ and $1\leq i\leq k$.
Consider $H'\in\mathcal{L}_r(n,m:\overline{e}_i)$
chosen uniformly at random. Let $E^*$
be the set of $r$-sets $e^*$ of $[n]$ such that $|e^*\cap e_i|\geq 2$.
Suppose that $n\to\infty$.
Then
\begin{equation*}
\mathbb{P}\bigl[E^*\cap H'\neq\emptyset\bigr]=
\frac{(m-i+1) \binom{r}{2} \binom{n-r}{r-2}}{N}
+O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\end{equation*}
\end{lemma}
\begin{proof}
Fix an $r$-set $e^*\in E^*$.
Let $\mathcal{L}_r(n,m:\overline{e}_i, e^*)$
be the set of all the hypergraphs in $\mathcal{L}_r(n,m:\overline{e}_i)$
which contain the edge $e^*$. Let
\begin{align*}
\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)=
\mathcal{L}_r(n,m:\overline{e}_i)-\mathcal{L}_r(n,m:\overline{e}_i, e^*).
\end{align*}
Thus, we have
\begin{align}
\mathbb{P}[e^*\in H']&
= \frac{|\mathcal{L}_r(n,m:\overline{e}_i, e^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i)|}\notag\\
&= \frac{|\mathcal{L}_r(n,m:\overline{e}_i, e^*)|}
{|\mathcal{L}_r(n,m:\overline{e}_i, e^*)|+|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)|}\notag\\
&= \biggl(1+ \frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e^*)|}\biggr)^{\!-1}.\label{e10.3}
\end{align}
Let $G\in\mathcal{L}_r(n,m:\overline{e}_i, e^*)$ and $S(G)$ be the set
of all ways to move the edge $e^*$ to an $r$-set of $[n]$ distinct from $e^*$ and $e_i$,
of which no two vertices are in any remaining edges of $G$.
Call the new graph~$G'$.
By the same proof as Lemma~\ref{l10.1}, we have
\[
S(G)=\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]
\biggl(1+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr).
\]
Conversely, let $G'\in\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)$
and let $S'(G')$ be the set of all ways to move one edge in $E(G')-\{e_1,\ldots,e_{i-1}\}$
to $e^*$ to make the resulting graph in
$\mathcal{L}_r(n,m:\overline{e}_i, e^*)$. In order to find the expected number of $S'(G')$,
we need to apply the same switching way to $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)$
with a simple analysis.
Likewise, let $E^{**}$ be the set of $r$-sets $e^{**}$ of $[n]$ such that $|e^{**}\cap e^*|\geq 2$
and fix an $r$-set $e^{**}\in E^{**}$.
Let $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*, {e}^{**})$
be the set of all the hypergraphs in $\mathcal{L}_r(n,m:\overline{e}_i,\overline{e}^*)$
which contain the edge $e^{**}$ and $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,\overline{e}^{**})=
\mathcal{L}_r(n,m:\overline{e}_i,\overline{e}^*)-\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,{e}^{**})$.
By the exactly same analysis above, we also have
\begin{align}\label{e10.4}
\mathbb{P}[e^{**}\in G']
&= \biggl(1+ \frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,\overline{e}^{**})|}{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,{e}^{**})|}\biggr)^{\!-1}
\end{align}
For any hypergraph in $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,{e}^{**})$,
by the same proof as Lemma~\ref{l10.1}, we also have $\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]\bigl(1+O\bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\bigr)\bigr)$
ways to move the edge $e^{**}$ to an $r$-set of $[n]$ distinct from ${e}_i$, ${e}^*$ and ${e}^{**}$,
of which no two vertices are in any remaining edges. Similarly, there are at most $m-i+1$ ways
to switch a hypergraph from $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,\overline{e}^{**})$ to
$\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*,{e}^{**})$.
As the equation shown in~\eqref{e10.4}, we have
$\mathbb{P}[e^{**}\in G']
= O\bigl( \frac{m}{N}\bigr)$. Note that $|E^{**}|=\sum_{i=2}^{r-1} \binom{r}{i} \binom{n-r}{r-i}=O\bigl( \binom{r}{2} \binom{n-r}{r-2}\bigr)$,
then $\mathbb{P}[E^{**}\cap G'\neq\emptyset]=O\bigl( \frac{r^4m}{n^2}\bigr)$ and
the expected number of $S'(G')$ is $(m-i+1)\bigl(1-O\bigl( \frac{r^4m}{n^2}\bigr)\bigr)$.
Thus, we have
\begin{align*}
& \frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e^*)|}
= \frac{|S(G)|}{|S'(G')|}\\
&{\qquad}= \frac{N- \binom{r}{2}m \binom{n-2}{r-2}}{m-i+1}
\biggl(1+O\Bigl( \frac{r^4m}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr)\\
&{\qquad}= \frac{N}{m-i+1}\biggl(1+O\Bigl( \frac{r^4m}{n^2}+\frac{r^6m^2}{n^3}\Bigr)\biggr).
\end{align*}
As the equation shown in~\eqref{e10.3}, we also have
\begin{align}\label{e10.5}
\mathbb{P}[e^*\in H']= \frac{m-i+1}{N}
\biggl(1+O\Bigl( \frac{r^4m}{n^2}+\frac{r^6m^2}{n^3}\Bigr)\biggr).
\end{align}
By inclusion-exclusion,
\begin{align}\label{e10.6}
\sum_{e^*\in E^*}\mathbb{P}[e^*\in H']-\sum_{e_1^*\in E^*, e_2^*\in E^*,e_1^*\cap e_2^*\neq\emptyset}\mathbb{P}[e_1^*,e_2^*\in H']\leq \mathbb{P}\bigl[E^*\cap H'\neq\emptyset\bigr]\leq \sum_{e^*\in E^*}\mathbb{P}[e^*\in H'].
\end{align}
Since $|E^*|=\sum_{i=2}^{r-1} \binom{r}{i} \binom{n-r}{r-i}= \binom{r}{2} \binom{n-r}{r-2}+O\bigl( \frac{r^3n^{r-3}}{(r-3)!}\bigr)$,
as the equation shown in~\eqref{e10.5}, we have
\begin{align}\label{e10.7}
\sum_{e^*\in E^*}\mathbb{P}[e^*\in H']=
\frac{(m-i+1) \binom{r}{2} \binom{n-2}{r-2}}{N}
+O\Bigl(\frac{r^6m^2}{n^3}\Bigr)
\end{align}
because $O\bigl( \frac{m}{N} \binom{r}{2} \binom{n-2}{r-2}\bigl( \frac{r^4m}{n^2}
+\frac{r^6m^2}{n^3}\bigr)\bigr)=
O\bigl(\frac{r^6m^2}{n^3}\bigr)$ and
$O\bigl( \frac{m}{N} \frac{r^3n^{r-3}}{(r-3)!}\bigr)=O\bigl( \frac{r^6m}{n^3}\bigr)$.
Consider
$\sum_{e_1^*\in E^*, e_2^*\in E^*, e_1^*\cap e_2^*\neq\emptyset}\mathbb{P}[e_1^*,e_2^*\in H']$ in the
equation~\eqref{e10.6}.
Note that $H'\in \mathcal{L}_r(n,m:\overline{e}_i)$, then $|e_1^*\cap e_2^*|=1$.
Let $\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)$,
$\mathcal{L}_r(n,m:\overline{e}_i, {e}_1^*, \overline{e}_2^*)$,
$\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, e_2^*)$ and $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)$
be the set of all linear
hypergraphs in $\mathcal{L}_r(n,m:\overline{e}_i)$ which contain both
$e_1^*$ and $e_2^*$, only contain $e_1^*$, only contain $e_2^*$ and neither of them,
respectively. Thus, we have
\begin{align}
&\mathbb{P}[e_1^*,e_2^*\in H']\notag\\&
= \frac{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i)|}\notag\\
&= \frac{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}
{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|+|\mathcal{L}_r(n,m:\overline{e}_i, {e}_1^*, \overline{e}_2^*)|+|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, e_2^*)|+|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)|}\notag\\
&= \biggl(1+ \frac{|\mathcal{L}_r(n,m:\overline{e}_i, {e}_1^*, \overline{e}_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}+ \frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, e_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}+ \frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}\biggr)^{\!-1}\label{e10.8}.
\end{align}
By the similar analysis above, we have
\begin{align}
\frac{|\mathcal{L}_r(n,m:\overline{e}_i, {e}_1^*,
\overline{e}_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}&\geq \frac{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}{m-i+1}\biggl(1+O\Bigl( \frac{r^4}{n^2}+\frac{r^6m^2}{n^3}\Bigr)\biggr), \notag\\
\frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, e_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}
&\geq \frac{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}{m-i+1}\biggl(1+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr). \label{e10.9}
\end{align}
For any hypergraph in $\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)$, we move
$e_1^*$ and $e_2^*$ away in two steps by
the similar switching operations in Section~\ref{s:5}.
For $e_1^*$ (resp. $e_2^*$), by the same proof as Lemma~\ref{l10.1},
there are $\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]
\bigl(1+O\bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\bigr)\bigr)$ ways to move $e_1^{*}$
(resp. $e_2^*$) to an $r$-set of $[n]$ distinct from ${e}_i$, $e_1^*$ and $e_2^*$
such that the resulting graph is in $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)$.
Similarly, there are at most $2\binom{m-i+1}{2}$ ways
to switch a hypergraph from $\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)$ to
$\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)$.
Thus, we have
\begin{align}\label{e10.10}
\frac{|\mathcal{L}_r(n,m:\overline{e}_i, \overline{e}_1^*, \overline{e}_2^*)|}{|\mathcal{L}_r(n,m:\overline{e}_i, e_1^*, e_2^*)|}&\geq \frac{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]^2}{2\binom{m-i+1}{2}}\biggl(1+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr).
\end{align}
By equations~\eqref{e10.8}--\eqref{e10.10} and note that there are
$O\bigl( \frac{r^3 n^{2r-4}}{(r-2)!^2}\bigr)$
ways to choose the pair $\{e_1^*,e_2^*\}$ such that $|e_1^*\cap e_i|\geq 2$, $|e_2^*\cap e_i|\geq 2$ and $|e_1^*\cap e_2^*|=1$,
then we have
\begin{align}\label{e10.11}
\sum_{e_1^*\in E^*, e_2^*\in E^*,e_1^*\cap e_2^*\neq\emptyset}\mathbb{P}[e_1^*,e_2^*\in H']
=O\biggl( \frac{r^3 n^{2r-4}}{(r-2)!^2} \frac{m^2}{N^2}\biggr)=O\Bigl( \frac{r^7 m^2}{n^4}\Bigr)=O\Bigl( \frac{r^6m^2}{n^3}\Bigr).
\end{align}
To complete the proof of Lemma~\ref{l10.2}, add together the equations~\eqref{e10.6},~\eqref{e10.7} and~\eqref{e10.11}.
\end{proof}
By Lemmas~\ref{l10.1} and~\ref{l10.2}, we have
\begin{lemma}\label{l10.3}
Assume $m=o(r^{-3}n^{ \frac32})$ and $1\leq i\leq k$. Then\\
$(a)$\ Let $H\in\mathcal{L}_r(n,m:{e}_i)$. The number of $e_i$-displacements is
\[
\biggl[N- \binom{r}{2}m \binom{n-2}{r-2}\biggr]\biggl(1+O\Bigl( \frac{r^4}{n^2}
+ \frac{r^6m^2}{n^3}\Bigr)\biggr).
\]
$(b)$\ Consider $H'\in\mathcal{L}_r(n,m:\overline{e}_i)$ chosen uniformly at random.
The expected number of legal $e_i$-replacements is
\[
(m-i+1)\biggl[1- \frac{(m-i+1) \binom{r}{2} \binom{n-r}{r-2}}{N}
+O\Bigl(\frac{r^6m^2}{n^3}\Bigr)\biggr].
\]
$(c)$
\begin{align*}
\frac{|\mathcal{L}_r(n,m:\overline{e}_i)|}{|\mathcal{L}_r(n,m:{e}_i)|}
&= \frac{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}{m-i+1}\\
&{\qquad}\times\biggl(1+ \frac{(m-i+1) \binom{r}{2} \binom{n-r}{r-2}}{N}
+O\Bigl( \frac{r^4}{n^2}+\frac{r^6m^2}{n^3}\Bigr)\biggr).
\end{align*}
\end{lemma}
By Lemma~\ref{l10.3}(c), we have
\begin{align*}
\mathbb{P}[K&\subseteq H]\\
&=\prod_{i=1}^{k} \biggl(1+
\frac{\mathbb{P}[e_1,\ldots,e_{i-1}\in H,e_i\notin H]}{\mathbb{P}[e_1,\ldots,e_i\in H]}\biggr)^{\!\!-1}\\
&=\prod_{i=1}^{k} \frac{m-i+1}{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}
\biggl(1- \frac{(m-i+1) \binom{r}{2} \binom{n-r}{r-2}}{N}
+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr)\\
&=\prod_{i=1}^{k} \frac{m-i+1}{\bigl[N- \binom{r}{2}m \binom{n-2}{r-2}\bigr]}
\exp\biggl[- \frac{(m-i+1) \binom{r}{2} \binom{n-r}{r-2}}{N}
+O\Bigl( \frac{r^4}{n^2}+ \frac{r^6m^2}{n^3}\Bigr)\biggr]\\
&= \frac{[m]_k}{N^k}\exp\biggl[ \frac{[r]_2^2k^2}{4n^2}
+O\Bigl( \frac{r^4k}{n^2}+ \frac{r^6m^2k}{n^3}\Bigr)\biggr],
\end{align*}
since $k=o\bigl(\frac{n^3}{r^6m^2}\bigr)$.
\section*{Acknowledgement}\label{s:11}
Fang Tian
was partially supported by the National Natural Science Foundation of China
(Grant No.~11871377) and China Scholarship Council [2017]3192, and is now a
visiting research fellow at the Australian National University. Fang Tian is
immensely grateful to Brendan D. McKay for giving her the opportunity to learn from him,
and thanks him for his problem and useful discussions.
|
2,869,038,154,877 | arxiv | \section{Introduction}
In this paper we shall study the motion in a 3D composite galaxy
model described by the potential
\begin{equation}
{{V}_{t}}\left( x,y,z \right)={{V}_{g}}\left(x,y,z \right)
+{{V}_{h}}\left(x,y,z \right) ,
\end{equation}
where
\begin{equation}
{{V}_{g}}\left(x,y,z \right) =\frac{\upsilon _{0}^{2}}{2}\ln \left(
{{x}^{2}} -\lambda {{x}^{3}}+ \alpha {{y}^{2}}+b{{z}^{2}}+c_{b}^{2}
\right),
\end{equation}
while
\begin{equation}
{{V}_{h}}\left(x,y,z\right)
=\frac{-{{M}_{h}}}{{{\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}+c_{h}^{2} \right)}^{1/2}}}.
\end{equation}
Potential {Equation~}(2) describes a triaxial elliptical galaxy
with a bulge and a small asymmetry introduced by the term
$-\lambda x^3$, $\lambda \ll 1$ (see \citealt{Binney2008}). The
parameters $\alpha${ and} $b$ describe the flattening of the
galaxy, while $c_b$ is the scale length of the bulge of the
galaxy. The parameter $\upsilon _0$ is used for the consistency of
the galactic units. To this potential we add a spherical dark
halo, described by the potential {Equation}~(3). Here $M_h$ and
$c_h$ are the mass and the scale length of the dark halo
component, respectively.
The aim of this article is twofold: (i) {T}o investigate the motion in
the potential {Equation}~(1) and to determine the role played by
the halo on the character of orbits. In particular, we are
interested {in} connect{ing} the percentage of
chaotic orbits, as well as the degree of chaos, with the
physical parameters, such as the mass and the scale length of the
dark halo component. (ii) To introduce, use and check a new fast
indicator, which is the total angular momentum $L_{\rm tot}$, of the
3D orbits, in order to obtain a reliable criterion to distinguish
between ordered and chaotic orbits.
The outcomes of the present research are mainly
based on the numerical integration of the equations of
motion
\begin{eqnarray}
\ddot{x}&=&-\frac{\partial \ V_t(x,y,z)}{\partial x}, \nonumber \\
\ddot{y}&=&-\frac{\partial \ V_t(x,y,z)}{\partial y}, \\
\ddot{z}&=&-\frac{\partial \ V_t(x,y,z)}{\partial z}\nonumber ,
\end{eqnarray}
where the dot indicates derivatives with respect to time. The
Hamiltonian {of} the potential {Equation}~(1){ is}
{written as}
\begin{equation}
H=\frac{1}{2}\left(p_x^2+p_y^2+p_z^2\right)+V_t(x,y,z)=h_3,
\end{equation}
where $p_x$, $p_y$ and $p_z$ are the momenta per unit mass
conjugate to $x$, $y$ and $z${,} respectively, while $h_3$ is the
numerical value of the Hamiltonian.
In this article, we use a system of galactic units, where the unit
of length is $1$\,kpc, the unit of mass is $2.325 \times 10^7
M_{\odot}$ and the unit of time is $0.97748 \times 10^8$\,yr.
The velocity unit is $10$\,km~s$^{-1}$, while $G$ is equal to
unity. The energy unit (per unit mass) is
$100$\,{km$^{2}$s$^{-2}$}. In the above units
we use the values: $\upsilon _0=15$, $c_b =2.5$, $\alpha =1.5$,
$b=1.8${ and} $\lambda =0.03$, while $M_h${ and} $c_h$ are treated as
parameters. Orbit calculations are based on the numerical
integration of the equations of motion{ given in} {Equation}~(4). This
was made using a Bulirsh - Stoer method in double precision and
the accuracy of the calculations was checked by the constancy of
the energy integral {Equation}~(5), which was conserved up to
the twelfth significant figure.
The article is organized as follows: In Section~2 we introduce a new
dynamical parameter{ and} present
results for the 2D system. In Section~3 we study the character of
orbits in the 3D system{ and make a} comparison with
other indicators. Finally, in
Section~4, we present a discussion and the conclusions of this
research.
\section{A new indicator: The character of motion in the 2D system}
The value of the total angular momentum for a star of mass $m=1$
moving in a 3D orbit is
\begin{equation}
L_{\rm tot}=\sqrt{L_x^2 + L_y^2 + L_z^2} ,
\end{equation}
where $L_x$, $L_y$,{ and} $L_z$ are the three components of angular
momentum along the $x$, $y$ and $z$ ax{e}s{, respectively,} given by
\begin{eqnarray}
L_x &=& y\dot{z} - \dot{y}z, \nonumber \\
L_y &=& z\dot{x} - \dot{z}x, \\
L_z &=& x\dot{y} - \dot{x}y\nonumber \, .
\end{eqnarray}
For {the} 2D system we {describe} in {Equation}~(7),
$z=\dot{z}=0$, that is $L_{\rm tot}$ reduces to $L_z$. Here, we
must note that the total angular momentum {Equation}~(6) is
conserved only for a spherical system. The same is true for all
the three components of the angular momentum. On the other hand,
in axially symmetric galactic models only the $L_z$ component of
the angular momentum is conserved. In this research, we shall use
the plot of the $L_{\rm tot}$ vs. time in order to distinguish
regular from chaotic motion.
Our next step is to study the properties of the 2D dynamical
system, which comes from potential {Equation}~(1) if we
set $z=0$. The corresponding 2D Hamiltonian {is
written as}
\begin{equation}
H_2=\frac{1}{2}\left(p_x^2+p_y^2\right)+V_t(x,y)=h_2 ,
\end{equation}
where $h_2$ is the numerical value of the Hamiltonian. We do this in
order to use the results obtained for the 2D model in the study of
the more complicated 3D model, which will be presented in the next
{s}ection.
\begin{figure}
\vs\vs \centering
\includegraphics[width=70mm]{Fig-1a.pdf}~~~
\includegraphics[width=70mm]{Fig-1b.pdf}
\includegraphics[width=70mm]{Fig-1c.pdf}~~~
\includegraphics[width=70mm]{Fig-1d.pdf}
\caption{ \baselineskip 3.6mm The $(x,p_x)$ phase plane when (a)
$M_h=0$, $h_2=516$; (b)
$M_h=10\,000$, $h_2=-226$; (c)
$M_h=20\,000$, $h_2=-1007$ and
(d)
$M_h=30\,000$, $h_2=-1788$. The values of all other parameters are
given in {the }text. }
\end{figure}
\begin{figure}
\vs\vs
\centering
\includegraphics[width=70mm]{Fig-2a.pdf}~~~
\includegraphics[width=70mm]{Fig-2b.pdf}
\includegraphics[width=70mm]{Fig-2c.pdf}~~~
\includegraphics[width=70mm]{Fig-2d.pdf}
\caption{\baselineskip 3.6mm The $(x,p_x)$ phase plane, when
$M_h=10\,000$ and (a)
$c_h=10.5$, $h_2=-135$; (b) $c_h=13$, $h_2=-55$; (c)
$c_h=15.5$, $h_2=11$ and (d)
$c_h=18$, $h_2=68$. The values of all other parameters are given
in the text.}
\end{figure}
Figure~1(a)--(d)
shows the $(x,p_x)$,
$\left(y=0,p_y>0\right)$ phase plane for four different values of
the mass of the halo. The values of all other parameters are
$\upsilon _0=15$, $c_b=2.5$, $\alpha =1.5$, $b=1.8$,
$\lambda=0.03$,{ and} $c_h=8$. The values of the energy $h_2$ were
chosen so that in all phase planes $x_{\rm max} \simeq 10$.
Figure~1(a) shows the phase plane when the system has no halo
component, that is when $M_h=0$. The value of $h_2$ is 516. One
can see that almost all {of }the phase plane is covered by a
chaotic sea. The regular regions consist of {a} small set of
islands produced by secondary resonances. Figure~1(b) shows the
phase plane when $M_h=10\,000$, while $h_2=- 226$. As we{ can}
see, the majority of the phase plane is covered by chaotic orbits.
There are also two considerable regular regions inside the chaotic
sea. These belong to invariant curves produced by
quasi-periodic orbits which are characteristic of{ a} 1:1
resonance. There are also regular regions produced by
quasi-periodic orbits{ which are} characteristic of {a} 2:3
resonance. In the outer part we can see some regular regions
produced by quasi-periodic orbits characteristic of the 2:1
resonance. Some small islands produced by secondary resonances are
also embedded in the chaotic sea. Figure~1(c) is similar to
Figure~1(a) but when $M_h=20\,000$ and $h_2=-1007$. Here, the
chaotic region is much smaller, while the majority of orbits are
regular. The most prominent characteristic of this phase plane is
the presence of many small islands produced by secondary
resonances. Figure~1(d) shows the phase plane when $M_h=30\,000${
and} $h_2=-1788$. Here we{ only} see a small chaotic layer, while
the rest of the phase plane is covered by regular orbits. The
characteristic of this phase plane is that a considerable part of
the regular orbits are box orbits. Secondary resonances are also
observed.
Therefore, our numerical results suggest that the chaotic regions
in our 2D composite galactic dynamical system described by the
Hamiltonian {Equation~}(8) strongly depend on the mass
of the halo component. The mass of the halo acts as a catalyst on
the asymmetry of the galaxy and drastically reduces the percentage
of the chaotic orbits. Thus, one can conclude that massive
spherical dark halos can act as chaos
controllers in galaxies showing small asymmetries.
\begin{figure*}
\centering
\includegraphics[width=68mm]{Fig-3a.pdf}~~
\includegraphics[width=68mm]{Fig-3b.pdf}
\caption{\baselineskip 3.6mm (a)
A plot of the area $A\%$ covered by chaotic orbits vs. $M_h$ and
(b)
A plot of the area $A\%$ covered by chaotic orbits vs. $c_h$. The
values of all {the }other parameters are given in text.}
\end{figure*}
Figure~2(a)--(d) is similar to Figure~1(a)--(d), when $M_h$ is
10\,000 while $c_h$ is treated as a parameter. All other
parameters are as{ shown} in Figure~1. Here again, the values of
the energy $h_2$ were chosen so that in all phase planes $x_{\rm
max} \simeq 10$. In Figure~2(a) we have $c_h=10.5${ and}
$h_2=-135$. Here the phase plane has a large chaotic region, while
one also observes considerable areas of regular motion. In
Figure~2(b), where the values of $c_h$ and $h_2$ are 13 and
--55{,} respectively, the chaotic sea increases, while the regular
region decreases. In the phase plane shown in Figure~2(c), we have
taken $c_h=15.5${ and} $h_2=11$. Obviously, the chaotic sea is
larger than that shown in Figure~2(b). On the other hand, the
regular region is smaller than that given in Figure~2(b). Finally,
in the results presented in Figure~2(d) we have chosen $c_h=18${
and} $h_2=68$. Here the chaotic sea is even larger, while the
regular regions are smaller than those shown in Figure~2(c).
The conclusion is that, for a given mass of the dark halo
component, the percentage of chaotic orbits increases as the scale
length of the halo increases. In other words, the numerical
experiments indicate that one would expect to observe less
chaos in asymmetric triaxial galaxies surrounded by dense halos,
while the chaotic orbits would increase in similar galaxies
surrounded {by} less dense spherical halo components.
Figure~3(a)
shows the percentage of the phase
plane $A\%$ covered by chaotic orbits as a function of the mass of
the dark halo, for two different values of $x_{\max}$. The values
of the parameters are $\upsilon_0 = 15$, $c_b = 2.5$, $\alpha =
1.5$, $b = 1.8$, $\lambda = 0.03$ and $c_h = 8$. We see that
$A\%$ decreases exponentially as $M_h$ increases. Figure~3(b)
shows a plot {of} $A\%$ and $c_h$. The values of the parameters
are $\upsilon_0 = 15$, $c_b = 2.5$, $\alpha = 1.5$, $b = 1.8$,
$\lambda = 0.03$ and $M_h = 10\,000$. Here we see that $A\%$
increases linearly as $c_h$ increases. The authors would like to
make {it }clear that $A\%$ is estimated on a completely empirical
basis by measuring the area in the $(x,p_x)$ phase plane occupied by
chaotic orbits.
Figure~4(a) and (b) show{s} a plot of the Lyapunov Characteristic
Exponent (LCE) (see \citealt{Lichtenberg1992}) vs. $M_h$ {and}
$c_h${,} respectively. The values of the parameters for
Figure~4(a) {are the same }as in Figure~3(a) and for Figure~4(b)
are {the same }as in Figure~3(b). One can see, in Figure~4(a),
that the LCE decreases exponentially {when} $M_h$ increases, while
in Figure~4(b) we see that the LCE increases exponentially {when}
$c_h$ increases. Here we must point out that it is well known that
the LCE has different values in each chaotic component (see
\citealt{Saito1979}). {Since} we have regular regions{ in all
cases} and only a large chaotic sea, we calculate the average
value of the LCE in each case by taking thirty orbits with
different initial conditions in the chaotic sea. In all cases, the
calculated values {of} the LCEs were different in the fourth
decimal point in the same chaotic sea.
\begin{figure*}
\centering
\includegraphics[width=68mm]{Fig-4a.pdf}
\includegraphics[width=68mm]{Fig-4b.pdf}
\vspace{-3mm}
\caption{\baselineskip 3.6mm (a)
A plot of the LCE vs. $M_h$ and (b)
A plot of the LCE vs. $c_h$. The values of all {the }other
parameters are given in the text.}
\vs
\centering
\includegraphics[width=66mm]{Fig-5a.pdf}
\includegraphics[width=66mm]{Fig-5b.pdf}
\includegraphics[width=66mm]{Fig-5c.pdf}
\includegraphics[width=66mm]{Fig-5d.pdf}
\vspace{-3mm}
\caption{\baselineskip 3.6mm (a)
An orbit in the 2D potential, (b)
Corresponding LCE, (c)
$P(f)$ indicator and (d)
$L_{\rm tot}$ indicator. The motion {is} chaotic. See text for
details.}
\end{figure*}
In the following, we shall investigate the regular or chaotic
character of orbits in the 2D Hamiltonian {Equation~}(8) using the
new dynamical indicator $L_{\rm tot}$. In order to see the
effectiveness of the new method, we shall compare the results with
two other indicators, the classical method of the LCE and the
method $P(f)$ used by \cite{Karanis2007}. This method uses the
Fast Fourier Transform (F.F.T{.}) of a series of time intervals,
each one representing the time that elapsed between two successive
points on the Poincar\`{e} $(x,p_x)$ phase plane for 2D systems,
while for 3D systems they take two successive points on the plane
$z=0$.
Figure~5(a) shows an orbit with initial conditions $x_0=-1.0${
and} $y_0=p_{x0}=0$, while the value of $p_y$ is always found from
the energy integral for all orbits. The values of all other
parameters and energy are the same as in Figure~1(a). One observes
in Figure~5(b) that the LCE, which was computed for a period of
$10^5$ time units, has a value of about 0.18{,} indicating chaotic
motion. The same result is shown by the $P(f)$ indicator, which is
given in Figure~5(c). Figure~5(d) shows a plot of the $L_{\rm
tot}$ vs. time for a time interval of 100 time units. We see that
the diagram is highly asymmetric. Furthermore, one observes large
deviations between the maxima and also large deviations between
the minima in the $[L_{\rm tot}, t]$ plot. The above
characteristics suggest that the corresponding orbit is chaotic.
\begin{figure
\vs \centering
\includegraphics[width=66mm]{Fig-6a.pdf}
\includegraphics[width=66mm]{Fig-6b.pdf}
\includegraphics[width=66mm]{Fig-6c.pdf}
\includegraphics[width=66mm]{Fig-6d.pdf}
\begin{minipage}[]{75mm}
\caption{ Similar {to} Fig.~5(a)--(d). The motion is
regular.}\end{minipage}
\end{figure}
Figure~6(a) shows an orbit with initial conditions $x_0=8.8$ and
$y_0=p_{x0}=0$. The values of all other parameters and energy are
{the same }as in Figure~1(b). As we{ can} see, this is a
quasi-periodic orbit. Therefore the LCE of this orbit goes to
zero, as is clearly seen in Figure~6(b). The $P(f)$ indicator in
Figure~6(c) shows a small number of peaks, also indicating regular
motion. The plot of the $L_{\rm tot}$ given in Figure~6(d) is now
quasi-periodic, with symmetric peaks, indicating regular motion.
\begin{figure}[h!!]
\centering
\includegraphics[width=67mm]{Fig-7a.pdf}~~~~
\includegraphics[width=67mm]{Fig-7b.pdf}
\includegraphics[width=67mm]{Fig-7c.pdf}~~~~
\includegraphics[width=67mm]{Fig-7d.pdf}
\begin{minipage}[]{75mm}
\caption{ Similar {to} Fig.~6(a)--(d) for a regular
orbit.}\end{minipage}
\vs\vs \centering
\includegraphics[width=67mm]{Fig-8a.pdf}~~~~
\includegraphics[width=67mm]{Fig-8b.pdf}
\includegraphics[width=67mm]{Fig-8c.pdf}~~~~
\includegraphics[width=67mm]{Fig-8d.pdf}
\begin{minipage}[]{70mm}
\caption{ Similar {to} Fig.~5(a)--(d) for a chaotic
orbit.}\end{minipage}\vs
\end{figure}
Figure~7(a)--(d) is similar to Figure~6(a)--(d) for an orbit with
initial conditions $x_0=-9.36${ and} $y_0=p_{x0}=0$, while the
values of all other parameters and energy are {the same }as in
Figure~2(a). As we{ can} see, the orbit is quasi-periodic and this
fact is indicated by all three dynamical parameters. On the
contrary, the orbit shown in Figure~8(a) has initial conditions
$x_0=10${ and} $y_0=p_{x0}=0$ {while} the values of all other
parameters and energy are {the same }as in Figure~2(d). The orbit
looks chaotic and this is indicated by the LCE, the $P(f)$ and the
$L_{\rm tot}$ shown in Figure~8(b), 8(c) and 8(d), respectively.
A large number of orbits in the 2D system were calculated for
different values of the parameters. All {of the} numerical results
suggest that the $L_{\rm tot}$ is a fast and reliable dynamical
parameter and can be safely used in order to distinguish ordered
from chaotic motion. There is no doubt that the $L_{\rm tot}$
method is much faster than the LCE, because it needs only about a
hundred time units, while the LCE needs about a hundred thousand
time units. {T}he $P(f)$ indicator needs several thousand time
units in order to give reliable results. Furthermore, it needs the
computation of the phase plane of the system, while the $L_{\rm
tot}$ {only }needs the calculations of the corresponding orbit.
\section{The character of motion in the 3D model}
We shall now proceed to investigate the regular or chaotic
behavior of the orbits in the 3D Hamiltonian {Equation}~(5).
The regular or chaotic nature of the 3D orbits is found as
follows: {W}e choose initial conditions
$\left(x_0,p_{x0},z_0\right)$, $y_0=p_{z0}=0$, such {that}
$\left(x_0,p_{x0}\right)$ is a point on the phase plane of the 2D
system. The point $\left(x_0,p_{x0}\right)$ lies inside the
limiting curve
\begin{equation}
\frac{1}{2}p_x^2+V_t(x)=h_2,
\end{equation}
which is the curve containing all the invariant curves of the 2D
system. We choose $h_3=h_2$ and the value of $p_{y0}$, for all
orbits, is obtained from the energy integral {Equation}~(5). Our
numerical experiments show that orbits with initial conditions
$\left(x_0,p_{x0},z_0\right)$, $y_0=p_{z0}=0$, such as
$\left(x_0,p_{x0}\right)$ which is a point in the chaotic regions
of Figures~1(a)--(d) and 2(a)--(d) for all permissible values of
$z_0$, produce chaotic orbits.
Our next step is to study the character of orbits with initial
conditions $\left(x_0,p_{x0},z_0\right),y_0=p_{z0}=0$, such as
$\left(x_0,p_{x0}\right)${ which} is a point in the regular
regions of Figure~1(a)--(d) and Figure~2(a)--(d). It was found
that in all cases the regular or chaotic character of the above 3D
orbits depends strongly on the initial value $z_0$. Orbits with
small{ values} of $z_0$ are regular, while for large values of
$z_0$ they change their character and become chaotic. The general
conclusion, which is based on the results derived from a large
number of orbits, is that orbits with values of $z_0 \geq 0.75$
are chaotic, while orbits with values of $z_0<0.75$ are regular.
Figure~9(a) shows the LCE of the 3D system, as a function of the
mass of{ the} halo, for a large number of chaotic orbits when
$c_h=8$. The values of the parameters are {shown} in Figure~4(a).
We see that the LCE decreases exponentially as the mass of the
halo increases. Figure~9(b) shows {the} LCE as a function of $c_h$
when $M_h=10\,000$. The values of the parameters are {shown} in
Figure~4(b). Here the LCE increases exponentially as $c_h$
increases. The above results suggest that the degree of chaos
decreases in asymmetric triaxial galaxies with more dense and more
massive halo components.
Figure~10(a)--(d) is similar to Figure~8(a)--(d) but for a 3D
orbit. The orbit shown in Figure~10(a) looks chaotic. Initial
conditions are $x_0=2.0$, $p_{x0}=0${ and} $z_0=0.5$. Remember
that all orbits have $y_0=p_{z0}=0$, while the value of $p_{y0}$
is always found from the energy integral. The values of all other
parameters and energy $h_3$ are {the same }as in Figure~2(b). The
LCE shown in Figure~10(b) assures the chaotic character of the
orbit. The $P(f)$ given in Figure~10(c) also suggests chaotic
motion. The same conclusion comes from the $L_{\rm tot}$, which is
shown in Figure~10(d). Figure~11(a)--(d) is similar to
Figure~10(a)--(d) but for a quasi-periodic 3D orbit. The initial
conditions are $x_0=5.0$, $p_{x0}=0${ and} $z_0=0.1$. The values
of all other parameters and energy $h_3$ are {the same }as in
Figure~1(c). Here one observes that all three detectors support
the regular character of orbit{s}.
\begin{figure}[h!!!]
\vs \centering
\includegraphics[width=66mm]{Fig-9a.pdf}~~~~
\includegraphics[width=66mm]{Fig-9b.pdf}
\vs
\begin{minipage}[]{130mm}
\caption{\baselineskip 3.6mm
Similar to Fig.~4(a)--(b) for the 3D
potential. The values of all other parameters are given in the
text.}\end{minipage}
\vs\vs
\centering
\vs \vs
\includegraphics[width=58mm]{Fig-10a.pdf}
\includegraphics[width=68mm]{Fig-10b.pdf}
\vs
\includegraphics[width=66mm]{Fig-10c.pdf}~~~~
\includegraphics[width=66mm]{Fig-10d.pdf}
\caption{\baselineskip 3.6mm (a)
An orbit in the 3D potential, (b)
The corresponding LCE, (c)
The $P(f)$ indicator and (d)
The $L_{\rm tot}$ indicator. The motion {is} chaotic. See {the
}text for details.}
\end{figure}
\begin{figure
\centering
\includegraphics[width=55mm]{Fig-11a.pdf}~~~~~
\includegraphics[width=66mm]{Fig-11b.pdf}
\vs\vs
\includegraphics[width=65mm]{Fig-11c.pdf}~~~~~~
\includegraphics[width=65mm]{Fig-11d.pdf}
\begin{minipage}[]{80mm}
\caption{ Similar {to} Fig.~10(a)--(d). The motion is
regular.}\end{minipage}
\vs \vs \centering
\vs
\includegraphics[width=58mm]{Fig-12a.pdf}~~~~~
\includegraphics[width=66mm]{Fig-12b.pdf}
\begin{minipage}[]{120mm}
\caption{\baselineskip 3.6mm (a)
A 3D chaotic orbit and (b)
The corresponding
$L_{\rm tot}$. See text for details.}\end{minipage}\vs\vs\vs\vs
\end{figure}
\begin{figure}[h!!]
\vs\centering
\includegraphics[width=55mm]{Fig-13a.pdf}~~~~~
\includegraphics[width=68mm]{Fig-13b.pdf}
\begin{minipage}[]{90mm}
\caption{ Similar to Fig.~12(a)--(b) for a quasi-periodic
orbit.}\end{minipage}
\end{figure}
Two more examples of 3D orbits are given in Figure~12(a) and (b)
and Figure~13(a) and (b). Here we present only the orbit and the
$L_{\rm tot}$. In Figure~12(a) and (b) the orbit has initial
conditions $x_0=-1.0$, $p_{x0}=0$ and $z_0=0.5$. The values of all
other parameters and energy $h_3$ are {shown} in Figure~2(c). The
motion is chaotic. In Figures~13(a) and (b) the orbit has initial
conditions $x_0=-9.55$, $p_{x0}=0$ and $z_0=0.1$. The values of
all other parameters and energy $h_3$ are as{ shown} in
Figure~2(d). The motion is regular.
The main conclusion for the study of the 3D model is that
the $L_{\rm tot}$ indicator can give reliable and very fast
results for the character of the orbits. There is no doubt
that the $L_{\rm tot}$ is much faster than the two
other indicators used in this research. Therefore, we can
say that this indicator is a very useful tool for a quick study of
the character of orbits in galactic potentials.
\section{Discussion and conclusions}
In this article, we have studied the regular {and} chaotic character
of motion in a 3D galactic potential. The potential describes the
motion in a triaxial elliptical galaxy with a small asymmetry
surrounded by a dark halo component. Dark halos may have a
variety of shapes (see \citealt{Ioka2000, Olling2000,
Oppenheimer2001, McLin2002, Penton2002, Steidel2002, Wechsler2002,
Papadopoulos2006, Caranicolas2010}). In this investigation we have
used a spherical dark halo component.
In order to distinguish between regular and chaotic motion, we
have introduced and used a new fast indicator{,} the $L_{\rm
tot}$. The validity of the results given by the new indicator was
checked using the LCE and an earlier method used by
\cite{Karanis2007}. We started from the 2D system and then
extended the results to the 3D potential.
The main conclusions of this research are the following:
\begin{itemize}
\item[(1)] The percentage of chaotic orbits decreases as the mass
of the spherical halo increases. Therefore, the mass of the halo
can be considered as an important physical quantity, acting as a
controller of chaos in galaxies showing small asymmetries.
\item[(2)] We expect to observe a smaller fraction of chaotic
orbits in asymmetric triaxial galaxies with a dense spherical
halo, while the fraction of chaotic orbits would increase in
asymmetric triaxial galaxies surrounded by less dense spherical
halo components.
\item[(3)] It was found that the LCE in both the 2D and the 3D
models decreases as the mass of {the }halo increases, while the
LCE increases as the scale length $c_h$ of the halo increases.
This means that not only the percentage of chaotic orbits, but
also the degree of chaos is affected by the mass or the scale
length of the spherical halo component.
\item[(4)] The $L_{\rm tot}$ gives fast and reliable results
regarding the nature of motion, both in 2D and 3D galactic
potentials. For all calculated orbits{,} the results given by the
$L_{\rm tot}$ coincide with the outcomes obtained using the LCE or
the $P(f)$ method used by \cite{Karanis2007}. The advantage of the
$L_{\rm tot}$ is that{ it} is faster than the above two methods.
\end{itemize}
\begin{acknowledgements}
Our thanks go to an anonymous referee for his useful suggestions and comments.
\end{acknowledgements}
|
2,869,038,154,878 | arxiv | \section{Introduction}
Properties of the spectrum of the Laplace-Beltrami operator on a manifold are closely related
to the properties of the underlying classical dynamical system. For
example ergodicity of the geodesic flow on the unit tangent bundle $T_1 X$ of a compact
Riemannian manifold $X$ implies quantum ergodicity.
Namely, for any complete orthonormal sequence of eigenfunctions $\phi_j \in
L^2(X)$ to the Laplace operator $\Delta$ with eigenvalues $\lambda_j \nearrow
\infty$ one has (see \cite{Shn74,Shn93,Zel87,CdV,HMR})
\begin{gather}
\lim_{N \to \infty} \frac{1}{N} \sum_{j \leq N} | \langle \phi_j,
A \phi_j \rangle - \int_{T_1^*X} \sigma_A(\xi) d \mu_L(\xi)|^2 =0,
\end{gather}
for any zero order pseudodifferential operator $A$,
where integration is with respect to the normalized Liouville measure $\mu_L$ on the
unit-cotangent bundle $T_1^* X$, and $\sigma_A$ is the principal symbol of
$A$. Quantum ergodicity is equivalent to the existence of a subsequence
$\phi_{j_k}$ of counting density one such that
\begin{gather}
\lim_{k \to \infty} \langle \phi_{j_k}, A \phi_{j_k} \rangle = \int_{T_1^*X} \sigma_A(\xi) d \mu_L(\xi).
\end{gather}
In particular, $A$ might be a smooth function on $X$ and the above implies that
the sequence
\begin{gather}
|\phi_{j_k}(x)|^2 dV_g
\end{gather}
converges to the normalized Riemannian measure $d V_g$ in the weak topology
of measures. For bundle-valued geometric operators like the Dirac
operator acting on sections of a spinor bundle or the
Laplace-Beltrami operator the corresponding Quantum ergodicity for
eigensections is known in a precise way to relate to the ergodicity
of the frame flow on the corresponding manifold \cite{JS}; see also
\cite{BG04.1,BG04.2,BO06}.
This paper deals with a situation in which the frame flow is not ergodic,
namely the case of K\"ahler manifolds. In this case the conclusions in \cite{JS}
do not hold since there is a huge symmetry algebra acting on the space of
differential forms. This algebra is the universal enveloping algebra of a
certain Lie superalgebra that is generated by the Lefschetz operator, the
complex differentials and their adjoints. On the level of harmonic forms this symmetry
is responsible for the rich structure of the cohomology of K\"ahler manifolds
and can be seen as the main ingredient for the Lefschetz theorems. Here we are
interested in eigensections with non-zero eigenvalues, that is in the spectrum
of the Laplace-Beltrami operator acting on
the orthogonal complement of the space of harmonic forms. The action of the
Lie superalgebra on the orthogonal complement of the space of harmonic forms
is much more complicated than on the space of harmonic forms where it
basically becomes the action of $sl_2(\field{C})$. In this paper we classify
all finite dimensional unitary representations of this algebra and determine
the asymptotic distribution of these representations in the eigenspaces.
Since the typical irreducible representation of the algebra decomposes into
four irreducible representation for $sl_2(\field{C})$ this shows that
eigenspaces to the Laplace-Beltrami operator have multiplicities.
An important observation in our treatment is that the universal enveloping algebra of this
Lie superalgebra is
generated by two commuting subalgebras, one of which is isomorphic to the universal
enveloping algebra of $sl_2(\field{C})$. This $sl_2(\field{C})$-action is generated
by an operator $L_t$ and its adjoint $L_t^*$ which is going to be defined in
section \ref{tlef}. This operator can be interpreted as the Lefschetz operator
in the directions of the frame bundle which are orthogonal to the frame flow.
However, $L_t$ is not an endomorphism of vector bundles, but it acts as a
pseudodifferential operator
of order zero.
Guided by this result we tackle the question of quantum ergodicity for the
Laplace-Beltrami operator on $(p,q)$-forms. Unlike in the case of ergodic
frame flow it turns out that there might be different quantum limits of
eigensections on the space of co-closed $(p,q)$-forms because of the presence
of the Lefschetz operator. Our main results establishes quantum ergodicity
for the Dirac operator and the Laplace Beltrami operator if one takes the
Lefschetz symmetry into account and under the assumption that the $U(m)$-frame
flow is ergodic. For example our analysis shows that in case of an
ergodic $U(m)$-frame flow for any
complete sequence of co-closed primitive $(p,q)$-forms there is a density
one subsequence which converges to a state which is an extension of the
Liouville measure and can be explicitly given.
For the Spin$^\field{C}$-Dirac operators we show that quantum ergodicity
does not hold for K\"ahler manifolds of complex dimension greater than one.
Thus, negatively curved
Spin-K\"ahler manifolds provide examples of manifolds with ergodic geodesic flow
where quantum ergodicity does not hold for the Dirac operator.
Our analysis shows that there are certain invariant subspaces for the Dirac operator
in this case and we prove quantum ergodicity for the Dirac operator restricted to these subspaces
provided that the $U(m)$-frame flow is ergodic.
\section{K\"ahler manifolds}
Let $(X,\omega,J)$ be a K\"ahler manifold of real dimension $n=2m$. Let $g$ be the metric,
$h=g + \,\mathrm{i}\, \omega$ be the hermitian metric, and $\omega$ the symplectic form.
As usual let $J$ be the complex structure.
A $k$-frame $(e_1,\ldots,e_k)$ for the cotangent space at some point $x \in X$
is called unitary if it is unitary with respect to the hermitian inner product
induced by $h$. Hence, a $k$-frame $(e_1,\ldots,e_k)$ is unitary iff
$(e_1,J e_1,e_2,J e_2,\ldots,e_k,J e_k)$ is orthonormal with respect to $g$.
A unitary $m$-frame at a point $x \in X$ is an ordered orthonormal basis
for $T_x^* X$ viewed as a complex vector space.
Clearly, the group $U(m)$
acts freely and transitively on the set of unitary $m$-frames. The bundle
$U_m X$ of unitary $m$-frames is therefore a $U(m)$-principal fiber bundle.
Let $T_1^* X$ be the unit cotangent bundle with bundle projection $\pi$.
Then projection onto the first vector makes $U_m X$ a principal $U(m-1)$-bundle
over $T_1^* X$.
{\footnotesize
$$
\begindc{0}[7]
\obj(10,15)[A]{$U_m X$}
\obj(40,15)[B]{$X$}
\obj(25,15)[C]{$T_1^* X$}
\mor{A}{C}{$U(m-1)$}
\mor{C}{B}{$S^{2m-1}$}
\cmor((10,13)(11,11)(15,10)(25,10)(35,10)(39,11)(40,13)) \pup(25,8){$U(m)$}
\enddc
$$
}
Transporting covectors parallel with respect to the Levi-Civita
connection extends the Hamiltonian flow on $T_1^* X$ to a flow on $U_m X$
which we call the $U(m)$-frame flow (in the literature it is also referred to
as the restricted frame flow). This is indeed a flow on $U_m X$
since $J$ is covariantly constant and therefore unitary frames are parallel
transported into unitary ones. This flow is the appropriate replacement for
the $SO(2m)$-frame flow for K\"ahler manifolds as it can be shown to be ergodic
in some cases, whereas the $SO(2m)$-frame flow never is ergodic for K\"ahler manifolds
Suppose that $X$ is a negatively-curved K\"ahler manifold
\cite{Borel}. We summarize results that can be found in \cite{Br82,
BrG, BrP}. We refer the reader to \cite{BuP, JS, Br82} and
references therein for discussion of frame flows on general
negatively-curved manifolds. Note that the frame flow is {\em not}
ergodic on negatively-curved K\"ahler manifolds, since the almost
complex structure $J$ is preserved. This is the only known example
in negative curvature when the geodesic flow is ergodic, but the
frame flow is not. In fact, given an orthonormal $k$-frame
$(e_1,\ldots,e_k)$, the functions $(e_i,J e_j), 1\leq i,j\leq k$ are
first integrals of the frame flow.
However, the following proposition was proved in \cite{BrG}:
\begin{pro}\label{erg:flow}
Let $X$ be a compact negatively-curved K\"ahler manifold of {\em
complex} dimension $m$. Then the $U(m)$-frame flow is ergodic on
$U_m X$ when $m=2$, or when $m$ is odd.
\end{pro}
\section{The Hodge Laplacian and the Lefschetz decomposition}
Let $\wedge^* X$ be the complex vector bundle $\wedge^* T_\field{C}^* X$,
where $T_\field{C}^* X$ is the complexification of the co-tangent bundle.
Then the Lefschetz
operator $L : C^\infty(X;\wedge^* X) \to C^\infty(X;\wedge^* X)$
is defined by exterior multiplication with the K\"ahler form $\omega$, i.e.
$L=\omega \wedge$. Its adjoint $L^*$ is then given by interior multiplication
with $\omega$. Is is well known that
\begin{gather}
[L^*,L]:=H=\sum_k (m-r) P_r,
\end{gather}
where $P_r$ is the orthoprojection onto $C^\infty(\wedge^r X)$,
and $L,L^*,H$ define a representation of $sl_2(\field{C})$ which commutes
with the Laplace operator $\Delta=2\Delta_{\bar \partial}=2\Delta_{\partial}$.
The decomposition into irreducible representations on the level of harmonic
forms is called the Lefschetz decomposition. We will refer to this decomposition
as the Lefschetz decomposition in general.
Note that since the Lefschetz operator commutes with $\Delta$ each eigenspace
decomposes into a direct sum of irreducible subspaces for the
$sl_2(\field{C})$ action.
The operators $L,L^*,H,\partial,\bar\partial,\partial^*,\bar\partial^*,\Delta$
satisfy the following relations (see e.g. \cite{Ballmann})
\begin{gather}
[L,\bar\partial^*]=-i \partial,\; [L^*,\partial]=i \bar \partial^*,\; \nonumber
[L^*,\bar \partial]=-i \partial^*,\; [L,\partial^*]=i \bar \partial,\\ \nonumber
[L^*,L]=H, \; [H,L]=-2 L,\; [H,L^*]=2 L^*,\\ \nonumber
\{\partial,\partial\}= \{\bar\partial,\bar\partial\}=
\{\partial^*,\partial^*\}= \{\bar\partial^*,\bar\partial^*\}=0, \\ \label{superrelations}
[L,\bar\partial]=[L,\partial]=[L^*,\bar\partial^*]=[L^*,\partial^*]=0,\\ \nonumber
\{\partial,\bar\partial\}=\{\partial,\bar\partial^*\}= \{\bar\partial,\partial^*\}=0,\\ \nonumber
\{\partial,\partial^*\}= \{\bar\partial,\bar\partial^*\}=\frac{1}{2}\Delta. \nonumber
\end{gather}
Thus, the operators form a Lie superalgebra with central element $\Delta$
(see also \cite{FrGrRe99}).
Let $\Delta^{-1}|_{\mathrm{ker} \Delta^\perp}$ the inverse of the Laplace operator
on the orthocomplement of the kernel of $\Delta$. We view this as an operator defined
in $L^2(X,\wedge^* X)$ by defining it to be zero on $\mathrm{ker}\Delta$ and write
$\Delta^{-1}$ slightly abusing notation.
\subsection{The transversal Lefschetz decomposition} \label{tlef}
The operator $Q:=2 \Delta^{-1} \bar \partial \partial$ is a partial isometry with
initial space $\mathrm{Rg}(\bar \partial^*) \cap \mathrm{Rg}(\partial^*) $ and final
space $\mathrm{Rg}(\bar \partial) \cap \mathrm{Rg}(\partial)$. Hence,
$Q^* Q$ is the orthoprojection onto $\mathrm{Rg}(\bar \partial^*) \cap \mathrm{Rg}(\partial^*) $
and $QQ^*$ is the orthoprojection onto $\mathrm{Rg}(\bar \partial) \cap
\mathrm{Rg}(\partial)$. From the above relations one gets
\begin{gather}
[L,Q]=0,\\
[L,Q^*]=2 i \Delta^{-1}(\bar \partial \bar \partial^* - \partial^* \partial),\\
[Q^*,Q]=-2 \Delta^{-1}(\bar \partial \bar \partial^* - \partial^* \partial),
\end{gather}
from which one finds that
\begin{gather}
[L-iQ,Q^*]=[L-iQ,Q]=0.
\end{gather}
We define the transversal Lefschetz operator $L_t$ by
\begin{gather}
L_t:=L-i Q.
\end{gather}
Then clearly $L_t^*=L^*+i Q^*$ and one gets that
\begin{gather}
[L_t^*,L_t]=H_t=H+[Q^*,Q],\\
[H_t,L_t]=-2 L_t, \quad [H_t,L_t^*]=-2 L_t^*,\\
[\partial,L_t]=[\partial^*,L_t]=[\bar\partial,L_t]=[\bar\partial^*,L_t]=0,
\end{gather}
and hence, also the transversal Lefschetz operators defines an action of
$\mathrm{sl}_2(\field{C})$ on $L^2(X,\wedge^* X)$. Unlike the Lefschetz
operator the transversal Lefschetz operator commutes with the holomorphic
and antiholomorphic codifferentials.
Denote by $\mathfrak{g}$ the Lie-superalgebra generated by
$a,\bar a,L,H,a^*,\bar a^*,L^*$ and relations
\begin{gather}
[L,\bar a^*]=-i a,\; [L^*,a]=i \bar a^*,\;\nonumber
[L^*,\bar a]=-i a^*,\; [L,a^*]=i \bar a,\\ \nonumber
[L^*,L]=H, \; [H,L]=-2 L,\; [H,L^*]=2 L^*,\\ \nonumber
\{a,a\}= \{\bar a,\bar a\}= \label{superrelations2}
\{a^*,a^*\}= \{\bar a^*,\bar a^*\}=0,\\
[L,\bar a]=[L,a]=[L^*,\bar a^*]=[L^*, a^*]=0,\\ \nonumber
\{\bar a,a\}=\{\bar a,a^*\}= \{a,\bar a^*\}=0,\\\nonumber
\{a,a^*\}= \{\bar a,\bar a^*\}=1.
\end{gather}
The subspace of odd elements is spanned by $a,a^*,\bar a, \bar a^*$, the
subspace of even elements is spanned by $L,L^*$ and $H$.
In the following we will denote by $\mathcal{U}(\mathfrak{g})$ the universal enveloping algebra of this
Lie-superalgebra viewed as a unital $*$-algebra, i.e. the unital $*$-algebra generated
by the symbols $\{L,L^*,H,a,a^*,\bar a,\bar a^*\}$ and the above relations.
The relations (\ref{superrelations2}) are obtained from the relations
(\ref{superrelations}) by
sending $a$ to $\sqrt{2} \Delta^{-1/2} \partial$ and $\bar a$ to $\sqrt{2}
\Delta^{-1/2} \bar \partial$. Therefore, we obtain a $*$-representation of the
Lie-superalgebra $\mathfrak{g}$
on the orthogonal complement of the kernel of $\Delta$.
\subsection{The representation theory of $\mathcal{U}(\mathfrak{g})$}
The calculations in the previous section used the relations in
$\mathcal{U}(\mathfrak{g})$ only. Hence, they remain valid if we
regard $Q=\bar a a$ and $L_t:=L-i Q$ as elements in the abstract
$*$-algebra $\mathcal{U}(\mathfrak{g})$. Hence, $L_t,L_t^*$ generate
a subalgebra in $\mathcal{U}(\mathfrak{g})$ which is canonically
isomorphic to the universal enveloping algebra of $\mathrm{sl}_2(\field{C})$
and which we therefore denote by $\mathcal{U}(\mathrm{sl}_2(\field{C}))$. Note that
$\mathcal{U}(\mathrm{sl}_2(\field{C}))$ commutes with $a,\bar a, a^*$ and $\bar a^*$.
Since
$\mathcal{U}(\mathfrak{g})$ is generated by two commuting
subalgebras the representation theory for
$\mathcal{U}(\mathfrak{g})$ is very simple. The $*$-subalgebra
$\mathcal{A}$ generated by $a$ and $\bar a$ has the following
canonical representation on $\wedge^{*} \field{C}^2 \cong
\field{C}^4$. For an orthonormal basis $\{e,\bar e\}$ of $\field{C}^2$ define
the action of $a$ by exterior multiplication by $\,\mathrm{i}\, e$, and the
action of $\bar a$ by exterior multiplication by $\,\mathrm{i}\, \bar e$. It is
easy to see that all non-trivial finite dimensional irreducible
$*$-representations of $\mathcal{A}$ are unitarily equivalent to
this representation.
Note that the equivalence classes of finite dimensional irreducible $*$-representations of
$\mathcal{U}(\mathrm{sl}_2(\field{C}))$ are labeled by the non-negative integers.
Denote the Verma-module for the Spin-$\frac{n}{2}$ representation by $V_n$
and the distinguished highest weight vector in $V_n$ by $h$.
Remember that $V_n$ is spanned by vectors of the form $L_t^k h$ with $k=0,\ldots,n$
and we have $L_t^* h=0$ and $H_t h = n h$.
Now define an action of $\mathcal{U}(\mathfrak{g})$ on
$H_n:=V_n \otimes \wedge^{*,*} \field{C}^2$ by
\begin{gather}\nonumber
L_t (v \otimes w) = (L_t v) \otimes w,\\ \nonumber
L_t^* (v \otimes w) = (L_t^* v) \otimes w,\\\nonumber
H_t (v \otimes w) = (H_t v) \otimes w,\\
a (v \otimes w)= v \otimes a w,\\\nonumber
\bar a (v \otimes w)= v \otimes \bar a w,\\\nonumber
a^* (v \otimes w)= v \otimes a^* w,\\\nonumber
\bar a^* (v \otimes w)= v \otimes \bar a^* w,\nonumber
\end{gather}
Clearly, this defines a $*$-representation
of $\mathcal{U}(\mathfrak{g})$ on $H_n$.
\begin{theorem}
The representations $H_n$ are irreducible and pairwise
inequivalent. Any non-trivial finite dimensional irreducible $*$-representation
of $\mathcal{U}(\mathfrak{g})$ is unitary equivalent to some $H_n$.
\end{theorem}
\begin{proof}
Since $\mathcal{U}(\mathfrak{g})$ is generated by two commuting subalgebras
$\mathcal{U}(\mathrm{sl}_2(\mathbb{C}))$ and $\mathcal{A}$ any
irreducible $*$-representation of is also an irreducible $*$-representation
of $\mathcal{U}(\mathrm{sl}_2(\mathbb{C})) \otimes \mathcal{A}$. If it is
finite dimensional it is therefore a tensor product of two finite dimensional
irreducible representations of $\mathcal{U}(\mathrm{sl}_2(\mathbb{C}))$ and $\mathcal{A}$.
\end{proof}
\begin{cor}
Any non trivial finite dimensional irreducible
$*$- representation of $\mathcal{U}(\mathfrak{g})$
decomposes into a direct sum of $4$ equivalent modules for the $\mathrm{sl}_2(\mathbb{C})$
action defined by $L_t,L_t^*,H_t$.
\end{cor}
If $h_n$ is a highest weight vector of $V_n$ then the kernel of $L_t^*$
in the representation $H_n$ is given by $h_n \otimes \wedge^* \field{C}^2$. Using
the unitary basis $e,\bar e$ for $\field{C}^2$ as before we see that the vectors
$$h_n \otimes 1, h_n \otimes e, h_n \otimes \bar e $$ are in the kernel of
$L^*$. Moreover,
\begin{gather}
H (h_n \otimes 1) = (n-1)(h_n \otimes 1),\\
H (h_n \otimes e) = n (h_n \otimes e),\\
H (h_n \otimes \bar e) = n (h_n \otimes \bar e).
\end{gather}
Therefore, in the decomposition of $H_n$ into irreducibles of the
$\mathrm{sl}_2(\mathbb{C})$ action defined by $L,L^*,H$ the representations
$V_n$ occur with multiplicity at least $2$ and the representation $V_{n-1}$
occurs with multiplicity at least $1$.
The vector $h_n \otimes (e \wedge \bar e)$ has weight $n+1$ and therefore,
there must be another representation of highest weight greater or equal than
$n+1$ occurring.
Since
$$
4 \dim V_{n} - 2 \dim V_n - \dim V_{n-1}=\dim V_{n+1}
$$
this shows that as a module for the $\mathrm{sl}_2(\mathbb{C})$
action defined by $L,L^*,H$ we have $H_n=V_{n+1} \oplus V_{n} \oplus V_{n} \oplus V_{n-1}$.
\begin{cor}\label{simplificator1}
Every non-trivial finite dimensional irreducible $*$- representation of $\mathcal{U}(\mathfrak{g})$
is as a module for the $\mathrm{sl}_2(\mathbb{C})$
action defined by $L,L^*,H$ unitarily equivalent to the direct sum
$V_{n+1} \oplus V_{n} \oplus V_{n} \oplus V_{n-1}$. By convention
$V_{-1}=\{0\}$.
\end{cor}
\begin{cor} \label{simplificator}
Let $V$ and $W$ be two finite dimensional $\mathcal{U}(\mathfrak{g})$ modules.
Then $V$ and $W$ are unitarily equivalent if and only if they are equivalent as modules
for the $sl_2(\mathbb{C})$ action defined by $L,L^*,H$.
\end{cor}
\subsection{The model representations}
There is another very natural representation $\rho$ of the $*$-algebra
$\mathcal{U}(\mathfrak{g})$ which is important for our purposes. This representation will be referred to
as the model representation and can be described as follows.
Let us view $\field{C}^{m} \cong \mathbb{R}^{2m}$ as a real
vector space with complex structure $J$.
Let $\{e_i\}_{i=1,\ldots,m}$ be the standard unitary basis in $\mathbb{R}^{2m}$. Then in
the complexification $\mathbb{R}^{2m} \otimes \field{C} = \mathbb{C}^{2m}$
we define
\begin{gather}
w_i= e_i - \,\mathrm{i}\, J e_i,\\
\bar w_i=e_i + \,\mathrm{i}\, J e_i.
\end{gather}
We define $\rho(L)$ to be the operator of exterior multiplication by
$\omega=\frac{\,\mathrm{i}\,}{2}\sum_{i=1}^m w_i \wedge \bar w_i$ on the space $\wedge^* \field{C}^{2m} =
\bigoplus_{p,q} \wedge^{p,q} \field{C}^{2m}$. Let $\pi(L^*)$ be its adjoint, namely the
operator of interior multiplication by $\omega$. Let $\rho(a)$
be the operator of exterior multiplication by
$\frac{\,\mathrm{i}\,}{\sqrt{2}} w_1$ and
$\rho( \bar a)$ be the operator of exterior multiplication by
$\frac{\,\mathrm{i}\,}{\sqrt{2}} \bar w_1$.
The operators $\rho(a^*)$ and $\rho(\bar a^*)$ are defined as the adjoints
of $\rho(a)$ and $\rho(\bar a)$.
This defines a representation $\rho$
of $\mathcal{U}(\mathfrak{g})$ on $\wedge^* \field{C}^{2m}$.
This representation
decomposes into a sum of irreducibles. Note that $\rho(L_t)=\rho(L-\,\mathrm{i}\, \bar a a)$ is given by
exterior multiplication by $\frac{\,\mathrm{i}\,}{2}\sum_{i=2}^m w_i \wedge \bar w_i$.
The restriction of $\rho$ to the two subalgebras generated by
$\rho(L),\rho(L^*),\rho(H)$ and $\rho(L_t),\rho(L^*_t),\rho(H_t)$ define
representations of $sl_2(\mathbb{C})$. Since the maximal eigenvalue of $H$ is $m$,
only representations of highest weight $k$ with $k \leq m$ can occur in the decomposition of the model
representation with respect to the $sl_2(\mathbb{C})$-action by $\rho(L),\rho(L^*),\rho(H)$.
Consequently, by Cor \ref{simplificator1} in the decomposition of the model representation into irreducible representations
only the representations $H_k$ with $k \leq m$ can occur.
\section{Asymptotic decomposition of Eigenspaces}
Since the action of $\mathcal{U}(\mathfrak{g})$ commutes with the Laplace
operator $\Delta$ on forms each eigenspace
$$V_{\lambda} = \{\phi \in \wedge^* X: \Delta \phi = \lambda
\phi\}$$ with $\lambda \not=0$
is a $\mathcal{U}(\mathfrak{g})$-module and can be decomposed into
a direct sum of irreducible $\mathcal{U}(\mathfrak{g})$-modules.
In the previous section we classified all irreducible $*$-representations
of $\mathcal{U}(\mathfrak{g})$ and found that they are isomorphic
to $H_k$ for some non-negative integer $k$.
Therefore, we may define the function $m_k(\lambda)$ as
\begin{gather}
m_k(\lambda):=\{\mbox{the multiplicity of} \;
H_k \; \mbox{in}\;\; V_{\lambda} \},
\end{gather}
so that
\begin{gather}
V_\lambda \cong \bigoplus_{k = 0}^{\infty} m_k(\lambda) H_k
\end{gather}
\begin{theorem}\label{main1}
Let $X$ be any compact K\"ahler manifold of complex dimension $m$.
Then in the decomposition of the eigenspaces of the Laplace-Beltrami operator $\Delta$ into irreducible
representations of $\mathcal{U}(\mathfrak{g})$
the proportion of irreducible summands of
type $H_k$ in $L^2(X; \wedge^* X)$ is in average the same as the
proportion of such irreducibles in the model representation of
$\mathcal{U}(\mathfrak{g})$ on $\wedge^* {\mathbb C}^{2m}$:
\begin{equation}\label{asymp:mult}
\frac{1}{N(\lambda)} \sum_{j: \lambda_j \leq \lambda} m_k(\lambda_j)
\sim \frac{1}{\mathrm{dim}(\wedge^*{\field{C}^{2m}})} m_k(\wedge^* {\mathbb C}^{2m}) ,
\end{equation}
where $N(\lambda) = \mathrm{tr} \Pi_{[0,\lambda]}$ and
$\Pi_{[0, \lambda]}$ is the spectral projection of the Laplace-Beltrami operator $\Delta$.
\end{theorem}
We recall that $N(\lambda) \sim \frac{rk(E) vol(X)}{(4 \pi)^m
\Gamma(m + 1)} \lambda^{m}$ for the Laplacian on a bundle $E \to X$
of rank $rk(E)$ over a manifold $X$ of real dimension $2m$.
Note that apart from the fact that we are not dealing with a group
but with a Lie superalgebra the action
is neither on $X$, nor on $T^*X$, but rather on
the total space of the vector bundle $\pi^*(\wedge^* X) \to T_1^* X$ . The action there
leaves the fibers invariant and therefore it is rather different from a group action
on the base manifold. The above theorem
thus falls outside the scope of the equivariant Weyl laws of
articles such as \cite{BH1, BH2, GU, HR, TU}. In fact its conclusion is rather different from the conclusions in these
articles as in our case only a fixed number
of types of irreducible representations may occur.
\begin{proof}
For a compact K\"ahler manifold $\mathcal{U}(\mathfrak{g})$ acts
by pseudodifferential operators on $C^\infty(X;\wedge^* X)$. Therefore, the symbol
map defines an action of $\mathcal{U}(\mathfrak{g})$ on each fiber of the bundle
$\pi^* (\wedge^* X) \to T_1^* X$. The representation of $\mathcal{U}(\mathfrak{g})$ on each
fiber is easily seen to be equivalent to the model representation.
Since the maximal eigenvalue of $H$, acting on $L^2(X;\wedge^* X)$, is $m$, only representations
of highest weight $k$ with $k \leq m$ can occur in the decomposition of
$L^2(X;\wedge^* X)$ into irreducible subspaces
with respect to the $\mathrm{sl}_2(\field{C})$-action by $L^*,L,H$. Again, by
Cor \ref{simplificator1} types $H_k$ with $k > m$
cannot occur in the decomposition with respect to the
$\mathcal{U}(\mathfrak{g})$-action.
Let $P_k$ be the orthogonal projection onto the type $H_k$ in $L^2(X;\wedge^* X)$.
Then $P_k$ is actually a pseudodifferential operator of order $0$.
Namely, the quadratic Casimir operator $\mathcal{C}$ of the
$\mathrm{sl}_2(\field{C})$-action by $L^*_t,L_t,H_t$ given by
\begin{gather}
\mathcal{C} = L^*_t L_t + L_t L_t^* + \frac{1}{2} H^2,
\end{gather}
is a pseudodifferential operator of order $0$. On a subspace
of type $H_k$ it acts like multiplication by $\frac{k^2}{2}+k$.
Therefore, if $Q$ is a real polynomial that is equal to $1$
at $\frac{k^2}{2}+k$ and equal to $0$ at $\frac{l^2}{2}+l$ for any integer $l \not=k$
between $0$ and $m$ it follows that $P_k=Q(\mathcal{C})$.
Thus, $P_k$ is a pseudodifferential operator of order $0$ and its principal
symbol at $\xi$ projects onto the subspace in the fiber $\pi^*_\xi( \wedge^* X)$ which is spanned
by the representations of type $H_k$. Therefore, for every $\xi$:
\begin{gather}
\frac{1}{\mathrm{dim}(H_k)} \mathrm{tr}(\sigma_{P_k}(\xi))=m_k(\wedge^* {\mathbb C}^{2m}).
\end{gather}
Applying Karamatas Tauberian theorem to the heat trace expansion
\begin{gather}
\mathrm{tr}(P_k e^{- t \Delta}) = (4\pi)^{-m} \mathrm{Vol}(X)
\left(\int_{T_1^* X} \mathrm{tr}(\sigma_{P_k}(\xi)) d\xi \right) t^{m} + O(t^{m-\frac{1}{2}}).
\end{gather}
gives
\begin{gather}
\frac{1}{N(\lambda)} \sum_{j: \lambda_j \leq \lambda}
\mathrm{tr}(\Pi_{[0,\lambda]} P_k) \sim m_k(\wedge^* {\mathbb C}^{2m})
\mathrm{dim}(H_k) \frac{1}{\mathrm{dim}(\wedge^*{\field{C}^{2m}})}.
\end{gather}
After dividing by $\mathrm{dim}(H_k)$ this reduces to the statement of the theorem.
\end{proof}
\begin{rem}
A natural question is whether, for generic K\"ahler metrics, the eigenspaces of
the Laplace-Beltrami operator are irreducible representations of the Lie superalgebra $\mathfrak{g}$
and of complex conjugation.
Such irreducibility is suggested by the heuristic principle of `no accidental degeneracies',
i.e. in generic cases, degeneracies of eigenspaces should be entirely due to symmetries
(see \cite{Zel90} for some results and references).
Cor. \ref{cor53} would then suggest that for a generic K\"ahler manifold the
spectrum of $\Delta$ on the space of primitive co-closed $(p,q)$-forms
should be simple for fixed $p$ and $q$.
\end{rem}
\section{Quantum ergodicity for the Laplace-Beltrami operator}
We will now investigate the question of quantum ergodicity for the Laplace-Beltrami operator
on a compact K\"ahler manifold $X$ and we keep the notations from the previous sections.
As shown in \cite{JS} this question is intimately related to the ergodic decomposition
of the tracial state on the $C^*$-algebra $C(X;\pi^*\wedge ^* X)$. The transversal Lefschetz
decomposition plays an important role here.
\subsection{Ergodic decomposition of the tracial state}
On the space of $(p,q)$-forms denote by $P_{p,q}$ the projection onto the space
of transversally-primitive forms, i.e. onto the kernel of $L_t^*$. Let $P_{p,q,k}$
be the projection onto the range of $L_t^k P_{p-k,q-k}$.
The operators
\begin{gather}
P_1=P_{\partial \bar \partial}=4 \Delta^{-2} \partial \bar \partial \bar \partial^* \partial^*=Q Q^*,\\
P_2=P_{\partial^* \bar \partial^*}=4 \Delta^{-2}\partial^* \bar \partial^* \bar \partial \partial=Q^*Q,\\
P_3=P_{\partial \bar \partial^*}=4 \Delta^{-2}\partial \bar \partial^* \bar \partial \partial^* ,\\
P_4=P_{\partial^* \bar \partial}=4 \Delta^{-2}\partial^* \bar \partial \bar \partial^* \partial
\end{gather}
are projections onto the ranges of the corresponding operators.
We have
\begin{gather}
P_H+\sum_{i=1}^4 P_i=1
\end{gather}
where $P_H$ is the finite dimensional projection onto the space of harmonic forms.
Using the transversal Lefschetz decomposition we obtain a further decomposition
\begin{gather}
\sum_{k=0}^{\mathrm{min}(p,q)} P_{p,q,k} P_H + \sum_{k=0}^{\mathrm{min}(p,q)}\sum_{i=1}^4
P_{p,q,k} P_i=1
\end{gather}
where each of the subspaces onto which $P_{p,q,k} P_i$ projects is invariant under the Laplace
operator.
Note that the principal symbols of these projections are invariant projections
in $C(T_1^*X,\pi^*\mathrm{End}(\wedge^{p,q} X))$ and the above relation gives rise to a
decomposition of the
tracial state $\omega_{tr}$ on $C(T_1^*X,\pi^*\mathrm{End}(\wedge^{p,q} X))$ defined by
\begin{gather}
\omega_{tr}(a):=\frac{1}{\mathrm{rk}(\wedge^{p,q} X)} \int_{T_1^*X} \mathrm{tr}(a(\xi)) d\xi
\end{gather}
into invariant states. Thus, the tracial state is not ergodic. However,
if the $U(m)$-frame flow is ergodic this decomposition turns out to be ergodic.
\begin{pro} \label{proerg}
Suppose that the $U(m)$-frame flow on $U_m X$ is ergodic. Let $P$ be
one of the projections
\begin{gather*}
P_{p,q,k} P_i,\\
1 \leq i \leq 4,\\
0 \leq k \leq \mathrm{min}(p,q).
\end{gather*}
Then the state $\omega_P$ on $C(T_1^*X;\pi^{*}\mathrm{End}(\wedge^{p,q} X))$
defined by $\omega_P(a):=c_P \omega_{tr}(\sigma_P a)$ is ergodic.
Here $c_P=\omega_{tr}(\sigma_P)^{-1}$.
\end{pro}
\begin{proof}
The bundle $\wedge^{p,q} X$ can be naturally identified with the associated
bundle $U_m X \times_{\hat \rho_1} \wedge^{p,q} \mathbb{C}^{2m}$, where $\hat\rho_1$ is the
representation of $U(m)$ on
$$
\wedge^{p,q} \mathbb{C}^{2m} = \wedge^p \mathbb{C}^m \otimes \wedge^q \overline{\mathbb{C}}^m.
$$
obtained from the canonical representation on $\mathbb{C}^m$.
The pull back $\pi^* \wedge^{p,q} X$ of $\wedge^{p,q} X$ can analogously be
identified with the associated bundle
\begin{gather}
U_m X \times_{\hat \rho} \wedge^{p,q} \mathbb{C}^{2m},
\end{gather}
where $\hat \rho$ is the restriction of $\hat \rho_1$ to the subgroup $U(m-1)$.
Since the first vector in $\mathbb{C}^m$ is invariant under the action of
$U(m-1)$ we have the decomposition
$$
\wedge^{p,q} \mathbb{C}^{2m} = \wedge^{p,q} \mathbb{C}^{2m-2} \oplus
\wedge^{p-1,q} \mathbb{C}^{2m-2} \oplus \wedge^{p,q-1} \mathbb{C}^{2m-2}
\oplus \wedge^{p-1,q-1} \mathbb{C}^{2m-2}
$$
into invariant subspaces. The projections onto these subspaces in each
fiber is exactly given by the principal symbols $\sigma_{P_i}$ of the
projections $P_i$. The representation of $U(m-1)$ on $\wedge^{p',q'} \mathbb{C}^{2m-2}$
may still fail to be irreducible. However, it is an easy exercise in representation theory
(c.f. \cite{MR1153249}, Exercise 15.30, p. 226) to show that the kernel of
$\sigma_{L_t^*}$ in each fiber is an irreducible representation of $U(m-1)$. Thus,
$\sigma_P$ projects onto a sub-bundle $F$ of $\pi^* \wedge^{p,q} X$ that is associated
with an irreducible representation $\rho$ of $U(m-1)$, i.e.
\begin{gather}
F \cong U_m X \times_\rho V_\rho.
\end{gather}
This identification intertwines the $U(m)$-frame flow on $U_m X$ and the flow $\beta_t$.
To show that the state $\omega_P$ is ergodic it is enough to show that
any positive $\beta_t$-invariant element $f$ in
$\sigma_P L^\infty(T_1^*X,\pi^* \mathrm{End} \wedge^{p,q} X) \sigma_P$
is proportional to $\sigma_P$ (see \cite{JS}, Appendix). Under the above
identification $f$ gets identified with a function $\hat f \in L^\infty(U_m X; V_\rho)$
which satisfies
\begin{gather}
\hat f(x g) =\rho(g) \hat f(x) \rho(g)^{-1}, \quad x \in U_m X, g \in U(m-1).
\end{gather}
If such a function is invariant under the $U(m)$-frame flow it follows
from ergodicity of the $U(m)$-frame flow
that it is constant almost everywhere. So almost everywhere $\hat f(x)= M$,
where $M$ is a matrix. By the above transformation rule $M$ commutes
with $\rho(g)$. Since $\rho$ is irreducible it follows that $M$ is a multiple
of the identity matrix. Thus, $\hat f$ is proportional to the identity
and consequently, $f$ is proportional to $\sigma_P$.
\end{proof}
Applying the abstract theory developed in \cite{MR1384146} the same argument
as in \cite{JS} can be applied to obtain
\begin{theorem}
Let $P$ be one of the projections
\begin{gather*}
P_{p,q,k} P_i,\\
1 \leq i \leq 4,\\
0 \leq k \leq \mathrm{min}(p,q).
\end{gather*}
and let $(\phi_j)$ be an orthonormal basis in $\mathrm{Rg}(P)$ with
\begin{gather}
\Delta \phi_j = \lambda_j \phi_j,\\ \nonumber
\lambda_j \nearrow \infty.
\end{gather}
If the $U(m)$-frame flow on $U_m X$is ergodic, then
quantum ergodicity holds in the sence that
\begin{gather}
\frac{1}{N} \sum_{j=1}^N |\langle \phi_j,A \phi_j \rangle - \omega_P(\sigma_A)| \to 0,
\end{gather}
for any $A \in \Psi\mathrm{DO}_{cl}^0(X,\wedge^{p,q} X)$.
\end{theorem}
Since for co-closed forms primitivity and transversal primitivity are
equivalent there is a natural gauge condition that manages without the above
heavy notation.
\begin{cor} \label{cor53}
Let $\phi_j$ be a complete sequence of primitive co-closed $(p,q)$-forms
such that
\begin{gather}
\Delta \phi_j = \lambda_j \phi_j,\\ \nonumber
\lambda_j \nearrow \infty.
\end{gather}
Then, if the $U(m)$-frame flow on $U_m X$is ergodic,
quantum ergodicity holds in the sence that
\begin{gather}
\frac{1}{N} \sum_{j=1}^N |\langle \phi_j,A \phi_j \rangle - \omega_P(\sigma_A)| \to 0,
\end{gather}
for any $A \in \Psi\mathrm{DO}_{cl}^0(X,\wedge^{p,q} X)$, where $P=P_{p,q,0} P_2$ is
the orthogonal projection onto the space of primitive co-closed $(p,q)$-forms.
\end{cor}
\section{Quantum ergodicity for Spin$^\field{C}$-Dirac operators}
In this section we consider the quantum ergodicity for Dirac type operators rather than Laplace operators.
The complex structure on K\"ahler manifolds gives rise to the so-called canonical and anti-canonical Spin$^\field{C}$-
structures. The spinor bundle of the latter can be canonically identified with the bundle $\wedge^{0,*} X$
in such a way that the Dirac operator gets identified with the so-called Dolbeault Dirac operator. Other Spin$^\field{C}$-
structures (e.g. the canonical one) can then be obtained by twisting with a holomorphic line bundle. Let us quickly describe
the construction of the twisted Dolbeault operator.
Let $L$ be a holomorphic line bundle. Then the twisted Dolbeault complex is given by
$$
\begindc{0}[7]
\obj(15,10)[A]{$\ldots$}
\obj(25,10)[B]{$\wedge^{0,k-1}X \otimes L$}
\obj(40,10)[C]{$\wedge^{0,k}X \otimes L$}
\obj(50,10)[D]{$\ldots$}
\mor{A}{B}{$\bar \partial$}
\mor{B}{C}{$\bar \partial$}
\mor{C}{D}{$\bar \partial$}
\enddc
$$
This is an elliptic complex and
the twisted Dolbeault Dirac operator is defined by
\begin{gather}
D=\sqrt{2}(\bar \partial + \bar \partial^*).
\end{gather}
As mentioned above this operator is the Dirac operator of a Spin$^{\field{C}}$-structure
on $X$ where the spinor bundle is identified with $S=\wedge^{0,*} X \otimes L$. Note that
Spin structures on $X$ are in one-one correspondence
with square roots of the canonical bundle $K=\wedge^{n,0} T X$, i.e.
with holomorphic line bundles $L$ such that $L \otimes L = K$.
In this case the Dirac operator $D$ is
exactly the twisted Dolbeault Dirac operator.
The twisted Dolbeault Dirac operator is a first order elliptic formally self-adjoint differential operator. It is therefore
self-adjoint on the domain $H^1(X;\wedge^{0,*} X \otimes L)$ of sections in the first Sobolev space. As $D$
is a first order differential operator its spectrum is unbounded from both sides.
The Dolbeault Laplace operator is given by $2(\bar \partial \bar \partial^* +
\bar \partial^* \bar \partial) = D^2$ and will be denoted by $\Delta^L$.
The Hodge decomposition is
\begin{gather}
C^\infty(X;\wedge^{0,k} X \otimes L)=\\ \nonumber
\mathrm{ker}(\Delta^L_k) \oplus
\bar \partial C^\infty(X;\wedge^{0,k-1} X \otimes L) \oplus \bar
\partial^* C^\infty(X;\wedge^{0,k+1} X \otimes L).
\end{gather}
Note that the Dirac operator leaves $\mathrm{ker}(\Delta^L_k)$
invariant since it commutes with $\Delta^L$. Moreover, $D$ maps $\bar
\partial C^\infty(X;\wedge^{0,k-1} X \otimes L)$ to $\bar
\partial^* C^\infty(X;\wedge^{0,k} X \otimes L)$ and $\bar
\partial^* C^\infty(X;\wedge^{0,k} X \otimes L)$ to $\bar \partial
C^\infty(X;\wedge^{0,k-1} X \otimes L)$. Therefore, the subspaces
\begin{gather}
\mathcal{H}^k=\bar \partial C^\infty(X;\wedge^{0,k-1} X \otimes L)
\oplus \bar \partial^* C^\infty(X;\wedge^{0,k} X \otimes L)
\end{gather}
are invariant subspaces for the Dirac operator. The orthogonal
projections $\Pi_k$ onto the closures of $\mathcal{H}^k$ are clearly zero
order pseudodifferential operators which commute with the Dirac operator.
Let $\overline{\Psi\mathrm{DO}_{cl}^0(X;S)}$ be the norm closure of the
$*$-algebra of zero order pseudodifferential operators in $\mathcal{B}(L^2(X,S))$.
Then the symbol map extends to an isomorphism
\begin{gather}
\overline{\Psi\mathrm{DO}_{cl}^0(X;S)}/\mathcal{K} \cong C(T_1^* X,\pi^* \mathrm{End}(S)).
\end{gather}
By theorem 1.4 in \cite{JS} $\overline{\Psi\mathrm{DO}_{cl}^0(X;S)}$ is invariant
under the automorphism group $\alpha_t(A):=e^{-i (\Delta^L)^{1/2} t} A e^{+i (\Delta^L)^{1/2} t}$
and the induced flow $\beta_t$ on $C(T_1^* X,\pi^* \mathrm{End}(S))$
is the extension of the geodesic flow defined by parallel translation along the fibers.
As in the analysis for the Laplace-Beltrami operator we have to consider the tracial state
\begin{gather}
\omega_{tr}(a)=\frac{1}{\mathrm{rk}(S)} \int_{T_1^*X} \mathrm{tr}(a(\xi)) d \xi,
\end{gather}
As already remarked in \cite{JS} this state
is not ergodic with respect to $\beta_t$ since it has a decomposition
\begin{gather} \label{decompspec}
\omega_{tr}(a)=\frac{1}{2} \omega_+(a) + \frac{1}{2} \omega_-(a),
\end{gather}
where
\begin{gather}
\omega_\pm(a)=\omega_{tr}((1 \pm \sigma_{\mathrm{sign}(D)}) a)
\end{gather}
On Spin$^{\field{C}}$-manifolds with ergodic frame flows the states
$\omega_\pm$ were shown in \cite{JS} to be ergodic. On K\"ahler manifolds
of complex dimension greater than one they are not ergodic since we have a further decomposition
\begin{gather}
\omega_\pm(a)=\sum_{k} \omega_\pm(\sigma_{\Pi_k} a)
\end{gather}
into invariant states.
\begin{pro}\label{pro:dirac}
Suppose that the $U(m)$-frame flow on $U_m X$ is ergodic. Then the states
$\omega_\pm^k:= c_k \omega_\pm(\sigma_{\Pi_k} a)$
are ergodic with respect to the group $\beta_t$. Here $c_k:=\omega_\pm(\sigma_{\Pi_k})^{-1}$.
\end{pro}
\begin{proof}
Let $R$ be one of the projections $\frac{1 \pm \mathrm{sign}(D)}{2}\Pi_k$ and let
$\sigma_R$ be its principal symbol. Hence, $\sigma_R$ is a projection in
\begin{gather}
C(T_1^*X,\pi^* \mathrm{End}(S)) \cong C(T_1^*X,\pi^* \mathrm{End}(\wedge^{0,*}X)).
\end{gather}
We need to show that $a \to \omega_{tr}(\sigma_R)^{-1}\omega_{tr}(\sigma_R a)$
is an ergodic state. As in the proof of Proposition \ref{proerg} this is equivalent to showing that
any positive element in $L^\infty(T_1^*X,\pi^*\mathrm{End}(\wedge^{0,k}X)) \sigma_R$
is proportional to $\sigma_R$.
A positive element in $L^\infty(T_1^*X,\pi^* \mathrm{End}(\wedge^{0,k}X)) \sigma_R$
is also in $$\sigma_R L^\infty(T_1^*X,\pi^* \mathrm{End}(\wedge^{0,k}X)) \sigma_R=
L^\infty(T_1^*X,\mathrm{End} F),$$ where $F$ is the sub-bundle of $\pi^*\wedge^{0,k}X$
onto which $\sigma_R$ projects. Since $\sigma_R$ is $\beta_t$-invariant the flow
clearly restricts to a flow on the sub-bundle $\mathrm{End}F$ of
$\pi^* \mathrm{End}(\wedge^{0,k}X)$. We will show that
under the stated assumptions an invariant element in $L^\infty(T_1^*X,\mathrm{End}F)$
is proportional to the identity in $L^\infty(T_1^*X,\mathrm{End}F)$.
Note that $\pi^* \wedge^{0,k} X$ is naturally identified with an associated
bundle
\begin{gather}
\pi^* \wedge^{0,k} X \cong U_m X \times_{\wedge^k \tilde \rho} \wedge^k
\overline \field{C}^{m},
\end{gather}
where $\tilde \rho$ is the restriction of the anticanonical representation
of $U(m)$ on $\bar \field{C}^m$ to $U(m-1)$. Here we view $U_m X$
as a $U(m-1)$-principal fiber bundle over $T_1^* X$.
Note that $\wedge^k \tilde \rho$ is not irreducible but splits into a direct
sum of two irreducible representations. This corresponds to the splitting
$\wedge^k (\bar\field{C}^{m-1} \oplus \bar\field{C})=\wedge^{k-1} \bar\field{C}^{m-1} \oplus
\wedge^{k} \bar\field{C}^{m-1}$. Under the above correspondence the projections
onto the sub-representations are exactly the principal symbols of the projections
onto $\mathrm{Rg}(\bar \partial)$ and $\mathrm{Rg}(\bar \partial^*)$. One
finds that $F$ is associated with a representation $\rho$ of $U(m-1)$
\begin{gather}
F \cong U_m X \times_{\rho} V_\rho,
\end{gather}
where
$\rho$ is equivalent to $\wedge^{k} \hat \rho$ and $\hat \rho$ is the anticanonical
representation of $U(m-1)$. Therefore, $\rho$ is irreducible.
Hence, elements in $f \in L^\infty(T_1^*X,\mathrm{End}F)$ can be identified
with functions $\hat f \in L^\infty(U_m X,\mathrm{End}V_\rho)$ that satisfy the
transformation property
\begin{gather}
\hat f(x g)=\rho(g) \hat f(x) \rho(g)^{-1}, \quad x \in U_m X, g \in U(m-1).
\end{gather}
This identification intertwines the pullback of the frame flow with
$\beta_t$. Now exactly in the same way as in the proof of Proposition
\ref{proerg} we conclude that an invariant element in $L^\infty(T_1^*X,\mathrm{End}F)$
must be a multiple of the identity. Thus, the corresponding state is ergodic.
\end{proof}
The above theorem gives rise to an ergodic decomposition of the tracial state
on the $C^*$-algebra of continuous sections of $\pi^* \mathrm{End}(S)$
which is different from the decomposition obtained from Prop. \ref{proerg}. The advantage
of this decomposition is that it is more suitable to study quantum ergodicity for the
Dirac operator. Namely, the decomposition (\ref{decompspec}) corresponds to the
splitting into negative energy and positive energy subspaces of the Dirac operator. Thus, if we
are interested in quantum limits of eigensections with positive energy we need to
decompose the state $\omega_+$ into ergodic components. This is achieved by Prop. \ref{pro:dirac}.
In the same way as in \cite{JS} one obtains
\begin{theorem}
Let $X$ be a K\"ahler manifold and let $L$ be a holomorphic line bundle.
Let $D$ be the associated $Spin^{\field{C}}$-Dirac operator and let $L^2_+(X,S)$
be the positive spectral subspace of $D$. Suppose that $(\phi_j)$ is an orthonormal
basis in $\Pi_k L^2_+(X,S)$ such that
\begin{gather}
D \phi_j = \lambda_j \phi_j,\\ \nonumber
\lambda_j \nearrow \infty.
\end{gather}
If the $U(m)$-frame flow on $U_m X$ is ergodic, then
\begin{gather}
\frac{1}{N} \sum_{j=1}^N |\langle \phi_j,A \phi_j \rangle - \omega_k(\sigma_A)| \to 0,
\end{gather}
for any $A \in \Psi\mathrm{DO}_{cl}^0(X,S)$. Here $\omega_k$ is the state on
$C(T_1^* X,\pi^* \mathrm{End}(S))$ defined by
\begin{gather}
\omega_k(a)= C \int_{T_1^*X} \mathrm{tr}
\left((1+\sigma_{\mathrm{sign}(D)}(\xi))\sigma_{\Pi_k}(\xi) a(\xi)\right) d \xi,
\end{gather}
where integration is with respect to the normalized Liouville measure
and $C$ is fixed by the requirement that $\omega_k(1)=1$.
\end{theorem}
This shows that quantum ergodicity for the Dirac operators holds only after
taking the symmetry $\Pi_k$ into account.
The states $\omega_k$ differ for different $k$. Therefore, Dirac operators on a K\"ahler manifolds
of complex dimension greater than one are never quantum ergodic in the sense of \cite{JS}.
\bigskip
|
2,869,038,154,879 | arxiv | \section{Introduction}
The goal of this paper is to generalise Toffoli's fundamental results about reversible computation.
There are several motivations for an interest in reversible computation, some related to physics and engineering, others of a more algebraic nature.
The engineering perspective notes that destroying information creates entropy and thus heat, which is bad for circuitry and wasteful\cite{engineering}.
The physics perspective notes that fundamental physical processes such as inelastic collisions and quantum processes are reversible\cite{physics}.
An algebraic perspective notes that we can say more about groups than about semigroups owing to the existence of inverses.
Toffoli \cite{toff80,tofflncs} introduced what later became known as the Toffoli gate, a simple yet universal basic element for
computation with reversible binary logic.
This paper is mostly concerned with the generalisation of this logic to multi valued logics.
Our main result is the generalisation of Toffoli gates to arbitrary arities and the demonstration that the multivalued logics of odd arity are
in some sense more powerful.
We use language based upon the function algebras of \cite{pk79}, also known as clones\cite{szendrei}.
Mal'cev \cite{malcev} introduced the concept of an iterative algebra, a generalised clone.
If an iterative algebra includes the projections $\pi_i^n : (x_1,\ldots,x_n) \mapsto x_i$, then it is a \emph{clone}.
In the next sections we will introduce some similar terminology to deal with arbitrary mappings, in particular bijections on $A^n$.
Later we will see that \emph{linear term algebras}, the reduct of iterative algebras that do not have the $\Delta$ operator, play an important role.
We introduce mappings and operations upon them, to show that there are several ways to talk about compositional closure.
In particular we show that the algebra of functions has a finite signature of finitary operations (Theorem \ref{thmRevcloneEquiv}).
We look at the ways that a given set of functions can generate a closed system of mappings, then at ways of
embedding other mappings into these closed systems.
We talk about \emph{realising} one mapping in a multiclone generated by some other mappings.
We show that all invertible mappings on a set of odd order can be
generated by four small Toffoli gates and using a definition of generation that is important from
an engineering perspective, show that these are enough for all arities.
We close by looking at the various forms of functional closure that arise in the discussion.
\section{Function Terminology}
\label{secterm}
Given a set $A$, ${\mathcal O}(A) = \{f:A^n\rightarrow A\vert n \in \mathbb N\}$ is the set of all functions on $A$.
We will use the notation of \cite{lehtonen}.
The operations of an iterative algebra are $\tau,\zeta,\Delta,\nabla,*$, defined as follows. Let $f$ be an $n$-ary function, $g$ an $m$-ary function.
The operations $\tau$ and $\zeta$ permute variables, $(\tau f)(x_1,\ldots,x_n) = f(x_2,x_1,x_3,\ldots,x_n)$
and $(\zeta f)(x_1,\ldots,x_n) = f(x_2,x_3,\ldots,x_n,x_1)$, with both being the identity on unary functions.
The operation $\Delta$ identifies variables, so $(\Delta f)(x_1,\ldots,x_{n-1}) = f(x_1,x_1,x_2,\ldots,x_{n-1})$,
while $\Delta$ is the identity on unary functions.
The operation $\nabla$ introduces a dummy variable, $(\nabla f)(x_0,x_1,\ldots,x_n) = f(x_1,x_2,\ldots,x_n)$.
Lastly we have composition, so
\[(f*g)(x_1,\ldots,x_{n+m-1}) = f(g(x_1,\ldots,x_m),x_{m+1},\ldots,x_{n+m-1}).\]
The \emph{full iterative algebra} is $({\mathcal O}(A); \tau,\zeta,\Delta,\nabla,*)$
and an iterative algebra on $A$ is a subuniverse of this.
If an iterative algebra includes the projections $\pi_i^n : (x_1,\ldots,x_n) \mapsto x_i$, then it is a \emph{clone}.
The \emph{full linear term algebra} is $({\mathcal O}(A); \tau,\zeta,\nabla,*)$
and a linear term algebra on $A$ is a subuniverse of this.
We use $S_A$ to denote the permutations of a set $A$, $S_n$ to denote the set of bijections on $\{1,\ldots,n\}$ and $S_\mathbb N = \cup_{n\in\mathbb N} S_n$.
We multiply permutations from left to right, so $(1 2 3) (1 2) = (1)(2 3)$, thus in order to
avoid difficulties, when we write permutations as functions, we introduce clarifying parentheses.
Thus if $\alpha = (1 2 3)$ and $\beta = (1 2)$ then $(\alpha \beta)(1) = 1^{\alpha \beta}= 1$ but $\alpha \circ \beta (1) = \alpha(\beta(1)) = 3$.
Let $A$ be a set. An \emph{$(n,m)$-map} is a function $f:A^n\rightarrow A^m$.
We call $n$ the \emph{arity}, $m$ the \emph{co-arity}.
We write $M_{n,m}(A) = \{f: A^n\rightarrow A^m\}$.
Then $M(A) = \bigcup_{n,m} M_{n,m}(A)$ is the set of all maps on $A$.
A \emph{function} is an $(n,1)$-map. An $(n,n)$-map, with arity equal to co-arity, will be called \emph{balanced}.
Furthermore, we define $B_{n,m}(A)$ as the set of all $(n,m)$-maps that are bijections, so for $A$ finite $n\neq m$ implies
that $B_{n,m}(A)=\emptyset$ and the bijective maps are balanced.
We will write $B_n(A)$ for the balanced bijections of arity $n$.
Similarly we write $B(A) = \bigcup_{n,m} B_{n,m}(A)$ is the set of all bijections on $A$, $B(A)=\bigcup_{n} B_{n}(A)$ if $A$ is finite.
For example the map $f(x,y) = (x+y,x-y)$ on an abelian group is a balanced $(2,2)$-map, it is a bijection if the group is of odd order.
\begin{defn}
Let $f:A^ m \rightarrow A^n$ an $(m,n)$-map,
$\theta \in\{1,\ldots,n\}^t$ without repetitions.
Define $(s(\theta,f))_j = f_{\theta_j}$, the $\theta_j$th component of $f$.
Let $i\in \{1,\dots,m\}$ and $a\in A$.
Then define
\begin{align*}
k(i,a,f)(x_1,\dots,x_{m-1}) = f(x_1,\ldots,x_{i-1},a,x_i,\ldots,x_{m-1}).
\end{align*}
We extend $k$ to tuples.
Let $\theta = \{i_1<i_2<\dots <i_r\} \subseteq \{1,\dots,m\}$, $\{1,\dots,m\} \setminus \theta = \{j_1<\dots<j_{m-r}\}$.
Define $k(\theta,a,f)(x_1,\dots,x_{m-r}) = f(y)$ with $y_i = a$ if $i\in \theta$, $y_{j_l} = x_{l}$ otherwise.
Let $a_1,\dots, a_r \in A$ and define
$k(\theta,(a_1,\dots,a_r),f)(x_1,\dots,x_{m-r}) = f(y)$ with $y_{i_r} = a_r$ if $i\in \theta$, $y_{j_l} = x_{l}$ otherwise.
We can then write $f_{\theta^\prime,\theta}^o = s(\theta,k(\theta^\prime,o,f))=s(\theta,k(\theta^\prime,(o,\ldots,o),f))$.
Then $f_{\theta^\prime,\theta}^o$ has arity $m-\vert \theta^\prime\vert$ and co-arity $\vert \theta\vert$.
\end{defn}
For example, take the map $f(x,y) = (x+y,x-y)$ on $\mathbb Z_7$,
then $f_{(2),(1)}^5 (z) = s((1),k((2),(5),f))(z) = z+5$ is a unary function on $\mathbb Z_n$.
\subsection{Operations}
In this section we look at the operations that allow us to combine mappings.
Our main result is that all these combinations can be written as a finite number of finitary operations.
\begin{defn}
\label{defops}
Let $f \in M_{n,s}(A)$, $g\in M_{m,t}(A)$ be mappings.
We define $i_1 \in {B}_1(A)$ with $i_1(x)=x$. We write $i_n \in B_n(A)$ for the identity function on $A^n$.
We place two mappings next to one another to form a wider mapping.
\begin{align*}
f\oplus g: A^{n+m} \rightarrow &A^{s+t}\\
(x_1,\ldots x_{n+m}) \mapsto (&f_1(x_1,\ldots,x_n),\ldots,f_s(x_1,\ldots,x_n),\\
&g_1(x_{n+1},\ldots,x_{n+m}),\ldots,g_t(x_{n+1},\ldots,x_{n+m}))
\end{align*}
We compose mappings.
For $k\leq n$, $k\leq t$,
\begin{align*}
\circ_k(f,g):A^{n+m-k} \rightarrow & A^{s+t-k} \\
(x_1,\ldots,x_{n+m-k}) \mapsto
&(f_1(g_1(x_{1},\ldots,x_{m}),\ldots,g_k(x_{1},\ldots,x_{m}),x_{m+1},\ldots,x_{m+n-k}),\\
& \vdots \\
& \hspace{1mm}f_s(g_1(x_{1},\ldots,x_{m}),\ldots,g_k(x_{1},\ldots,x_{m}),x_{m+1},\ldots,x_{m+n-k}),\\
& \hspace{1mm}g_{k+1}(x_{1},\ldots,x_{l}), \ldots,g_t(x_{1},\ldots,x_{m}))
\end{align*}
For $k\geq 2$, $\circ_k$ is a partial operation.
We identify variables.
\begin{align*}
&\Delta f : A^{n-1} \rightarrow A^s\\
&\Delta f(x_1,\ldots x_{n-1}) = f(x_1,x_1,\ldots,x_{n-1})
\end{align*}
and define $\Delta f = f $ if $f$ has arity 1.
We introduce dummy variables.
\begin {align*}
&\nabla f : A^{n+1} \rightarrow A^s \\
&\nabla f (x_1,\dots,x_{n+1}) = f(x_2,\dots,x_{n+1})
\end {align*}
The following operations allow us to permute the inputs and outputs of a given mapping.
\begin{align*}
\tau f (x_1,\ldots,x_n) =
(&f_1(x_2,x_1,x_3,\ldots,x_n),\\
&f_2(x_2,x_1,x_3,\ldots,x_n),\\
&\hspace{8mm}\vdots\\
&f_s(x_2,x_1,x_3,\ldots,x_n))
\end{align*}
\begin{align*}
\zeta f (x_1,\ldots,x_n) =
(&f_1(x_2,x_3,\ldots,x_n,x_1),\\
&f_2(x_2,x_3,\ldots,x_n,x_1),\\
&\hspace{8mm}\vdots\\
&f_s(x_2,x_3,\ldots,x_n,x_1))
\end{align*}
\begin{align*}
\bar\tau f (x_1,\ldots,x_n) =
(&f_2(x_1,x_2,x_3,\ldots,x_n),\\
&f_1(x_1,x_2,x_3,\ldots,x_n),\\
&\hspace{8mm}\vdots\\
&f_s(x_1,x_2,x_3,\ldots,x_n))
\end{align*}
\begin{align*}
\bar\zeta f (x_1,\ldots,x_n) =
(&f_2(x_1,x_2,x_3,\ldots,x_n),\\
&f_3(x_1,x_2,x_3,\ldots,x_n),\\
&\hspace{8mm}\vdots\\
&f_s(x_1,x_2,x_3,\ldots,x_n),\\
&f_1(x_1,x_2,x_3,\ldots,x_n)).
\end{align*}
In the case that $f$ has arity 1, $\tau f = \zeta f = f$.
In the case that $f$ has co-arity 1, $\bar\tau f = \bar\zeta f = f$.
\end{defn}
For example, in the case that $n=t$, $f \circ_n g$ is the composition that we would expect:
\begin{align*}
f \circ_n g = (&f_1(g_1(x_1,\ldots,x_m),\ldots,g_n(x_1,\ldots,x_m)),\\
&\hspace{10mm}\vdots\\
&f_s(g_1(x_1,\ldots,x_m),\ldots,g_n(x_1,\ldots,x_m)))
\end{align*}
Note that there are many relations amongst these operations.
For instance let $f$ have co-arity $n$, then $\bar\tau f = (\tau i_n) \circ_n f$ and $\bar\zeta f = (\zeta i_n) \circ_n f$.
The $s$ operator introduced above allows us to say $s((2,\dots,n+1),\nabla f) = f$.
We can look at a general form of mapping composition corresponding to clones.
\begin{defn}
A collection of mappings on $A$ is called a \emph{multiclone} if it is closed under
the operations $\{i_1,\oplus,\Delta,\nabla,\tau,\zeta\}$ and the partial operations $\{\circ_i\vert i\in \mathbb N\}$.
Let $F\subseteq M(A)$ be a collection of mappings.
Then we write $C_\Delta(F)$ for the closure under the operations,\ i.e. the \emph{multiclone generated by} $F$.
\end{defn}
However in order to focus on the realm of bijective mappings,
we are predominately interested in closure without the $\Delta$ and $\nabla$ operations.
\begin{defn}
A collection of mappings on $A$ is called a \emph{read once multiclone} if it is closed under
the operations $\{i_1,\oplus,\tau,\zeta\}$ and the partial operations $\{\circ_i\vert i\in \mathbb N\}$.
Let $F\subseteq M(A)$ be a collection of mappings.
Then we write $C(F)$ for the closure under the operations,\ i.e. the \emph{read once multiclone generated by} $F$.
\end{defn}
These operations reflect what \cite{toff80} calls a \emph{combinatorial network},
as well as corresponding to the idea of a read once function as used in in \cite{coucierolehtonen}.
In particular, every output is used at most once as an input to another mapping,
avoiding the duplication of variables or so-called fan-out,
where one data signal is spread to two or more outputs. The mapping
$\phi_n \in M_{1,n}(A)$ with $\phi_n(a)=(a,\dots,a)$ is a fan out mapping.
Also, no input or output is thrown away.
As a connection between these concepts, we have the following.
\begin{lemma}
\label{lemmfan}
Let $F$ be a read once multiclone. Then $F$ is a multiclone iff $\phi_2(x)=(x,x)$, $g(x,y)=y \in F$.
\end{lemma}
Proof:
$(\Rightarrow)$: We know that $i_2=i_1\oplus i_1\in F$. We know $F$ is a multiclone,
so $\Delta i_2 \in F$. But $\Delta i_2(x_2)=(x_2,x_2) = \phi_2(x_2)$ so $\phi_2 \in F$.
Similarly $\nabla i_1 = g$.
$(\Leftarrow)$:
Suppose $\phi_2\in F$. Let $h\in F\cap M_{n,m}$ with $n \geq 2$.
Then
\begin{align*}
(h \circ_2 \phi_2) &= h(x_1,x_1,\ldots,x_{n-1})\\
&= \Delta h(x_1,\ldots x_{n-1})
\end{align*}
so $F$ is closed under $\Delta$.
Let $h\in F\cap M_{n,m}$.
We calculate that $h \circ_1 g \in M_{n+2-1,m+1-1}=M_{n+1,m}$
\begin{align*}
(h \circ_1 g)(x_1,\dots,x_{n+1}) &= h(g_1(x_1,x_2),x_3,\dots,x_{n+1}) \\
& = h(x_2,\dots,x_{n+1} \\
& = \nabla h(x_1,\dots,x_{n+1})
\end{align*}
so $F$ is also closed under $\nabla$, thus $F$ is a multiclone.
\hfill$\Box$
If we allow tuples with repetitions in the definition of $s(I,f)$, then this result shows that
a read one multiclone $F$ will give a multiclone $S(F)$ because $s((1,1),i_1)$ and $s((2),i_2)$
are the two functions used in this lemma.
\begin{defn}
A read once multiclone is called a \emph{reversible clone} or \emph{revclone} if each element is bijective.
\end{defn}
Note that multiclones have the fan-out mapping by the previous Lemma,
which is not surjective and thus not bijective.
So no confusion can arise by the use of the phrase reversible
clone rather than reversible read once multiclone.
Note that $i_1\oplus i_1\oplus \ldots\oplus i_1=i_{n}$.
We will also use $\oplus$ to concatenate tuples,
$(x_1,\dots,x_n)\oplus(y_1,\dots,y_m) = (x_1,\dots,x_n,y_1,\dots,y_m)$.
Let $\alpha \in S_n$ be a permutation.
Then let $\pi_\alpha$ be the function
$\pi_\alpha (x_1,\ldots,x_n) = (x_{\alpha^{-1}1},\ldots, x_{\alpha^{-1}n})$.
\begin{example}
The set $\{\pi_\alpha\vert \alpha \in S_n,\,n\in \mathbb N\}$ is a reversible clone, in fact it is the minimal reversible clone.
\end{example}
There exist well known reversible clones.
\begin{example}
Let $A$ be a field. Then the collection of linear mappings on vector spaces over $A$ form a multiclone.
The collection of invertible linear mappings $\cup_{n\in \mathbb N} GL(n,A)$ forms a reversible clone, but not a multiclone.
The permutation matrices form the smallest subrevclone.
\end{example}
\begin{defn}
Let $f,g$ be mappings, $f$ have arity $n$, $g$ have co-arity $m$.
Let $k = min(m,n)$. Define $f \bullet g = f \circ_k g$.
\end{defn}
\begin{thm}\label{thmRevcloneEquiv}
Let $F$ be a collection of mappings on a set $A$. Then the following are equivalent:
\begin{enumerate}
\item $F$ is a read once multiclone.
\item $F$ is closed under $\{i_1,\oplus,\tau,\zeta\} \cup\{ \circ_k\vert k\in \mathbb N\}$.
\item $F$ is closed under $\{\oplus\}\cup\{ \pi_\alpha\vert \alpha \in S_\mathbb N\} \cup\{ \circ_k \vert k \in \mathbb N\}$.
\item $F$ is closed under $\{i_1,\oplus,\tau,\zeta,\bullet \}$.
\end{enumerate}
\end{thm}
Proof: The second case is a writing of the definition of the first.
We demonstrate that the others are equivalent by showing that each collection of operations can be built from the others as terms.
$3\Rightarrow 2:$
In this signature, $i_1$ is $\pi_\alpha$ for $\alpha$ the identity permutation on $\{1\}$.
$\oplus$ is the same, as are $\circ_k$.
Let $f \in M_{m,n}$.
We see that $\tau f = f \circ_m \pi_{(1,2)}$, the permutation $(1,2)$ acting upon $\{1,\ldots,m\}$,
and $\zeta f = f \circ_m \pi_{(m,\ldots,2,1)}$, the permutation $(m,\ldots,2,1)$ acting upon $\{1,\ldots,m\}$.
$2\Rightarrow 4:$
From the definition of $\bullet$, we know that it can be written in terms of $\circ_k$ for any given arguments.
$4\Rightarrow 3:$
Let $\alpha$ be a permutation on $\{1,\ldots,n\}$.
Let $\alpha = \alpha_1\alpha_2\ldots\alpha_k$ for $\alpha_i \in \{(1,2),(1,2,\ldots,n)\}$.
Let $\beta_i = \tau$ if $\alpha_i=(1,2)$,
otherwise $\beta_i=\zeta$.
Then $\pi_\alpha=\beta_k\bullet\ldots\bullet\beta_1\bullet (i_1\oplus \ldots \oplus i_1)$ with $i_1$ repeated $n$ times.
Let $f,g$ be mappings, $f\in M_{n,s}(A)$, $g\in M_{m,t}(A)$, $k\leq t$, $k\leq n$.
Then we claim that
\begin{align}
f\circ_k g = (f \oplus i_{m-k} ) \bullet \pi_\alpha \bullet (g \oplus i_{n-k}) \label{eqfullcomp}
\end{align}
with
\[\alpha = \pmtv[1\ldots {k}{k+1}\ldots {t}{t+1}\ldots{t+n-k}] {{k+1}{n+1} {t}{t+n-k} {t+1}{k+1} {t+n-k}{n} }.\]
First note that because of the structure of the permutation $\alpha$,
\begin{align*}
(\pi_\alpha \bullet (g \oplus i_{n-k}))_j =
\begin{cases}
g_j(x_1,\ldots,x_m) & j \leq k\\
x_{t-k+j} & k < j \leq n \\
g_{k+j-n}(x_1,\ldots,x_m) & n < j \leq t+n-k
\end{cases}
\end{align*}
We can then calculate the right hand side of $(\ref{eqfullcomp})$. For $j\leq s$, the $j$th entry is:
\begin{align*}
&f_j\bullet \pi_\alpha \bullet (g \oplus i_{n-k})(x_1,\ldots,x_{n+m-k}) = \\
& \hspace{15mm}f_j(g_1(x_1,\ldots,x_m),\ldots,g_k(x_1,\ldots,x_m),x_{m+1},\ldots,x_{m+n-k})
\end{align*}
while for $s<j\leq s+t-k$, the $j$th entry is:
\begin{align*}
( \pi_\alpha \bullet (g \oplus i_{n-k}))_{n+j-s}(x_1,\ldots,x_{n+m-k}) = g_{k+j-s}(x_1,\ldots,x_m)
\end{align*}
By comparing entries, we see that the left hand side and the right hand side agree at all entries and are thus equal.
\hfill$\Box$
This enables us to easily see that the operations of a read once multiclone do not destroy reversibility,
so we know that $B(A)$ is the largest reversible clone on $A$.
\begin{lemma}
\label{lemmaBclosed}
Let $F\subseteq B(A)$. Then $C(F) \subseteq B(A)$.
\end{lemma}
Proof:
We only need to check that $B(A)$ is a read once multiclone.
The operations $i_1,\tau,\zeta$ are invertible.
Let $f,g\in B(A)$.
The inverse of $f\oplus g $ is $f^{-1} \oplus g^{-1}$.
If the coarity of $g$ is the same as the arity of $f$, then $(f \bullet g)^{-1} = g^{-1} \bullet f^{-1}$.
If the coarity of $g$ is less than the coarity of $f$, then note that $f\bullet g = f \bullet (g\oplus i_s)$ where $s$
is the difference between the arity of $f$ and the coarity of $g$. Then the coarity of $g\oplus i_s$ equals the arity of $f$
and we are done. Similarly if the arity of $f$ is less that the coarity of $g$.
Thus all of the operations of a read once multiclone map bijections to bijections, so we are done.
\hfill$\Box$
We know that there is a finite signature for read once multiclones,
so we can apply the standard tools and techniques of universal algebra,
such as the following.
\begin{cor}
\label{corAlgLattice}
Let $A$ be a set. Then the set of read once multiclones on $A$, ordered by inclusion, is an algebraic lattice.
The set of reversible clones on $A$, ordered by inclusion, is an algebraic lattice.
\end{cor}
Proof: The set of read once multiclones on $A$ is the set of subuniverses of the algebra $(M(A); i_1,\oplus,\tau,\zeta,\bullet)$.
These operations are all finitary, so
by \cite{BS}[Cor 3.3] we have our result.
Similarly reversible clones are subuniverses of $(B(A); i_1,\oplus,\tau,\zeta,\bullet)$ and we are done.
\hfill$\Box$
We note also that the definition of a proper multiclone adds only two unary operations, so we obtain the following
as a corollary to Theorem \ref{thmRevcloneEquiv} and Corollary \ref{corAlgLattice}.
\begin{cor}
Let $F$ be a collection of mappings on a set $A$. Then the following are equivalent:
\begin{enumerate}
\item $F$ is a multiclone.
\item $F$ is closed under $\{i_1,\oplus,\tau,\zeta,\Delta,\nabla\} \cup\{ \circ_k\vert k\in \mathbb N\}$.
\item $F$ is closed under $\{\oplus,\Delta,\nabla\}\cup\{ \pi_\alpha\vert \alpha \in S_\mathbb N\} \cup\{ \circ_k \vert k \in \mathbb N\}$.
\item $F$ is closed under $\{i_1,\oplus,\tau,\zeta,\Delta,\nabla,\bullet \}$.
\end{enumerate}
The set of multiclones, ordered by inclusion, is an algebraic lattice.
\end{cor}
\section{Realisation}
\label{secrealisation}
In this section we look at the ways in which one set of mappings can be found within another.
\begin{thm}
\label{thmToffFund}
Let $A$ be a finite set.
For all $g:A^m \rightarrow A^n$ there exists some integer $r = \max(m,n+\lceil\log_{\vert A\vert} max_{a\in A^n} \vert g^ {-1} (a)\vert \rceil)$,
an invertible $f:A^{r} \rightarrow A^{r}$, some $o\in A$
and $\theta_1=(1,\ldots,m), \theta_2 = (1,\ldots,n)$ such that $g = f_{\theta_1,\theta_2}^o$.
The value of $r$ can always be found such that $\max(m,n) \leq r \leq n+m$ and these bounds are achieved.
\end{thm}
Proof:
Select $o\in A$ arbitrary but fixed.
For each $a \in A^n$, let $x^{(1)},\ldots,x^{(k)} \in A^m$ be a lexicographical ordering of $g^{-1}(a)$.
Let $b^{(1)},\ldots,b^{(k)}$ be the first $k$ elements of $A^{r-n}$ in lexicographical order.
Define $f(x^{(i)}_{1},\ldots,x^{(i)}_{m},o,\ldots,o) = (a_1,\ldots,a_n,b^{(i)}_{1},\ldots,b^{(i)}_{(r-n)})$ for each $i \in \{1,\ldots,k\}$.
Now $f$ is a partial injective map on $A^{r}$, thus it can be completed to a bijection of $A^{r}$.
For any $z \in A^m$, let $b=g(z_1,\ldots,z_m)$.
Then $(z_1,\ldots,z_m) \in g^{-1}(b)$, so
$f(z_1,\ldots,z_m,o,\ldots,o) = (b_1,\ldots,b_n,c_1,\ldots,c_m)$ for some $c_1,\ldots,c_m\in A$.
Thus
$f^o_{\theta_1,\theta_2}(z) =s(\theta_2,k(\theta_1,(o,\ldots,o),f)(z) = b = g(z)$,
showing $f_{\theta_1,\theta_2}^o = g$.
The average value of $\vert g^{-1}(a)\vert$ is $\frac{\vert A\vert^m}{\vert A\vert^n} = \vert A\vert^{m-n}$.
Thus we know that $\lceil\log_{\vert A\vert} max_{a\in A^n} \vert g^ {-1} (a)\vert\rceil \geq (m-n)$.
This value can be minimised if all $\vert g^{-1}(a)\vert$ are equal to the average value,
so $\vert g^{-1}(a)\vert = \vert A\vert^{m-n}$, so the minimum value of $r$ is $\max(m,n)$.
The maximum value is obtained if $g$ is constant, so there is some $a\in A^n$ such that $g^{-1}(a) = A^m$.
In this case,
$\log_{\vert A\vert} max_{a\in A^n} \vert g^ {-1} (a)\vert = m$ and thus $r=n+m$.
\hfill$\Box$
\begin{cor}[ \cite{toff80} Theorem 4.1]
\label{corToffFund}
Let $A$ be a finite set.
For all $g:A^m \rightarrow A^n$ there exists $r \leq n$, invertible $f:A^{m+r} \rightarrow A^{m+r}$, $o\in A$
and $\theta_1 \in \{1,\ldots,m+r\}^m$, $\theta_2 \in \{1,\ldots,m+r\}^n$ such that $g = f_{\theta_1,\theta_2}^o$.
\end{cor}
Proof:
Let $r_0$ be the value of $r$ obtained in the theorem above.
There are two possibilities for $r_0$. If $r=m$ then we take $r=0<n$ in this theorem and we are done.
If $r_0=n+\lceil\log_{\vert A\vert} max_{a\in A^n} \vert g^ {-1} (a)\vert \rceil$ then
note that $g^ {-1} (a) \subseteq A^m$ so $\vert g^ {-1} (a)\vert \leq \vert A\vert ^m $.
So $r_0 \leq n+m$ so $r \leq n$ in this theorem and we are done.
\hfill$\Box$
\subsection{Forms of Realisation}
We now look at several ideas about generation, corresponding to different forms of closure.
While the simplest definition talks about the multiclone generated from a set of mappings,
Toffoli introduces some interesting ideas about more general, engineering inspired forms of realisation.
The following is a translation of his definitions to the
language we have developed here.
\begin{defn}
Let $A$ be a set, $F\subseteq M(A)$.
Then $K(F)$ is the set of all functions we can obtain from $F$ by inserting constants.
Hence $K(F)$ the closure under all possible applications of $k$, including none.
$S(F)$ is the set of all submappings of mappings in $F$.
Then $S(F)$ is the closure under all possible applications of $s$.
$R(F)$ is the collection of bijections in $F$, i.e. $R(F) = F \cap B(A)$.
\end{defn}
\begin{defn}
Let $F$ be a collection of mappings on $A$.
A mapping $g\in M(A)$ is said to be \emph{isomorphically realised} by $F$\cite[p. 8]{toff80},
or \emph{realized without ancilla bits} \cite{yangetal}, if $g\in C(F)$.
Let $A$ be a set, $F \subseteq M(A)$ and $g \in M_{m,n}(A)$.
\begin{itemize}
\item We say $g$ is \emph{realised by} $F$ if there exists some $f\in C(F)$,
$f\in M_{l,k}(A)$ and $a_1,\ldots,a_{l-m}\in A$ such that for all $i \in \{1,\ldots,n\}$ and for all $x_1,\ldots,x_m\in A$:
\begin{align}
g_i(x_1,\ldots,x_m) = f_i(x_1,\ldots,x_m,a_1,\ldots,a_{l-m}) \label{eqrealize}
\end{align}
Equivalently, we can say $g \in SKC(F)$.
\item We say $g$ is \emph{realised with no garbage} if there exists some $f \in M_{l,n}(A)\cap C(F)$ satisfying $(\ref{eqrealize})$,
equivalently $g\in KC(F)$.
Otherwise $g$ is \emph{realised with garbage}.
\item We say $g$ is \emph{realised with no constants} if there exists some $f \in M_{m,k}(A)\cap C(F)$ satisfying $(\ref{eqrealize})$,
equivalently $g\in SC(F)$.
Otherwise $g$ is \emph{realised with constants}.
\end{itemize}
\end{defn}
A mapping is isomorphically realised iff it is realised with no constants and no garbage.
This language allows us to rephrase Corollary \ref{corToffFund} as:
\begin{cor}
For all $g\in M_{m,n}(A)$, $g$ is realised by $B_{m+n}(A)$.
\end{cor}
There is a special class of realisations that have constants and garbage,
but in some way do not use them up, the garbage representing the constants.
\begin{defn}
Let $A$ be a set, $F \subseteq M(A)$ and $g \in M_{m,n}(A)$.
Then $g$ is \emph{realised with temporary storage by} $F$ if there exists some $f\in C(F)$,
$f\in M_{l,k}(A)$ and $a_1,\ldots,a_{l-m}\in A$ such that $n+l-m=k$ and
\begin{align}
\forall i \in \{1,\ldots,n\}:&\, g_i(x_1,\ldots,x_m) = f_i(x_1,\ldots,x_m,a_1,\ldots,a_{l-m})\label{eqt1}\\
\forall i \in \{1,\ldots,l-m\}&\, \forall x_1,\ldots,x_m \in A:\nonumber\\
&\, f_{n+i}(x_1,\ldots,x_m,a_1,\ldots,a_{l-m}) = a_i \label{eqt2}
\end{align}
\end{defn}
\begin{defn}
Let $A$ be a set, $F \subseteq M(A)$ and $g \in M_{m,n}(A)$.
Then $g$ is \emph{realised with strong temporary storage by} $F$ if there exists some $f\in C(F)$, $f\in M_{l,k}(A)$
and $a_1,\ldots,a_{l-m}\in A$ such that $n+l-m=k$ and
\begin{align*}
\forall i \in \{1,\ldots,n\}:&\, g_i(x_1,\ldots,x_m) = f_i(x_1,\ldots,x_m,a_1,\ldots,a_{l-m})\\
\forall i \in \{1,\ldots,l-m\}&\, \forall x_1,\ldots,x_m\in A:\nonumber\\
&\, f_{n+i}(x_1,\ldots,x_m,a_1,\ldots,a_{l-m}) = a_i\\
\forall b_1,\ldots,b_m\in A:&\, (x_1,\ldots,x_{l-m}) \mapsto (f_{n+1}(b_1,\ldots,b_m,x_1,\ldots,x_{l-m}),\ldots\\
&\hspace{12mm}\ldots f_{k}(b_1,\ldots,b_m,x_1,\ldots,x_{l-m})) \in B_{l-m}(A)
\end{align*}
\end{defn}
Let $F\subseteq M(A)$ be some mappings on $A$.
Then let $T(F)$ be the set of mappings realised with temporary storage by $F$,
$T_S(F)$ the set of mappings realised with strong temporary storage by $F$.
These concepts are introduced because it allows us to use the garbage again in the next function, as we
know what it looks like. Thus the extra inputs and outputs are not garbage, as they are appropriately recycled and
reduce waste information. In the engineering perspective, this reduces waste heat in implementations.
Let's look at an example.
Let $A=\mathbb Z_n$, $f(x,y) = (2x+y,xy)$.
Then the map $g(x)=2x$ is realised with temporary storage by $\{f\}$, since $g(x)=f_1(x,0)$ and $f_2(x,0)=0$ for all $x$.
However $g$ is not realised with strong temporary storage, because $y\mapsto f_2(0,y) \not\in B_1(A)$.
\begin{lemma}
Let $A$ be finite, $F \subseteq B(A)$. Then $T_S(F) \subseteq T(F) \subseteq B(A)$.
\end{lemma}
Proof:
The first inclusion is clear, as the definition of strong temporary storage is stricter than the definition of temporary storage.
Let $g \in T(F)$. This means there is some $f \in C(F)$, $a_1,\dots,a_{l-m}\in A$ satisfying the requirements $(\ref{eqt1})$ and $(\ref{eqt2})$ above.
We know that $f\in B(A)$ because $F \subseteq B(A)$. Thus $f$ is balanced, so $l=k$ and thus $n=m$.
By $(\ref{eqt2})$, $f$ fixes $\{(x_1,\ldots,x_m,a_1\dots,a_{l-m}) \vert x_1,\dots,x_m\in A\}$ as a set.
Because $f$ is a bijection, $f$ is a permutation of the first $m$ entries, so $g$ is a permutation in $B(A)$.
Thus $T(F) \subseteq B(A)$ and we are done.
\hfill$\Box$
\section{The Functions of a Revclone}
In Toffoli's ``Fundamental Theorem,'' our Corollary \ref{corToffFund}, he shows that all maps and
thus all functions can be realised by the full reversible clone.
If we look at clones, this means that the clone of all functions on $A$ can be realised, i.e.\ ${\mathcal O}^A \subseteq KS(B(A))$.
Let $F \subseteq M(A)$. We call $S_1(F)=\{s((1),f)\vert f\in F\} =\{f_1 \vert f \in F\}$ the \emph{function set} of $F$.
\begin{thm}
\label{thmfunctions}
Let $F$ be a read once multiclone. Then $S_1(F)$ is a linear term algebra with projections.
\end{thm}
Proof: The operations $\tau$ and $\zeta$ act the same in the revclone and the linear term algebra.
Let $f,g\in F$, then
$\nabla (f_1) = (\bar \tau (i_1\oplus f))_1$.
Also $f_1 * g_1 = (f \circ_1 g)_1$. Thus $F$ being closed as a multiclone means that $S_1(F)$ is closed as a linear term algebra.
Moreover, the constants $\pi_\alpha \in F$ mean that all projections are also in $S_1(F)$,
$\pi^n_{\alpha^{-1}1} = (\pi_\alpha)_1$.
\hfill$\Box$
This also applies for revclones. Note that we obtain the $ \nabla$ operation ``for free'' from the other operators.
In general we see the following.
\begin{cor}
Let $F$ be a multiclone. Then $S_1(F)$ is a clone.
\end{cor}
Proof: A multiclone is a read once multiclone closed under the $\Delta$ and $\nabla$ operations on multiclones.
By Theorem \ref{thmfunctions} we have $\nabla$ operating on $S_1(F)$.
For any $f_1 \in S_1(F)$, $\Delta (f_1) = (\Delta f)_1$ and we have clone closure.
\hfill$\Box$
This justifies the use of the expression ``clone'' in our expression multiclone.
Note that the mapping $F\mapsto \{f_1 \vert f \in F\}$ is not injective.
Let $A=\{0,1,2,3,4\}$,
$C = \bigcup_n GL(n,\mathbb Z_5) \subseteq B(A)$ a reversible clone,
\begin{align*}
D=\bigcup_n \{-1,1\}SGL(n,\mathbb Z_5) = \{f\in C\vert \det f \in \{-1,1\}\}
\end{align*}
the subclone.
$D$ is a proper subrevclone of $C$, by the determinate.
Both $GL(n,\mathbb Z_5)$ and $D\cap B_n(A)$ are transitive on $\mathbb Z_5^n \setminus \{(0,0,0)\}$
so both revclones have function sets that are
$\bigcup_n \{f:A^n\rightarrow A \vert f(x_1,\ldots,x_n)=\sum a_ix_i,\, a_i \in \mathbb Z_5,\, (a_1,\ldots,a_n)\neq 0\}$,
the clone of nonzero linear forms on $\mathbb Z_5$.
\begin{cor}
Let $F$ be a revclone.
Then $f\in S_1(F)$, $f$ of arity $m$ implies that for all $ a \in A,\,\vert f^{-1}(a)\vert = \vert A\vert^{(m-1)}$.
\end{cor}
This is a corollary of Theorem \ref{thmToffFund} with $n=1$.
\section{Some results about Realisation}
\label{secgeneration}
Here we collect some results on reversible mappings.
Much of this is based upon section 5 in \cite{toff80}, with the extension to Theorem \ref{ThmOdd}.
\begin{defn}
Let $\alpha$ be a permutation of $A$, $o\in A$ some constant, $n\in \mathbb N$.
For $n>1$, let
\begin{align*}
TG(n,\alpha,o)(x_1,\ldots,x_n)_i &= x_i \hspace{10mm} i < n\\
TG(n,\alpha,o)(x_1,\ldots,x_n)_n &= \begin{cases}
\alpha(x_n) &\mbox{ if } x_1=\ldots=x_{n-1}=o \\
x_n &\mbox{ otherwise}
\end{cases}
\end{align*}
Let $TG(1,\alpha,o)(x)_1 = \alpha(x)$.
Then $TG(n,\alpha,o) \in {B}_n$ is an invertible mapping, the \emph{Toffoli Gate} induced by $n$, $\alpha$ and $o$.
\end{defn}
In \cite{toff80}, this mapping is defined for $A=\mathbb Z_2$ with $\alpha$ swapping 0 and 1, $o=1$, written as $\theta^{(n)} = TG(n,(0\, 1),1)$.
\begin{defn}
A permutation $f \in {B}_n$ is \emph{elementary} if there exist $x,y\in A^n$ such that $f(x)=y,f(y)=x$ and all other elements of $A^n$ are fixed.
An elementary permutation is \emph{atomic} if $x$ and $y$ only differ in one entry, i.e.\ $x_i=y_i$ for all $i$ except one.
\end{defn}
\begin{lemma}
Let $n\in \mathbb N$, let $o\in A$. Let $f\in {B}_n(A)$ be atomic.
Then $f$ can be isomorpically realised by $\{TG(n,\alpha,o)\vert \alpha \in S_A\} \cup \{TG(1,\alpha,o)\vert \alpha \in S_A\}$.
\end{lemma}
Proof:
Suppose $f$ exchanges $x,y$ which differ at position $i$.
Then $\pi_{(i,n)} \circ_n f \circ_n \pi_{(i,n)}$ is atomic, exchanging two vectors that differ at position $n$.
Wlog suppose $f$ is of this form.
Let $\alpha_i := (o,x_i)$ be a transposition of $A$.
Let $\beta_i = TG(1,\alpha_i,o)$ for $i<n$, $\beta_n$ the identity.
Then $\beta = \beta_1\oplus\ldots\oplus\beta_n \in {B}_n$ is an involution, generated by $TG(1,\alpha_i,o)$ as a revclone.
Note that $\beta(x)_i = \beta(y)_i = o$ for $i <n$.
Note that $f(\beta(z)) = \beta(z)$ unless $\beta(z) = x$ respectively $y$.
But $\beta(z) = x$(resp.\ $y$) iff $z_i = o$ for all $i<n$ and $z_n=x_n$ (resp.\ $y_i$).
So $\beta \bullet f \bullet \beta$ fixes all elements of $A^n$ except $(o,\ldots,o,x_n)$ and $(o,\ldots,o,y_n)$, which it exchanges.
Thus $\beta \bullet f \bullet \beta = TG(n,(x_n,y_n),o)$,
so $f$ is in the revclone generated by $\{TG(n,\alpha,o)\vert \alpha \in S_A\} \cup \{TG(1,\alpha,o)\vert \alpha \in S_A\}$.
\hfill$\Box$
\begin{thm}[\cite{toff80} Thm 5.1]
Any $f \in {B}_n(A)$ can be isomorphically realised by atomic permutations.
\end{thm}
Proof:
Any permutation can be written as a product of elementary permutations. So wlog, let $f$ be elementary,
exchanging $x$ and $y$.
Define a sequence $(a^{(i)} \vert 1\leq i \leq n+1)$ such that
for all $k\in \{1,\ldots n+1\}$, $(a^{(k)})_i = y_i$ if $i<k$, $(a^{(k)})_i = x_i$ otherwise.
Then $a^{(1)}=x$, $a^{(n+1)}=y$
Note that $a^{(i)}=a^{(i+1)}$ or they differ in position $i$.
Thus $f_i \in {B}_n(A)$ exchanging $a^{(i)}$ and $a^{(i+1)}$ is an atomic permutation.
Now the permutation $f= f_1\bullet f_2 \bullet \ldots\bullet f_{n-1} \bullet f_n \bullet f_{n-1} \bullet \ldots \bullet f_1$
is a product of atomic permutations,
so we are done.
\hfill$\Box$
\begin{cor}
\label{corrBnGen}
Let $A$ be finite, say $A=\{1,\dots,k\}$. Then ${B}_n(A)$ is realised by $\{TG(n,\alpha,1)\vert \alpha \in \{(1,2),(1,\dots,k)\}\} \cup
\{(1,2),(1,\dots,k)\}$.
\end{cor}
Proof: We remind ourselves that $TG(1,\alpha,1)=\alpha$ and thus see that the four gates listed here
generate $\{TG(n,\alpha,1)\vert \alpha \in S_A\} \cup \{TG(1,\alpha,1)\vert \alpha \in S_A\}$.
By the above Lemma, this is enough to generate all atomic permutations and thus by the above Theorem,
enough to generate $B_n(A)$.
\hfill$\Box$
This is a correct form of the result hoped for in Conjecture 1 of \cite{yangetal}.
The result that they had conjectured can be demonstrated to be false using calculations in GAP \cite{GAP4} for $A$ of order 5.
Interestingly enough, for $A$ of even order, $n\geq 2$, the conjecture might hold, as calculations in GAP for small values find no contradiction.
So we obtain the following conjecture that a smaller generating set might suffice.
\begin{conj}
For $A=\{1,\dots,k\}$ even, $n\geq 2$, \[B_n(A) = C(\{(1,\dots,k), TG(n,(1,\dots,k),1)\})\].
\end{conj}
Below we will see that a small generating set exists for $A$ of odd order.
It is worth noting that the main result in \cite{yangetal} is also subtly wrong for $n=1$, as they do not have enough permutations,
so the permutation $(0\,1)$ is not generated.
They have found an interesting property of ternary logic that any permutation of $\{0,1,2\}$ (written in $\mathbb Z_3$) can be written as $(n,n+1,n+1+1)$
or $(n,n+2,n+2+2)$, leading to their minimal generating set for $n\geq 2$.
The following results lets us calculate precisely what the $n$-ary part of a revclone looks like, given
the generators.
\begin {thm}
\label{thmFbar}
Let $F \subseteq B(A)$, $n\in \mathbb N$.
Let
\begin{align*}
\bar F = &\{ \bar f \vert f \in F,\,\mbox{ arity } f =m < n,\, \bar f = f\oplus i_{n-m}\}\\
&\cup\{f \in F\vert \mbox{ arity }f=n\}
\cup \{\pi_\alpha \vert \alpha \in S_n\}.
\end{align*}
Then the group $(\langle \bar F \rangle ; \circ_n)$ is equal to $(C(F) \cap B_n(A); \circ_n)$.
\end {thm}
Proof:
Both groups are subgroups of $(B_n(A);\circ_n)$, so we need only prove they are equal as sets.
We proceed by induction.
For the case $n=1$, $\bar F$ consists of the permutations of $A$ within $F$.
Then the subgroup of $S_A$ generated by these permutations is precisely $C(F) \cap B_1(A) = C(F) \cap S_A$.
Then we proceed with our induction.
Suppose our claim holds up to $n-1$.
$(\subseteq):$ Every element $\bar f = f \oplus i_m$ of $\bar F$ is within $C(F)$.
The arity of all elements of $\bar F$ is $n$ so they are all within $B_n(A)$. So this inclusion holds.
$(\supseteq):$
Let $f\in C(F) \cap B_n(A)$, so $f$ is either in $\bar F$ or is a term of the form:
\begin{enumerate}
\item $f = g \oplus h$ for some $g,h \in C(F)$, both of smaller arity than $n$.
\item $f = g \circ_k h$ for some $g,h \in C(F)$, both of smaller arity than $n$, $k<n$.
\item $f = g \circ_k h$ for some $g,h \in C(F)$, exactly one of them in $B_n(A)$, so the other one has arity $k$, $k<n$.
\item $f = g \circ_n h$ for some $g,h \in C(F)\cap B_n(A)$.
\end{enumerate}
In case 1, let $m$ be the arity of $g$, so $h$ has arity $n-m$.
We know from our induction hypothesis that $g$ is a product $g= \bar g_1\circ_m\ldots\circ_m\bar g_l$ of
elements of the form $\bar g_j = f_j\oplus i_{m_j}$ for some $f_j\in F$ with arity less than $m$,
$\bar g_j = f_j$ for some $f_j\in F$ with arity $m$,
or $\bar g_j = \pi_{\alpha_j}$ for some
permutation $\alpha_j\in S_m$.
Then let $\phi_j = f_j \oplus i_{m_j + n-m} = f_j \oplus i_{m_j} \oplus i_{n-m}\in \bar F$ respectively
$\phi_j = f_j \oplus i_{ n-m}\in \bar F$ respectively
$\phi_j = \pi_{\alpha_j}\oplus i_{n-m} = \pi_{\beta_j}$ where $\beta_j \in S_n$
is equal to $\alpha_j$ on $\{1,\ldots,m\}$ and
fixes the elements $\{m+1,\ldots,n\}$.
Then $\phi_1\circ_n\ldots \circ_n \phi_l = g\oplus i_{n-m}$, so $g\oplus i_{n-m} \in \langle \bar F \rangle$.
Similarly we can write $h\oplus i_m$ as an element of $\langle \bar F \rangle$.
Let $\delta$ be the permutation of $\{1,\ldots,n\}$ defined by
\[\delta = \pmtv[1\dots {m}{(m+1)}\dots {n}] {{1}{(n-m)} {m}{n} {(m+1)}{1} {n}{(n-m)} }.\]
Then
\[\pi_{\delta^{-1}} \circ_n (h\oplus i_m) \circ_n \pi_\delta \circ_n (g\oplus i_{n-m}) = g \oplus h\]
Thus we see that $f \in (\langle \bar F \rangle ; \circ_n)$.
In case 2 we proceed similarly.
In this case $g$ is $m$-ary and $h$ is $n-m+k$-ary.
We use the same techniques as above to show that $g\oplus i_{n-m}, h\oplus i_{m-k} \in \langle \bar F \rangle$.
Let $\delta$ be the permutation of $\{1,\ldots,n\}$ given by
\[\delta = \pmtv[1\ldots {k}{(k+1)}\ldots {(n-m+k)}{(n-m+k+1)}\ldots {n}] {{1}{1} {k}{k} {(k+1)}{(m+1)} {(n-m+k)}{n}{(n-m+k+1)}{(k+1)} {n}{m} }\]
so $\pi_\delta \in \bar F$.
Then
\begin{align*}
&(g\oplus i_{n-m}) \circ_n \pi_\delta \circ_n (h\oplus i_{m-k})(x_1,\ldots,x_n)\\
& = (g\oplus i_{n-m}) \circ_n \pi_\delta (h_1(x_1,\ldots,x_{n-m+k}),\ldots,h_{n-m+k}(x_1,\ldots,x_{n-m+k}),\\
& \hspace{30mm}x_{n-m+k+1},\dots,x_n)\\
& = (g\oplus i_{n-m})(h_1(x_1,\ldots,x_{n-m+k}),\ldots,h_{k}(x_1,\ldots,x_{n-m+k}),\\
& \hspace{30mm}x_{n-m+k+1},\dots,x_n,\\
& \hspace{30mm} h_{k+1}(x_1,\ldots,x_{n-m+k}),\dots,h_{n-m+k}(x_1,\ldots,x_{n-m+k})) \\
&= (g_1(h_1(x_1,\ldots,x_{n-m+k}),\ldots,h_{k}(x_1,\ldots,x_{n-m+k}),x_{n-m+k+1},\dots,x_n),\\
& \hspace {20mm} \vdots\\
& \hspace {6mm} g_m(h_1(x_1,\ldots,x_{n-m+k}),\ldots,h_{k}(x_1,\ldots,x_{n-m+k}),x_{n-m+k+1},\dots,x_n),\\
& \hspace {6mm} h_{k+1}(x_1,\ldots,x_{n-m+k}),\dots,h_{n-m+k}(x_1,\ldots,x_{n-m+k})) \\
& = g \circ_k h (x_1,\dots,x_n)
\end{align*}
Thus we see that $f \in (\langle \bar F \rangle ; \circ_n)$.
In case 3 we have $k < n$ then one of the terms is of lower arity,
so as above we can write $g \oplus i_{n-k}$ (respectively $h\oplus i_{n-k}$)
as an element of $B_n(A)$ and have $f = (g \oplus i_{n-k}) \circ_n h$ (respectively $f=g \circ_n (h\oplus i_{n-k})$).
Then we have the next case.
In case 4 we have two terms of arity $n$ but of strictly lower term complexity.
We use induction on term complexity.
For the initial case, we look at the trivial terms $t \in C(F)\cap B_n(A)$.
These are either elements of $F$ or the permutations $\pi_\alpha$ for $\alpha \in S_n$.
In both these cases, $t \in \bar F$ so $t\in \langle \bar F \rangle$.
Now we look at $f = g \circ_n h$. Both $g$ and $h$ have lower term complexity than $f$.
Thus we know that $g,h \in \langle \bar F \rangle$.
Thus $f = g \circ_n h\in \langle \bar F \rangle$ and we are done.
\hfill$\Box$
We can apply this to obtain a generalisation of one of Toffoli's results.
\begin{cor}[\cite{toff80} Thm 5.2]
\label{thmtoffeven}
Let $A$ be of even order, $n\in \mathbb N$.
Then ${B}_n(A)$ is not isomorphically realised by $\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i<n\}$ for any $o\in A$
\end{cor}
Proof:
For $n=1$ the set of generators is empty, so we only have the identity permutation.
For $n=2$ and $\vert A \vert = 2$, simple calculations show that the generated revclone is of order $8$, while
$B_2(A)$ is of order $24$.
We carry on for the other cases.
Using the terminology in Theorem \ref{thmFbar}, we note that $\overline{TG}(i,\alpha,o) = TG(i,\alpha,o) \oplus i_{n-i}$.
The action of $\tau = TG(i,\alpha,o) \oplus i_{n-i}$ on $A^n$, when not identity, is of the form
\[\tau (o,\ldots,o,a,a_1,\ldots,a_{n-i}) = (o,\ldots,o,\alpha(a),a_1,\ldots,a_{n-i}) \]
If we write $\alpha$ as a product of involutions $\alpha = \alpha_1\ldots\alpha_k$ we can use the expression above to see that there will be
$k \vert A\vert^{n-i}$ involutions when we write the action of $\tau$ on $A^n$.
Thus $\overline{TG}(i,\alpha,o) = TG(i,\alpha,o) \oplus i_{n-i}$ is an even permutation in $B_n(A)$.
The action of $\pi_\beta$ for $\beta \in S_n$ acting on $A^n$ can also be calculated.
If $\beta$ is an involution in $S_n$, say $\beta = (ij)$,
then $\pi_\beta$ acts nontrivially on $(a_1\ldots a_i \ldots a_j \ldots a_n)$ with $a_i \neq a_j$. There are
$\frac{\vert A\vert (\vert A \vert -1)}{2} (\vert A\vert )^{n-2}$ such tuple pairs. This number is
even when $\vert A\vert$ is even (except in the case $n=\vert A \vert = 2$ which we dealt with above),
From Theorem \ref{thmFbar}, we know that the arity $n$ part of $C(\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i<n\})$ is generated
by precisely these permutations, which are all even.
Thus for some $a,b\in A$, $a\neq b$, $TG(n,(a,b),a)$ is not in the generated revclone
and thus $B_n(A)$ is not isomorphically realised by $\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i<n\}$.
\hfill$\Box$
The following has been noted using examples in GAP\cite{GAP4}.
\begin{conj}
Let $A$ be of even order, $n\geq 3$.
Then $\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i<n, o\in A\}$ generates a subgroup of $B_n$ isomorphic to
the alternating group on $A^n$, except for $\vert A \vert = 2$, $n=3$.
\end{conj}
The restriction shown in Corollary \ref{thmtoffeven} does not hold for $A$ odd, as we see in the following result.
Moreover, it shows that we only need use the Toffoli gates of arity 1 and 2 to obtain all bijections on $A$.
\begin{thm}
\label{ThmOdd}
Let $A $ be of odd order.
Then $B(A)$ is isomorphically realised by
\[\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i<3\} = S_A \cup \{TG(2,\alpha,o) \vert \alpha \in S_A\}\]
for any $o\in A$.
\end{thm}
Proof:
Let $\vert A \vert = k$, wlog $A=\{1,\ldots,k\}$, $o=1$.
We proceed by induction on $n$. Our start is for $n=2$, given by the hypothesis.
We assume we have shown the claim up to $n$.
We first show that we can obtain $TG(n+1,(1 2),o)$ from $\{TG(i,\alpha,o) \vert \alpha \in S_A,\, i\leq n\}$.
Let $\gamma =(n\; n+1)$ acting on $\{1,\ldots,n+1\}$.
Define
\begin{align*}
\Sigma_1 = (TG&(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \pi_{\gamma} \bullet (TG(n,(1 2),1) \oplus i_1)\\
&\bullet \pi_{\gamma} \bullet (i_{n-1} \oplus TG(2,(1 2),1) )\\
&\bullet (TG(n,(1\ldots k),1) \oplus i_1)
\end{align*}
We calculate.
\begin{align*}
&\Sigma_1(x_1,\ldots,x_{n+1}) \\
= & \begin{cases}
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\ldots \bullet (i_{n-1} \oplus TG(2,(1 2),1) )(x_1,\ldots,x_{n}^{(1\ldots k)},x_{n+1})\\ & \mbox{ if } x_1=\ldots=x_{n-1}=1 \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\dots \bullet (i_{n-1} \oplus TG(2,(1 2),1) )(x_1,\ldots,x_{n},x_{n+1})\\ & \mbox{ otherwise}
\end{cases}\\
= & \begin{cases}
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots& \bullet \pi_{\gamma}(x_1,\ldots,x_{n}^{(1\ldots k)},x_{n+1}^{(12)})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots& \bullet \pi_{\gamma}(x_1,\ldots,x_{n}^{(1\ldots k)},x_{n+1})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots& \bullet \pi_{\gamma}(x_1,\ldots,x_{n},x_{n+1}^{(12)})\\ & \hspace{8mm}\mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n=1\\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots& \bullet \pi_{\gamma}(x_1,\ldots,x_{n},x_{n+1})\\ & \hspace{8mm}\mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n\neq 1
\end{cases} \\
= &\begin{cases}
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots &\bullet (TG(n,(1 2),1) \oplus i_1)(x_1,\ldots,x_{n-1},x_{n+1}^{(12)},x_{n}^{(1\ldots k)})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots &\bullet (TG(n,(1 2),1) \oplus i_1)(x_1,\ldots,x_{n-1},x_{n+1},x_{n}^{(1\ldots k)})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots &\bullet (TG(n,(1 2),1) \oplus i_1)(x_1,\ldots,x_{n-1},x_{n+1}^{(12)},x_{n})\\ & \hspace{8mm}\mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n=1\\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet \dots &\bullet (TG(n,(1 2),1) \oplus i_1)(x_1,\ldots,x_{n-1},x_{n+1},x_{n})\\ &\hspace{8mm} \mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n\neq 1
\end{cases}\\
\end{align*}
\begin{align*}
= &\begin{cases}
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\pi_{\gamma}(x_1,\ldots,x_{n-1},x_{n+1}^{(12)(12)},x_{n}^{(1\ldots k)})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1,x_n=k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\pi_{\gamma}(x_1,\ldots,x_{n-1},x_{n+1}^{(12)},x_{n}^{(1\ldots k)})\\ & \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\pi_{\gamma}(x_1,\ldots,x_{n-1},x_{n+1}^{(12)},x_{n})\\ & \hspace{8mm}\mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n=1\\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) \bullet &\pi_{\gamma}(x_1,\ldots,x_{n-1},x_{n+1},x_{n})\\ & \hspace{8mm}\mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n\neq 1
\end{cases}\\
= &\begin{cases}
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) &(x_1,\ldots,x_{n-1},x_{n}^{(1\ldots k)},x_{n+1}) \\ & \hspace{8mm} \mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) &(x_1,\ldots,x_{n-1},x_{n}^{(1\ldots k)},x_{n+1}^{(12)}) \\ & \hspace{8mm} \mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq k \\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) &(x_1,\ldots,x_{n-1},x_{n},x_{n+1}^{(12)}) \\ & \hspace{8mm} \mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n=1\\
(TG(n,(1\ldots k)^{-1},1) \oplus i_1) &(x_1,\ldots,x_{n-1},x_{n},x_{n+1}) \\ & \hspace{8mm} \mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n\neq 1
\end{cases}\\
= &\begin{cases}
(x_1,\ldots,x_{n-1},x_{n},x_{n+1}) & \mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=k \\
(x_1,\ldots,x_{n-1},x_{n},x_{n+1}^{(12)}) & \mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq k \\
(x_1,\ldots,x_{n-1},x_{n},x_{n+1}^{(12)}) & \mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n=1\\
(x_1,\ldots,x_{n-1},x_{n},x_{n+1}) & \mbox{ if some } x_1,\ldots,x_{n-1} \neq 1, x_n\neq 1
\end{cases}
\end{align*}
We see that $\Sigma_1$ is nonidentity iff:
\begin{enumerate}
\item Not all of $x_1\ldots,x_{n-1}$ are 1 and $x_n=1$, then $x_{n+1} \mapsto x_{n+1}^{(12)}$, or
\item $x_1=\ldots=x_{n-1}=1$ and $x_n\neq k$, then $x_{n+1} \mapsto x_{n+1}^{(12)}$
\end{enumerate}
For $m \in \{2,\ldots,k-1\}$, let
\begin{align*}
\sigma_m = (TG(n,(1 m),1) \oplus i_1) \bullet (i_{n-1} \oplus TG(2,(1 2),1)) \bullet(TG(n,(1 m),1) \oplus i_1)
\end{align*}
We calculate
\begin{align*}
&\sigma_m(x_1,\ldots,x_{n+1}) \\
&= \begin {cases}
(TG(n,(1 m),1) \oplus i_1) \bullet (i_{n-1} \oplus &TG(2,(1 2),1))(x_1,\ldots,x_n^{(1m)},x_{n+1}) \\& \hspace{8mm}\mbox{ if } x_1=\ldots=x_{n-1}=1 \\
(TG(n,(1 m),1) \oplus i_1) \bullet (i_{n-1} \oplus &TG(2,(1 2),1))(x_1,\ldots,x_n,x_{n+1}) \\& \hspace{8mm}\mbox{ if some } x_1,\ldots, x_{n-1}\neq 1
\end {cases} \\
&= \begin {cases}
(TG(n,(1 m),1) \oplus i_1) (x_1,\ldots,&x_n^{(1m)},x_{n+1}^{(12)}) \\&\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=m\\
(TG(n,(1 m),1) \oplus i_1) (x_1,\ldots,&x_n^{(1m)},x_{n+1}) \\&\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq m\\
(TG(n,(1 m),1) \oplus i_1) (x_1,\ldots,&x_n,x_{n+1}^{(12)}) \\&\mbox{ if some } x_1,\ldots, x_{n-1}\neq 1, x_n=1\\
(TG(n,(1 m),1) \oplus i_1) (x_1,\ldots,&x_n,x_{n+1}) \\&\mbox{ if some } x_1,\ldots, x_{n-1}\neq 1, x_{n} \neq 1
\end {cases} \\
& = \begin {cases}
(x_1,\ldots,x_n,x_{n+1}^{(12)}) &\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n=m\\
(x_1,\ldots,x_n,x_{n+1}) &\mbox{ if } x_1=\ldots=x_{n-1}=1, x_n\neq m\\
(x_1,\ldots,x_n,x_{n+1}^{(12)}) &\mbox{ if some } x_1,\ldots, x_{n-1}\neq 1, x_n=1\\
(x_1,\ldots,x_n,x_{n+1}) &\mbox{ if some } x_1,\ldots, x_{n-1}\neq 1, x_{n} \neq 1
\end {cases}
\end{align*}
Once again this function is almost always identity, $\sigma_m$ is nonidentity iff:
\begin{enumerate}
\item Not all of $x_1\ldots,x_{n-1}$ are 1 and $x_n=1$, then $x_{n+1} \mapsto x_{n+1}^{(12)}$, or
\item $x_1=\ldots=x_{n-1}=1$ and $x_n= m$, then $x_{n+1} \mapsto x_{n+1}^{(12)}$.
\end{enumerate}
Then define
\begin{align*}
\Sigma_2 = \sigma_{k-1} \bullet \dots \bullet\sigma_{2}
\end{align*}
We have then that $\Sigma_2$ is nonidentity iff
\begin{enumerate}
\item Not all of $x_1,\ldots,x_{n-1}$ are 1 and $x_n=1$, then $x_{n+1} \mapsto x_{n+1}^{(12)^{k-2}}=x_{n+1}^{(12)}$ because $k$ and thus $k-2$ are odd, or
\item $x_1=\ldots=x_{n-1}=1$ and $x_n \in \{2,\ldots,k-1\}$, then $x_{n+1} \mapsto x_{n+1}^{(12)}$.
\end{enumerate}
We see that $\Sigma_2 \bullet \Sigma_1$ is identity unless one of the factors is nonidentity. There are three cases:
\begin{enumerate}
\item Both are nonidentity by their first case. Then not all of $x_1,\ldots,x_{n-1}$ are 1 and $x_n=1$, so
\begin{align*}
\Sigma_2 \bullet \Sigma_1 (x_1\ldots x_{n+1})
& = \Sigma_2(x_1,\ldots,x_{n+1}^{(12)}) \\
& = (x_1,\ldots,x_{n+1}^{(12)(12)}) \\
& = (x_1,\ldots,x_{n+1})
\end{align*}
so $\Sigma_2 \bullet \Sigma_1$ is the identity.
\item Both are nonidentity by their second case, so $x_1=\ldots=x_{n-1}=1$ and $x_n \in \{2,\ldots,k-1\}$,
\begin{align*}
\Sigma_2 \bullet \Sigma_1 (x_1\ldots x_{n+1})
& = \Sigma_2(x_1,\ldots,x_{n+1}^{(12)}) \\
& = (x_1,\ldots,x_{n+1}^{(12)(12)}) \\
& = (x_1,\ldots,x_{n+1})
\end{align*}
so $\Sigma_2 \bullet \Sigma_1$ is the identity.
\item Only $\Sigma_1$ is nonidentity by the second case, so $x_1=\ldots=x_{n}=1$, then
\begin{align*}
\Sigma_2 \bullet \Sigma_1 (x_1\ldots x_{n+1})
& = \Sigma_2(x_1,\ldots,x_{n+1}^{(12)}) \\
& = (x_1,\ldots,x_{n+1}^{(12)}) \\
& = (x_1,\ldots,x_{n+1}^{(12)})
\end{align*}
so we have the only case that $\Sigma_2 \bullet\Sigma_1$ is nonidentity.
\end{enumerate}
Thus we see that $\Sigma_2 \bullet\Sigma_1 = TG(n+1,(1 2),1)$.
We now look at $TG(n+1,(1 \ldots k),1)$.
Because the group generated by $(1 \ldots k)$ is cyclic of odd order, the homomorphism $x\mapsto x^2$ of this group is an automorphism.
Thus there exists some $\beta$ such that $\beta^2 = (1 \ldots k)$. This will also be a $k$-cycle, write $\beta = (\beta_1 \ldots \beta_k)$.
Let $k=2l+1$.
Define $\alpha = (\beta_1 \beta_k)(\beta_2 \beta_{k-1}) \ldots (\beta_l \beta_{l+2})$.
Then $\alpha \beta \alpha = \beta^{-1}$, so $\beta\alpha\beta^{-1}\alpha=\beta^2=(1\ldots k)$.
Let $\gamma = (n\,n+1)$ as a permutation on $\{1,\ldots,n+1\}$. Now define
\begin{align*}
\begin{split}
\Sigma = \pi_\gamma \bullet (TG(n,\alpha,1) \oplus i_1)\bullet\pi_\gamma \bullet (i_{n-1} \oplus TG(2,\beta^{-1},1))
\bullet\pi_\gamma \hspace{10mm}\\
\bullet(TG(n,\alpha,1) \oplus i_1)\bullet\pi_\gamma\bullet (i_{n-1} \oplus TG(2,\beta,1))
\end{split}
\end{align*}
We calculate.
\begin{align*}
&\Sigma(x_1,\ldots,x_{n+1}) \\
&= \begin{cases}
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}^\beta) & x_n=1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}) & x_n\neq 1 \\
\end{cases} \\
&= \begin{cases}
\pi_\gamma \bullet \dots \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1}^\beta,x_n) & x_n=1 \\
\pi_\gamma \bullet \dots \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1},x_n) & x_n\neq 1
\end{cases} \\
&= \begin{cases}
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_{n+1}^{\beta\alpha},x_n) & x_1=\ldots=x_n=1 \\
\pi_\gamma \bullet \dots \bullet\pi_\gamma (x_1,\ldots,x_{n+1}^\beta,x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_{n+1}^\alpha,x_n) & x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_{n+1},x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases} \\
&= \begin{cases}
\pi_\gamma \bullet \dots \bullet (i_{n-1} \oplus TG(2,\beta^{-1},1)) &(x_1,\ldots,x_n,x_{n+1}^{\beta\alpha}) \\& x_1=\ldots=x_n=1 \\
\pi_\gamma \bullet \dots \bullet (i_{n-1} \oplus TG(2,\beta^{-1},1)) &(x_1,\ldots,x_n,x_{n+1}^\beta) \\& \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
\pi_\gamma \bullet \dots \bullet (i_{n-1} \oplus TG(2,\beta^{-1},1)) &(x_1,\ldots,x_n,x_{n+1}^\alpha) \\& x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
\pi_\gamma \bullet \dots \bullet (i_{n-1} \oplus TG(2,\beta^{-1},1)) &(x_1,\ldots,x_n,x_{n+1}) \\& \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases}\\
&= \begin{cases}
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}^{\beta\alpha\beta^{-1}}) & x_1=\ldots=x_n=1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}^{\beta\beta^{-1}}) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}^\alpha) & x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
\pi_\gamma \bullet \dots \bullet \pi_\gamma (x_1,\ldots,x_n,x_{n+1}) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases}\\
&= \begin{cases}
\pi_\gamma \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1}^{\beta\alpha\beta^{-1}},x_n) & x_1=\ldots=x_n=1 \\
\pi_\gamma \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1},x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
\pi_\gamma \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1}^\alpha,x_n) & x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
\pi_\gamma \bullet (TG(n,\alpha,1) \oplus i_1) (x_1,\ldots,x_{n+1},x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases}\\
&= \begin{cases}
\pi_\gamma (x_1,\ldots,x_{n+1}^{\beta\alpha\beta^{-1}\alpha},x_n) & x_1=\ldots=x_n=1 \\
\pi_\gamma (x_1,\ldots,x_{n+1},x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
\pi_\gamma (x_1,\ldots,x_{n+1}^{\alpha\alpha},x_n) & x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
\pi_\gamma (x_1,\ldots,x_{n+1},x_n) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases}\\
&= \begin{cases}
(x_1,\ldots,x_n,x_{n+1}^{\beta\alpha\beta^{-1}\alpha}) & x_1=\ldots=x_n=1 \\
(x_1,\ldots,x_n,x_{n+1}) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n=1 \\
(x_1,\ldots,x_n,x_{n+1}) & x_1=\ldots=x_{n-1}=1,\, x_n\neq 1 \\
(x_1,\ldots,x_n,x_{n+1}) & \mbox{ some } x_1,\ldots,x_{n-1} \neq 1,\,x_n\neq 1
\end{cases}\\
\end{align*}
\begin{align*}
&= \begin{cases}
(x_1,\ldots,x_n,x_{n+1}^{(1\ldots k)}) & x_1=\ldots=x_n=1 \\
(x_1,\ldots,x_n,x_{n+1}) & \mbox{otherwise }
\end{cases}
\end{align*}
Thus we see that $\Sigma = TG(n+1,(1,\ldots,k),1)$.
We have shown that we can realise the two Toffoli gates that generate all of $\{TG(n+1,\alpha,o)\vert \alpha \in S_A\}$.
Thus by Corollary \ref{corrBnGen}, $B_{n+1}(A)$ is realised for all $n$, so by induction, all of $B(A)$ is realised.
\hfill$\Box$
Thus we have a small generating set for the bijections.
\begin{cor}
Let $A $ be of odd order $k$.
Let $\alpha = (1\ldots k), \beta=(1 2) \in S_A$.
Then $B(A)$ is
realised by $\{\alpha,\beta,TG(2,\alpha,1), TG(2,\beta,1) \}$.
\end{cor}
We see that when $A$ is of odd order, $TG(i+1,\alpha,1)$ is in the revclone generated by $\{TG(i,\alpha,1)\vert \alpha \in S_A\}$,
which is not the case for even order $A$.
We see a distinct property looking at closure with temporary storage in the following.
Note that $TG(n,\alpha,o) = s((2,3,\ldots,n+1),k(1,o,TG(n+1,\alpha,o)$ so $TG(n,\alpha,o)$
is realised with strong temporary storage from $TG(n+1,\alpha,o)$.
We see that closure under $T_S$ is a stronger form of generation.
\begin{thm}[\cite{toff80} Thm 5.3]
\label{thmtoffsts}
Let $A$ be a set, $o\in A$. For every $n$, for every $\alpha$, $TG(n,\alpha,o)$ can be realised with strong temporary storage by
$\{TG(i,\alpha,o) \vert i \leq 3,\, \alpha \in S_A\}$.
\end{thm}
Proof:
We proceed by induction. Assume that we can construct $TG(n-1,\alpha,o)$ for all $\alpha$.
Let $p\in A$ some non-$o$ element, $\beta = (o\,p)$ be an involution of $A$.
Define $f\in B_{n+1}(A)$ by
\begin{align*}
f = ( TG(n-1,\beta,o) \oplus i_2) \bullet (i_{n-2} \oplus TG(3,\alpha,o) ) \bullet (TG(n-1,\beta,o) \oplus i_2)
\end{align*}
Then
\begin{align*}
&f(x_1,\dots,x_{n+1}) \\
&= \begin{cases}
( TG(n-1,\beta,o) \oplus i_2) \bullet (i_{n-2} \oplus TG(3,\alpha,o) )(x_1,\dots,x_{n-1}^\beta,x_n,x_{n+1})\\
\hspace{50mm} x_1=\dots=x_{n-2}=o\\
( TG(n-1,\beta,o) \oplus i_2) \bullet (i_{n-2} \oplus TG(3,\alpha,o) )(x_1,\dots,x_{n-1},x_n,x_{n+1})\\
\hspace{50mm} \mbox{otherwise}
\end{cases}\\
\end{align*}
\begin{align*}
&= \begin{cases}
( TG(n-1,\beta,o) \oplus i_2)(x_1,\dots,x_{n-1}^\beta,x_n,x_{n+1}^\alpha)\\
\hspace{30mm} x_1=\dots=x_{n-2}=x_{n-1}^\beta=x_n=o\\
( TG(n-1,\beta,o) \oplus i_2)(x_1,\dots,x_{n-1}^\beta,x_n,x_{n+1})\\
\hspace{30mm} x_1=\dots=x_{n-2}=o \wedge (x_{n-1}^\beta \neq o \vee x_n \neq o)\\
( TG(n-1,\beta,o) \oplus i_2)(x_1,\dots,x_{n-1},x_n,x_{n+1}^\alpha)\\
\hspace{30mm} x_j\neq o \exists j\leq n-2,x_{n-1}=x_n=o\\
( TG(n-1,\beta,o) \oplus i_2)(x_1,\dots,x_{n-1},x_n,x_{n+1}) \hspace{10mm} \mbox{otherwise}
\end{cases}\\
&= \begin{cases}
(x_1,\dots,x_{n-1},x_n,x_{n+1}^\alpha) & x_1=\dots=x_{n-2}=x_{n-1}^\beta=x_n=o\\
(x_1,\dots,x_{n-1},x_n,x_{n+1}) & x_1=\dots=x_{n-2}=o \,\wedge \\& \hspace{4mm}(x_{n-1}^\beta \neq o \vee x_n \neq o)\\
(x_1,\dots,x_{n-1},x_n,x_{n+1}^\alpha) & x_j\neq o \exists j\leq n-2,x_{n-1}=x_n=o\\
(x_1,\dots,x_{n-1},x_n,x_{n+1})& \mbox{otherwise}
\end{cases}
\end{align*}
Note that $f_i(x_1,\dots,x_{n+1}) =x_i$ for all $i\leq n$.
Let $g$ be the reduct
\begin{align*}
g(x_1,\dots,x_{n}) = s((1,2,\ldots,n-2,n,n+1),k(n-1,p,f))
\end{align*}
From above we obtain, knowing that $p^\beta=o$ and $p\neq o$
\begin{align*}
&k(n-1,p,f) (x_1,\dots,x_{n})\\
&= f(x_1,\ldots,x_{n-2},p,x_{n-1},x_n) \\
&= \begin{cases}
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}^\alpha) & x_1=\dots=x_{n-2}=p^\beta=x_{n-1}=o\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}) & x_1=\dots=x_{n-2}=o \,\wedge\\ &\hspace{4mm}(p^\beta \neq o \vee x_{n-1} \neq o)\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}^\alpha) & x_j\neq o \exists j\leq n-2,\\ &\hspace{4mm} p=x_{n-1}=o \mbox{ [contradiction]}\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}) & \mbox{otherwise}
\end{cases} \\
&= \begin{cases}
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}^\alpha) & x_1=\dots=x_{n-2}=x_{n-1}=o\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}) & x_1=\dots=x_{n-2}=o \wedge ( x_{n-1} \neq o)\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}) & \mbox{otherwise}
\end{cases}\\
&= \begin{cases}
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}^\alpha) & x_1=\dots=x_{n-2}=x_{n-1}=o\\
(x_1,\dots,x_{n-2},p,x_{n-1},x_{n}) & \mbox{otherwise}
\end{cases}
\end{align*}
Thus
\begin{align*}
g(x_1,\dots,x_{n})
&= s((1,2,\ldots,n-2,n,n+1),k(n-1,p,f))(x_1,\dots,x_{n}) \\
&= \begin{cases}
(x_1,\dots,x_{n-2},x_{n-1},x_{n}^\alpha) & x_1=\dots=x_{n-2}=x_{n-1}=o\\
(x_1,\dots,x_{n-2},x_{n-1},x_{n}) & \mbox{otherwise}
\end{cases}\\
&= TG(n,\alpha,o)
\end{align*}
Note that $g_{n-1}(x_1,\ldots,x_{n+1}) = x_{n-1}$, a trivial permutation, so we have strong temporary storage.
\hfill$\Box$
Thus we see some essential differences between isomorphic realisation and realisation with (strong) temporary storage.
In the next section we will further investigate the differences between certain types of realisation and closure.
\section{Various Closures}
The following result shows us how the various class and closure operators
that we have seen above relate and thus
we will be able to determine a lot of information about the
relationships between various types of closure.
\begin{thm}
$KK=K$, $SS=S$, $CC=C$, $C_\Delta C_\Delta=C_\Delta$, $SC=CS$, $KC=CK$,
$C_\Delta K=KC_\Delta$, $C_\Delta S=SC_\Delta$ and $SK=KS$.
\end{thm}
Proof:
The $K$ and $S$ operators are idempotent as a simple implication of the definitions.
The $C$ and $C_\Delta$ operators are idempotent because they are algebraic closure operations.
To show that $SC=CS$ we show that all multiclone operations commute with $s$.
We use the operation set $\{\oplus,\pi_\alpha,\circ_k\vert \alpha \in S_A,\, k\in \mathbb N\}$.
We start with the inclusion $SC \subseteq CS$.
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $r\in \mathbb N$, $I \in \{1,\dots,m+p\}^r$, $\alpha \in S_m$.
Let $J\subseteq \{1,\dots,r \}$ such that $j\in J$ iff $I_j \leq m$.
Write $J=\{j_1,\dots,j_t\}$. Then let $I^\prime=(I_{j_i}\vert i \in (1,\dots,t)) \in \{1,\dots,m\}^t$.
Let $\{1,\dots,r\} \setminus J = \{\bar j_1,\dots,\bar j_u\}$.
Then let $I^{\prime\prime} = (I_{\bar j_i}-m \vert i \in (1,\dots,u)) \in \{1,\dots,p\}^u$.
Finally define $\beta \in S_r$ with
\[ \beta: i \mapsto h \mbox{ such that } \begin{cases}
h=j_i & i\leq t \\
h = \bar j_{i-t} & i > t
\end{cases}
\] which can be written as
\[\beta = \pmtv[1\ldots {t}{t+1}\ldots {r}] {{1}{j_1} {t}{j_t} {t+1}{\bar j_1} {r}{\bar j_u} }\]
Note that $i\leq t$ iff $I_{\beta(i)} \leq m$ and that $I_i = I^\prime_{\beta^{-1}i}$ if $I_i\leq m$,
$I_i=I^{\prime\prime}_{(\beta^{-1}i)-t}+m$ otherwise.
Then we claim that
\begin{align*}
s(I,f\oplus g) &= \pi_\beta (s(I^\prime,f) \oplus s(I^{\prime\prime},g))
\end{align*}
The left hand side is
\begin{align*}
s(I,f\oplus g)_i &=
\begin{cases}
(f)_{I_i} & I_i \leq m \\
(g)_{I_i-m} & I_i > m
\end{cases}
\end{align*}
The right hand side is
\begin{align*}
\pi_\beta (s(I^\prime,f) \oplus s(I^{\prime\prime},g))_i
&= (s(I^\prime,f) \oplus s(I^{\prime\prime},g))_{\beta^{-1}i} \\
&=\begin{cases}
s(I^\prime,f)_{\beta^{-1}i} & {\beta^{-1}i}\leq t \\
s(I^{\prime\prime},g)_{(\beta^{-1}i)-t} & {\beta^{-1}i} > t
\end{cases}\\
&=\begin{cases}
(f)_{I^\prime_{\beta^{-1}i}} & {\beta^{-1}i}\leq t \\
(g)_{I^{\prime\prime}_{(\beta^{-1}i)-t}} & {\beta^{-1}i} > t
\end{cases}\\
&=\begin{cases}
(f)_{I_{i}} & I_i\leq m \\
(g)_{I_{i}-m} & I_{i} > m
\end{cases}
\end{align*}
which shows our claim.
It is a simple calculation that
\begin{align*}
s(I,\pi_\alpha f) &= s(\alpha^{-1}(I),f)
\end{align*}
where $\alpha^{-1}$ acts upon the entries in $I$, so $(\alpha^{-1}I)_i = \alpha^{-1} (I_i)$.
For composition, we use a similar argument to the $\oplus$ case above.
Assume that $I$ is increasing. Let $I^\prime = (I_1,\dots,I_t)$ with $I_t \leq m$, $I_{t+1}>m$.
Let $I^{\prime\prime} = (1,\dots,k)\oplus(I_{t+1}-m+k,\dots,I_r-m+k)$.
We claim that
\begin{align*}
s(I,f\circ_k g) &= s(I^\prime,f) \circ_k s(I^{\prime\prime},g)
\end{align*}
The left hand side is
\begin{align*}
s(I,f\circ_k g)_i &=
\begin{cases}
f_{I_i} \circ_k g & i \leq t\\
g_{I_i-m+k} & i > t
\end{cases}
\end{align*}
The right hand side is
\begin{align*}
(s(I^\prime,f) \circ_k s(I^{\prime\prime},g))_i &=
\begin{cases}
s(I^\prime,f)_i \circ_k s(I^{\prime\prime},g) & i \leq t\\
s(I^{\prime\prime},g)_{i-t+k} & i > t
\end{cases}\\
&=
\begin{cases}
f_{I^\prime_i} \circ_k g & i \leq t \;\;\mbox{because of the first $k$ entries in }I^{\prime\prime}\\
g_{I^{\prime\prime}_{i-t+k}} & i > t
\end{cases}\\
&=
\begin{cases}
f_{I_i} \circ_k g & i \leq t \\
g_{I_{i}-m+k} & i > t
\end{cases}
\end{align*}
Which is what we wanted.
If $I$ is not increasing, then there is a permutation $\beta \in S_r$ such that $\pi_\beta I$ is increasing.
Then
\[s(I,f \circ_k g) = \pi_{\beta^{-1}} s(\pi_\beta I,f \circ_k g)
= \pi_{\beta^{-1}}( s(I^\prime,f) \circ_k s(I^{\prime\prime},g))
\]
Thus we see that $SC \subseteq CS$.
For the converse, i.e.\ $CS \subseteq SC$, we use similar techniques.
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $t,u\in \mathbb N$, $I^\prime \in \{1,\dots,m\}^t$, $I^{\prime\prime} \in \{1,\dots,p\}^u$,
$\alpha \in S_t$.
Let $I=I^\prime \oplus (I^{\prime\prime}+m)$, i.e.\ $I_i = I^\prime_i$ for $i\leq t$, $I_i = I^{\prime\prime}_{i-t}+m$ for $i>t$.
Then it is a simple calculation to see that
\begin{align*}
s(I^\prime,f) \oplus s(I^{\prime\prime},g) = s(I,f\oplus g)
\end{align*}
Let $I \in \{1,\dots,m\}^t$, define $I^\alpha_i = I_{\alpha^{-1}i}$.
Then we see that
\begin{align*}
\pi_\alpha s (I,f) = s(I^\alpha,f)
\end{align*}
Let $\beta \in S_p$ be defined by $\beta^{-1}(i) = I^{\prime\prime}_i$ for $i\leq u$,
with the rest filled in to make it a permutation. Because $I^{\prime\prime}$ contains no repeats,
we know this can be done.
Let $k \leq \min(n,u)$.
Then we claim that
\begin{align*}
s(I^\prime,f) \circ_k s(I^{\prime\prime},g) = s(I^\prime \oplus (m+1,\dots,m+u-k),f \circ_k (\pi_\beta \circ_p g))
\end{align*}
For $i \leq t$ we have
\begin{align*}
&(s(I^\prime,f) \circ_k s(I^{\prime\prime},g))_i(x_1,\dots,x_{l+n-k}) \\
&= f_{I^\prime_i} \circ_k s(I^{\prime\prime},g) (x_1,\dots,x_{l+n-k})\\
&= f_{I^\prime_i}(g_{I^{\prime\prime}_1}(x_1,\dots,x_{l}),\dots,g_{I^{\prime\prime}_k}(x_1,\dots,x_{l}),x_{l+1},\dots,x_{l+n-k}) \\
&= f_{I^\prime_i}(g_{\beta^{-1}1}(x_1,\dots,x_{l}),\dots,g_{\beta^{-1}k}(x_1,\dots,x_{l}),x_{l+1},\dots,x_{l+n-k}) \\
&= (s(I^\prime \oplus (m+1,\dots,m+u-k),f \circ_k (\pi_\beta \circ_p g)))_i(x_1,\dots,x_{l+n-k})
\end{align*}
while for $t < i \leq m+u-k$ we have
\begin{align*}
&(s(I^\prime,f) \circ_k s(I^{\prime\prime},g))_i(x_1,\dots,x_{l+n-k}) \\
&= g_{I^{\prime\prime}_{k+i-t}} (x_1,\dots,x_l) \\
&= g_{\beta^ {-1}(k+i-t)} (x_1,\dots,x_l) \\
&= (\pi_\beta \circ_p g)_{k+(i-t)}(x_1,\dots,x_l) \\
&= (f \circ_k (\pi_\beta \circ_p g))_{m+i-t}(x_1,\dots,x_{l+n-k}) \\
&= s(I^\prime \oplus (m+1,\dots,m+u-k),f \circ_k (\pi_\beta \circ_p g))_i(x_1,\dots,x_{l+n-k})
\end{align*}
Thus we have that $CS \subseteq SC$ and thus $SC=CS$.
Similarly we show that $KC=CK$.
We start with $KC \subseteq CK$.
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $i \in \{1,\dots,n+l\}$, $a\in A$.
It is a simple application of the definitions to see that
\begin{align*}
k(i,a,f \oplus g) &= \begin{cases}
k(i,a,f) \oplus g & i \leq n\\
f \oplus k(i-n,a,g) & i > n
\end{cases}
\end{align*}
Let $f\in M_{n,m}(A)$, $\alpha \in S_m$, $i \in \{1,\dots,n\}$, $a\in A$.
It is a simple calculation that
\begin{align*}
k(i,a,\pi_\alpha f) &= \pi_\alpha k(i,a,f)
\end{align*}
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $k \leq \min(n,p)$, $i \in \{1,\dots,n+l\}$, $a\in A$.
Applying the definitions gives us
\begin{align*}
k(i,a,f \circ_k g) &= \begin{cases}
f \circ_k k(i,a,g) & i \leq n\\
k(i+k-n,a,f) \circ_k g & i > n
\end{cases}
\end{align*}
For $CK \subseteq KC$ we proceed as follows.
Note that we must also include the nonapplication of $k$ as a case here.
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $i_1 \in \{1,\dots,n\}$, $i_2\in \{1,\dots,l\}$, $a_1,a_2\in A$.
Then we claim that
\begin{align*}
k(i_1,a_1,f) \oplus g &= k(i_1,a_1,f\oplus g)\\
f \oplus k(i_2,a_2,g) &= k(i_2+n,a_2,f\oplus g)\\
k(i_1,a_1,f) \oplus k(i_2,a_2,g) &= k(i_1,a_1,k(i_2+n,a_2,f\oplus g))
\end{align*}
The first two follow by simple application of the definitions.
The third claim combines these two.
As we saw above, $\pi_\alpha$ commutes with $k$.
Let $f\in M_{n,m}(A)$, $g\in M_{l,p}(A)$, $k \leq \min(n,p)$, $i_1 \in \{1,\dots,n\}$, $i_2\in \{1,\dots,l\}$, $a_1,a_2\in A$.
Let $\beta \in S_n$ be $(i_1,i_1+1,\dots,n)$. Then we claim that
\begin{align*}
f \circ_k k(i_2,a_2,g) &= k(i_2,a_2,f\circ_k g) \\
k(i_1,a_1,f) \circ_k g &=
\begin{cases}
k(l+n-k,a_1,(f \circ_n \pi_\beta) \circ_k g) & i_1 \leq k \\
k(i_1+l-k,a_1,f \circ_k g) & i_1 > k
\end{cases}\\
k(i_1,a_1,f) \circ_k k(i_2,a_2,g) &=
\begin{cases}
k(i_2,a_2,k(l+n-k,a_1,(f \circ_n \pi_\beta) \circ_k g)) & i_1 \leq k \\
k(i_2,a_2,k(i_1+l-k,a_1,f \circ_k g)) & i_1 > k
\end{cases}
\end{align*}
The first claim is direct from the definitions.
The second requires some more work. Let $i\in \{1,\dots,m+p-k\}$.
\begin{align*}
&(k(i_1,a_1,f) \circ_k g )_i(x_1,\ldots,x_{n+l-k-1})\\
&= \begin{cases}
(k(i_1,a_1,f)_i \circ_k g)(x_1,\ldots,x_{n+l-k-1}) & i\leq m\\
g_{i-m+k}(x_1,\dots,x_l) & i > m
\end{cases}\\
&= \begin{cases}
f_i(g_1(x_1,\dots,x_l),\dots,g_{i_1-1}(\dots),a_1,g_{i_1}(\dots),\dots\\
\hspace{18mm}\dots,g_k(x_1,\dots,x_l),x_{l+1},\dots,x_{n+l-k-1}) & i_1 \leq k,\, i\leq m\\
f_i(g_1(x_1,\dots,x_l),\dots,g_k(x_1,\dots,x_l),x_{l+1},\dots\\
\hspace{18mm}\dots,x_{i_1-k+l-1},a_1,x_{i_1+l-k},\dots,x_{n+l-k-1}) & i_1 > k,\, i\leq m\\
g_{i-m+k}(x_1,\dots,x_l) & i > m
\end{cases}\\
&= \begin{cases}
(f\circ_n\pi_\beta)_i(g_1(x_1,\dots,x_l),\dots,g_{i_1-1}(\dots),g_{i_1}(\dots),\dots\\
\hspace{18mm}\dots,g_k(x_1,\dots,x_l),x_{l+1},\dots,x_{n+l-k-1},a_1) & i_1 \leq k,\, i\leq m\\
k(i_1+l-k,a_1,f\circ_k g)_i(x_1,\ldots,x_{n+l-k-1}) & i_1 > k,\, i\leq m\\
g_{i-m+k}(x_1,\dots,x_l) & i > m
\end{cases}\\
&= \begin{cases}
k(l+n-k,a_1,(f \circ_n \pi_\beta)\circ_k g)_i(x_1,\ldots,x_{n+l-k-1}) & i_1 \leq k,\, i\leq m\\
k(i_1+l-k,a_1,f\circ_k g)_i(x_1,\ldots,x_{n+l-k-1}) & i_1 > k,\, i\leq m\\
g_{i-m+k}(x_1,\dots,x_l) & i > m
\end{cases}\\
&= \begin{cases}
k(l+n-k,a_1,(f \circ_n \pi_\beta)\circ_k g)_i(x_1,\ldots,x_{n+l-k-1}) & i_1 \leq k\\
k(i_1+l-k,a_1,f\circ_k g)_i(x_1,\ldots,x_{n+l-k-1}) & i_1 > k
\end{cases}
\end{align*}
which is what we wanted.
The third case is the combination of the first two cases.
So every expression in $CK$ can be written in $KC$, so we obtain $CK=KC$.
We see that $SK=KS$ because it makes no difference whether the inputs are fed constants and then some outputs are ignored,
or some outputs are ignored and then some inputs are fed constants.
To see this formally,
let $f\in M_{n,m}(A)$, $1\leq i\leq n$, $r\in \mathbb N$, $I\in \{1,\ldots,m\}^r$ and $a\in A$.
\begin{align*}
s(I,k(i,a,f))(x_1,\ldots,x_{n-1}) &= s(I,f(x_1,\ldots,x_{i-1},a,x_i,\ldots,x_{n-1})) \\
&= (f_j(x_1,\ldots,x_{i-1},a,x_i,\ldots,x_{n-1}) \vert j\in I) \\
&= (k(i,a,f_j)\vert j\in I)(x_1,\ldots,x_{n-1}) \\
&= k(i,a,s(I,f))
\end{align*}
Lastly, we show that $C_\Delta$ is well behaved by showing that $\Delta$ and $\nabla$ commute with $S$ and $K$.
Let $f\in M_{n,m}(A)$, $r\in \mathbb N$, $I\in \{1,\ldots,n\}^r$ and $a\in A$.
\begin{align*}
s(I,\Delta f)(x_1,\dots,x_{n-1}) &= s(I,f(x_1,x_1,x_2,\ldots,x_{n-1}))\\
&= (f_j(x_1,x_1,x_2,\ldots,x_{n-1}) \vert j\in I)\\
&= \Delta (f_j(x_1,x_2,\ldots,x_{n-1}) \vert j\in I)\\
&= \Delta s(I,f)(x_1,\dots,x_{n-1})\\
s(I,\nabla f)(x_1,\dots,x_{n+1}) &= s(I, f(x_2,\dots,x_{n+1})) \\
&= (f_j(x_2,\dots,x_{n+1})\vert j\in I) \\
&= \nabla s(I,f)(x_1,\dots,x_{n+1})
\end{align*}
So we see that $C_\Delta S = S C_\Delta$.
Now let $f\in M_{n,m}(A)$, $1\leq i\leq n$ and $a\in A$.
\begin{align*}
&k(i,a,\Delta f)(x_1,\dots,x_{n-2})\\ &=
(\Delta f)(x_1,\dots,x_{i-1},a,x_i,\dots,x_{n-2}) \\
&= f(x_1,x_1,x_2,\dots,x_{i-1},a,x_i,\dots,x_{n-2}) \\
&= \begin{cases}
f(a,a,x_1,\ldots,x_{n-2}) & i=1 \\
f(x_1,x_1,a,x_2,\ldots,x_{n-2}) & i=2 \\
f(x_1,x_1,x_2,\ldots,x_{i-1},a,x_i,\ldots x_{n-2}) & i>2
\end{cases}\\
&= \begin{cases}
k(1,a,k(1,a,f)) & i=1 \\
\Delta k(i+1,a,f) & i\geq2
\end{cases}\\
\end {align*}
\begin{align*}
k(i,a,\nabla f)(x_1,\dots,x_n) &= (\nabla f)(x_1,\dots,x_{i-1},a,x_i,\dots,x_n) \\
&=
\begin{cases}
(\nabla f)(a,x_1,\dots,x_n) & i=1\\
(\nabla f)(x_1,x_2,\dots,x_{i-1},a,x_i,\dots,x_n) & i>1
\end{cases}\\
&=
\begin{cases}
f(x_1,\dots,x_n) & i=1\\
f(x_2,\dots,x_{i-1},a,x_i,\dots,x_n) & i>1
\end{cases}\\
&=
\begin{cases}
f(x_1,\dots,x_n) & i=1\\
\nabla k(i-1,a,f)(x_1,\dots,x_n) & i>1
\end{cases}
\end{align*}
Thus we have $KC_\Delta \subseteq C_\Delta K$.
For the converse we calculate.
\begin{align*}
&\hspace{-10mm}k(i,a,f) (x_1,\ldots,x_{n-1})\\
&= \begin{cases}
f(a,x_1,x_2,\ldots,x_{n-1}) & i=1 \\
f(x_1,a,x_2,\ldots,x_{n-1}) & i=2 \\
f(x_1,x_2,\ldots,x_{i-1},a,x_i,\ldots,x_{n-1}) & i>2
\end{cases} \\
&\hspace{-10mm}\Rightarrow \Delta k(i,a,f) (x_1,\ldots,x_{n-2})\\
&= \begin{cases}
f(a,x_1,x_1,x_2\ldots,x_{n-2}) & i=1 \\
f(x_1,a,x_1,x_2,\ldots,x_{n-2}) & i=2 \\
f(x_1,x_1,x_2,\ldots,x_{i-2},a,x_{i-1},\ldots,x_{n-2}) & i>2
\end{cases} \\
&= \begin{cases}
k(2,a,\Delta(f\bullet \pi_{(1\,2\,3)}))(x_1,\dots,x_{n-2}) & i=1 \\
k(2,a,\Delta (f\bullet \pi_{(2\,3)}))(x_1,\dots,x_{n-2}) & i=2 \\
k(i-1,a,\Delta f)(x_1,\dots,x_{n-2}) & i>2
\end{cases} \\
&\hspace{-10mm}\nabla k(i,a,f)(x_1,\dots,x_n)\\ &= k(i,a,f)(x_2,\dots,x_n)\\
&= f(x_2,\dots,x_i,a,x_{i+1},\dots,x_n)\\
&= (\nabla f)(x_1,\dots,x_i,a,x_{i+1},\dots,x_n) \\
&= k(i+1,a,\nabla f)(x_1,\dots,x_n)
\end{align*}
So we see that
$K C_\Delta \subseteq C_\Delta K \subseteq K C_\Delta$ so they are equal.
\hfill$\Box$
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=.6]
\node (20) at (2,4) {$KSC(F)$};
\node (21) at (0,1) {$T(F)$};
\node (22) at (0,-1) {$T_S(F)$};
\node (10) at (2,-3) {$C(F)$};
\node (KC) at (1,0) {$KC(F)$};
\node (SC) at (3,0) {$SC(F)$};
\node (RKSC) at (-1,2) {$RKSC(F)$};
\node (RT) at (-2,-1) {$RT(F)$};
\node (RTS) at (-2,-3) {$RT_S(F)$};
\node (RC) at (0,-5) {$RC(F)$};
\node (KSCD) at (5,6) {$KSC_\Delta(F)$};
\node (KCD) at (4,2) {$KC_\Delta(F)$};
\node (SCD) at (7,2) {$SC_\Delta(F)$};
\node (CD) at (5,-1) {$C_\Delta(F)$};
\draw (20) -- (21) -- (22) -- (10);
\draw (20) -- (KC) -- (10);
\draw (20) -- (SC) -- (10)--(RC);
\draw (20)--(RKSC)--(RT)--(RTS)--(RC);
\draw (21)--(RT);
\draw (22)--(RTS);
\draw (KSCD)--(20);
\draw (KSCD)--(KCD)--(CD);
\draw (KSCD)--(SCD)--(CD)--(10);
\draw (KCD)--(KC);
\draw (SCD)--(SC);
\end{tikzpicture}
\begin{tikzpicture}[scale=.7]
\node (20) at (2,4) {$KSC(F)$};
\node (22) at (-1,-2) {$T_S(F)=RT_S(F)$};
\node (10) at (2,-4) {$C(F)=RC(F)$};
\node (KC) at (1,0) {$KC(F)$};
\node (SC) at (3,0) {$SC(F)$};
\node (RKSC) at (-1,2) {$RKSC(F)$};
\node (RT) at (-2,0) {$T(F)=RT(F)$};
\draw (22) -- (10);
\draw (20) -- (KC) -- (10);
\draw (20) -- (SC) -- (10);
\draw (20)--(RKSC)--(RT)--(22);
\end{tikzpicture}
\end{center}
\caption{On the left we have inclusion of the various closure operations applied to a set of maps $F \subseteq M(A)$.
When we are looking at a set $F \subseteq B(A)$ and are not interested in $\Delta$, we get the inclusions on the right.}
\label{figureinclusion}
\end{figure}
Let $F\subseteq M(A)$ be a collection of mappings.
Then $g \in SKC(F)$ is equivalent to saying that $g$ is realised by $F$.
We see that we have some inclusions amongst the various closure operations introduced above,
obtaining the inclusion diagram in Figure \ref{figureinclusion}.
We know that many of these inclusions are strict.
For $A$ of even order, we know from Corollary \ref{thmtoffeven} and Theorem \ref{thmtoffsts} that $C(F)$ is strictly included in $T_S(F)$.
On the other hand, for specific classes of $F$, some closure classes fall together.
If $F \subseteq B(A)$, then $RC(F)=C(F)$, $RT_S(F) = T_S(F)$ and $RT(F)=T(F)$.
Thus we obtain the second inclusion diagram, where we also omit the $\Delta$ operator.
One of the general problems is to look at the ways in which we can
define these various classes as a form of closure via a Galois connection.
This has been preliminarily investigated in \cite{jerebek} for $C(F)$ and $T(F)$ from reversible $F$.
\section{Conclusion}
We have presented a language based upon clone theory in order to discuss systems of mappings $A^m\rightarrow A^n$ on a set $A$.
We have then written Toffoli's model of reversible computation as a theory of mapping composition, finding that larger sets of
states leads to a more complex system than binary reversible logic.
We see that the ideas can be used more generally. We are now confronted with the spectrum of questions that arise naturally
in clone theory and related fields of general algebra, such as the size and structure of the lattice of multiclones and revclones on a given set.
The largest challenge is to determine a suitable combinatorial structure to be used for invariance type results of the Pol-Inv type for the
five natural types of closure.
It seems probable that the results in \cite{aaronson,jerebek} will be of use to develop such a theory.
\section{Acknowledgments}
I would like to thank Erhard Aichinger for many interesting conversations and suggestions about developing
and presenting this paper.
|
2,869,038,154,880 | arxiv | \section{Introduction}
A \emph{$\delta$--hyperbolic space} is a geodesic metric space where geodesic triangles are \emph{$\delta$--slim}: the $\delta$--neighborhood of any two sides of a geodesic triangle contains the third side. Such spaces were introduced by Gromov in~\cite{col:Gromov87} as a coarse notion of negative curvature for geodesic metric spaces and since then have evolved into an indispensable tool in geometric group theory.
There is a classification of isometries of $\delta$--hyperbolic metric spaces analogous to the classification of isometries of hyperbolic space $\HH^{n}$ into elliptic, hyperbolic and parabolic. Of these, hyperbolic isometries have the best dynamical properties and are often the most desired. For example, typically they can be used to produce free subgroups in a group acting on a $\delta$--hyperbolic space~\cite[5.3B]{col:Gromov87}, see also~\cite[III.$\Gamma$.3.20]{bk:BH99}. Another application is to show that a certain element does not have fixed points in its action on some set. Indeed, if the set naturally sits inside of a $\delta$--hyperbolic metric space and the given element acts as a hyperbolic isometry then it has no fixed points (in a strong sense). This strategy has been successfully employed for the curve complex of a surface and for the free factor complex of a free group by several authors~\cite{ar:CLM12,ar:CP12-1,DTHyp,ar:Fujiwara15,un:Gultepe, Hshort,ar:Mangahas13,ar:Taylor14}.
We consider the situation of a group acting on finitely many $\delta$--hyperbolic spaces and produce a sufficient condition that guarantees the existence of a single element in the group that is a hyperbolic isometry for each of the spaces. Of course, a necessary condition is that for each of the spaces there is some element of the group that is a hyperbolic isometry. Thus we are concerned with when we may reverse the quantifiers: $\forall \exists \leadsto \exists\forall$. Our main result is the following theorem.
\begin{restate}{Theorem}{th:constructing hyperbolic actions}
Suppose that $\{ X_{i} \}_{i = 1,\ldots, n}$ is a collection of $\delta$--hyperbolic spaces, $G$ is a group and for each $i = 1,\ldots, n$ there is a homomorphism $\rho_{i} \colon\thinspace G \to \Isom(X_{i})$ such that:
\begin{enumerate}
\item there is an element $f_{i} \in G$ such that $\rho_{i}(f_{i})$ is hyperbolic; and
\item for each $g \in G$, either $\rho_{i}(g)$ has a periodic orbit or is hyperbolic.
\end{enumerate}
Then there is an $f \in G$ such that $\rho_{i}(f)$ is hyperbolic for all $i = 1,\ldots,n$.
\end{restate}
\begin{remark} After the completion of this paper we have been alerted that Theorem \ref{th:constructing hyperbolic actions} should follow from random walk techniques developed in \cite{BH10} and \cite{MT}. Here we provide an elementary and constructive proof.
\end{remark}
Essentially, we assume that there are no parabolic isometries and that elliptic isometries are relatively tame.
As an application of our main theorem we prove a conjecture of Handel and Mosher which exactly involves the same type of quantifier reversing: $\forall \exists \leadsto \exists\forall$. Consider a finitely generated subgroup $\mathcal{H}<\IA_{N}(\ZZ/3)<\Out(F_N)$ and a maximal $\mathcal{H}$--invariant filtration of $F_{N}$, the free group of rank $N$, by free factor systems
\[\emptyset = \mathcal{F}_{0} \sqsubset \mathcal{F}_{1} \sqsubset \cdots \sqsubset \mathcal{F}_{m} = \{[ F_{N}] \}\]
(see Section \ref{sec:application}). Handel and Mosher prove that for each multi-edge extension $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ there exists some $\varphi_{i} \in \mathcal{H}$ that is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$~\cite[Theorem~D]{HMIntro}. They conjecture that there exists a single $\varphi \in \mathcal{H}$ that is irreducible with respect to each multi-edge extension $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$. We show that this is indeed the case.
\begin{restate}{Theorem}{th:application}
For each finitely generated subgroup $\mathcal{H} < \IA_{N}(\ZZ/3)<\Out(F_N)$ and each maximal $\mathcal{H}$--invariant filtration by free factor systems $\emptyset = \mathcal{F}_{0} \sqsubset \mathcal{F}_{1} \sqsubset \cdots \sqsubset \mathcal{F}_{m} = \{[ F_{N}] \}$, there is an element $\varphi \in \mathcal{H}$ such that for each $i = 1,\ldots,m$ such that $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ is a multi-edge extension, $\varphi$ is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$.
\end{restate}
Our paper is organized as follows. Section~\ref{sec:spaces} contains background on $\delta$--hyperbolic spaces and their isometries. In Section~\ref{sec:actions} we generalize a construction of the first author and Pettet from~\cite{ar:CP12-1} that is useful to constructing hyperbolic isometries. This result is Theorem~\ref{th:uniform hyperbolic}. We examine certain cases that will arise in the proof of the main theorem to see how to apply Theorem~\ref{th:uniform hyperbolic} in Section~\ref{sec:neighborhoods}. The proof of Theorem~\ref{th:constructing hyperbolic actions} constitutes Section~\ref{sec:simultaneous}. The application to $\Out(F_{N})$ appears in Section~\ref{sec:application}.
\subsection*{Acknowledgements.} We would like to thank Lee Mosher and Camille Horbez for useful discussions. We are grateful to Camille Horbez for informing us about his work with Vincent Guirardel~\cite{un:GH}. We thank the referee for a careful reading and for providing useful suggestions. The second author thanks Ilya Kapovich and Chris Leininger for guidance and support.
\section{Background on $\delta$--hyperbolic spaces}\label{sec:spaces}
In this section we recall basic notions and facts about $\delta$--hyperbolic spaces, their isometries and their boundaries. The reader familiar with these topics can safely skip this section with the exception of Definition~\ref{def:independent}. References for this section are \cite{col:AlonsoEtAl91}, \cite{bk:BH99} and \cite{col:KB02}.
\subsection{$\delta$--hyperbolic spaces}\label{subsec:spaces}
We recall the definition of a $\delta$--hyperbolic space given in the Introduction.
\begin{definition}\label{def:hyperbolic}
Let $(X,d)$ be a geodesic metric space. A geodesic triangle with sides $\alpha$, $\beta$ and $\gamma$ is \emph{$\delta$--slim} if for each $x \in \alpha$, there is some $y \in \beta \cup \gamma$ such that $d(x,y) \leq \delta$. The space $X$ is said to be \emph{$\delta$--hyperbolic} if every geodesic triangle is $\delta$--slim.
\end{definition}
There are several equivalent definitions that we will use in the sequel. The first of these is insize. Let $\Delta$ be the geodesic triangle with vertices $x$, $y$ and $z$ and sides $\alpha$ from $y$ to $z$, $\beta$ from $z$ to $x$ and $\gamma$ from $x$ to $y$. There exist unique points $\hat{\alpha} \in \alpha$, $\hat{\beta} \in \beta$ and $\hat{\gamma} \in \gamma$, called the \emph{internal points} of $\Delta$, such that:
\begin{equation*}
d(x,\hat{\beta}) = d(x,\hat{\gamma}), \,
d(y,\hat{\gamma}) = d(y,\hat{\alpha}) \text{ and }
d(z,\hat{\alpha}) = d(z,\hat{\beta}).
\end{equation*}
The \emph{insize} of $\Delta$ is the diameter of the set $\{ \hat{\alpha},\hat{\beta},\hat{\gamma}\}$.
Another notion makes use of the so-called \emph{Gromov product}:
\begin{equation}\label{eq:gp}
\GP{x}{y}_{w} = \frac{1}{2}(d(x,w) + d(w,y) - d(x,y)).
\end{equation}
The Gromov product is said to be \emph{$\delta$--hyperbolic \textup{(}with respect to $w \in X$\textup{)}} if for all $x,y,z \in X$:
\[ \GP{x}{z}_{w} \geq \min\left\{ \GP{x}{y}_{w}, \GP{y}{z}_{w} \right\} - \delta. \]
\begin{proposition}[{\cite[Proposition~2.1]{col:AlonsoEtAl91}, \cite[III.H.1.17 and III.H.1.22]{bk:BH99}}]\label{prop:delta-hyp}
The following are equivalent for a geodesic metric space $X$:
\begin{enumerate}
\item There is a $\delta_{1} \geq 0$ such that every geodesic triangle in $X$ is $\delta_{1}$--slim, i.e., $X$ is $\delta_{1}$--hyperbolic.
\item There is a $\delta_{2} \geq 0$ such that every geodesic triangle in $X$ has insize at most $\delta_{2}$.
\item There is a $\delta_{3} \geq 0$ such that for some
\textup{(}equivalently any\textup{)} $w \in X$, the Gromov product is $\delta_{3}$--hyperbolic.
\end{enumerate}
\end{proposition}
Henceforth, when we say $X$ is a $\delta$--hyperbolic space we assume that $\delta$ is large enough to satisfy each of the above conditions.
\subsection{Boundaries}\label{subsec:boundaries}
There is a useful notion of a boundary for a $\delta$--hyperbolic space that plays the role of the ``sphere at infinity'' for $\HH^{n}$. This space is defined using equivalence classes of certain sequences of points in $X$ and the Gromov product. Fix a basepoint $w \in X$.
\begin{definition}\label{def:boundary}
We say a sequence $(x_{n}) \subseteq X$ \emph{converges to infinity} if $\GP{x_{i}}{x_{j}}_{w} \to \infty$ as $i,j \to \infty$. Two such sequences $(x_{n})$, $(y_{n})$ are equivalent if $\GP{x_{i}}{y_{j}}_{w} \to \infty$ as $i,j \to \infty$. The \emph{boundary of $X$}, denoted $\partial X$, is the set of equivalence classes of sequences $(x_{n}) \subseteq X$ that converge to infinity.
\end{definition}
One can show that the notion of ``converges to infinity'' and the subsequent equivalence relation do not depend on the choice of basepoint $w \in X$~\cite{col:KB02}.
The definition of the Gromov product in~\eqref{eq:gp} extends to boundary points $\hat{x}, \hat{y} \in \partial X$ by:
\[ \GP{\hat{x}}{\hat{y}}_{w} = \inf \{ \liminf_{n} \GP{x_{n}}{y_{n}}_{w} \} \]
where the infimum is over sequences $(x_{n}) \in \hat{x}$, $(y_{n}) \in \hat{y}$. If $y \in X$ then we set:
\[ \GP{\hat{x}}{y}_{w} = \inf \{ \liminf_{n} \GP{x_{n}}{y}_{w} \} \]
where the infimum is over sequences $(x_{n}) \in \hat{x}$. For $x \in X$, the Gromov product $\GP{x}{\hat{y}}_{w}$ is defined analogously. Let $\overline{X} = X \cup \partial X$.
We will make use of the following properties of the Gromov product on $\overline{X}$.
\begin{proposition}[{\cite[Lemma~4.6]{col:AlonsoEtAl91}, \cite[III.H.3.17]{bk:BH99}}]\label{prop:extended gp}
Let $X$ be a $\delta$--hyperbolic space.
\begin{enumerate}
\item If $x,y \in \overline{X}$ then $\GP{x}{y}_{w} = \infty \iff x = y \in \partial X$.
\item If $\hat{x} \in \partial X$ and $(x_{n}) \subseteq X$ then $\GP{\hat{x}}{x_{n}}_{w} \to \infty$ as $n \to \infty \iff (x_{n}) \in \hat{x}$.
\item If $\hat{x},\hat{y} \in \partial X$ and $(x_{n}) \in \hat{x}$, $(y_{n}) \in \hat{y}$ then:
\[ \GP{\hat{x}}{\hat{y}}_{w} \leq \liminf_{n} \GP{x_{n}}{y_{n}}_{w} \leq \GP{\hat{x}}{\hat{y}}_{w} - 2\delta. \]
\item If $x,y,z \in \overline{X}$ then:
\[ \GP{x}{z}_{w} \geq \min\left\{ \GP{x}{y}_{w}, \GP{y}{z}_{w} \right\} - \delta. \]
\end{enumerate}
\end{proposition}
\begin{proposition}[{\cite[Proposition~4.8]{col:AlonsoEtAl91}}]\label{prop:basis}
The following collection of subsets of $\overline{X}$ forms a basis for a topology:
\begin{enumerate}
\item $B(x,r) = \{ y \in X \mid d(x,y) < r \}$, for each $x \in X$ and $r > 0$; and
\item $N(\hat{x},k) = \{ y \in \overline{X} \mid \GP{\hat{x}}{y}_{w} > k \}$ for each $\hat{x} \in \partial X$ and $k > 0$.
\end{enumerate}
\end{proposition}
\subsection{Isometries}\label{subsec:isometries}
As mentioned in the Introduction, there is a classification of isometries of a $\delta$--hyperbolic space $X$ into elliptic, parabolic and hyperbolic~\cite[8.1.B]{col:Gromov87}. We will not make use of parabolic isometries and so do not give the definition here.
\begin{definition}\label{def:isometries}
An isometry $f \in \Isom(X)$ is \emph{elliptic} if for any $x \in X$, the set $\{ f^{n}x \mid n \in \ZZ \}$ has bounded diameter.
An isometry $f \in \Isom(X)$ is \emph{hyperbolic} if for any $x \in X$ there is a $t > 0$ such that $t \abs{m-n} \leq d(f^{m}x,f^{n}x)$ for all $m,n \in \ZZ$. In this case, one can show, the sequence $(f^{n}x) \subseteq X$ converges to infinity and the equivalence class it defines in $\partial X$ is independent of $x \in X$. This point in $\partial X$ is called the \emph{attracting fixed point} of $f$. The \emph{repelling fixed point} of $f$ is the attracting fixed point of $f^{-1}$ and is represented by the sequence $(f^{-n}x) \subseteq X$.
\end{definition}
The action of a hyperbolic isometry $f \in \Isom(X)$ on $\overline{X}$ has ``North-South dynamics.''
\begin{proposition}[{\cite[8.1.G]{col:Gromov87}}]\label{prop:ns}
Suppose that $f \in \Isom(X)$ is a hyperbolic isometry and that $U_{+}, U_{-} \subset \overline{X}$ are disjoint neighborhoods of the attracting and repelling fixed points of $f$ respectively. There exists an $N \geq 1$ such that for $n \geq N$:
\begin{equation*}
f^{n}(\overline{X} - U_{-}) \subseteq U_{+} \text{ and }
f^{-n}(\overline{X} - U_{+}) \subseteq U_{-}.
\end{equation*}
\end{proposition}
We will make use of the following definition.
\begin{definition}\label{def:independent}
Suppose $X$ is a $\delta$--hyperbolic space and $f,g \in \Isom(X)$ are hyperbolic isometries. Let $A_{+}$, $A_{-}$ be the attracting and repelling fixed points of $f$ in $\partial X$ and let $B_{+}$, $B_{-}$ be the attracting and repelling fixed points of $g$ in $\partial X$. We say $f$ and $g$ are \emph{independent} if:
\[ \{A_{+},A_{-}\} \cap \{B_{+},B_{-}\} = \emptyset. \]
Hyperbolic isometries that are not independent are said to be \emph{dependent}.
\end{definition}
\section{A recipe for hyperbolic isometries}\label{sec:actions}
In this section we prove the principal tool used in the proof of the main result of this article, producing a single element in the given group that is hyperbolic for each action. The idea is to start with elements $f$ and $g$ that are hyperbolic for different actions and then combine them into a single element $f^{a}g^{b}$ that is hyperbolic for both actions. A theorem of the first author and Pettet shows that if $g$ does not send the attracting fixed point of $f$ to the repelling fixed point, then $f^{a}g$ is hyperbolic in the first action for large enough $a$. We can reverse the roles to get that $fg^{b}$ is hyperbolic in the second action for large enough $b$. In order to simultaneously work with powers for both $f$ and $g$, we need a uniform version of this result. That is the content of the next theorem, which generalizes Theorem~4.1 in~\cite{ar:CP12-1}.
\begin{theorem}\label{th:uniform hyperbolic}
Suppose $X$ is a $\delta$--hyperbolic space and $f \in \Isom(X)$ is a hyperbolic isometry with attracting and repelling fixed points $A_{+}$ and $A_{-}$ respectively. Fix disjoint neighborhoods $U_{+}$ and $U_{-}$ in $\overline{X}$ for $A_{+}$ and $A_{-}$ respectively. Then there is an $M \geq 1$ such that if $m \geq M$ and $g \in \Isom(X)$ then $f^{m}g$ is a hyperbolic isometry whenever $g U_{+} \cap U_{-} = \emptyset$.
\end{theorem}
The proof follows along the lines of Theorem~4.1 in \cite{ar:CP12-1}. In the following two lemmas we assume the hypotheses of Theorem~\ref{th:uniform hyperbolic}. The first lemma is obvious in the hypothesis of Theorem~4.1 in \cite{ar:CP12-1} but requires a proof in this setting.
\begin{lemma}\label{lem:uniform hyperbolic 1}
Given a point $x \in U_{+} \cap X$ there are constants $t >0 $ and $C \geq 0$ such that if $g \in \Isom(X)$ is such that $g U_{+} \cap U_{-} = \emptyset$ then $d(x,f^{m}gx) \geq mt - C$ for all $m \geq 0$.
\end{lemma}
\begin{proof}
Let $A = \{ f^{n}x | n \in \ZZ\}$ and for $z \in X$ let \[d_{z} = \inf\{ d(x',z) \mid x' \in A\}.\] As $f$ is a hyperbolic isometry, there is a constant $\tau \geq 1$ such that:
\[ \frac{1}{\tau}\abs{m-n} \leq d(f^{m}x,f^{n}x) \leq \tau\abs{m-n}. \] This shows that for any $z \in X$ the set $\pi_{z} = \{ x' \in A \mid d(x',z) = d_{z} \}$ is nonempty and finite.
\medskip \noindent {\bf Claim 1:} {\it There is a constant $D \geq 0$ such that for any $z \in X$ and $x_{z} \in \pi_{z}$:}
\[ d(x,z) \geq d(x,x_{z}) + d(x_{z},z) - D. \]
\begin{proof}[Proof of Claim 1]
Fix a point $x_{z} \in \pi_{z}$ and geodesics $\alpha$ from $x_{z}$ to $x$, $\beta$ from $z$ to $x_{z}$ and $\gamma$ from $z$ to $x$. Let $\Delta$ be the geodesic triangle formed with these segments and $\hat{\alpha} \in \alpha$, $\hat{\beta} \in \beta$ and $\hat{\gamma} \in \gamma$ be the internal points of $\Delta$. These points satisfy the equalities:
\begin{align*}
d(z,\hat{\beta}) = d(z,\hat{\gamma}) &= a \\
d(x,\hat{\gamma}) = d(x,\hat{\alpha}) & = b \\
d(x_{z},\hat{\alpha}) = d(x_{z},\hat{\beta}) & = c
\end{align*}
As insize of geodesic triangles is bounded by $\delta$ in a $\delta$--hyperbolic space, we have that $d(\hat{\alpha},\hat{\beta}), d(\hat{\beta},\hat{\gamma}), d(\hat{\gamma},\hat{\alpha}) \leq \delta$. By the Morse lemma~\cite[III.H.1.7]{bk:BH99}, there is a constant $R$, only depending on $\tau$ and $\delta$, and a point $y \in A$ such that $d(\hat{\alpha},y) \leq R$. Thus we have that:
\[ d(z,y) \leq d(z,\hat{\beta}) + d(\hat{\beta},\hat{\alpha}) + d(\hat{\alpha},y) \leq a + \delta + R.\]
As $x_{z} \in \pi_{z}$ we have:
\[a + c = d(x_{z},z) \leq d(z,y) \leq a + \delta + R\]
and so $c \leq \delta + R$. Letting $D = 2\delta + 2R$ we compute:
\begin{align*}
d(x,z) & = a + b \\
&= (b+c) + (a+c) - 2c \\
&\geq d(x,x_{z}) + d(x_{z},z) - D.\qedhere
\end{align*}
\end{proof}
\medskip \noindent {\bf Claim 2:} {\it There is a constant $M_{0} \in \ZZ$ such that if $z \notin U_{-}$ and $f^{m}x \in \pi_{z}$ then $m \geq M_{0}$.}
\begin{proof}[Proof of Claim 2]
Let $x_{z} = f^{m}x \in \pi_{z}$ and without loss of generality assume that $m \leq 0$. Using the constant $D$ from Claim 1 we have:
\begin{align*}
\GP{x_{z}}{z}_{x} & = \frac{1}{2}\left(d(x,x_{z}) + d(x,z) - d(x_{z},z)\right) \\
& \geq d(x,x_{z}) - D/2.
\end{align*}
Suppose that $i \leq m$ and let $\alpha$ be a geodesic from $f^{i}x$ to $x$. The Morse lemma implies that there is an $y \in \alpha$ such that $d(x_{z},y) \leq R$. Therefore:
\begin{align*}
d(x,x_{z}) + d(x_{z},f^{i}x) &\leq d(x,y) + d(y,f^{i}x) + 2R \\
&= d(x,f^{i}x) + 2R.
\end{align*}
Hence for such $i$ we have:
\begin{align*}
\GP{x_{z}}{f^{i}x}_{x} &= \frac{1}{2}\left(d(x,x_{z}) + d(x,f^{i}x) - d(x_{z},f^{i}x)\right) \\
& \geq d(x,x_{z}) - R.
\end{align*}
This shows that $\GP{x_{z}}{A_{-}}_{x} \geq d(x,x_{z}) - R - 2\delta$ and so for $K = \max\{ D/2,R + 2\delta \}$ we have:
\[ \GP{z}{A_{-}}_{x} \geq \min \left\{ \GP{x_{z}}{z}_{x},\GP{x_{z}}{A_{-}}_{x} \right\} - \delta \geq d(x,x_{z}) - K - \delta \]
As $z \notin U_{-}$, the Gromov product $\GP{z}{A_{-}}_{x}$ is bounded independently of $z$ and hence $d(x,x_{z})$ is also bounded.
\end{proof}
Now we will finish the proof of the lemma. Fix a point $x_{g} \in \pi_{gx}$. Clearly we have $f^{m}x_{g} \in \pi_{f^{m}gx}$ for $m \geq 0$. As $gx \notin U_{-}$, by Claim 2 we have $x_{g} = f^{M_{0}+n}x$ for some $n \geq 0$ and therefore:
\begin{align*}
d(x,f^{m}x_{g}) = d(x,f^{M_{0}+n+m}x) &\geq d(x,f^{m+n}x) - d(x,f^{M_{0}}x) \\
&\geq \frac{1}{\tau}m - \tau \abs{M_{0}}.
\end{align*}
As $f^{m}x_{g} \in \pi_{f^{m}gx}$, Claim 1 implies:
\begin{align*}
d(x,f^{m}gx) &\geq d(x,f^{m}x_{g}) + d(f^{m}x_{g},f^{m}gx) - D \\
& \geq \frac{1}{\tau}m - (\tau\abs{M_{0}} + D).
\end{align*}
Since the constants $\tau$, $D$ and $M_{0}$ only depend on $f$, $x$ and the open neighborhoods $U_{+}$ and $U_{-}$, the lemma is proven.
\end{proof}
The next lemma replaces Lemma~4.3 in~\cite{ar:CP12-1} and its proof is a small modification of the proof there.
\begin{lemma}\label{lem:uniform hyperbolic 2}
Fix $x \in X \cap U_{+}$ and for $m \geq 0$ let $\alpha_{m}$ be a geodesic connecting $x$ to $f^{m}gx$. Then there is an $\epsilon \geq 0$ and $M_{1} \geq 0$ such that for $m \geq M_{1}$ the concatenation of the geodesics $\alpha_{m} \cdot f^{m}g\alpha_{m}$ is a $(1,\epsilon)$-quasi-geodesic.
\end{lemma}
\begin{proof}
Let $d_{m} = d(x,f^{m}gx)$.
As $gU_{+} \cap U_{-} = \emptyset$ we have $U_{+} \cap g^{-1}U_{-} = \emptyset$ and so the Gromov product $\GP{g^{-1}f^{-m}x}{f^{m}x}_{x}$ is bounded independent of $g$ and $m \geq M_{1}$ for some constant $M_{1}$. Indeed, by Proposition~\ref{prop:basis} there is a $k \geq 0$ such that $N(A_{+},k) \subseteq U_{+}$ and $M_{1} \geq 0$ such that $f^{-m}x \in U_{-}$ and $f^{m}x \in N(A_+,k + 2\delta)$ for $m \geq M_{1}$. Hence $\GP{A_{+}}{g^{-1}f^{-m}x}_{x} \leq k$ and so $\GP{g^{-1}f^{-m}x}{f^{m}x}_{x} \leq k + \delta$ as:
\begin{equation*}
\min\{\GP{A_{+}}{f^{m}x}_{x}, \GP{g^{-1}f^{-m}x}{f^{m}x}_{x}\} - \delta \leq \GP{A_{+}}{g^{-1}f^{-m}x}_{x} \leq k
\end{equation*}
for $m \geq M_{1}$.
By making $M_{1}$ larger, we can assume that for $m \geq M_{1}$ we have $f^{m}(\overline{X} - U_{-}) \subseteq N(A_+,k+4\delta)$ by Proposition~\ref{prop:ns}. Since $gx, x \notin U_{-}$, we have that $f^{m}gx,f^{m}x \in N(A_+,k+4\delta)$ and so $\GP{f^{m}xg}{f^{m}x}_{x} \geq k+3\delta$. Hence $\GP{g^{-1}f^{-m}x}{f^{m}gx}_{x} \leq k + 2\delta$ as:
\begin{multline*}
\min \left\{ \GP{g^{-1}f^{-m}x}{f^{m}gx}_{x},\GP{f^{m}gx}{f^{m}x}_{x} \right\} - \delta \leq \GP{g^{-1}f^{-m}x}{f^{m}x}_{x} \leq k + \delta.
\end{multline*}
Therefore for $C = k + 2\delta$ and $m \geq M_{1}$ we have:
\begin{align*}
d(x,f^{m}gf^{m}gx) &= d(g^{-1}f^{-m}x,gf^{m}x) \\
& \geq d(g^{-1}f^{-m}x,x) + d(x,f^{m}gx) - 2C \\
& = 2d_{m} - 2C.
\end{align*}
The proof now proceeds exactly as that of Lemma~4.3 in~\cite{ar:CP12-1}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{th:uniform hyperbolic}] Using lemmas~\ref{lem:uniform hyperbolic 1} and \ref{lem:uniform hyperbolic 2} the proof of Theorem~\ref{th:uniform hyperbolic} proceeds exactly like that of Theorem~4.1 in~\cite{ar:CP12-1}. We repeat the argument here.
Fix $x \in U_{+} \cap X$, and let $t >0$ and $C \geq 0$ be the constants from Lemma~\ref{lem:uniform hyperbolic 1} ,and $\epsilon > 0$ and $M_{1} \geq 0$ be the constants from Lemma~\ref{lem:uniform hyperbolic 2}. For $m \geq M_{1}$ we set $L_{m} = d(x,f^mgx) \geq mt - C$. As in Lemma~\ref{lem:uniform hyperbolic 2}, let $\alpha_{m} \colon\thinspace [0,L_{m}] \to X$ be a geodesic connecting $x$ to $f^mgx$, and let $\beta_{m} = \alpha_{m} \cdot f^mg\alpha_{m}$. Then define a path $\gamma \colon\thinspace \mathbb{R} \to X$ by:
\[ \gamma = \cdots (f^mg)^{-1}\beta_{m} \bigcup_{\alpha_{m}} \beta_{m} \bigcup_{f^mg\alpha_{m}} f^mg\beta_{m} \bigcup_{(f^mg)^2\alpha_{m}} (f^mg)^2\beta_{m} \cdots \]
See Figure~\ref{fig:concatentation}.
\begin{figure}[ht]
\begin{tikzpicture}[scale=1.75]
\tikzstyle{vertex} =[circle,draw,fill=black,thick, inner sep=0pt,minimum size= 0.75 mm]
\draw[thick] (-3.5,0.5) -- (-3,0) -- (-2,1) (-1,0) -- (0,1) -- (1,0) -- (2,1) -- (3,0) -- (3.5,0.5);
\draw[thick,red] (-2,1) -- (-1,0);
\draw[thick,blue] (-2.9,-0.1) -- (-2,0.8) -- (-1.1,-0.1);
\draw[thick,blue] (-1.9,1.1) -- (-1,0.2) -- (-0.1,1.1);
\node at (-2,0.1) {$(f^mg)^{-1}\beta_{m}$};
\node at (-1,0.7) {$\beta_{m}$};
\node[vertex,label={below:$(f^mg)^{-1}x$}] at (-3,0) {};
\node[vertex,label={below:$f^mgx$}] at (-1,0) {};
\node[vertex,label={below:$(f^mg)^{3}x$}] at (1,0) {};
\node[vertex,label={below:$(f^mg)^{5}x$}] at (3,0) {};
\node[vertex,label={above:$x$}] at (-2,1) {};
\node[vertex,label={above:$(f^mg)^{2}x$}] at (0,1) {};
\node[vertex,label={above:$(f^mg)^{4}x$}] at (2,1) {};
\end{tikzpicture}
\caption{The path $\gamma$ in the proof of Theorem~\ref{th:uniform hyperbolic}.}\label{fig:concatentation}
\end{figure}
By Lemma~\ref{lem:uniform hyperbolic 2}, $\gamma$ is an $L_{m}$--local $(1,\epsilon)$--quasi-geodesic and hence for $m$ large enough, $\gamma$ is a $(\lambda',\epsilon')$--quasi-geodesic from some $\lambda' \geq 1$ and $\epsilon' \geq 0$ (see~\cite[III.H.1.7 and III.H.1.13]{bk:BH99} or \cite[Theorem~4.4]{ar:CP12-1}).
Let $N$ be such that $t = \frac{1}{\lambda'}L_{m}N - \epsilon' > 0$. Then for any $k \neq\ell \in \ZZ$ we have
\begin{equation*}
d((f^mg)^{Nk}x,(f^mg)^{N\ell}x) \geq \frac{1}{\lambda'}L_{m}N\abs{k - \ell} - \epsilon' \geq t \abs{k - \ell}.
\end{equation*}
Thus $(f^{m}g)^{N}$ is hyperbolic and therefore so is $f^{m}g$.
\end{proof}
We conclude this section with an application of Theorem~\ref{th:uniform hyperbolic} to dependent hyperbolic isometries (Theorem~\cite[Theorem~4.1]{ar:CP12-1} would suffice as well).
\begin{proposition}\label{prop:dependent hyperbolic}
Suppose $X$ is a $\delta$--hyperbolic space and $f,g \in \Isom(X)$ are dependent hyperbolic isometries. There is an $N \geq 0$ such that if $n \geq N$ then $fg^{n}$ is hyperbolic.
\end{proposition}
\begin{proof}
Let $A_{+}$, $A_{-}$, $B_{+}$, $B_{-} \in \partial X$ be the attracting and repelling fixed points for $f$ and $g$ respectively. Then $fB_{+} \neq B_{-}$ as one of these points is fixed by $f$. Thus there are neighborhoods $V_{+}$ and $V_{-}$ for $B_{+}$ and $B_{-}$ respectively in $\overline{X}$ such that $fV_{+} \cap V_{-} = \emptyset$. Let $N$ be the constant from Theorem~\ref{th:uniform hyperbolic} applied to this set-up after interchanging the roles of $f$ and $g$. Hence $g^{n}f$, and therefore the conjugate $fg^{n}$ as well, is hyperbolic when $n \geq N$.
\end{proof}
\section{Finding neighborhoods}\label{sec:neighborhoods}
We now need to understand when we can find neighborhoods satisfying the hypotheses of Theorem~\ref{th:uniform hyperbolic} for all powers (or at least lots of powers) of a given $g$. There are two cases that we examine: first when $g$ has a fixed point and second when $g$ is hyperbolic.
\begin{proposition}\label{prop:elliptic}
Suppose $X$ is a $\delta$--hyperbolic space and $f \in \Isom(X)$ is a hyperbolic isometry with attracting and repelling fixed points $A_{+}$ and $A_{-}$ in $\partial X$. Suppose $g \in \Isom(X)$ has a fixed point and consider a sequence of elements $(g_{k})_{k \in \NN} \subseteq \I{g}$. Then either:
\begin{enumerate}
\item there are disjoint neighborhoods $U_{+}$ and $U_{-}$ of $A_{+}$ and $A_{-}$ respectively and a constant $M \geq 1$ such that if $k \geq M$ then $g_{k}U_{+} \cap U_{-} = \emptyset$; or \label{item:elliptic 1}
\item there is a subsequence $(g_{k_{n}})$ so that $g_{k_{n}}A_{+} \to A_{-}$\label{item:elliptic 2}.\end{enumerate}
Further, if $gA_{-} = A_{-}$ then \eqref{item:elliptic 1} holds.
\end{proposition}
\begin{proof}
Let $p \in X$ be such that $gp = p$. Thus $g_{k}p = p$ for all $k \in \NN$.
Fix a system of decreasing disjoint neighborhoods $U_{-}^{k}$ of $A_{-}$ and $U_{+}^{k}$ of $A_{+}$ indexed by the natural numbers so that:
\begin{align*}
\GP{x}{A_{+}}_{p} \geq k + \delta & \mbox{ for } x \in U_{+}^{k} \mbox{, and}\\
\GP{x}{A_{-}}_{p} \geq k + \delta & \mbox{ for } x \in U_{-}^{k}.
\end{align*}
This implies that for any two points $x,x'\in U_{+}^{k}$
we have that
\[ \GP{x}{x'}_{p} \geq \min \{ \GP{x}{A_{+}}_{p},\GP{x'}{A_{+}}_{p} \} - \delta \geq k. \]
Likewise for any two points $y,y' \in U_{-}^{k}$ we have that $\GP{y}{y'}_p\geq k$.
For each $n \in \NN$, define $I_n = \{ k \in \NN \mid g_kU_+^n \cap U_-^n \neq \emptyset \}$. If $I_n$ is a finite set for some $n$, then \eqref{item:elliptic 1} holds for the neighborhoods $U_{-}=U_-^n$ and $U_{+}=U_+^n$ where $M = \max I_{n} + 1$.
Otherwise, there is a strictly increasing sequence $(k_{n})_{n \in \NN}$ such that $k_{n} \in I_{n}$. Hence, for each $n \in \NN$, there is an element $x_{n} \in U_{+}^{n}$ such that $g_{k_{n}}x_{n} \in U_{-}^{n}$. In particular,
\begin{equation}\label{eq:finite then fix 1}
\GP{g_{k_{n}}x_{n}}{A_{-}}_{p}\geq n+\delta.
\end{equation}
On the other hand, since $x_{n} \in U_{+}^{n}$ and $g_{k_{n}}$ fixes the point $p$, we have
\begin{align}\label{eq:finite then fix 2}
\GP{g_{k_{n}}x_{n}}{g_{k_{n}}A_{+}}_{p} & = \GP{g_{k_{n}}x_{n}}{g_{k_{n}}A_{+}}_{g_{k_{n}}p}\notag \\
&= \GP{x_{n}}{A_{+}}_p \geq n+\delta.
\end{align}
Combining \eqref{eq:finite then fix 1} and \eqref{eq:finite then fix 2}, we get $\GP{g_{k_{n}}A_{+}}{A_{-}}_{p}\geq n$ for any $n \in \NN$. Hence \eqref{item:elliptic 2} holds.
Now suppose that $gA_{-} = A_{-}$. As $A_{+} \neq A_{-}$, there is a constant $D \geq 0$ such that $\GP{f^{-k}p}{f^{k}p}_{p} \leq D$ for all $k \in \NN$.
For any $n \in \ZZ$, we have that $\GP{f^{-k}p}{g^{n}f^{-k}p}_{p} \to \infty$ as $k \to \infty$. In particular, for each $n \in \ZZ$, there is a constant $K_{n} \geq 0$ such that $\GP{f^{-k}p}{g^{n}f^{-k}p}_{p} \geq D + \delta$ for $k \geq K_{n}$. Therefore $\GP{g^{n}f^{-k}p}{f^{k}p}_{p} \leq D + \delta$ for $k \geq K_{n}$ as:
\[ \GP{f^{-k}p}{f^{k}p}_{p} \geq \min\left\{\GP{f^{-k}p}{g^{n}f^{-k}p}_{p},\GP{g^{n}f^{-k}p}{f^{k}p}_{p} \right\} - \delta. \] As $gp = p$, we have $\GP{f^{-k}p}{g^{n}f^{k}p}_{p} = \GP{g^{-n}f^{-k}p}{f^{k}p}_{p}$ and so we see that $\GP{f^{-k}p}{g^{n}f^{k}p}_{p} \leq D + \delta$ for $k \geq K_{-n}$. This shows that \eqref{item:elliptic 2} cannot hold if $gA_{-} = A_{-}$.
\end{proof}
\begin{proposition}\label{prop:independent hyperbolic}
Suppose $X$ is a $\delta$--hyperbolic space and $f, g \in \Isom(X)$ are independent hyperbolic isometries. There are disjoint neighborhoods $U_{+}$ and $U_{-}$ of $A_{+}$ and $A_{-}$ and an $N \geq 1$ such that if $k \geq N$ then $g^{k}U_{+} \cap U_{-} = \emptyset$.
\end{proposition}
\begin{proof}
Let $A_{+}$, $A_{-}$, $B_{+}$, $B_{-} \in \partial X$ be the attracting and repelling fixed points for $f$ and $g$ respectively. As $f$ and $g$ are independent, the set $\{A_{-}, A_{+}, B_{-}, B_{+}\}$ consists of 4 distinct points. Take mutually disjoint open neighborhoods $U_{-}, U_{+}, V_{-}, V_{+}$ of $A_{-}, A_{+}, B_{-}, B_{+}$ respectively. North-South dynamics of the action of $g$ on $\overline{X}$ implies that there exist a $N \geq 1$ such that $g^{k}(\overline{X} - V_{-})\subset V_{+}$ for all $k \ge N$. In particular, $g^{k}U_{+} \subseteq V_{+}$ and since $V_{+} \cap U_{-}=\emptyset$ we see that $g^{k}U_{+} \cap U_{-} = \emptyset$ for $k \geq N$.
\end{proof}
\section{Simultaneously producing hyperbolic isometries}\label{sec:simultaneous}
We can now apply the above propositions via a careful induction to prove the main result.
\begin{theorem}\label{th:constructing hyperbolic actions}
Suppose that $\{ X_{i} \}_{i = 1,\ldots, n}$ is a collection of $\delta$--hyperbolic spaces, $G$ is a group and for each $i = 1,\ldots, n$ there is a homomorphism $\rho_{i} \colon\thinspace G \to \Isom(X_{i})$ such that:
\begin{enumerate}
\item there is an element $f_{i} \in G$ such that $\rho_{i}(f_{i})$ is hyperbolic; and
\item for each $g \in G$, either $\rho_{i}(g)$ has a periodic orbit or is hyperbolic.
\end{enumerate}
Then there is an $f \in G$ such that $\rho_{i}(f)$ is hyperbolic for all $i = 1,\ldots,n$.
\end{theorem}
\begin{proof}
We will prove this by induction. The case $n=1$ obviously holds by hypothesis.
For $n \geq 2$, by induction there is an $f\in G$ such that for $i = 1,\ldots, n-1$ the isometry $\rho_{i}(f) \in \Isom (X_{i})$ is hyperbolic. For $i = 1,\ldots,n-1$, let $A_{+}^{i}, A_{-}^{i} \in \partial X_{i}$ be the attracting and repelling fixed points of the hyperbolic isometry $\rho_{i}(f)$. By hypothesis, there is a $g \in G$ so that $\rho_{n}(g) \in \Isom(X_{n})$ is hyperbolic. Let $B_{+},B_{-} \in \partial X_{n}$ be the attracting and repelling fixed points of the hyperbolic isometry $\rho_{n}(g)$. Our goal is to find $a,b \in \NN$ so that $\rho_{i}(f^{a}g^{b})$ is hyperbolic for each $i = 1,\ldots,n$.
We begin with some simplifications. If $\rho_{n}(f) \in \Isom(X_{n})$ is hyperbolic then there is nothing to prove, so assume that $\rho_{n}(f)$ has a periodic orbit, and so after replacing $f$ by a power we have that $f$ has a fixed point. By replacing $g$ with a power if necessary, we can assume that for $i = 1,\ldots, n-1$ the isometry $\rho_{i}(g)$ is either the identity or has infinite order. In fact, we can assume that $\rho_{i}(g)$ has infinite order. Indeed, if $\rho_{i}(g)$ is the identity, then for all $a,b \in \NN$ we have $\rho_{i}(f^{a}g^{b}) = \rho_{i}(f^{a})$, which is hyperbolic by the inductive hypothesis. Hence any powers for $f$ and $g$ that work for all other indices between $1$ and $n-1$ necessarily work for this index $i$ as well. Again, by replacing $g$ with a power if necessary, we can assume that for each $i = 1, \ldots,n-1$ either $\rho_{i}(g)A_{-}^{i} = A_{-}^{i}$ or $\rho_{i}(g^{b})A_{-}^{i} \neq A_{-}^{i}$ for each $b \in \ZZ - \{0\}$. Finally, replacing $g$ with a further power necessary, we can assume that for each $i = 1, \ldots,n-1$ if $\rho_i(g)$ is not hyperbolic, then it has a fixed point. Analogously, by replacing $f$ with a power if necessary, we can assume that the isometry $\rho_{n}(f)$ has infinite order and that either $\rho_{n}(f)B_{-} = B_{-}$ or $\rho_{n}(f^{a})B_{-} \neq B_{-}$ for $a \in \ZZ - \{0\}$.
There are various scenarios depending on the dynamics of the isometries $\rho_{i}(g)$ and $\rho_{n}(f)$.
Let $E \subseteq \{1,\ldots,n-1\}$ be the subset where the isometries $\rho_{i}(g)$ has a fixed point. Let $H = \{1,\ldots,n-1\} - E$; this is of course the subset where $\rho_{i}(g)$ is hyperbolic. For $i \in H$, let $B_{+}^{i}, B_{-}^{i} \in \partial X_{i}$ be the attracting and repelling fixed points of the hyperbolic isometry $\rho_{i}(g)$. We further identify the subset $H' \subseteq H$ where $\rho_{i}(f)$ and $\rho_{i}(g)$ are independent.
We first deal with the spaces where $\rho_{i}(g)$ is hyperbolic. To this end, fix $i \in H$.
If $i \in H'$, then by Proposition~\ref{prop:independent hyperbolic} there are disjoint neighborhoods $U_{+}^{i},U_{-}^{i} \subset \overline{X_{i}}$ of $A_{+}^{i}$ and $A_{-}^{i}$ respectively and an $N_{i}$ so that for $k \geq N_{i}$ we have $\rho_{i}(g^{k})U_{+}^{i} \cap U_{-}^{i} = \emptyset$. Applying Theorem~\ref{th:uniform hyperbolic} with the neighborhoods $U_{+}$ and $U_{-}$, there is a $M_{i}$ so that for $a \geq M_{i}$ and $b \geq N_{i}$ the element $\rho_{i}(f^{a}g^{b})$ is hyperbolic.
If $i \in H - H'$ then, by Proposition~\ref{prop:dependent hyperbolic}, for each $a \in \NN$ there is a constant $C_{i}(a) \geq 0$ such that the isometry $\rho_{i}(f^{a}g^{b})$ is hyperbolic if $b \geq C_{i}(a)$.
To create a uniform statements in the sequel, for $i \notin H'$ (including $i \in E$), set $C_{i}(a) = 0$ for all $a \in \NN$. Also, set $M_{i} = N_{i} = 0$ for $i \in H - H'$.
Summarizing the situation for far, we let $\mathsf{M_{0}} = \max\{ M_{i} \mid i \in H \}$ and $\mathsf{N_{0}} = \max\{ N_{i} \mid i \in H \}$. Then, at this point, we know that if $i \in H$, $a \geq \mathsf{M_{0}}$ and $b \geq \mathsf{N_{0}}$ then the element $\rho_{i}(f^{a}g^{b})$ is hyperbolic so long as $b \geq C_{i}(a)$.
Next we deal with the spaces where $\rho_{i}(g)$ has a fixed point. To this end, fix $i \in E$.
Let $E' \subseteq E$ be the subset where condition~\eqref{item:elliptic 1} of Proposition~\ref{prop:elliptic} holds using $\rho_{i}(g_{k}) = \rho_{i}(g^{\mathsf{N_{0}}+k})$. The analysis here is similar to the the case when $i \in H'$. By assumption, for $i \in E'$, there are disjoint neighborhoods $U_{+}^{i},U_{-}^{i} \subset \overline{X_{i}}$ of $A_{+}^{i}$ and $A_{-}^{i}$ respectively and an $N_{i}$ so that for $k \geq N_{i}$ we have $\rho_{i}(g_{k})U_{+}^{i} \cap U_{-}^{i} = \emptyset$. Applying Theorem~\ref{th:uniform hyperbolic} with the neighborhoods $U_{+}^{i}$ and $U_{-}^{i}$, there is a $M_{i}$ so that for $a \geq M_{i}$ the element $\rho_{i}(f^{a}g^{b})$ is hyperbolic if $b \geq N_{i}$.
To summarize again, let $\mathsf{M_{1}} = \max\{ M_{i} \mid i \in H \cup E' \}$ and $\mathsf{N_{1}} = \max\{ N_{i} \mid i \in H \cup E' \}$. Then at this point, if $i \in H \cup E'$, $a \geq \mathsf{M_{1}}$ and $b \geq \mathsf{N_{1}}$ then the element $\rho_{i}(f^{a}g^{b})$ if hyperbolic so long as $b \geq C_{i}(a)$.
It remains to deal with $E - E'$; enumerate this set by $\{i_{1},\ldots,i_{\ell}\}$. As condition~\eqref{item:elliptic 1} of Proposition~\ref{prop:elliptic} does not hold for $\rho_{i_{1}}(g_{k}) = \rho_{i_{1}}(g^{\mathsf{N_{0}} + k})$ acting on $X_{i_{1}}$, there is a subsequence $(g^{k_{n}}) \subseteq (g^{\mathsf{N_{0}}+k})$ such that $\rho_{i_{1}}(g^{k_{n}})A_{+}^{i_{1}} \to A_{-}^{i_{1}}$. By iteratively passing to subsequences of $(g^{k_{n}})$, we can assume that for all $i \in E - E'$, either the sequence of points $(\rho_{i}(g^{k_{n}})A_{+}^{i}) \subseteq \partial X_{i}$ converges or is discrete.
Notice that for $i \in E - E'$, the the final statement of Proposition~\ref{prop:elliptic} implies that $\rho_{i}(g)A_{-}^{i} \neq A_{-}^{i}$. Coupling this with one of our earlier simplifications, we have that $\rho_{i}(g^{b})A_{-}^{i} \neq A_{-}^{i}$ for all $b \in \ZZ - \{0\}$. Hence, there is a $K \in \NN$ such that for any $i \in E - E'$ the sequence $(g^{K+k_{n}})$ satisfies either: $\rho_{i}(g^{K+k_{n}})A_{+}^{i} \to p_{i} \neq A_{-}^{i}$ or $(\rho_{i}(g^{K+k_{n}})A_{+}^{i}) \subset \partial X_{i}$ is discrete. Indeed, suppose $\rho_{i}(g^{k_{n}})A_{+}^{i} \to p_{i}$ (nothing new is being claimed in the discrete case). If $p_{i} \notin \{ \rho_{i}(g^{k})A_{-}^{i} \}_{k \in \ZZ}$, then neither is $\rho_{i}(g^{K})p_{i}$ for any $K \in \NN$ so $\rho_{i}(g^{K+k_{n}})A_{+}^{i} \to \rho_{i}(g^{K})p_{i} \neq A_{-}^{i}$. Else, if $p_{i} = \rho_{i}(g^{K_{i}})A_{-}^{i}$, then for $K \neq -K_{i}$ we have $\rho_{i}(g^{K+k_{n}})A_{+}^{i} \to \rho_{i}(g^{K + K_{i}})A_{-}^{i} \neq A_{-}^{i}$. So by taking $K \in \NN$ to avoid the finitely many such $-K_{i}$ we see that the claim holds. Without loss of generality, we can assume that $K \geq \mathsf{N_{1}}$.
Hence for each $i \in E - E'$, by Proposition~\ref{prop:elliptic}, there are disjoint neighborhoods $U_{+}^{i}, U_{-}^{i} \subset \overline{X}$ of $A_{+}^{i}$ and $A_{-}^{i}$ respectively and an $N_{i}$ so that for $n \geq N_{i}$ we have $\rho_{i}(g^{K+k_{n}})U_{+}^{i} \cap U_{-}^{i} = \emptyset$. Applying Theorem~\ref{th:uniform hyperbolic} with the neighborhoods $U_{+}^{i}$ and $U_{-}^{i}$, there is a $M_{i}$ so that for $a \geq M_{i}$ the element $\rho_{i}(f^{a}g^{K+k_{n}})$ is hyperbolic if $n \geq N_{i}$.
Putting all of this together, let $\mathsf{M_{2}} = \max\{M_{i} \mid 1 \leq i \leq n-1 \}$ and let $\mathsf{N_{2}}= \max\{N_{i} \mid i \in E - E' \}$. Thus for all $i = 1,\ldots,n-1$, if $a \geq \mathsf{M_{2}}$, and $n \geq \mathsf{N_{2}}$ then $\rho_{i}(f^{a}g^{K + k_{n}})$ is hyperbolic so long as $K + k_{n} \geq C_{i}(a)$. (Notice that $K + k_{n} \geq K \geq \mathsf{N_{1}}$ by assumption.)
We now work with the action on the space $X_{n}$. Interchanging the roles of $f$ and $g$ and arguing as above using Proposition~\ref{prop:elliptic} to the sequence of isometries $(\rho_{n}(f^{\ell}))$ we either obtain a subsequence $(f^{\ell_{m}}) \subseteq (f^{\ell})$ and constants $\mathsf{M_{3}}$ and $\mathsf{N_{3}}$ so that $\rho_{n}(f^{\ell_{m}}g^{b})$ is hyperbolic if $m \geq \mathsf{M_{3}}$ and $b \geq \mathsf{N_{3}}$.
Fix some $m \geq\mathsf{M_{3}}$ large enough so that $a = \ell_{m} \geq \mathsf{M_{2}}$ and let $\mathsf{C} = \max\{C_{i}(a) \mid 1 \leq i \leq n-1\}$. Now for $n \geq \mathsf{N_{2}}$ large enough so that $b = K + k_{n} \geq \max\{ \mathsf{C},\mathsf{N_{3}} \}$ we have that $\rho_{i}(f^{a}g^{b})$ is hyperbolic for $i=1,\ldots,n$ as desired.
\end{proof}
\section{Application to $\Out(F_{N})$}\label{sec:application}
Let $F_N$ be a free group of rank $N\ge2$. A \emph{free factor system} of $F_{N}$ is a finite collection $\mathcal{A}=\{[A_{1}],[A_{2}],\ldots,[A_{K}] \}$ of conjugacy classes of subgroups of $F_N$, such that there exist a free factorization
\[
F_N=A_1\ast\cdots\ast A_{K}\ast B
\]
where $B$ is a (possibly trivial) subgroup, called a \emph{cofactor}. There is a natural partial ordering among the free factor systems: $\mathcal{A}\sqsubseteq\mathcal{B}$ if for each $[A] \in \mathcal{A}$ there is a $[B] \in \mathcal{B}$ such that $gAg^{-1} < B$ for some $g\in F_N$. In this case, we say that $\mathcal{A}$ is \emph{contained} in $\mathcal{B}$ or $\mathcal{B}$ is an \emph{extension} of $\mathcal{A}$.
Recall, the \emph{reduced rank} of a subgroup $A < F_{N}$ is defined as \[\rerank(A) = \min\{0,\rank(A) - 1\}.\] We extend this to a free factor systems by addition: \[\rerank(\mathcal{A}) = \sum_{k=1}^{K} \rerank(A_{k})\] where $\mathcal{A}=\{[A_{1}],[A_{2}],\ldots,[A_{K}] \}$. An extension $\mathcal{A} \sqsubseteq
\mathcal{B}$ is called a \emph{multi-edge extension} if $\rerank(\mathcal{B}) \geq \rerank(\mathcal{A}) + 2$.
The group $\Out(F_{N})$ naturally acts on the set of free factor systems as follows. Given $\mathcal{A}=\{[A_{1}],[A_{2}],\ldots,[A_{K}] \}$, and $\varphi\in \Out(F_N)$ choose a representative $\Phi\in\Aut(F_N)$ of $\varphi$, a realization $F_N=A_1\ast\cdots\ast A_K\ast B$ of $\mathcal{A}$ and define $\varphi (\mathcal{A})$ to be the free factor system $\{[\Phi(A_1)],\ldots, [\Phi(A_K)]\}$. Given a free factor system $\mathcal{A}$ consider the subgroup $\Out(F_N;\mathcal{A})$ of $\Out(F_N)$ that stabilizes the free factor system $\mathcal{A}$. The group $\Out(F_N;\mathcal{A})$ is called the \emph{outer automorphism group of $F_N$ relative to $\mathcal{A}$}, or the \emph{relative outer automorphism group} if the free factor system $\mathcal{A}$ is clear from context. If $\mathcal{A} = \{[A]\}$, there is a well-defined restriction homomorphism $\Out(F_{N};\mathcal{A}) \to \Out(A)$ we denote by $\varphi \mapsto \varphi\mid_{A}$~\cite[Fact~1.4]{HMpart1}.
For a subgroup $\mathcal{H} < \Out(F_N)$ and $\mathcal{H}$--invariant free factor systems $\mathcal{F}_1\sqsubseteq\mathcal{F}_2$, we say that $\mathcal{H}$ is \emph{irreducible with respect to the extension $\mathcal{F}_1\sqsubseteq\mathcal{F}_2$} if for any $\mathcal{H}$--invariant free factor system $\mathcal{F}$ such that $\mathcal{F}_1\sqsubseteq \mathcal{F} \sqsubseteq \mathcal{F}_2$ it follows that either $\mathcal{F}=\mathcal{F}_1$ or $\mathcal{F}=\mathcal{F}_2$. We sometimes say that $\mathcal{H}$ is \emph{relatively irreducible} if the extension is clear from the context. The subgroup $\mathcal{H}$ is \emph{relatively fully irreducible} if each finite index subgroup $\mathcal{H}' < \mathcal{H}$ is relatively irreducible. For an individual element $\varphi\in \Out(F_{N})$, we say that $\varphi$ is relatively (fully) irreducible if the cyclic subgroup $\langle\varphi\rangle$ is relatively (fully) irreducible.
In close analogy with Ivanov's classification of subgroups of mapping class groups \cite{Iva}, in a series of papers Handel and Mosher gave a classification of finitely generated subgroups of $\Out(F_N)~$\cite{HMIntro, HMpart1, HMpart2, HMpart3, HMpart4}.
\begin{theorem}[{\cite[Theorem~D]{HMIntro}}]\label{HMMain} For each finitely generated subgroup $\mathcal{H} < \IA_{N}(\ZZ/3) < \Out(F_N)$, each maximal $\mathcal{H}$--invariant filtration by free factor systems $\emptyset = \mathcal{F}_{0} \sqsubset \mathcal{F}_{1} \sqsubset \cdots \sqsubset \mathcal{F}_{m} = \{[ F_{N}] \}$, and each $i = 1,...,m$ such that $\mathcal{F}_{i-1}\sqsubset\mathcal{F}_{i}$ is a multi-edge extension, there exists $\varphi\in\mathcal{H}$ which is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$.
\end{theorem}
Here, $\IA_{N}(\ZZ/3)$ is the finite index subgroup of $\Out(F_N)$ which is the kernel of the natural surjection \[
p \colon\thinspace \Out(F_N)\to\ H^{1}(F_N,\mathbb{Z}/3)\cong GL(N,\mathbb{Z}/3).
\] For elements in $\IA_{N}(\ZZ/3)$, irreducibility is equivalent to full irreducibility hence in the above statement we can also conclude that $\varphi$ is fully irreducible~\cite[Theorem~B]{HMIntro}.
Handel and Mosher conjecture that there is a single $\varphi\in \mathcal{H}$ which is (fully) irreducible for each multi-edge extension $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$~\cite[Remark following Theorem~D]{HMIntro}. The goal of this section is to prove this conjecture. Invoking theorems of Handel--Mosher and Horbez--Guirardel, this is (essentially) an immediate application of Theorem~\ref{th:constructing hyperbolic actions}. We state the set-up and their theorems now.
\begin{definition}\label{def:rff}
Let $\mathcal{A}$ be a free factor system of $F_{N}$. The \emph{complex of free factor systems of $F_{N}$ relative to $\mathcal{A}$}, denoted $\mathcal{FF}(F_{N};\mathcal{A})$, is the geometric realization of the partial ordering $\sqsubseteq$ restricted to proper free factor systems that properly contain $\mathcal{A}$.
\end{definition}
If $\mathcal{A} = \{[A_{1}],[A_{2}],\ldots,[A_{K}] \}$ is a free factor system for $F_{N}$, its \emph{depth} is defined as:
\[ {\rm D}_{\mathcal{FF}}(\mathcal{A}) = (2N-1) - \sum_{k=1}^{K} \bigl(2\rank(A_{k}) -1\bigr) \]
The free factor system $\mathcal{A}$ is \emph{nonexceptional} if ${\rm D}_{\mathcal{FF}}(\mathcal{A}) \geq 3$.
\begin{theorem}[{\cite[Theorem~1.2]{un:HM-relative}}]\label{hyperbolicity}
For any nonexceptional free factor system $\mathcal{A}$ of $F_{N}$, the complex $\mathcal{FF}(F_{N};\mathcal{A})$ is positive dimensional, connected and $\delta$--hyperbolic.
\end{theorem}
Although the group $\Out(F_N)$ does not act on $\mathcal{FF}(F_{N};\mathcal{A})$, the natural subgroup $\Out(F_N;\mathcal{A})$ associated to the free factor system $\mathcal{A}$ acts on $\mathcal{FF}(F_{N};\mathcal{A})$ by simplicial isometries. In a companion paper Handel and Mosher characterize the elements of $\Out(F_N;\mathcal{A})$ that act as a hyperbolic isometry of $\mathcal{FF}(F_{N};\mathcal{A})$:
\begin{theorem}[\cite{un:HM-relative2}]\label{loxod} For any nonexceptional free factor system $\mathcal{A}$ of $F_N$, $\varphi\in\Out(F_N;\mathcal{A})$ acts as a hyperbolic isometry on $\mathcal{FF}(F_{N};\mathcal{A})$ if and only if $\varphi$ is fully irreducible with respect to $\mathcal{A} \sqsubset \{[F_{N}]\}$.
\end{theorem}
\begin{remark} \label{mainremark} An alternative proof of Theorem \ref{loxod} is given by Guirardel and Horbez in \cite{un:GH} using the description of the boundary of the relative free factor complex. Further, with a slight modification of the definition of the relative free factor complex, both Handel and Mosher and Guirardel and Horbez can additionally prove that the theorem holds for the only remaining multi-edge configuration which is when $\mathcal{A}=\{[A_1],[A_2],[A_3]\}$ and $F_N=A_1\ast A_2\ast A_3$. Yet another proof of Theorem \ref{loxod} when the cofactor is non-trivial is given by Radhika Gupta in \cite{un:Gupta} using dynamics on relative outer space and relative currents.
\end{remark}
We are now ready to prove our application:
\begin{theorem}\label{th:application}
For each finitely generated subgroup $\mathcal{H} < \IA_{N}(\ZZ/3)<\Out(F_N)$ and each maximal $\mathcal{H}$--invariant filtration by free factor systems $\emptyset = \mathcal{F}_{0} \sqsubset \mathcal{F}_{1} \sqsubset \cdots \sqsubset \mathcal{F}_{m} = \{[ F_{N}] \}$, there is an element $\varphi \in \mathcal{H}$ such that for each $i = 1,\ldots,m$ such that $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ is a multi-edge extension, $\varphi$ is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$.
\end{theorem}
\begin{proof}
Let $I$ be the subset of indices $i$ such that $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ is a multi-edge extension.
Given $i \in I$, since $\mathcal{H} < \IA_{N}(\ZZ/3)$, each component of $\mathcal{F}_{i-1}$ and $\mathcal{F}_i$ is $\mathcal{H}$--invariant~\cite[Lemma~4.2]{HMpart2}. Moreover, by the argument at the beginning of Section~2.1 in~\cite{HMpart4}, since $\mathcal{H}$ is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ (this follows from maximality of the filtration) there is precisely one component $[B_{i}] \in \mathcal{F}_{i}$ that is not a component of $\mathcal{F}_{i-1}$. Let $\widehat{\mathcal{A}}_{i}$ be the maximal subset of $\mathcal{F}_{i-1}$ such that $\widehat{\mathcal{A}}_{i} \sqsubset \{[B_{i}]\}$. Notice that this extension is again multi-edge, indeed $\rerank(B_{i}) - \rerank(\widehat{\mathcal{A}}_{i}) = \rerank(\mathcal{F}_{i}) - \rerank(\mathcal{F}_{i-1})$. The system $\widehat{\mathcal{A}}_{i}$ can be represented by $\{ [A_{i,1}],\ldots, [A_{i,K_{i}}] \}$ where $A_{i,k} < B_{i}$ for each $k$. Let $\mathcal{A}_{i}$ be the free factor system in the subgroup $B_{i}$ consisting of the conjugacy classes in $B_{i}$ of the subgroups $A_{i,k}$. Then a given $\varphi \in \mathcal{H}$ is irreducible with respect to $\widehat{\mathcal{A}}_{i} \sqsubset \{[B_{i}]\}$, equivalently $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ as the remaining components are the same, if and only if the restriction $\varphi\mid_{B_{i}} \in \Out(B_{i};\mathcal{A}_{i})$ is irreducible relative to $\mathcal{A}_{i}$.
For $i \in I$, let $X_{i} = \mathcal{FF}(B_{i};\mathcal{A}_{i})$ and consider the action homomorphism $\rho_{i} \colon\thinspace \mathcal{H} \to \Isom(X_{i})$ defined by $\rho_{i}(\varphi) = \varphi\mid_{B_{i}}$. These spaces are $\delta$--hyperbolic for some $\delta$ by Theorem~\ref{hyperbolicity} and by the above discussion and Theorem~\ref{loxod}, $\rho_{i}(\varphi)$ is a hyperbolic isometry if $\varphi \in \mathcal{H}$ is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$. If $\rho_{i}(\varphi)$ is not irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$, then $\rho_{i}(\varphi)$ fixes a point in $X_{i}$. By Theorem~\ref{HMMain}, for each $i \in I$, there exist some $\varphi_{i} \in\mathcal{H}$ that is irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ and hence $\rho_{i}(\varphi_{i})$ is a hyperbolic isometry.
We are now in the model situation of Theorem~\ref{th:constructing hyperbolic actions}. We conclude that there is a $\varphi \in \mathcal{H}$ such that $\rho_{i}(\varphi)$ is a hyperbolic isometry for all $i \in I$. By the above discussion, this means that $\varphi$ is (fully) irreducible with respect to $\mathcal{F}_{i-1} \sqsubset \mathcal{F}_{i}$ for each $i \in I$ as desired.
\end{proof}
|
2,869,038,154,881 | arxiv | \section{Introduction}
\label{sec:intro}
A contact manifold $(M,\xi)$ is a $2n+1$ dimensional manifold
equipped with a completely non-integrable distribution of rank $2n$,
called a contact structure. Complete non-integrability of $\xi$
can be expressed by the non-vanishing property
$$
\lambda \wedge (d\lambda)^n \neq 0
$$
for a $1$-form $\lambda$ which
defines the distribution, i.e., $\ker \lambda = \xi$. Such a
$1$-form $\lambda$ is called a contact form associated to $\xi$.
Associated to the given contact form $\lambda$, there is the unique vector field $X_\lambda$ named the Reeb vector field,
determined by
$$
X_{\lambda} \rfloor \lambda \equiv 1, \quad X_{\lambda} \rfloor d\lambda \equiv 0.
$$
In relation to the study of pseudo-holomorphic curves, one considers an
endomorphism $J: \xi \to \xi$ with $J^2 = -id|_\xi$ and regard $(\xi,J)$ as
a complex vector bundle. In the presence of the contact form $\lambda$,
one usually considers the set of $J$ that is compatible to $d\lambda$ in
the sense that the bilinear form $g_\xi = d\lambda(\cdot, J\cdot)$ defines
a Hermitian vector bundle $(\xi,J, g_\xi)$ on $M$. We call the triple $(M,\lambda,J)$
a \emph{contact triad}.
Motivated by the already well-established Gromov's theory of pseudo-holomorphic
curves, in his study of the Weinstein's conjecture in 3 dimension for over-twisted contact
structure, Hofer \cite{hofer} (and others follow) studied pseudo-holomorphic curves in
the symplectization $W = {\mathbb R}_+ \times M$ with symplectic form $\omega = d(r \lambda)$
or in ${\mathbb R} \times M$ with $\omega = d(e^{s} \lambda)$ with $r = e^{s}$
with cylindrical almost complex structure given by
$$
J = J_0 \oplus J,\quad TW \cong {\mathbb R}\left\{\frac{\partial}{\partial s}\right\} \oplus {\mathbb R}\{X_\lambda\} \oplus \xi
$$
where $J_0: {\mathbb R}\left\{\frac{\partial}{\partial s}\right\} \oplus {\mathbb R}\{X_\lambda\} \to {\mathbb R}\left\{\frac{\partial}{\partial s}\right\} \oplus {\mathbb R}\{X_\lambda\}$
is the endomorphism determined by
$
J_0\frac{\partial}{\partial s} = X_\lambda, \, J_0 X_\lambda = -\frac{\partial}{\partial s}.
$
A map $u=(a, w): (\Sigma,j) \to W={\mathbb R} \times M$ is $J$-holomorphic if and only if its components
$a$ and $u$ satisfy the equations
$$
\begin{cases}
\pi_\lambda\left(\frac{\partial w}{\partial \tau}\right)
+ J(w) \pi_\lambda\left(\frac{\partial w}{\partial t}\right) = 0 \\
w^*\lambda\circ j = da.
\end{cases}
$$
Aimed at a better understanding of the contact manifold itself instead of its symplectization,
we focus on looking at the smooth maps $w:\dot\Sigma\to M$ from the (punctured) Riemann surface
$(\dot\Sigma, j)$ to the contact manifold $M$ itself.
By decomposing the tangent bundle into $TM=\xi\oplus {\mathbb R}\{X_\lambda\}$ and denoting the projection
to $\xi$ by $\pi$, one can further decompose $d^\pi w := \pi dw = \partial^\pi w + {\overline \partial}^\pi w$
into the $J$-linear and anti-$J$-linear part as $w^*\xi$-valued
$1$-forms on the punctured Riemann surface $\dot\Sigma$.
We start with the maps $w$ satisfying just ${\overline \partial}^\pi w = 0$, which is
a nonlinear degenerate elliptic equation.
\begin{defn}[Contact Cauchy-Riemann Map] Let $(M,\lambda, J)$ be a contact triad
and let $(\dot\Sigma,j)$ be a (punctured) Riemann surface.
We call any smooth map $w: \dot\Sigma \to M$ a \emph{contact Cauchy-Riemann map}
if it satisfies ${\overline \partial}^\pi w = 0$.
\end{defn}
To maximize the advantage of using the tensor calculus in the analytic study of
the contact Cauchy-Riemann maps in the present paper, we use the \emph{contact triad connection}
the present authors introduced in \cite{oh-wang1} associated to the contact triad $(M,\lambda,J)$:
The contact triad connection in particular preserves the triad metric defined by
$$
g = g_\xi+ \lambda \otimes \lambda.
$$
In Section \ref{sec:connection}, we recall further properties of the connection proved in \cite{oh-wang2},
which will enable us to perform tensor computations in an efficient way.
Denote by $\nabla$ the contact triad connection on $M$ and $\nabla^\pi$ the associated
Hermitian connection on the Hermitian vector bundle $(\xi,d\lambda|_\xi,J)$,
various symmetry properties carried by the connections $\nabla$ and $\nabla^\pi$
enable us to derive the following precise formulae concerning the second covariant differential of
$w$ and the Laplacian of the $\pi$-harmonic energy density function
for any contact Cauchy-Riemann map $w$.
\begin{thm}[Fundamental Equation]
Let $w$ be a contact Cauchy-Riemann map. Then
$$
d^{\nabla^\pi}(d^\pi w) = -w^*\lambda\circ j \wedge\left( \frac{1}{2}({\mathcal L}_{X_\lambda}J)\, \partial^\pi w\right).
$$
\end{thm}
Define the $\xi$-component of the standard harmonic energy density function by
$
e^\pi : = |d^\pi w|^2 := |\pi dw|^2
$, and further introduce the following
\begin{defn}\label{defn:pi-harmonic-energy} For any smooth map $w: \dot\Sigma \to M$,
$$
E_{(\lambda,J)}^\pi(w,j): = \frac{1}{2} \int_{\dot \Sigma} e^\pi=\frac{1}{2} \int_{\dot \Sigma} |d^\pi w|^2
$$
and call it the \emph{$\pi$-harmonic energy} of the smooth map $w$.
\end{defn}
Since we do not vary $j, \, J$ in the present paper, we will just denote $E^\pi(w) := E_{(\lambda,J)}^\pi(w,j)$
from now on.
\begin{thm}
Let $w$ be a contact Cauchy-Riemann map. Then
\begin{eqnarray*}
- \frac{1}{2} \Delta e^\pi & = & |\nabla^\pi(\partial^\pi w)|^2 + K\, |\partial^\pi w|^2
+ \langle \mbox{\rm Ric}^{\nabla^\pi}(w) \partial^\pi w, \partial^\pi w\rangle\\
&{}& + \langle \delta^{\nabla^\pi} \left((w^*\lambda \circ j) \wedge ({\mathcal L}_{X_\lambda}J) \partial^\pi w\right), \partial^\pi w \rangle
\end{eqnarray*}
where $K$ is the Gaussian curvature of the given K\"ahler metric $h$ on $(\dot\Sigma,j)$
and $\textup{Ric}^{\nabla^\pi}(w)$ is the Ricci curvature operator of the contact Hermitian connection
$\nabla^\pi$ along the map $w$.
\end{thm}
Notice that due to the dimension reason, the contact Cauchy-Riemann map itself is not an elliptic system. To conduct geometric analysis, we augment
the equation ${\overline \partial}^\pi w = 0$ by another equation
$$
d(w^*\lambda \circ j) = 0,
$$
and define
\begin{defn}[Contact Instanton] Let $(\dot\Sigma, j)$ be a (punctured) Riemann surface as above.
We call a pair $(j,w)$ of $j$ a complex structure on $\dot\Sigma$ and a map $w:\dot \Sigma \to M$ a
\emph{contact instanton} if they satisfy
\begin{equation}\label{eq:contact-instanton}
{\overline \partial}^\pi w = 0, \, \quad d(w^*\lambda \circ j) = 0.
\end{equation}
\end{defn}
We would like to point out that the system \eqref{eq:contact-instanton} (for a fixed $j$)
forms an elliptic system, which is a natural elliptic twisting of the
Cauchy-Riemann equation ${\overline \partial}^\pi w = 0$. (We refer to \cite{oh:sigmamodel} for the elaboration
of this point of view.)
Another point which is worthwhile to point out is that while the first part of the equation involves
first derivatives, the second part of the equation involves second derivatives of $w$.
Therefore it is not enough to have $W^{2,2}$ a priori estimate to get a classical
solution out of a weak solution, and hence establishing at least $W^{3,2}$ coercive estimate
is crucial to start the bootstrapping arguments.
The following a priori local $C^k$-estimates for such a map is derived by using tensorial calculations with the help of the contact triad connection.
\begin{thm}\label{thm:local-higher-regularity}
Let $(\dot \Sigma,j)$ be a be a punctured Riemann surface possibly with
empty set of punctures. Let $w: \dot\Sigma \to M$ satisfy \eqref{eq:contact-instanton} and
$
\quad \|dw\|_{C^0} < \infty
$
on $\dot\Sigma$. For any open domains $D_1$ and $D_2$ in $\dot\Sigma$ such that $\overline{D}_1\subset \textup{Int}(D_2)$,
$$
\|dw\|^2_{W^{k,2}(D_1)}\leq {\mathcal P}(k; D_1,D_2)\left(\|dw\|^2_{L^2(D_2)}, \|dw\|^4_{L^4(D_2)}\right)
$$
for any contact instanton $w$, where ${\mathcal P}(k; D_1,D_2)(s, t)$ is some polynomial function of $s, \, t$
up to $0, \, \ldots, k$ of degree at most $2k + 4$ depending also on $D_1$, $D_2$ and $\|\textup{Ric}^{\nabla^\pi}\|_{C^k}$,
$\|{\mathcal L}_{X_\lambda}J\|_{C^k}$ and $\|K\|_{C^k;D_2}$, but independent of $w$.
In particular, any weak solution of \eqref{eq:contact-instanton} in
$W^{1,4}_{\textup{loc}}$ automatically becomes the classical solution.
\end{thm}
We also establish the following global $W^{k,2}$-estimates in terms of
$|d^\pi w| \in L^2 \cap L^4$ and $\|w^*\lambda\|_{C^0}< \infty$
on $\dot \Sigma$.
\begin{thm} Let $(\dot\Sigma,j)$ and $w$ satisfying \eqref{eq:contact-instanton} on $\dot \Sigma$ as above.
If $|d^\pi w| \in L^2 \cap L^4$ and $\|w^*\lambda\|_{C^0}< \infty$ on $\dot \Sigma$, then
$$
\int_{\dot \Sigma} |(\nabla)^{k+1}(dw)|^2 \leq \int_{\dot \Sigma} J_{k}'(d^\pi w, w^*\lambda).
$$
Here $J'_{k+1}$ a polynomial function of the norms of the covariant derivatives
of $d^\pi w, \, w^*\lambda$ up to $0, \, \ldots, k$ with degree at most $2k + 4$
whose coefficients depend on
$$
\|K\|_{C^k}, \|R^\pi\|_{C^k}, \|{\mathcal L}_{X_\lambda}J\|_{C^k}, \, \|w^*\lambda\|_{C^0}.
$$
\end{thm}
We refer to Theorem \ref{thm:higher-regularity} and \ref{thm:local-higher-regularity}
and discussions around them for further expounding of these estimates.
Equipping each puncture with a cylindrical end, for simpleness,
we just look at $w: [0,\infty) \times S^1\to M$ that satisfies
\eqref{eq:contact-instanton}.
There are two natural asymptotic invariants $T$ which we name the
\emph{asymptotic contact action} and $Q$ which we name the
\emph{asymptotic contact charge} defined as
\begin{eqnarray*}
T & := & \frac{1}{2}\int_{[0,\infty) \times S^1} |d^\pi w|^2 + \int_{\{0\}\times S^1}(w|_{\{0\}\times S^1})^*\lambda\\
Q & : = & \int_{\{0\}\times S^1}((w|_{\{0\}\times S^1})^*\lambda\circ j).
\end{eqnarray*}
For the asymptotic behavior of the contact instanton map near each puncture of the punctured
Riemann surface, we provide the following asymptotic convergence result under a suitable finite energy hypothesis.
For this study, it is an important ingredient to classify the \emph{massless instanton}
(i.e., $E^\pi(w) = 0$) on the cylinder ${\mathbb R} \times S^1$ equipped with the standard complex
structure $j$. This is where the main difference between $Q = 0$ and $Q \neq 0$ occurs.
\begin{prop}\label{prop:massless} Let $w: {\mathbb R} \times S^1 \to M$ be a massless contact instanton. Then
there exists a leaf of the Reeb foliation such that
we can write $w_\infty(\tau,t)= \gamma(-Q\, \tau + T\, t)$, where $\gamma$ is a parameterization
of the leaf satisfying $\dot \gamma = X_\lambda(\gamma)$.
In particular, if $T \neq 0$, $\gamma$ is a \emph{closed} Reeb orbit of $X_\lambda$ with period $T$.
In addition if $Q = 0$, $w_\infty$ is $\tau$-translational invariant.
If $T = 0$ and $Q \neq 0$, the leaf may not be a closed leaf but a non-compact
immersed image of ${\mathbb R}$ in general.
\end{prop}
We refer readers to Theorem \ref{thm:subsequence} and around for
more precise assumption for the following theorem.
\begin{thm}\label{thm:subsequence-intro} Let $w$ be any contact instanton on $[0,\infty) \times S^1$ with finite
$\pi$-harmonic energy
$$
E^\pi(w) = \frac{1}{2} \int_{[0,\infty) \times S^1} |d^\pi w|^2 < \infty,
$$
and finite gradient bound
$$
\|dw\|_{C^0;[0,\infty) \times S^1} < \infty.
$$
Then for any sequence $s_k\to \infty$, there exists a subsequence, still denoted by $s_k$, and a
massless instanton $w_\infty(\tau,t)$ (i.e., $E^\pi(w_\infty) = 0$)
on the cylinder ${\mathbb R} \times S^1$ such that
$$
\lim_{k\to \infty}w(s_k + \tau, t) = w_\infty(\tau,t)
$$
uniformly on $K \times S^1$ for any given compact set $K\subset{\mathbb R}$.
Furthermore if $Q = 0$ and $T \neq 0$, where $w_\infty(\tau, t) \equiv \gamma(T\, t)$ for some
closed Reeb orbit $\gamma$ of period $T$, the convergence is exponentially fast.
\end{thm}
Theorem \ref{thm:subsequence-intro} (and Proposition \ref{prop:massless}) generalizes Hofer's subsequence
convergence result, which was proved in \cite{hofer} and roughly corresponds to the exact case
(i.e., $Q = 0$ in our setting) in the context of symplectization.
Further, it is well-known from \cite{HWZ96,HWZ98, HWZplane} that, when $(\gamma,T)$ is a
nondegenerate (more general, Morse-Bott type) Reeb orbit, the limit $z$ does not depend on the choice of subsequences and
the convergence is exponentially fast. In this paper, motivated by the method from \cite{mundet-tian}, \cite{oh:book},
we provide a new way, called the \emph{three-interval method}, of proving the
exponential decay for the nondegenerate limiting Reeb orbit when the charge $Q=0$
(and its Morse-Bott analogue in the sequel \cite{oh-wang2}).
From the technical point of view, using the global and canonical tensorial calculations for
the energy density estimates established (see Part \ref{part:analysis}),
we establish not only the a priori estimates in contact manifolds without getting
into symplectization, but also use coordinate-free tensorial calculation
to prove the exponential decay unlike as in \cite{HWZ96,HWZ98, HWZplane}
(as well as \cite{HWZ4}, \cite{bourgeois} for the Morse-Bott case) in which
coordinate calculations using some special coordinates. This tensorial calculation shows its power when combined with
the usage of contact triad connection from \cite{oh-wang1} and
the three-interval method as presented in Section \ref{sec:3-interval} (and \cite{oh-wang2} for the Morse-Bott case).
We would also like to remark that the possibility of applying such three-interval method
to the solution of our contact instanton equations or pseudo-holomorphic curves in symplectization,
highly depends on our new interpretation of the convergence behavior under the
$\pi$-harmonic energy in cylindrical ends, see Section \ref{sec:reeborbits}.
To be specific, the uniform convergence in $\tau$ on any given compact set is the key observation which
seems to have not been exploited in such limiting analysis given before (e.g., \cite{hofer}).
Actually, as pointed out before, for a much more general family of such type geometric PDEs,
with this general fact being observed, the three-interval method applies without any trouble
whenever the limiting operator is of nondegenerate (or Morse-Bott) type and hence the exponential decay follows.
Also from our asymptotic analysis for \eqref{eq:contact-instanton}, the new phenomenon of the appearance of the
`spiraling' instantons along the `rotating' Reeb orbits is observed when the charge is nonzero.
\medskip
As addressed before, our original motivation to study this new elliptic system lies in our attempt
to directly handle the contact manifold itself without taking its symplectization, hoping
that it may give rise to an invariant of contact manifold itself rather than that of
symplectization. Indeed the question if two contact manifolds having symplectomorphic symplectization are
contactomorphic or not was addressed in the book by Cieliebak and Eliashberg
(p. 239 \cite{ciel-eliash}).
In this regard, Courte \cite{courte} recently provided a construction of two contact manifolds that have
symplectomorphic symplectizations which are not contactomorphic (actually, even not diffeomorphic).
With the asymptotic behavior being well-understood in the future, it would be interesting to
see whether our study leads to a construction of genuinely contact
topological quantum invariants of the Gromov-Witten or Floer theoretic type
that can be used to investigate the following kind of question.
(See \cite{courte} where a similar question was explicitly stated.)
\begin{ques}\label{ques:courte} Does there exist contact structures $\xi$ and $\xi'$ on a closed
manifold $M$ that have the same classical invariants and are not contactomorphic,
but whose symplectizations are
(exact) symplectomorphic?
\end{ques}
We would like also to
recall the celebrated result by Ruan \cite{ruan} in symplectic geometry where
using the Gromov-Witten invariant, he discovered
a pair of algebraic surfaces which have the same classical invariants
but whose products with $S^2$ are not symplectically deformation equivalent.
\medskip
We end the introduction section with the following historical recount on the equation \eqref{eq:contact-instanton}.
After the preliminary version of the present paper was posted in the arXiv e-print,
the authors were informed that the equation \eqref{eq:contact-instanton} was first mentioned
by Hofer in p.698 of \cite{hofer-survey}. Then its expected application to the Weinstein conjecture
for dimension $3$ was pursued by Abbas-Cielibak-Hofer in \cite{ACH} and Abbas \cite{abbas},
as well as by Bergmann in \cite{bergmann, bergmann2}.
We would like to point out that their equations correspond to our instanton equations of vanishing charge, i.e., $Q=0$,
and the new asymptotic behavior we addressed here is somehow related to the \emph{compactification} difficulty exposed in \cite{bergmann}.
\bigskip
\part{Geometric analysis of contact Cauchy-Riemann maps}
\label{part:analysis}
In this part, we exercise canonical tensorial calculations involving
the $\pi$-harmonic energy density exploiting various defining properties of
the contact triad connection, and establish the basic local a priori $C^k$
estimates for $k \geq 2$ under the presence of the derivative bound and
the bound for the $\pi$-harmonic energy for the contact Cauchy-Riemann map
and the contact instanton maps.
\section{Review of the contact triad connection}
\label{sec:connection}
A $2n+1$ dimensional smooth manifold $M$ equipped with a co-dimension $1$ distribution $\xi$ is called
a contact manifold, if $\xi$ is the kernel of some locally defined $1$-form $\lambda\in \Omega^1(M)$ with the property that
$\lambda\wedge(d\lambda)^n$ is nowhere vanishing. If $\xi$ is co-oriented, one can choose a global
$1$-form $\lambda$ so that $\ker \lambda = \xi$.
As an immediate consequence from the definition of such $1$-form,
$(\xi, d\lambda|_\xi)$ becomes a symplectic vector bundle
over $M$ of rank $2n$, and we call $\xi$ the \emph{contact structure} or \emph{contact distribution}.
The $1$-form $\lambda$ is called a \emph{contact form} of $\xi$, and it defines the \emph{Reeb vector field} $X_\lambda$
by requiring $X_\lambda\rfloor d\lambda\equiv 0$ together with the normalization condition $\lambda(X_\lambda)\equiv 1$.
With the contact $1$-form $\lambda$ being chosen, there comes a natural splitting of the tangent bundle
$$
TM=\xi\oplus_{\lambda}{\mathbb R}\{X_\lambda\}.
$$
We denote by $\pi_\lambda$ the projection from $TM$ to $\xi$ induced by such splitting, and
$\Pi_\lambda: TM\to TM$ the idempotent
associated to $\pi_\lambda$, i.e., the endomorphism of $TM$ satisfying
${\Pi_\lambda}^2 = \Pi_\lambda$, $\opname{Im} \Pi_\lambda = \xi$, $\ker \Pi_\lambda = {\mathbb R}\{X_\lambda\}$.
We will omit $\lambda$ in the notations whenever there is no danger of confusion.
From the definition of the contact structure, the contact distribution $\xi$
is maximally nonintegrable. The famous Weinstein conjecture states that
every Reeb vector field $X_\lambda$ must have at least one \emph{closed} integral orbit.
We remark that the choice of contact forms is not unique, and for any positive (or negative) function $f\in C^\infty(M)$,
$f\cdot \lambda$ is also a contact $1$-form.
Throughout this paper, we fix the contact form $\lambda$. We introduce a Hermitian structure
to the symplectic vector bundle $(\xi, d\lambda|_\xi)$, i.e., a complex structure $J\in End(\xi)$ with $J^2=-id|_\xi$,
and $d\lambda|_\xi(J\cdot, J\cdot)=d\lambda|_\xi(\cdot, \cdot)$, where the latter is called the compatibility condition for complex structures.
We extend $J$ to an endomorphism of $TM$ by introducing $\tilde JX_\lambda=0$, and we omit $\tilde{}$ if there
occurs no confusion.
We call the triple $(M, \lambda, J)$ a \emph{contact triad} and equip it with the Riemannian metric
$
g=d\lambda|_\xi(\cdot, J\cdot)+\lambda\otimes\lambda
$
which we call the \emph{contact triad metric}.
Denote by $g_\xi:=d\lambda|_\xi(\cdot, J\cdot)$ the Hermitian inner product of the Hermitian bundle $\xi\to M$.
We remark that with the contact triad metric, a contact triad carries the same information as a contact metric manifold (see \cite{blair}).
We remark that since ${\mathcal L}_{X_\lambda}\lambda=0={\mathcal L}_{X_\lambda}d\lambda$,
the Reeb vector field is a Killing vector field with respect to the triad metric if and only if ${\mathcal L}_{X_\lambda}J=0$.
In general, this is an extra requirement, and for contact manifolds of dimension $3$, it is equivalent to the Sasakian condition.
\medskip
In this section, we review the properties of the \emph{contact triad connection} associated to every contact triad $(M, \lambda, J)$
introduced by the authors in \cite{oh-wang1}, where
the existence and uniqueness of such connection were proved.
The contact triad connection enables us to derive several \emph{identities} related to
the energy density formula for contact Cauchy-Riemann maps in coordinate-free forms,
which forms the main content of Part \ref{part:analysis}.
\begin{thm}[Contact Triad Connection \cite{oh-wang1}]\label{thm:connection}
There exists a unique affine connections $\nabla$ associated to
every contact triad $(M,\lambda,J)$ satisfying the following properties:
\begin{enumerate}
\item $\nabla$ is a metric connection of the triad metric, i.e., $\nabla g=0$;
\item The torsion tensor of $\nabla$ satisfies $T(X_\lambda, \cdot)=0$;
\item $\nabla_{X_\lambda} X_\lambda = 0$, and $\nabla_Y X_\lambda\in \xi$ for any $Y\in \xi$;
\item $\nabla^\pi := \pi \nabla|_\xi$ defines a Hermitian connection of the vector bundle
$\xi \to M$ with the Hermitian structure $(d\lambda|_\xi, J)$;
\item The $\xi$ projection of the torsion $T$, denoted by $T^\pi: = \pi T$ satisfies the following property:
\begin{equation}\label{eq:TJYYxi}
T^\pi(JY,Y) = 0
\end{equation}
for all $Y$ tangent to $\xi$;
\item For $Y\in \xi$,
$$
\partial^\nabla_Y X_\lambda = \frac12(\nabla_Y X_\lambda- J\nabla_{JY} X_\lambda)=0.
$$
\end{enumerate}
We name $\nabla$ the contact triad connection.
\end{thm}
The contact triad connection $\nabla$ canonically induces
a Hermitian connection for the Hermitian vector bundle $(\xi,J,g_\xi)$
with $g_\xi = d\lambda(\cdot, J \cdot)|_\xi$.
We denote this vector bundle connection by $\nabla^\pi$ and call it the \emph{contact Hermitian connection},
which will be used to derived $\pi$-energy estimates in later sections.
Recall that the leaf space of Reeb foliations of the contact triad $(M,\lambda,J)$
canonically carries a (non-Hausdorff) almost K\"ahler structure which we denote by
$
(\widehat M,\widehat{d\lambda},\widehat J).
$
We would like to note that Axioms (4) and (5)
are nothing but properties of the canonical connection
on the tangent bundle of the (non-Hausdorff) almost
K\"ahler manifold $(\widehat M, \widehat{d\lambda},\widehat J_\xi)$
lifted to $\xi$. (In fact, as in the almost K\"ahler case, vanishing of $(1,1)$-component
also implies vanishing of $(2,0)$-component and hence the torsion automatically
becomes $(0,2)$-type.)
On the other hand, Axioms (1), (2), (3)
indicate this connection behaves like the Levi-Civita connection when the Reeb direction $X_\lambda$ get involved.
Axiom (6) is an extra requirement to connect the information in $\xi$ part and $X_\lambda$ part,
which uniquely pins down the desired connection.
Moreover, the following fundamental properties of the contact triad connection was
proved in \cite{oh-wang1}, which will be used to perform our tensorial calculations.
\begin{cor}\label{cor:connection}
Let $\nabla$ be the contact triad connection. Then
\begin{enumerate}
\item For any vector field $Y$ on $M$,
\begin{equation}
\nabla_Y X_\lambda = \frac{1}{2}({\mathcal L}_{X_\lambda}J)JY;\label{eq:YX-triad-connection}
\end{equation}
\item $\lambda(T|_\xi)=d\lambda$.
\end{enumerate}
\end{cor}
We end this section with the following
\begin{rem}In \cite{oh-wang1}, the authors considered an ${\mathbb R}$-family of affine connections
associated the contact triad $(M, \lambda, J)$ with the Condition (6) in Theorem \ref{thm:connection}
being modified to
$$
\partial^\nabla_Y X_\lambda = \frac12(\nabla_Y X_\lambda- J\nabla_{JY} X_\lambda)=cY.
$$
The results corresponding to Corollary \ref{cor:connection} in this case become
\begin{enumerate}
\item For any vector field $Y$ on $Q$,
\begin{eqnarray*}
\nabla_Y X_\lambda =-\frac{1}{2}cJY+\frac{1}{2}({\mathcal L}_{X_\lambda}J)JY;
\end{eqnarray*}
\item $\lambda(T|_\xi)=(1+c)d\lambda$.
\end{enumerate}
The contact triad connection corresponds to the one with $c=0$ and the connection with the simplest torsion
in the family corresponds to $c=-1$.
\end{rem}
\section{The contact Cauchy-Riemann maps}
\label{sec:CRmap}
Denote by $(\dot\Sigma, j)$ a punctured Riemann surface (including closed Riemann surfaces without punctures).
\begin{defn}A smooth map $w:\dot\Sigma\to M$ is called a \emph{contact Cauchy-Riemann map}
(with respect to the contact triad $(M, \lambda, J)$), if $w$ satisfies the following Cauchy-Riemann equation
$$
{\overline \partial}^\pi w:={\overline \partial}^{\pi}_{j,J}w:=\frac{1}{2}(\pi dw+J\pi dw\circ j)=0.
$$
\end{defn}
Recall that for a fixed smooth map $w:\dot\Sigma\to M$,
$(w^*\xi, w^*J, w^*g_\xi)$ becomes a Hermitian vector bundle
over the punctured Riemann surface $\dot\Sigma$, which introduces a Hermitian bundle structure on
$Hom(T\dot\Sigma, w^*\xi)\cong T^*\dot\Sigma\otimes w^*\xi$ over $\dot\Sigma$,
with the inner product given by
$$
\langle \alpha\otimes \zeta, \beta\otimes\eta \rangle =h(\alpha,\beta)g_\xi(\zeta, \eta),
$$
where $\alpha, \beta\in\Omega^1(\dot\Sigma)$, $\zeta, \eta\in \Gamma(w^*\xi)$,
and $h$ is the K\"ahler metric on the punctured Riemann surface $(\dot\Sigma, j)$.
Let $\nabla^\pi$ be the contact Hermitian connection.
Combine the pulling-back of this connection and the Levi-Civita connection of the Riemann surface,
we get a Hermitian connection for the bundle $T^*\dot\Sigma\otimes w^*\xi\to \dot\Sigma$,
which we will still denote by $\nabla^\pi$ by a slight abuse of notation.
The smooth map $w$ has an associated $\pi$-harmonic energy density
defined as the norm of the section $d^\pi w:=\pi dw$ of $T^*\dot\Sigma\otimes w^*\xi\to \dot\Sigma$.
In other words, it is a function $e^\pi(w):\dot\Sigma\to {\mathbb R}$ defined by
$
e^\pi(w)(z):=|d^\pi w|^2(z).
$
(Here we use $|\cdot|$ to denote the norm from $\langle\cdot, \cdot \rangle$ which should be clear from the context.)
Similar to the standard Cauchy-Riemann maps for almost Hermitian manifold (i.e., the pseudo-holomorphic curves),
we have
\begin{prop}\label{prop:energy-omegaarea}
Fix a K\"ahler metric $h$ of $(\dot\Sigma,j)$,
and consider a smooth map $w:\dot\Sigma \to M$, then we have
\begin{enumerate}
\item $e^\pi(w):=|d^\pi w|^2 = |\partial^\pi w| ^2 + |{\overline \partial}^\pi w|^2$;
\item $2\, w^*d\lambda = (-|{\overline \partial}^\pi w|^2 + |\partial^\pi w|^2) \,dA $
where $dA$ is the area form of the metric $h$ on $\dot\Sigma$;
\item $w^*\lambda \wedge w^*\lambda \circ j = - |w^*\lambda|^2\, dA$.
\end{enumerate}
As a consequence, if $w$ is a contact Cauchy-Riemann map, i.e., ${\overline \partial}^\pi w=0$, then
\begin{equation}\label{eq:onshell}
|d^\pi w|^2 = |\partial^\pi w| ^2 \quad \text{and}\quad w^*d\lambda = \frac{1}2|d^\pi w|^2 \,dA.
\end{equation}
\end{prop}
\begin{proof} The proofs of (1), (2) are exactly the same as the case of pseudo-holomorphic maps
in symplectic manifolds with replacement of $dw$ by $d^\pi w$ and the symplectic
form by $d\lambda$ and so omitted. (See e.g., Proposition 7.19 \cite{oh:book} for the
statements and their proofs in the symplectic case corresponding the statements (1), (2) here.)
Statement (3) follows from the
definition of the Hodge star operator which shows that for any $1$-form $\beta$ on the Riemann surface
$*\beta=-\beta\circ j$, and we take $\beta=w^*\lambda$.
\end{proof}
Notice that the contact Cauchy-Riemann map itself is \emph{not} an elliptic system
since the symbol is of rank $2n$ which is $1$ dimension lower than $TM$.
Here enters the closedness condition $d(w^*\lambda\circ j)=0$
leading to an elliptic system
\begin{equation}\label{eq:instanton}
{\overline \partial}^\pi w=0, \quad d(w^*\lambda\circ j)=0.
\end{equation}
We name a solution of this system of equations a \emph{contact instanton} which is the main object that we are going to study in the paper.
To illustrate the effect of the closedness condition on the behavior of
contact instantons, we look at them on \emph{closed}
Riemann surface and prove the following classification
result. The following proposition is stated by Abbas as a part of \cite[Proposition 1.4]{abbas}. For readers' convenience,
we separate this part for closed contact instanton (which is named as the homologically
perturbed pseudo-holomorphic curve in \cite{abbas}) and give
a proof somewhat different therefrom.
\begin{prop} Assume $w:\Sigma\to M$ is a smooth contact instanton from a closed Riemann surface.
Then
\begin{enumerate}
\item If $g(\Sigma)=0$, $w$ can only be a constant map;
\item If $g(\Sigma)\geq 1$, $w$ is either a constant or has its locus of its image
is a \emph{closed} Reeb orbit.
\end{enumerate}
\end{prop}
\begin{proof}Since for contact Cauchy-Riemann maps, Proposition \ref{prop:energy-omegaarea} implies
that $|d^\pi w|^2=d(2w^*\lambda)$.
By Stokes' formula, we get $d^\pi w=0$ if the domain is a closed Riemann surface,
and further, $dw=w^*\lambda \otimes X_\lambda$, i.e., $w$ must have its image contained in
a single leaf of the (smooth) Reeb foliation.
Another consequence of the vanishing $d^\pi w=0$
is $dw^*\lambda = 0$.
Now this combined with the equation $d(w^*\lambda\circ j)=0$, which is equivalent to $\delta w^*\lambda=0$,
implies that $w^*\lambda$ (so is $*w^*\lambda$) is a harmonic $1$-form on the Riemann surface $\Sigma$.
If the genus of $\Sigma$ is zero, $w^*\lambda = 0$ by the Hodge's theorem.
This proves the statement (1).
Now assume $g(\Sigma) \geq 1$. Suppose $w$ is not a constant map. Since $\Sigma$ is compact and connected,
$w(\Sigma)$ is compact and connected. Furthermore recall $w(\Sigma)$ is contained in a single leaf
of the Reeb foliation which we denote by
${\mathcal L}$. We take a parametrisation $\gamma: {\mathbb R} \to {\mathcal L} \subset M$ such that $\dot \gamma = X_\lambda(\gamma(t))$.
By the classification of compact one dimensional manifold, the image $w(\Sigma)$ is homeomorphic to the unit
closed interval or the circle. For the latter case, we are done.
For the former case, we denote by $I = \omega(\Sigma)$ which is contained in the leaf ${\mathcal L}$
We slightly extend the interval $I$ to $I' \subset {\mathcal L}$ so that $I'$ still becomes an
embedded interval contained in ${\mathcal L}$. The preimage $\gamma^{-1}(I')$ is a disjoint union of
a sequence of intervals $[\tau_k, \tau_{k+1}]$ with $\cdots < \tau_{-1} < \tau_0 < \tau_1 < \cdots$ for
$k \in {\mathbb Z}$. Fix any single interval, say,
$[\tau_0, \tau_1] \subset {\mathbb R}$.
We denote by $\gamma^{-1}: I' \to [\tau_0,\tau_1] \subset {\mathbb R}$ the inverse of
the parametrisation $\gamma$ restricted to $[\tau_0,\tau_1]$. Then by construction
$\gamma^{-1}(I) \cap [\tau_0,\tau_1] \subset (\tau_0,\tau_1)$.
Now we denote by $t$ the standard coordinate function of ${\mathbb R}$ and
consider the composition $f: = \gamma^{-1} \circ w: \Sigma \to {\mathbb R}$. It follows that $f$ defines
a smooth function on $\Sigma$ satisfying
$$
\gamma \circ f = w
$$
on $\Sigma$ by construction. Then recalling $\dot \gamma = X_\lambda(\gamma)$, we obtain
$$
w^*\lambda = f^*(\gamma^*\lambda) = f^*(dt) = df.
$$
Therefore $\Delta f = \delta df = \delta w^*\lambda = 0$, i.e., $f$
is a harmonic function on the closed surface $\Sigma$ and so must be a constant function.
This in turn implies $w^*\lambda = 0$. Then $dw = d^\pi w + w^*\lambda X_\lambda(w) = 0 + 0 = 0$
i.e., $w$ is a constant map which contradicts to the standing hypothesis. Therefore
the map $w$ must be constant unless the image of $w$
wraps up a closed Reeb orbit.
Combining the above discussion, we have finished the proof.
\end{proof}
\section{Tensorial calculations for geometric energy density function}
\label{sec:calculations}
In this section, we use the contact triad connection to derive some identities related to the $\pi$-harmonic energy for
contact Cauchy-Riemann maps.
Our derivation is based on \emph{coordinate-free} tensorial calculations.
The contact triad connection fits well for this mission which will be seen clearly in this section.
We start with by looking at the (Hodge) Laplacian of the $\pi$-harmonic energy density function.
Standard Weitzenb\"ock formula applied to
$\nabla^\pi$ the pull-back of the contact Hermitian connection for $w^*\xi$ as well as
for $T^*\dot\Sigma \otimes w^*\xi$ as described in Section \ref{sec:CRmap} provides us the following formula
\begin{equation}\label{eq:bochner-weitzenbock-e}
-\frac{1}{2}\Delta e^\pi(w)=|\nabla^\pi(d^\pi w) |^2-\langle \Delta^{\nabla^\pi} d^\pi w, d^\pi w\rangle
+K\cdot |d^\pi w|^2+\langle \textup{Ric}^{\nabla^\pi}(d^\pi w), d^\pi w\rangle,
\end{equation}
where $e^\pi:=e^\pi(w)$, $K$ is the Gaussian curvature of $\dot\Sigma$,
and $\textup{Ric}^{\nabla^\pi}$ is the Ricci tensor of the vector bundle $w^*\xi$
with respect to the connection $\nabla^\pi$.
For the readers' convenience, we give the proof of this formula in Appendix \ref{appen:weitzenbock},
and freely apply the notations used in Section \ref{appen:weitzenbock}.
Up to now, our $w$ is an arbitrary smooth map which is not required to be a contact Cauchy-Riemann map.
For contact Cauchy-Riemann maps, we would like to examine how $d^\pi w(=\partial^\pi w)$
is far away from a $w^*\xi$-valued harmonic $1$-form. We remark that any standard $J$-holomorphic map $u$
in an almost K\"ahler manifold is always a $u^*TM$-valued harmonic $1$-form with respect to the canonical connection
(see \cite{oh:book} for its proof). For the current contact case, we explicitly calculate out
the difference caused by Reeb projection now.
We start with the following calculation of $d^{\nabla^\pi}d^\pi w$ for any smooth map $w$.
\begin{lem}\label{lem:FE-autono} Let $w: \dot\Sigma \to M$ be any smooth map.
As a two-form with values in $w^*\xi$,
$d^{\nabla^\pi} (d^\pi w)$ has the expression
\begin{equation}\label{eq:dnabladpiw}
d^{\nabla^\pi} (d^\pi w)= T^\pi(\Pi dw, \Pi dw)+ w^*\lambda \wedge \left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, Jd^\pi w\right)
\end{equation}
where $T^\pi$ is the torsion tensor of $\nabla^\pi$.
\end{lem}
\begin{proof}
For given $\xi_1, \, \xi_2 \in \Gamma(T\Sigma)$, we evaluate
\begin{eqnarray*}
&{}& d^{\nabla^\pi} (d^\pi w)(\xi_1, \xi_2)\\
&=&(\nabla^\pi_{\xi_1}(\pi dw))(\xi_2)-(\nabla^\pi_{\xi_2}(\pi dw))(\xi_1)\\
&=&\left(\nabla^\pi_{\xi_1}(\pi dw(\xi_2))-\pi dw(\nabla_{\xi_1}\xi_2)\right)
-\left(\nabla^\pi_{\xi_2}(\pi dw(\xi_1))
-\pi dw\left(\nabla_{\xi_2}\xi_1\right)\right)\\
&=& \pi\Big(\left(\nabla_{\xi_1}( dw(\xi_2))-\nabla_{\xi_1}(\lambda(dw(\xi_2))X_\lambda)\right)
-\left(\nabla_{\xi_2}( dw(\xi_1))-\nabla_{\xi_2}(\lambda(dw(\xi_1))X_\lambda)\right)\\
&{}&- dw \left(\nabla_{\xi_1}\xi_2-\nabla_{\xi_1}\xi_2\right)\Big)\\
&=&\pi \left(\nabla_{\xi_1}(dw(\xi_2))-\nabla_{\xi_2}(dw(\xi_1))-[dw(\xi_1), dw(\xi_2)]\right)\\
&{}&-\nabla_{\xi_1}(\lambda(dw(\xi_2))X_\lambda)
+\nabla_{\xi_2}(\lambda(dw(\xi_1))X_\lambda)\Big)\\
&=& \pi(T(dw(\xi_1), dw(\xi_2))-\lambda(dw(\xi_2))\nabla_{\xi_1}X_\lambda-\xi_1[\lambda(dw(\xi_2))]X_\lambda\\
&{}& +\lambda(dw(\xi_1))\nabla_{\xi_2}X_\lambda + \xi_2[\lambda(dw(\xi_1))]X_\lambda\Big) \\
&=&\pi(T(dw(\xi_1), dw(\xi_2)))-\lambda(dw(\xi_2))\nabla_{\xi_1}X_\lambda+\lambda(dw(\xi_1))\nabla_{\xi_2}X_\lambda\\
&=&T^\pi(\Pi dw(\xi_1), \Pi dw(\xi_2))\\
&{}&+\frac{1}{2}\lambda(dw(\xi_2))J({\mathcal L}_{X_\lambda}J)\pi dw(\xi_1)-\frac{1}{2}\lambda(dw(\xi_1))J({\mathcal L}_{X_\lambda}J)\pi dw(\xi_2)\\
&=&T^\pi(\Pi dw(\xi_1), \Pi dw(\xi_2))\\
&{}&-\frac{1}{2}\lambda(dw(\xi_2))({\mathcal L}_{X_\lambda}J)J\pi dw(\xi_1)+\frac{1}{2}\lambda(dw(\xi_1))({\mathcal L}_{X_\lambda}J)J\pi dw(\xi_2).
\end{eqnarray*}
Here we used \eqref{eq:YX-triad-connection} and Axiom (3) for the last second equality.
Rewrite the above result as
$$
d^{\nabla^\pi} (d^\pi w)= T^\pi(\Pi dw, \Pi dw)+ w^*\lambda \wedge \left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, Jd^\pi w\right)
$$
for any $w$, and we have finished the proof.
\end{proof}
We would like to remark that readers should not get confused at the wedge product we used here, which is
the wedge product for forms in the normal sense, i.e., $(\alpha_1\otimes\zeta)\wedge\alpha_2=(\alpha_1\wedge\alpha_2)\otimes\zeta$
for $\alpha_1, \alpha_2\in \Omega^*(P)$ and $\zeta$ is a section of $E$, with the one we defined in Appendix \ref{appen:forms}.
As an immediate corollary of the previous lemma applied to the contact Cauchy-Riemann maps,
we derive the following formula and name it the \emph{fundamental equation}.
This is the contact analogue to the symplectic case in \cite[Proposition 7.27]{oh:book}.
\begin{thm}[Fundamental Equation]\label{thm:Laplacian-w}
Let $w$ be a contact Cauchy-Riemann map, i.e., a solution of ${\overline \partial}^\pi w=0$.
Then
\begin{equation}\label{eq:Laplacian-w}
d^{\nabla^\pi} (d^\pi w) =d^{\nabla^\pi} (\partial^\pi w)= -w^*\lambda\circ j \wedge\left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, \partial^\pi w\right).
\end{equation}
\end{thm}
\begin{proof} The first equality follows since
$d^\pi w=\partial^\pi w$ for the solution $w$. Also notice that being a contact Cauchy-Riemann map, it follows that
$$
T^\pi(\Pi dw,\Pi dw) = T^\pi(\partial^\pi w, \partial^\pi w) = 0,
$$
which is due to the torsion $T^\pi |_\xi$ is of $(0,2)$-type, in particular, has vanishing $(1,1)$-component.
Further we write \eqref{eq:dnabladpiw} as
\begin{eqnarray*}
d^{ \nabla^\pi}(d^\pi w) & = & w^*\lambda
\wedge \left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, J\partial^\pi w\right)\\
& = & w^*\lambda \wedge \left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, \partial^\pi w\right) \circ j\\
&=&-w^*\lambda\circ j \wedge\left(\frac{1}{2} ({\mathcal L}_{X_\lambda}J)\, \partial^\pi w\right),
\end{eqnarray*}
where we use the identity $J\partial^\pi w = \partial^\pi w \circ j$.
\end{proof}
\begin{cor}[Fundamental Equation in Isothermal Coordinates]
Let $(\tau,t)$ be an isothermal coordinates.
Write $\zeta := \pi \frac{\partial w}{\partial \tau}$
as a section of $w^*\xi \to M$. Then
\begin{equation}\label{eq:main-eq-a}
\nabla_\tau^\pi \zeta + J \nabla_t^\pi \zeta
- \frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J)\zeta + \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J)J\zeta =0.
\end{equation}
\end{cor}
\begin{proof}
We denote $\eta = \pi \frac{\partial w}{\partial t}$. By the isothermality of the coordinate $(\tau,t)$, we
have $J \frac{\partial}{\partial \tau} = \frac{\partial}{\partial t}$. Using the $(j,J)$-linearity of $d^\pi w$, we derive
$$
\eta = dw^\pi (\frac{\partial}{\partial t}) = dw^\pi ( j\frac{\partial}{\partial \tau}) = J dw^\pi (\frac{\partial}{\partial \tau}) = J\zeta.
$$
Now we evaluate each side of \eqref{eq:Laplacian-w} against $(\frac{\partial}{\partial\tau}, \frac{\partial}{\partial t})$.
For the left hand side, we get
$$
\nabla^\pi_\tau\eta -\nabla^\pi_t \zeta = \nabla^\pi_\tau J\zeta -\nabla^\pi_t \zeta = J\nabla^\pi_\tau \zeta -\nabla^\pi_t \zeta.
$$
For the right hand side, we get
\begin{eqnarray*}
&{}& \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J) J\eta
-\frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J) J\zeta\\
& = & - \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J) \zeta
-\frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J) J\zeta
\end{eqnarray*}
where we use the equation $\eta = J \zeta$ for the equality. By setting them equal and applying $J$
to the resulting equation using the fact that ${\mathcal L}_{X_\lambda}J$ is anti-commuting with $J$, we
have obtained the equation.
\end{proof}
We are going to use the fundamental equation under the cylindrical coordinate $(\tau, t)\in [0,\infty)\times S^1$ to derive the exponential decay at cylindrical ends
in Part \ref{part:exponential}.
\begin{rem}
The fundamental equation in cylindrical coordinates is nothing but the linearization of the
contact Cauchy-Riemann equation in the direction $\frac{\partial}{\partial\tau}$.
\end{rem}
The following lemmas regarding the calculation of
$\langle \Delta^{\nabla^\pi} d^\pi w, d^\pi w\rangle$ for contact Cauchy-Riemann maps
$d^\pi w=\partial^\pi w$ will be needed in the next section.
\begin{lem}\label{lem:2delta}
For any smooth map $w$, we have
\begin{eqnarray*}
\langle d^{\nabla^\pi} \delta^{\nabla^\pi}\partial^\pi w, \partial^\pi w\rangle=\langle \delta^{\nabla^\pi}d^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle.
\end{eqnarray*}
As a consequence,
\begin{eqnarray}
\langle \Delta^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle=2\langle \delta^{\nabla^\pi}d^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle.\label{eq:2Delta}
\end{eqnarray}
\end{lem}
\begin{proof}
\begin{eqnarray}
\langle \delta^{\nabla^\pi}d^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle &=&-\langle *d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle\nonumber\\
&=&-\langle d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w, *\partial^\pi w\rangle\nonumber\\
&=&-\langle d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w, -\partial^\pi w\circ j\rangle\label{eq:dstard1}\\
&=&\langle d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w, J \partial^\pi w\rangle\nonumber\\
&=&-\langle J d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w, \partial^\pi w\rangle\nonumber\\
&=&-\langle d^{\nabla^\pi} * d^{\nabla^\pi} J\partial^\pi w, \partial^\pi w\rangle\label{eq:dstard3}\\
&=&-\langle d^{\nabla^\pi} * d^{\nabla^\pi} \partial^\pi w\circ j, \partial^\pi w\rangle\nonumber\\
&=&\langle d^{\nabla^\pi} * d^{\nabla^\pi} * \partial^\pi w, \partial^\pi w\rangle\label{eq:dstard5}\\
&=&\langle d^{\nabla^\pi} \delta^{\nabla^\pi}\partial^\pi w, \partial^\pi w\rangle.\nonumber
\end{eqnarray}
Here for \eqref{eq:dstard1} and \eqref{eq:dstard5}, we use $*\alpha=-\alpha\circ j$ for any $1$-form $\alpha$.
For \eqref{eq:dstard3}, we use the connection is $J$-linear.
\end{proof}
\begin{lem}
For any contact Cauchy-Riemann map $w$,
\begin{eqnarray*}
-\Delta^{\nabla^\pi}d^\pi w
&=&
\delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w]\\
&=&-*\langle (\nabla^\pi({\mathcal L}_{X_\lambda}J))\partial^\pi w, w^*\lambda\rangle\\
&&
-*\langle ({\mathcal L}_{X_\lambda}J)\nabla^\pi\partial^\pi w, w^*\lambda\rangle
-*\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, \nabla w^*\lambda\rangle.
\end{eqnarray*}
\end{lem}
\begin{proof}The first equality immediately follows from the fundamental equation Theorem \ref{thm:Laplacian-w} for contact Cauchy-Riemann maps.
For the second equality, we calculate by writing
\begin{eqnarray*}
\delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w]=
-*d^{\nabla^\pi}*[({\mathcal L}_{X_\lambda}J)\partial^\pi w\wedge (*w^*\lambda)],
\end{eqnarray*}
and then from the definition of the Hodge $*$ (see Appendix \ref{appen:forms}) to $*[({\mathcal L}_{X_\lambda}J)\partial^\pi w\wedge (*w^*\lambda)]$,
we further get
\begin{eqnarray*}
&&\delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w]\\
&=&-*d^{\nabla^\pi}\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, w^*\lambda\rangle\\
&=&-*\langle (\nabla^\pi({\mathcal L}_{X_\lambda}J))\partial^\pi w, w^*\lambda\rangle
-*\langle ({\mathcal L}_{X_\lambda}J)\nabla^\pi\partial^\pi w, w^*\lambda\rangle
-*\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, \nabla w^*\lambda\rangle.
\end{eqnarray*}
\end{proof}
We would like to remark that, here in the above lemma $\langle\cdot, \cdot\rangle$ denotes the inner product induced from $h$, i.e.,
$\langle\alpha_1\otimes\zeta, \alpha_2\rangle:=h(\alpha_1, \alpha_2)\zeta$,
for any $\alpha_1, \alpha_2\in \Omega^k(P)$ and $\zeta$ a section of $E$,
which should not be confused with the inner product of the vector bundles.
We end this section by writing the Weitzenb\"ock formula \eqref{eq:bochner-weitzenbock-e}
into
\begin{eqnarray}\label{eq:e-pi-weitzenbock}
-\frac{1}{2}\Delta e^\pi(w)&=&|\nabla^\pi (\partial^\pi w)|^2+K|\partial^\pi w|^2+\langle \textup{Ric}^{\nabla^\pi} (\partial^\pi w), \partial^\pi w\rangle\nonumber\\
&{}&+\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle
\end{eqnarray}
for contact Cauchy-Riemann maps, whose derivation has already been shown from the context above.
\section{A priori estimates for contact instantons}
In this section, we derive some basic total energy density estimates for contact instantons. These estimates are important for the derivation of local regularity and
$\epsilon$-regularity needed for the compactification of certain moduli space (though we are not going to provide the compactification in this paper).
\subsection{$W^{2,2}$-estimates}
Recall from the last section that we have derived the following identity
\begin{eqnarray}\label{eq:e-pi-weitzenbock}
-\frac{1}{2}\Delta e^\pi(w)&=&|\nabla^\pi (\partial^\pi w)|^2+K|\partial^\pi w|^2+\langle \textup{Ric}^{\nabla^\pi} (\partial^\pi w), \partial^\pi w\rangle\nonumber\\
&{}&+\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle,
\end{eqnarray}
and the first entry in the last Laplacian term $\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle$
can be decomposed as
\begin{eqnarray}\label{eq:delta-ident}
&{}&\delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w]\nonumber\\
&=&-*\langle (\nabla^\pi({\mathcal L}_{X_\lambda}J))\partial^\pi w, w^*\lambda\rangle
-*\langle ({\mathcal L}_{X_\lambda}J)\nabla^\pi\partial^\pi w, w^*\lambda\rangle
-*\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, \nabla w^*\lambda\rangle.\nonumber\\
\end{eqnarray}
Hence we get the bound for the last term $\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle$
by
\begin{eqnarray*}
&{}&|\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle|\\
&\leq&
\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}|dw|^4\\
&{}&+|\langle ({\mathcal L}_{X_\lambda}J)\nabla^\pi(\partial^\pi w), w^*\lambda\rangle||dw|
+|\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, \nabla w^*\lambda\rangle||dw|.
\end{eqnarray*}
We further bound the last two terms of \eqref{eq:delta-ident} as
\begin{eqnarray*}
|\langle ({\mathcal L}_{X_\lambda}J)\nabla^\pi(\partial^\pi w), w^*\lambda\rangle||dw|&\leq&
\|{\mathcal L}_{X_\lambda}J\|_{C^0(M)}|\nabla^\pi (\partial^\pi w)||dw|^2\\
&\leq&\frac{1}{2c}|\nabla^\pi (\partial^\pi w)|^2+\frac{c}{2}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|dw|^4\\
\text{and }\quad \quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad&{}&\\
|\langle ({\mathcal L}_{X_\lambda}J)\partial^\pi w, \nabla w^*\lambda\rangle||dw|
&\leq&\frac{1}{2c}|\nabla w^*\lambda|^2+\frac{c}{2}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|dw|^4
\end{eqnarray*}
similarly. Here $c$ is any positive constant which will be determined later.
Above all, we get the upper bound for
\begin{eqnarray}\label{eq:laplacian-pi-upper}
&{}&|\langle \delta^{\nabla^\pi}[(w^*\lambda\circ j)\wedge ({\mathcal L}_{X_\lambda}J)\partial^\pi w], \partial^\pi w\rangle|\nonumber\\
&\leq&
\frac{1}{2c}\left(|\nabla^\pi (\partial^\pi w)|^2+ |\nabla w^*\lambda|^2\right)
+\left(c\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}\right)|dw|^4\nonumber\\
\end{eqnarray}
\medskip
Now we consider contact instantons which are Cauchy-Riemann maps satisfying $\delta w^*\lambda=0$.
By using the Bochner-Weitzenb\"ock formula (for forms on Riemann surface), we get the following inequality
\begin{equation}\label{eq:e-lambda-weitzenbock}
-\frac{1}{2}\Delta|w^*\lambda|^2=|\nabla w^*\lambda|^2+K|w^*\lambda|^2-\langle \Delta (w^*\lambda), w^*\lambda\rangle.
\end{equation}
Write
$$
\Delta (w^*\lambda)=d\delta (w^*\lambda)+\delta d(w^*\lambda),
$$
in which the first term vanishes since $w$ satisfies the contact instanton equations.
Then
\begin{eqnarray*}
\langle \Delta (w^*\lambda), w^*\lambda\rangle&=&\langle \delta d(w^*\lambda), w^*\lambda\rangle\\
&=&-\frac{1}{2}\langle *d|\partial^\pi w|^2, w^*\lambda\rangle\\
&=&-\langle *\langle \nabla^\pi \partial^\pi w, \partial^\pi w\rangle, w^*\lambda\rangle.
\end{eqnarray*}
Similarly as the previous estimates for the Laplacian term of $\partial^\pi w$, we can bound
\begin{eqnarray}\label{eq:laplacian-lambda-upper}
|-\langle \Delta (w^*\lambda), w^*\lambda\rangle|&=&|\langle *\langle \nabla^\pi \partial^\pi w, \partial^\pi w\rangle, w^*\lambda\rangle|\nonumber\\
&\leq& |\nabla^\pi \partial^\pi w||dw|^2\nonumber\\
&\leq& \frac{1}{2c}|\nabla^\pi \partial^\pi w|^2+\frac{c}{2}|dw|^4.
\end{eqnarray}
At last, we calculate the total energy density which is defined as $e(w):=|dw|^2=e^\pi(w)+|w^*\lambda|^2$.
Sum \eqref{eq:e-pi-weitzenbock} and \eqref{eq:e-lambda-weitzenbock}, and then apply the estimates \eqref{eq:laplacian-pi-upper} and
\eqref{eq:laplacian-lambda-upper} respectively, we have the following inequality for the total energy density
\begin{eqnarray}\label{eq:laplace-e-derivative}
-\frac{1}{2}\Delta e(w)
&\geq& \left(1-\frac{1}{c}\right)|\nabla^\pi(\partial^\pi w)|^2+\left(1-\frac{1}{2c}\right)|\nabla w^*\lambda|^2\nonumber\\
&{}&- \left(c\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+\frac{c}{2}+\|\textup{Ric}\|_{C^0(M)} \right)e^2
+Ke\nonumber\\
\\
&\geq& - \left(c\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+\frac{c}{2}+\|\textup{Ric}\|_{C^0(M)} \right)e^2+Ke,\nonumber
\end{eqnarray}
for any $c>1$. We fix $c=2$ and get the following
\begin{thm}For contact instanton $w$, we have the following total energy density $e=e(w)$ estimate
$$\Delta e\leq Ce^2+\|K\|_{L^\infty(\dot\Sigma)}e,$$
where $C=2\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+\|\textup{Ric}\|_{C^0(M)}+1$ is a positive constant independent of $w$.
\end{thm}
\medskip
Now rewrite $\eqref{eq:laplace-e-derivative}$ into
\begin{eqnarray}\label{eq:laplace-higherderivative}
&{}&\left(1-\frac{1}{c}\right)|\nabla^\pi(\partial^\pi w)|^2+\left(1-\frac{1}{2c}\right)|\nabla w^*\lambda|^2\nonumber\\
&\leq&-\frac{1}{2}\Delta e(w)
+\left(c\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+\frac{c}{2}+\|\textup{Ric}\|_{C^0(M)} \right)e^2
-Ke.\nonumber\\
\end{eqnarray}
We want to get the coercive $L^2$ bound for $\nabla dw$, which contains two parts given below according to the decomposition
$dw=d^\pi w+w^*\lambda\otimes X_\lambda$.
\begin{equation}\label{eq:|nabladw|2-1}
|\nabla dw|^2 = |\nabla(d^\pi w)+ \nabla(w^*\lambda\otimes X_\lambda)|^2
\leq 2 |\nabla(d^\pi w)|^2 + 2 |\nabla(w^*\lambda\otimes X_\lambda)|^2.
\end{equation}
For the first term on the right of \eqref{eq:|nabladw|2-1}, we can write
\begin{eqnarray}
|\nabla(d^\pi w)|^2&=&|\nabla^\pi (d^\pi w)|^2+|\langle \nabla(d^\pi w), X_\lambda\rangle|^2\nonumber\\
&=&|\nabla^\pi (d^\pi w)|^2+\frac{1}{4}|\langle d^\pi w, ({\mathcal L}_{X_\lambda}J)J d^\pi w\rangle|^2\label{eq:nabla-dpi1}\\
&\leq&|\nabla^\pi (d^\pi w)|^2+\frac{1}{4}|{\mathcal L}_{X_\lambda}J|^2|d^\pi w|^4\nonumber\\
&\leq&|\nabla^\pi (d^\pi w)|^2+\frac{1}{4}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|d^\pi w|^4,\nonumber
\end{eqnarray}
where \eqref{eq:nabla-dpi1} comes from the metric property of the contact triad connection together with
\eqref{eq:YX-triad-connection}.
For the second term on the right of \eqref{eq:|nabladw|2-1}, we have
\begin{eqnarray*}
|\nabla(w^*\lambda\otimes X_\lambda)|^2&=&|(\nabla w^*\lambda)\otimes X_\lambda+(w^*\lambda)\otimes\nabla X_\lambda|^2\\
&=&|\nabla w^*\lambda|^2+|w^*\lambda|^2|\frac{1}{2}({\mathcal L}_{X_\lambda}J)Jd^\pi w|^2\\
&\leq&|\nabla w^*\lambda|^2+\frac{1}{4}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|w^*\lambda|^2|d^\pi w|^2.
\end{eqnarray*}
Sum them and go back to \eqref{eq:|nabladw|2-1}, we get
\begin{eqnarray}\label{eq:higherorder1}
|\nabla(dw)|^2&\leq& 2|\nabla^\pi (d^\pi w)|^2+2|\nabla w^*\lambda|^2\nonumber\\
&{}&+\frac{1}{2}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|d^\pi w|^4+\frac{1}{2}\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|w^*\lambda|^2|d^\pi w|^2.
\end{eqnarray}
Hence from this, we have
\begin{eqnarray*}
|\nabla(dw)|^2&\leq&
\frac{2}{1-\frac{1}{c}}\left[\left(1-\frac{1}{c}\right)|\nabla^\pi(\partial^\pi w)|^2+\left(1-\frac{1}{2c}\right)|\nabla w^*\lambda|^2 \right]\\
&{}&+\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}|d w|^4
\end{eqnarray*}
and combine it with \eqref{eq:laplace-higherderivative}, we get
\begin{eqnarray*}
& {}& |\nabla(dw)|^2\\
&\leq&
\left[(\frac{2c^2}{c-1}+1)\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+\frac{2c}{c-1}
\left(\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+\frac{c}{2}+\|\textup{Ric}\|_{C^0(M)} \right)\right]|dw|^4\\
&{}&-\frac{2c\cdot K}{c-1}|dw|^2
+\frac{c}{1-c}\Delta e
\end{eqnarray*}
for any constant $c>1$. We still take $c=2$ and get the following coercive estimate for contact instantons
\begin{equation}\label{eq:higher-derivative}
|\nabla(dw)|^2\leq C_1|dw|^4-4K|dw|^2-2\Delta e
\end{equation}
where
$$
C_1:=9\|{\mathcal L}_{X_\lambda}J\|^2_{C^0(M)}+4\|\nabla^\pi({\mathcal L}_{X_\lambda}J)\|_{C^0(M)}+4\|\textup{Ric}\|_{C^0(M)}+4
$$
denotes a constant.
The local regularity follows from the following proposition, which is a consequence of use of cut-off function.
We give the proof in Appendix \ref{appen:local-coercive}
\begin{prop}\label{prop:coercive-L2}
For any pair of open domains $D_1$ and $D_2$ in $\dot\Sigma$ such that $\overline{D}_1\subset \textup{Int}(D_2)$,
$$
\|\nabla(dw)\|^2_{L^2(D_1)}\leq C_1(D_1, D_2)\|dw\|^2_{L^2(D_2)}+C_2(D_1, D_2)\|dw\|^4_{L^4(D_2)}
$$
for any contact instanton $w$,
where $C_1(D_1, D_2)$, $C_2(D_1, D_2)$ are some constants which depend on $D_1$, $D_2$, $(M, \lambda, J)$, but independent of $w$.
\end{prop}
\subsection{$W^{k,2}$ estimates for $k \geq 3$}
Starting from the above $W^{2,2}$-estimate, we proceed the higher $W^{k,2}$-estimate
inductively. For this purpose, consider the decomposition
$$
dw = d^\pi w + w^*\lambda\otimes X_\lambda
$$
and estimate $|\nabla^{k+1} d^\pi w|$ and $|\nabla^k(w^*\lambda\otimes X_\lambda)|$
inductively by alternatively bootstrapping staring from $k=0$ as for the case of
$|\nabla dw|$ in the previous subsection.
We start with estimating
$$
\frac{1}{2}\Delta |(\nabla^\pi)^k d^\pi w|^2 =
- |\nabla^\pi((\nabla^\pi)^k d^\pi w)|^2 + \langle (\nabla^\pi)^*\nabla^\pi((\nabla^\pi)^k d^\pi w),
\nabla^k(d^\pi w) \rangle
$$
similarly as for $\frac{1}{2}\Delta e^\pi = \frac{1}2 \Delta |d^\pi w|^2$. Rewriting this and
then combining the general Weitzenb\"ock formula (see (C.7) Appendix \cite{freed-uhlen}
e.g.) applied to $(\nabla^\pi)^k d^\pi w$, we obtain
\begin{eqnarray}\label{eq:|nablakdpiw|2}
|\nabla^\pi((\nabla^\pi)^k d^\pi w)|^2 & = &
- \frac{1}{2}\Delta |(\nabla^\pi)^k d^\pi w|^2 + \langle \Delta^\pi((\nabla^\pi)^k d^\pi w),
(\nabla^\pi)^k d^\pi w\rangle \nonumber \\
& {} & - \langle \widetilde R (\nabla^\pi)^k d^\pi w, (\nabla^\pi)^k d^\pi w \rangle
\end{eqnarray}
where $\widetilde R$ is a zero-order operator acting on the sections of $w^*\xi \otimes T^*\dot \Sigma$
which depends only on the curvature of the pull-back connection
$\nabla^\pi = w^*\nabla^\pi$ and the Levi-Civita connection of $(\dot \Sigma,h)$. In particular,
$\widetilde R = \widetilde R(dw,dw)$ quadratically depends on $dw$.
Therefore we have derived
\begin{eqnarray*}
\int_\Sigma |\nabla^\pi((\nabla^\pi)^k d^\pi w)|^2 & = &
\int_\Sigma \langle \Delta^\pi((\nabla^\pi)^k d^\pi w), (\nabla^\pi)^k d^\pi w\rangle \\
&{}& - \int_\Sigma \langle \widetilde R (\nabla^\pi)^k d^\pi w, (\nabla^\pi)^k d^\pi w \rangle.
\end{eqnarray*}
Obviously the last two terms are bounded by the norm $\|dw\|_{k,2}^2$.
It remains to examine the integral
$$
\int_\Sigma \langle \Delta^\pi((\nabla^\pi)^k d^\pi w), (\nabla^\pi)^k d^\pi w\rangle.
$$
Recalling $\Delta^\pi = d^{\nabla^\pi}\delta^{\nabla^\pi} + \delta^{\nabla^\pi}d^{\nabla^\pi}$,
we rewrite
\begin{eqnarray*}
\int_\Sigma \langle \Delta^\pi((\nabla^\pi)^k d^\pi w), (\nabla^\pi)^k d^\pi w\rangle
= \int_\Sigma |d^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2 + \int_\Sigma |\delta^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2.
\end{eqnarray*}
On the other hand, we compute
\begin{equation}
d^{\nabla^\pi}((\nabla^\pi)^k d^\pi w)).
\end{equation}
For this purpose, we quote the following lemma
\begin{lem} For any $\xi$-valued $1$-form $\alpha$,
\begin{equation}\label{eq:dnablapinablazeta}
d^{\nabla^\pi}(\nabla^\pi \alpha) = \nabla^\pi(d^{\nabla^\pi} \alpha) + \widetilde R \, \alpha
\end{equation}
for some zero-order operator $\widetilde R: \Lambda^1(T^*M) \to \Lambda^2(T^*M)$
depending only on the curvature as above.
Equivalently
\begin{equation}\label{eq:dnablapinablazeta2}
\left[d^{\nabla^\pi},\nabla^\pi \right]\alpha = \widetilde R \, \alpha
\end{equation}
for the commutator $[\cdot, \cdot]$.
\end{lem}
Applying this to $d^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))$ iteratively, we derive
\begin{equation}\label{eq:k-commute}
|d^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2 \leq |(\nabla^\pi)^k(d^{\nabla^\pi}d^\pi w)|^2
+ G_k(|d^\pi w|, |w^*\lambda|)
\end{equation}
where $G_k$ is a polynomial function of $|d^\pi w|$, $|w^*\lambda|$ and their
covariant derivatives up to order $k$. And applying the fundamental equation \eqref{eq:Laplacian-w} to
$d^{\nabla^\pi}d^\pi w$, the term itself has the bound
$$
|(\nabla^\pi)^k(d^{\nabla^\pi}d^\pi w)|^2 \leq H_k(|d^\pi w|, |w^*\lambda|)
$$
for a polynomial function $H_k$ of the form $G_k$.
Similarly, we obtain
$$
|\delta^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2 \leq I_k(|d^\pi w|, |w^*\lambda|)
$$
for similar polynomial function $I_k$
since $|\delta^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2=|d^{\nabla^\pi}((\nabla^\pi)^k d^\pi w))|^2$.
We now summarize the above computations into
\begin{prop}\label{prop:higher-pi-regularity} Let $w: \dot\Sigma \to M$ be a contact instanton. Then
\begin{equation}\label{eq:k|nablapidw|}
\int_{\dot \Sigma} |(\nabla^\pi)^{k+1}(d^\pi w)|^2
\leq \int_{\dot \Sigma} J_k(|\partial^\pi w|, |w^*\lambda|)
\end{equation}
for a polynomial function $J_k$ of $|d^\pi w|, \, |w^*\lambda|$ its covariant derivatives
up to $0, \, \ldots, k$ of degree at most $2k + 4$.
\end{prop}
Next we compute
\begin{eqnarray*}
\nabla^{k+1}(w^*\lambda X_\lambda) = \nabla^{k+1}(w^*\lambda)\, X_\lambda + \sum_{l = 1}^{k+1}
\left(\begin{matrix}k+1\\lambda} \def\La{\Lambda \end{matrix}\right) \nabla^l(w^*\lambda) \,\nabla^{k+1-l}X_\lambda
\end{eqnarray*}
Here we recall the formula $\nabla X_\lambda$ in \eqref{eq:YX-triad-connection}. Therefore it follows that
$$
\left|\sum_{l = 1}^{k+1} \nabla^l(w^*\lambda) \nabla^{k+1-l}X_\lambda\right| \leq L_k(|d^\pi w|, |w^*\lambda|)
$$
for a polynomial function $L_k$ similar to $J_k$. We write
\begin{eqnarray*}
|\nabla^{k+1}(w^*\lambda)|^2 & = & |\nabla(\nabla^k(w^*\lambda))|^2 \\
& = & - \frac12\Delta |\nabla^k(w^*\lambda))|^2 +
\langle \nabla^*\nabla ((\nabla^k(w^*\lambda)), \nabla^k(w^*\lambda)\rangle.
\end{eqnarray*}
Applying the Weitzenb\"ock formula for the $1$-form on $\dot\Sigma$, we obtain
$$
\nabla^*\nabla ((\nabla^k(w^*\lambda)) = - \Delta (\nabla^k(w^*\lambda)) - K\, \nabla^k(w^*\lambda).
$$
Therefore we have obtained
\begin{eqnarray*}
&{}&\int_\Sigma \langle \nabla^*\nabla ((\nabla^k(w^*\lambda)), \nabla^k(w^*\lambda)\rangle \\
& = &
- \int_\Sigma |d (\nabla^k(w^*\lambda))|^2 + |\delta (\nabla^k(w^*\lambda))|^2
- \int_\Sigma R\, |\nabla^k(w^*\lambda)|^2.
\end{eqnarray*}
By applying similar arguments considering the commutators $[d, \nabla^k]$, $[\delta, \nabla^k]$
and the equations $dw^*\lambda = \frac{1}{2}|d^\pi w|^2\, dA$ from Proposition \ref{prop:energy-omegaarea}
and $\delta w^*\lambda = 0$, we have derived
\begin{prop}\label{prop:higher-w*lambda-regularity} Let $w: \dot\Sigma \to M$ be a contact instanton. Then
$$
\int_{\Sigma} |\nabla^{k+1}w^*\lambda|^2 \leq L_k(|\partial^\pi w|, |w^*\lambda|)
$$
for a polynomial function $L_k$ of $|d^\pi w|, \, |w^*\lambda|$ its covariant derivatives
up to $0, \, \ldots, k$ of degree at most $2k+3$.
\end{prop}
Now combining Propositions \ref{prop:higher-pi-regularity}, \ref{prop:higher-w*lambda-regularity},
we derive
\begin{thm}\label{thm:higher-regularity} Then
\begin{equation}\label{eq:k|nablapidw|}
\int_{\Sigma} |(\nabla)^{k+1}(dw)|^2 \leq \int_{\Sigma} J_{k}'(|\partial^\pi w|, |w^*\lambda|).
\end{equation}
Here $J'_{k+1}$ a polynomial function of covariant derivatives
of $|d^\pi w|, \, |w^*\lambda|$ up to $0, \, \ldots, k$ with degree at most $2k + 4$
whose coefficients are bounded by
$$
\|K\|_{C^k;\operatorname{supp} K}, \|R^\pi\|_{C^k}, \|{\mathcal L}_{X_\lambda}J\|_{C^k}.
$$
\end{thm}
\begin{proof} It remains to check the second statement, which itself follows expressing
the bound of $\|dw\|_{k,2}^2$ inductively staring from $k = 1$, i.e., Proposition
\ref{prop:coercive-L2}. This finishes the proof.
\end{proof}
Again similar inductive argument gives rise to the following local higher
regularity estimates.
\begin{thm}\label{thm:local-higher-regularity}
Let $(\dot \Sigma,j)$ be a be a punctured Riemann surface possibly with
empty set of punctures. Let $w: \dot\Sigma \to M$ satisfy
$
{\overline \partial}^\pi w=0, \quad d(w^*\lambda \circ j) = 0, \quad \|dw\|_{C^0} < \infty
$
on $\dot\Sigma$. For any open domains $D_1$ and $D_2$ in $\dot\Sigma$ such that $\overline{D}_1\subset D\subset \textup{Int}(D_2)$
for some domain $D$,
$$
\|dw\|^2_{W^{k,2}(D_1)}\leq {\mathcal P}(k; D_1,D_2)\left(\|dw\|^2_{L^2(D_2)}, \|dw\|^4_{L^4(D_2)}\right)
$$
for any contact instanton $w$, where ${\mathcal P}(k; D_1,D_2)$ is some polynomial function given
which are independent of $\|dw\|_{C^0}$.
\end{thm}
In particular, any weak solution of \eqref{eq:instanton} in
$W^{1,4}_{\textup{loc}}$ automatically lies in $W^{3,2}_{\textup{loc}}$ and becomes the classical solution, and also smooth.
\section{Asymptotic behavior of contact instantons}
\label{sec:reeborbits}
In this section, we study the asymptotic behavior of the solutions of the contact instanton equations \eqref{eq:instanton}
from the Riemann surface $(\dot\Sigma, j)$ associated with a metric $h$ with \emph{cylindrical ends}.
To be precise, we assume there exists a compact set $K_\Sigma\subset \dot\Sigma$,
such that $\dot\Sigma-\textup{Int}(K_\Sigma)$ is disjoint union of punctured disks
each of which is isometric to the half cylinder $[0, \infty)\times S^1$ or $(-\infty, 0]\times S^1$, where
the choice of positive or negative cylinders depends on the assignment of positive punctures or negative punctures.
We denote by $\{p^+_i\}_{i=1, \cdots, l^+}$ the positive punctures, and by $\{p^-_j\}_{j=1, \cdots, l^-}$ the negative punctures.
Here $l=l^++l^-$. Denote by $\phi^{\pm}_i$ such isometries from cylinders to disks.
We first state the assumption for the study of puncture behaviors.
\begin{defn}Let $\dot\Sigma$ by a punctured Riemann surface with punctures
$\{p^+_i\}_{i=1, \cdots, l^+}\cup \{p^-_j\}_{j=1, \cdots, l^-}$ equipped
with a metric $h$ with \emph{cylindrical ends} outside a compact subset $K_\Sigma$ such that
$\dot\Sigma-\textup{Int}(K_\Sigma)$ is isometric to the union of half cylinders as described above.
Assume
$w: \dot \Sigma \to M$ be any smooth map. We define
\begin{equation}\label{eq:endenergy}
E^\pi(w) = E^\pi_{(\lambda,J;\dot\Sigma,h)}(w) = \frac{1}{2} \int_{\dot \Sigma} |d^\pi w|^2
\end{equation}
where the norm is taken in terms of the given metric $h$ on $\dot \Sigma$ and the triad metric on $M$.
\end{defn}
We put the following hypotheses in our asymptotic study of the finite
energy contact instanton maps $w$:
\begin{hypo}\label{hypo:basic}
Let $h$ be the metric on $\dot \Sigma$ given as above.
Assume $w:\dot\Sigma\to M$ satisfies the contact instanton equations \eqref{eq:instanton},
and
\begin{enumerate}
\item $E^\pi_{(\lambda,J;\dot\Sigma,h)}(w)<\infty$;
\item $\|d w\|_{C^0(\dot\Sigma)} <\infty$.
\end{enumerate}
\end{hypo}
Throughout this section, we just work locally near one puncture, i.e., on
$D^\delta(p) \setminus \{p\}$. By taking the associated conformal coordinates $\phi^+ = (\tau,t)
:D^\delta(p) \setminus \{p\} \to [0, \infty)\times S^1 \to $ such that $h = d\tau^2 + dt^2$,
we only look at the map $w$ defined on the half cylinder $[0, \infty)\times S^1\to M$
without loss of generality from now on.
The above finite $\pi$-energy hypothesis implies
\begin{equation}\label{eq:hypo-basic-pt}
\int_{[0, \infty)\times S^1}|d^\pi w|^2 \, d\tau \, dt <\infty, \quad \|d w\|_{C^0([0, \infty)\times S^1)}<\infty
\end{equation}
in this coordinates.
Let $w$ be as in Hypothesis \ref{hypo:basic}. We can associate two
natural asymptotic invariants for each puncture (here we only look at one positive puncture for simple notations)
defined by
\begin{eqnarray}
T & := & \frac{1}{2}\int_{[0,\infty) \times S^1} |d^\pi w|^2 + \int_{\{0\}\times S^1}(w|_{\{0\}\times S^1})^*\lambda\label{eq:TQ-T}\\
Q & : = & \int_{\{0\}\times S^1}((w|_{\{0\}\times S^1})^*\lambda\circ j).\label{eq:TQ-Q}
\end{eqnarray}
\begin{rem}\label{rem:TQ}
For any contact instanton $w$, since $\frac{1}{2}|d^\pi w|^2\, dA=d(w^*\lambda)$, by Stokes' formula,
$$
T = \frac{1}{2}\int_{[s,\infty) \times S^1} |d^\pi w|^2 + \int_{\{s\}\times S^1}(w|_{\{s\}\times S^1})^*\lambda, \quad
\text{for any } s\geq 0.
$$
In the mean time, since $d(w^*\lambda\circ j)=0$, the integral
$$
\int_{\{s \}\times S^1}(w|_{\{s \}\times S^1})^*\lambda\circ j, \quad
\text{for any } s \geq 0
$$
does not depend on $s$ whose common value is nothing but $Q$.
\end{rem}
We call $T$ the \emph{asymptotic contact action}
and $Q$ the \emph{asymptotic contact charge} of the contact instanton $w$ at the given puncture.
\medskip
For a given contact instanton $w: [0, \infty)\times S^1\to M$, define maps
$w_s: [-s, \infty) \times S^1 \to M$ by
$
w_s(\tau, t) = w(\tau + s, t)$.
For any compact set $K\subset {\mathbb R}$, there exists some $s_0$ large such that every $s\geq s_0$,
$K\subset [-s, \infty)$. For such $s\geq s_0$, we also get an $[s_0, \infty)$-family of maps by defining $w^K_s:=w_s|_{K\times S^1}:K\times S^1\to M$.
The asymptotic behavior of $w$ at infinity can be understood by studying the limiting of the sequence of maps
$\{w^K_s:K\times S^1\to M\}_{s\in [s_0, \infty)}$, for any compact set $K\subset {\mathbb R}$.
First of all,
it is easy to check that under the Hypothesis \ref{hypo:basic}, the family
$\{w^K_s:K\times S^1\to M\}_{s\in [s_0, \infty)}$ satisfies the following
\begin{enumerate}
\item ${\overline \partial}^\pi w^K_s=0$, $d((w^K_s)^*\lambda\circ j)=0$, for every $s\in [s_0, \infty)$
\item $\lim_{s\to \infty}\|d^\pi w^K_s\|_{L^2(K\times S^1)}=0$
\item $\|d w^K_s\|_{C^0(K\times S^1)}\leq \|d w\|_{C^0([0, \infty)\times S^1)}<\infty$.
\end{enumerate}
From (1) and (3) together with the compactness of the target manifold $M$ (which provides the uniform $L^2(K\times S^1)$ bound)
and Theorem \ref{thm:local-higher-regularity}, we have
$$
\|w^K_s\|_{W^{3,2}(K\times S^1)}\leq C_{K;(3,2)}<\infty,
$$
for some constant $C_{K;(3,2)}$ independent of $s$.
Then by the compactness of the embedding of $W^{3,2}(K\times S^1)$ into $C^2(K\times S^1)$, we
get $\{w^K_s:K\times S^1\to M\}_{s\in [s_0, \infty)}$ is sequentially compact.
Therefore, for any sequence $s_k \to \infty$, there exists a subsequence, still denote by $s_k$,
and some limit (may depend on the subsequence $\{s_k\}$) $w^K_\infty\in C^2(K\times S^1, M)$, such that
$$
w^K_{s_k}\to w^K_{\infty}, \quad \text {as } k\to \infty,
$$
in $C^2(K\times S^1, M)$-norm sense.
Hence further together with (2), we get
$$
dw^K_{s_k}\to dw^K_{\infty} \quad \text{and} \quad dw^K_\infty=(w^K_\infty)^*\lambda\otimes X_\lambda,
$$
as well as both $(w^K_\infty)^*\lambda$ are $(w^K_\infty)^*\lambda\circ j$ are harmonic $1$-forms by taking (1) into consideration.
Notice that these limiting maps $w^K_\infty$ have common extension $w_\infty: {\mathbb R}\times S^1\to M$
by the nature of the diagonal argument as taking a sequence of compact sets $K$
in the way one including another and exhausting ${\mathbb R}$.
Then $w_\infty$ is $C^2$ (actually $C^\infty$) and satisfies
$$
\|d w_\infty\|_{C^0({\mathbb R}\times S^1)}\leq \|d w\|_{C^0([0, \infty)\times S^1)}<\infty
$$
and
$$
dw_\infty=(w_\infty)^*\lambda \otimes X_\lambda.
$$
Also notice that both $(w_\infty)^*\lambda$ and $(w_\infty)^*\lambda\circ j$ are bounded harmonic $1$-forms on ${\mathbb R}\times S^1$,
and hence they must be written into the forms
$$
(w_\infty)^*\lambda=a\,d\tau+b\,dt, \quad (w_\infty)^*\lambda\circ j=b\,d\tau-a\,dt,
$$
where $a$, $b$ are some constants.
Now we show that such $a$ and $b$ are actually related to $T$ and $Q$ as
$$
a=-Q, \quad b=T.
$$
By taking an arbitrary point $r\in K$, since $w_\infty|_{\{r\}\times S^1}$ is the limiting of some sequence $w_{s_k}|_{\{r\}\times S^1}$
in $C^2$ sense,
we have
\begin{eqnarray*}
b=\int_{\{r\}\times S^1}(w_\infty|_{\{r\}\times S^1})^*\lambda
&=&\int_{\{r\}\times S^1}\lim_{k\to \infty}(w_{s_k}|_{\{r\}\times S^1})^*\lambda\\
&=&\lim_{k\to \infty}\int_{\{r\}\times S^1}(w_{s_k}|_{\{r\}\times S^1})^*\lambda\\
&=&\lim_{k\to \infty}\int_{\{r+s_k\}\times S^1}(w|_{\{r+s_k\}\times S^1})^*\lambda\\
&=&\lim_{k\to \infty}(T-\frac{1}{2}\int_{[r+s_k, \infty)\times S^1}|d^\pi w|^2)\\
&=&T-\lim_{k\to \infty}\frac{1}{2}\int_{[r+s_k, \infty)\times S^1}|d^\pi w|^2\\
&=&T;\\
\\
-a=\int_{\{r\}\times S^1}(w_\infty|_{\{r\}\times S^1})^*\lambda\circ j
&=&\int_{\{r\}\times S^1}\lim_{k\to \infty}(w_{s_k}|_{\{r\}\times S^1})^*\lambda\circ j\\
&=&\lim_{k\to \infty}\int_{\{r\}\times S^1}(w_{s_k}|_{\{r\}\times S^1})^*\lambda\circ j\\
&=&\lim_{k\to \infty}\int_{\{r+s_k\}\times S^1}(w|_{\{r+s_k\}\times S^1})^*\lambda\circ j\\
&=&Q.
\end{eqnarray*}
Here in the derivation, we used Remark \ref{rem:TQ}.
As we have already seen, and since the connectedness of $[0,\infty) \times S^1$, the image of $w_\infty$ is contained in
a single leaf of the Reeb foliation. Let $\gamma: {\mathbb R} \to M$ be a parametrisation of
the leaf so that $\dot \gamma = X_\lambda(\gamma)$,
then can write $w_\infty(\tau, t)=\gamma(s(\tau, t))$, where
$s:{\mathbb R}\times S^1\to {\mathbb R}$ and $s=-Q\,\tau+T\,t+c_0$ since $ds=-Q\,d\tau+T\,dt$, where $c_0$ is some constant.
Hence we can see, if $T\neq 0$,
$\gamma$ is a closed orbit of period $T$.
If $T=0$ but $Q\neq 0$, we can only conclude that $\gamma$ is some Reeb trajectory parameterized by $\tau\in {\mathbb R}$.
Of course, if both $T$ and $Q$ vanish, $w_\infty$ is a constant map.
\medskip
From the above, we have given the proof of the following theorem, which includes a special case, $Q=0$, $T\neq 0$ and $K=\{0\}$ that was given in \cite[Theorem 31]{hofer}.
Besides we look at two constants $T$ and $Q$,
we also emphasize the uniform convergence strengthened here which becomes an important ingredient in the
proof of exponential convergence later in Part \ref{part:exponential}, as well as for the case of Morse-Bott situation the proof provided by the authors in \cite{oh-wang2} .
\begin{thm}[Subsequence Convergence]\label{thm:subsequence}
Let $w:[0, \infty)\times S^1\to M$ satisfy the contact instanton equations \eqref{eq:instanton}
and hypothesis \eqref{eq:hypo-basic-pt}.
Then for any sequence $s_k\to \infty$, there exists a subsequence, still denoted by $s_k$, and a
massless instanton $w_\infty(\tau,t)$ (i.e., $E^\pi(w_\infty) = 0$)
on the cylinder ${\mathbb R} \times S^1$ such that
$$
\lim_{k\to \infty}w(s_k + \tau, t) = w_\infty(\tau,t)
$$
in $C^l(K \times S^1, M)$ sense for any $l$, where $K\subset [0,\infty)$ is an arbitrary compact set.
Further, we can write $w_\infty(\tau,t)= \gamma(-Q\, \tau + T\, t)$, where $\gamma$ is some Reeb trajectory,
and for the case of $Q=0$ or $T\neq 0$, $\gamma$ is a \emph{closed} Reeb orbit of $X_\lambda$ with period $T$.
\end{thm}
From the previous theorem, we get the following corollary immediately.
\begin{cor}\label{cor:tangent-convergence}Let $w:[0, \infty)\times S^1\to M$ satisfy the contact instanton equations \eqref{eq:instanton}
and hypothesis \eqref{eq:hypo-basic-pt}.
Then
\begin{eqnarray*}
&&\lim_{s\to \infty}\left|\pi \frac{\partial w}{\partial\tau}(s+\tau, t)\right|=0, \quad
\lim_{s\to \infty}\left|\pi \frac{\partial w}{\partial t}(s+\tau, t)\right|=0\\
&&\lim_{s\to \infty}\lambda(\frac{\partial w}{\partial\tau})(s+\tau, t)=-Q, \quad
\lim_{s\to \infty}\lambda(\frac{\partial w}{\partial t})(s+\tau, t)=T
\end{eqnarray*}
and
$$
\lim_{s\to \infty}|\nabla^l dw(s+\tau, t)|=0 \quad \text{for any}\quad l\geq 1.
$$
All the limits are uniform for $(\tau, t)$ on $K\times S^1$ with compact $K\subset {\mathbb R}$.
\end{cor}
\begin{proof}If any of the above limit doesn't hold uniformly (we take $|\pi \frac{\partial w}{\partial\tau}(s+\tau, t)|$ for example), then there exists some $\epsilon_0>0$ and a sequence
$k\to \infty$, $(\tau_j, t_j)\in K\times S^1$ such that $|\pi \frac{\partial w}{\partial\tau}(s_k+\tau_j, t_j)|\geq \epsilon_0$.
Then we can take subsequence limiting $(\tau_j, t_j)\to (\tau_0, t_0)$ such that
$|\pi \frac{\partial w}{\partial\tau}(s_k+\tau_0, t_0)|\geq \frac{1}{2}\epsilon_0$ for $k$ large enough.
However, from Theorem \ref{thm:subsequence}, we can take subsequence of $s_k$ such that
$w(s_k+\tau, t)$ converges to $\gamma(-Q\, \tau + T\, t)$ in a neighborhood of $(\tau_0, t_0)\in K\times S^1$,
in $C^\infty$ sense. Here $\gamma$ is some Reeb trajectory. Then we get $\lim_{s\to \infty}|\pi \frac{\partial w}{\partial\tau}(s_k+\tau_0, t_0)|=0$ and get contradiction.
Once we establish this uniform decay, the last higher order decay result is
an immediate consequence of the local pointwise higher order a priori estimates, Theorem \ref{thm:local-higher-regularity}.
\end{proof}
\part{Asymptotic behavior of charge vanishing contact instantons}
\label{part:exponential}
In this part, we further study the contact instantons \emph{with vanishing charge}.
To be specific, assume $w:\dot\Sigma\to M$ is a contact instanton, i.e., satisfies \eqref{eq:instanton}, and
$Q_i=0=Q_j$, for every $i=1, \cdots, l^+$, $j=1, \cdots, l^-$, where $Q_i$, $Q_j$ denote the charges
at the punctures $p_i$, $p_j$, where the notations here are the ones we introduced in Section \ref{sec:reeborbits}.
In other words, we require that
$w^*\lambda\circ j$ is an exact $1$-form when restricted to each cylindrical ends.
We would like to remark that the charge vanishing contact instantons are equivalent to the
homological pseudo-holomorphic curves considered by Hofer \cite{hofer-survey} and by Abbas-Cieliebak-Hofer \cite{ACH}.
The results in the part are basically not new and was studied in \cite{hofer}, \cite{HWZ96, HWZ98}, \cite{HWZplane}.
However, to prove the exponential decay of charge vanishing contact instantons at cylindrical ends to non-degenerate Reeb orbits,
we provide a completely different and new method, named the \emph{three-interval method},
which can also be applied to general evolution type geometric PDEs.
\section{Linearization operator of Reeb orbits}
\label{sec:prelim}
As the beginning, we would like to study the linearization of the equation $\dot x = X_\lambda(x)$
along a closed Reeb orbit, which is the (subsequence) limit of charge vanishing contact instantons
as proved in Section \ref{sec:reeborbits}.
The materials in this section are mostly standard and well-known
results in contact geometry. (See Appendix \cite{wendl} for the exposition that is
the closest to the one given in this section.) Since our systematic usage of
the contact triad connection in the exposition gives rise to some explicit
useful formulae occurring in this study which will be
also important in our proof of exponential convergence,
we provide precise statements and details of proofs of the results
that will be relevant to the study of later sections.
Let $\gamma$ be a closed Reeb orbit of period $T > 0$. In other words,
$\gamma: {\mathbb R} \to M$ is a periodic solution of $\dot \gamma = X_\lambda(\gamma)$ with period $T$, thus satisfying $\gamma(T) = \gamma(0)$.
We will call the pair $(T,z)$ a Reeb orbit of period $T$ instead for a such
closed orbit $\gamma$ of period $T$ by writing $z(t) = \gamma(Tt)$ for a loop parameterized over
the unit interval $S^1= [0,1]/\sim$.
Denote the Reeb flow $\phi^t := \phi^t_{X_\lambda}$ of the Reeb vector field $X_\lambda$,
we can write $\gamma(t) = \phi^t(\gamma(0))$ for any Reeb trajectory $\gamma$.
In particular $p:= \gamma(0)$ is a fixed point of the diffeomorphism $\phi^T$ when $\gamma$ is a closed Reeb
orbit of period $T$. Since ${\mathcal L}_{X_\lambda}\lambda = 0$, the contact diffeomorphism $\phi^T$ canonically induces the isomorphism
$$
\Psi_p : = d\phi^T(p)|_{\xi_p}: \xi_p \to \xi_p
$$
which is the linearized Poincar\'e return map $\phi^T$ restricted to $\xi_p$
via the splitting $T_pM = \xi_p \oplus {\mathbb R}\{X_\lambda(p)\}$.
\begin{defn} We say a Reeb orbit with period $T$, $(T,z)$, is \emph{nondegenerate}
if the linearized return map $\Psi_z:\xi_p \to \xi_p$ with $p = z(0)$ has no eigenvalue 1.
\end{defn}
Denote ${\operatorname{Cont}}(M, \xi)$ the set of contact $1$-forms with respect to the contact structure $\xi$ and ${\mathcal L}(M)=C^\infty(S^1,M)$
the space of loops $z: S^1 = {\mathbb R} /{\mathbb Z} \to M$.
Let ${\mathcal L}^{1,2}(M)$ be the $W^{1,2}$-completion of ${\mathcal L}(M)$.
We would like to consider some Banach vector bundle ${\mathcal L}$ over the Banach manifold
$(0,\infty) \times {\mathcal L}^{1,2}(M) \times {\operatorname{Cont}}(M,\xi)$ whose fiber at $(T, z, \lambda)$ is given by $L^2(z^*TM)$.
We consider the assignment
$$
\Upsilon: (T,z,\lambda) \mapsto \dot z - T \,X_\lambda(z)
$$
which is a section of ${\mathcal L}$. Then $(T,z,\lambda) \in \Upsilon^{-1}(0)$ if and only if there exists some
Reeb orbit $\gamma: {\mathbb R} \to M$ with period $T$, such that $z(\cdot)=\gamma(T\cdot)$.
We also denote $DX_\lambda: \Omega^0(\xi) \to \Omega^0(\xi)$ the covariant derivative of $X_\lambda$ induced from
the contact triad connection $\nabla$ to highlight its aspect as a linear operator, whenever
we feel convenient. The following derivation of the linearization of $\Upsilon$ is a routine exercise.
Since it is not essential to our purpose in this paper, we omit its derivation only by stating the
final result.
\begin{lem}\label{lem:full-linearization} For any torsion free connection,
\begin{eqnarray*}
d{(T,z, \lambda)}\Upsilon(a,Y, B) = \frac{D Y}{dt} - T DX_\lambda(z)(Y)-aX_\lambda-T\delta_{\lambda}X_\lambda(B),
\end{eqnarray*}
where $a\in {\mathbb R}$, $Y\in T_z{\mathcal L}^{1,2}(M)=W^{1,2}(z^*TQ)$, $B\in T_\lambda {\operatorname{Cont}}(M, \xi)$ and the last term $\delta_{\lambda}X_\lambda$ is some linear operator.
\end{lem}
We remark that the contact triad connection we use in this paper is \emph{not} torsion-free. However, when
$(T,z, \lambda)\in \Upsilon^{-1}(0)$, i.e., $z(\cdot)= \gamma(T \cdot)$ for some $\gamma$ which is a Reeb
orbit with period $T$ with respect to contact $1$-form $\lambda$, the torsion Axiom (2) in Definition
\ref{thm:connection} is already enough to derive Lemma \ref{lem:full-linearization}. From now on, we use the contact triad connection through out this section.
We recall the readers that the linearization at $(T,z, \lambda)\in \Upsilon^{-1}(0)$
actually doesn't depend of the choice of connections.
In this paper we only need to look at the linearization restricted to subspace $W^{1,2}(z^*\xi)$
for fixed $(T, \lambda)$. Denote the corresponding operator by
$$
\Upsilon_{T, \lambda}=\Upsilon(T, \cdot, \lambda).
$$
We have the following characterization of the nondegeneracy condition.
\begin{prop}\label{prop:nondegeneracy} A closed Reeb orbit $\gamma$ with period $T$ is nondegenerate if and only if
the $\xi$ projection of the linearization restricted to $W^{1,2}(z^*\xi) $, i.e.,
$$
d^\pi_{z}\Upsilon_{T, \lambda}|_{W^{1,2}(z^*\xi) }
:=\pi d_{z}\Upsilon_{T, \lambda}|_{W^{1,2}(z^*\xi) }: W^{1,2}(z^*\xi) \to L^2(z^*\xi)
$$
is surjective, where $z(\cdot):=\gamma(T\cdot): S^1\to M$.
\end{prop}
The rest of the section will be occupied by the proof of this proposition.
From Lemma \ref{lem:full-linearization} and Corollary \ref{cor:connection}, we compute
\begin{eqnarray*}
d^\pi_{z}\Upsilon_{T, \lambda}|_{W^{1,2}(z^*\xi) }(\zeta) & = & \pi \frac{D\zeta}{dt} - T \cdot \pi DX_\lambda(z)(\zeta)\\
& = & \frac{D^\pi \zeta}{dt} - \frac{T}{2} ({\mathcal L}_{X_\lambda}J) J\zeta.
\end{eqnarray*}
Since from Axiom (3) of Definition \ref{thm:connection}, the image of $d_{z}\Upsilon_{T, \lambda}|_{W^{1,2}(z^*\xi) }$
automatically in $\xi$.
\begin{lem}\label{lem:DXlambda} Let $DX_\lambda(z) = \nabla_{(\cdot)} X_\lambda: z^*TM \to z^*TM$ be the covariant derivative
of $X_\lambda$ with respect to the pull-back connection $z^*\nabla$ of the contact triad connection.
Consider a Reeb orbit $(T,z)$ i.e., a map $z: S^1 \to M$ satisfying $\dot z = T X_\lambda(z)$
with $z(1) = z(0)$. Then
$$
DX_\lambda(z)(Y)
= \frac{1}{2}({\mathcal L}_{X_\lambda}J(z))J(z) Y
$$
for any section $Y \in \Omega^0(z^*TM)$.
\end{lem}
\begin{proof} By definition, we have
$$
DX_\lambda(z)(Y) = \nabla_Y X_\lambda
$$
and then apply \eqref{eq:YX-triad-connection}, which proves the equality.
\end{proof}
Now recall the following whose proof is also given in Lemma 5.2
of he arXiv version of \cite{oh-wang1}.
\begin{lem}[Lemma 6.2 \cite{blair}]\label{lem:symmetry}
Both $({\mathcal L}_{X_\lambda}J) J$ and ${\mathcal L}_{X_\lambda}J$ are pointwise symmetric
with respect to the triad metric of $(M,\lambda, J)$.
\end{lem}
Combining the above discussion, we have derived
\begin{prop}\label{prop:self-adjoint} The linear operator
$$
J d^\pi_{z}\Upsilon_{T, \lambda} = J \pi\left(\frac{D}{d t} - DX_\lambda(z)\right)
= J \frac{D^\pi}{d t} - \frac{T}{2} ({\mathcal L}_{X_\lambda}J) : L^2(z^*\xi) \to L^2(z^*\xi)
$$
is a self-adjoint operator. In particular, we obtain
\begin{equation}\label{eq:index}
\operatorname{Index} d^\pi_{z}\Upsilon_{T, \lambda} = \operatorname{Index} J d^\pi_{z}\Upsilon_{T, \lambda} = 0.
\end{equation}
\end{prop}
Finally we are ready to prove the above analytic characterization of
the nondegeneracy. By Proposition \ref{prop:self-adjoint},
the subjectivity of $d^\pi_{z}\Upsilon_{T, \lambda}$ is equivalent to the injectivity of the operator.
In fact, we prove the following characterization of kernel elements of the linearization
map $d^\pi_{z}\Upsilon_{T, \lambda}$ in terms of the eigenvectors of the linear map
$\Psi_p: \xi_p \to \xi_p$ where $\Psi_p= d\phi^{T}(p)|_{\xi_p}$.
\begin{prop} Let $p = z(0)$ be a fixed point of $\phi^T: M \to M$ lying in the given Reeb
orbit $(T,z)$. Then there exists a one-one correspondence
$$
v \in \xi_p \mapsto \eta; \quad \eta(t): = d\phi^{tT}(v), \quad t \in [0,1]
$$
between the set of eigenvectors $v$ of
$\Psi_\gamma = d\phi^T|_{\xi_p}: \xi_p \to \xi_p$ with eigenvalue 1 and the set of solutions $\eta$ to
$\frac{D\eta}{dt} - \frac{T}{2} ({\mathcal L}_{X_\lambda}J) \eta = 0$.
\end{prop}
\begin{proof} Recall that any closed Reeb orbit of period $T$ has the form
$z(t) = \phi^{tT}(p)$ for a fixed point $p$ of $\phi^{tT}$.
Suppose $\eta$ is a solution to $0= d_z^\pi\Upsilon_{T,\lambda}(\eta) = \frac{D\eta}{dt} - \frac{T}{2} ({\mathcal L}_{X_\lambda}J) J \eta$.
We consider the one-parameter family
$$
v(t) = (d\phi^{tT})^{-1} (\eta(t))
$$
of tangent vectors at $p \in M$, and so $\eta(t) = d\phi^{tT}(v(t))$.
We compute $\nabla_t\eta(t)$ by considering the map
$\Gamma(s,t) = \phi^{tT}(\alpha(s,t))$ such that $\alpha(0,t) \equiv p$ and $\frac{\partial}{\partial s}\Big|_{s=0}\alpha(s, t)=v(t)$ . Then
we compute
$$
\frac{\partial \Gamma}{\partial s} = d\phi^{tT}\left(\frac{\partial \alpha}{\partial s}\right), \quad
\frac{\partial \Gamma}{\partial t}(s,t) = T X_\lambda(\Gamma(s,t)) + d\phi^{tT}\left(\frac{\partial \alpha}{\partial t}(s,t)\right)
$$
and so
\begin{eqnarray*}
\nabla_t \eta
& = & \frac{D}{dt} \frac{\partial \Gamma}{\partial s}\Big|_{s=0}
= \frac{D}{ds} \frac{\partial \Gamma}{\partial t}\Big|_{s=0}\\
& = & T \frac{D}{ds}( X_\lambda(\Gamma(s,t))\Big|_{s=0}
+d\phi^{tT}\left(\frac{D}{\partial s}\Big|_{s=0}\frac{\partial \alpha}{\partial t}(s,t)\right)\\
& = &T \frac{D}{ds}( X_\lambda(\Gamma(s,t))\Big|_{s=0}
+d\phi^{tT}\left(\frac{D}{\partial t}\frac{\partial \alpha}{\partial s}\Big|_{s=0}(s,t)\right)\\
& = & T \frac{D}{ds}( X_\lambda(\Gamma(s,t))\Big|_{s=0}
+d\phi^{tT}\left(v'(t)\right).
\end{eqnarray*}
Here the second and the fourth equalities follow from the torsion property of the triad connection
$$
T\left(\frac{\partial \Gamma}{\partial t}\Big|_{s=0} , \frac{\partial \Gamma}{\partial s}\Big|_{s=0}\right)
= T(d\phi^{tT} v(t), X_\lambda(\phi^{tT}(p)) = 0.
$$
The first term of the farthest right becomes
$$
T \frac{D}{ds}( X_\lambda(\Gamma(s,t))\Big|_{s=0} = \frac{T}{2}({\mathcal L}_{X_\lambda}J) J \eta.
$$
Therefore we have derived
$$
v'(t)
= (d\phi^{tT})^{-1}\left(\nabla_t^\pi\eta- \frac{T}{2}({\mathcal L}_{X_\lambda}J)J \eta \right) = 0.
$$
by the hypothesis that $\eta$ satisfies the equation
$\frac{D\eta}{dt} - \frac{T}{2} ({\mathcal L}_{X_\lambda}J) J \eta = 0$. Therefore we have
$$
v(1) = v(0), i.e., (d\phi^T)^{-1}(\eta(1)) = \eta(0).
$$
Since $\eta(0) = \eta(1)$, it implies that $J \eta(0)$ is an eigenvector of eigenvalue 1
if $\eta(0) \neq 0$.
Conversely suppose that
$v$ is an eigenvector of $\phi^{tT}: \xi_p \to \xi_p$.
Then the above computation of $v'$ applied to constant function $v(t) \equiv v$
proves that the vector field $t \mapsto d\phi^{tT}(v)$ satisfies
$\nabla_t^\pi\eta- \frac{T}{2}({\mathcal L}_{X_\lambda}J)J \eta = 0$.
This finishes the proof.
\end{proof}
This proposition in particular finishes the proof of the statement that
$d^\pi_{z}\Upsilon_{T, \lambda}|_{W^{1,2}(z^*\xi) }$ is surjective
if and only if $\Psi_\gamma = d\phi^T|_{\xi_p}$
has no eigenvalue 1 and so finish the proof of Proposition \ref{prop:nondegeneracy}.
Now we add the following additional nondegeneracy hypothesis
of the relevant Reeb orbits.
\begin{hypo}[Nondegeneracy]\label{hypo:nondegeneracy}
Assume that the $T$-periodic orbit in Hypothesis \ref{hypo:basic} is nondegenerate:
If $z(\cdot) = \gamma(T\cdot)$ for a nondegenerate $T$-periodic Reeb orbit $\gamma$ on $M$, we denote the
$S^1$-family of rotations of the loop $z: S^1 \to M$ by
$$
Z = \{z_\theta \in C^1(S^1,M) \mid z_\theta(t): = z(t - \theta), \, \theta \in S^1\}.
$$
\end{hypo}
Since the linearization
operator of the Reeb orbit $z$
$$
A_z: W^{1,2}(z^*\xi)\rightarrow L^2(z^*\xi),
$$
has the form
\begin{eqnarray}\label{eq:Az-eta}
A_z(\eta) & = & Jd(\Upsilon_T)_z(\eta) = J \frac{D\eta}{dt} -T JD X_\lambda(\eta) \nonumber\\
& = & J \frac{D\eta}{dt} - \frac{1}{2}T({\mathcal L}_{X_\lambda}J)\eta.
\end{eqnarray}
Nondegeneracy hypothesis of $z$ implies $\ker A_z = \{0\}$
and then since the Fredholm index of $A_z$ is zero its cokernel is also trivial.
We note that the operator $A_z:W^{1,2}(z^*\xi)\rightarrow L^2(z^*\xi)$ is a self-adjoint unbounded operator
and so has spectral decomposition with the minimal eigenvalue having a positive gap from $0$.
\section{Asymptotic behavior of the operators}
\label{sec:operator}
The aim of this section is to make preparation for the proof of the exponential decay of any (charge vanishing) contact instanton $w$
to some (non-degenerate) Reeb orbit
by using the three-interval method which is given in the next section.
All assumptions in this section can be achieved by looking at the subsequence convergence we have derived from Section \ref{sec:reeborbits}.
\medskip
In this section, we study a sequence of maps $w_k:[0,1]\times S^1\to M$ such that $w_k(\tau,t) \to w_\infty(\tau,t)$
uniformly in $C^l$ sense
where $w_\infty$ is a $\tau$-translational invariant map satisfying $w_\infty(\tau,t) \equiv z(t)$
for a Reeb orbit $z:S^1\to M$.
Here we can look at arbitrary $l\geq 2$, which we fix throughout this section.
First, we look at the Banach bundle $\pi_{{\mathcal E}}:{\mathcal E}_k\to [0,1]$ with fibers
${\mathcal E}_k|_\tau:=L^2(S^1, w_k(\tau, \cdot)^*\xi)$. With the assumption of the convergence of $w_k$ to $z$, we have
\begin{lem}When $k$ is large enough, $\pi_{{\mathcal E}}:{\mathcal E}_k\to [0,1]$ is a Banach bundle with global trivialization
$$\Phi_k:{\mathcal E}_k\to L^2(S^1, z^*\xi)\times [0,1].$$
\end{lem}
\begin{proof}
For any $\zeta_\tau\in {\mathcal E}_k$ with $\pi_{{\mathcal E}}(\zeta_\tau)=\tau$,
define $\Phi_k:{\mathcal E}_k \to L^2(S^1, z^*\xi)\times [0,1]$ by
$$
\Phi_k(\zeta_\tau)(t)=(\Pi(t)\zeta_\tau(t), \tau),
$$
where $\Pi$ is the parallel transport of the vector bundle $\xi\to M$
from $w_k(\tau, \cdot)$ to $z(\cdot)$ via the connection $\nabla^\pi$ along the shortest geodesic.
By the convergence $w_k \to z$, $w_k(\tau,\cdot)$ lies in a small neighborhood of $z$ in $C^l(S^1, M)$,
and by the compactness of the image of $z$ in $M$. In particular $|w_k(\tau,t) - z(t)| \leq \iota_g$ for all
$(\tau,t) \in [0,1] \times S^1$ and $k$ and so $\Pi$ is uniquely defined. Using the $C^1$-closeness,
it is easy to check that $\Phi_k$ defines a bundle isomorphism and so define a trivialization of ${\mathcal E}_k$.
\end{proof}
This trivialization $\Phi_k$ canonically induces one on the induced bundle ${\mathcal L}({\mathcal E}_k)$
over $[0,1]$ with fibers ${\mathcal L}({\mathcal E}_k)|_\tau= \mathbb L(L^2({\mathcal E}_k|_\tau))$, the set of
bounded linear operators on $L^2(S^1, w_k(\tau, \cdot)^*\xi)$.
Now, in particular, we are interested in
$A_k$, where $A_k(\tau)$ can be considered as the linear operator mapping $L^2(S^1, w_k(\tau, \cdot)^*\xi)$
to itself defined by
$$
A_k(\tau)\zeta(t):=J(w_k(\tau, t))\nabla^\pi_t\zeta(t)-\frac{1}{2}\lambda(\frac{\partial w_k}{\partial t})({\mathcal L}_{X_\lambda}J)\zeta(t)
+\frac{1}{2}\lambda(\frac{\partial w_k}{\partial \tau})({\mathcal L}_{X_\lambda}J)J\zeta(t)
$$
for each $\zeta \in L^2(S^1, w_k(\tau, \cdot)^*\xi)$.
Next we prove
\begin{prop}\label{prop:bb2}
$$
\|\Phi_k\circ A_k\circ (\Phi_k)^{-1}(\tau)-A_\infty\|_{{\mathcal L}(L^2(S^1, z^*\xi))} \to 0
$$
uniformly for $\tau\in [0,1]$.
\end{prop}
\begin{proof} The proof is an immediate consequence of the following geometric convergence result
\begin{lem}\label{lem:1/4lambda1} As $k \to \infty$,
$$
\|(\Pi^{-1} A_{\tau_k} \Pi - A_{z_\infty})\| \to 0
$$
$C^2$ uniformly over $\tau \in [0,1]$ as a differential operator acting on $\Gamma(z_\infty^*\xi)$.
\end{lem}
\begin{proof} We recall
$$
A_{\tau} = J \nabla^\pi_t + B
$$
where $B$ is the zero-order operator on $w(\tau,\cdot)^*\xi$ by
\begin{equation}\label{eq:B=}
B \eta = - \frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda} J)\eta + \frac{1}{2} \lambda(\frac{\partial w}{\partial\tau}) ({\mathcal L}_{X_\lambda} J) J\eta
\end{equation}
for $\eta \in w(\tau,\cdot)^*\xi$. Then using the $J$-linearity of
the Hermitian connection $\nabla^\pi$, we compute
\begin{eqnarray}\label{eq:Pi-1Atau}
(\Pi^{-1} A_{\tau} \Pi)\eta & = & \Pi^{-1}(J \nabla^\pi_t + B)(\Pi \eta)\nonumber\\
& = & J \Pi^{-1}\nabla^\pi_t (\Pi \eta) + \Pi^{-1} B \Pi(\eta).
\end{eqnarray}
Now for given $\tau$, we consider the map
$$
\Gamma(s,t)= \textup{exp}_{z_\tau(t)}(s E(z_\tau(t), w(\tau,t)))
$$
where we recall the definition $E(x,y) = \textup{exp}_x^{-1} y$.
Then by definition of the parallel transport, we have
$$
\Pi \eta(t) = \Xi(1,t)
$$
where $\Xi= \Xi(s,t)$ is the solution to the ordinary differential equation
$$
\nabla^\pi_s \Xi = 0, \quad \Xi(0,t) = \eta(t)
$$
which defines the parallel transport of $\eta(t)$
along the geodesic
$$
s \mapsto \Gamma(s,t): = \textup{exp}_{z_\tau(t)}(s E(z_\tau(t), w(\tau,t))).
$$
Now we write
\begin{equation}\label{eq:Pi-1}
\Pi^{-1}\nabla^\pi_t(\Pi \eta) - \nabla^\pi_{\dot z_\tau}\eta
= \int_0^1 \frac{d}{ds} \left(\Pi_s^{-1}(\nabla^\pi_t(\Pi_s \eta))\right)\, ds
\end{equation}
where $\Pi_s$ is the parallel transport from $z_\tau$ to $\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))$.
Then we compute
\begin{eqnarray*}
\frac{d}{ds} \left(\Pi_s^{-1}\nabla^\pi_t(\Pi_s \eta)\right)
&= & \Pi_s^{-1}\nabla^\pi_s (\nabla^\pi_t(\Pi_s \eta))) \\
& = & \Pi_s^{-1}\left(\nabla^\pi_t \nabla^\pi_s(\Pi_s \eta)\right)
+ \Pi_s^{-1} R^\pi\left(\frac{\partial \Gamma}{\partial s},
\frac{\partial \Gamma}{\partial t}\right) (\Pi_s \eta) \\
& = & \Pi_s^{-1} R^\pi\left(\frac{\partial \Gamma}{\partial s}, \frac{\partial \Gamma}{\partial t}\right) (\Pi_s \eta).
\end{eqnarray*}
But we note
\begin{eqnarray*}
\frac{\partial \Gamma}{\partial s} & = & d_2\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))(E(z_\tau,w(\tau,\cdot))\\
\frac{\partial \Gamma}{\partial t} & = &
D_1\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))\left(\frac{\partial z_\tau}{\partial t}\right) +
d_2\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))\left(s\frac{\partial E(z_\tau,w(\tau,\cdot))}{\partial t}\right).
\end{eqnarray*}
The constants $C, \, C'$ appearing in the computations below may vary place by place
but always depend only on the triad $(M,\lambda, J)$ and the $C^1$-bound of $w$.
Using the equality $|E(x,y)|= d(x,y)$ when $d(x,y)$ is less than injective radius, it follows that
\begin{eqnarray*}
|d_2\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))(E(z_\tau,w(\tau,\cdot))| & \leq & C |E(z_\tau,w(\tau,\cdot))| \leq C \|d(z_\tau,w(\tau,\cdot))\|_{C^0}\\
\left|D_1\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))\left(\frac{\partial z_\tau}{\partial t}\right)\right|
& \leq & C \left|\frac{\partial z_\tau}{\partial t}\right| \leq C T\|X_\lambda\|_{C^0} = C T.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
&{}& \left|D_1\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))\left(s\frac{\partial E(z_\tau,w(\tau,\cdot))}{\partial t}\right)\right|
= \\
&{}& \left|D_1\textup{exp}_{z_\tau}(s E(z_\tau,w(\tau,\cdot)))s
\left(D_1(E(z_\tau,w(\tau,\cdot))\frac{\partial z_\tau}{\partial t} +
d_2(E(z_\tau,w(\tau,\cdot))\frac{\partial w}{\partial t}\right)\right|.
\end{eqnarray*}
It follows from the standard Jacobi field estimate \cite{katcher}
\begin{eqnarray*}
\left|D_1(E(z_\tau,w(\tau,\cdot))\frac{\partial z_\tau}{\partial t} +
d_2(E(z_\tau,w(\tau,\cdot))\frac{\partial w}{\partial t}\right| & \leq & C \left(
\left|\frac{\partial z_\tau}{\partial t}\right| + \left|\frac{\partial w}{\partial t}\right|\right) \\
& \leq & C'T
\end{eqnarray*}
where the second inequality comes since $\frac{\partial z_\tau}{\partial t} = TX_\lambda(z_\tau)$ and
$
\left|\frac{\partial w}{\partial t}\right| \to T |X_\lambda| = T
$
uniformly as $\tau \to \infty$. In summary, we have derived
$$
\left|\frac{\partial \Gamma}{\partial s}\right| \leq C \|d(z_\tau,w(\tau,\cdot))\|_{C^0}, \quad
\left|\frac{\partial \Gamma}{\partial t}\right| \leq C'T.
$$
Therefore we have established
$$
\left|\Pi_s^{-1} R^\pi\left(\frac{\partial \Gamma}{\partial s}, \frac{\partial \Gamma}{\partial t}\right) (\Pi_s \eta)\right|
\leq C C' T \|d(z_\tau,w(\tau,\cdot))\|_{C^0} |\eta|.
$$
Substituting this into \eqref{eq:Pi-1Atau}, we have obtained
\begin{eqnarray*}
|(\Pi^{-1} A_{\tau} \Pi \eta) - A_{z_\tau} \eta|
& \leq & |\Pi^{-1}\nabla^\pi_t(\Pi \eta) - \nabla^\pi_{\dot z_\tau}\eta|
+ |\Pi^{-1} B \Pi(\eta) - DX_\lambda(z_\tau)\eta| \\
& = & o(\tau)|\eta|.
\end{eqnarray*}
Here we also use the convergence $|\Pi^{-1} B \Pi(\eta) - DX_\lambda(z_\tau)\eta| = o(\tau)|\eta|$
from the expression of $B$ \eqref{eq:B=} and the convergence
$$
\lambda(\frac{\partial w}{\partial \tau}) \to 0, \quad \lambda(\frac{\partial w}{\partial t}) \to T
$$
from Corollary \ref{cor:tangent-convergence}, where the first convergence to zero follows from the assumption $Q = 0$.
This finishes the proof of the lemma.
\end{proof}
\end{proof}
Here we denote by $A_\infty$ the linearized operator $A_z$ introduced in Section \ref{sec:prelim}.
and $\nabla^z_\tau$ is the covariant derivative with respect to the trivial connection on
Now let us assume $\zeta_k$ is a section of the vector bundle $w_k^*\xi\to [0,1]\times S^1$ such that
$\zeta_k(\tau)\in L^2(S^1, w_k(\tau, \cdot)^*\xi)$ for each $\tau\in [0,1]$.
Then we can consider $\zeta_k$ as a section of ${\mathcal E}_k\to [0,1]$.
We also use $\nabla^\pi_\tau$ to denote the covariant derivative of the Banach bundle ${\mathcal E}_k\to [0,1]$
induced form the pull-back connection $w^*\nabla^\pi$ in the direction of $\frac{\partial}{\partial \tau}$
for every $k$.
After these preparation, we look at the following evolution equation
$$
\nabla^\pi_\tau \zeta_k + A_k \zeta_k =0
$$
on $[0,1]\times S^1$. We rewrite it into
$$
\Phi_k \nabla^\pi_\tau(\Phi_k^{-1}\eta_k)+\Phi_k\circ A_k (\Phi_k^{-1}\eta_k)=0
$$
for the section $\zeta_k = \Phi_k \circ \eta_k$ as a section of $\pi^*{\mathcal E}_k$
the pull-back bundle $\pi^*{\mathcal E}_\infty \to [0,1] \times S^1$
under the projection map $\pi: [0,1] \times S^1 \to S^1$.
\begin{lem} Denote by $\frac{d}{d \tau}$ the covariant derivative
with respect to the trivial connection on $\pi^*{\mathcal E}_\infty = [0,1] \times {\mathcal E}_\infty$. Then
$$
\|\Phi_k \nabla^\pi_\tau \Phi_k^{-1} - \frac{d}{d \tau}\| \to 0
$$
as $k \to \infty$ where the norm is the norm taken as the operator $W^{1,2}(\pi^*{\mathcal E}_\infty))$
to $L^2(\pi^*{\mathcal E}_\infty))$.
\end{lem}
\begin{proof} The proof is similar to that of Proposition \ref{prop:bb2} and so omitted.
\end{proof}
Now if we suppose that $\eta_k$ converges to a section $\eta_\infty = \eta_\infty(\tau)(t)$
in $C^1$ on $[0,1] \times S^1$, then we get the following equation
$$
\frac{d \eta_\infty}{d\tau} +A_\infty\eta_\infty=0
$$
on $\pi^* {\mathcal E}_\infty$ by the $C^1$-convergence of $w(\tau,t) \to z(t)$.
\section{The three-interval method in the application of proving exponential decay at cylindrical ends}
\label{sec:3-interval}
The three-interval method is based on the following so-called the three-interval lemma
which we learn from \cite[Lemma 9.4]{mundet-tian}.
\begin{lem}[Three-interval]\label{lem:three-interval}
For any sequence of nonnegative numbers $x_k$, $k=0, 1, \cdots, N$,
if there exists some fixed constant $\gamma\in (0, \frac{1}{2})$ such that
the inequality
$$
x_k\leq \gamma(x_{k-1}+x_{k+1})
$$
holds for all $1\leq k\leq N-1$.
Then we have
\begin{equation}\label{eq:3-inter-ineq}
x_k\leq x_0\xi^{-k}+x_N\xi^{-(N-k)},
\end{equation}
for all $k=0, 1, \cdots, N$, where $\xi:=\frac{1+\sqrt{1-4\gamma^2}}{2\gamma}$.
\end{lem}
\begin{rem}\label{rem:three-interval}
\begin{enumerate}
\item If we write $\gamma=\gamma(c):=\frac{1}{e^c+e^{-c}}$ for some $c>0$, then the conclusion
can be written into the exponential form
$$
x_k\leq x_0e^{-ck}+x_Ne^{-c(N-k)}.
$$
\item
For any bounded infinite sequence $\{x_k\}_{k=0, 1, \cdots}$ with the three-interval inequality \eqref{eq:3-inter-ineq}
holding for every $k$,
then we have
$$
x_k\leq x_0e^{-ck},
$$
which is the case we are going to use for our purpose.
\end{enumerate}
\end{rem}
In our consideration below, roughly speaking, we are going to look at
$$
x_k:=\|\pi\frac{\partial w}{\partial\tau}\|^2_{{L^2([k+1,k+2]\times S^1)}}=\|\pi\frac{\partial w_k}{\partial\tau}\|^2_{L^2([1,2]\times S^1)},
$$
where $w_k(\tau, t):=w(k+\tau, t)$ on $[0,1]\times S^1$ are defined as in Section
\ref{sec:reeborbits}.
\begin{lem}If for such $x_k:=\|\pi\frac{\partial w}{\partial\tau}\|^2_{{L^2([k+1,k+2]\times S^1)}}$,
we have proved that $x_k\leq x_0e^{-ck}$ for any $k=0, 1, \cdots$, where $c>0$ is some constant.
Then
$$
|(\nabla^\pi)^l\pi\frac{\partial w}{\partial \tau}(\tau, t)|\leq C_l e^{-c_l \tau},
$$
where $C_l$ and $c_l$ are some constants depending on $l$ and $(M, \lambda, J)$, but independent of $w$.
\end{lem}
\begin{proof} We first note that the section $\zeta:=\pi\frac{\partial w}{\partial \tau}$ satisfies the fundamental equation \eqref{eq:main-eq-a}
$$
\nabla_\tau^\pi \zeta + J \nabla_t^\pi \zeta
- \frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J)\zeta + \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J)J\zeta =0
$$
on $[0,\infty) \times S^1$ in the isothermal coordinates $(\tau,t)$.
We apply the local elliptic bootstrapping for the elliptic operator
$$
\nabla^\pi_\tau+J\nabla^\pi_t : W^{l+1, 2}(w^*\xi)\to W^{l, 2}(w^*\xi)
$$
to $\zeta$ inductively over $\ell \in {\mathbb N}$ for the pair of the compact domains
$D_1 = [k+1,k+2] \times S^1 \subset D_2 = [k+1/2,k+5/2] \times S^1$ for each $k \in {\mathbb Z}_{\geq 0}$
and derive the conclusion.
The uniform $C^\infty$ convergence results obtained in Corollary \ref{cor:tangent-convergence}
then give rise to the uniformity of the constants $C_\ell$ independent of $k$. This finishes the proof
of the independence of $C_\ell$ on the map $w$.
\end{proof}
\medskip
In the rest of this section, we are going to apply the three-interval method to prove the assumption of the above lemma, i.e., $x_k\leq x_0e^{-ck}$, for some $c>0$.
\smallskip
First, if the bounded infinite sequence $\{x_k\}$ defined above satisfies the three-interval inequality \eqref{eq:3-inter-ineq}
for some choice of $0 < \delta < 1$, then
by using Lemma \ref{lem:three-interval}, in particular Remark \ref{rem:three-interval} (2), we are done.
Now assume that \eqref{eq:3-inter-ineq} is not satisfied by every triple $(k-1, k, k+1)$.
We collect all the triples that reverse the direction of the inequality.
If such triples are finitely many, in another word, \eqref{eq:3-inter-ineq} holds after some large $k_0$,
then we will still get the exponential estimate as we want.
Otherwise, there are infinitely many such triples, which we enumerate
by $\{(l_k-1, l_k, l_k+1)\}$, such that
\begin{equation}\label{eq:against-3interval}
x_{l_k}>\gamma(2\delta)(x_{l_k-1}+x_{l_k+1}).
\end{equation}
Before we deal with this case, we first remark that
this hypothesis in particular implies $\pi\frac{\partial w}{\partial \tau} \not \equiv 0$ on
$[l_k,l_k+3]\times S^1$, i.e.,
$\|\pi\frac{\partial w_{l_k}}{\partial t}\|_{L^\infty([0,3]\times S^1)}\neq 0$.
Recall that $\zeta$ satisfies the fundamental equation
\eqref{eq:main-eq-a}
$$
\nabla_\tau^\pi \zeta + J \nabla_t^\pi \zeta
- \frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J)\zeta + \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J)J\zeta=0.
$$
Follow the notation in Section \ref{sec:operator}
$$
A(\tau)(\zeta):=J \nabla_t^\pi \zeta
- \frac{1}{2} \lambda(\frac{\partial w}{\partial t})({\mathcal L}_{X_\lambda}J)\zeta + \frac{1}{2} \lambda(\frac{\partial w}{\partial \tau})({\mathcal L}_{X_\lambda}J)J\zeta,
$$
we rewrite it as
\begin{equation}\label{eq:main-eq-a1}
\nabla^\pi_\tau \zeta+A(\tau)(\zeta)=0.
\end{equation}
Since \eqref{eq:main-eq-a1} is a linear equation of $\zeta$, we divide it by
$\|\pi\frac{\partial w_{l_k}}{\partial \tau}\|_{L^\infty([0,3]\times S^1)}$, which can not vanish
by the hypothesis as we remarked before, and get a sequence of new equations
$$
\nabla^\pi_\tau \zeta^{l_k}+A(\tau)(\zeta^{l_k})=0
$$
with $\zeta^{l_k}$ defined by
$$
\zeta^{l_k}:=\frac{\pi\frac{\partial w}{\partial\tau}|_{[l_k, l_k+3]\times S^1}}{\|\pi\frac{\partial w_{l_k}}{\partial \tau}\|_{L^\infty([0,3]\times S^1)}}.
$$
Next we translate them to be a sequence of sections defined on $[0,3]\times S^1$ as
$$
\overline\zeta_{l_k}(\tau, t):=\zeta^{l_k}_{l_k}(\tau, t):=\zeta^{l_k}(l_k+\tau, t).
$$
We have now
\begin{eqnarray}
\|\overline \zeta_{l_k}\|_{{L^\infty([0,3]\times S^1)}}&=&1\nonumber\\
\nabla_\tau\overline\zeta_{l_k}+ A_{l_k}(\tau)(\overline\zeta_{l_k})
&=&0\label{eq:nablabarzeta}\\
\|\overline \zeta_{l_k}\|^2_{L^2([1,2]\times S^1)}&\geq&\gamma(2\delta)
(\|\overline \zeta_{l_k}\|^2_{L^2([0,1]\times S^1)}+\|\overline \zeta_{l_k}\|^2_{L^2([2,3]\times S^1)}).\nonumber
\end{eqnarray}
Here $A_{l_k}$ is defined as $A_{l_k}(\tau):=A(l_k+\tau)$ as operators from $L^2(w_{l_k}(\tau, \cdot)^*\xi)$ to itself.
By applying Theorem \ref{thm:subsequence} and further taking subsequence (but we still denote by $\{l_k\}$),
we obtain a (non-degenerate) Reeb orbit
$\gamma$ such that of $w_{l_k}(\tau, t)\to \gamma(T\,t)$ uniformly on $[0,3]\times S^1$.
Now by using the uniform $L^\infty$-bound of the sequence $\overline\zeta_k$, we can apply Proposition \ref{prop:bb2}
in Section \ref{sec:operator}, and show that
\eqref{eq:nablabarzeta} lead to
\begin{equation}\label{eq:xibarinfty-zeta}
\nabla^\pi_\tau \overline\zeta_\infty+A_\infty (\overline\zeta_\infty)=0,
\end{equation}
where $A_\infty$ is the linearization operator along the limiting Reeb orbit $\gamma$
($A_z$ in Section \ref{sec:operator}), and $\overline\zeta_\infty(\tau, \cdot)$ is a nonzero section of $z^*\xi\to S^1$ for every $\tau\in [0,3]$.
From the fact that $\overline\zeta_\infty$ as the subsequence limit of $\overline\zeta_k$, it also follows
\begin{equation}\label{eq:xibarinfty-ineq-zeta}
\|\overline\zeta_\infty\|^2_{L^2([1,2]\times S^1)}\geq\gamma(2\delta)
(\|\overline\zeta_\infty\|^2_{L^2([0,1]\times S^1)}+\|\overline\zeta_\infty\|^2_{L^2([2,3]\times S^1)}).
\end{equation}
Since $A_\infty$ is assumed to be a
(unbounded) self-adjoint operator on $L^2(S^1, \gamma^*\xi)$ with its domain $W^{1,2}(S^1, \gamma^*\xi)$,
let $\{e_i\}$ be its orthonormal eigen-basis of $L^2(S^1, \gamma^*\xi)$ with respect to $A_\infty$, and
we consider the eigen-function expansion of
$$
\overline\zeta_\infty(\tau) = \sum_{i=1}^\infty a_i(\tau)\, e_i
$$
for each $\tau \in [0,1]$, where $e_i$ are the eigen-functions associated to the eigenvalue $\lambda_i$ with
$$
0 < \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_i \leq \cdots \to \infty.
$$
By plugging such $\overline\zeta_\infty$ into \eqref{eq:xibarinfty-zeta}, we derive that
$$
a_i'(\tau)+\lambda_ia_i(\tau)=0, \quad i=1, 2, \cdots
$$
and further it follows that
$$
a_i(\tau)=c_ie^{-\lambda_i\tau}, \quad i=1, 2, \cdots
$$
which indicates that
$$
\|a_i\|^2_{L^2([1,2])}=\gamma(2\lambda_i)(\|a_i\|^2_{L^2([0,1])}+\|a_i\|^2_{L^2([2,3])}).
$$
Notice that
\begin{eqnarray*}
\|\overline\zeta_\infty\|^2_{L^2([k, k+1]\times S^1)}&=&\int_{[k,k+1]}\|\overline\zeta_\infty(\tau)\|^2_{L^2(S^1)}\,d\tau\\
&=&\int_{[k,k+1]}\sum_i |a_i(\tau)|^2 d\tau\\
&=&\sum_i\|a_i\|^2_{L^2([k,k+1])},
\end{eqnarray*}
and by the decreasing property of $\gamma$, we get
$$
\|\overline \zeta_\infty\|^2_{L^2([1,2]\times S^1)}< \gamma(2\delta) (\|\overline \zeta_\infty\|^2_{L^2([0,1]\times S^1)}+\|\overline \zeta_\infty\|^2_{L^2([2,3]\times S^1)}).
$$
Since $\overline\zeta_\infty\not \equiv 0$, this contradicts to \eqref{eq:xibarinfty-ineq-zeta}, if we choose $0 < \delta < \lambda_1$ at the beginning. This finishes the proof.
\medskip
We end this section by the following remark.
\begin{rem}
In \cite{oh-wang2}, the authors generalize the three-interval method
so that it also applies to the case with an exponentially decayed perturbation term
added to the evolution equation (the fundamental equation in the current case).
The differential inequality method of proving the exponential decay of that case
is presented in \cite{robbin-salamon}.
\end{rem}
\section{Exponential decay of the Reeb component}
In this section, we prove the exponential decay of the Reeb component $w^*\lambda$ of any contact instanton with vanishing charge.
\medskip
We define a complex-valued function
$$
\theta(\tau,t) = \left(w^*\lambda(\frac{\partial}{\partial t})-T\right) + \sqrt{-1}\left(w^*\lambda(\frac{\partial}{\partial\tau})\right).
$$
Then using the relation $*dw^*\lambda=|\zeta|^2$ from Proposition \ref{prop:energy-omegaarea} and the equation $d(w^*\lambda\circ j)=0$, we
notice that $\theta$ satisfies the equations
\begin{equation}\label{eq:atatau-equation}
{\overline \partial} \theta =\mu, \quad \mu
=\frac{1}{2}|\zeta|^2 + \sqrt{-1}\cdot 0,
\end{equation}
where ${\overline \partial}=\frac{1}{2}\left(\frac{\partial}{\partial \tau}+\sqrt{-1}\frac{\partial}{\partial t}\right)$ the standard Cauchy-Riemann operator for the standard complex structure $J_0=\sqrt{-1}$.
Notice that from previous section we have already
established the $C^\infty$ exponential decay of $\mu$.
The exponential decay of $\theta$ follows from the following lemma.
(See Remark \ref{rem:ms-3} below for more about this lemma.)
\begin{lem}\label{lem:exp-decay-lemma}
Suppose the complex-valued functions $\theta$ and $\mu$ defined on $[0, \infty)\times S^1$
satisfy
$$
{\overline \partial} \theta = \mu
$$
and
\begin{eqnarray*}
\|\mu\|_{L^2(S^1)}+\left\|\nabla\mu\right\|_{L^2(S^1)}&\leq& Ce^{-\delta \tau} \quad \text{ for some constants } C, \delta>0\\
\lim_{\tau\rightarrow +\infty}\theta &=& 0,\\
\end{eqnarray*}
then $\|\theta\|_{L^2(S^1)}\leq \overline{C}e^{-\delta \tau}$
for some constant $\overline{C}$.
\end{lem}
\begin{proof}
Denote by $\eta:=\frac{\partial\theta}{\partial\tau}$, then \eqref{eq:atatau-equation} indicates
$J_0\eta=J_0\mu+\frac{\partial\theta}{\partial t}$, and with direct calculations, we can write
\begin{equation}\label{eq:2-order}
\frac{\partial^2}{\partial\tau^2}|\eta|^2=2|J_0\frac{\partial}{\partial t}\eta|^2+o(1)|\eta|^2+\frac{\partial}{\partial t}(\cdots),
\end{equation}
where by $(\cdots)$ we mean some functions can be calculated explicitly but we omit here for it is not used.
Since we have
$$
\int_{S^1}J_0(\eta-\mu)=\int_{S^1}\frac{\partial \theta}{\partial t}=0,
$$
by Wirtinger's inequality it follows
$$
\int_{S^1}|J_0(\frac{\partial}{\partial t}\eta-\frac{\partial}{\partial t}\mu)|^2\geq \int_{S^1}|J_0(\eta-\mu)|^2.
$$
Hence we get
\begin{eqnarray*}
2\int_{S^1}|J_0\frac{\partial}{\partial t}\eta|^2+2\int_{S^1}|J_0\frac{\partial}{\partial t}\mu|^2&\geq& \int_{S^1}|J_0(\frac{\partial}{\partial t}\eta-\frac{\partial}{\partial t}\mu)|^2\\
&\geq&\int_{S^1}|J_0(\eta-\mu)|^2\geq
\frac{1}{2}\int_{S^1}|\eta|^2-\int_{S^1}|\mu|^2,
\end{eqnarray*}
and further
$$
2\int_{S^1}|J_0\frac{\partial}{\partial t}\eta|^2\geq \frac{1}{2}\int_{S^1}|\eta|^2-2\int_{S^1}|J_0\frac{\partial}{\partial t}\mu|^2-\int_{S^1}|\mu|^2.
$$
Notice that the last two terms on the right hand side exponentially decays since the assumption.
After integrating \eqref{eq:2-order} over $S^1$, we have now
\begin{eqnarray*}
\frac{\partial^2}{\partial\tau^2}\int_{S^1}|\eta|^2\geq (\frac{1}{2}+o(1))\int_{S^1}|\eta|^2+\text{some exponentially decay terms}.
\end{eqnarray*}
Then the exponential decay of
$\int_{S^1}|\eta|^2$ follows from some standard tricks of dealing with differential inequality with
an exponential decay term (e.g., see \cite{robbin-salamon} for details of the derivation). Together
with $\lim_{\tau\rightarrow +\infty}\theta=0$, the exponential decay of $\int_{S^1}|\theta|^2$ follows.
\end{proof}
\begin{rem}\label{rem:ms-3}
We emphasize that this baby model of degenerate type (the section we look at here is contained in
the eigenspace of eigenvalue $1$ of the full linearized operator given in Section \ref{sec:prelim})
already indicates some important ideas that need to be clarified in the more general setting named
the \emph{Morse-Bott} case, where the linearized operator along the Reeb orbit has a nontrivial
kernel in the tangent plane of the Reeb foliated submanifold.
The idea of looking at the derivatives ($\frac{\partial \theta}{\partial\tau}$ here) instead of the original
section itself ($\theta$ here) is one essential point when dealing with the asymptotic estimates for degenerate directions.
Unlike this baby model, for which the original Wirtinger's inequality can be easily derived and directly applied,
the general Morse-Bott case needs much more subtle considerations, which is mainly due to the lack of suitable
geometric coordinates to conduct analysis.
In \cite{HWZ4}, \cite{bourgeois}, the authors there use special coordinates around one Reeb orbit to prove
the exponential decay of Morse-Bott case. In the midst of complicated coordinate calculations, it is hard to
see the geometry behind the coordinate calculations.
On the other hand, the present authors construct
the \emph{canonical geometric coordinates} in terms of the center of mass for loops in contact manifolds
and provide a new proof of the exponential decay for the Morse-Bott case (with the help of the three-interval
method developed in this paper), which makes the geometric meaning become much clearer. Moreover,
we believe this geometric observation will lead to a much better understanding of contact instantons in contact manifolds
(or pseudo-holomorphic curves in symplectizations) that related many standard issues such as compactification and gluing, etc.
\end{rem}
The exponential decay in $C^\infty$-sense follows from the elliptic bootstrapping by keeping
applying Lemma \ref{lem:exp-decay-lemma}.
\section{$C^0$ exponential convergence of $w$ (and of $a$)}
\label{sec:C0exponential}
\subsection{$C^0$ exponential convergence of the map $w$}
The following is the main proposition we prove here.
\begin{prop}\label{prop:czero-convergence}
Under Hypothesis \ref{hypo:basic}, for any contact instanton $w$ with vanishing charge,
there exists a unique Reeb orbit $z(\cdot)=\gamma(T\cdot):S^1\to M$ with period $T>0$, such that
$$
\|d(w(\tau, \cdot), z(\cdot))
\|_{C^0(S^1)}\rightarrow 0,
$$
as $\tau\rightarrow +\infty$.
\end{prop}
\begin{proof}
We start with claiming that for each $t\in S^1$, $w(\cdot, t)$ is a Cauchy sequence.
If this claim is not true, then there exist some $t_0\in S^1$ and some constant $\epsilon>0$, sequences $\{\tau_k\}$, $\{p_k\}$ such that
$$
d(w(\tau_{k+p_k}, t_0), w(\tau_k, t_0))\geq\epsilon.
$$
Then from the continuity of $w$ in $t$, there exists some $l>0$ small such that
$$
d(w(\tau_{k+p_k}, t), w(\tau_k, t))\geq\frac{\epsilon}{2}, \quad |t-t_0|\leq l.
$$
Hence
\begin{eqnarray*}
&{}&\int_{S^1}d(w(\tau_{k+p_k}, t), w(\tau_k, t))\ dt\\
&=&\int_{|t-t_0|\leq l}d(w(\tau_{k+p_k}, t), w(\tau_k, t))\ dt+\int_{|t-t_0|>l}d(w(\tau_{k+p_k}, t), w(\tau_k, t))\ dt\\
&\geq&\int_{|t-t_0|\leq l}d(w(\tau_{k+p_k}, t), w(\tau_k, t))\ dt \geq \epsilon l.
\end{eqnarray*}
On the other hand, we compute
\begin{eqnarray*}
&{}&\int_{S^1}d(w(\tau_{k+p_k}, t), w(\tau_k, t))\ dt\\
&\leq&\int_{S^1}\int_{\tau_k}^{\tau_{k+p_k}}\left|\frac{\partial w}{\partial s}(s, t)\right|\ ds\ dt\\
&=&\int_{\tau_k}^{\tau_{k+p_k}}\int_{S^1}\left|\frac{\partial w}{\partial s}(s, t)\right|\ dt\ ds\\
&\leq&\int_{\tau_k}^{\tau_{k+p_k}}\left(\int_{S^1}\left|\frac{\partial w}{\partial s}(s, t)\right|^2\,
dt\right)^{\frac{1}{2}}\ ds\\
&\leq&\int_{\tau_k}^{\tau_{k+p_k}}Ce^{-\delta s}\ ds\\
&=&\frac{C}{\delta}(1-e^{-(\tau_{k+p_k}-\tau_k)})e^{-\tau_k} \leq \frac{C}{\delta}e^{-\tau_k}.
\end{eqnarray*}
Hence we can take $\tau_k$ large and get contradiction.
Now by using the subsequence convergence from Theorem \ref{thm:subsequence},
we can pick an arbitrary subsequence $\{\tau_k\}$ and $z\in Z$ such that
$$
w(\tau_k, t)\to z(t), \quad k\to \infty
$$
uniformly in $t$.
Then immediately from the fact that $w(\cdot, t)$ is a Cauchy sequence for any $t$, we get for any $t\in S^1$,
$$
w(\tau, t)\to z(t), \quad \tau\to \infty.
$$
What left to show is just this convergence is uniform in $t$, i.e., it is in $C^{0}(S^1)$ sense.
Assume this is not true. Then there exist some $\epsilon>0$ and some sequence $(\tau_k, t_k)$ such that
$$
d(w(\tau_k, t_k), z(t_k))\geq2\epsilon.
$$
Since $t_k\in S^1$, we can further take subsequence, still denote by $t_k$, such that $t_k\to t_0\in S^1$.
We can take $k$ large such that $d(z(t_k), z(t_0))\leq \frac{1}{2}\epsilon$.
We also look at
\begin{eqnarray*}
d(w(\tau, t_k), w(\tau, t_0)) \leq \int_{t_0}^{t_k}\left|\frac{\partial w}{\partial t}(\tau, s)\right|\, ds
\leq(t_k-t_0)\|d w\|_{C^0},
\end{eqnarray*}
and so we can make it less than $\frac{1}{2}\epsilon$ by taking $k$ large.
On the other hand, we have
\begin{eqnarray*}
d(w(\tau_k, t_0), z(t_0))&\geq&
d(w(\tau_k, t_k), z(t_k))-d(w(\tau_k, t_k), w(\tau_k, t_0))\\
&{}&-d(z(t_k), z(t_0))\\
&\geq&2\epsilon-\frac{1}{2}\epsilon-\frac{1}{2}\epsilon = \epsilon,
\end{eqnarray*}
which gives contradiction to the pointwise convergence.
This finishes the proof.
\end{proof}
\subsection{$C^0$ exponential convergence of $a$ in the symplectization case}
Finally we relate our general study of contact Cauchy-Riemann map to
the special exact case, i.e. the case of maps $(a,w)$ into
the symplectization ${\mathbb R} \times M$.
In other words, we prove the $C^0$ convergence of $a$, assuming that
$$
w^*\lambda\circ j=da.
$$
In this case, we have
$$
w^*\lambda(\frac{\partial}{\partial t})=\frac{\partial a}{\partial \tau}, \quad w^*\lambda(\frac{\partial}{\partial\tau})= - \frac{\partial a}{\partial t}
$$
and the pair $(a,w)$ satisfies the standard pseudo-holomorphic curve equation
$$
{\overline \partial}^\pi w = 0, \quad w^*\lambda \circ j = da.
$$
\begin{prop}\label{prop:czero-convergence2}
There exists some constant $C_0$, such that
$$
\|a(\tau, \cdot)-T\tau-C_0\|_{C^0(S^1)}\rightarrow 0,
$$
as $\tau\rightarrow +\infty$.
\end{prop}
\begin{proof}
Define $b(\tau, t)=a(\tau, t)-T\tau$. Then we have
\begin{eqnarray*}
\frac{\partial b}{\partial \tau}= w^*\lambda(\frac{\partial}{\partial t})-T &\to& 0,\\
\frac{\partial b}{\partial t}= -w^*\lambda(\frac{\partial}{\partial\tau})&\to& 0,
\end{eqnarray*}
as $\tau\rightarrow +\infty$ in $C^0(S^1)$ topology.
Define $\alpha(\tau)=\int_{S^1}b(\tau, t)dt$ and $\tilde{b}(\tau)=b(\tau, t)-\alpha(\tau)$.
Then
$$|\alpha'(\tau)|=\left|\int_{S^1}\frac{\partial b}{\partial \tau} dt\right| \leq \int_{S^1}
\left|w^*\lambda(\frac{\partial}{\partial t})-T \right|dt \leq Ce^{-\delta \tau},$$
which indicates that $\alpha(\tau)$ is a Cauchy sequence. Then there exists some
constant $C_0$ such that $\alpha(\tau)\rightarrow C_0$ as $\tau\rightarrow +\infty$.
On the other hand, notice here $\int_{S^1}\tilde{b}(\tau, t)dt=0$ and then for any $\tau$,
there exists some point $t_0\in S^1$ such that $\tilde{b}(\tau, t_0)=0$.
Then for any $\tau$,
$$
|\tilde{b}(\tau, t)|=|\tilde{b}(\tau, t)-\tilde{b}(\tau, t_0)|\leq
|t-t_0|\left\|\frac{\partial \tilde{b}}{\partial t}\right\|_{C^0(S^1)}
\leq\left\|\frac{\partial \tilde{b}}{\partial t}\right\|_{C^0(S^1)}
= \left\|\frac{\partial b}{\partial t}\right\|_{C^0(S^1)},
$$
and thus
$$\|\tilde{b}(\tau, \cdot)\|_{C^0(S^1)}\leq \left\|\frac{\partial b}{\partial t}\right\|_{C^0(S^1)}
=\left\|w^*\lambda(\frac{\partial}{\partial\tau})\right\|_{C^0(S^1)} \rightarrow 0$$
as $\tau\rightarrow +\infty$. Hence
$$
\|a(\tau,\cdot)-T\tau-C_0\|_{C^0(S^1)}\leq\|\tilde{b}(\tau, \cdot)\|_{C^0(S^1)}+|\alpha(\tau)-C_0|\rightarrow 0,
$$
as $\tau\rightarrow +\infty$.
We are done with the proof.
\end{proof}
Now the $C^0$ exponential decay immediately follows
from the $C^\infty$ exponential decay of $dw$ proved in previous sections and the $C^0$ convergence,
Proposition \ref{prop:czero-convergence} and Proposition \ref{prop:czero-convergence2}.
\begin{thm}
There exist some constants $C>0$, $\delta>0$ and $\tau_0$ large such that for any $\tau>\tau_0$,
\begin{eqnarray*}
\|d\left( w(\tau, \cdot), z(\cdot) \right) \|_{C^0(S^1)} &\leq& C\, e^{-\delta \tau}\\
\|a(\tau, \cdot)-T\tau-C_0\|_{C^0(S^1)} & \leq & C\, e^{-\delta \tau}
\end{eqnarray*}
\end{thm}
\begin{proof}
For any $\tau<\tau_+$, similarly as in previous proof,
\begin{eqnarray*}
d(w(\tau, t), w(\tau_+, t))\leq \int^{\tau_+}_{\tau}\left| \frac{\partial w}{\partial \tau}(s,t) \right|\,ds
\leq \frac{C}{\delta}e^{-\delta \tau}.
\end{eqnarray*}
Take $\tau_+\rightarrow +\infty$ and using the $C^0$ convergence of $w$ part, i.e.,
Proposition \ref{prop:czero-convergence}, we get
$$d(w(\tau, t), z(t))\leq\frac{C}{\delta}e^{-\delta \tau}.$$
This proves the first inequality.
Similarly, we have
$$|(a(\tau_+, t)-T\tau_-C_0) -(a(\tau, t)-T\tau-C_0) |\leq \int^{\tau_+}_{\tau}
\left| w^*\lambda(\frac{\partial}{\partial t})(s, t)-T \right|ds\leq \frac{C}{\delta}e^{-\delta \tau},
$$
where the last inequality comes from the $C^0$ exponential decay of
$\left| w^*\lambda(\frac{\partial}{\partial t})(s, \cdot)-T \right|$ in \ref{prop:czero-convergence2}.
By taking $\tau_+\rightarrow +\infty$, we are done with the proof of the second inequality.
\end{proof}
\part{Appendix}
\section{The Weitzenb\"ock formula for vector valued forms}\label{appen:weitzenbock}
In this appendix, we recall the standard Weitzenb\"ock formulas applied to our
current circumstance. A good exposition on the general Weitzenb\"ock formula is
provided in Appendix of \cite{freed-uhlen}.
Assume $(P, h)$ is a Riemannian manifold of dimension $n$ with metric $h$, and $D$ is the Levi-Civita connection.
Let $E\to P$ be any vector bundle with inner product $\langle\cdot, \cdot\rangle$,
and assume $\nabla$ is a connection of $E$ which is compatible with $\langle\cdot, \cdot\rangle$.
For any
vector bundle $E$-valued form $s$, calculate the (Hodge) Laplacian of the energy density
of $s$ and we get
\begin{eqnarray*}
-\frac{1}{2}\Delta|s|^2=|\nabla s|^2+\langle Tr\nabla^2 s, s\rangle,
\end{eqnarray*}
where for $|\nabla s|$ we mean the induced norm in the vector bundle $T^*P\otimes E$, i.e.,
$|\nabla s|^2=\Sigma_{i}|\nabla_{E_i}s|^2$ with $\{E_i\}$ an orthonormal frame of $TP$.
$Tr\nabla^2$ denotes the connection Laplacian, which is defined as
$Tr\nabla^2=\Sigma_{i}\nabla^2_{E_i, E_i}s$,
where $\nabla^2_{X, Y}:=\nabla_X\nabla_Y-\nabla_{\nabla_XY}$.
Denote $\Omega^k(E)$ the space of $E$-valued $k$-forms on $P$. The connection $\nabla$
induces an exterior derivative by
\begin{eqnarray*}
d^\nabla&:& \Omega^k(E)\to \Omega^{k+1}(E)\\
d^\nabla(\alpha\otimes \zeta)&=&d\alpha\otimes \zeta+(-1)^k\alpha\wedge \nabla\zeta.
\end{eqnarray*}
It is not hard to check that for any $1$-forms, equivalently one can write
$$
d^\nabla\beta (v_1, v_2)=(\nabla_{v_1}\beta)(v_2)-(\nabla_{v_2}\beta)(v_1),
$$
where $v_1, v_2\in TP$.
We extend the Hodge star operator to $E$-valued forms by
\begin{eqnarray*}
*&:&\Omega^k(E)\to \Omega^{n-k}(E)\\
*\beta&=&*(\alpha\otimes\zeta)=(*\alpha)\otimes\zeta
\end{eqnarray*}
for $\beta=\alpha\otimes\zeta\in \Omega^k(E)$.
Define the Hodge Laplacian of the connection $\nabla$ by
$$
\Delta^{\nabla}:=d^{\nabla}\delta^{\nabla}+\delta^{\nabla}d^{\nabla},
$$
where $\delta^{\nabla}$ is defined by
$$
\delta^{\nabla}:=(-1)^{nk+n+1}*d^{\nabla}*.
$$
The following lemma is important for the derivation of the Weitzenb\"ock formula.
\begin{lem}\label{lem:d-delta}Assume $\{e_i\}$ is an orthonormal frame of $P$, and $\{\alpha^i\}$ is the dual frame.
Then we have
\begin{eqnarray*}
d^{\nabla}&=&\Sigma_{i}\alpha^i\wedge \nabla_{e_i}\\
\delta^{\nabla}&=&-\Sigma_{i}e_i\rfloor \nabla_{e_i}.
\end{eqnarray*}
\end{lem}
\begin{proof}Assume $\beta=\alpha\otimes \zeta\in \Omega^k(E)$, then
\begin{eqnarray*}
d^{\nabla}(\alpha\otimes \zeta)&=&(d\alpha)\otimes \zeta+(-1)^k\alpha\wedge\nabla\zeta\\
&=&\Sigma_{i}\alpha^i\wedge \nabla_{e_i}\alpha\otimes\zeta+(-1)^k\alpha\wedge\nabla\zeta.
\end{eqnarray*}
On the other hand,
\begin{eqnarray*}
\Sigma_{i}\alpha^i\wedge \nabla_{e_i}(\alpha\otimes\zeta)&=&
\Sigma_{i}\alpha^i\wedge\nabla_{e_i}\alpha\otimes\zeta+\alpha^i\wedge\alpha\otimes\nabla_{e_i}\zeta\\
&=&\Sigma_{i}\alpha^i\wedge \nabla_{e_i}\alpha\otimes\zeta+(-1)^k\alpha\wedge\nabla\zeta,
\end{eqnarray*}
so we have proved the first one.
For the second equality, we compute
\begin{eqnarray*}
\delta^{\nabla}(\alpha\otimes\zeta)&=&(-1)^{nk+n+1}*d^{\nabla}*(\alpha\otimes\zeta)\\
&=&(\delta\alpha)\otimes\zeta+(-1)^{nk+n+1}*(-1)^{n-k}(*\alpha)\wedge\nabla\zeta\\
&=&-\Sigma_{i}e_i\rfloor \nabla_{e_i}\alpha\otimes\zeta+\Sigma_{i}(-1)^{nk-k+1}*((*\alpha)\wedge\alpha^i)\otimes\nabla_{e_i}\zeta\\
&=&-\Sigma_{i}e_i\rfloor \nabla_{e_i}\alpha\otimes\zeta-\Sigma_{i}e_i\rfloor \alpha\otimes\nabla_{e_i}\zeta\\
&=&-\Sigma_{i}e_i\rfloor \nabla_{e_i}(\alpha\otimes\zeta).
\end{eqnarray*}
and then we are done with this lemma.
\end{proof}
\begin{thm}[Weitzenb\"ock Formula]\label{thm:weitzenbock}
Assume $\{e_i\}$ is an orthonormal frame of $P$, and $\{\alpha^i\}$ is the dual frame. Then
when apply for any vector bundle $E$-valued forms
\begin{eqnarray*}
\Delta^{\nabla}=-Tr\nabla^2+\Sigma_{i,j}\alpha^j\wedge (e_i\rfloor R(e_i,e_j)\cdot)
\end{eqnarray*}
where $R$ is the curvature tensor of the bundle $E$ with respect to the connection $\nabla$.
\end{thm}
\begin{proof}Since the right hand side of the equality is independent of the choice of orthonormal basis,
and it is a pointwise formula,
we can take the normal coordinates $\{e_i\}$ at a point $p\in P$ (and $\{\alpha^i\}$ the dual basis), i.e., $h_{ij}:=h(e_i, e_j)(p)=\delta_{ij}$ and $dh_{i,j}(p)=0$,
and prove such formula holds at $p$ for such coordinates. For the Levi-Civita connection, the condition $dh_{i,j}(p)=0$
of the normal coordinate is equivalent to let
$\Gamma^k_{i,j}(p):=\alpha^k(D_{e_i}e_j)(p)=0$.
For $\beta\in \Omega^k(E)$, using Lemma \ref{lem:d-delta} we calculate
\begin{eqnarray*}
\delta^{\nabla}d^{\nabla}\beta&=&-\Sigma_{i,j}e_i\rfloor \nabla_{e_i}(\alpha^j\wedge\nabla_{e_j}\beta)\\
&=&-\Sigma_{i,j}e_i\rfloor (D_{e_i}\alpha^j \wedge\nabla_{e_j}\beta+\alpha^j\wedge \nabla_{e_i}\nabla_{e_j}\beta).
\end{eqnarray*}
At the point $p$, the first term vanishes, and we get
\begin{eqnarray*}
\delta^{\nabla}d^{\nabla}\beta(p)&=&-\Sigma_{i,j}e_i\rfloor (\alpha^j\wedge \nabla_{e_i}\nabla_{e_j}\beta)(p)\\
&=&-\Sigma_i\nabla_{e_i}\nabla_{e_i}\beta(p)+\Sigma_{i,j}\alpha^j\wedge (e_i\rfloor\nabla_{e_i}\nabla_{e_j}\beta)(p)\\
&=&-\Sigma_i\nabla^2_{e_i, e_i}\beta(p)+\Sigma_{i,j}\alpha^j\wedge (e_i\rfloor\nabla_{e_i}\nabla_{e_j}\beta)(p).
\end{eqnarray*}
Also,
\begin{eqnarray*}
d^{\nabla}\delta^{\nabla}\beta&=&-\Sigma_{i,j}\alpha^i\wedge \nabla_{e_i}(e_j\rfloor \nabla_{e_j}\beta)\\
&=&-\Sigma_{i,j}\alpha^i\wedge (e_j\rfloor \nabla_{e_i}\nabla_{e_j}\beta)
-\Sigma_{i,j}\alpha^i\wedge ((D_{e_i}e_j) \rfloor \nabla_{e_j}\beta),
\end{eqnarray*}
for which, the point $p$, the second term vanishes.
Now we sum the two parts $d^\nabla\delta^\nabla$ and $\delta^\nabla d^\nabla$ and get
$$
\Delta^{\nabla}\beta(p)=-\Sigma_i\nabla^2_{e_i, e_i}\beta(p)
+\Sigma_{i,j}\alpha^j\wedge (e_i\rfloor R(e_i,e_j)\beta)(p).
$$
\end{proof}
In particular, when acting on zero forms, i.e., sections of $E$, the second term on the right hand side vanishes, and there is
$$
\Delta^{\nabla}=-Tr\nabla^2.
$$
When acting on full rank forms, the above also holds by easy checking.
When $\beta\in \Omega^1(E)$, which is the case we use in this paper, there is the following
\begin{cor}
For $\beta=\alpha\otimes \zeta\in \Omega^1(E)$, the Weizenb\"ock formula can be written as
$$
\Delta^{\nabla}\beta=-\Sigma_i\nabla^2_{e_i, e_i}\beta
+\textup{Ric}^{D*}(\alpha)\otimes\zeta+\textup{Ric}^{\nabla}\beta,
$$
where $\textup{Ric}^{D*}$ denotes the adjoint of $\textup{Ric}^D$, which acts on $1$-forms.
In particular, when $P$ is a surface, we have
\begin{eqnarray}\label{eq:bochner-weitzenbock}
\Delta^{\nabla}\beta&=&-\Sigma_i\nabla^2_{e_i, e_i}\beta
+K\cdot\beta+\textup{Ric}^{\nabla}(\beta)\nonumber\\
-\frac{1}{2}\Delta|\beta|^2&=&|\nabla \beta|^2-\langle \Delta^\nabla \beta, \beta\rangle +K\cdot|\beta|^2+\langle\textup{Ric}^{\nabla}(\beta), \beta\rangle.
\end{eqnarray}
where $K$ is the Gaussian curvature of the surface $P$, and $\textup{Ric}^{\nabla}(\beta):=\alpha\otimes \Sigma_{i, j}R(e_i, e_j)\zeta$.
\end{cor}
\section{Wedge product of vector-valued forms}
\label{appen:forms}
In this section, we continue with the settings from Appendix \ref{appen:weitzenbock}.
To be specific, we assume $(P, h)$ is a Riemannian manifold of dimension $n$ with metric $h$, and denote by $D$
the Levi-Civita connection. $E\to P$ is a vector bundle with inner product $\langle \cdot, \cdot\rangle$
and $\nabla$ is a connection of $E$ which is compatible with $\langle \cdot, \cdot\rangle$.
We would like to remark that we include this section for the sake of completeness of the vector valued forms, while
the content of this appendix is not used in any section of this paper.
Actually one can derive the exponential decay using the differential inequality method from the formulas we provide here.
We leave the proof to readers who have interests.
The wedge product of forms can be extended to $E$-valued forms by defining
\begin{eqnarray*}
\wedge&:&\Omega^{k_1}(E)\times \Omega^{k_2}(E)\to \Omega^{k_1+k_2}(E)\\
\beta_1\wedge\beta_2&=&\langle \zeta_1, \zeta_2\rangle\,\alpha_1\wedge\alpha_2,
\end{eqnarray*}
where $\beta_1=\alpha_1\otimes\zeta_1\in \Omega^{k_1}(E)$ and $\beta_2=\alpha_2\otimes\zeta_2\in \Omega^{k_2}(E)$
are $E$-valued forms.
\begin{lem}\label{lem:inner-star}
For $\beta_1, \beta_2\in \Omega^k(E)$,
$$
\langle \beta_1, \beta_2\rangle=*(\beta_1\wedge *\beta_2).
$$
\end{lem}
\begin{proof}
Write $\beta_1=\alpha_1\otimes\zeta_1$ and $\beta_2=\alpha_2\otimes\zeta_2$,
\begin{eqnarray*}
*(\beta_1\wedge *\beta_2)&=&*\big((\alpha_1\otimes\zeta_1)\wedge ((*\alpha_2)\otimes\zeta_2)\big)\\
&=&*(\langle\zeta_1, \zeta_2\rangle\,\alpha_1\wedge *\alpha_2)\\
&=&\langle\zeta_1, \zeta_2\rangle\,*(\alpha_1\wedge *\alpha_2)\\
&=&\langle\zeta_1, \zeta_2\rangle\, h(\alpha_1, \alpha_2)\\
&=&\langle \beta_1, \beta_2\rangle
\end{eqnarray*}
\end{proof}
The following lemmas essentially use the compatibility of $\nabla$ with the inner product $\langle \cdot, \cdot\rangle$.
\begin{lem}
$$
d(\beta_1\wedge\beta_2)=d^\nabla\beta_1\wedge \beta_2+(-1)^{k_1}\beta_1\wedge d^\nabla\beta_2,
$$
where $\beta_1\in \Omega^{k_1}(E)$ and $\beta_2\in \Omega^{k_2}(E)$
are $E$-valued forms.
\end{lem}
\begin{proof}
We write $\beta_1=\alpha_1\otimes\zeta_1$ and $\beta_2=\alpha_2\otimes\zeta_2$ and calculate
\begin{eqnarray*}
d(\beta_1\wedge\beta_2)&=&d(\langle \zeta_1, \zeta_2\rangle\,\alpha_1\wedge\alpha_2)\\
&=&d\langle \zeta_1, \zeta_2\rangle\wedge\alpha_1\wedge\alpha_2+\langle \zeta_1, \zeta_2\rangle\,d(\alpha_1\wedge\alpha_2)\\
&=&\langle \nabla\zeta_1, \zeta_2\rangle\wedge\alpha_1\wedge\alpha_2+\langle \zeta_1, \nabla\zeta_2\rangle\wedge\alpha_1\wedge\alpha_2\\
&{}&+\langle \zeta_1, \zeta_2\rangle\,d\alpha_1\wedge\alpha_2+(-1)^{k_1}\langle \zeta_1, \zeta_2\rangle\,\alpha_1\wedge d\alpha_2.
\end{eqnarray*}
While
\begin{eqnarray*}
d^\nabla\beta_1\wedge \beta_2&=&d^\nabla(\alpha_1\otimes\zeta_1)\wedge(\alpha_2\otimes\zeta_2)\\
&=&(d\alpha_1\otimes\zeta_1+(-1)^{k_1}\alpha_1\wedge\nabla\zeta_1)\wedge(\alpha_2\otimes\zeta_2)\\
&=&\langle\zeta_1, \zeta_2\rangle\,d\alpha_1\wedge\alpha_2+\langle\nabla\zeta_1, \zeta_2\rangle\wedge\alpha_1\wedge\alpha_2,
\end{eqnarray*}
and similar calculation shows that
$$
(-1)^{k_1}\beta_1\wedge d^\nabla\beta_2=
(-1)^{k_1}\langle \zeta_1, \zeta_2\rangle\,\alpha_1\wedge d\alpha_2+\langle \zeta_1, \nabla\zeta_2\rangle\wedge\alpha_1\wedge\alpha_2,
$$
sum them up and we get equality we want.
\end{proof}
\begin{lem}\label{lem:metric-property}
Assume $\beta_0\in \Omega^k(E)$ and $\beta_1\in \Omega^{k+1}(E)$, then we have
$$
\langle d^\nabla \beta_0, \beta_1\rangle-(-1)^{n(k+1)}\langle \beta_0, \delta^\nabla\beta_1\rangle
=*d(\beta_0\wedge *\beta_1).
$$
\end{lem}
\begin{proof}
We calculate
\begin{eqnarray*}
*d(\beta_0\wedge *\beta_1)&=&*\big(d^\nabla\beta_0\wedge*\beta_1+(-1)^k\beta_0\wedge (d^\nabla*\beta_1)\big)\\
&=&\langle d^\nabla\beta_0, \beta_1\rangle
+(-1)^n*\big(\beta_0\wedge *(*d^\nabla *\beta_1\big)\\
&=&\langle d^\nabla\beta_0, \beta_1\rangle
-(-1)^{n(k+1)}\langle\beta_0, \delta^\nabla\beta_1\rangle.
\end{eqnarray*}
\end{proof}
\section{Local coercive estimates}
\label{appen:local-coercive}
In this appendix, we
give the proof of Proposition \ref{prop:coercive-L2} which we restate here.
\begin{prop}
For any open domains $D_1$ and $D_2$ in $\dot\Sigma$ satisfying $\overline{D_1}\subset D\subset \textup{Int}(D_2)$ for some domain $D$,
$$
\|\nabla(dw)\|^2_{L^2(D_1)}\leq C_1(D_1, D_2)\|dw\|^2_{L^2(D_2)}+C_2(D_1, D_2)\|dw\|^4_{L^4(D_2)}
$$
for any contact instanton $w$,
where $C_1(D_1, D_2)$, $C_2(D_1, D_2)$ are some constants given in \eqref{eq:for-coeff}, which are independent of $w$.
\end{prop}
\begin{proof}
For the pair of given domains $D_1$ and $D_2$, we choose a smooth cut-off function $\chi:D_2\to {\mathbb R}$ such that
$\chi\geq 0$ and
$\chi\equiv 1$ on $\overline{D_1}$, $\chi\equiv 0$ on $D_2-D$.
Multiplying $\chi$ to \eqref{eq:higher-derivative} and integrating over $D_2$, we get
\begin{eqnarray*}
\int_{D}|\nabla(dw)|^2&\leq&\int_{D_2}\chi|\nabla(dw)|^2\\
&\leq&C_1\int_{D_2}\chi|dw|^4-4\int_{D_2}K\chi|dw|^2-2\int_{D_2}\chi\Delta e\\
&\leq&C_1\int_{D_2}|dw|^4+4\|K\|_{L^\infty(\dot\Sigma)}\int_{D_2}|dw|^2-2\int_{D_2}\chi\Delta e.
\end{eqnarray*}
We now deal with the last term $\int_{D_2}\chi \Delta e$.
Since
\begin{eqnarray*}
\chi\Delta e\, dA&=&*(\chi \Delta e)=\chi *\Delta e
=-\chi d*de\\
&=&-d(\chi *de)+d\chi\wedge (*de),
\end{eqnarray*}
we get
$$
\int_{D_2}\chi\Delta e\, dA=\int_{D_2}d\chi\wedge (*de)
$$
by integrating the identity over $D_2$ and applying the Stokes' formula by the vanishing of $\chi$ on $D_2-D$, in particular
on $\partial D_2$.
To deal with the right hand side, we have
\begin{eqnarray*}
&{}&|\int_{D_2}d\chi\wedge(*de)|
=|\int_{D_2}\langle d\chi, de\rangle \,dA|
\leq\int_{D_2}|\langle d\chi, de\rangle \,dA|\\
&\leq&\int_{D_2}|d\chi||de|\,dA
=\int_{D}|d\chi||de|\,dA
\leq\|d\chi\|_{C^0(D)}\int_{D}|de|\,dA.
\end{eqnarray*}
Notice also
\begin{eqnarray*}
|de|&=&|d\langle dw, dw\rangle|=2|\langle \nabla(dw), dw\rangle|\leq 2|\nabla (dw)||dw|\\
&\leq&\frac{1}{\epsilon}|\nabla(dw)|^2+\epsilon|dw|^2.
\end{eqnarray*}
Then we can sum all the estimates above and get
\begin{eqnarray*}
\int_{D}|\nabla(dw)|^2
&\leq& \frac{2\|d\chi\|_{C^0(D)}}{\epsilon}\int_{D}|\nabla(dw)|^2\\
&{}&+\left(4\|K\|_{L^\infty(\dot\Sigma)}+2\|d\chi\|_{C^0(D)}\epsilon\right)\int_{D_2}|dw|^2\\
&{}&+C_1\int_{D_2}|dw|^4.
\end{eqnarray*}
We take $\epsilon>2\|d\chi\|_{C^0(D)}$, say $\epsilon=4\|d\chi\|_{C^0(D)}$,
then
\begin{eqnarray}\label{eq:for-coeff}
&{}&\int_{D_1}|\nabla(dw)|^2\leq\int_{D}|\nabla(dw)|^2\nonumber\\
&\leq&\left(8\|K\|_{L^\infty(\dot\Sigma)}+16\|d\chi\|^2_{C^0(D)}\right)\int_{D_2}|dw|^2
+2C_1\int_{D_2}|dw|^4\nonumber\\
\end{eqnarray}
and we are done with the proof.
\end{proof}
\bigskip
\textbf{Acknowledgements:} For their valuable feedbacks, we thank the audience of our talks on
this topic in the seminars of various institutions.
Rui Wang sincerely thanks Bohui Chen for
numerous mathematical discussions as well as for his continuous encouragement.
|
2,869,038,154,882 | arxiv | \section{\bf Introduction}\label{sect-1}
\subsection{Weyl geometry} Let $N$ be a smooth manifold of dimension $n\ge3$. Let $\nabla$ be a torsion free
connection on the tangent bundle $TN$ of $N$ and let $g$ be a semi-Riemannian metric on $N$. Then the triple
$\mathcal{W}:=(N,g,\nabla)$ is said to be a {\it Weyl manifold} if there exists a smooth $1$-form $\phi\in C^\infty(T^*N)$ so
that:
\begin{equation}\label{eqn-1.a}
\nabla g=-2\phi\otimes g\,.
\end{equation}
Weyl geometry \cite {W22} is linked with conformal geometry. If $f\in C^\infty(N)$,
let
$g_1:=e^{2f}g$ be a conformally equivalent metric. If
$\mathcal{W}=(N,g,\nabla)$ is a Weyl manifold, then
the triple $(N, g_1,\nabla)$ is again a Weyl manifold where the associated
$1$-form is given by taking $\phi_1:=\phi - df$. The transformation of the pair
$(g,\phi)\rightarrow(g_1,\phi_1)$ is called a {\it gauge transformation}. Properties of the Weyl geometry that
are invariant under gauge transformations are called {\it gauge invariants}.
Let $\nabla^{g}$ be the Levi-Civita connection of $g$. There exists a
conformally equivalent metric
$g_1$ locally so that $\nabla=\nabla^{g_1}$ if and only if $d\phi=0$; if $d\phi=0$, such a class exists
globally if and only if the associated de Rham cohomology class $[\phi]$ vanishes.
\subsection{Affine and Riemannian geometry} We say that the pair
$\mathcal{A}:=(N,\nabla)$ is an {\it affine manifold} if
$\nabla$ is a torsion free connection on $TN$. Similarly, we say that the pair $\mathcal{N}:=(N,g)$ is a {\it
semi-Riemannian manifold} if $g$ is a semi-Riemannian metric on $N$. Weyl geometry lies
between affine geometry and semi-Riemannian geometry. Every Weyl manifold gives rise both to an underlying
affine manifold $(N,\nabla)$ and to an underlying semi-Riemannian manifold $(N,g)$; Equation
(\ref{eqn-1.a}) provides the link between these two structures.
Since the Levi-Civita connection $\nabla^{g}$ is torsion free and since $\nabla^g g=0$, the triple $(N,g,\nabla^{g})$ is a Weyl
manifold. There are, however, examples with $d\phi\ne0$, so Weyl geometry is more general than
semi-Riemannian geometry or even than conformal semi-Riemannian geometry.
\subsection{Curvature} The {\it curvature operator} $\mathcal{R}$ of a torsion free
connection $\nabla$ is the element of
$\otimes^2T^*N\otimes\operatorname{End}(TN)$ which is defined by:
$$\mathcal{R}(x,y)z := (\nabla_x\nabla_y-\nabla_y\nabla_x-\nabla_{[x,y]})z\,.$$
The $g$-associated {\it curvature tensor} is given by using the metric to lower an index:
$$R(x,y,z,w):=g(\mathcal{R}(x,y)z,w)\,.$$
We have the following identities:
\begin{eqnarray}
&&R(x,y,z,w)+R(y,x,z,w)=0,\quad\text{and}\label{eqn-1.b}\\
&&R(x,y,z,w)+R(y,z,x,w)+R(z,x,y,w)=0\label{eqn-1.c}\,.
\end{eqnarray}
The relation of Equation (\ref{eqn-1.c}) is called the {\it Bianchi identity}.
The {\it Ricci tensor}
$\operatorname{Ric} := \operatorname{Ric}(R) := \operatorname{Ric}(\mathcal{R})$ is defined by setting:
$$
\operatorname{Ric}(x,y):=\operatorname{Tr}\{z\rightarrow (\mathcal{R}(z,x)y\} \,.
$$
The tensor $\operatorname{Ric}(\mathcal{R})$ does not depend on the metric $g$ and is a gauge invariant. Let $g_{ij}:=g(e_i,e_j)$
give the components of the metric tensor $g$ relative to a local frame $\{e_i\}$ for $TN$. Let $g^{ij}$ be the components of the
inverse matrix
$g^{-1}$ relative to the dual frame $\{e^i\}$ on $T^*N$. We adopt the {\it Einstein convention} and sum over repeated indices.
We then express:
$$\operatorname{Ric}(x,y) =g^{ij}R(e_i,x,y,e_j)\,.$$
We contract again to define the {\it scalar curvature} of $R$ with respect to $g$
by setting:
$$\tau_g:=\operatorname{Tr}_g\operatorname{Ric}:=g^{ij}\operatorname{Ric}(e_i,e_j)\,.$$
We can define another Ricci-type tensor $\operatorname{Ric}^\star:= \operatorname{Ric}^\star(R)$ by
setting:
$$
\operatorname{Ric}^\star(x,y):=g^{ij}R(x,e_i,e_j,y).
$$
We let $R_{ij}$ and $R^\star_{ij}$ be the components of these two tensors:
$$R_{ij} := \operatorname{Ric}(e_i,e_j)\quad\text{and}\quad R^\star_{ij} := \operatorname{Ric}^\star(e_i,e_j)\,.$$
We may decompose any (0,2)-tensor $\theta$ in the form $\theta=S\theta+\Lambda\theta$ where $S\theta$ and $\Lambda\theta$ are
the symmetrization and the anti-symmetrization, respectively, of $\theta$. We have the following result (see, for example, the
discussion in
\cite{GNU10,W22}):
\begin{theorem}\label{thm-1.1}
Let $(N,g,\nabla)$ be a Weyl manifold. Let $R=R(\nabla)$. Then we have
\begin{eqnarray}
&&R(x,y,z,w)+R(x,y,w,z)=\textstyle\frac2n\{\operatorname{Ric}(y,x) - \operatorname{Ric}(x,y)\}g(z,w) \nonumber\\
&&\qquad= - \tfrac4n\, (\Lambda\operatorname{Ric})(x,y)\,g(z,w),\label{eqn-1.e}\\
&&S\operatorname{Ric}^\star = S\operatorname{Ric},\quad \Lambda
\operatorname{Ric}^\star=\textstyle\frac{n-4}n\Lambda \operatorname{Ric},\quad\text{and}\quad
\tau_g =\operatorname{Tr}_g\operatorname{Ric}^\star\,.
\end{eqnarray}
\end{theorem}
\begin{remark}\rm
If $\nabla=\nabla^{g}$ arises from semi-Riemannian geometry, then one has an additional symmetry:
\begin{equation}\label{eqn-1.f}
R(x,y,z,w) + R(x,y,w,z)=0 \,.
\end{equation}
\end{remark} |
2,869,038,154,883 | arxiv | \section{Introduction}
\noindent Structured light beams have wide applications in technologies such as lithography, nanoscopy, spectroscopy, optical tweezers and quantum cryptography~\cite{grosjean:07,hell:07,he:95,paterson:01,molina:07}. Among these, beams with helical phase fronts $\exp{(i\ell\phi)}$, where $\ell$ is an integer number and $\phi$ is the azimuthal angle in polar coordinates, are of particular interest since they can be used for classical~\cite{gibson:04,wang:12,siddharth:13} and quantum communications~\cite{molina:07}. These beams carry a well-defined value of optical orbital angular momentum (OAM) $\ell\hbar$ per photon along the propagation direction. Due to these proposed applications, there are fervent attempts to design innovative devices to generate such beams. Until now, possible solutions include spiral phase plates~\cite{beijersbergen:94}, computer-generated holograms imprinted onto spatial light modulators (holographic approach)~\cite{bazhenov:92,ngcobo:13}, mode converters (cylindrical lenses)~\cite{allen:92}, $q$-plates (nonuniform liquid crystal plates)~\cite{marrucci:06,karimi:09}, and some types of OAM-sorters~\cite{berkhout:10,mohammad:13}. These solutions are practical and widely used in various experimental realizations, and are implemented both in classical and quantum regimes. However, with the exception of mode converter and a hologram with an intensity mask~\cite{leach:05,eliot:13}, the above methods do not generate a pure Laguerre-Gauss mode~\cite{karimi:07,dennis:09}. In some cases, the reverse process can be used to \emph{detect} the spectrum of OAM of an unknown beam, where each mode is coupled to a single mode optical fiber (SMOF) after its azimuthal phase dependence has been flattened. Such a method, was first introduced by Mair et al.~\cite{mair:01} in the quantum domain and then used commonly in the classical regime. This technique might sound accurate, but as we will show, its shortcoming is that the OAM bandwidth that can be measured has a bias that depends on the characteristics of the beam. As a consequence, the bias for an unknown beam cannot be removed. Moreover, the detection efficiency for high OAM modes can be extremely low, making it seem like those components are very weak. This issue is particularly important for those experiments that rely on a high detection efficiency, for example, experiments that aim at maximizing the heralding efficiency, or at closing a detection loophole~\cite{brunner:10,dada:11}, or at characterizing a state by measuring each of its OAM components separately~\cite{torres:03,pires:10}. In this letter, we study projective measurements based on phase-flattening followed by coupling into a SMOF. We examine our theoretical model experimentally for various mode projections, and we verify the trends in coupling efficiencies.\newline
\section{Theoretical analysis}
In our analysis we use Laguerre-Gauss (LG) modes, which are characterized by two indices: the radial index $p$ (nonnegative integer) and the azimuthal number $\ell$ (integer), which are associated to the number of radial nodes and to the OAM value, respectively. The LG modes are a complete and orthonormal family of solutions of the paraxial wave equation, i.e. (in Dirac notation) $\braket{p',\ell'}{p,\ell}=\delta_{p',p}\,\delta_{\ell',\ell}$, and in the position representation at the pupil they are given by
\begin{align}
\label{eq:lgs}
\mbox{LG}_{p,\ell}(r,\phi)&:=\sqrt{\frac{ 2^{|\ell|+1}p!}{\pi w_0^2\,(p+|\ell|)!}}\,\left(\frac{r}{w_0}\right)^{|\ell|} e^{-\frac{r^2}{w_0^2}} L_{p}^{|\ell|}\left(\frac{2r^2}{w_0^2}\right)\,e^{-i\ell\phi},
\end{align}
where $r, \phi$ are the transverse cylindrical coordinates, $w_0$ is the beam waist radius at the pupil and $L_{p}^{\ell}(.)$ is the generalized Laguerre polynomial. The devices listed above can generate LG modes with limited fidelity. The most convenient and commonly used method is the holographic approach, with an embedded intensity masking. However, a \emph{mode-cleaning filter cavity} can be used to increase fidelity of the generated mode~\cite{granata:10,carbone:13}.
\subsection{Projecting on LG modes}
To perform a projective measurement, the mode $\mbox{LG}_{p,\ell}$ (in our case generated by an SLM) is imaged onto a \emph{different} conjugate mode, $\mbox{LG}_{p',\ell'}^{*}$, and the resulting field is propagated and coupled into a SMOF in the far-field, which selects only the near Gaussian component. Imaging onto an SLM is described by taking the product of the two modes, i.e. $\mbox{LG}_{p,\ell}(\mathbf r_\bot)\,\mbox{LG}_{p',\ell'}^{*} (\mathbf r_\bot)$ where $\mathbf r_\bot$ stands for the transverse coordinates. The far-field distribution becomes a polynomial-Gaussian function given by a 2D-Fourier transform:
\begin{align}
\label{eq:2dFF}
{\cal F}_{p,\ell}\left(\rho,\varphi\right)={\cal{FT}}\left[\mbox{LG}_{p,\ell}(\mathbf r_\bot) {\mbox{LG}_{p',\ell'}^*(\mathbf r_\bot)}\right],
\end{align}
where ${\cal{FT}}$ stands for the 2D-Fourier transform, and $\rho$ and $\varphi$ are the cylindrical coordinates in the far field.
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{fig1.pdf}
\caption{(Color online) Intensity of the $p=0$ (left) and $p=1$ (right) modes at the input to the fiber. As an effect of diffraction, local maxima at the periphery gain intensity as $p$ and $|\ell|$ increase. Here $\rho$ is in units of $a_0=(\sqrt{2}\lambda f)/(\pi w_0)$, which is the natural scaling factor in the far-field of the lens.}
\label{fig:fig1}
\end{figure}
The fact that a SMOF only supports the $\mathrm{TEM}_{00}$ mode limits this technique to the case in which $\ell'=\ell$. Moreover, as oscillating radial phases would alter the coupling to the SMOF, we also choose $p'=p$. Due to the absence of any angular dependence after the phase flattening stage, the 2D-Fourier transform ${\cal FT}$ can be simplified into the Hankel transform of order zero, i.e.
%
\begin{align}
\mathcal F_{p,\ell}(\rho,\varphi)=\frac{2\pi e^{\frac{i\pi}{\lambda f}\rho^2}}{i\lambda f}\int_0^\infty rdr|\mathrm{LG}_{p,\ell}(\mathbf{r}_\bot)|^2J_0\left(\frac{2\pi}{\lambda f}r\rho\right).
\end{align}
In Fig.~\ref{fig:fig1} we show some examples of transverse intensity at the fiber for several values of $p$ and $\ell$. Notice that the beams have a Gaussian-like shape with local maxima at the periphery, which give rise to a ringed pattern in the transverse plane. As $|\ell|$ and $p$ become larger, the beam intensity distribution moves to the outer rings. This is related to the effect that a larger phase-flattened doughnut beam is turned into a smaller and weaker central spot at the far field, which has been studied and discussed for special cases in \cite{slussarenko:09,gabriel:07}.
The coupling efficiency to a SMOF, then, is given by the overlap of the Gaussian mode supported by the fiber and the far-field distribution calculated in \eqref{eq:2dFF}:
\begin{align}\label{eq:projection}
\eta^{\ell}=\frac{2}{\pi \sigma^2}\left|\int_{0}^{\infty}\rho\,d\rho\int_{0}^{2\pi}d\varphi\,\,{\cal F}_{\ell}\left(\rho,\varphi\right)\,e^{-\frac{\rho^2}{\sigma^2}}\right|^2,
\end{align}
where $\sigma$ is the beam waist radius of the SMOF Gaussian mode. It is worth mentioning that the mode of a SMOF can be approximated with a Gaussian beam. Here we give the results for $p=0$ and $p=1$:
\begin{align}
\eta_0^\ell&=\frac{|\ell|!^2}{(2|\ell|)!}A^{2|\ell|+1}B\\
\eta_1^\ell&=\frac{(|\ell|+1)!|\ell|!}{4(2+3|\ell|)(2|\ell|)!}A^{2|\ell|+1}B(A^2+B^2(|\ell|+1))^2,
\end{align}
where $A=2/(1+\frac{\sigma^2}{a_0^2})$ and $B=2/(1+\frac{a_0^2}{\sigma^2})$, and $a_0=(\sqrt{2}\lambda f)/(\pi w_0)$ is the natural scaling factor at the fiber. Notice that it is only the ratio $\sigma/a_0$ that matters, as it should be. These results are shown in Fig.~\ref{fig:fig2}, where it is possible to see that the highest coupling efficiency for different modes is achieved for different values of the waist at the fiber, which can be tuned by adjusting the focal length of the Fourier lens.\newline
\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{fig2.pdf}
\caption{(Color online) Coupling efficiency for projective measurements for the modes in Fig.~\ref{fig:fig1}. For a given choice of optics, the coupling efficiency shows a bias dependent on the order of the transverse modes. The horizontal axis is scanned by changing $w_0$, as $a_0$ is inversely proportional to $w_0$. The shaded box indicates the region limited by the active area of the SLMs (see experimental section).}
\label{fig:fig2}
\end{figure}
\subsection{Projecting on spiral modes}
An alternate and less desirable solution that we explore only theoretically is to project onto a purely spiral field $e^{i\ell\varphi}$ (which can be implemented with a pitchfork hologram on an SLM), whereby the effect is to simply cancel out the spiral phase from an initial LG mode, in which case the field at the fiber is given by
$
{\cal F}_{p,\ell}\left(\rho,\varphi\right)={\cal{FT}}\left[\mbox{LG}_{p,\ell}(\mathbf r_\bot) e^{i\ell\phi}\right].
$
This equation can be solved analytically, recall that the Eq.~(\ref{eq:2dFF}) has an analytical solution only once a value of $p$ is specified. However, for this specific case the coupling efficiency is given by
\begin{align}\label{eq:projectionflattening}
\eta_{p}^{\ell}=\left|{\mathcal{N}_p^{\ell}\sum_{j=0}^{p}(-1)^{j}{p\choose j}a_j^{\ell} f_j^{p,\ell}}\right|^2
\end{align}
with
\begin{align}\label{eq:projectiondefinition}\nonumber
f_j^{p,\ell}=&\prod_{k=1}^p\bigl(|\ell|+k+k\,H(j-k)\bigr)\cr
a_j^{\ell}=&\frac{\sqrt{2\pi}\,\sigma/a_0}{\left(1+(\sigma/a_0)^2\right)^{\frac{|\ell|}{2}+j+1}},
\end{align}
where $\mathcal{N}_p^{\ell}=\sqrt{\frac{2^{|\ell|+1}}{\pi (p+|\ell|)!\,p!}}\,\Gamma\left(\frac{|\ell|}{2}+1\right)$ is the normalization function. $H(j-k)$ in $f_j^{p,\ell}$ is the unit step function: its value is 0 for $j<k$ and $1$ for $j\geq k$, and $\Gamma$ is the gamma function, respectively. As was expected, the coupling efficiency $\eta_{p}^{\ell}$ depends on the ratio between the beam waist radius of the SMOF $\sigma$ and the size of the field at the fiber position $a_0$.\newline
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig3.png}
\caption{(Color online) Experimental setup for generating and detecting photon transverse states. A linearly polarized HeNe laser beam is spatially cleaned with two lenses and a pinhole. A half-wave plate (HWP) optimizes the first order of diffraction on SLM$_\text{A}$, since SLMs are polarization dependent. The mode $\mbox{LG}_{p,\ell}(\mathbf{r}_\bot)$ produced by SLM$_\text{A}$ is then projected on the mode $\mbox{LG}_{p,\ell}(\mathbf{r}_\bot)^{*}$ on SLM$_\text{B}$. The resulting far field is coupled into a single mode optical fiber (SMOF). We implement two $4f$-system with unit magnification and a microscope objective to image SLM$_\text{A}$ on SLM$_\text{B}$ and SLM$_\text{B}$ on the microscope objective. Irises are used to select the first order of diffraction at the far-field plane of SLMs, where higher order of diffraction are well separated.}
\label{fig:fig3}
\end{figure}
\section{Experimental results}
In order to verify the above theory, we prepared an experimental setup (Fig.~\ref{fig:fig3}) in which we examined the projective measurement method for different sets of transverse modes with varying beam sizes. A linearly polarized light beam of a HeNe laser is spatially cleaned, and illuminates the first of two Pluto HOLOEYE SLMs (SLM$_A$), to generate the initial $\mbox{LG}_{p,\ell}(\mathbf{r}_\bot)$ mode. This is then imaged on a second SLM (SLM$_B$) via a $4f$-system with unit magnification, where the mode is projected onto $\mbox{LG}_{p,\ell}(\mathbf{r}_\bot)^*$. We used intensity masking to encode transverse modes with high fidelity~\cite{eliot:13,vincenzo:13}. The product field is finally coupled to a SMOF with mode diameter of $\simeq4.8\,\mu$m and a numerical aperture $\mathrm{NA}=0.12$ at the far-field of a $20\times$ microscope objective ($f=9$ mm and $\mathrm{NA}=0.40$). In order to normalize the coupling efficiency for different modes, we used a Newport power meter with two read out heads to record both the coupling efficiency and the power of the field just before the fiber simultaneously. Recall that due to the intensity masking different modes have different generation and detection efficiencies, for more details see Ref.~\cite{eliot:13,flamm:13}. An automatic program optimized the center of the holograms on the both SLMs and the coupling efficiency with the SMOF. The pixel size and active area of the SLMs were $8\,\mu$m and $15.36\,\mbox{mm}\times8.64\,\mbox{mm}$. These characteristics set the limits of the range of beam waists and mode numbers that could be investigated.
Figure~\ref{fig:fig4} shows the experimental results, to be compared with the coupling efficiency shown in Fig.~\ref{fig:fig2}. Aside from an overall multiplicative efficiency of about $50\%$ (which comprises reflection and scattering by microscope objective and fiber), the observed data (Fig.~\ref{fig:fig4}) and the theoretical model (Fig.~\ref{fig:fig2}) agree, especially in those regions where the SLMs resolution and active area do not affect the quality of the generated and projected beams. The region below 0.2 $\sigma/a_0$ is limited by resolution, as too few pixels are used. On the opposite end of the horizontal axis the beams eventually fall out of the the active area. These regions are indicated by a shaded area in the figures. Due to truncation mainly induced by the microscope objective, there a small deviation for the case of $p=1$ at large beam waist size with respect to the theoretical calculation. However, the spread of these curves is an indication that the coupling efficiency differs for different initial modes and that therefore the spectrum that is ultimately measured is likely not representative of the true OAM distribution of the beam. We can deduce that projective measurement methods should be used with care. A possible solution could be to calibrate the coupling efficiency for different transverse modes and then post-process the measurement data, but even in this case, if the radial distribution of the initial field is unknown, the bias may not be removable, as the radial decomposition depends on the waist that is chosen for the modes. Of course it is also true that a linear superposition of LG beams leads to an inaccurate result, since the projective measurement gives a bias among projection of different pure OAM states. \newline
\begin{figure}[!t]
\includegraphics[width=\columnwidth]{fig4.pdf}
\caption{(Color online) Experimentally measured overall coupling efficiency for the modes shown in Fig.~\ref{fig:fig1}: (left) $\ell=0\dots5$ and (right) $\ell=0\dots2$. The shaded regions indicate a domain in which the effective beam size exceeds the active area of the SLM, resulting in unreliable data.}
\label{fig:fig4}
\end{figure}
\section{Conclusions}
In conclusion, we studied the efficiency of projective measurement as a method to characterize the transverse mode of a light beam. Our analysis can be summarized in two important messages. The first is that although the coupling efficiency is modal- and beam waist-dependent, the bias that is induced might be removed in post-processing after a careful calibration. Of course, issues may arise in the context of an experiment aimed at violating Bell's inequalities: post-processed results could be regarded as an artificial manipulation of the data, and detection-related loopholes might be called into consideration. The second message builds on the fact that the radial modal content of the initial beam depends on the waist $w_0$ that is chosen for the decomposition, and the optimal choice (i.e. the one that results in the least number of radial modes) for a general beam could be found only after further measurements. However, as one then should have to calibrate for this optimal size, this is clearly not an ideal procedure, especially in the context of quantum optics, where there might be a scarce number of photons available). It is our hope that this work will motivate the search of new and better measurement techniques for OAM and more generally for the transverse radial modes of a light beam.\newline
\section{Acknowledgments}
The authors thank Prof. Gerd Leuchs for fruitful discussions. H.Q., F.M.M., E.K. and R.W.B. acknowledge the support of the Canada Excellence Research Chairs (CERC) Program. M.J.P. and R.W.B. were supported by the DARPA InPho program. M.J.P. supported by the royal society.
|
2,869,038,154,884 | arxiv | \section{Introduction}
From the optical spectra of AGNs, one can generally distinguish two
main types: those that show broad emission lines (BLAGNs) and those
that show only narrow emission lines (NLAGNs). With an absolute
magnitude M$_V\geq -23$, the local BLAGNs are called Seyfert~1 (Sy1),
while the NLAGNs are called Seyfert~2 (Sy2). In the literature, one
can also find other types of Seyfert galaxies: Sy1.2, Sy1.5, Sy1.8,
and Sy1.9, all of them being some sort of Sy1 and consequently
BLAGNs. NLAGNs may also come into the form of Low Ionization Nuclear
Emission-line Regions (LINERs).
Phenomenologically, it is unclear why AGNs come in different types.
Based on spectral variation, the Narrow Line Region (NLR) is thought
to be located farther out from the central Black Hole accretion disk
than the Broad Line Region (BLR), and to be spatially much more
extended. Within the unification model \citep{ant93}, which assumes
all AGNs to be intrinsically the same, the BLR in NLAGNs is hidden
behind an optically thick torus of gas and dust. Consistent with
this model, many observations of NLAGNs have revealed hidden BLRs
through polarized spectroscopy \citep{ant02}. However, not all
NLAGNs observed with this technique have shown such structures
\citep{tra01,tra03,lao03,shu07}, suggesting that in some NLAGNs the
BLR might simply be absent. This last finding is consistent with
alternative models in which the accretion rate and, consequently,
AGN luminosity plays a direct role in determining the presence of
the BLR \citep{nic00,nic03,eli06}.
One possible way how to solve this dilemma is to explore the
connection of AGNs with their environment. According to the
unification model, for example, one does not expect to find any
differences in the number of AGNs in different environments.
Unfortunately, such studies are usually controversial. While some
authors found Sy2 to inhabit richer environments than Sy1
\citep{rob98}, others claimed the opposite \citep[and references
therein]{sch01}. Recently, \citet{kou06} have found no differences
in the environment of Sy1 and Sy2 over large scales ($\lesssim$ 1
Mpc), but a higher fraction of Sy2 with close companion than Sy1 in
systems with spatial scales smaller than 100 kpc (using H$_0$=70 km
s$^{-1}$ Mpc$^{-1}$). These results agree with \citet{sor06} who
found two times more Sy2 than Sy1 in local ($\lesssim$ 100 kpc)
environment.
To explore further the possible conection between AGN activity and
environment we have undertaken a new survey to determine the nature
and frequency of nuclear activity in two different samples of
Compact Groups of galaxies (CGs): the Hickson Compact Groups
\citep[HCGs]{hic82} and a sample of CGs from the Updated Zwicky
Catalog \citep[UZC-CG]{foc02}. Previous studies on CGs revealed a
high percentage of low luminosity AGNs in these structures, but very
few luminous ones \citep{coz98,coz00,mar06,mar07}. Having in hand a
statistically significant sample of CGs with complete information on
the nuclear activity of their members, allows us to better quantify
the frequency of BLAGNs in these systems.
\section{Data and Results}
\label{sec2}
Among the HCGs, we have selected the groups with redshifts
$z\leq0.045$, having a surface brightness $\mu_G\leq24.4$. These
criteria provided us with a statistically complete sample of 283
galaxies in 65 groups. We have obtained medium resolution
spectroscopy for 238 of these galaxies. The spectra of 71 galaxies
come from previous observations made by \citet{coz98,coz00,coz04}.
The remaining 167 galaxies were observed by our group using four
different telescopes: the 2.5m NOT\footnote{ALFOSC is owned by the
IAA and operated at the Nordic Optical Telescope (NOT), under
agreement between IAA and the NBIfAFG of the Astronomical
Observatory of Copenhagen.} in ``El Roque de los Muchachos'' (RM,
Spain), the 2.2m in Calar Alto (CAHA\footnote{The Centro
Astron{\'o}mico Hispano Alem\'an is operated jointly by the
Max-Planck Institut fur Astronomie and the IAA-CSIC.}, Spain), the
2.12m in San Pedro M{\'a}rtir (SPM, Mexico) and the 1.5m telescope
in Sierra Nevada Observatory (OSN, Spain).
For all the galaxies, broad components search and activity
classification were done only after template subtraction, to correct
for absorption features produced by the underlying stellar
population. Detailed of the template subtraction method used can be
found in \citet{coz98,coz04}. Preliminary results have already been
published in \citet{mar07}. Complete description of observations,
reduction and analysis method will be published elsewhere.
For the UZC-CG sample, we have collected spectra from three
spectroscopic archives: the Sloan Digital Sky Survey (SDSS-DR4), the
Z-Machine and the FAST Spectrograph Archives. We have found spectra
for all the galaxy members of 215 groups (720 galaxies): 210 spectra
are from the SDSS, 187 from FAST and 323 from Z-Machine
\citep{mar06}. Because
the Z-Machine spectra have too low S/N ratios to measure broad
components, we restrict our analysis to the 397 spectra found in the
SDSS and FAST databases. Spectra from the SDSS survey
were template subtracted using \citet[H05]{hao05} eigenspectra and
their PCA method. No correction was applied to the galaxies in the
FAST sample, due to the non availability of suitable spectra to be used
as templates.
Emission lines were found in 153 of the 238 galaxies in the HCG
sample (64\%) and 274 of the 397 galaxies (69\%) in the UZC-CG
sample. The identification of broad components was done by fitting a
multi-components Gaussian on the emission lines, using the IRAF
task NGAUSSFIT. The FWHM of [SII] (or [OIII] when the [SII] lines
were too faint or noisy) have been used to model the narrow
components of the H$_{\alpha}$ and [NII] lines. When a broad fourth
component was necessary, it was centered on H$_{\alpha}$. A $\chi^2$
criterion, as described in H05, was applied to choose the fourth
component parameters, establishing in this way its presence and
characteristics. Examples of fitting plots for three BLAGNs are
shown in Fig~\ref{fig1}.
Following \citet{ost89} we classified BLAGNs galaxies having FWHM
$\ge 500$ km s$^{-1}$. Our analysis revealed only 1 BLAGN in the HCG
sample and 8 in the UZC-CG sample. For each of these galaxies we give,
in Table ~\ref{tbl1} the FWHM of the broad
component and activity classification according to
\citet{ost89}: a Sy1.9 shows a broad component only in H$_{\alpha}$,
while a Sy1.8 shows also a weak broad component in H$_{\beta}$. None
of our BLAGNs are classified as Sy1, Sy1.2 or Sy1.5. Based on our
analysis, BLAGNs represent only 1\% (1/153) of emission line
galaxies in the HCG sample and 3\% (8/274) in the UZC-CG one.
To classify NLAGNs we used the diagnostic diagram
based on the four most intense emission lines: H$_{\beta}$,
[OIII]5007\AA\ , H$_{\alpha}$, and [NII]6583\AA\ and criteria
similar to those employed by \citet{kew06}. Galaxies located above
the theoretical maximum sequence for star formation are classified
as AGNs. We also distinguished between Sy2 and LINERs using the
classical upper limit log([OIII]5007\AA/H$_{\beta}$) $<$ 0.4 for
LINER. Both CG samples are rich in galaxies having only
[NII]6583\AA\ and H$_{\alpha}$, we classified these cases as Low
Luminosity AGNs (LLAGNs) when log([NII]/H$_{\alpha}$)$>-0.1$
\citep{coz98,sta06}.
A summary of the nuclear classification for the AGN galaxies in our
two samples are presented in Table~\ref{tbl2}. For each sample
(column 1) we give the number of Sy2, LINERs and LLAGNs (columns 2
to 4), which, put together, constitute the total NLAGN populations.
In column 5 we report the fractions of BLAGNs over NLAGNs and in
column 6 the fraction of Sy1 over Sy2, considering all BLAGNs as Sy1
like.
\subsection{Lower ratio of BLAGNs to NLAGNs in CGs than in the field}
The fraction of BLAGNs over NLAGNs in our two CG samples is
extremely low: 1\% for the HCGs and 6\% for the UZC-CGs. Also
noticeably low are the ratios of Sy1 to Sy2: 4\% in the HCG and 19\%
in the UZC-CG. To realize how low these ratios are one has to
compare with what is usually found in other surveys.
The mean H$_{\alpha}$ luminosity of both NLAGNs and BLAGNs in our two
samples is about $10^{39}$ erg s$^{-1}$, which is typical of the faint
end of the luminosity function of AGNs. This value is comparable with
the mean H$_{\alpha}$ luminosity of AGNs observed in the local
universe by \citet[HFS97]{ho97a}. Except for some galaxies in Virgo,
all the galaxies in the HFS97 sample are located in low density
environments (either loose groups or isolated). In Table~\ref{tbl2} we
compare their results (395 galaxies, excluding the galaxies in Virgo)
with ours. In the HFS97 sample, the BLAGNs were classified as such by
\citet{ho97b}, based on the detection of a broad H$_{\alpha}$
component. To be consistent with our definition, all these galaxies
were classified as Sy1. Also for comparison sake, the narrow emission
lines galaxies in the HFS97 sample were reclassified using the
criteria described in sect.~\ref{sec2}.
The fraction BLAGN/NLAGN in the HFS97 sample is 22\% and the ratio
Sy1/Sy2 is 61\%. There is consequently a clear deficiency of BLAGNs
in CGs. This also appears as an extremely large difference in the
number of Sy1 as compared to Sy2 galaxies. This phenomenon is quite
intriguing considering that there is no deficit of AGNs as a whole
in CGs: 46\% AGNs in the HCG, 51\% in the UZC-CG compared to 44\% in
the HFS97 sample.
Comparable ratios (Sy1/Sy2 $\backsim$ 60\%) were obtained by
\citet[SRR06]{sor06} in the field, with a slight increase in ``loose
groups'' (Sy1/Sy2 $\backsim$ 69\%). In the nearby (z$<$0.33) sample
of SDSS AGN galaxies covering four orders of luminosity and similar
environments as SRR06, H05 determined a ratio BLAGN/NLAGN of 43\%
and a ratio Sy1/Sy2 of 54\%. Assuming BLAGNs are slightly favored at
higher luminosity these high ratios are comparable to those found by
HFS97.
\section{Discussion}
\subsection{Quantifying biases and detection limits}
Our results suggest there is an important deficit of BLAGNs in our
two CG samples as compared to similar surveys in the field. This
result confirm the tendency first encountered by
\citet{coz98,coz00}. To verify that the lack of BLAGNs in CGs can not
be induced by differences in observation, reduction or
analysis methods we have investigated thoroughly these
possibilities.
Comparison of the UZC-CG sample with the survey made by H05 is safe,
because our SDSS data derive from the same telescope, reduction and
analysis methods (including template subtraction) as theirs. The ratio
BLAGN/NLAGN is 8\% in our sample compared to 43\% for the sample of
H05, which is already a huge difference.
A possible effect due to difference in spectral resolution can also be
excluded. \citet{ho97b} have used high resolution (2.5 \AA) spectra,
but made tests with two others low resolution set-ups (5\AA\ and
10\AA) obtaining similar results. These are comparable to our own
observations: CAHA and OSN (4\AA), NOT and SPM (8\AA), SDSS (3.5\AA)
and FAST (6\AA). Both H05 and SRR06 have 3.5\AA.
The S/N continuum levels of the different surveys are also
comparable. On average the S/N of AGNs in our spectra is 60 with a
maximum of the order of 120. This is comparable to H05 and SRR06
spectra (they both used SDSS). HFS97 did not published their values.
However their BLAGNs rates are comparable to those of H05 and SRR06,
suggesting this is not an issue.
There is no evidence either for a higher galaxy contamination (the
amount of galaxy falling into the aperture) in our samples. Taking
into account the slit aperture and distances of the host galaxies in
each sample we find medians of 1 kpc and 1.3 kpc for the HCG and
UZC-CG, respectively. Although the median for HFS97 is lower (0.5
kpc) than for H05 (7 kpc) the results are similar. Obviously,
template subtraction (like we also did) alleviates the
differences. We may note also that no relation is observed, in any
of these surveys (including ours), between the frequency of BLAGNs
and the redshift of the galaxies where they are found, which means
that nearby galaxies are not more likely BLAGNs than remote ones.
To test if our low number of Sy1 could be due to a difference in
morphologies \citep{sch01}, we have divided our two samples and the
HFS97 one in three morphology classes: E for early-type galaxies
(E-S0), Se for early-type spirals (S0a-Sbc), and Sl for late-type
spirals (Sc and later). For homogeneity sake, all the morphologies
have been taken from the Hyperleda database \citep{pat03}. In
Table~\ref{tbl3} we give for each morphology class the fraction of
galaxy and the ratios BLAGN/NLAGN and Sy1/Sy2. There are no BLAGNs
in late-type spirals in any sample. In the HFS97 sample, the ratio
of BLAGN/NLAGN is marginally higher in the E class while the ratio
Sy1/Sy2 is significantly higher, which indicates a definitive
increase in BLAGNs in early-type galaxies. In the two CG samples we
almost see an inverse trend: the ratios of BLAGN/NLAGN and Sy1/Sy2
are both larger in the Se class than in the
E one. Moreover there is a definite rise
in the number of early-type galaxies in CGs. Following the HFS97
trend, this should have produced more BLAGNs in CGs instead of less.
This eliminates a difference in morphologies as a possible
explanation.
We also reject the hypothesis of lower sensitivity. Comparing the
median luminosity in H$\alpha$ of the different types of galaxies in
our samples with those in the HFS97 sample, lower sensitivity would
have translated into higher values in our samples. This is not
observed. In the HFS97 sample the median H$\alpha$ luminosity of the
NLAGNs is log(L$_{H\alpha} =38.72 $ ergs s$^{-1}$). Our values are
comparable: 38.69 for the HCG and 38.79 for the UZC-CG.
Finally we have determined the detection limits in our samples as in
\citet{ho97b}. Different simulations were performed adding to each
set-up spectra a grid of synthetic spectra with broad gaussian
components of various widths and amplitudes centered on H$\alpha$. We
then applied our template and extraction analysis to deduce the
following limits. For the CAHA and OSN spectra, broad components
equivalent to 15\% or higher of the total blended flux in
H$\alpha+$[NII] were recovered. Using medians of AGN blended flux and
redshfit this transforms into a detection limit in H$\alpha$ broad
luminosity of $3.5\times10^{38}$ ergs s$^{-1}$. We find slightly
higher fraction (20\%) for the NOT and SPM spectra, equivalent to a
detection limit in luminosity of $4.0\times10^{38}$ ergs
s$^{-1}$. Only three BLAGNs in the HFS97 sample have a luminosity
lower than these limits. Obviously, the lack of BLAGNs encountered in
our samples cannot be explain by a higher detection limits in our
samples.
There seem to be no obvious observational biases or differences in
reduction and analysis methods capable of reproducing the lack of
BLAGNs in CGs as compared to lower density environments. It is
consequently reasonable to conclude that this phenomenon must be
related to the environment typical of CGs.
\subsection{The disappearance of BLRs in CGs}
In the unification model for AGNs, a torus of matter is assumed to
be responsible for hiding the BLR from the observer. To be
consistent with our analysis, this mechanism should be much more
efficient in CGs. However, this assumption goes against the evidence
of tidal stripping: in CGs the infall of gas in the disk seem to be
stopped, generally diminishing the amount of star formation
\citep{coz98,coz00,ver01,ros07,dur08}. At the same time, the number
of early-type galaxies in CGs is observed to increase. Therefore, a
possible reason why no BLRs appear in AGNs in CGs may be because any
amount of gas that has reached the center was consumed into stars,
building larger bulges \citep{car99}. Alternatively, the bulges of
galaxies in CGs may have grown without gas, through dry mergers
\citep{coz07}.
The fact that the average luminosity of the AGNs in CGs is low is
another argument in favor of the dissolution hypothesis for the BLR.
According to recent results obtained by reverberation mapping, the
size of the BLR in AGNs is correlated to the optical luminosity at
5100\AA\ \citep{kas05}. It is consequently possible to imagine the
size of the BLR shrinking almost to zero at some low threshold
luminosity \citep{eli06}. The luminosity at 5100\AA\ in our samples
range from log($\lambda$L$_{\lambda}$(5100\AA)) 40.7 to 43.1 (in
units of erg s$^{-1}$); comparing with data of \citet{pet04} we are
in the lower luminosity part of their distribution, where few
objects with broad lines have been observed. We also are at the
lower limit where no hidden BLRs have been found
\citep{shu07,bia07}. Using the relation log(L$_{bol}$/L$_{Edd}$),
most of our galaxies are below -1.37, which suggests that broad
features may simply not exist in these LLAGNs.
According to \citet{nic00} and \citet{nic03} low accretion rates
rather than smaller mass black holes are responsible in explaining
the absence of BLRs in Low Luminosity AGNs which is fully compatible
with our observations.
\section{Conclusion}
Based on the above statistics, we confirm that there is a remarkable
deficiency of BLAGNs as compared to NLAGNs in CGs. This result
suggests that BLRs in AGN CGs are directly affected by tidal or
group interaction effects, which make them shrink below detection or
completely disappear.
In CGs environment, galaxies are undergoing morphological
transformations and the main mechanisms for such transformations are
tidal interactions and mergers \citep{coz07}. Our analysis suggests
that the combined effects of these two mechanisms also result in an
important decrease in the amount of gas that can reach the nucleus
to form a BLR in AGNs.
\acknowledgments We are grateful to Lei Hao for making available her
eigenspectra and to Jaime Perea for his PCA software and helpful
discussion. MAM acknowledges Ministerio de Educacion y Ciencia for
financial support grant FPU AP2003-4064. MAM and AdO are partially
supported by spanish research projects AYA2006-1325 and TIC114. RC was
partially supported by the CONACyT, under grant
No. 47282. P.F. acknowledges financial contribution from MIUR and from
the contract ASI-INAF I/023/05/0. We thank the referee for
constructive comments. We also thank the TAC of the Observatorio
Astron\'omico Nacional at San Pedro M\'artir for time allocations. We
thank the SDSS collaboration for providing the extraordinary database
and processing tools that made part of this work possible. The SDSS
Web Site is http://www.sdss.org/. We acknowledge also the usage of the
Hyperleda database (http://leda.univ-lyon1.fr).
|
2,869,038,154,885 | arxiv | \section{Introduction}
Arising as the leading saddle point for the gravitational Euclidean path integral with a positive cosmological constant, the sphere plays a dominant role in the study of quantum gravity in de Sitter space \cite{PhysRevD.15.2752, Gibbons:1978ji, Christensen:1979iy,Fradkin:1983mq, Allen:1983dg,Taylor:1989ua,GRIFFIN1989295, Mazur:1989ch, Vassilevich:1992rk, Volkov:2000ih, Anninos:2020hfj,Polchinski:1988ua}. From the static patch point of view, quadratic fluctuations around the sphere saddle contribute to the 1-loop corrections to the Gibbons-Hawking de Sitter horizon entropy. Recently, envisioning a 1-loop test for microscopic models for de Sitter horizon, the authors in \cite{Anninos:2020hfj} derived an integral formula for 1-loop sphere path integrals in terms of Harish-Chandra characters for the de Sitter group.
For example, the character formula for a scalar field with generic mass $m^2$ on $S^{4}$ (with radius set to 1) is \cite{Anninos:2020hfj}
\begin{align}\label{massive char}
\log Z_\text{PI}=\int_0^\infty \frac{dt}{2t}\frac{1+e^{-t}}{1-e^{-t}} \,\chi_\Delta(t)\,,
\end{align}
where
\begin{align}\label{Characters}
\chi_\Delta(t)=\frac{e^{-\Delta t}+e^{-\bar{\Delta} t}}{(1-e^{-t})^3}
\end{align}
is the Harish-Chandra character of the isomatry group $SO(1,4)$ of $dS_{4}$. The scaling dimension $\Delta$ is related to the mass $m^2$ through $m^2=(\Delta-2)(\bar{\Delta}-2)$ and $\bar{\Delta}=3-\Delta$. As explained in \cite{Anninos:2020hfj}, $\chi(t)$ captures massive scalar quasinormal modes in a de Sitter static patch. With the formula \eqref{massive char} we can compute exact 1-loop contribution by a massive scalar to de Sitter horizon entropy.
The present work goes back to the starting point of the derivation of the character formula, that is, the 1-loop sphere path integral itself
\begin{align}\label{PI general}
Z_\text{PI} =\int \mathcal{D}\phi \, e^{-S[\phi]}
\end{align}
where $S[\phi]$ is the quadratic action of the field $\phi$. Typically, such an object is expressed in terms of functional determinants of kinetic operators. A massive real scalar has
\begin{align}\label{intro scalar det}
Z_\text{PI}=\det\left(-\nabla^2+m^2\right)^{-1/2}
\end{align}
which can be massaged into the formula \eqref{massive char} as shown in \cite{Anninos:2020hfj}. However, the computation becomes more subtle and intricate for fields with spin $s\geq 1$. For instance, it took decades of work \cite{PhysRevD.15.2752, Gibbons:1978ji, Christensen:1979iy,Fradkin:1983mq, Allen:1983dg,Taylor:1989ua,GRIFFIN1989295, Mazur:1989ch, Vassilevich:1992rk, Volkov:2000ih,Polchinski:1988ua} before the correct generalization of \eqref{intro scalar det} was obtained for a massless spin-2 field on $S^4$.
To generalize the formula \eqref{massive char} for wider classes of field contents, one must first obtain their correct functional determinant expressions. This is the main motivation for this work. In this paper we perform detailed derivations starting from manifestly local covariant path integrals on $S^{d+1}$. Our goals are twofold: 1. address subtlties of 1-loop sphere path integrals and clarify previous computations; 2. generalize all known results to any massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin $s\geq 0$ in any dimensions $d\geq 2$.
As an illustration, starting from \eqref{PI general} with the free massless spin-$s$ Fronsdal action, a lengthy derivtion in section \ref{Massless PI} leads to the following expression (equation \eqref{HS final}) for massless higher-spin theories
\begin{align}\label{intro HS PI}
Z^\text{HS}_\text{PI}=
i^{-P}
\frac{\gamma^{\text{dim}\,G}}{\text{Vol($G$)}_\text{can}} \prod_s \left( (d+2s-2)(d+2s-4)\right)^{\frac{N^\text{KT}_{s-1}}{2}}
\frac{\det\nolimits'_{-1}\left|-\nabla_{(s-1)}^2-\lambda_{s-1,s-1}\right|^{1/2}}{ \det\nolimits'_{-1}\left|-\nabla_{(s)}^2-\lambda_{s-2,s}\right|^{1/2}}
\end{align}
Here we highlight a few features of this expression:
\begin{itemize}
\item In the ratio of ghost and physical functional determinants, $-\nabla_{(s)}^2$ is the spin-$s$ symmetric transverse traceless (STT) Laplacians on $S^{d+1}$ and $\lambda_{n,s}$ are its eigenvalues. Their relevant properties are summarized in appendix \ref{STSH}. The prime denotes omission of zero modes of the Laplace operators. The subscript -1 is related to the contribution from longitudinal modes, which was obtained in \cite{Anninos:2020hfj} by demanding the absence of logarithmic divergence for odd $d+1$. Here we obtain these by a direct path integral computation.
\item The second factor is associated with the group $G$ of trivial gauge transformations. $\gamma$ is related to the coupling constant of the theory, while $\text{Vol}(G)_\text{can}$ is what was called canonical group volume in \cite{Anninos:2020hfj}. It was emphasized in \cite{Donnelly:2013tia} that the inclusion of this factor was crucial for consistency with locality and unitarity. The peculiar factor $\prod_s \left((d+2s-2)(d+2s-4)\right)^{\frac{N^\text{KT}_{s-1}}{2}}$ arises when we relate the metric on the space of trivial gauge transformations induced by the path integral measure to the canonical metric to be defined precisely below.
\item The phase factor $i^{-P}$ is present only for fields with spin $s\geq 2$, whose origin is the negative conformal modes that render the Euclidean path integral divergent. The standard prescription \cite{Gibbons:1978ac} is to Wick rotate the problematic conformal modes in field space so that the integrals converge. Polchinski later \cite{Polchinski:1988ua} found that on $S^{d+1}$ this procedure led to a finite number of $i$ factors (with $P=d+3$ in that case) that could leave the Euclidean path integral positive, negative or imaginary depending on the dimensions.
\end{itemize}
Analogous expressions are derived for any massive (equation \eqref{massive general}), shift-symmetric (equation \eqref{shift general}) and partially massless (equation \eqref{general s t}) totally symmetric tensor fields of arbitrary spin $s\geq 0$ in any dimensions $d\geq 2$. With these precise expressions one could then derive a formula analogous to \eqref{massive char} for more general representations. As for most quantities in quantum field theory, objects such as \eqref{intro HS PI} are UV-divergent. The recipe for exactly evaluating these formal expressions and their UV divergences were discussed in \cite{Anninos:2020hfj}.
Finally, although this work is primarily motivated by the study of de Sitter thermodynamics, sphere partition functions are of interest in a broad range of contexts such as string theory, Chern-Simons theory, supersymmetry, AdS/CFT correspondence, conformal field theory, as well as entanglement entropy in quantum field theories. We anticipate that the 1-loop results contained in this paper will be relevant in these contexts as well.
\paragraph{Plan of the paper:} We first review the computations for massless spin-1 and spin-2 fields in sections \ref{massless spin1} and \ref{massless spin2}. We then turn to our complete derivation for massless fields of arbitrary integer spins in section \ref{Massless PI}. In section \ref{Massive PI}, we study fields with generic mass. In sections \ref{shift sym} and \ref{PM fields}, we study general shift-symmetric fields and partially massless fields respectively. We conclude in section \ref{conclusion}. All conventions are summarized in appendix \ref{convention}. Relevant properties of the STT Laplacians on $S^{d+1}$ and their eigenfunctions are collected in appendix \ref{STSH}. The higher spin invariant bilinear form is reviewed in appendix \ref{cubic}.
\section{Review of massless vectors}\label{massless spin1}
We start with a pedagogical review of the case of massless vectors. The object of interest is the 1-loop approximation to the full Euclidean path integral
\begin{align}\label{full vector PI}
Z_\text{PI} =\frac{1}{\text{Vol}(\mathcal{G})}\int \mathcal{D} A^a \mathcal{D}\Phi \, e^{-S_E[A^a,\Phi]}
\end{align}
for a theory that involves a collection of massless vector (for example U(1) or Yang-Mills) gauge fields interacting with some matter fields, denoted as $A\indices{^a_\mu}$ and collectively as $\Phi$ respectively, living on $S^{d+1}$.
\paragraph{$U(1)$ with a complex scalar}
The simplest example involves a single U(1) gauge field $A_\mu$ interacting with a complex scalar $\phi$ (studied in \cite{Allen:1983dg}):
\begin{align}\label{U1 ex}
S_E[A, \phi]=\int_{S^{d+1}} \bigg[ \frac{1}{4\mathrm{g}^2} F_{\mu\nu}F^{\mu\nu}+ D_{\mu}\phi (D^\mu \phi)^* +m^2 \phi \phi^*\bigg],
\end{align}
where
\begin{align}
F_{\mu\nu}\equiv\partial_\mu A_\nu-\partial_\nu A_\mu,\qquad D_\mu \phi \equiv (\partial_\mu -i A_\mu)\phi
\end{align}
are the field strength and the covariant derivative of the scalar. This action is invariant under the local U(1) gauge transformations
\begin{align}
\phi(x) \to e^{i\alpha (x)}\phi(x),\quad A_\mu(x)\to A_\mu(x)+\partial_\mu \alpha(x).
\end{align}
The normalization adopted here is to emphasize the presence of the coupling constant $\mathrm{g}$. In this convention $\mathrm{g}$ does not show up in the gauge transformation.
\paragraph{Yang-Mills}
Another example is Yang-Mills (YM) theory with a Lie algebra
\begin{align}\label{Lie al}
[L^a,L^b]=f^{abc}L^c
\end{align}
generated by some standard basis of anti-hermitian matrices and $f^{abc}$ is real and totally antisymmetric. The YM action is
\begin{align}\label{YM ex}
S_E[A, \phi]= \frac{1}{4\mathrm{g}^2}\int_{S^{d+1}} \text{Tr}F^2= \frac{1}{4\mathrm{g}^2}\int_{S^{d+1}} F_{\mu\nu}^a F^{a, \mu\nu},
\end{align}
where the curvature is $F_{\mu\nu}\equiv\partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu,A_\nu]$ with $A_\mu=A_\mu^a L^a$. Here the overall normalization for the trace (or Killing form) is {\it defined} such that the generators $L^a$ are unit normalized:
\begin{align}\label{trace def}
\text{Tr}(L^a L^b)\equiv\delta^{ab}.
\end{align}
For SU(2) YM, $L^a=-\frac{i \sigma^a}{2}$ satisfying $[L^a,L^b]=\epsilon^{abc}L^c$, and the trace \eqref{trace def} would be $ \text{Tr}\equiv -2 \text{tr}$ with tr being the matrix trace. The YM action is invariant under the non-linear gauge transformations $\alpha=\alpha^a L^a$
\begin{align}
A_\mu \to A_\mu+\partial_\mu \alpha+ [A_\mu,\alpha].
\end{align}
In both the $U(1)$ and YM examples, the corresponding path integral in \eqref{full vector PI} clearly overcounts gauge equivalent configurations. A factor $\text{Vol}(\mathcal{G})$ is thus inserted in \eqref{full vector PI} to quotient out configurations connected by gauge transformations. This factor is formally the volume of the space of gauge transformations $\mathcal{G}$ (the measure with respect to which the volume is defined will be discussed in later subsection) and is theory dependent. For the U(1) example, $\text{Vol}(\mathcal{G})$ is simply a path integral over a single local scalar field
\begin{align}
\text{Vol}(\mathcal{G})_\text{U(1)}=\int \mathcal{D}\alpha,
\end{align}
while for SU(2) YM it would be a path integral over $3$ local scalar fields
\begin{align}
\text{Vol}(\mathcal{G})_\text{SU(2)}=\int \mathcal{D}\alpha_1\mathcal{D}\alpha_2\mathcal{D}\alpha_3.
\end{align}
More generally, $\text{Vol}(\mathcal{G})$ is an integral over $N=\text{dim} \,G$ local scalar fields for a gauge group $G$.
\paragraph{1-loop approximation}
Now, suppose the equation of motion admits the trivial solution $A\indices{^a_\mu}=0=\Phi$, around which we perform a saddle point approximation for \eqref{full vector PI}. Then at the quadratic (1 loop) level the vector and matter fields decouple:
\begin{gather}
Z^\text{1-loop}_\text{PI} =Z^{\delta A}_\text{PI} Z^{\delta \Phi}_\text{PI}.
\end{gather}
In the following, we focus on the vector part of the 1-loop path integral (with $A^a$ understood as the fluctuations around the background)
\begin{align}\label{spin 1 1loop}
Z^{\delta A}_\text{PI}=\frac{1}{\text{Vol}(\mathcal{G})}\int \prod_{a=1}^{\text{dim}\,G} \mathcal{D} A^a \, e^{-\sum_{a=1}^{\text{dim}\,G} S_E[A^a]}.
\end{align}
where $S_E[A^a]$ is simply a Maxwell action
\begin{align}\label{Maxwell action}
S_E[A^a]= \frac{1}{4\mathrm{g}^2} \int_{S^{d+1}} F^a_{\mu\nu}F^{a, \mu\nu}, \quad F_{\mu\nu} =\partial_\mu A^a_\nu-\partial_\nu A^a_\mu\, .
\end{align}
A careful analysis of the Euclidean path integral for the U(1) theory on arbitrary manifolds has been presented in \cite{Donnelly:2013tia}, where the authors point out the importance of taking care of zero modes, large gauge transformations and non-trivial bundles for consistency with locality and unitarity.
In the following we will express $Z^0_\text{PI}$ in terms of functional determinants and highlight the relevant subtleties in our case of $S^{d+1}$ along the way. Before doing so we would like to make a last comment. Even though the problem is largely reduced to that of a free Maxwell theory on $S^{d+1}$, the knowledge of the parent theory \eqref{full vector PI} to which we performed the 1-loop approximation is required for the full determination of the \eqref{spin 1 1loop}. In particular, the group volume factor $\text{Vol}(\mathcal{G})$ will eventually lead to a residual group volume factor in the final result, whose value depends on the specific gauge group (e.g. $U(1)$ or $SU(N)$) and couplings of the gauge field to the matter fields.
\subsection{Transverse vector determinant and Jacobian}
\subsubsection*{Geometric approach and change of variables}
Since the path integrations over $A^a$ in \eqref{spin 1 1loop} are decoupled, we can focus on one of the factors, and we will suppress the index $a$. Traditional ways to proceed include Faddeev-Popov or BRST gauge fixing (as done in \cite{Donnelly:2013tia} for example). Here instead we take the ``geometric approach'' \cite{Babelon:1979wd,Mazur:1989by,Bern:1990bh}, which manifests its advantages when we deal with massless higher spin fields later. In this approach one changes the field variables by decomposing
\begin{align}\label{spin 1 CoV}
A_\mu = A^T_\mu+\partial_\mu \chi
\end{align}
where $A^T_\mu$ is the transverse or on-shell part of $A_\mu$ satisfying $\nabla^\mu A^T_\mu=0$, and $\chi$ is the longitudinal or pure gauge part of $A_\mu$. Since $S^{d+1}$ is compact, the scalar Laplacian has a normalizable constant $(0,0)$ mode, which must be excluded from the path integration for the change of variables \eqref{spin 1 CoV} to be unique
\begin{align}
\mathcal{D}A=J\,\mathcal{D}A^T\mathcal{D}' \chi
\end{align}
where prime denotes the exclusion of the $(0,0)$ mode. We will find the Jacobian $J$ for the change of variables \eqref{spin 1 CoV} below.
\subsubsection*{Action for $A^T_\mu$}
Because the gauge invariance of the action, $\chi$ simply drops out upon substituting \eqref{spin 1 CoV}
\begin{align}
S_E[A^T,\chi]=\frac{1}{2\mathrm{g}^2} \int_{S^{d+1}} [A^T_\mu (-\nabla_{(1)}^2 +d) A_T^\mu ]
\end{align}
where $-\nabla_{(1)}^2$ is the trasnverse Laplacian on $S^{d+1}$. Now we expand $A^T_\mu$ in terms of spin-1 transverse spherical harmonics (see appendix \ref{STSH} for their basic properties):
\begin{align}
A^T_\mu = \sum_{n=1}^\infty c_{n,1} f_{n,\mu}
\end{align}
and the integration measure in our convention is
\begin{align}
\mathcal{D}A^T = \prod_{n=1}^\infty \frac{dc_{n,1}}{\sqrt{2\pi}\mathrm{g}}.
\end{align}
Performing the path integration over these modes we have
\begin{align}
\int \mathcal{D} A e^{-S_E[A]}= J \, \det(-\nabla_{(1)}^2 +d)^{-1/2} \int \mathcal{D}' \chi
\end{align}
\subsubsection*{Jacobian}
We find the Jacobian $J$ by requiring consistency with the normalization condition
\begin{align}\label{spin 1 norm}
1=\int \mathcal{D}A \, e^{-\frac{1}{2\mathrm{g}^2} (A,A)}=\int J \,\mathcal{D}A^T\mathcal{D}' \chi \, e^{-\frac{1}{2\mathrm{g}^2} (A^T+\nabla \chi,A^T+\nabla \chi)}.
\end{align}
Since $A^T$ is transverse, we have
\begin{align}
(A^T+\nabla \chi,A^T+\nabla \chi)= (A^T,A^T)+ (\nabla \chi,\nabla \chi).
\end{align}
We can then path integrate $A^T$ trivially. We expand $\chi$ in terms of scalar spherical harmonics:
\begin{align}
\chi = \sum_{n=1}^\infty c_{n,0} f_n
\end{align}
with path integration measure
\begin{align}
\mathcal{D}'\chi = \prod_{n=1}^\infty \frac{dc_{n,0}}{\sqrt{2\pi}\mathrm{g}}.
\end{align}
Plugging this into \eqref{spin 1 norm} results in
\begin{align}\label{vector Jac}
J = \det\nolimits'(-\nabla_{(0)}^2)^{1/2},
\end{align}
where the prime denotes the omission of the constant $(0,0)$ mode.
\subsection{Residual group volume}\label{s 1 vol}
Let us go back to the full 1-loop path integral \eqref{spin 1 1loop}. So far we have
\begin{align}
Z^{\delta A}_\text{PI}=\frac{\int \prod_{a=1}^{\text{dim} G} \mathcal{D}'\chi^a}{\text{Vol}(\mathcal{G})} \left(\frac{ \det'(-\nabla_{(0)}^2)^{1/2}}{\det(-\nabla_{(1)}^2 +d)^{1/2}} \right)^{\text{dim} G}
\end{align}
where we have restored the color index $a$. Now we focus on the factor
\begin{align}
\frac{\int \prod_{a=1}^{\text{dim} G} \mathcal{D}'\chi^a}{\text{Vol}(\mathcal{G})}.
\end{align}
As explained above, the factor $\text{Vol}(\mathcal{G})$ is theory dependent and is formally an integral over $N={\text{dim} G}$ \textbf{local} scalar fields
\begin{align}
\text{Vol}(\mathcal{G}) =\int \prod_{n=1}^{\text{dim} G} \mathcal{D}\alpha_n .
\end{align}
In particular, the integral includes integrations over constant scalar modes. As explained in \cite{Donnelly:2013tia}, the inclusion of zero modes is crucial for consistency with locality and unitarity. Thus, this factor does not cancel completely with the integrations over $\chi$, leaving a factor
\begin{align}\label{vec group fac}
\frac{\int \prod_{a=1}^{\text{dim} G} \mathcal{D}'\chi^a}{\text{Vol}(\mathcal{G})}=\frac{1}{\text{Vol}(G)_\text{PI}},\quad \text{Vol}(G)_\text{PI} \equiv \int \prod_{a=1}^{\text{dim}\,G} \frac{d\alpha^a_0}{\sqrt{2\pi}\mathrm{g}}.
\end{align}
where $\alpha^a_0$ is the expansion coefficient of the $(0,0)$ mode of $\alpha^a$ ($a$ is the color index)
\begin{align}
\alpha^a = \sum_{n=0}^\infty \alpha^a_{n} f_n.
\end{align}
These constant scalar modes correspond to the gauge transformations that leave the background $A^\mu=0$ invariant, or equivalently whose linear part is trivial. If the original full theory contains matter fields such as \eqref{U1 ex}, these act non-trivially on the latter. $G$ is therefore the group of \textbf{global} symmetries of the theory and $\text{Vol}(G)_\text{PI}$ is the volume of $G$. Note that the precise value of $\text{Vol}(G)_\text{PI}$ depends on the metric on $G$. We have been using a specific choice of metric
\begin{align}\label{YM PI metric}
ds_\text{PI}^2 = \frac{1}{2\pi \mathrm{g}^2} \int_{S^{d+1}} \text{Tr}(\delta \alpha \delta \alpha)
\end{align}
induced by our convention for the path integral measure. Note that had we normalized the generators $L^a$ in a different way: $L^a \to \lambda L^a$ (or equivalently choosing a different overall normalization for the trace in the action \eqref{YM ex}: $\text{Tr} \to \lambda^2 \text{Tr}$), the path integral describes the same physics if we rescale $\mathrm{g}\to \lambda \mathrm{g}$. In particular, the metric \eqref{YM PI metric} remains the same. We want to relate the volume $\text{Vol}(G)_\text{PI}$ measured in this metric to a ``canonical volume'' $\text{Vol}(G)_\text{can}$, defined as follows. A general group element in $G$ takes the form
\begin{align}
e^{\theta \cdot \hat{L}}=e^{\theta^a \hat{L}^a}
\end{align}
where $\hat{L}^a$ are unit-normalized. We define $\text{Vol}(G)_\text{can}$ to be the volume of the space spanned by $\theta$. In our convention, $L^a$ are unit-normalized, and therefore the relation between the metric \eqref{YM PI metric} (restricted to the subspace of trivial gauge transformations) and the canonical metric is simply
\begin{align}\label{pi metric can}
ds_\text{PI}^2 = \frac{1}{2\pi \mathrm{g}^2} \sum_{a}(d\alpha_0^a)^2= \frac{1}{2\pi \mathrm{g}^2} \sum_{a}\left(\frac{d\theta^a}{f_0}\right)^2= \frac{\text{Vol}(S^{d+1})}{2\pi \mathrm{g}^2} ds_\text{can}^2,\quad ds_\text{can}^2 \equiv d\theta \cdot d\theta.
\end{align}
Thus we can express the group volume as
\begin{align}\label{PI volume can}
\text{Vol}(G)_\text{PI} = \left( \frac{\text{Vol}(S^{d+1})}{2\pi \mathrm{g}^2}\right)^\frac{\text{dim}(G)}{2}\text{Vol}(G)_\text{can} = \bigg( \frac{\text{Vol}(S^{d-1})}{d \mathrm{g}^2}\bigg)^\frac{\text{dim}(G)}{2}\text{Vol}(G)_\text{can},
\end{align}
where we have used $\text{Vol}(S^{d+1})=\frac{2\pi}{d}\text{Vol}(S^{d-1})$ in the last step. The canonical volume $\text{Vol}(G)_\text{can}$ so defined is evidently independent of the coupling. To summarize, the full 1-loop path integral is
\begin{align}
Z^{\delta A}_\text{PI}=&Z_\text{G}Z_\text{Char}\nonumber\\
Z_\text{G}=&\frac{\gamma^{\text{dim} G}}{\text{Vol}(G)_\text{can}},\quad \gamma=\frac{\mathrm{g}}{\sqrt{(d-2)\text{Vol}(S^{d-1})}}\nonumber\\
Z_\text{Char}=&\left(d(d-2)\right)^{\frac{1}{2}{\text{dim} G}}\left(\frac{ \det'(-\nabla_{(0)}^2)}{\det(-\nabla_{(1)}^2 +d)}\right)^{\frac{1}{2}{\text{dim} G}}
\end{align}
The notation $Z_\text{Char}$ emphasizes that this part can be re-written in terms of the $SO(1,d+1)$ characters as explained in \cite{Anninos:2020hfj}. In retrospect, the coupling dependence of the result is precisely encoded in the group volume factor $\text{Vol}(G)_\text{PI}$. In the $G=U(1)$ example, $\text{Vol}(G)_\text{can}=\text{Vol}(U(1))_c =2\pi$, and the full 1 loop vector path integral is therefore
\begin{align}
Z^{U(1)}_\text{PI}=\frac{ \mathrm{g}}{ \sqrt{2\pi\text{Vol}(S^{d+1})}} \frac{ \det'(-\nabla_{(0)}^2)^{1/2}}{\det(-\nabla_{(1)}^2 +d)^{1/2}},
\end{align}
which reproduces eq.(2.6) in \cite{Giombi:2015haa}. For $G=SU(2)$, $\text{dim}\, G=3$ and $\text{Vol}(G)_\text{can}=16\pi^2$, and thus
\begin{align}
Z^{SU(N)}_\text{PI}=\frac{1}{16\pi^2}\bigg( \frac{2\pi \mathrm{g}^2}{ \text{Vol}(S^{d+1})}\bigg)^\frac{3}{2}\left(\frac{ \det'(-\nabla_{(0)}^2)}{\det(-\nabla_{(1)}^2 +d)} \right)^{3/2}.
\end{align}
As a concrete example for an exact evaluation of such a formal expression, with the recipe in \cite{Anninos:2020hfj} one can find for the 1-loop path integral for $SU(4)$ Yang-Mills on $S^5$
\begin{align} \label{MaxwellS5}
\log Z_{\rm PI}^{SU(4), \,S^5} = 15\left( \frac{9 \pi }{8} \frac{\ell^5}{\epsilon^5}-\frac{5 \pi }{8}\frac{\ell^3}{\epsilon^3}-\frac{7 \pi }{16} \frac{\ell}{\epsilon}\right)+ \log \frac{\left(\mathrm{g}/\sqrt{\ell}\right)^{15}}{{\frac{1}{6}(2\pi)^9}} + 15 \left( \, \frac{5 \, \zeta (3)}{16 \, \pi ^2} +\frac{3 \, \zeta (5)}{16 \, \pi ^4} \, \right)
\end{align}
where we have restored the sphere radius $\ell$ and $\epsilon$ is the UV regulator in heat kernel regularization.
\subsubsection*{Local gauge algebra, global symmetry and invariant bilinear form}
For the later discussions on spin 2 and massless higher spin fields, and to make connection with the work in \cite{Joung:2013nma}, we offer another perspective for the non-abelian case.
\paragraph{Local gauge algebra}
Recall that the original Yang-Mills action \eqref{YM ex} is invariant under the full non-linear infinitesimal gauge transformations
\begin{gather}
\delta_\alpha A_\mu =\delta^{(0)}_\alpha A_\mu +\delta^{(1)}_\alpha A_\mu \nonumber\\
\delta^{(0)}_\alpha A_\mu= \partial_\mu \alpha ,\qquad \delta^{(1)}_\alpha A_\mu= [A_\mu,\alpha].
\end{gather}
Here the superscript $(n)$ denotes the power in fields. This generates an algebra
\begin{align}\label{YM local gauge algebra}
\delta_\alpha \delta_{\alpha'}A_\mu- \delta_{\alpha'}\delta_\alpha A_\mu=\delta_{[[\alpha,\alpha']]}A_\mu
\end{align}
where we have defined a bracket $[[\cdot,\cdot ]]$ on the space of gauge parameters, which in our convention is equal to the negative of the matrix commutator\footnote{One should keep in mind that $[[\cdot,\cdot]]$ is defined using the gauge transformations of $A_\mu$, whose precise form depends on the normalization conventions, while the commutator on the right hand side is the matrix commutator $[A,B]=AB-BA$. Had we normalized $A_\mu$ canonically, so that the action takes the form $- \frac{1}{4}\int_{S^{d+1}} \text{Tr}F^2$, the gauge transformations will be instead $\delta_\alpha A_\mu =\partial_\mu \alpha+\mathrm{g} [A_\mu,\alpha]$ and the local gauge algebra will become $[\delta_\alpha, \delta_{\alpha'}]=\delta_{-\mathrm{g}[\alpha,\alpha']}$ and the bracket will read $[[\bar{\alpha},\bar{\alpha}']]=-\mathrm{g}[\alpha,\alpha']$.}
\begin{align}\label{YM bracket}
[[\alpha,\alpha']]=-[\alpha,\alpha'].
\end{align}
\paragraph{Global symmetry algebra from the gauge algebra}
The constant $(0,0)$ modes $\bar{\alpha}$ generate background ($A_\mu=0$) preserving gauge transformations satisfying
\begin{align}
\delta^{(0)}_{\bar{\alpha}}=0,
\end{align}
which form a subalgebra $\mathfrak{g}$ of the local gauge algebra, with the bracket $[[\cdot,\cdot]]$ naturally inherited from the local gauge algebra
\begin{align}
[[\bar{\alpha},\bar{\alpha}']]=-[\bar{\alpha},\bar{\alpha}'].
\end{align}
This global symmetry algebra $\mathfrak{g}$ is clearly isomorphic to the original Lie algebra \eqref{Lie al}. On $\mathfrak{g}$, the path integral metric \eqref{YM PI metric} corresponds to the bilinear form with a specific normalization:
\begin{align}\label{vec PI bi form}
\bra{\bar{\alpha}}\ket{\bar{\alpha}'}_\text{PI}=\frac{1}{2\pi \mathrm{g}^2} \int_{S^{d+1}} \bar{\alpha}^a \bar{\alpha}'^{ a}=\frac{\text{Vol}(S^{d+1})}{2\pi \mathrm{g}^2} \bar{\alpha}^a \bar{\alpha}'^{ a}.
\end{align}
We define a theory independent ``canonical'' invariant bilinear form $\bra{\cdot}\ket{\cdot}_\text{c}$ on $\mathfrak{g}$ as follows.
\begin{enumerate}
\item Pick a basis $M^a$ of $\mathfrak{g}$ such that they satisfy the same commutation relation as $L^a$: $[[M^a,M^b]]=f^{abc}M^c$. This fixes the relative normalizations of $M^a$.
\item Fix the overall normalization of $\bra{\cdot}\ket{\cdot}_\text{c}$ by requiring $M^a$ to be unit-normalized:
\begin{align}
\bra{M^a}\ket{M^b}_\text{c}=\delta^{ab}
\end{align}
\end{enumerate}
In the current case, this means that we should take $M^a=L^a$ and
\begin{align}
\bra{\alpha}\ket{\alpha'}_\text{c}= \bar{\alpha}^a \bar{\alpha}'^{ a}.
\end{align}
Comparing this with \eqref{vec PI bi form}, we see that the path integral and canonical metrics are related as in \eqref{pi metric can}, leading to the same result \eqref{PI volume can}.
\section{Review of massless spin 2}\label{massless spin2}
Next we review the computation for linearized Einstein gravity on $S^{d+1}$, which has a long and dramatic history \cite{Gibbons:1978ji, Christensen:1979iy,Fradkin:1983mq, Allen:1983dg,Taylor:1989ua,GRIFFIN1989295, Mazur:1989ch, Vassilevich:1992rk, Volkov:2000ih,Polchinski:1988ua}. Expanding the gravitational path integral with the Einstein-Hilbert action $\frac{1}{16\pi G_N}\int_{S^{d+1}} (2\Lambda-R)$ around the $S^{d+1}$ saddle: $g_{\mu\nu}=g^{S^{d+1}}_{\mu\nu}+h_{\mu\nu}$ up to quadratic order, we obtain the Euclidean path integral
\begin{align}\label{s2pathintegral}
Z_\text{PI}=\frac{1}{\text{Vol($\mathcal{G}$)}}\int \mathcal{D}h \, e^{-S[h]}
\end{align}
where the action for a massless spin-2 particle on $S^{d+1}$ is
\begin{align}\label{eq:spin2action}
S[h] = \frac{1}{2\mathrm{g}^2}\int_{S^{d+1}}h^{\mu\nu}\bigg[ (-\nabla^2+2)h_{\mu\nu}+2 \nabla_{(\mu}\nabla^\lambda h_{\nu) \lambda}+g_{\mu\nu}(\nabla^2 h\indices{_\lambda^\lambda}-2\nabla^{\sigma}\nabla^\lambda h_{\sigma \lambda})+(D-3)g_{\mu\nu}h\indices{_\lambda^\lambda}\bigg],
\end{align}
where $\mathrm{g}=\sqrt{32\pi G_N}$. \eqref{eq:spin2action} is invariant under the linearized diffeomorphisms\footnote{The insertion of the factor $\frac{1}{\sqrt{2}}$ is for later convenience.}
\begin{align}\label{lin diff}
h_{\mu\nu}\to h_{\mu\nu}+\sqrt{2}\nabla_{(\mu}\Lambda_{\nu)}=h_{\mu\nu}+\frac{1}{\sqrt{2}}(\nabla_{\mu} \Lambda_{\nu}+\nabla_{\nu} \Lambda_{\mu}).
\end{align}
The factor $\text{Vol($\mathcal{G}$)}$ is the volume of the space of diffeomorphisms, which is a path integral over a local vector field $\alpha_\mu$
\begin{align}\label{vol g local vec}
\text{Vol}(\mathcal{G}) =\int \mathcal{D}\alpha.
\end{align}
This factor is inserted in \eqref{s2pathintegral}to compensate for the over-counting of gauge equivalent orbits connected by \eqref{lin diff}.
\subsubsection*{Change of variables}
As in the case of massless vectors, we decompose $h_{\mu\nu}$ as
\begin{align}\label{spin 2 CoV}
h_{\mu\nu}=h_{\mu\nu}^\text{TT} +\frac{1}{\sqrt{2}} (\nabla_{\mu} \xi_{\nu}+\nabla_{\nu} \xi_{\mu})+\frac{g_{\mu\nu}}{\sqrt{d+1}}\tilde{h}
\end{align}
where $h_{\mu\nu}^\text{TT}$ is the transverse-traceless part of $h_{\mu\nu}$ satisfying $\nabla^\lambda h_{\lambda \mu}=0=h\indices{^\lambda_\lambda}$, $\xi_{\mu}$ is the pure gauge part of $h_{\mu\nu}$, and $\tilde{h}$ is the trace of $h_{\mu\nu}$. For \eqref{spin 2 CoV} to be unique, we require $\xi_{\nu}$ to be orthogonal to all Killing vectors (KVs) on $S^{d+1}$
\begin{align}\label{xi con}
(\xi,\xi^\text{KV})=0,\qquad \nabla_{\mu} \xi^\text{KV}_{\nu}+\nabla_{\nu} \xi^\text{KV}_{\mu} = 0
\end{align}
and $\tilde{h}$ to be orthogonal to divergence of the rest of all conformal Killing vectors (CKVs)
\begin{align}\label{tilde h con}
(\tilde{h},\nabla \cdot\xi^\text{CKV})=0,\qquad \nabla_{\mu} \xi^\text{CKV}_{\nu}+\nabla_{\nu} \xi^\text{CKV}_{\mu} = \frac{1}{2(d+1)}g_{\mu\nu}\nabla^{\lambda}\xi^\text{CKV}_{\lambda}.
\end{align}
The path integral measure then becomes
\begin{align}
\mathcal{D}h = J \, \mathcal{D}h^\text{TT}\mathcal{D}'\xi \mathcal{D}'\tilde{h}
\end{align}
where the Jacobian $J$ will be found below. The primes indicate that we exclude the integrations over the $(1,1)$ and $(1,0)$ modes excluded due to conditions \eqref{xi con} and \eqref{tilde h con}.
\subsection{Transverse tensor and trace mode determinants}
\subsubsection*{Action for $h^{TT}_{\mu\nu}$}
Due to the gauge invariance \eqref{lin diff}, we have
\begin{align}
S[h]=S[h^\text{TT}+\tilde{h}]=S[h^\text{TT}]+S[\tilde{h}].
\end{align}
$S[h^\text{TT}]$ can be easily obtained as
\begin{align}
S[h^\text{TT}]=\frac{1}{2\mathrm{g}^2}\int_{S^{d+1}}h^\text{TT}_{\mu\nu}(-\nabla_{(2)}^2+2)h_\text{TT}^{\mu\nu}.
\end{align}
where $-\nabla_{(2)}^2$ is the spin-2 STT Laplacian. The integration over $h^\text{TT}$ thus gives
\begin{align}\label{spin2TTaction}
Z^\text{TT}_h = \int \mathcal{D}h^\text{TT} \, e^{-S[h^\text{TT}]}= \det (-\nabla_{(2)}^2+2)^{-1/2}.
\end{align}
\subsubsection*{Action for $\tilde{h}$ and the conformal factor problem}
Similarly, after a bit more work, the quadratic action for $\tilde{h}$ can be obtained as
\begin{align}\label{spin2tildehaction}
S[\tilde{h}]=&-\frac{d(d-1)}{2(d+1)\mathrm{g}^2}\int_{S^{d+1}} \tilde{h}(-\nabla_{(0)}^2-(d+1))\tilde{h}\nonumber\\
=&-\frac{d(d-1)}{2(d+1)\mathrm{g}^2}\sum_{n\neq 1} (n(n+d)-(d+1))c_{n,0}^2
\end{align}
where in the second line we have inserted the mode expansion
\begin{align}
\tilde{h}=\sum_{n\neq 1}c_{n,0} f_n,\qquad ( f_n, f_m)=\delta_{n,m}.
\end{align}
Here the sum runs over the spectrum of the scalar Laplacian except the $(1,0)$ modes, which corresponds to the CKVs. Notice that \eqref{spin2tildehaction} has a wrong overall sign for all positive modes of the operator $-\nabla_{(0)}^2-(d+1)$. This is the well-known conformal factor problem \cite{Gibbons:1978ac} in Euclidean gravity method. We follow the standard prescription: we replace $c_{n,0}\to ic_{n,0}$, \footnote{The sign in front of the $i$ is a matter of convention.} for all $n\geq 2$, which leads to the change in the path integral measure
\begin{align}
\mathcal{D}'\tilde{h}=\prod_{n\neq 1}\frac{dc_{n,0}}{\sqrt{2\pi}\mathrm{g}}\to\bigg( \prod_{n=2}^\infty i\bigg)\prod_{n\neq 1}\frac{dc_{n,0}}{\sqrt{2\pi}\mathrm{g}}=i^{-d-3}\left(\prod_{n=0}^\infty i\right) \prod_{n\neq 1}\frac{dc_{n,0}}{\sqrt{2\pi}\mathrm{g}}.
\end{align}
The factor in the last step runs through the spectrum of $-\nabla_{(0)}^2$ and is thus a local infinite constant that can be absorbed into bare couplings. Doing this the path integral becomes
\begin{align}
Z_{\tilde{h}}=&\int \mathcal{D}'\tilde{h}\, e^{S[\tilde{h}]}= i^{-d-3} Z^+_{\tilde{h}}Z^-_{\tilde{h}} \nonumber\\
Z^+_{\tilde{h}} =& \int \mathcal{D}^+\tilde{h}\, e^{-\frac{d(d-1)}{2(d+1)\mathrm{g}^2}\int_{S^{d+1}} \tilde{h}(-\nabla_{(0)}^2-(d+1))\tilde{h}}\nonumber\\
Z^-_{\tilde{h}} =& \int \mathcal{D}^-\tilde{h}\, e^{\frac{d(d-1)}{2(d+1)\mathrm{g}^2}\int_{S^{d+1}} \tilde{h}(-\nabla_{(0)}^2-(d+1))\tilde{h}}
\end{align}
where $\pm$ indicate the contribution from positive and negative modes respectively. The overall phase factor $ i^{-d-3}$ was first obtained by Polchinski \cite{Polchinski:1988ua}. Later we will see the generalization of this phase factor for all massless higher spin fields.
\subsection{Jacobian}\label{spin 2 Jacobian}
Again, we find the Jacobian $J$ by requiring consistency with the normalization condition
\begin{align}
1=\int\mathcal{D}h \, e^{-\frac{1}{2\mathrm{g}^2}(h,h)}.
\end{align}
Since $h^\text{TT}$ is transverse and traceless, we have
\begin{align}
(h,h)=(h^\text{TT},h^\text{TT})+(\sqrt{2}\nabla \xi +\frac{g \tilde{h}}{\sqrt{d+1}},\sqrt{2}\nabla \xi +\frac{g \tilde{h}}{\sqrt{d+1}}).
\end{align}
To proceed we separate $\xi_\mu=\xi'_\mu+\xi^{\text{CKV}}_\mu$, where $\xi^{\text{CKV}}_\mu$ is a linear combination of the CKVs and $\xi'_\mu$ is the part of $\xi_\mu$ that is orthogonal to the CKVs, that is $(\xi',\xi^{\text{CKV}})=0$. Note that while $g\tilde{h}$ is orthogonal to $\xi^{\text{CKV}}_\mu$ because of \eqref{tilde h con}, $g\tilde{h}$ and $\nabla \xi'$ are not orthogonal to each other. To remove the off-diagonal terms, we shift
\begin{align}
\tilde{h}' = \tilde{h}+\sqrt{\frac{2}{d+1}}\nabla^\lambda \xi'_\lambda.
\end{align}
Since it is just a shift, the Jacobian is trivial. It is then easy to compute
\begin{align}
(\sqrt{2}\nabla \xi +\frac{g \tilde{h}}{\sqrt{d+1}},\sqrt{2}\nabla \xi +\frac{g \tilde{h}}{\sqrt{d+1}})=(\tilde{h}',\tilde{h}')+\frac{1}{2}(K \xi' ,K \xi' )+2(\nabla \xi^\text{CKV},\nabla \xi^\text{CKV})
\end{align}
where we have defined the differential operator
\begin{align}
(K\xi)_{\mu\nu}\equiv\nabla_\mu \xi_\nu+\nabla_\nu \xi_\mu-\frac{2}{d+1}g_{\mu\nu}\nabla^\lambda \xi_\lambda.
\end{align}
Now the integrations over $h^\text{TT}$ and $\tilde{h}'$ become trivial. To proceed, we first simplify
\begin{align}
(K \xi' ,K \xi' ) = 2 \int_{S^{d+1}} \bigg[ {\xi'}^\nu \Big( -\nabla^2 -d\Big) \xi'_\nu -{\xi'}^\nu \Big(\frac{d-1}{d+1} \nabla_\nu \nabla^\lambda \xi'_\lambda\Big)\bigg].
\end{align}
Then we decompose $\xi'$ into its transverse and longitudinal parts: $\xi'_\nu =\xi_\nu^T +\nabla_\nu \sigma$. Once again this change of variables leads to a Jacobian factor which is easily found as before. With this decomposition we can further simplify
\begin{align}
\frac{1}{2}(K \xi' ,K \xi' ) = S[\xi^T]+\frac{2d}{d+1} S[\sigma],
\end{align}
where
\begin{align}
S[\xi^T]=\int_{S^{d+1}} \xi^T_{\nu}(-\nabla^2_{(1)}-d)\xi_T^{\nu},\qquad S[\sigma]=\int_{S^{d+1}} \sigma(-\nabla_{(0)}^2)(-\nabla_{(0)}^2-(d+1))\sigma.
\end{align}
We therefore arrive at
\begin{align}
\begin{split}\label{spin 2 Jac}
J=&\frac{W^+_\sigma}{Y^\text{T}_\xi Y^+_\sigma}\frac{1}{Y^\text{CKV}_\xi} \\
Y^\text{T}_\xi=&\int\mathcal{D}' \xi^T \, e^{-\frac{1}{2\mathrm{g}^2} (\xi^T,(-\nabla^2_{(1)}-d)\xi^T)} \\
Y^+_\sigma =&\int\mathcal{D}^+\sigma\, e^{-\frac{1}{2\mathrm{g}^2}\frac{2d}{d+1}( \sigma,(-\nabla_{(0)}^2)(-\nabla_{(0)}^2-(d+1))\sigma)}\\
W^+_\sigma=&\int\mathcal{D}^+\sigma \,e^{-\frac{1}{2\mathrm{g}^2}( \sigma, (-\nabla_{(0)}^2)\sigma)}\\
Y^\text{CKV}_\xi=&\int\mathcal{D} \xi^\text{CKV}\, e^{-\frac{1}{\mathrm{g}^2}(\nabla \xi^\text{CKV},\nabla \xi^\text{CKV})}
\end{split}
\end{align}
Here $W^+_\sigma$ is the Jacobian corresponding to the change of variables $\{\xi'_\nu\} \to \{\xi_\nu^T +\nabla_\nu \sigma\}$. The +'s denote the positive modes for the operator $(-\nabla_{(0)}^2-(d+1))$. \footnote{The zero modes of the operator $(-\nabla_{(0)}^2-(d+1))$ are excluded because $\sigma$ satisfies $(\sigma,f_0)=0=(\sigma,\nabla \xi^\text{CKV})$.}
\subsection{Residual group volume}
As in the massless vector case, we have a factor
\begin{align}
\frac{\int \mathcal{D}'\xi}{\text{Vol($\mathcal{G}$)}}
\end{align}
in the path integral. We recall from \eqref{vol g local vec} that Vol($\mathcal{G}$) is a path integral over a local vector field. This does not cancel completely with the integration over $\xi_\mu$, and we are left with a factor (restoring the label $a$ for degenerate modes with same quantum number $(1,1)$)
\begin{align}
\frac{\int \mathcal{D}'\xi}{\text{Vol($\mathcal{G}$)}}=\frac{1}{\text{Vol}(G)_\text{PI}},\quad \text{Vol}(G)_\text{PI} \equiv \int \prod_{a=1}^{\frac{(d+1)(d+2)}{2}} \frac{d\alpha^{(a)}_{1,1}}{\sqrt{2\pi}\mathrm{g}}.
\end{align}
where $\alpha^{(a)}_{1,1}$ is the expansion coefficient in the expansion
\begin{align}
\alpha_\mu = \sum_{n=1}^\infty \alpha_{n,1} f_{n,\mu}+ \sum_{n=1}^\infty \alpha_{n,0} \hat{T}_{n, \mu}^{(0)}.
\end{align}
These $(1,1)$ modes are diffeomorphisms that leave the background $S^{d+1}$ metric invariant, so they in fact correspond to the Killing vectors of $S^{d+1}$. $G$ is therefore the isometry group $SO(d+2)$ of $S^{d+1}$. As in the massless vector case, we want to relate $\text{Vol}(G)_\text{PI}$ to a canonical volume, following the argument in section \ref{s 1 vol}.
\paragraph{Local gauge algebra}
Recall that the original Einstein-Hilbert action is invariant under non-linear diffeomorphisms generated by any vector field $\alpha=\frac{1}{\sqrt{2}}\alpha^\mu \partial_\mu$, which reads
\begin{align}
\delta_\alpha h_{\mu\nu}=&\delta^{(0)}_\alpha h_{\mu\nu}+\delta^{(1)}_\alpha h_{\mu\nu}+O(h^2)\nonumber\\
\delta^{(0)}_\alpha h_{\mu\nu}=&\frac{1}{\sqrt{2}}(\nabla_{\mu} \alpha_{\nu}+\nabla_{\nu} \alpha_{\mu})\nonumber\\
\delta^{(1)}_\alpha h_{\mu\nu}=&\frac{1}{\sqrt{2}}(\alpha^\rho\nabla_\rho h_{\mu\nu}+\nabla_\mu \alpha^\rho h_{\rho\nu}+\nabla_\nu \alpha^\rho h_{\mu\rho}),
\end{align}
where the superscript $(n)$ again denotes the power in fields. This generates the algebra
\begin{align}\label{diff algebra}
[\delta_\alpha,\delta_{\alpha'}]=\delta_{[[\alpha,\alpha']]}.
\end{align}
In this case, the bracket is proportional to the usual Lie derivative\footnote{If we had worked with canonical normalization, obtained by replacing $h_{\mu\nu}\to \mathrm{g} h_{\mu\nu}$, the bracket will read instead $[[\alpha,\alpha']] =-\frac{\mathrm{g}}{\sqrt{2}}[\alpha,\alpha']_L=-\sqrt{16\pi G_N} [\alpha,\alpha']_L$. This relation can be viewed as a \textit{definition} of the Newton constant $G_N$ in any gauge theory with a massless spin 2 field.}
\begin{align}\label{Einstein bracket}
[[\alpha,\alpha']] =-\frac{1}{\sqrt{2}}[\alpha,\alpha']_L, \qquad [\alpha,\alpha']_L= (\alpha^\mu\partial_\mu \alpha'^\nu-\alpha'^\mu\partial_\mu \alpha^\nu)\partial_\nu.
\end{align}
\paragraph{Isometry algebra from the local gauge algebra}
The background ($S^{d+1}$) preserving gauge transformations or isometries generated by the Killing vectors satisfying
\begin{align}
\delta^{(0)}_{\bar{\alpha}}=0
\end{align}
and form a subalgebra of the local gauge algebra, which inherits a bracket from the latter
\begin{align}\label{so(d+2) bracket}
[[\bar{\alpha},\bar{\alpha}']]=-\frac{1}{\sqrt{2}}[\bar{\alpha},\bar{\alpha}']_L.
\end{align}
To define the canonical volume, we again first find a set of generators $M_{IJ}$ that satisfy the standard $so(d+2)$ commutation relation under the bracket \eqref{so(d+2) bracket}:
\begin{align}
[[M_{IJ},M_{KL}]]=\eta_{JK}M_{IL}-\eta_{JL}M_{IK}+\eta_{IL}M_{JK}-\eta_{IK}M_{JL}.
\end{align}
One such basis is $M_{IJ}=-\sqrt{2}(X_I\partial_{X^J}-X_J\partial_{X^I})$ where $X^I X_I=1,X^I\in \mathbb{R}^{d+2},I=1\cdots d+2$ are the coordinates of on $S^{d+1}$ represented in the ambient space. Its norm in the invariant bilinear form induced by the path integral is (it suffices to consider only one of the generators)
\begin{align}
\bra{M_{12}}\ket{M_{12}}_\text{PI}=\frac{1}{2\pi \mathrm{g}^2} \int_{S^{d+1}}(M_{12})^{IJ} (M_{12})_{IJ} =\frac{2}{2\pi \mathrm{g}^2} \int_{S^{d+1}} (X_1^2+X_2^2)=\frac{2}{2\pi \mathrm{g}^2} \frac{2}{d+2}\text{Vol}(S^{d+1}).
\end{align}
Since the canonical bilinear form is defined such that $\bra{M_{12}}\ket{M_{12}}_\text{c}=1$, the path integral metric on $G$ is related to the canonical metric as
\begin{align}
ds_\text{PI}^2= \frac{2}{2\pi \mathrm{g}^2} \frac{2}{d+2}\text{Vol}(S^{d+1}) ds_\text{can}^2=\frac{1}{8\pi G_N} \frac{\text{Vol}(S^{d-1})}{d(d+2)} ds_\text{can}^2
\end{align}
where we have used $\text{Vol}(S^{d+1})=\frac{2\pi}{d}\text{Vol}(S^{d-1})$ and substituted $\mathrm{g}=\sqrt{32\pi G_N}$ in the last step. Therefore
\begin{align}
\text{Vol}(G)_\text{PI} =\bigg(\frac{1}{8\pi G_N} \frac{\text{Vol}(S^{d-1})}{d(d+2)} \bigg)^{\frac{(d+1)(d+2)}{4}} \text{Vol}(G)_\text{can} .
\end{align}
The canonical volume $\text{Vol}(G)_\text{can}=\text{Vol}(SO(d+2))_\text{can}$ is well-known:\footnote{This follows from the fact that $SO(n+1)/SO(n)=S^{n}$, which implies that $\text{Vol}(SO(n+1))=\text{Vol}(SO(n))\text{Vol}(S^n)$}
\begin{align}
\text{Vol}(SO(d+2))_c = \prod_{n=1}^{d+1}\text{Vol}(S^n) = \prod_{n=1}^{d+1} \frac{2\pi^{\frac{n+1}{2}}}{\Gamma(\frac{n+1}{2})}
\end{align}
\subsection{Final result}
So far we have
\begin{align}\label{spin 2 inter}
Z_\text{PI}=\frac{i^{-d-3}}{\text{Vol}(G)_\text{PI}} \bigg(\frac{Z^\text{TT}_h}{Y^\text{T}_\xi}\bigg)\bigg(\frac{Z^+_{\tilde{h}} W^+_\sigma}{ Y^+_\sigma}\bigg)\frac{Z^-_{\tilde{h}}}{Y^\text{CKV}_\xi} .
\end{align}
Note that the factor
\begin{align}
\frac{Z^\text{TT}_h}{Y^\text{T}_\xi} = \frac{\det'(-\nabla_{(1)}^2-d)^{1/2}}{\det(-\nabla_{(2)}^2+2)^{1/2}}
\end{align}
is the usual ratio of determinants. Next, the factors in the second bracket in \eqref{spin 2 inter} cancel up to an infinite product
\begin{align}
\frac{Z^+_{\tilde{h}} W^+_\sigma}{ Y^+_\sigma} =& \int \mathcal{D}^+\tilde{h} \,e^{-\frac{d-1}{4\mathrm{g}^2}\int_{S^{d+1}} \tilde{h}^2} =\prod_{n=2}^\infty \Big( \frac{d-1}{2}\Big)^{-\frac{D^{d+2}_{n,0}}{2}}=\Big( \frac{d-1}{2}\Big)^{\frac{d+3}{2}}\prod_{n=0}^\infty \Big( \frac{d-1}{2}\Big)^{-\frac{D^{d+2}_{n,0}}{2}},
\end{align}
where in the last line we have complete the product so that it runs through the spectrum of the scalar Laplacian. The infinite product can then be absorbed into bare couplings. Finally, the factors in the last bracket in \eqref{spin 2 inter} can be explicitly evaluated to be
\begin{align}
Z^-_{\tilde{h}}=\bigg(\frac{1}{d(d-1)}\bigg)^{1/2},\quad Y^\text{CKV}_\xi = 2^{-\frac{d+2}{2}}.
\end{align}
Putting everything together, we conclude
\begin{align}
Z_\text{PI}=&Z_G Z_\text{Char}, \nonumber\\
Z_G=&i^{-d-3}\frac{\gamma^{\frac{(d+1)(d+2)}{2}}}{ \text{Vol}(SO(d+2))_c },\quad \gamma=\sqrt{\frac{8\pi G_N}{\text{Vol}(S^{d-1})}}\nonumber\\
Z_\text{Char}=&\bigg(d(d+2) \bigg)^{\frac{(d+1)(d+2)}{4}} \frac{(d-1)^{\frac{d+2}{2}}}{(2d)^{1/2}}\frac{\det'(-\nabla_{(1)}^2-d)^{1/2}}{\det(-\nabla_{(2)}^2+2)^{1/2}}.
\end{align}
As a check, we note that except for the inclusion of the phase factor $i^{-d-3}$, for $d=3$ we agree exactly with the 1-loop part of (5.43) in \cite{Volkov:2000ih}.\footnote{Note that our expression agrees with the first line of (5.43) in \cite{Volkov:2000ih}, while the authors made an error in evaluating the determinants, so their second line is incorrect, as already noted in \cite{Anninos:2020hfj}.} In this case one finds \cite{Anninos:2020hfj}
\begin{align}
\log Z_{\rm PI}^{S^4} = &\log\left( \frac{i^{-6}}{\frac{2}{3}(2\pi)^6}\left(\frac{8\pi G_N}{4\pi \ell^2}\right)^5 \right) +\frac{8}{3} \frac{1}{\epsilon^4} \ell^4 -\frac{32}{3} \frac{1}{\epsilon^2} \ell^2 -\frac{571}{45} \log\left(\frac{2 e^{-\gamma}}{\epsilon} L\right) \nonumber\\
&- \frac{571}{45} \log\left(\frac{\ell}{L}\right)
+\frac{715}{48} -\log 2 - \frac{47}{3} \zeta'(-1) +\frac{2}{3} \zeta'(-3)
\end{align}
where the sphere radius $\ell$ has been restored. $\epsilon$ is the UV regulator in heat kernel regularization and $L$ is a renormalization scale in the minimal subtraction scheme \cite{Anninos:2020hfj}.
\section{Massless higher spin}\label{Massless PI}
Now we are ready for the 1-loop path integrals for higher spin (HS) theories on $S^{d+1}$. One of the motivations of considering these theories lies in their relevance in holographic proposals \cite{Anninos:2011ui,Anninos:2012ft,Anninos:2013rza,Anninos:2017eib} in de Sitter space. Although the equations of motion for these theories have been constructed \cite{Vasiliev:1990en,Vasiliev:2003ev,Bekaert:2005vh}, the full actions from which these are derived remain elusive.\footnote{See also \cite{Boulanger:2015ova,Sleight:2017pcz} for arguments against the existence for consistent interacting HS theories.} However, since the interactions are at least cubic, their 1-loop partition functions (around the trivial saddle) decouple into a product of free partition functions.
\begin{align}\label{Vasiliev PI}
Z^\text{HS}_\text{PI}=\prod_{s} Z^{(s,m^2=0)}_\text{PI},
\end{align}
where $Z^{(s,m^2=0)}_\text{PI}$ is the 1-loop path integral for a massless spin-$s$ field to be described below. The precise range over $s$ in the product depends on the specific higher spin theory we are interested in. In AdS, the determinant expressions for $Z^{(s,m^2=0)}_\text{PI}$ are obtained in \cite{Gaberdiel:2010ar} and \cite{Gupta:2012he}, which are subsequently used in 1-loop tests of HS/CFT dualities \cite{Giombi:2013fka,Giombi:2014iua,Giombi:2016pvg,Gunaydin:2016amv}. In the following, we perform a careful computation for $Z^{(s,m^2=0)}_\text{PI}$ on $S^{d+1}$, whose early stage has some overlap with \cite{Gaberdiel:2010ar}. In fact, the following can be viewed as a derivation for the AdS case as well, except that the latter does not contain the subtleties of phases and group volume that appear on $S^{d+1}$.\footnote{This is because the modes that cause these subtleties are non-normalizable in AdS and are excluded from the beginning.}
\subsection{Operator formalism}
It is much simpler to carry out the entire computation in terms of generating functions, which significantly simplifies tensor manipulations. Here we adopt the convention of \cite{Sleight:2017cax} but on $S^{d+1}$. In this formalism, the tensor structure of a totally symmetric spin-$s$ field $\phi_{\mu_1 \cdots \mu_s}$ in $S^{d+1}$ is encoded in a constant auxiliary ($d+1$)-dimensional vector $u^\mu$:
\begin{align}
\phi_{(s)}(x)=\phi_{\mu_1 \cdots \mu_s}(x) \to \phi_s(x,u)\equiv \frac{1}{s!} \phi_{\mu_1 \cdots \mu_s}(x) u^{\mu_1} \cdots u^{\mu_s}.
\end{align}
In the following we will suppress the position argument $x$, and interchangeably refer to a rank-$s$ tensor with $\phi_{(s)}$ or its generating function $\phi_s(u)$. Since the original covariant derivative $\nabla_\mu$ acts on both $\phi_{\mu_1 \cdots \mu_s}$ and $u^\mu$, we modify the covariant derivative as
\begin{align}
\nabla_\mu \to \nabla_\mu +\omega\indices{_\mu ^a ^b}u_a \frac{\partial}{\partial u^b},
\end{align}
where $u^a =e\indices{_\mu^a}u^\mu$ with vielbein $e\indices{_\mu^a}(x)$ and $\omega\indices{_\mu ^a ^b}$ is the spin connection. With this modification the actions of covariant derivatives on $u^\mu$ offset each other, and we can work as if no derivative is acting on $u^\mu$. In the following we will only work in the contracted variables $u^\mu=e\indices{^\mu_a}u^a$ and the associated derivative $\partial_{u^\mu}=e\indices{_\mu^a}\partial_{u^a}$. As a consequence of vielbein postulate we have
\begin{align}
[\nabla_\mu,u^\nu]=0=[\nabla_\mu,\partial_{u^\nu}].
\end{align}
In this formalism all tensor manipulations are translated to an operator calculus. For instance, tensor contraction:
\begin{align}
\phi_{\mu_1 \cdots \mu_s} \chi^{\mu_1 \cdots \mu_s} =s! \phi_s(\partial_u)\chi_s(u).
\end{align}
In particular, the inner product \eqref{inner} is represented as
\begin{align}
( \phi_s, \chi_s ) =s! \int_{S^{d+1}} \phi_s(\partial_u)\chi_s(u).
\end{align}
List of operations:
\begin{align}\label{op list}
\text{divergence: }& \nabla \cdot \partial_u, & \text{sym. gradient: }& u\cdot \nabla, & \text{Laplacian: }& \nabla^2,\nonumber \\
\text{sym. metric: }& u^2, & \text{trace: }& \partial_u^2, & \text{spin: }& u\cdot \partial_u.
\end{align}
One of the biggest advantages of this formalism is that we can work algebraically with these operators without explicitly referring to the tensor. For example, to define the de Donder operator, we can either state explicitly its action on a spin-$s$ field $\phi_{(s)}$
\begin{align}
\hat{D}\phi_{(s)}=\hat{D}\phi_{\mu_1 \cdots \mu_s} = \nabla^\lambda \phi_{\mu_1 \cdots \mu_{s-1}\lambda} -\frac{1}{2} \nabla_{(\mu_1}\phi\indices{_{\mu_2 \cdots \mu_{s-1})\lambda}^\lambda}
\end{align}
or simply in terms of its generating function
\begin{align}
\hat{D}(\nabla,u,\partial_u)=& \nabla \cdot \partial_u -\frac{1}{2}(u\cdot \nabla) (\partial_u^2).
\end{align}
In the following we will use these two kinds of notations interchangeably.
On $S^{d+1}$, the operators \eqref{op list} satisfy the following operator algebra
\begin{align}
[\nabla_\mu, \nabla_\nu] =& u_\mu\partial_{u^\nu}-u_\nu \partial_{u^\mu} \label{algebra1}\\
[\nabla^2 , u\cdot\nabla] =& u\cdot \nabla (2u\cdot\partial_u +d)-2u^2 \nabla\cdot \partial_u \label{algebra2}\\
[\nabla\cdot \partial_u , \nabla^2] =& (2u\cdot\partial_u +d) \nabla\cdot \partial_u-2 u\cdot \nabla \partial_u^2\label{algebra3}\\
[\nabla\cdot \partial_u , u\cdot\nabla]=& \nabla^2 + u\cdot \partial_u (u\cdot \partial_u+d-1)-u^2 \partial_u^2\label{algebra4} \\
[\nabla\cdot \partial_u ,u^2] =& 2u\cdot \nabla\label{algebra5}\\
[ \partial_u^2 ,u\cdot\nabla] =& 2 \nabla\cdot \partial_u\label{algebra6}\\
[\partial_u^2,u^2]=& 2(d+1+2u\cdot \partial_u)\label{algebra7}
\end{align}
where we have denoted $\partial_u^2 \equiv \partial_u \cdot \partial_u, u^2 \equiv u\cdot u$.
\subsection{Fronsdal action on $S^{d+1}$}
The 1-loop partition function for a free bosonic spin-$s$ massless gauge field on $S^{d+1}$ is
\begin{align}\label{eq:Epathintegral}
Z^{(s)}_\text{PI}=\frac{1}{\text{Vol($\mathcal{G}_{s}$)}}\int \mathcal{D}\phi_{(s)}\,e^{-S[\phi_{(s)}]}
\end{align}
where the quadratic Fronsdal action \cite{Fronsdal:1978rb} in the operator language is given by
\begin{align}\label{eq:opaction}
S[\phi_{(s)}] = \frac{s!}{2\mathrm{g}_s^2}\int_{S^{d+1}} \phi_s (\partial_u)\Big(1-\frac{1}{4}u^2 \partial_u^2 \Big)\hat{\mathcal{F}}_s (\nabla,u,\partial_u) \phi_s (u)
\end{align}
with $\hat{\mathcal{F}}_s(\nabla,u,\partial_u) $ is the Fronsdal operator
\begin{align}
\hat{\mathcal{F}}_s(\nabla,u,\partial_u) =&-\nabla^2+M_s^2-u^2 \partial_u^2 +u\cdot \nabla \hat{D}(\nabla,u,\partial_u)\\
\hat{D}(\nabla,u,\partial_u)=& \nabla \cdot \partial_u -\frac{1}{2}(u\cdot \nabla) (\partial_u^2),
\end{align}
where
\begin{align}\label{mass para}
M_s^2=s-(s-2)(s+d-2)
\end{align}
and $\hat{D} $ is the de Donder operator. An $s$-dependent factor $\mathrm{g}_s^2$ is inserted as an overall factor. Canonical normalization corresponds to setting $\mathrm{g}_s=1$. We will choose a particular value for $\mathrm{g}_s$ when we discuss the issue of group volume. (\ref{eq:opaction}) is invariant under the gauge transformations
\begin{align}\label{eq:opgauge}
\phi_s (u)\mapsto \phi_s (u) + \frac{1}{\sqrt{s}}u\cdot\nabla\Lambda_{s-1}(u).
\end{align}
In this off-shell formalism $\phi_s (u)$ satisfies a double-tracelessness condition (trivial for $s\leq 3$)
\begin{align}\label{eq:opdouble-tracelessness}
(\partial_u^2)^2 \phi_s (u)=0 ,
\end{align}
which implies that the gauge parameter $\Lambda_{(s-1)}$ must be traceless (imposed even for $s=3$)
\begin{align}
\partial_u^2 \Lambda_{s-1}(u)=0 .
\end{align}
The division by the gauge group volume $\text{Vol($\mathcal{G}_{s}$)}$ in \eqref{eq:Epathintegral} compensates for the overcounting of gauge equivalent configurations connected by \eqref{eq:opgauge}.
\subsubsection*{Change of variables}
To proceed, we change field variables
\begin{align}\label{opsdisplit}
\phi_s (u) = \phi^\text{TT}_s (u)+ \frac{1}{\sqrt{s}}u\cdot \nabla \xi_{s-1}(u) +\frac{1}{\sqrt{2s(s-1)(d+2s-3)}}u^2 \chi_{s-2}(u).
\end{align}
Here $\phi_{(s)}^\text{TT}$ is the transverse traceless piece of $\phi_{(s)}$ for which
\begin{align}
\nabla\cdot\partial_u \phi^\text{TT}_s (u)=&0= \partial_u^2 \phi^\text{TT}_s (u).
\end{align}
Next, $\xi_{(s-1)}$ is the symmetric traceless spin-($s-1$) gauge parameters which are required to be orthogonal to all spin-($s-1$) Killing tensors $\epsilon_{(s-1)}^{KT}$ (which generate trivial gauge transformations) so that it is uniquely fixed:
\begin{align}\label{xi condition}
\partial_u^2 \xi_{s-1}(u) =&0,\quad (\xi_{(s-1)}, \epsilon_{(s-1)}^{KT})=0
\end{align}
Finally, $\chi_{(s-2)}$ is the spin-($s-2$) piece which carries all the trace information of $\phi_{(s)}$. The double-tracelessness condition (\ref{eq:opdouble-tracelessness}) implies that $\chi_{(s-2)}$ is traceless:
\begin{align}\label{chi traceless}
\partial_u^2 \chi_{s-2}(u) =&0.
\end{align}
Note that if $\epsilon^{CKT}_{(s-1)}$ is a conformal Killing tensor (CKT) satisfying
\begin{align}\label{CKeq}
\hat{K}_{s} (\nabla,u,\partial_u)\epsilon^{CKT}_{ s-1}(u)\equiv u\cdot \nabla \epsilon^{CKT}_{ s-1}(u)-\frac{u^2}{d+2s-3}(\nabla \cdot \partial_u )\epsilon^{CKT}_{ s-1}(u)=0,
\end{align}
then any new set of variables related by the transformation
\begin{align}
\xi_{s-1}(u)&\to\xi_{s-1}(u)+\epsilon^{CKT}_{s-1}(u)\\
\chi_{s-2}(u)&\to\chi_{s-2}(u)-\frac{u^2}{d+2s-3}(\nabla \cdot \partial_u )\epsilon^{CKT}_{ s-1}(u)
\end{align}
will result in the same $\phi_{(s)}$. To uniquely fix $\chi_{(s-2)}$, we thus impose
\begin{align}\label{eq:gauge}
( \chi_{(s-2)},(\nabla \cdot\epsilon^{CKT})_{(s-2)} )=0
\end{align}
for all the spin-($s-1$) CKTs $\epsilon^{CKT}_{(s-1)}$. The path integral measure then becomes
\begin{align}\label{sdimeasure}
\mathcal{D}\phi_s = J_{(s)} \mathcal{D}\phi^\text{TT}_{(s)}\mathcal{D}'\xi_{(s-1)} \mathcal{D}'\chi_{(s-2)}
\end{align}
where the Jacobian $J_{(s)}$ will be found below. The primes indicate that we exclude the $(s-1,m)$ ($0\leq m\leq s-1$) modes excluded due to conditions \eqref{xi condition} and \eqref{eq:gauge}.
\subsection{Quadratic actions for $\phi^\text{TT}_{(s)}$ and $\chi_{(s-2)}$}
\subsubsection*{Action for $S[\phi_{(s)}^\text{TT}] $}
The quadratic action for $\phi^\text{TT}_{(s)}$ is
\begin{align}
S[\phi_{(s)}^\text{TT}] =\frac{s!}{2\mathrm{g}_s^2} \int_{S^{d+1}} \phi^\text{TT}_s (\partial_u) ( -\nabla_{(s)}^2+M_s^2 )\phi^\text{TT}_s (u)=\frac{1}{2\mathrm{g}_s^2}( \phi_{(s)}^\text{TT}, ( -\nabla_{(s)}^2+M_s^2) \phi_{(s)}^\text{TT} ).
\end{align}
which leads to the path integral
\begin{align}
Z^{(s)}_{\phi^\text{TT}}=\int \mathcal{D}\phi_{(s)}^\text{TT}\, e^{-\frac{1}{2\mathrm{g}_s^2} ( \phi_{(s)}^\text{TT}, ( -\nabla_{(s)}^2+M_s^2) \phi_{(s)}^\text{TT} )}=\det(-\nabla_{(s)}^2+M_s^2)^{-1/2}
\end{align}
\subsubsection*{Action for $S[\chi_{(s-2)}]$ and the HS conformal factor problem}
From (\ref{eq:opaction}) we have
\begin{align}\label{eq:op pre tilde phi action}
S[\chi_{(s-2)}] =& \frac{(s-2)!}{8\mathrm{g}_s^2} \int_{S^{d+1}} \chi_{s-2}(\partial_u)(\partial_u^2)\Big(1-\frac{1}{4}u^2 \partial_u^2 \Big)\hat{\mathcal{F}}_s (\nabla,u,\partial_u) u^2 \chi_{s-2}(u)\nonumber\\
=&-\frac{(s-2)!(d+2s-5)}{8(d+2s-3)\mathrm{g}_s^2}\int_{S^{d+1}}\chi_{s-2}(\partial_u)(\partial_u^2) \hat{\mathcal{F}}_s (\nabla,u,\partial_u) u^2 \chi_{s-2}(u)
\end{align}
where we have used \eqref{algebra7} and the tracelessness of $ \chi_{(s-2)}$ \eqref{chi traceless}. Using the operator algebras, one easily finds that
\begin{align}
&\hat{\mathcal{F}}_s (\nabla,u,\partial_u) u^2 \nonumber\\
=&u^2 \Big(- \nabla^2-s(-1+d+s)+2\Big)+ u^2 (u\cdot\nabla)( \nabla\cdot \partial_u)-(d+2s-5)(u\cdot\nabla)^2+\cdots
\end{align}
where and henceforth $\cdots$ denotes terms that will not contribute because of the tracelessness condition \eqref{chi traceless}: $\partial_u^2 \chi_{s-2}(u) =0$ or $\chi_{s-2}(\partial_u) u^2 =0$. Then we have
\begin{align}
(\partial_u^2)\hat{\mathcal{F}}_s (\nabla,u,\partial_u) u^2 = &4(d+2s-4) \Big(- \nabla^2-(s-1)(s+d-2)-1\Big) \nonumber\\
&-2(d+2s-7) (u\cdot\nabla)( \nabla\cdot \partial_u)+\cdots.
\end{align}
Defining the differential operator
\begin{align}\label{Qop}
\hat{\mathcal{Q}}(\nabla,u,\partial_u)\equiv &2\frac{d+2s-4}{d+2s-3}\Big(-\nabla^2 -(s-1)(s+d-2)-1\Big)-\frac{d+2s-7}{d+2s-3}(u\cdot\nabla) (\nabla\cdot\partial_u),
\end{align}
the quadratic action for $\chi_{(s-2)}$ is simply
\begin{align}
S[\chi_{(s-2)}] =&-\frac{d+2s-5}{4\mathrm{g}_s^2} (\chi_{(s-2)}, \hat{\mathcal{Q}} \chi_{(s-2)} ).
\end{align}
To proceed, we expand $\chi_{(s-2)}$ (see appendix \ref{STSH} for the properties of the induced symmetric traceless spherical harmonics)
\begin{align}
\chi_{(s-2)} = \sum_{m=0}^{s-2} A_{s-2,m} \hat{T}_{s-2, (s-2)}^{(m)}+ \sum_{m=0}^{s-2} \sum_{n=s}^\infty A_{n,m} \hat{T}_{n, (s-2)}^{(m)},
\end{align}
where the modes $(n,m)=(s-1,m), 0\leq m\leq s-2$ are excluded because of the condition \eqref{eq:gauge}. It is easy to verify that $\hat{\mathcal{Q}}$ is negative for the modes in the first sum and positive in the second. This is the HS generalization of the conformal factor problem. To make the integrals converge, we replace $A_{n,m}\to iA_{n,m}$ for $0\leq m\leq s-2, s\leq n<\infty$, leading to the change in the path integral measure
\begin{align}
\mathcal{D}'\chi_{(s-2)} = \prod_{m=0}^{s-2} \frac{dA_{s-2,m}}{\sqrt{2\pi}\mathrm{g}_s}\prod_{m=0}^{s-2} \prod_{n=s}^\infty \frac{dA_{n,m}}{\sqrt{2\pi}\mathrm{g}_s}\to \bigg(\prod_{m=0}^{s-2} \prod_{n=s}^\infty i^{D^{d+2}_{n,m}}\bigg) \prod_{m=0}^{s-2} \frac{dA_{s-2,m}}{\sqrt{2\pi}\mathrm{g}_s}\prod_{m=0}^{s-2} \prod_{n=s}^\infty \frac{dA_{n,m}}{\sqrt{2\pi}\mathrm{g}_s}.
\end{align}
We complete the product so that it runs through the spectrum for the unconstrained spin-($s-2$) Laplacian, i.e.
\begin{align}
\prod_{m=0}^{s-2} \prod_{n=s}^\infty i^{D^{d+2}_{n,m}}=i^{-N^\text{CKT}_{s-2}-N^\text{CKT}_{s-1} +N^\text{KT}_{s-1}}\prod_{m=0}^{s-2} \prod_{n=s-2}^\infty i^{D^{d+2}_{m,n}}
\end{align}
where $N^\text{CKT}_{s}=\sum_{m=0}^{s} D^{d+2}_{s,m}$ and $N^\text{KT}_{s}=D^{d+2}_{s,s}$ are the number of spin-$s$ CKTs and spin-$s$ KTs respectively. The local infinite product can then be absorbed into bare couplings. The remaining phase factor is the HS generalization of the Polchinski's phase. We can then write the path integral over $\chi_{(s-2)}$ as
\begin{align}\label{phis-2}
Z^{(s)}_\chi=&i^{-N^\text{CKT}_{s-2}-N^\text{CKT}_{s-1} +N^\text{KT}_{s-1}} Z^{(s)}_{\chi^+}Z^{(s)}_{\chi^-}\nonumber\\
Z^{(s)}_{\chi^+}=&\int \mathcal{D}^+\chi_{(s-2)}\,e^{-\frac{d+2s-5}{4\mathrm{g}_s^2}( \chi_{(s-2)},\hat{\mathcal{Q}}\chi_{(s-2)})}\nonumber\\
Z^{(s)}_{\chi^-}=&\int\mathcal{D}^-\chi_{(s-2)}\, e^{\frac{d+2s-5}{4\mathrm{g}_s^2}( \chi_{(s-2)},\hat{\mathcal{Q}}\chi_{(s-2)})}
\end{align}
where the superscripts $\pm$ denotes integrations over the positive (negative) modes of $\hat{\mathcal{Q}}$.
\subsection{Jacobian}
Again, we find the Jacobian in \eqref{sdimeasure} by the normalization condition
\begin{align}\label{snorm}
\int \mathcal{D} {\phi}_{(s)} \, e^{-\frac{1}{2\mathrm{g}_s^2}( {\phi}_{(s)},{\phi}_{(s)})}=1.
\end{align}
We plug in (\ref{sdimeasure}) and \eqref{opsdisplit} to find $J_{(s)}$. Notice that $\phi^\text{TT}_{(s)}$ is orthogonal to $g\chi_{(s-2)}$ and $\nabla \xi_{(s-1)}$ with respect to the inner product $( \cdot , \cdot )$; on the other hand, when $\xi$'s are orthogonal to the spin-($s-1$) CKTs (denoted as $\xi'$), $g\chi_{(s-2)}$ and $\nabla \xi'_{(s-1)}$ are not orthogonal, and we remove the off-diagonal terms by shifting
\begin{align}\label{furtherCoV}
\chi'_{s-2}(u) =\chi_{s-2}(u) +\sqrt{\frac{s(s-1)}{2(d+2s-3)}}(\nabla \cdot \partial_u ) \xi'_{s-1}(u).
\end{align}
The Jacobian corresponding to this shift is trivial. We then have
\begin{align}
( \phi_{(s)},\phi_{(s)} )= ( \phi^\text{TT}_{(s)},\phi^\text{TT}_{(s)} )+( \chi'_{(s-2)},\chi'_{(s-2)})+\frac{1}{s}( \hat{K}_{s}\xi'_{(s-1)} , \hat{K}_{s} \xi'_{(s-1)} )+\frac{1}{s}( \nabla\xi^{CKT}_{(s-1)} , \nabla\xi^{CKT}_{(s-1)} ).
\end{align}
where $\hat{K}_{s}$ is the operator appearing in \eqref{CKeq}. It is useful to note that acting on any symmetric traceless tensor $\epsilon_{(s-1)}$,
\begin{align}\label{K iden}
\partial_u^2 \hat{K}_{s}(\nabla,u,\partial_u)\epsilon_{s-1}(u)=0=\hat{K}_{s}(\nabla,\partial_u,u)\epsilon_{s-1}(\partial_u)u^2.
\end{align}
The path integrals over $\phi^\text{TT}_{(s)}$ and $\chi'_{(s-2)}$ are trivial, and therefore $J_{(s)}$ can be expressed as
\begin{align}
J_{(s)}^{-1} =& Y^{(s)}_{\xi'} Y^{(s)}_{\xi^\text{CKT}} \\
Y^{(s)}_{\xi'}\equiv &\int \mathcal{D}\xi'_{(s-1)}\, e^{-\frac{1}{2s\mathrm{g}_s^2}( K\xi'_{(s-1)} , K\xi'_{(s-1)} )}\\
Y^{(s)}_{\xi^\text{CKT}} \equiv&\int \mathcal{D}\xi^{CKT}_{(s-1)}\, e^{-\frac{1}{2s\mathrm{g}_s^2}( \nabla\xi^{CKT}_{(s-1)} , \nabla\xi^{CKT}_{(s-1)} )}.
\end{align}
\subsubsection*{Expressing $Y_{\xi'}$ in terms of functional determinants}
To proceed, we use the operator algebra and \eqref{K iden} and simplify
\begin{align}
\frac{1}{s}( \hat{K}_{s}\xi'_{(s-1)} , \hat{K}_{s} \xi'_{(s-1)} ) =& ( \xi'_{(s-1)}, \Big( -\nabla_{(s-1)}^2-(s-1)(s+d-2) \Big)\xi'_{(s-1)} ) \nonumber\\
&+\frac{d+2s-5}{d+2s-3}( \xi'_{(s-1)},-\nabla \nabla\boldsymbol{\cdot}\xi'_{(s-1)}).
\end{align}
We then perform the change of variables
\begin{align}\label{xiCoV}
\xi'_{(s-1)} = {\xi'}^{\text{TT}}_{(s-1)}+\hat{K}_{s-1}\sigma_{(s-2)},
\end{align}
where ${\xi'}^{\text{TT}}_{(s-1)}$ is the transverse traceless part of ${\xi'}_{(s-1)}$, $\sigma_{(s-2)}$ is a spin-($s-2$) symmetric traceless field and the differential operator $\hat{K}_{s-1}(\nabla,u,\partial_u)$ is defined in \eqref{CKeq}. We require $\sigma_{(s-2)}$ to be orthogonal to the kernel of $\hat{K}_{s-1}$, i.e. the spin-($s-2$) CKTs. Also, ${\xi'}^{\text{TT}}_{(s-1)}$ and $\sigma_{(s-2)}$ are automatically orthogonal to the spin-($s-1$) CKTs.
Plugging in these, we have two decoupled pieces
\begin{align}\label{gh inter}
\frac{1}{s}( \hat{K}_{s}\xi'_{(s-1)} , \hat{K}_{s} \xi'_{(s-1)} ) =& S[ {\xi'}_{(s-1)}^\text{TT}]+S[\sigma_{(s-2)}].
\end{align}
Here the first term is the ghost action
\begin{align}
S[ {\xi'}_{(s-1)}^\text{TT}]=& ( {\xi'}_{(s-1)}^\text{TT}, \Big( -\nabla_{(s-1)}^2+m_{s-1,s}^2+M_{s-1}^2\Big){\xi'}_{(s-1)}^\text{TT} ) \label{gh STT}
\end{align}
with $M_{s-1}^2$ as defined in \eqref{mass para} and we have defined
\begin{align}\label{PM mass}
m_{s,t}^2=(s-1-t)(d+s+t-3),
\end{align}
which is exactly the mass for a partially massless field with spin-$s$ and depth $t$ for $0\leq t\leq s-1$. The second term in \eqref{gh inter} is the action of a spin-$(s-2)$ field
\begin{align}
S[\sigma_{(s-2)}]=& ( \hat{K}_{s-1}\sigma_{(s-2)}, \hat{\mathcal{P}}\hat{K}_{s-1}\sigma_{(s-2)}) \label{sigma}\\
\hat{\mathcal{P}}(\nabla,u,\partial_u)=&-\nabla_{(s-1)}^2-(s-1)(s+d-2)-\frac{d+2s-5}{d+2s-3}(u\cdot\nabla)( \nabla\cdot\partial_u)
\end{align}
To proceed, we commute $\hat{\mathcal{P}}$ and $\hat{K}_{s-1}$. This requires the relation
\begin{align}
\nabla_{(s-1)}^2 \hat{K}_{s-1}(\nabla,u,\partial_u) - \hat{K}_{s-1}(\nabla,u,\partial_u)\nabla_{(s-2)}^2 = (d+2s-4)u\cdot \nabla+\cdots
\end{align}
and the commutator
\begin{align}
&[(u\cdot \nabla )(\nabla\cdot \partial_u),\hat{K}_{s-1}(\nabla,u,\partial_u)] \nonumber\\
=& [(u\cdot \nabla )(\nabla\cdot \partial_u),u\cdot\nabla]-\frac{1}{d+2s-5}[(u\cdot \nabla )(\nabla\cdot \partial_u), u^2 \nabla \cdot \partial_u]
\end{align}
which can be computed using
\begin{align}
[(u\cdot \nabla )(\nabla\cdot \partial_u),u\cdot\nabla] =& (u\cdot \nabla ) \Big(\nabla^2 +(s-2)(s+d-3) \Big)+\cdots \\
[(u\cdot \nabla )(\nabla\cdot \partial_u), u^2 \nabla \cdot \partial_u]=&2(u\cdot \nabla )^2 (\nabla\cdot \partial_u)+\cdots
\end{align}
where and henceforth $\cdots$ denotes terms that will not contribute to \eqref{sigma} because of the tracelessness condition \eqref{K iden} of the operator $\hat{K}_{s-1}$. We have also used the fact that $u\cdot \partial_u \sigma_{s-2}(u)=(s-2)\sigma_{s-2}(u)$. To briefly summarize,
\begin{align}
& \hat{\mathcal{P}}(\nabla,u,\partial_u)\hat{K}_{s-1}(\nabla,u,\partial_u) \nonumber\\
=& \hat{K}_{s-1} (\nabla,u,\partial_u)\hat{\mathcal{P}}(\nabla,u,\partial_u)-(d+2s-4)u\cdot \nabla\nonumber\\
&+\frac{d+2s-5}{d+2s-3}(u\cdot \nabla )\bigg[ \Big(-\nabla^2 -(s-2)(s+d-3) \Big)+\frac{2}{d+2s-5}(u\cdot \nabla ) (\nabla\cdot \partial_u)\bigg]+\cdots .
\end{align}
Now, observe that because of \eqref{K iden}, $u\cdot \nabla$ can be replaced by the operator $\hat{K}_{s-1}$
\begin{align}
u\cdot \nabla = \hat{K}_{s-1}(\nabla,u,\partial_u) +\cdots
\end{align}
up to trace terms that do not contribute to \eqref{sigma}. Therefore we have
\begin{align}
\hat{\mathcal{P}}(\nabla,u,\partial_u)\hat{K}_{s-1}(\nabla,u,\partial_u) = \hat{K}_{s-1}(\nabla,u,\partial_u)\hat{\mathcal{W}}(\nabla,u,\partial_u)+\cdots,
\end{align}
with
\begin{align}
\hat{\mathcal{W}}(\nabla,u,\partial_u)=&\hat{\mathcal{P}}(\nabla,u,\partial_u)-(d+2s-4)\nonumber\\
&+\frac{d+2s-5}{d+2s-3}\bigg[ \Big(-\nabla^2 -(s-2)(s+d-3) \Big)+\frac{2}{d+2s-5}(u\cdot \nabla ) (\nabla\cdot \partial_u)\bigg]
\end{align}
Amazingly, one can show that this operator is exactly equal to $\hat{\mathcal{Q}}$ defined in \eqref{Qop}, that is $\hat{\mathcal{W}}(\nabla,u,\partial_u)=\hat{\mathcal{Q}}(\nabla,u,\partial_u)$. So we have found
\begin{align}
S[\sigma_{(s-2)}]=( \hat{K}_{s-1}\sigma_{(s-2)}, \hat{K}_{s-1}\hat{\mathcal{Q}}\sigma_{(s-2)})= (\sigma_{(s-2)}, \hat{K}_{s-1}^\dagger\hat{K}_{s-1}\hat{\mathcal{Q}}\sigma_{(s-2)}).
\end{align}
To conclude, we have
\begin{align}
Y^{(s)}_{\xi'}=&\frac{Y^{(s)}_{\xi^\text{TT}}Y^{(s)}_{\sigma^+}}{ W^{(s)}_{\sigma^+}} \label{Yxi}\\
Y^{(s)}_{\xi^\text{TT}}\equiv&\int\mathcal{D} {\xi'}^{\text{TT}}_{(s-1)}\, e^{-\frac{1}{2\mathrm{g}_s^2} ( {\xi'}_{(s-1)}^\text{TT}, \Big( -\nabla_{(s-1)}^2+m_{s-1,s}^2+M_{s-1}^2 \Big){\xi'}_{(s-1)}^\text{TT} )}\\
Y^{(s)}_{\sigma^+}\equiv &\int \mathcal{D}^+\sigma_{(s-2)}\, e^{-\frac{1}{2\mathrm{g}_s^2}( \sigma_{(s-2)}, K^\dagger K \hat{\mathcal{Q}}\sigma_{(s-2)})}\\
W^{(s)}_{\sigma^+}=&\int \mathcal{D}^+\sigma_{(s-2)}\, e^{-\frac{1}{2\mathrm{g}_s^2}( \sigma_{(s-2)}, K^\dagger K\sigma_{(s-2)})}
\end{align}
Here the superscript $+$ emphasizes the fact that we are integrating over modes orthogonal to the spin-($s-1$) and spin-($s-2$) CKTs. In particular, this is the part of spectrum that coincides with the ``+'' integral in \eqref{phis-2}. Here $(W^{(s)}_{\sigma^+})^{-1}$ is the Jacobian associated with the change of variables \eqref{xiCoV}.
\subsection{Residual group volume}
Recall that after the change of variables \eqref{opsdisplit}, the integration over the pure gauge modes $\xi$ decoupled from the $\phi^\text{TT}_{(s)}$ and $\chi_{(s-2)}$ path integrals, and we are left with a factor (we have restored the label $a$ for degenerate modes with same quantum number $(s-1,s-1)$)
\begin{align}
\frac{\int\mathcal{D}' \xi_{(s-1)}}{\text{Vol($\mathcal{G}_{s}$)}}=\frac{1}{\text{Vol($G_{s}$)}},\quad \text{Vol($G_{s}$)} =\int \prod_{a=1}^{N_{s-1}^\text{KT}}\frac{d\alpha^{(a)}_{s-1,s-1}}{\sqrt{2\pi}\mathrm{g}_s}.
\end{align}
due to the integration over the spin-($s-1$) Killing tensor modes. This leads to a product in the original path integral \eqref{Vasiliev PI}:
\begin{align}\label{HS PI vol}
\text{Vol($G$)}_\text{PI}=\prod_{s} \text{Vol($G_{s}$)}.
\end{align}
HS symmetries typically form infinite dimensional groups. Therefore there is an issue of making sense of \eqref{HS PI vol}, which we are not going to attempt in this paper.
\paragraph{HS invariant bilinear form}
Instead, we are going to do a more modest task. As in the warm-up examples, the volume $\text{Vol($G$)}_\text{PI}$ is defined with a particular metric, namely
\begin{align}\label{hs PI metric}
ds_\text{PI}^2=\frac{1}{2\pi } \sum_{s} \frac{1}{\mathrm{g}_s^2} d\alpha_{s-1,s-1}^2.
\end{align}
Again we want to express this in terms of a canonical metric with respect to which we define a canonical volume $\text{Vol($G$)}_\text{can}$. There are however complications compared to the massless spin-1 and spin-2 cases:
\begin{enumerate}
\item As opposed to the case for Yang-Mills or Einstein gravity, we do not know the full nonlinear actions for Vasiliev theories that give rise to the interacting equations of motion and the full nonlinear gauge transformations in the metric-like formalism. This implies that we do not know the full local HS gauge algebra. Fortunately, the global part of the algebra does not require this knowledge, but only the lowest order ones, which only requires the information of the cubic couplings.
\item Another complication is that since HS symmetries mix different spins, the HS invariant bilinear form depends on the relative normalizations of fields withe different spins in the action. Once this is fixed, the bilinear form is uniquely determined up to an overall normalization.
\end{enumerate}
All of these have been worked out in the case of a negative cosmological constant \cite{Joung:2013nma}. To go to the case of a positive cosmological constant is a simple matter of analytic continuation. In appendix \ref{cubic}, we translate the relevant results from \cite{Joung:2013nma} to the case of $S^{d+1}$. The final result is that upon choosing
\begin{align}
\mathrm{g}_s^2=s!,
\end{align}
the HS invariant bilinear form is determined to be
\begin{align}\label{HS can form}
\bra{\bar{\alpha}_1}\ket{\bar{\alpha}_2}_\text{can} =\frac{8\pi G_N}{\text{Vol}(S^{d-1})}\sum_{s} (d+2s-2)(d+2s-4) \bra{\bar{\alpha}_{1,(s-1)}}\ket{\bar{\alpha}_{2,(s-1)}}_\text{PI}
\end{align}
where the overall normalization is again fixed by requiring the canonical spin-2 generators to be unit-normalized with respect to \eqref{HS can form}. This implies that the group volume \eqref{HS PI vol} is related to the canonical volume as
\begin{align}\label{HS can vol}
\text{Vol($G$)}_\text{PI}=\text{Vol($G$)}_\text{can}\prod_s \left( \frac{\text{Vol}(S^{d-1})}{8\pi G_N}\frac{1}{(d+2s-2)(d+2s-4)}\right)^{\frac{N^\text{KT}_{s-1}}{2}}.
\end{align}
\subsection{Final result}
So far we have
\begin{align}
Z^\text{HS}_\text{PI}=& \frac{i^{-P}}{\text{Vol($G$)}_\text{PI}}\prod_{s}
\Bigg(\frac{Z^{(s)}_{\phi^\text{TT}}}{Y^{(s)}_{\xi^\text{TT}}}\Bigg)\Bigg( \frac{Z^{(s)}_{\chi^+}W^{(s)}_{\sigma^+}}{Y^{(s)}_{\sigma^+}}\Bigg)\Bigg( \frac{Z^{(s)}_{\chi^-}}{ Y^{(s)}_{\xi^\text{CKT}} }\Bigg)
\end{align}
where $P=\sum_s (N^\text{CKT}_{s-2}+N^\text{CKT}_{s-1} -N^\text{KT}_{s-1})$. In the infinite product, the first factor is the usual ratio of determinants of physical and ghost operators
\begin{align}
\frac{Z^{(s)}_{\phi^\text{TT}}}{Y^{(s)}_{\xi^\text{TT}}}=\frac{\det'(-\nabla_{(s-1)}^2+m_{s-1,s}^2+M_{s-1}^2)^{1/2}}{ \det(-\nabla_{(s)}^2+M_s^2)^{1/2}} .
\end{align}
In the second factor, $Z^+_{\chi}, W^+_\sigma, Y^+_{\sigma}$ run over the exact same spectrum and cancel almost completely up to an infinite constant
\begin{align}
\frac{Z^+_{\chi}W^+_\sigma}{Y^+_{\sigma}}=&\int \mathcal{D}\chi^+_{(s-2)}\,e^{-\frac{d+2s-5}{4\mathrm{g}_s}( \chi^+_{(s-2)},\chi^+_{(s-2)})}\nonumber\\
=&\frac{\int \mathcal{D}\chi_{(s-2)}\,e^{-\frac{d+2s-5}{4\mathrm{g}_s}( \chi_{(s-2)},\chi_{(s-2)})}}{\int \mathcal{D}\chi^0_{(s-2)}\mathcal{D}\chi^-_{(s-2)}\,e^{-\frac{d+2s-5}{4\mathrm{g}_s}( \chi_{(s-2)},\chi_{(s-2)})}}
\end{align}
where in the denominator $\chi^0_{(s-2)}$ denotes the modes excluded due to \eqref{eq:gauge}. The infinite constant in the numerator is a path integral over the entire spectrum of an unconstrained spin-$(s-2)$ symmetric traceless field and therefore can be absorbed into bare couplings. To proceed, we plug in explicit mode expansions
\begin{align}
\chi^0_{(s-2)}=&\sum_{m=0}^{s-1} A_{s-1,m} \hat{T}_{s-1, (s-2)}^{(m)},\quad \chi^-_{(s-2)}=&\sum_{m=0}^{s-2} A_{s-2,m} \hat{T}_{s-2, (s-2)}^{(m)},\quad \xi^{CKT}_{(s-1)} =&\sum_{m=0}^{s-2} A_{s-1,m}\hat{T}_{s-1, (s-1)}^{(m)},
\end{align}
which lead to
\begin{align}
\frac{Z^{(s)}_{\chi^+}W^{(s)}_{\sigma^+}}{Y^{(s)}_{\sigma^+}}=& \prod_{m=0}^{s-2} \prod_{n=s-2}^{s-1} \bigg[\frac{ 2}{d+2s-5}\bigg]^{D^{d+2}_{n,m}/2}\\
Z^{(s)}_{\chi^-}=&\prod_{m=0}^{s-2} \Big[\frac{2}{(d+2s-5)m_{s+1,m}^2}\Big]^{\frac{D^{d+2}_{s-2,m}}{2}}\\
Y^{(s)}_{\xi^\text{CKT}}=&\prod_{m=0}^{s-2} \bigg[ \frac{2 m^2_{s,m}}{d+2 s-5}\bigg]^{-\frac{D^{d+2}_{s-1,m}}{2}}.
\end{align}
We therefore have
\begin{align}\label{finite fac}
\Bigg( \frac{Z^{(s)}_{\chi^+}W^{(s)}_{\sigma^+}}{Y^{(s)}_{\sigma^+}}\Bigg)\Bigg( \frac{Z^{(s)}_{\chi^-}}{ Y^{(s)}_{\xi^\text{CKT}} }\Bigg)=&\prod_{m=0}^{s-2}(m_{s+1,m}^2)^{-\frac{D^{d+2}_{s-2,m}}{2}} \prod_{m=0}^{s-2} ( m^2_{s,m})^{\frac{D^{d+2}_{s-1,m}}{2}}.
\end{align}
Together with the determinant factor, this can be further written as
\begin{align}
\Bigg(\frac{Z^{(s)}_{\phi^\text{TT}}}{Y^{(s)}_{\xi^\text{TT}}}\Bigg)\Bigg( \frac{Z^{(s)}_{\chi^+}W^{(s)}_{\sigma^+}}{Y^{(s)}_{\sigma^+}}\Bigg)\Bigg( \frac{Z^{(s)}_{\chi^-}}{ Y^{(s)}_{\xi^\text{CKT}} }\Bigg)
=\frac{\det\nolimits'_{-1}|-\nabla_{(s-1)}^2-\lambda_{s-1,s-1}|^{1/2}}{ \det\nolimits'_{-1}|-\nabla_{(s)}^2-\lambda_{s-2,s}|^{1/2}}.
\end{align}
Here the subscript $-1$ means that we extend the eigenvalue product from $n=s$ to $n=-1$. The primes denote omission of the zero modes from the determinants. In the numerator we omitted the $n=s-1$ mode while in the denominator we omitted the $n=s-2$ mode.\footnote{Originally $\lambda_{n,s}$ and $D^{d+2}_{n,s}$ were defined only for $n\geq s$, which are now extended to all $n\in \mathbb{Z}$.} To obtain this expression we used the relation
\begin{align}\label{eigen PM}
\lambda_{t-1,s}+M_s^2 =-m_{s,t}^2
\end{align}
and the fact that $D^{d+2}_{s-1,t}=-D^{d+2}_{t-1,s}$ (implying $D^{d+2}_{s-1,s}=0$). This extension of the eigenvalue product from $n=s$ to $n=-1$ is exactly the prescription described in \cite{Anninos:2020hfj}. Putting everything together, we finally obtain the expression
\begin{align}\label{HS final}
Z^\text{HS}_\text{PI}=&Z_\text{G}Z_\text{Char}\nonumber\\
Z_\text{G}=&i^{-P} \frac{\gamma^{\text{dim}\,G}}{\text{Vol($G$)}_\text{can}},\qquad Z_\text{Char}=\prod_s Z^{(s)}_\text{Char}\nonumber\\
Z^{(s)}_\text{Char}=& \left( \frac{(d+2s-2)(d+2s-4)}{M^4}\right)^{\frac{N^\text{KT}_{s-1}}{2}}\frac{\det\nolimits'_{-1}\left|\frac{-\nabla_{(s-1)}^2-\lambda_{s-1,s-1}}{M^2}\right|^{1/2}}{ \det\nolimits'_{-1}\left|\frac{-\nabla_{(s)}^2-\lambda_{s-2,s}}{M^2}\right|^{1/2}},
\end{align}
with
\begin{gather}
P=\sum_s (N^\text{CKT}_{s-2}+N^\text{CKT}_{s-1} -N^\text{KT}_{s-1}), \qquad \gamma=\sqrt{\frac{8\pi G_N}{\text{Vol}(S^{d-1})}},\qquad \text{dim}\,G =\sum_s N^\text{KT}_{s-1}
\end{gather}
Note that we have restored the dimensionful parameter $M$. As noted in \cite{Anninos:2020hfj}, the factor $(d+2s-2)(d+2s-4)$ gets nicely canceled after evaluating the character integrals for the determinants.
\section{Massive fields}\label{Massive PI}
Now let us turn to fields with generic masses. In this case we do not have a group volume factor, and thus no coupling dependence. We will work with canonical normalizations.
\subsection{Massive scalars and vectors}
\paragraph{Massive scalars}
The path integral for a scalar $\phi$ with mass $m^2>0$ is simply
\begin{align}
Z^{(s=0,m^2)}_\text{PI}=\int \mathcal{D}\phi\, e^{-\frac{1}{2}\int_{S^{d+1}}\phi (-\nabla^2+m^2)\phi}=\det(-\nabla^2+m^2)^{-1/2}
\end{align}
\paragraph{Massive vectors}
Massive vectors are described by the Proca action
\begin{align}\label{Proca action}
S[A]=&\int_{S^{d+1}} \Big(\frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{m^2}{2} A_\mu A^\mu\Big).
\end{align}
Similar to the massless case, to proceed we make a change of variables \eqref{spin 1 CoV} with Jacobian \eqref{vector Jac}, so that the action becomes
\begin{gather}
S[A]=S[A^T]+S[\chi]\nonumber\\
S[A^T]=\frac{1}{2}(A^T, (-\nabla_{(1)}^2+m^2+d)A^T),\quad S[\chi] =\frac{m^2}{2}(\chi,(-\nabla_{(0)}^2)\chi).
\end{gather}
For $m^2>0$ that corresponds to unitary de Sitter representations, the result is
\begin{align}
Z^{(s=1,m^2)}_\text{PI}=\det(-\nabla_{(1)}^2+m^2+d)^{-1/2} (m^2)^{1/2}=\det\nolimits_{-1}(-\nabla_{(1)}^2+m^2+d)^{-1/2}.
\end{align}
The presence of the factor $(m^2)^{1/2}$ originates from the fact that the $(0,0)$ mode is excluded from the integration over the longitudinal mode. In the last equality we again note that the multiplication of the factor $(m^2)^{1/2}$ is equivalent to extending the product to $n=-1$.
\subsection{Massive spin 2 and beyond}
\subsubsection{Massive $s=2$}
The action for a free massive spin-2 field on $S^{d+1}$ is (see for example \cite{Higuchi:1986py})
\begin{align}\label{massivespin2action}
S[h] = &\frac{1}{2}\int_{S^{d+1}}h^{\mu\nu}\bigg[ (-\nabla^2+2)h_{\mu\nu}+2\nabla_{(\mu}\nabla^\lambda h_{\nu) \lambda}+g_{\mu\nu}(\nabla^2 h\indices{_\lambda^\lambda}-2\nabla^{\sigma}\nabla^\lambda h_{\sigma \lambda})+(d-2)g_{\mu\nu}h\indices{_\lambda^\lambda}\nonumber\\
&+m^2(h_{\mu\nu}-g_{\mu\nu}h\indices{_\lambda^\lambda})\bigg].
\end{align}
If we put $m=0$ we recover the action \eqref{eq:spin2action} (with $\mathrm{g}=1$) for linearized gravity. To proceed, we again change the variables \eqref{spin 2 CoV}. It is convenient to further decompose $\xi_\mu$ into its transverse and longitudinal parts: $\xi_\mu=\xi_\mu^T+\nabla_\mu \sigma$, so that the full decomposition for $h_{\mu\nu}$ is
\begin{align}\label{spin 2 full CoV}
h_{\mu\nu}=&h_{\mu\nu}^\text{TT} +\frac{1}{\sqrt{2}} (\nabla_{\mu} \xi^T_{\nu}+\nabla_{\nu} \xi^T_{\mu}) +\sqrt{2}\nabla_\mu\nabla_\nu \sigma +\frac{g_{\mu\nu}}{\sqrt{d+1}} \tilde{h} .
\end{align}
For this decomposition to be unique, we impose
\begin{align}\label{new xi con}
(\xi^T,f_{1,(1)})=0\quad , \quad (\tilde{h},f_1)=0\quad \text{and} \quad (\sigma,f_0)=0.
\end{align}
The first two constraints are equivalent to \eqref{xi con} and \eqref{tilde h con} while the last one ensures $\nabla_\mu \sigma\neq 0$. With a slight modification of the steps in section \ref{spin 2 Jacobian}, the Jacobian for the \eqref{spin 2 full CoV} is obtained as
\begin{align}
\begin{split}\label{spin 2 full Jac}
\mathcal{D}h =& J \mathcal{D}h^\text{TT}\mathcal{D}'\xi^T\mathcal{D}^+\sigma \mathcal{D}'\tilde{h}\\
J=&\frac{1}{Y^\text{T}_\xi Y^+_\sigma Y^\text{CKV}_\xi} \\
Y^\text{T}_\xi=&\int\mathcal{D}' \xi^T \,e^{-\frac{1}{2} (\xi^T,(-\nabla^2_{(1)}-d)\xi^T)} \\
Y^+_\sigma =&\int\mathcal{D}^+\sigma \,e^{-\frac{1}{2}\frac{2d}{d+1}( \sigma,(-\nabla_{(0)}^2)(-\nabla_{(0)}^2-(d+1))\sigma)}\\
Y^0_\sigma=&\int\mathcal{D}^0 \sigma\, e^{-(\sigma,(-\nabla_{(0)}^2)(-\nabla_{(0)}^2-d) \sigma)}=\int\mathcal{D}^0 \sigma \,e^{-(d+1)(\sigma, \sigma)}.
\end{split}
\end{align}
Here $\mathcal{D}^+\sigma$ ($\mathcal{D}^0\sigma$) involves integrations over only the positive (zero) modes for the operator $(-\nabla_{(0)}^2-(d+1))$. After substituting \eqref{spin 2 full CoV} the action decouples into
\begin{align}
S[h]=S[h^{TT}]+S[\xi^{T}]+S[\sigma,\tilde{h}].
\end{align}
The quadratic actions for $h^\text{TT}$ and $\xi^T$ are simply
\begin{align}\label{mass TT action}
S[h^\text{TT}]=\frac{1}{2}\int_{S^{d+1}}h^\text{TT}_{\mu\nu}(-\nabla_{(2)}^2+m^2+2)h_\text{TT}^{\mu\nu},
\end{align}
and
\begin{align}\label{mass transverse action}
S[\xi^T]=\frac{m^2}{2} ( \xi^T , (-\nabla_{(1)}^2 -d)\xi^T)
\end{align}
respectively. Since $\sigma$ and $\tilde{h}$ are not orthogonal, they mix in the action
\begin{align}\label{spin 2 scalar action}
S[\sigma,\tilde{h}] =& -\frac{(d-1)d}{2(d+1)}( \tilde{h}, (-\nabla_{(0)}^2+(d+1)(\frac{ m^2}{d-1}-1))\tilde{h} ) -\sqrt{\frac{2}{d+1}} d m^2 ( \nabla_{(0)}^2 \sigma , \tilde{h}) \nonumber \\
&+\frac{m^2}{2} ( \nabla \sigma , (-\nabla_{(0)}^2 -d)\nabla \sigma) -\frac{m^2}{2} ( \nabla_{(0)}^2 \sigma,\nabla_{(0)}^2 \sigma ).
\end{align}
To diagonalize $S[\sigma,\tilde{h}]$, we make a shift (with a trivial Jacobian)\footnote{Because of the constraints \eqref{new xi con}, the $(0,0)$ and $(1,0)$ modes do not mix in \eqref{spin 2 scalar action}}
\begin{align}\label{spin 2 shift}
\sigma'=\sigma-\frac{1}{\sqrt{2(d+1)}}\tilde{h}
\end{align}
for all scalar modes $f_n$ with $n\geq 2$, so that $S[\sigma,\tilde{h}] =S[\sigma',\tilde{h}] =S[\sigma'] +S[\tilde{h}]$, with
\begin{align}\label{spin 2 de scalar action}
S[\sigma'] =-d m^2 ( \sigma',-\nabla_{(0)}^2 \sigma' )\quad \text{and} \quad S[\tilde{h}]=\frac{d(m^2-(d-1))}{2(d+1)}( \tilde{h},(-\nabla_{(0)}^2-(d+1)) \tilde{h} ).
\end{align}
Notice that $S[\sigma']$ and $S[\tilde{h}]$ vanishes identically when $m^2 =0$ and $m^2 =d-1$ respectively. These are the cases when we have gauge symmetries. The massless case has already been discussed in section \ref{massless spin2}. The case of $m^2 =d-1$ will be considered in section \ref{PM fields}.
Depending on the precise value of $m^2>-2(d+2)$,\footnote{This is the range where the kinetic operator in \eqref{mass TT action} is positive definite. The case $m^2<-2(d+2)$ will be considered when we discuss the shift-symmetric spin-2 fields in section \ref{shift sym}.} some of the modes in \eqref{mass transverse action} and \eqref{spin 2 de scalar action} might acquire an overall negative sign. We Wick rotate the negative modes, absorbing local infinite constants into bare couplings. This will induce a phase factor. Below we give a summary for different cases (n.m. stands for negative modes):
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{ |c|c|c|c|c| }
\hline
Range of $m^2$ & n.m. in $S[\xi^T]$ & n.m. in $S[\sigma']$ &n.m. in $S[\tilde{h}]$ & Phase \\
\hline
$-2(d+2)<m^2<0$ &$f_{n,\mu}, n\geq 1$ & None & $f_n, n\geq 2$ & $i^{-D^{d+2}_{1,1}-D^{d+2}_{1,0}}=i^{-\frac{(d+3)(d+2)}{2}}$\\
\hline
$0<m^2<d-1$ & None & $f_n,n\geq 1$ & $f_n,n\geq 2$ & $i^{-2D^{d+2}_{0,0}-D^{d+2}_{1,0}}=i^{-d-4}$\\
\hline
$m^2>d-1$ & None & $f_n,n\geq 1$ & $f_0$ & $i^{0}=1$\\
\hline
\end{tabular}
\end{center}
The last case ($m^2>d-1$) is precisely the case when the corresponding de Sitter representations are unitary.\footnote{Principal series for $m^2>(\frac{d}{2})^2$ and complementary series for $d-1<m^2<(\frac{d}{2})^2$ \cite{Basile:2016aen}.} We will focus on this case from now on.
Putting everything together, we have
\begin{align}
Z^{(s=2,m^2)}_\text{PI}=Z^\text{TT}_h\bigg(\frac{Z^\text{T}_\xi}{Y^\text{T}_\xi}\bigg)\bigg(\frac{Z^+_{\sigma'}Z^+_{\tilde{h}}}{ Y^+_\sigma}\bigg)\bigg(\frac{Z^0_{\sigma'}}{Y^0_\sigma}\bigg)Z^-_{\tilde{h}}
\end{align}
Here $Z^\text{TT}_h,Z^\text{T}_\xi,Z^\pm_{\sigma'},Z^0_{\sigma'},Z^\pm_{\tilde{h}}$ are the path integrals with actions \eqref{mass TT action}, \eqref{mass transverse action} and \eqref{spin 2 de scalar action}. The labels $\pm$ and 0 denote the positive (negative) and zero modes for the scalar operator $-\nabla_{(0)}^2-(d+1)$. Every factor can be easily evaluated:
\begin{align}
\begin{split}
Z^\text{TT}_h =& \det (-\nabla_{(2)}^2+m^2+2)^{-1/2} \\
\frac{Z^\text{T}_\xi}{Y^\text{T}_\xi} =& \int\mathcal{D}' \xi^T\, e^{-\frac{m^2}{2} (\xi^T,\xi^T)}\\
\frac{Z^+_{\sigma'}Z^+_{\tilde{h}}}{ Y^+_\sigma}=&\int\mathcal{D}^+\sigma' \,e^{-\frac{m^2}{2} ( \sigma', \sigma' )}\int\mathcal{D}^+\tilde{h}\,e^{-\frac{d(m^2-(d-1))}{2}( \tilde{h}, \tilde{h} )}\\
\frac{Z^0_{\sigma'}}{Y^0_\sigma}=&\int\mathcal{D}^0\sigma' \,e^{-\frac{d m^2}{2} ( \sigma', \sigma' )}\\
Z^-_{\tilde{h}}=&\int\mathcal{D}^-\tilde{h} \,e^{-\frac{d(m^2-(d-1))}{2}( \tilde{h}, \tilde{h} )}.
\end{split}
\end{align}
Observe that all factors but $Z^\text{TT}_h$ can be combined in the following way:
\begin{align}
\bigg(\frac{Z^\text{T}_\xi}{Y^\text{T}_\xi}\bigg)\bigg(\frac{Z^+_{\sigma'}Z^+_{\tilde{h}}}{ Y^+_\sigma}\bigg)\bigg(\frac{Z^0_{\sigma'}}{Y^\text{CKV}_\xi}\bigg)Z^-_{\tilde{h}}=&\frac{\int\mathcal{D} \xi \,e^{-\frac{m^2}{2} (\xi,\xi)}\int\mathcal{D}\tilde{h} \,e^{-\frac{d(m^2-(d-1))}{2}( \tilde{h}, \tilde{h} )}}{\int\mathcal{D}^0\sigma'\, e^{-\frac{ (m^2-(d-1))}{2} ( \sigma', \sigma' )}\int\mathcal{D}^0 \xi^T \,e^{-\frac{m^2}{2} (\xi^T,\xi^T)}}.
\end{align}
In the numerator, the path integrations are over local unconstrained fields and thus can be absorbed into bare couplings. In the denominator $\mathcal{D}^0 \xi^T$ denotes integration over the modes $f_{1,\mu}$. The integrals in the denominator can be easily evaluated. To conclude, we have
\begin{align}
Z^{(s=2,m^2)}_\text{PI}=&\det (-\nabla_{(2)}^2+m^2+M_2^2)^{-1/2}(m^2-m_{2,0}^2)^{\frac{D^{d+2}_{1,0}}{2}}(m^2-m_{2,1}^2)^{\frac{D^{d+2}_{1,1}}{2}}\nonumber\\
=&\det\nolimits_{-1} (-\nabla_{(2)}^2+m^2+M_2^2)^{-1/2}
\end{align}
where we recall that $m_{s,t}^2$ is defined in \eqref{PM mass}.
\subsubsection{Massive arbitrary spin $s\geq 1$}
In principle, one starts with the full manifestly local and covariant action \cite{Zinoviev:2001dt}, which involves a tower of spin $t< s$ Stueckelberg fields, and repeat the derivation above. However, having worked out the cases for $s=1,2$, the pattern is clear. For a free massive spin-$s$ field, its path integral is simply
\begin{align}\label{massive general}
Z^{(s,m^2)}_\text{PI}=\det\nolimits_{-1} \left(\frac{-\nabla_{(s)}^2+m^2+M_s^2}{M^2}\right)^{-1/2}.
\end{align}
Note that we have restored the dimensionful parameter $M$. Recall that the scaling dimension $\Delta$ is related to the mass $m^2$ as
\begin{align}\label{mass dim}
m^2 =(\Delta+s-2)(d+s-2-\Delta)
\end{align}
so that
\begin{align}
\lambda_{n,s}+m^2+M_s^2=(n+\Delta)(d+n-\Delta)=\left(n+\frac{d}{2}\right)^2-\left(\Delta-\frac{d}{2}\right)^2.
\end{align}
The requirement that $\lambda_{n,s}+m^2+M_s^2$ is positive for all $n\geq -1$ is equivalent to the unitary bounds on $\Delta$ \cite{Basile:2016aen}:
\begin{align}
\Delta=\frac{d}{2}+i \nu,\nu\in\mathbb{R} \quad \text{(Principal series)} \quad \text{or} \quad 1<\Delta <d-1\quad \text{(Complementary series)}
\end{align}
Outside of this bound, a finite number of $\lambda_{n,s}+m^2+M_s^2$ will become negative, which leads to the presence of some power of $i$, as we have seen in the $s=2$ case. Also, as we take $m^2 \to m_{s,t}^2$, \eqref{massive general} becomes ill-defined, signaling a gauge symmetry. The case of $t=s-1$ is the massless case discussed in section \ref{Massless PI}. We will comment on the general $(s,t)$ case in section \ref{PM fields}.
\section{Shift-symmetric fields}\label{shift sym}
In (A)dS space, when massive fields attain certain mass values, they can have shift symmetries \cite{Bonifacio:2018zex} that generalize the shift symmetry, galileon symmetry, and special galileon symmetry of massless scalars in flat space. In AdS, these theories are unitary; in dS, these theories do not fall into the classifications of dS UIRs \cite{Basile:2016aen}.\footnote{However, there are arguments (see e.g. \cite{Bonifacio:2018zex}) that they can be cured to become unitary.} In the following we study their 1-loop (free) path integrals on $S^{d+1}$, which contain analogous subtleties as the massless case, namely the phases and group volumes.
\subsection{Shift-symmetric scalars}
Let us start with a free scalar $\phi$ with generic mass $m$, with action
\begin{align}\label{massive s0 action}
S[\phi]=\frac{1}{2}\int_{S^{d+1}}\phi (-\nabla^2+m^2)\phi.
\end{align}
When $m^2$ takes values of the negative the eigenvalues of the scalar Laplacian $-\nabla_{(0)}^2$, i.e.
\begin{align}
m^2=-\lambda_{k,0}=-k(k+d)=m_{0,k+1}^2+M_0^2=m_{k+2,1}^2=-m_{2,k+1}^2\leq 0,\quad k\geq 0,
\end{align}
(recall that $m_{s,t}^2$ is defined in \eqref{PM mass}), the action is invariant under a shift symmetry (of level $k$ in the terminology of \cite{Bonifacio:2018zex})
\begin{align}
\delta \phi = f_k
\end{align}
where $f_k$ is the $(k,0)$ eigenmodes of $-\nabla^2_{(0)}$ with eigenvalue $\lambda_k$.
\subsubsection{$k=0$: massless scalars}
The simplest example is $k=0$, i.e. a massless scalar \cite{Allen:1985ux,Allen:1987tz}:
\begin{align}
S[\phi]=\frac{1}{2}\int_{S^{d+1}}\phi (-\nabla^2)\phi
\end{align}
which is invariant under a constant shift $\phi \to \phi +c$. The case with $d=3$ is of particularly interest because of its relevance in inflation. The path integral is simply
\begin{align}
Z^{(s=0,m^2=0)}_\text{PI} =\text{Vol}(G)_\text{PI}\det\nolimits'\left(-\nabla_{(0)}^2\right)^{-1/2}\, .
\end{align}
Here the prime denotes the omission of the constant $(0,0)$ mode from the functional determinant. The group volume factor is the integral over the constant mode
\begin{align}
\text{Vol}(G)_\text{PI}\equiv \int \frac{dA_{0,0}}{\sqrt{2\pi}}.
\end{align}
Unlike the case of massless gauge fields, the residual group volume is multiplying the determinant instead of being divided. As for massless gauge fields, $\text{Vol}(G)_\text{PI}$ depends on the non-linear completion of the theory. There will be a problem of relating $\text{Vol}(G)_\text{PI}$ to a canonical volume $\text{Vol}(G)_\text{can}$ and the determination of $\text{Vol}(G)_\text{can}$ itself. Also, we expect there will be a dependence on coupling constants of the interacting theory.
An example for which we can make sense of these issues is that of a compact scalar. They are scalars subject to the identification
\begin{align}
\phi \sim \phi +2\pi R\,,
\end{align}
that is, they take values on a circle of radius $R$. In this case the integration range for the (0,0) mode is restricted to the fundamental domain $0<A_{0,0}<2\pi R \sqrt{\text{Vol}(S^{d+1})}$ and therefore
\begin{align}
Z^{\text{compact scalar}}_\text{PI}=\sqrt{2\pi R^2\text{Vol}(S^{d+1})}\det\nolimits'(-\nabla_{(0)}^2)^{-1/2}.
\end{align}
Here the (inverse of) radius $R$ plays the role of the coupling constant.
\subsubsection{$k\geq 1$: tachyonic scalars}
For any $k\geq 1$, the scalar is tachyonic. See for example \cite{Bros:2010wa} and \cite{Epstein:2014jaa} for the study of such tachyonic scalars. The $k=1$ and $k=2$ cases are the dS analogs for the Galileon and special Galileon theories in flat space \cite{Bonifacio:2018zex} respectively. Note that the action is negative for all $(n,0)$ modes with $n<k$, and vanishes for the $(k,0)$ modes. To make sense of the path integral, we again perform Wick rotations for all $(n,0)$ modes with $n<k$, so that
\begin{align}
\int \mathcal{D}^{<k}\phi \, e^{-S_{<k}[\phi]} \to i^{\sum_{n=0}^{k-1}D^{d+2}_{n,0}} \int \mathcal{D}^{<k}\phi\, e^{S_{<k}[\phi]}=i^{\sum_{n=0}^{k-1}D^{d+2}_{n,0}}\prod_{n=0}^{k-1}|\lambda_{n,0}-\lambda_{k,0}|^{-1/2},
\end{align}
and interpret the integration over the $(k,0)$ modes as a residual group volume
\begin{align}
\text{Vol}(G_k)_\text{PI}\equiv \int \prod_{a=1}^{D^{d+2}_{k,0}}\frac{dA^{(a)}_{k,0}}{\sqrt{2\pi}}.
\end{align}
The modes with $n>k$ can be integrated as usual. The final result is
\begin{align}
Z^{(s=0,m_{k+2,1}^2)}_\text{PI}=i^{\sum_{n=0}^{k-1}D^{d+2}_{n,0}}\text{Vol}(G_k)_\text{PI}\det\nolimits'|-\nabla_{(0)}^2-\lambda_{k,0}|^{-1/2}\,.
\end{align}
Note that absolute value is taken in the determinant. The prime denotes the omission of the $(k,0)$ modes from the functional determinant. Same as the $k=0$ case, the residual group volume $\text{Vol}(G_k)_\text{PI}$ is multiplying the determinant instead of being divided. Again, $\text{Vol}(G_k)_\text{PI}$ should depend on the interaction structure of the parent theory. There will be a problem of relating $\text{Vol}(G_k)_\text{PI}$ to a canonical volume $\text{Vol}(G_k)_\text{can}$ and the determination of $\text{Vol}(G_k)_\text{can}$ itself.
\subsection{Shift-symmetric vectors}
When the mass takes values
\begin{align}
m^2=-\lambda_{k+1,1}-d=-(k+2)(k+d)=m_{k+3,0}^2=-m_{1,k+2}^2\leq -2d,\quad k\geq 0,
\end{align}
the Proca action \eqref{Proca action} is invariant under a level-$k$ shift symmetry generated by the $(k+1,1)$ modes
\begin{align}
\delta A_{\mu} = f_{k+1,\mu}.
\end{align}
Following analogous steps as for the scalars, it is straightforward to work out the path integral
\begin{align}
Z^{(s=1,m_{k+3,0}^2)}_\text{PI}=i^{\sum_{n=-1}^{k}D^{d+2}_{n,1}}\text{Vol}(G_{k+1,1})_\text{PI}\det'\nolimits_{-1}|-\nabla_{(1)}^2-\lambda_{k+1,1}|^{-1/2}
\end{align}
where the prime denotes the omission of the $(k+1,1)$ modes and
\begin{align}
\text{Vol}(G_{k+1,1})_\text{PI}\equiv\int\prod_{a=1}^{D^{d+2}_{k+1,1}} \frac{dA^{(a)}_{k+1,1}}{\sqrt{2\pi}}.
\end{align}
Note that in the phase factor we have used the fact that $D^{d+2}_{-1,1}=-D^{d+2}_{0,0}$ and $D^{d+2}_{0,1}=0$.
\subsection{Shift-symmetric spin $s\geq 2$}
\subsubsection{Shift-symmetric spin 2 fields}
The massive spin-2 action \eqref{massivespin2action} with
\begin{align}
m^2=-\lambda_{k+2,2}-2=m_{k+4,1}^2=-m_{2,k+3}^2\leq 2(d+2),\quad k\geq 0,
\end{align}
is invariant under a level-$k$ shift symmetry generated by the $(k+2,2)$ modes
\begin{align}
\delta h_{\mu\nu} = f_{k+2,\mu\nu}.
\end{align}
It is straightforward to work out the path integral
\begin{align}
Z^{(s=2,m_{k+4,1}^2)}_\text{PI}=i^{\sum_{n=-1}^{k+1}D^{d+2}_{n,2}}\text{Vol}(G_{k+2,2})_\text{PI}\det'\nolimits_{-1}|-\nabla_{(2)}^2-\lambda_{k+2,2}|^{-1/2}
\end{align}
where the prime denotes the omission of the $(k+2,2)$ modes and
\begin{align}
\text{Vol}(G_{k+2,2})_\text{PI}\equiv\int\prod_{a=1}^{D^{d+2}_{k+2,2}} \frac{dA^{(a)}_{k+2,2}}{\sqrt{2\pi}}.
\end{align}
Note that in the phase factor we have used the fact that $D^{d+2}_{-1,2}=-D^{d+2}_{1,0}$ and $D^{d+2}_{0,2}=-D^{d+2}_{1,1}$.
\subsubsection{Shift-symmetric arbitrary spins $s\geq 0$}
Now the pattern is clear. When the mass for a spin-$s$ field $\phi_{(s)}$ ($s\geq 0$) reaches the values
\begin{align}
m^2=-\lambda_{k+s,s}^2-M_s^2=m_{k+s+2,s-1}^2=-m_{s,s+k+1}^2,\quad k\geq 0,
\end{align}
there will be a level-$k$ shift symmetry generated by the $(k+s,s)$ modes
\begin{align}
\delta \phi_{(s)} = f_{k+s,(s)}.
\end{align}
The path integral is
\begin{align}\label{shift general}
Z^{(s,m_{k+s+2,s-1}^2)}_\text{PI}=i^{\sum_{n=-1}^{k+s-1}D^{d+2}_{n,s}}\text{Vol}(G_{k+s,s})_\text{PI}\det'\nolimits_{-1}\left|\frac{-\nabla_{(s)}^2-\lambda_{k+s,s}}{M^2}\right|^{-1/2}
\end{align}
where
\begin{align}
\text{Vol}(G_{k+s,s})_\text{PI}\equiv\int\prod_{a=1}^{D^{d+2}_{k+s,s}} \frac{M}{\sqrt{2\pi}}dA^{(a)}_{k+s,s}.
\end{align}
Note that we have restored the dimensionful parameter $M$. Such a shift-symmetric field can be thought of as the longitudinal mode decoupled from a massive spin-$(k+s+1)$ field as its mass approaches $m_{k+s+1,s}^2$. Note that for $k=0$, it can be thought of as the ghost part of the spin-$(s+1)$ massless path integral. We will see more connections of shift-symmetric fields with general partially massless fields in the next section.
\section{Partially massless fields}\label{PM fields}
In (A)dS space, there exist ``partial massless'' (PM) representations \cite{Higuchi:1986py,Zinoviev:2001dt,Deser:1983tm,DESER1984396,Brink:2000ag,Deser:2001pe,Deser:2001us, Deser:2001wx,Deser:2001xr, Skvortsov:2006at, Hinterbichler:2016fgl}. Except for the massless case, they are not unitary in AdS. In $dS_{d+1}$ with $d\geq 4$, they correspond to the unitary exceptional series representations, while for $d=3$ they correspond to the discrete series representations \cite{Basile:2016aen}. A PM spin-$s$ field of depth $t$ has a gauge symmetry\footnote{We adopt the convention that depth $t$ is equal to the spin of the gauge parameter.}
\begin{align}
\delta \phi_{(s)}=\nabla^{(s-t)}\xi_{(t)}+\cdots
\end{align}
where $\cdots$ stand for terms with fewer derivatives \cite{Hinterbichler:2016fgl}. The massless case corresponds to $t=s-1$. In the following we first work out the case of spin-2 depth-0 field. Then we will provide a general prescription for general PM fields.
\subsection{Spin-2 depth-0 field}
The action for a spin-2 depth-0 field is \eqref{massivespin2action} with mass
\begin{align}
m^2=m_{2,0}^2=d-1,
\end{align}
in which case there is a gauge symmetry
\begin{align}
\delta h_{\mu\nu}=\nabla_\mu\nabla_\nu \chi+g_{\mu\nu}\chi.
\end{align}
This can be seen by first substituting \eqref{spin 2 shift} into \eqref{spin 2 full CoV} so that the decomposition becomes
\begin{align}
h_{\mu\nu}=&h_{\mu\nu}^\text{TT} +\frac{1}{\sqrt{2}} (\nabla_{\mu} \xi^T_{\nu}+\nabla_{\nu} \xi^T_{\mu}) +\sqrt{2}\nabla_\mu\nabla_\nu \sigma' +\frac{1}{\sqrt{d+1}} \left(\nabla_\mu\nabla_\nu\tilde{h}+g_{\mu\nu}\tilde{h}\right)
\end{align}
and noting that $S[\tilde{h}]$ defined in \eqref{spin 2 de scalar action} vanishes identically for $m^2=d-1$. Spin-2 field with such a mass was first considered in \cite{Higuchi:1986py}. This gauge invariance implies that there is an integration
\begin{align}
\int\mathcal{D}'\tilde{h}
\end{align}
that must be canceled by a gauge group volume factor $\text{Vol}(\mathcal{G})$ divided by hand. To be consistent with locality, this gauge group factor must take the form of a path integral of a local scalar field $\alpha$
\begin{align}
\text{Vol}(\mathcal{G}) =\int \mathcal{D}\alpha.
\end{align}
Due to mismatch of modes excluded due to \eqref{new xi con}, we have a residual group volume
\begin{align}
\frac{ \int\mathcal{D}'\tilde{h}}{\text{Vol}(\mathcal{G})}=\frac{1}{\text{Vol}(G_{1,0})_\text{PI}},\quad \text{Vol}(G_{1,0})_\text{PI} \equiv \int\prod_{a=1}^{D^{d+2}_{1,0}} \frac{dA^{(a)}_{1,0}}{\sqrt{2\pi}}.
\end{align}
The rest of the computation proceeds as before, and the final result is
\begin{align}
Z^{(s=2,m_{2,0}^2)}_\text{PI}=\frac{i^{-1}}{\text{Vol}(G_{1,0})_\text{PI}}\frac{\det'_{-1} |-\nabla_{(0)}^2-(d+1)|^{1/2}}{\det\nolimits'_{-1} \left(-\nabla_{(2)}^2+d+1\right)^{1/2}}.
\end{align}
\subsection{General PM fields}
We now provide a prescription to obtain the path integral expression for a general spin-$s$ depth-$t$ field. First, take the spin-$s$ path integral \eqref{massive general} with generic mass and take the limit $m^2\to m_{s,t}^2$ while omitting the $(t-1,s)$ modes:
\begin{align}
Z^{(s,m^2\to m_{s,t}^2)}_\text{PI}\to i^{\sum_{m=-1}^{t-2}D^{d+2}_{m,s}}\det\nolimits'_{-1} |-\nabla_{(s)}^2-\lambda_{t-1,s}^2|^{-1/2}
\end{align}
where we have used \eqref{eigen PM}. The phases appear because the mode with $n=-1,0,\cdots, t-2$ becomes negative. Then we exchange $s$ and $t$ and flip $i\to -i$ to obtain another expression
\begin{align}
Z^{(t,m^2\to m_{t,s}^2)}_\text{PI}\to i^{-\sum_{m=-1}^{s-2}D^{d+2}_{m,t}}\det\nolimits'_{-1} |-\nabla_{(t)}^2-\lambda_{s-1,t}^2|^{-1/2}.
\end{align}
We propose that the final result is simply given by the ratio between these two expressions, divided by a group volume factor:
\begin{align}\label{general s t}
Z^{(s,m^2= m_{s,t}^2)}_\text{PI}=\frac{i^{\sum_{m=-1}^{t-2}D^{d+2}_{m,s}+\sum_{m=-1}^{s-2}D^{d+2}_{m,t}}}{\text{Vol}(G_{s-1,t})_\text{PI}}\frac{\det\nolimits'_{-1} \left|\frac{-\nabla_{(t)}^2-\lambda_{s-1,t}^2}{M^2}\right|^{1/2}}{\det\nolimits'_{-1} \left|\frac{-\nabla_{(s)}^2-\lambda_{t-1,s}^2}{M^2}\right|^{1/2}}
\end{align}
where
\begin{align}
\text{Vol}(G_{s-1,t})_\text{PI} \equiv \int\prod_{a=1}^{D^{d+2}_{s-1,t}} \frac{M^2}{\sqrt{2\pi}} dA_{s-1,t}
\end{align}
Note that we have restored the dimensionful parameter $M$. One can easily verify that \eqref{general s t} reduces to the massless case when $t=s-1$ and the spin-2 depth-0 case when $s=2,t=0$. The division by $Z^{(t,m^2\to m_{t,s}^2)}_\text{PI}$ can be thought of as the decoupling of the spin-$t$ level-$(s-1-t)$ shift-symmetric field from the massive spin-$s$ field as we take $m^2\to m_{s,t}^2$. Note that the ratio of determinants (without the extension to $n=-1$ modes) in \eqref{general s t} and the relations between PM and conformal higher spin partition functions were first discussed in \cite{Tseytlin:2013jya} for $S^4$ and \cite{Tseytlin:2013fca} for $S^6$.
As we stressed repeatedly, the determination of the group volume factor $\text{Vol}(G_{s-1,t})_\text{PI}$ requires knowledge of the interactions of the parent theory. In the current case, a natural class of parent theories would be the PM generalizations of higher spin theories \cite{Brust:2016zns}, which include a tower of PM gauge fields and a finite number of massive fields. These theories gauge the PM algebras studied in \cite{Joung:2015jza} and are holographic duals to $\Box^k$ CFTs \cite{Brust:2016gjy}. Their 1-loop path integrals would take the form
\begin{align}\label{PM HS PI}
Z^\text{PM HS}_\text{PI}=\frac{i^P}{\text{Vol}(G)_\text{PI}}\prod_{s,t}\frac{\det\nolimits'_{-1} \left|\frac{-\nabla_{(t)}^2-\lambda_{s-1,t}^2}{M^2}\right|^{1/2}}{\det\nolimits'_{-1} \left|\frac{-\nabla_{(s)}^2-\lambda_{t-1,s}^2}{M^2}\right|^{1/2}}
\end{align}
where
\begin{align}
P=\sum_{s,t}\left(\sum_{m=-1}^{t-2}D^{d+2}_{m,s}+\sum_{m=-1}^{s-2}D^{d+2}_{m,t}\right),\quad \text{Vol}(G)_\text{PI}=\prod_{s,t} \text{Vol}(G_{s-1,t})_\text{PI}
\end{align}
There will be analogous problem of relating $\text{Vol}(G)_\text{PI}$ to a canonical volume $\text{Vol}(G)_\text{can}$ (and making sense of the volume itself) as in the massless case, which will give us the dependence on the Newton's constant $G_N$. If we demand $\log Z^\text{PM HS}_\text{PI}$ to be consistent with a universal form as in the massless case \cite{Anninos:2020hfj}, we should take
\begin{align}\label{PM HS blinear}
\text{Vol($G$)}_\text{PI}=\text{Vol($G$)}_\text{can}\prod_{s,t} \left( \frac{\text{Vol}(S^{d-1})}{8\pi G_N}\frac{M^4}{(d+2s-2)(d+2t-2)}\right)^{\frac{D^{d+2}_{s-1,t}}{2}}
\end{align}
so that the factor $(d+2s-2)(d+2t-2)$ gets nicely canceled upon evaluating the character integrals for the determinants. To verify this, one has to repeat the analysis of \cite{Joung:2013nma} and appendix \ref{cubic} and express the PM HS invariant bilinear form in terms of the bilinear form induced by the path integral measure. Provided that \eqref{PM HS blinear} is valid, we note that except the phase and $\text{Vol($G$)}_\text{can}$, the expression \eqref{PM HS PI} becomes the inverse of itself upon exchanging $s$ and $t$. We leave the validation of \eqref{general s t}, \eqref{PM HS blinear} and the implication of the suggestive $s\leftrightarrow t$ symmetry for future work.
\section{Discussion and outlooks}\label{conclusion}
In this work, we derive the determinant expressions of the 1-loop path integrals for massive, shift-symmetric and (partially) massless symmetric tensor fields on $S^{d+1}$.
At the technical level, we have clarified subtleties arising from normalizable zero modes or negative modes on the sphere, and have made broad generalizations of all known results. One natural generalization of this work is to study path integrals involving fermionic massive, shift-symmetric and (partially) massless gauge fields. For instance, the free actions for massless fermionic fields are presented in \cite{Fang:1978wz}. Since fermionic fields are Grassman-valued, no Wick rotation is needed to make the path integral convergent. However, there is still a group volume factor corresponding to trivial fermionic gauge transformations, whose physical interpretations are more obscure than their bosonic counterparts, because the Grassman integrals are formally zero. Perhaps we need to combine bosonic and fermionic higher spin fields into a supersymmetric HS theory \cite{Sezgin:2012ag} so that we can make sense of the super-higher-spin group volume.
At the conceptual level, our results and their implications on de Sitter thermodynamics were discussed extensively in \cite{Anninos:2020hfj}. However, the interpretations of a number of features of 1-loop sphere partition functions require further investigations: First is the Polchinski's phase. While we generalize the original massless spin-2 result to other classes of fields, their physical interpretations remain elusive. One is tempted to say perhaps these phases indicate non-unitarity. While this seems to be natural for massive fields with masses outside the unitary bounds (including shift-symmetric fields), the phases are present for PM fields which are perfectly unitary irreducible representations. Without other physical inputs, it is not clear whether we should ignore or retain these phases. However, we stress that these phases deserve our attentions. One reason is that perhaps a better understanding of these phases will lead us to the correct statistical interpretation of the Euclidean path integral.\footnote{For example, one might guess that these $i$'s are precisely the $i$'s present in the inverse Laplace transform to extract microcanonical entropies from the partition function.} Another reason is that $S^{d+1}$ is only one of the many saddle points of the Euclidean gravitational path integral with a positive cosmological constant. If one considers other saddle points such as $S^2\times S^{d-1}$, since they have different amount of symmetries, after Wick rotating the conformal modes there will be \textbf{relative} phases between different saddle points.
The second remaining mystery is the residual group volume factor present for PM gauge fields and shift-symmetric fields. Such a factor is present for a manifestly local path integral and depends on the non-linear completion of the theory. Higher spin groups are typically infinite-dimensional and there is an issue of making sense of the group volume. The group volume may be more well-defined in theories gauging finite dimensional higher spin algebras studied in \cite{Joung:2015jza}.
The context in which both subtleties of phases and group volume are sharpest is in $d+1=3$ dimension \cite{Anninos:2020hfj}. In this case one can check that for any PM fields, the determinants for the on-shell kinetic operator and the ghost operator cancel completely, so that the group volume and phases are the \textbf{only} non-trivial contributions to the 1-loop path integral. Also, on $S^3$ there is an alternative formulation of massless HS gravity as a $SU(N)\times SU(N)$ Chern-Simons theory. As noted in \cite{Anninos:2020hfj}, one finds that their 1-loop results agree only if we identify the residual group volume with the $SU(N)\times SU(N)$ HS group volume, further supporting the claim that this factor depends on the interactions of the full theory. Also, the phases will match exactly for odd framing.
Finally, as mentioned at the end of the introduction, it would be interesting to apply our results to other contexts such as such as string theory, Chern-Simons theory, supersymmetry, AdS/CFT correspondence, conformal field theory, as well as entanglement entropy in quantum field theories. We leave these to future work.
\section*{Acknowledgments}
I am grateful to Dionysios Anninos, Frederik Denef and Zimo Sun for numerous stimulating discussions, at different stages of this research project. I also thank Manvir Grewal and Klaas Parmentier for their precious comments. Finally, I appreciate the reviewer whose suggestions helped improve and clarify this manuscript. AL was supported in part by the U.S. Department of Energy grant de-sc0011941.
|
2,869,038,154,886 | arxiv | \section{Introduction}
\IEEEPARstart{S}{ensitivity} factors play an important role in power systems operations and control. In transmission systems, linear sensitivity distribution factors are traditionally utilized in power systems analysis -- for e.g., contingency analysis, generation re-dispatch, and security assessment \cite{PS_DSE_2019}, just to mention a few. Injection shift factors~\cite{SensitivityPS_1968,Chen_2014} as well as power transfer distribution factors (PTDFs) allow grid operators to estimate line flows in real-time in response to changes in the (net) power injections.
Computation of these sensitivities typically relies on either \emph{model-based} or \emph{measurement-based} approaches. As an example of a model-based method, injection shift factors and the PTDF matrix for transmission systems are typically computed by leveraging the DC approximation \cite{Sauer_1981}; similarly, voltage sensitivities can be computed via linear approximations of the AC power flow equations (see, e.g.,~\cite{bolognani2015fast}). In both cases, model-based approaches require an accurate knowledge of the network topology (including line impedances), and are not dependent on specific operating points of the network \cite{Sauer_1981}. Measurement-based methods leverage data obtained from phase measurement units (PMUs) or Supervisory Control and Data Acquisition (SCADA) systems, to obtain estimates of the sensitivity matrix using, e.g., a least-squares approach or alternative estimation criteria. See, for example, the method proposed in \cite{Chen_2014} to compute sensitivity distribution factors. Notably, measurement-based methods do not require a knowledge of the topology and impedances, and they do not rely on pseudo-measurements obtained via power flow solutions.
Approaches based on the least-squares estimation criterion are effective only if one can collect measurements of the net power injections that are ``sufficiently rich''; that is, measurements that lead to a regression matrix that has full column rank \cite{Chen_2014, refA3, refA1}. In principle, the regression matrix may have full column rank when the perturbations of the net power injections can be properly designed by the grid operator; for example, by adopting the probing techniques of~\cite{bhela2019smart} at some of the nodes or all the nodes. However, an under-determined system may emerge when (i) perturbations may not be performed at a sufficient number of nodes (and, thus, the variations of powers are simply due to uncontrollable devices); (ii) changes in the power of uncontrollable loads and generation units located throughout the network may lead to correlated measurements~\cite{abdullah2013probabilistic}; and, (iii) when the power network is operating under dynamic conditions due to fluctuations introduced by intermittent renewable generation and uncontrollable loads~\cite{DhopleNoFuel}, the operator may not have time to collect enough measurements before the operating point of the network changes (and, thus, the sensitivities change). On the other hand, approaches based on the least-square method as \cite{Chen_2013, refA3, refA1, refA5} do not consider the case where the collected measurements have outliers that might lead to unreliable estimates.
To address these challenges, this paper proposes a robust nuclear norm minimization method~\cite{candes2010matrix,Mateos_sparse_2013} to estimate sensitivities from measurements, along with an online algorithm to solve the nuclear norm minimization method with streaming measurements. The proposed approach is
motivated by our observation that certain classes of sensitivity matrices can afford a low-rank approximation. For example:
\noindent $\bullet$ Figure \ref{fig:Intro_SVD} shows the singular values of the PTDF matrices for three different transmission networks; it can be seen that the PTDF matrix can be approximated by a low-rank matrix where only the dominant singular values are retained.
\noindent $\bullet$ Figure \ref{fig:Intro_SVD_2} shows the singular values of the sensitivity matrix for the voltage magnitudes with respect to the net injected powers for two different distribution networks; in particular, a real feeder from California was utilized (the feeder has 126 multi-phase nodes, with a total of 366 single-phase
points of connection, as described in \cite{Bernstein2018unified}).
Relative to existing methods based on the least-squares approach, the proposed method have the following contributions: (C1) obtains meaningful estimates of the sensitivity matrices with a smaller number of measurements and when the regression model is underdetermined (this is particularly important in time-varying conditions and in case of switches in the topology or switchgear); (C2) by leveraging sparsity-promoting regularization functions, the proposed estimator can identify faulty measurements; and, (C3) the low-rank approach can handle missing data and measurements collected at different rates. The proposed approach can be used to estimate various sensitivity coefficients in a power grid; for example, sensitivity injection shift factors~\cite{SensitivityPS_1968,Chen_2014,GISF_2017} in transmission systems.
To adapt to power networks increasingly operating under dynamic conditions (and, hence, having sensitivity matrices that change rapidly over time), the development of real-time algorithms that can estimate the sensitivity matrix on-the-fly from streaming measurements is presented in this paper. In particular, the paper proposes an online proximal-gradient method~\cite{Online-opti} to solve the nuclear norm minimization problem based on measurements collected from PMUs and SCADA systems at the second or sub-second level. In par with the broad literature on online optimization, convergence results in terms of dynamic regret~\cite{Hall15,Jadbabaie15} are offered in this case. We point out that the proposed algorithm is markedly different from the competing alternative~\cite{akhriev2018pursuit}, and relies on an online proximal-gradient method.
Lastly, it is also worth recognizing related works such as \cite{GISF_2017}, where the AC equations are perturbed in order to derive a closed-form expression of so-called ``generalized" injection shift factors. An approach to estimate dynamic distribution factors is introduced in \cite{DDF_2019}, where reduced-order models are used to derive dynamic injection shift factors and generator participation factors. An example of online convex optimization in power systems in presented in \cite{dataDrivenLR}, for the specific application of estimating load changes in the network.
\begin{figure}[t!]
\centering
\includegraphics[width=3.0in]{Fig_intro_R1_1.eps}
\vspace{-.4cm}
\caption{Singular values (ordered in decreasing order) of the power transfer distribution factors matrix for three different transmission networks.}
\vspace{-.3cm}
\label{fig:Intro_SVD}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=3.0in]{Fig_intro_R1_2.eps}
\vspace{-.4cm}
\caption{Singular values (ordered in decreasing order) of power-voltage magnitude sensitivity matrix for two different multi-phase distribution networks.}
\vspace{-.3cm}
\label{fig:Intro_SVD_2}
\end{figure}
The remainder of the paper is organized as follows. Section II describes the system model and the existing methods used to calculate the PTDF matrix. Section III presents the proposed low-rank approach. The proposed data-driven online estimation method is presented in Section IV. Test cases for transmission and distribution are provided in Section V. Finally, Section VI concludes the paper.
\section{Preliminaries}
As mentioned in the previous section, the proposed approach can be leveraged to estimate various sensitivity coefficients in a power grid. These include, for example, sensitivity injection shift factors~\cite{SensitivityPS_1968,Chen_2014} in transmission systems, and voltage sensitivities (with respect to power injections) in distribution networks~\cite{Rigoni18}. In the following, to clearly and concretely explain the proposed approach, we tailor the exposition to the estimation of the power transfer distribution factors matrix in transmission systems\footnote{\textit{Notation:} Upper-case (lower-case) boldface letters will be used for matrices (column vectors), and $(\cdot)^\top$ denotes transposition. For a given column vector $\mathbf{x} \in \mathbb{R}^n$, $\norm{\mathbf{x}} := \sqrt{\mathbf{x}^\top\mathbf{x}}$, and $\norm{\mathbf{x}}_1 := \sum_{i=1}^n | x_i |$. Given a matrix $\mathbf{X} \in \mathbb{R}^{n \times l}$, $\text{vec}(\mathbf{X}) \in \mathbb{R}^p$ denotes the column vectorized $\mathbf{X}$ with its columns stacked in order on top of one another and $p := nl$, $\norm{\mathbf{X}}_* := \sum_{i=1}^r \sigma_i (\mathbf{X})$, where $r$ is the rank of $\mathbf{X}$, and $\sigma_i$ represented the singular values of $\mathbf{X}$. A vector of ones is represented by $\mathbf{1}$ with the corresponding dimensions, and a vector of zeros is represented by $\mathbf{0}$ with the corresponding dimensions. $\mathcal{O}$ refers to the big O notation; that is, given two positive sequences $\{a_k\}_{k = 0}^\infty$ and $\{b_k\}_{k = 0}^\infty$, we say that $a_k = \mathcal{O}(b_k)$ is $\limsup_{k \rightarrow \infty} (a_k/b_k) < \infty$.}.
\subsection{System Model} \label{System_model}
Let $\mathcal{N}:= \{1, \dots, n\}$ be the set of nodes where generators and/or loads are located, and let $\mathcal{L}:= \{1, \dots,l\}$ be the set of transmission lines (or branches). Towards this, let $\Delta p_j \in \mathbb{R}$ represent a change in the net active power injection at node $j \in \mathcal{N}$, around a given point $p_j$; then, the vector capturing the change in the active power flow on the lines in response to the change of power $\Delta p_j$ can be approximated as $\mathbf{h}_j \Delta p_j$, where $\mathbf{h}_j \in \mathbb{R}^{l}$ represent the sensitivity coefficients~\cite{SensitivityPS_1968,Chen_2014}.
Discretize the temporal axis as $\{t_k = k T, k \in \mathbb{N}\}$, with $T$ as a given time interval. Let $\Delta \mathbf{p}_k := [\Delta p_{1 k}, \Delta p_{2 k}, \dots, \Delta p_{n k}]^\top$ be the vector of net active power changes collected at time instant $t_k$ at the $n$ nodes, and define the sensitivity matrix as $\mathbf{H}_k := [ \mathbf{h}_{1 k} \; \mathbf{h}_{2 k} \; \dots \; \mathbf{h}_{n k} ] \in \mathbb{R}^{l \times n}$. Then, the vector $\Delta \mathbf{f}_k \in \mathbb{R}^l$ representing the change in the power flow on the lines in the network due to $\Delta \mathbf{p}_k$ can be expressed by~\cite{SensitivityPS_1968,Chen_2014}
\begin{equation}
\Delta \mathbf{f}_k = \mathbf{H}_k \Delta \mathbf{p}_k,
\label{Vec-Ma_GSF}
\end{equation}
\noindent
where the entry $i,j$ of $\mathbf{H}_k$ represents the sensitivity injection shift factors~\cite{Chen_2014}. Overall, $\mathbf{H}_k $ can be thought as a proxy for the Jacobian of the map $\mathbf{f} = \mathcal{F}(\mathbf{p})$, which yields flows as a function of power injections, calculated at a given point.
By considering $m$ measurements\footnote{Here, we consider measurements taken at times $t_{k-m+1}, \ldots, t_{k}$ for exposition simplicity; however, one may use measurements collected at irregular intervals.} we can define the matrices $ \Delta \mathbf{F}_k = [\Delta \mathbf{f}_{k - m+1} \; \dots \; \Delta \mathbf{f}_k] \in \mathbb{R}^{l \times m}$, and $ \Delta \mathbf{P}_k = [\Delta \mathbf{p}_{k - m+1} \; \dots \; \Delta \mathbf{p}_{k}] \in \mathbb{R}^{n \times m}$. Then, the following linear system of equations can be written as
\begin{equation}
\Delta \mathbf{F}_k = \mathbf{H}_k \Delta \mathbf{P}_k .
\label{Matrix_GSF}
\end{equation}
\noindent
Based on~\eqref{Matrix_GSF}, the following subsection will review existing approaches based on the least-squares method as well as model-based approaches.
\vspace{.1cm}
\subsection{Existing Methods} \label{Existing_models}
\subsubsection{Least-squares estimation}
Assuming that $\Delta \mathbf{P}_k$ is known and measurements (or pseudo-measurements) of $\Delta \mathbf{F}_k$ are available, one possible way to estimate $\mathbf{H}_k$ is via a least-squares criterion. For example, a method similar to \cite{Chen_2014} can be used, where the injection shift factors for a branch were estimated using PMU measurements obtained in (near) real-time. In particular, borrowing the approach of~\cite{Chen_2014}, $\mathbf{H}_k$ can be obtained at time $t_k$ by solving:
\begin{equation}
\mathbf{H}_{LS,k} \in \arg \min_{\mathbf{H} \in \mathcal{H}} \norm{\Delta \mathbf{F}_k - \mathbf{H} \Delta \mathbf{P}_k}^2_F,
\label{ISF_LSE}
\end{equation}
where $\norm{\cdot}_F$ denotes the Frobenious norm, and $\mathcal{H}$ is a compact set ensuring that each entry $(i,j)$ of the matrix $\mathbf{H}$ satisfies the constraint $h_{min} \leq [\mathbf{H}]_{ij} \leq h_{max}$; that is, $\mathcal{H} = [h_{min}, h_{max}]^{l n}$ (in this case, $h_{min} = -1$ and $h_{max} = 1$). Alternatively, a weighted least-squares method can be utilized when the noise affecting $\Delta \mathbf{F}_k$ is colored or it is not identically distributed across lines. Notice that in~\eqref{ISF_LSE} there are $lm$ measurements and $ln$ unknowns. With this in mind, existing works such as~\cite{Chen_2014} generally assume that $m \geq n$ and that, the matrix $\Delta \mathbf{P}_k$ has full column rank; with these assumptions, one avoids an underdetermined system and, furthermore,~\eqref{ISF_LSE} has a unique solution. In principle, the matrix $\Delta \mathbf{P}_k$ can have full column rank when the perturbations $\{\Delta p_{j k}\}$ can be properly designed by the grid operator~\cite{bhela2019smart}, or when nodes are perturbed in a round-robin fashion. However, this is impractical in a realistic setting (if not infeasible), because the grid operator may not have access to controllable devices at each node of the network; moreover, changes in the power of uncontrollable loads and generation units located throughout the network contribute to $\{\Delta p_{j k}\}$, and this
may lead to correlated measurements
(therefore, system \eqref{Matrix_GSF} becomes underdetermined). In an underdetermined setting, only a minimum-norm solution would be available using a least-squares criterion, which may provide inaccurate estimates of $\mathbf{H}$ (as corroborated in the numerical results in Section~\ref{Simu}).
Before presenting the proposed method, we briefly mention a model-based approach.
\subsubsection{Model-based method}
For transmission systems a widely-used model-based approach to calculate the linear sensitivity distribution factors is based on the DC approximation \cite{Sauer_1981}. In particular, by letting $\mathbf{B} \in \mathbb{R}^{n \times n}$ represent the matrix of line series susceptances of the transmission system, one can calculate the changes in the phase angles $\Delta {\utwi{\theta}}$ by using the following relation:
\begin{equation}
\Delta \mathbf{p}_k = \mathbf{B} \Delta {\utwi{\theta}}_k.
\label{DC_flow_delta}
\end{equation}
Define $\mathbf{X} = \text{diag}(\{ -x_{ab} \}) \in \mathbb{R}^{l \times l}$, where $x_{ab}$ represents the line reactance between node $a \text{ and } b \in \mathbb{N}$, and let $\mathbf{A} \in \mathbb{R}^{l \times n}$ be the branch-bus incidence matrix. Then, using the DC power flow formulation, $\mathbf{f}_k$ can be expressed as the linear relation $\mathbf{f}_k = \mathbf{X}^{-1} \mathbf{A} {\utwi{\theta}}_k$~\cite{Wood_Wollwnberg}. If we want to express the active power flow perturbation $\Delta \mathbf{f}_k$ due to a change in the phase angles $\Delta {\utwi{\theta}}_k$, we can write $\Delta \mathbf{f}_k$ as $\Delta \mathbf{f}_k = \mathbf{X}^{-1} \mathbf{A} \Delta {\utwi{\theta}}_k$. By replacing this equation and \eqref{DC_flow_delta} in \eqref{Vec-Ma_GSF} we obtain the model-based relation for the sensitivity matrix as: $\mathbf{H} = \mathbf{X}^{-1} \mathbf{A} \mathbf{B}^{-1}$.
In order to guarantee that the inverse of $\mathbf{B}$ exists, we require that the DC power flow equations for the nodal power balances are linearly independent. Then, taking the node $1$ as the slack bus, and denoting as $\mathbf{B}_r\in \mathbb{R}^{(n-1) \times (n-1)}$ and $\mathbf{A}_r \in \mathbb{R}^{l \times (n-1)}$ the reduced matrices, the final sensitivity matrix is given by $\mathbf{H} = [ \mathbf{0} \quad \mathbf{X}^{-1} \mathbf{A}_r \mathbf{B}_r^{-1} ]$.
With the DC formulation, the sensitivity matrix factors depends only on the topology of the network, and are invariant to changes in the system operation point, such as line outages, load and generation perturbations, etc.
In the following, a low-rank method will be presented, which does not require knowledge of network topology or reactances.
\section{Low-Rank Approach}
\label{sec:lowrank}
In this section, we present an approach for the estimation of the matrix $\mathbf{H}$ with the following features: i) it leverages measurements of $\Delta \mathbf{F}_k$ and $\Delta \mathbf{P}_k$ obtained from phasor measurement units (PMUs), supervisory control and data acquisition (SCADA), or other similar sources, rather than relying on a network model; ii) it allows for obtaining meaningful estimates of $\mathbf{H}$ even when~\eqref{Matrix_GSF} is underdetermined by leveraging a low-rank approximation of $\mathbf{H}$; iii) when $\Delta \mathbf{P}_k$ is full column rank, it yields an estimation accuracy similar to the least-squares estimator and, iv) can handle missing measurements of flows on some lines (i.e., some entries of $\Delta \mathbf{F}_k$ may be missing).
For simplicity of exposition, we first consider the case where measurements are error-free (then, we consider noisy measurements as well as measurement outliers). Based on the model~\eqref{Matrix_GSF}, the nearly low-rank property of $\mathbf{H}$ motivates us to consider the following affine rank minimization problem (RMP) \cite{Recht_2010}:
\begin{subequations}
\label{eq:rank_min}
\begin{align}
\underset{\mathbf{H} \in \mathcal{H}}{\min} & \quad
\text{rank}(\mathbf{H}) \\
\text{s.t.} ~& \text{vec}(\Delta \mathbf{F}_k) = \mathcal{A}(\mathbf{H}),
\end{align}
\end{subequations}
\noindent where $\mathcal{H}$ is the convex compact set (in the simplest case, the Cartesian product of box constraints), $\text{vec}(\Delta \mathbf{F}_k) \in \mathbb{R}^p$ where $p:= lm$, denotes the vectorized $\Delta \mathbf{F}_k$, and the linear map $ \mathcal{A} : \mathbb{R}^{l \times n} \rightarrow \mathbb{R}^p $ is defined as: $\mathcal{A}(\mathbf{H}) = \textbf{A}_{\mathbf{P},k} \, \text{vec}(\mathbf{H})$, where $\text{vec}(\mathbf{H}) \in \mathbb{R}^d$, $d := ln$, and $\textbf{A}_{\mathbf{P},k}$ is a matrix of dimensions $p \times d$, appropriately built using the perturbations $\Delta \mathbf{P}_k$. Specifically, matrix $\mathbf{A}_{\mathbf{P},k}$ is the Kronecker product defined by $\mathbf{A}_{\mathbf{P},k} := \Delta \mathbf{P}_k^\top \otimes \mathbf{I}$, where $\mathbf{I}$ is the identity matrix of dimensions $l \times l$.
Unfortunately, the rank criterion in~\eqref{eq:rank_min} is in general NP-hard to optimize; nevertheless, drawing an analogy from compressed sensing to rank minimization, the following convex relaxation of the RMP~\eqref{eq:rank_min} can be utilized \cite{Recht_2010}:
\begin{subequations}
\label{eq:nuclear_norm_min}
\begin{align}
\underset{\mathbf{H}\in \mathcal{H}}{\min} & \; \norm{\mathbf{H}}_* \\
\text{s.t.~} & \text{vec}(\Delta \mathbf{F}_k) = \textbf{A}_{\mathbf{P},k} \, \text{vec}(\mathbf{H}),
\end{align}
\end{subequations}
where $\norm{\mathbf{H}}_* := \sum_i \sigma_i(\mathbf{H})$ is the nuclear norm of $\mathbf{H}$, with $\sigma_i(\mathbf{H})$ denoting the $i$th singular value of $\mathbf{H}$. Interestingly, it was shown in \cite{Recht_2010} that, if the constraints of \eqref{eq:nuclear_norm_min} are defined by a linear transformation that satisfies a restricted isometry property condition, the minimum rank solution can be recovered by the minimization of the nuclear norm over the linear space; see the necessary and sufficient condition in \cite{Recht_2010}.
The proposed methodology leverages the relaxation~\eqref{eq:nuclear_norm_min} to estimate the matrix $\mathbf{H}$ from measurements of $\Delta \mathbf{F}_k$ induced by the perturbations in the net power injections $\Delta \mathbf{P}_k$. Assuming that the measurements $\Delta \mathbf{F}_k$ are affected by a zero-mean Gaussian noise (instead of being noise-free), a pertinent relaxation of~\eqref{eq:nuclear_norm_min} amounts to the following convex program~\cite{candes2010matrix}:
\begin{equation}
\underset{\mathbf{H}\in \mathcal{H}}{\min}
\norm{\text{vec}(\Delta \mathbf{F}_k) - \textbf{A}_{\mathbf{P},k} \, \text{vec}(\mathbf{H})}^2_2 + \lambda \norm{\mathbf{H}}_*,
\label{noise_nuclear}
\end{equation}
\noindent
where $\lambda > 0$ is a given regularization parameter that is used to promote sparsity in the singular values of $\mathbf{H}$ (and, hence, to obtain a low-rank matrix $\mathbf{H}$).
\subsection{Robustness to outliers}
We further consider the case where some measurements of $\Delta \mathbf{F}_k$ may be corrupted by outliers. This can be due to, for example, faulty readings of PMUs and micro PMUs, communication errors, or malicious attacks. To this end, we augment the model~\eqref{Matrix_GSF} as $\Delta \mathbf{F}_k = \mathbf{H}_k \Delta {\mathbf{P}_k} + \mathbf{O}_k + \mathbf{E}_k$, where $\mathbf{E}_k$ is a matrix containing (small) measurement errors and $\mathbf{O}_k$ is a matrix containing measurement outliers~\cite{zhou2010stable,Mateos_sparse_2013}. When no outliers are present, $\mathbf{O}_k$ is a matrix with all zeros. Based on this augmented model, estimates of $\mathbf{H}_k$ and $\mathbf{O}_k$ can be sought by solving the following convex problem~\cite{zhou2010stable,Mateos_sparse_2013}:
\begin{multline}
\underset{\mathbf{H} \in \mathcal{H}, \mathbf{O} \in \mathcal{M}}{\min} \,
\norm{\text{vec}(\Delta \mathbf{F}_k) - \textbf{A}_{\mathbf{P},k} \, \text{vec}(\mathbf{H}) - \text{vec}(\mathbf{O})}^2_2 \\ + \lambda \norm{ \mathbf{H}}_* + \gamma \norm{\text{vec}(\mathbf{O})}_1,
\label{eq:noise_nuclear_outliers}
\end{multline}
where $\norm{\text{vec}(\mathbf{O})}_1 = \sum_{i} |[\text{vec}(\mathbf{O})]_i|$ is the $\ell_1$-norm of the vector $\text{vec}(\mathbf{O})$, $\gamma > 0$ is a sparsity-promoting coefficient, and $\mathcal{M}$ are box constraints of the form $o_{min} \leq [\mathbf{O}_{k}]_{ij} \leq o_{max}$. Notice that the $\ell_1$-norm is the closest convex surrogates to the cardinality function. Once \eqref{eq:noise_nuclear_outliers} is solved, the locations of nonzero entries in $\mathbf{O}$ reveal outliers across both lines and time; on the other hand,
the amplitudes quantify the magnitude of the anomalous measurement. It is important to notice that the parameters $\lambda$ and $\gamma$ control the tradeoff between fitting error, rank of $\mathbf{H}$,
and sparsity level of $\mathbf{O}$; in particular, when an estimate of the
variance of the measurement noise is available, one can follow guidelines for selection of $\lambda$ and $\gamma$ similar to the ones proposed in~\cite{zhou2010stable}.
\subsection{Missing measurements and measurements at different rates}
It is worth pointing out that the proposed methodology is applicable to the case where some measurements in $\Delta \mathbf{F}_k$ in~\eqref{noise_nuclear} are \emph{missing}. This may be due to communication failures or because measurements are collected at different rates. For the latter, one can take the highest measurement frequency (i.e., the $T$ smallest inter-arrival time) as a reference frame, and treat measurements that are received less frequently (i.e., with a larger inter-arrival time) as missing entries.
In this case, missing measurements are discarded from the least-squares term in~\eqref{eq:noise_nuclear_outliers}~\cite{Mateos_sparse_2013}. In particular, let $\Omega_k \subseteq \{1, 2, \ldots, p\}$ be a set indicating which measurements in the vector $\text{vec}(\Delta \mathbf{F}_k)$ are available at time $t_k$; for example, if the measurement for the line $1$ is missing at time $t_{k-m+1}$ and $t_{k-m+2}$, then $\Omega_k \subseteq \{2, 3, \ldots, l, l+2, \ldots, p\}$. Let $\mathcal{P}_{\Omega_k}$ be a time-varying vector sampling operator, which sets the entries of its vector argument not indexed by $\Omega_k$ to zero and leaves the other entries unchanged. Then,~\eqref{eq:noise_nuclear_outliers} can be reformulated as:
\begin{multline}
\underset{\mathbf{H} \in \mathcal{H}, \mathbf{O} \in \mathcal{M}}{\min} \,
\norm{\mathcal{P}_{\Omega_k}\left\{\text{vec}(\Delta \mathbf{F}_k) - \textbf{A}_{\mathbf{P},k} \, \text{vec}(\mathbf{H}) - \text{vec}(\mathbf{O}) \right\}}^2_2 \\ + \lambda \norm{ \mathbf{H}}_* + \gamma \norm{\text{vec}(\mathbf{O})}_1,
\label{eq:noise_nuclear_outliers_2}
\end{multline}
where, of course, missing measurements are not accounted for in the least-squares term.
\section{Data-Driven Online Estimation} \label{Online}
Based on $\Delta \mathbf{P}_k$ and $\Delta \mathbf{F}_k$, which collect measurements of new power injections and power flows acquired at time steps $k-m+1, \ldots, k$, an estimate of $\mathbf{H}_k$ can be obtained by solving the convex problem~\eqref{eq:noise_nuclear_outliers} using existing batch solvers for non-smooth convex optimization problems. When the power network is operating under dynamic conditions, for example, due to swings in the net power due to intermittent renewable generation and uncontrollable loads~\cite{DhopleNoFuel}, the sensitivity matrix $\mathbf{H}_k$ may rapidly change over time (since, in general, it depends on the current operating points~\cite{Chen_2014}); in these dynamic conditions, it may not be possible to solve~\eqref{eq:noise_nuclear_outliers} sufficiently fast due to underlying computational complexity considerations, and a solution of~\eqref{eq:noise_nuclear_outliers} generated by batch solvers can be outdated. That is, by the time the solution is produced, the operating conditions of the network (and, hence, $\mathbf{H}_k$) have changed.
This aspect motivates the development of an online algorithm that estimates $\mathbf{H}_k$ based on streams of measurements and identifies outliers ``on the fly,'' as explained in this section.
Measurements are assumed to arrive at times $\{t_k = k T, k \in \mathbb{N}\}$, with $T$ the inter-arrival time (e.g, $T$ could be one second or a few seconds~\cite{DhopleNoFuel}); suppose further that measurements are processed over a sliding window $\mathcal{T}_k = \{t_{k-m+1}, \ldots, t_{k}\}$. Then, at each instant $t_k$, the matrix $\mathbf{H}_k$ can be estimated via~\eqref{eq:noise_nuclear_outliers}, which is re-written here as the following time-varying problem~\cite{Online-opti}:
\begin{subequations}
\label{eq:timevarying_opt}
\begin{equation}
(\mathbf{H}_k^*, \mathbf{O}_k^*) \in \arg \underset{\mathbf{H}_k \in \mathbb{R}^{l \times n}, \mathbf{O}_k \in \mathbb{R}^{l \times m}} {\text{min}} \; f_k(\mathbf{H}_k, \mathbf{O}_k) , \hspace{.4cm} \forall \, k T
\end{equation}
where $f_k(\mathbf{H}_k, \mathbf{O}_k) := s_k(\mathbf{H}_k, \mathbf{O}_k)
+ g_k(\mathbf{H}_k, \mathbf{O}_k)$,
\begin{equation}
s_k(\mathbf{H}_k, \mathbf{O}_k) := \norm{{\Delta\mathbf{F}}_k - \mathbf{H}_k {\Delta \mathbf{P}}_k - \mathbf{O}_k}_F^2,
\end{equation}
\begin{align}
g_k(\mathbf{H}_k, \mathbf{O}_k) : = & \lambda_k \norm{\mathbf{H}_k}_* + \gamma_k \norm{\text{vec}(\mathbf{O}_k)}_1 \nonumber \\
& + \iota_{\mathcal{H}}(\mathbf{H}_k) + \iota_{\mathcal{M}}(\mathbf{O}_k),
\end{align}
\end{subequations}
with $\iota_{\mathcal{H}}(\mathbf{H})$ the set indicator function for the compact set $\mathcal{H}$ and
$\iota_{\mathcal{M}}(\mathbf{O})$ the set indicator function for the compact set $\mathcal{M}$. The goal posed here is to develop an online algorithm that can track a solution $\{\mathbf{H}_k^*, \mathbf{O}_k^*\}_{k \in \mathbb{N}}$ and the trajectory of optimal value functions $\{f_k^* := f_k(\mathbf{H}_k^*, \mathbf{O}_k^*)\}_{k \in \mathbb{N}}$ by processing measurements in a sliding window fashion. In the following, let $\mathbf{o}_k = \text{vec}(\mathbf{O}_k)$, and $\mathbf{x}_k = [\text{vec}(\mathbf{H}_k)^\top, \mathbf{o}_k^\top]^\top \in \mathcal{X}_k : = \mathcal{H} \times \mathcal{M}$ for brevity. Notice that $s_k(\mathbf{x}_k)$ is closed, convex and proper, with a $L_k$-Lipschitz continuous gradient at each time $t_k$; on the other hand, $g_k(\mathbf{x}_k)$ is a lower semi-continuous proper convex function. Lastly, the function attains a finite minimum at a certain $\mathbf{x}_k^*$. Given this particular structure of~\eqref{eq:timevarying_opt}, we propose to use an online proximal-gradient algorithm~\cite{Online-opti} to solve~\eqref{eq:timevarying_opt} under streams of measurements. Assuming that, because of communication delays and computational considerations, one step of the algorithm can be performed within an interval $T$ (which coincides with the inter-arrival rate of the measurements)\footnote{We stress that it may be possible to perform multiple proximal-gradient steps within an interval $T$, but we consider the case of one step to simplify the notation.}, the online proximal-gradient algorithm amounts to the sequential execution of the following step:
\begin{subequations}
\label{prox_update}
\begin{align}
\mathbf{y}_k & = \mathbf{x}_{k-1} - \alpha \nabla_\mathbf{x} s_k(\mathbf{x}_{k-1}) \label{prox_update_y}, \\
\mathbf{x}_{k} & = \text{prox}_{g_k,\mathcal{X}}^{\alpha} \{ \mathbf{y}_k \},\label{prox_update_x}
\end{align}
\end{subequations}
where $\alpha > 0 $ is the step size, and
the proximal operator is defined over the non-differentiable function $g_k$ as \cite{Beck_prox}
\begin{equation}
\text{prox}_{g}^{\alpha} \{ \mathbf{y} \} := \underset{\mathbf{x}} {\text{arg min}} \Big \{g(\mathbf{x}) + \frac{1}{2 \alpha} \norm{\mathbf{x} - \mathbf{y}}^2 \Big \} .
\end{equation}
Notice that, if we re-write the function $s_k$ as
\begin{equation}
s_k(\mathbf{x}_k) = \norm{\Delta \mathbf{f}_k - \mathbf{A_{Ps}}_{,k} \mathbf{x}_k}^2,
\end{equation}
where $\Delta \mathbf{f}_k = \text{vec}(\Delta \mathbf{F}_k)$, $\mathbf{A_{Ps}}_{,k} = [\mathbf{A_{P}}_{,k}, \mathbf{I}]$, and $\mathbf{x}_k$ defined as before, then, $\nabla_\mathbf{x} s_k$ is given by
\begin{equation}
\nabla_\mathbf{x} s_k(\mathbf{x}_k) = 2 \mathbf{A_{Ps}}_{,k}^\top \left( \mathbf{A_{Ps}}_{,k
} \mathbf{x}_k - \Delta \mathbf{f}_k \right).
\label{closed_nabla_s}
\end{equation}
Furthermore, one can notice that the proximal operator in~\eqref{prox_update_x} is separable across the two variables of interest $\mathbf{H}_k$ and $\mathbf{o}_k$, which can therefore be computed separately. In particular, one has that:
\begin{align}
\mathbf{H}_{k} & = \text{prox}_{\lambda_k \norm{\cdot}_* + \iota_{\mathcal{H}} }\{\mathbf{Y}_{H,k}\}\label{prox_update_H}, \\
\mathbf{o}_{k} & = \text{prox}_{\gamma_k \norm{\cdot}_1 + \iota_{\mathcal{M}} }\{\mathbf{y}_{o,k}\}\label{prox_update_o},
\end{align}
with $\mathbf{Y}_{H,k}$ and $\mathbf{y}_{o,k}$ extracted from the stacked vector $\mathbf{y}_k$ in~\eqref{prox_update_y}. Moreover,~\eqref{prox_update_o} admits a closed-form solution, which is given by:
\begin{equation}
\mathbf{o}_{k} = [\mathcal{S}_\gamma (\mathbf{y}_{o,k})]_{o_{min}}^{o_{max}},
\label{prox_norm1}
\end{equation}
where $[x]_a^b = \max\{\min\{x,b\},a\}$, and the thresholding operator $\mathcal{S}_\gamma$ is defined as:
\begin{align}
\mathcal{S}_\gamma (\mathbf{y}) & = \max\{|\mathbf{y}| - \gamma \mathbf{1}, \mathbf{0}\} \odot \text{sgn}(\mathbf{y}) \nonumber \\
& = \Bigg\{ \begin{matrix} \mathbf{y} - \gamma \mathbf{1},
& \textrm{~if~} \mathbf{y} \geq \gamma \mathbf{1}, \\
0, & \textrm{\,~~if~} |\mathbf{y}|< \gamma \mathbf{1}, \\ \mathbf{y} + \gamma \mathbf{1}, & \textrm{~~~if~} \mathbf{y} \leq -\gamma \mathbf{1}. \end{matrix}
\end{align}
With the previous definitions in place, the online proximal-gradient algorithm for the robust estimation of the sensitivity matrix is tabulated as Algorithm \ref{algo}.
\begin{algorithm}
\caption{Online robust estimation of sensitivity matrices}\label{algo}
\begin{algorithmic}[1]
\For {$k = m, m+1,\dots, $}
\State \textbf{[S1]} Collect $\Delta \mathbf{f}_k$ and $\Delta \mathbf{p}_k$
\State \textbf{[S2]} Build $\Delta \mathbf{F}_k$ and $\Delta \mathbf{P}_k$ based on $\{\Delta \mathbf{f}_k, \Delta \mathbf{p}_k\}_{k \in \mathcal{T}_k}$
\State \textbf{[S3]} Compute $\mathbf{y}_k$ via~\eqref{prox_update_y}
\State \textbf{[S4]} Update $\mathbf{H}_{k}$ via~\eqref{prox_update_H}
\State \textbf{[S5]} Update $\mathbf{o}_{k}$ via~\eqref{prox_norm1}
\State Go to \textbf{[S1]}
\EndFor
\end{algorithmic}
\end{algorithm}
We stress that, by using Algorithm \ref{algo}, we can estimate the sensitivity matrix robustly over a sliding window $\mathcal{T}_k$, this adapting to changing operational points of the power system.
In order to analyze the estimation accuracy of the Algorithm \ref{algo} performance, the dynamic regret metric is considered here; see, e.g. \cite{Hall15,Jadbabaie15,Online-opti}. In particular, it is defined as:
\begin{equation*}
\text{Reg}_k := \sum_{i=1}^k \left[ f_i(\mathbf{x}_i) - f_i(\mathbf{x}_i^*) \right],
\end{equation*}
where we recall the $f_i$ is the cost function in~\eqref{eq:noise_nuclear_outliers} (see also~\eqref{eq:timevarying_opt}).
The dynamic regret is an appropriate performance metric for time-varying problems with a cost that is convex, but not necessarily strongly convex~\cite{Online-opti}. To derive bounds on the dynamic regret, it is first necessary to introduce a ``measure'' of the temporal variability of \eqref{eq:timevarying_opt}. One possible measure is:
\begin{equation}
\omega_k := \norm{\mathbf{x}_k^*-\mathbf{x}_{k-1}^*},
\label{eq:omega}
\end{equation}
along with the so-called ``path length'':
\begin{equation}
\Omega_k := \sum_{i=1}^k \omega_i, \qquad \Bar{\Omega}_k := \sum_{i=1}^k \omega_i^2.
\label{eq:path_length}
\end{equation}
Recall that the least-squares term $s_k(\mathbf{x}_k)$ is closed, convex and proper, with a $L_k$-Lipschitz continuous gradient at each time $t_k$, and that $g_k(\mathbf{x}_k)$ is a lower semi-continuous proper convex function. Then, by using the definitions~\eqref{eq:omega}--\eqref{eq:path_length} and leveraging bounding techniques similar to~\cite{Inexact_prox_online}, the following result can be obtained.
\vspace{.1cm}
\begin{theorem}
\label{thm:convergence}
Suppose that the step size $\alpha$ is chosen such that $\alpha \leq 1/L$, with $L := \max\{L_k\}$. Then, the dynamic regret of Algorithm~\ref{algo} has the following limiting behavior:
\begin{equation}
\frac{1}{k} \text{Reg}_k = \mathcal{O}(1 + k^{-1} \Omega_k + k^{-1} \bar{\Omega}_k).
\end{equation}
\end{theorem}
\emph{Proof.} See the Appendix.
\vspace{.1cm}
\noindent Note that:
\noindent $\bullet$ When the sensitivity matrix changes over time, $\Omega_k$ and $\bar{\Omega}_k$ grow as $\mathcal{O}(k)$. Therefore, $(1/k)\text{Reg}_k = \mathcal{O}(1)$; that is, the sensitivity matrix can be estimated within a bounded error even in the considered online setting~\cite{Online-opti}.
\noindent $\bullet$ A no-regret result (i.e., $(1/k) \text{Reg}_k$ asymptotically goes to $0$) can not be obtained in general.
\noindent $\bullet$ If the sensitivity matrix is constant, then one trivially has that $\text{Reg}_k$ approaches $0$ asymptotically, thus recovering convergence results for the batch proximal-gradient method.
\vspace{.1cm}
\begin{remark}
In the paper, we assume that that $\Delta \mathbf{P}_k$ is either known or can be measured with negligible noise; on the other hand, $\Delta \mathbf{F}_k$ is noisy and may contain outliers. In principle, the proposed approach could be extended to handle noise and outliers in $\Delta \mathbf{P}_k$ by replacing the least-squares term with a total least squares (TLS) criterion; see, for example,~\cite{markovsky2007overview}. However, the resultant cost function is in this case nonconvex (because of bilinear terms); the challenges rely on the model for the trajectories of the critical points of the cost function.
\end{remark}
\section{Simulation Results} \label{Simu}
In this section, the estimation accuracy of the proposed methodology is assessed, for both a batch and online implementation. The following transmission networks are considered: i) Western Electricity Coordinating Council (WECC) 3-machine 9-bus transmission system~\cite{matpower_2019}, and ii) the synthetic South Carolina 500-bus transmission power system model~\cite{Birchfield}. In both cases, MATPOWER is utilized to compute the power-flow solutions~\cite{ matpower_2019}.
\subsection{Batch estimation} \label{transmission_ex}
Fluctuations in active power injection around a given operating point are simulated as in \cite{Chen_2014}. In particular, the injection at node $j$, denoted by $p_j$, is given by $p_j[k] = p_j^0[k] + \sigma_{N1} p_j^0[k] \eta_1 + \sigma_{N2} \eta_2$, where $p_j^0[k]$ is the nominal power injection at node $j$ at instant $k$, and ($\eta_1$, $\eta_2$) are random values, where $\eta_1 \sim \mathcal{N}(0, \sigma_{N1})$ and $\eta_2 \sim \mathcal{N}(0, \sigma_{N2})$ for standard deviations $\sigma_{N1} = \sigma_{N2} = 0.1$; see \cite{Chen_2014} for details. For each time $k$, we take the difference between consecutive line flow measurements to obtain $\Delta \mathbf{F}_k$. We obtained $\Delta \mathbf{P}_k$ by taking the differences between consecutive values of active power injections at each node. The batch optimization problems~\eqref{noise_nuclear} and~\eqref{eq:noise_nuclear_outliers} can also be solved efficiently using the proximal-gradient method (i.e., a batch version of Algorithm~\ref{algo}); see, for example,~\cite{cevher2014convex} (and references therein) for standard computational times of proximal-gradient methods. It can also be solved using CVX (available at: \texttt{http://cvxr.com/cvx}).
The performance of the proposed low-rank based approach is considered for both transmission networks; first, the batch method~\eqref{noise_nuclear} is evaluated, when a decrease of generation occurs at generator 2 for the 9-bus, and in generator 9 for the 500-bus transmission system. Figure \ref{fig:Transmision_9bus} and Figure \ref{fig:Transmision_500bus} compares the performance of the proposed method with the least-squares approach \cite{Chen_2014}. In this case, 10 trials were used, and the relative error (RE) with respect to the bus $i$ is defined by (we use this definition to be consistent with~\cite{Chen_2014})
\begin{equation*}
\text{RE}_i = \frac{\norm{\mathbf{h}_{ik} - \mathbf{h}_{ik}^*}}{\norm{\mathbf{h}_{ik}^*}},
\end{equation*}
where the actual sensitivity of the lines due the change of generation in bus $i$ is denoted as $\mathbf{h}_{ik}^*$, and $\mathbf{h}_{ik}$ specifies the column of the estimate sensitivity matrix $\mathbf{H}_k$ for the bus $i$ obtained form the DC model-based, least-squares or low-rank approaches. Figure \ref{fig:Transmision_9bus} shows that, when the set of measurements is less than 9 (the total number nodes in this case) the least-squares approach does not give an accurate estimation, because it is underdetermined. In the case of the proposed low-rank method, the median of the relative errors can be of just 3\% even when we have only 6-7 sets of measurements; the proposed method performs better than the model-based approach via DC approximation once we collect 8 measurements. When more than 9 measurements are collected, the proposed method and the least-squares approach have similar performance as expected. Figure~\ref{fig:Transmision_500bus} shows a similar behavior for the synthetic South Carolina 500-bus transmission systems, where our method is able to estimate the sensitivity matrix from 200 sets of measurements. In both cases, it is evident that the proposed approach provides accurate results with less measurements than the least-squares approach.
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_9_BC_LSE_R1.eps}}
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_9_BC_LR_R1.eps}}
\vspace{-.3cm}
\caption{Case 1 - 9-bus: Box plot of the relative error (RE) over 10 trials for the estimation of the sensitivity matrix under different number of measurements: (a) RE for the least square estimator and (b) RE for the low-rank approach.}
\vspace{-.3cm}
\label{fig:Transmision_9bus}
\end{figure}
\begin{figure}[!t]
\centering
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_500_BC_LSE_R1.eps}}
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_500_BC_LR_R1.eps}}
\vspace{-.3cm}
\caption{Case 1 - 500-bus: Box plot of the relative error (RE) over 10 trials for the estimation of the sensitivity matrix under different number of measurements: (a) RE for the least square estimator and (b) RE for the low-rank approach.}
\vspace{-.3cm}
\label{fig:Transmision_500bus}
\end{figure}
\begin{figure}[!ht]
\centering
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_9_outliers_LSE_R1.eps}}
\subfigure[]{\includegraphics[width=0.5\textwidth]{Fig_9_outliers_LR_R1.eps}}
\vspace{-.3cm}
\caption{Case 1 - 9-bus: Box plot of the relative error (RE) over 10 trials for the estimation of the sensitivity matrix with outliers under different number of measurements: (a) RE for the least square estimator and (b) RE for the low-rank approach.}
\vspace{-.3cm}
\label{fig:Figure_Tra_outliers}
\end{figure}
Further, in order to assess the performance of~\eqref{eq:noise_nuclear_outliers} in the case of outliers, we replicated the previous case for the 9-bus transmission system but with random outliers in the measurements. Figure \ref{fig:Figure_Tra_outliers} presents the results for the proposed method and the least-squares approach. Again, the proposed method outperforms the least-squares approach, and provides better estimates than the DC model-based method.
\subsection{Online Estimation}
As an example of an application of Algorithm~\ref{algo}, we consider an online robust estimation of the sensitivity matrix for the 9-bus transmission network. Relative to the test case presented at Section \ref{transmission_ex}, the nominal power injections at the nodes are now changing over time as in~\cite{dall2016optimal}. Figure \ref{fig:online_ouliers} shows the dynamic regret $(1/k) \text{Reg}_k$, when a window of 18 measurements is used. Based on Theorem~\ref{thm:convergence}, in the current setting the limiting behavior of $(1/k)\text{Reg}_k$ is $\mathcal{O}(1)$. Indeed, we can see that an asymptotic error is decreasing with the time index. Figure \ref{fig:online_changetopo} presents the cumulative sum of the relative error (RE) over $k$, i.e., $(1/k) \sum_{i=1}^k \mathrm{RE}_2$, for the online robust estimation of the sensitivity matrix in 9-bus transmission system, when there are changes of topology. In this case, we change the reactance of line 5 at $k = 400$, and the reactance of line 8 at $k=700$.
\begin{figure}[t]
\centering
\includegraphics[width=3.2in]{Fig_Reg_9_outliers.eps}
\vspace{-.2cm}
\caption{Evolution $(1/k) \sum_{i=1}^k [f(\mathbf{x}_i) - f(\mathbf{x}^*_i)]$ for the online robust estimation of the sensitivity matrix in 9-bus transmission system using LR (low-rank) method and LSE (least square estimation) approach.}
\label{fig:online_ouliers}
\vspace{-.3cm}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3in]{Fig_cumRE_9bus_changetop.eps}
\vspace{-.3cm}
\caption{Cumulative sum of the relative errors (RE) over $k$, i.e., $(1/k) \sum_{i=1}^k \mathrm{RE}_2$, for the online robust estimation of the sensitivity matrix in 9-bus transmission system, when there are changes of topology ($k = 400$ change reactance of line 5 and $k=700$ change reactance of line 8), using LR (low-rank) method and LSE (least square estimation) approach.}
\label{fig:online_changetopo}
\vspace{-.3cm}
\end{figure}
Reference~\cite{cevher2014convex} has shown that the proximal gradient method can be used to efficiently solve problems with thousands of variables, and each step can be performed in seconds or at the sub-second level. We used a computer with a processor Inter(R) Core(TM) i7-8850H CPU @ 2.60GHz, 32.0 GB of RAM, 64-bit Operating System. At each step of the proximal-gradient method, the computational time for the calculation of the gradients and the computation of the proximal operator (via CVX) were 0.0005 s, and 1.514 s for the 9-bus system. Lower computational times for the proximal operator can be obtained by utilizing a dedicated algorithm. Notice also that the proximal operator can be computed via SVD by removing the constraints on the entries of the matrix~\cite{cevher2014convex}; in this case the computational time was 0.019 s for the 9-bus system.
\section{Conclusion}
This paper proposed a method to estimate sensitivities in a power grid by leveraging a nuclear norm minimization approach as well as sparsity-promoting regularization functions. The proposed methodology is applicable to the estimation of various sensitivities at transmission level. Relative to a least-squares estimation method, the proposed approach allows to obtain meaningful estimates of the sensitivity matrix even when measurements are correlated. The method can identify outliers due to faulty sensors and is not deterred by missing measurements.
An online proximal-gradient algorithm was proposed to estimate sensitivity matrices on-the-fly and enable operators to maintain up-to-date information of sensitivities under dynamic operating conditions.
|
2,869,038,154,887 | arxiv | \section{Introduction}
Let $X$ and $K$ be two separable Hilbert spaces and ${\mathscr L}(K, X)$ the space of all bounded and linear operators from $K$ into $X$. We also denote by ${\mathscr L}_{HS}(K, X)$ the space of all Hilbert-Schmidt operators from $K$ into $X$.
The goal of this paper is to consider the existence of invariant measures for the following stochastic functional differential equation (SFDE) of retarded type on $X$,
\begin{equation}
\label{13/09/13(1)}
\begin{cases}
\displaystyle du(t) =\Lambda u(t)dt + \displaystyle\int^0_{-r} \alpha(\theta)\Lambda_1 u(t+\theta)d\theta dt + \Lambda_2 u(t-r)dt +\Psi(u_t)dt + \Sigma(u_t)dW(t),\,\,t\ge 0,\\
u(0)=\phi_0,\,\,\,\,u(\theta)=\phi_1(\theta),\,\,\,\theta\in [-r, 0],\,\,\,r>0,
\end{cases}
\end{equation}
where $\Lambda: {\mathscr D}(\Lambda)\subset X\to X$ generates an analytic semigroup of bounded linear operators $e^{t\Lambda}$, $t\ge 0$, on $X$, $u_t(\theta) := u(t+\theta)$, $\theta\in [-r, 0]$, $t\ge 0$, $\alpha\in L^2([-r, 0]; {\mathbb R})$ and $\Lambda_i: {\mathscr D}(\Lambda_i)\subset X\to X$ are linear, generally unbounded, operators on $X$ with ${\mathscr D}(\Lambda)\subset {\mathscr D}(\Lambda_i)$, $i=1,\,2$. Here ${\mathscr D}(\Lambda)$ is the domain of $\Lambda$ which is a Hilbert space under the usual graph norm. The initial value $\phi=(\phi_0, \phi_1)$ belongs to the product space ${\cal X} = M\times L^2([-r, 0]; {\mathscr D}(\Lambda))$ with $M=({\mathscr D}(\Lambda), X)_{1/2, 2}$ being the Lions' interpolation Hilbert space between ${\mathscr D}(\Lambda)$ and $X$, $W$ is a $K$-valued Wiener process and $\Psi: L^2([-r, 0]; {\mathscr D}(\Lambda))\to M$, $\Sigma: L^2([-r, 0]; {\mathscr D}(\Lambda))\to {\mathscr L}_{HS}(K, M)$ are appropriate nonlinear Lipschitz continuous mappings.
One of the most effective approaches in handling (\ref{13/09/13(1)}) is to lift it up to obtain a stochastic evolution equation without delay on ${\cal X}$. Precisely, we define a linear operator ${A}$ on ${\cal X}$ by
\begin{equation}
\label{13/09/13(2)}
\begin{split}
{\mathscr D}({A}) = \Big\{\phi=(\phi_0, \phi_1):\, &\phi_0=\phi_1(0)\in {\mathscr D}(\Lambda),\,\,\phi_1\in W^{1, 2}([-r, 0]; {\mathscr D}(\Lambda)),\\
&\hskip 30pt \Lambda\phi_0 + \displaystyle\int^0_{-r}\alpha(\theta)\Lambda_1\phi_1(\theta)d\theta + \Lambda_2 \phi_1(-r)\in M\Big\},
\end{split}
\end{equation}
and for any $\phi=(\phi_0, \phi_1)\in {\mathscr D}({A})$, let
\begin{equation}
\label{20/08/2013(20)}
{A}\phi= \Big(\Lambda\phi_0 + \displaystyle\int^0_{-r}\alpha(\theta)\Lambda_1\phi_1(\theta)d\theta + \Lambda_2 \phi_1(-r), \frac{d\phi_1(\theta)}{d\theta}\Big).
\end{equation}
It may be shown that ${A}$ generates a strongly continuous or $C_0$-semigroup $e^{t{A}}$, $t\ge 0$, on the space ${\cal X}$. As a consequence, we may lift up (\ref{13/09/13(1)}) onto ${\cal X}$ to consider a stochastic differential equation without delay on ${\cal X},$
\begin{equation}
\label{13/09/13(3)}
\begin{cases}
\displaystyle dU(t) ={A}U(t)dt + F(U(t))dt + {B}(U(t))dW(t),\,\,\,\,t\ge 0,\\
U(0)=(\phi_0, \phi_1)\in {\cal X},
\end{cases}
\end{equation}
where $U(t)= (u(t), u_t)$, $t\ge 0$ and $F: {\cal X}\to {\cal X}$ and ${B}: {\cal X}\to {\mathscr L}(K, {\cal X})$ are defined respectively by
\[
{F}:\, (\phi_0, \phi_1)\to (\Psi\phi_1, 0)\hskip 15pt\hbox{for any}\hskip 15pt (\phi_0, \phi_1)\in {\cal X},
\]
and
\[
{B}:\, (\phi_0, \phi_1)\to (\Sigma\phi_1, 0)\hskip 15pt\hbox{for any}\hskip 15pt (\phi_0, \phi_1)\in {\cal X}.\]
For instance, the equation (\ref{13/09/13(1)}) can be applied to a class of stochastic partial integrodifferential equations with distributed delays in the highest order partial derivatives of the form
\begin{equation}
\label{17/12/13(1)}
\begin{cases}
du(t, x) =\Delta u(t, x)dt + \displaystyle\int^0_{-r} \alpha(\theta)\Delta u(t+\theta, x)d\theta dt + \Sigma(u(t-r, x)) dW(t, x),\\
\hskip 250pt\,\,\,\,(t, x)\in [0, \infty)\times {\cal O},\\
u(0)=\phi_0,\,\,\,\,u(\theta)=\phi_1(\theta),\,\,\,\theta\in [-r, 0],\,\,\,r>0,
\end{cases}
\end{equation}
where $\Delta$ is the usual Laplacian operator, ${\cal O}$ is an open bounded subset of ${\mathbb R}^N$ with regular boundary $\partial{\cal O}$ and $\Sigma:\, {\mathbb R}\to {\mathscr L}(K, W^{1, 2}({\mathbb R}^N))$ is an appropriate nonlinear Lipschitz function.
In this work, we are concerned about the existence of invariant measures for the type of equation (\ref{13/09/13(3)}), which involves, in essence, a compactivity argument (Krylov-Bogoliubov theory). In finite dimensional spaces, the compactivity problem could be reduced to the investigation of boundedness for a solution of (\ref{13/09/13(3)}). However, the space ${\cal X}$ is infinite dimensional when $r>0$, and due to the absence of local compactness of ${\cal X}$, we need to exploit compact properties of the stochastic differential equation (\ref{13/09/13(3)}). The reader is referred to, e.g., \cite{gdpjz96} for a systematic statement about invariant measures of stochastic systems. In contrast with non time delay systems, the treatment here is complicated due to the infinite dimensional nature of delay problems. For instance, if both $\Lambda_1$ and $\Lambda_2$ in (\ref{13/09/13(1)}) are bounded on $X$ and $\Lambda$ generates a compact semigroup $e^{t\Lambda}$ on $t> 0$, it may be shown (see, e.g., \cite{kl09(2)}) that ${A}$ in (\ref{13/09/13(3)}) generates an eventually compact semigroup $e^{t{A}}$, i.e., $e^{t{A}}$ is compact for all $t>r$. In this case, the existence of an invariant measure of (\ref{13/09/13(3)}) is considered in \cite{jbovgsl09} when some further conditions on the diffusion term are imposed. When $\Lambda_1$ or $\Lambda_2$ in (\ref{13/09/13(1)}) is unbounded on $X$, it is generally untrue that the associated semigroup $e^{t{A}}$, $t\ge 0$, of the system (\ref{13/09/13(3)}) is eventually compact. For example, it is shown in
\cite{Gdbkkes85(2)} that the associated semigroup $e^{t{A}}$, $t\ge 0$, of the system (\ref{17/12/13(1)}) is never compact or eventually compact,
a fact which means the theory in \cite{jbovgsl09} cannot be applied to a system like (\ref{17/12/13(1)}). However, it is known in \cite{jj1991} that the semigroup $e^{t{A}}$, $t\ge 0$, associated with (\ref{17/12/13(1)})
is eventually norm continuous, i.e., $t\to e^{t{A}}$ is continuous with respect to the operator norm $\|\cdot\|$ on all $t\ge t_0$ for some $t_0>0$.
In this work, we will study the existence of invariant measures to Eq. (\ref{13/09/13(3)}) where ${A}$ generates an eventually norm continuous semigroup on ${\cal X}$. As a consequence, the theory established in the note is applied to a system like (\ref{17/12/13(1)}) to get an invariant measure. The author is also referred to \cite{es84} for a concrete example of noncompact but norm comtinuous semigroup on a Hilbert space.
\section{Main Results}
With some abuse of notation, we shall consider throughout the remainder of the work the mild solutions of the following stochastic evolution equation on a Hilbert space $H$,
\begin{equation}
\label{18/12/13(1)}
\begin{cases}
\displaystyle dy(t) = Ay(t)dt + F(y(t))dt + B(y(t))dW(t),\,\,\,\,t\ge 0,\\
y(0)=y_0\in H,
\end{cases}
\end{equation}
where $A$ generates a $C_0$-semigroup $e^{tA}$, $t\ge 0$, on $H$, $W$ is a $K$-valued cylindrical Wiener process and $F: H\to H$, $B:\, H\to {\mathscr L}_{HS}(K, H)$ are two globally Lipschitz mappings.
\begin{remark}\rm
We cannot establish existence of invariant measures by exploiting dissipativity property of the drift parts of the equation (cf. Ch. 6 in \cite{gdpjz96}) since this condition does not hold in our situation. Indeed, let us consider a real stochastic differential equation with point delay,
\begin{equation}
\label{18/12/13(160)}
\begin{cases}
\displaystyle dy(t) = by(t-r)dt + cy(t)dB(t),\,\,\,\,t\ge 0,\\
y(t)=0, \,\,\,t\in [-r, 0],
\end{cases}
\end{equation}
where $b,\,c\in {\mathbb R}$ and $B$ is a standard one-dimensional Brownian motion. Let ${\cal X} = {\mathbb R}\times L^2([-r, 0]; {\mathbb R})$. For arbitrarily given $a\ge 0$, we have
\[
\begin{split}
\langle (F-aI)(\phi) - (&F-aI)(\psi), \phi-\psi\rangle_{\cal X}\\
& = (\phi_0 -\psi_0)[\phi_1(-r) - 0] -a\Big[(\phi_0-\psi_0)^2 + \int^0_{-r} (\phi_1(\theta)-\psi_1(\theta))^2d\theta\Big]
\end{split}
\]
for any $\phi,\,\psi\in {\mathbb R}\times W^{1, 2}([-r, 0]; {\mathbb R})$.
We claim that for every $a\ge 0$, there exist $\phi,\,\psi\in {\cal X}$ such that the right-hand side in the previous expression is strictly positive. Indeed, let $\psi=(0, 0)$ and $\phi_0>0$, we can make the expression $\phi_0\cdot \phi_1(-r) - a(x^2_0 + \int^0_{-r}\phi_1(\theta)^2d\theta)$ as large as possible by choosing $\phi_1(-r)$ properly and, in the meanwhile, fixing the value $\phi_0$ and keeping $\int^0_{-r} \phi_1(\theta)^2d\theta$ unchanged.
\end{remark}
In the sequel, we impose the following conditions on (\ref{18/12/13(1)}).
\begin{enumerate}
\item[(H1)] The semigroup $e^{tA}$ is norm continuous on $[t_0, \infty)$ for some $t_0>0$;
\item[(H2)]
$B: H\to {\mathscr L}_{HS}(K, H)$ admits a factorization $B=LD$ such that $D: H\to {\mathscr L}_{HS}(K, H_1)$ is globally Lipschitz where $H_1$ is a seperable Hilbert space and $L\in {\mathscr L}(H_1, H)$ with $t\to e^{tA}L$ being norm continuous on $[0, t_0]$, e.g., $L$ is compact or $L=e^{t_0A}$ with $X=H$.
\item[(H3)]
$F: H\to H$ admits a factorization $F=UV$ such that $V: H\to H_2$ is globally Lipschitz where $H_2$ is a seperable Hilbert space and $U\in {\mathscr L}(H_2, H)$ with $t\to e^{tA}U$ being norm continuous on $[0, t_0]$.
\end{enumerate}
At the end of this work, a concrete class of stochastic delay heat equations satisfying (H1), (H2) and (H3) are shown how to use for us the theory in the work to practical problems. On this occasion, we only note that (H2) and (H3) may be satisfied when, for example, $L$ is compact or $L=e^{t_0A}$ with $H_1=H$ for (H2) and similarly for (H3).
\begin{lemma}
\label{25/12/13(10)}
Suppose that $A$ generates a norm continuous semigroup $e^{tA}$ on $[t_0, \infty)$ for some $t_0>0$. Let
\[
B_R(0) = \{x\in H:\, \|x\|_H\le R\},\hskip 20pt R>0,\]
then $e^{t_0A}B_R(0)$ is relatively compact in $H$ for any $R>0$.
\end{lemma}
\begin{proof}
For arbitrary $R>0$ and sequence $\{x_n\}_{n\ge 1}\in B_R(0)$, let us consider the set
\[
y_n(t) := e^{tA}x_n\in H,\hskip 15pt n\in {\mathbb R},\hskip 15pt t\in [t_0, t_0+1].\]
Since $\|e^{tA}\|\le Ce^{\mu t}$ for some constants $C>0$, $\mu\ge 0$ and all $t\ge 0$, it is easy to see that
\[
\max_{\substack{n\ge 1,\\ t\in [t_0, t_0+1]}}\|y_n(t)\|_H = \max_{\substack{n\ge 1,\\t\in [t_0, t_0+1]}}\|e^{tA}x_n\|_H \le Ce^{\mu (t_0+1)}R<\infty.\]
On the other hand, since $e^{tA}$ is norm continuous on $[t_0, \infty)$ and $\{x_n\}\subset B_R(0)$, we have for any $s,\,t\in [t_0, t_0+1]$ that
\[
\lim_{t\to s}\max_{n\ge 1}\|e^{tA}x_n-e^{sA}x_n\|_H \le \lim_{t\to s}\|e^{tA}-e^{sA}\|R\to 0.\]
By virtue of Ascoli-Alzel\`a Theorem, we thus have that $\{y_n(t)\}$, $n\in {\mathbb N}$, is relatively compact in $C([t_0, t_0+1], H)$. That is, there exists a subsequence, still denote it by $y_n(t)$, and the corresponding $x_n\in B_R(0)$, $n\ge 1$, such that $\{y_n(t)\}=\{e^{tA}x_n\}$ is convergent in $C([t_0, t_0+1]; H)$, i.e., there exists $y(t)\in C([t_0, t_0+1]; H)$ such that
\[
\lim_{n\to\infty}\max_{t\in [t_0, t_0+1]}\|y_n(t)-y(t)\|_H = \lim_{n\to\infty}\max_{t\in [t_0, t_0+1]}\|e^{tA}x_n-y(t)\|_H =0,\]
which particularly implies that
\[
\lim_{n\to\infty}\|e^{t_0 A}x_n-y(t_0)\|_H=0.\]
This means that $e^{t_0 A}B_R(0)$ is relatively compact in $H$. The proof is complete now.
\end{proof}
Let $y(t, y_0)$, $t\ge 0$, be the mild solution of Eq. (\ref{18/12/13(1)}). Firstly, let us consider the stochastic convolution
\[
u(t, y_0) := \int^{t}_0 e^{(t-s)A}B(y(s, y_0))dW(s),\hskip 20pt t\ge 0.\]
\begin{lemma}
\label{26/12/13(1)}
Suppose that the conditions (H1) and (H2) hold for some $t_0>0$.
For any $\varepsilon>0$ and $R>0$, there exists a compact set $S_{R, \varepsilon}\subset H$ such that
\[
{\mathbb P}(u(t_0, y_0)\in S_{R, \varepsilon})>1-\varepsilon\hskip 20pt \hbox{for all}\hskip 20pt \|y_0\|_H\le R.\]
\end{lemma}
\begin{proof}
Recall the factorization $B= LD$ in (H2) through Hilbert space $X$ and let us consider the set
\begin{equation}
\label{25/12/13(1)}
V := \{e^{sA}Lx:\, s\in [0, t_0], x\in X, \|x\|_X\le R\}\subset H,\hskip 20pt R>0.
\end{equation}
We show that $V$ is relatively compact in $H$. Indeed, let $\{v_n\}$, $n\in {\mathbb N}$, be an arbitrary sequence in $V$. Then there exist sequences $\{s_n\}\subset [0, t_0]$ and $\{x_n\}\subset X$ with $\|x_n\|_X\le R$, $n\in {\mathbb N}$, such that
\begin{equation}
\label{25/12/13(2)}
v_n = e^{s_nA}Lx_n,\hskip 20pt n\in {\mathbb N}.
\end{equation}
For the sequence $\{s_n\}_{n\ge 1}\subset [0, t_0]$, there exists a subsequence, still denote it by $\{s_n\}$, and a number $s_0\in [0, t_0]$ such that $s_n\to s_0$ as $n\to\infty$. Since $s\to e^{sA}L$ is norm continuous on $[0, t_0]$, by analogy with Lemma \ref{25/12/13(10)}, it is not difficult to see that the sequence $\{e^{s_0A}Lx_n\}$, $n\in {\mathbb N}$, is relatively compact in $H$. Thus there exists a subsequence of $\{x_n\}$, still denote it by $\{x_n\}$, such that $e^{s_0A}Lx_n\to z$ as $n\to \infty$ for some $z\in H$. In addition to the norm continuity of $t\to e^{tA}L$ on $[0, t_0]$, this further implies that
\[
\begin{split}
\|e^{s_{n}A}Lx_{n} - z\|_H &\le \|e^{s_{n}A}Lx_{n} - e^{s_0A}Lx_n\|_H + \|e^{s_{0}A}Lx_n -z\|_H\\
&\le R\|e^{s_{n}A}L - e^{s_0 A}L\| + \|e^{s_{0}A}Lx_n -z\|_X\\
&\to 0\hskip 15pt \hbox{as}\hskip 15pt n\to\infty.
\end{split}
\]
That is, $V$ is relatively compact. Therefore, there exists (e.g., see Ex. 3.8.13 (ii) in \cite{vib98}) a compact, self-adjoint, invertible operator $T\in {\mathscr L}(H)$ such that
\[
V\subset T(B_1(0))\]
where $B_1(0)=\{x\in H:\, \|x\|_H\le 1\}$.
Let $K(\lambda) = \lambda T(B_1(0))$ for any $\lambda>0$ and $T^{-1}\in {\mathscr L}(H)$ denote the bounded inverse of $T$. Then it is easy to see that
\[
u(t_0, y_0) \in K(\lambda) \iff \int^{t_0}_0 T^{-1}e^{(t_0-s)A}LD(y(s, y_0))dW(s)\in \lambda B_1(0),\]
which, together with the fact $\|e^{tA}\|\le Ce^{\mu t}$, $C>0$, $\mu\ge 0$, for all $t\ge 0$, immediately implies that
\[
\begin{split}
{\mathbb P}(u(t_0, y_0)\notin K(\lambda)) &\le {\mathbb P}\Big(\int^{t_0}_0 T^{-1}e^{(t_0-s)A}LD(y(s, y_0))dW(s)\notin \lambda B_1(0)\Big)\\
&\le \frac{1}{\lambda^2}{\mathbb E}\Big(\Big\|\int^{t_0}_0 T^{-1}e^{(t_0-s)A}LD(y(s, y_0))dW(s)\Big\|^2_H\Big)\\
&\le \frac{\|T^{-1}\|^2C^2e^{2\mu t_0}\|L\|^2}{\lambda^2}{\mathbb E}\Big(\int^{t_0}_0 \|D(y(s, y_0))\|^2_{HS}ds\Big)\\
&\le \frac{C_1}{\lambda^2}(1+ \|y_0\|^2_H)\to 0\hskip 15pt \hbox{as}\,\,\,\,\lambda\to\infty,
\end{split}
\]
for some $C_1=C_1(t_0)>0$. Here we use Th. 5.3.1 of \cite{gdpjz96} in the last inequality. The proof is thus complete.
\end{proof}
In an analogous manner, let us consider the integral
\[
v(t, y_0) := \int^{t}_0 e^{(t-s)A}F(y(s, y_0))ds,\hskip 20pt t\ge 0.\]
Then it is possible to establish the following results.
\begin{lemma}
\label{26/12/13(177)}
Suppose that the conditions (H1) and (H3) hold for some $t_0>0$.
For any $\varepsilon>0$ and $R>0$, there exists a compact set $S'_{R, \varepsilon}\subset H$ such that
\[
{\mathbb P}(v(t_0, y_0)\in S'_{R, \varepsilon})>1-\varepsilon\hskip 20pt \hbox{for all}\hskip 20pt \|y_0\|_H\le R.\]
\end{lemma}
Now we are in a position to state the main result in this work.
\begin{theorem}
\label{14/01/14(2)}
Suppose that the conditions (H1), (H2), (H3)
hold and for any $x\in H$ and $\varepsilon>0$, there exists $R>0$ such that for all $T>0$,
\begin{equation}
\label{26/12/13(70)}
\frac{1}{T}\int^T_0 {\mathbb P}(\|y(t, y_0)\|_H\ge R)dt < \varepsilon.
\end{equation}
Then there exists at least one invariant measure for the solution $y(t, y_0)$, $t\ge 0$, of (\ref{18/12/13(1)}).
\end{theorem}
\begin{proof}
Note that for the mild solution $y(t, y_0)$, $t\ge 0$, of (\ref{18/12/13(1)}), it is valid that
\begin{equation}
\label{26/12/13(10)}
y(t_0, y_0)= e^{t_0A}y_0 + \int^{t_0}_0 e^{(t_0 -s)A}F(y(s, y_0))ds + \int^{t_0}_0 e^{(t_0 -s)A}B(y(s, y_0))dW(s).
\end{equation}
By virtue of Lemma \ref{25/12/13(10)}, for any $R>0$ there exists a compact set $K_1(R)\subset H$ such that $e^{t_0A}x\in K_1(R)$ for all $\|x\|_H\le R$. On the other hand, by Lemmas \ref{26/12/13(1)} and \ref{26/12/13(177)} for any $R>0$ and $\varepsilon>0$, there exist compact sets $K_2(R, \varepsilon)$ and $K_3(R, \varepsilon)$ in $H$ such that
\[
{\mathbb P}\Big(\int^{t_0}_0 e^{(t_0-s)A}B(y(s, y_0))dW(s)\in K_2(R, \varepsilon)\Big)>1-\varepsilon,\]
and
\[
{\mathbb P}\Big(\int^{t_0}_0 e^{(t_0-s)A}F(y(s, y_0))ds\in K_3(R, \varepsilon)\Big)>1-\varepsilon.\]
Hence, we conclude from (\ref{26/12/13(10)}) that for $\|y_0\|_H<R$ and $\varepsilon>0$, there is
\begin{equation}
\label{26/12/13(20)}
{\mathbb P}\Big\{y(t_0, y_0)\in K_1(R)\cup K_2(R, \varepsilon)\cup K_3(R, \varepsilon)\Big\}\ge 1-\varepsilon.
\end{equation}
Let $t>t_0$, $K(R, \varepsilon) = K_1(R)\cup K_2(R, \varepsilon)\cup K_3(R, \varepsilon)$ and $p_t(x, dy)$ be the Markov transition probabilities of the solution $y(t, y_0)$, $t\ge 0$, of (\ref{18/12/13(1)}). Then it is easy to get that
\[
\begin{split}
{\mathbb P}(y(t, y_0)\in K(R, \varepsilon)) &= {\mathbb E}[p_{t_0}(y(t-t_0, y_0), K(R, \varepsilon))]\\
&\ge {\mathbb E}\big[p_{t_0}(y(t-t_0, y_0), K(R, \varepsilon)){\bf 1}_{\{\|y(t-t_0, y_0)\|_H\le R\}}\big],\hskip 20pt \forall\, t>t_0,
\end{split}
\]
which, together with (\ref{26/12/13(20)}), immediately implies that
\[
{\mathbb P}(y(t, y_0)\in K(R, \varepsilon))\ge (1-\varepsilon){\mathbb P}(\|y(t-t_0, y_0)\|_H\le R),\hskip 20pt \forall\, t>t_0.\]
This further yields that
\begin{equation}
\label{14/01/2014(1)}
\frac{1}{T}\int^{T+t_0}_{t_0}{\mathbb P}(y(t, y_0)\in K(R, \varepsilon))dt \ge \frac{1-\varepsilon}{T}\int^T_0 {\mathbb P}(\|y(t, y_0)\|_H\le R)dt.
\end{equation}
Choosing $R>0$ large enough and $\varepsilon>0$ small enough, we have by virtue of (\ref{26/12/13(70)}) and (\ref{14/01/2014(1)}) that the family
\[
\frac{1}{T}\int^{T+t_0}_{t_0}p_t(y_0, \cdot)dt,\,\,\,\, T>0,\]
is tight. According to the classic Krylov-Bogoliubov theory, there exists an invariant measure. The proof is complete now.
\end{proof}
\begin{example}\rm
Consider the stochastic reaction-diffusion equation with delay
\begin{equation}
\label{14/01/14(10)}
\begin{cases}
\displaystyle\frac{\partial y(t, x)}{\partial t} = \displaystyle\frac{\partial^2}{\partial x^2}y(t, x) + \displaystyle\int^0_{-r} \alpha(\theta) \frac{\partial^2}{\partial x^2}y(t+\theta, x)d\theta + \Sigma \dot W(t, x),\hskip 20pt t\ge 0,\hskip 15pt x\in {\mathbb R},\\
y(\theta, \cdot) = \phi_1(\theta, \cdot)\in W^{1, 2}({\mathbb R}),\hskip 15pt \theta\in [-r, 0],\\
y(0, x) = \phi_0(x)\in L^2({\mathbb R}),\hskip 20pt x\in {\mathbb R},
\end{cases}
\end{equation}
where $r>0$, $\alpha\in L^2([-r, 0]; {\mathbb R})$ and $\Sigma\in {\mathscr L}_{HS}(L^2({\mathbb R}), W^{1, 2}({\mathbb R}))$. Here $\dot W(t, x)$, $t\ge 0$, $x\in {\mathbb R}$, is a cylindrical space and time white noise.
We can formulate this equation by setting $H =K= L^2({\mathbb R})$, $M= W^{1, 2}({\mathbb R})$ and ${\cal X} = M\times L^2([-r, 0]; W^{2, 2}({\mathbb R}))$ with $\Lambda$ defined by
\[
\begin{split}
&\Lambda = \Lambda_1 = \frac{\partial^2}{\partial x^2},\\
&{\mathscr D}(\Lambda) = \Big\{y\in W^{2,2}({\mathbb R}):\,\, \lim_{x\to \pm\infty}y(x) =0,\,\,\lim_{x\to \pm\infty}\frac{dy(x)}{dx} =0\Big\},
\end{split}
\]
and
\[
B= (\Sigma, 0)\,\,\,\,\hbox{on}\,\,\,\,{\cal X}.\]
Then we have an abstract stochastic evolution equation without delay on ${\cal X}$,
\begin{equation}
\label{18/12/13(102)}
\begin{cases}
\displaystyle dY(t) = AY(t)dt + BdW(t),\,\,\,\,t\ge 0,\\
Y(0)=(\phi_0, \phi_1)\in {\cal X},
\end{cases}
\end{equation}
where $Y(t)= (y(t), y_t)$, $t\ge 0$, and $A$ generates a $C_0$-semigroup $e^{tA}$, $t\ge 0$, on ${\cal X}$. Note that the theory in \cite{jbovgsl09} cannot be applied to this example since $A$ does not generate an eventually compact semigroup (cf. \cite{Gdbkkes85(2)}). However, it is known (see \cite{mm02})
that the associated ${A}$ does generate an eventually norm continuous semigroup $e^{tA}$, $t\ge 0$. This means that we can still use the theory in the note to this example.
Indeed, on this occasion we may write (with some abuse of notation) that $\Sigma=i\circ \Sigma$ where $i:\, W^{1, 2}({\mathbb R})\to L^2({\mathbb R})$ is the canonical embedding from $W^{1, 2}({\mathbb R})$ into $L^2({\mathbb R})$, which is known to be a compact mapping. Hence, the operator $B$ admits a factorization satisfying the conditions in (H2). Further, we may conclude from Theorem \ref{14/01/14(2)} that if the mild solution of (\ref{14/01/14(10)}) are bounded in probability, then an invariant measure exists.
\end{example}
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\end{document} |
2,869,038,154,888 | arxiv | \subsection*{Methods: perturbations and the base flow}
Perturbations are defined as
\begin{equation}\label{eq:def_pert}
\uvph = \uvp - \Uvp, \,\,\, \php = \pp,
\end{equation}
to place focus on the deviation between the instantaneous flow $(\uvp$, $\pp)$ and the laminar fixed point $\Uvp$.
The laminar fixed point is the streamwise independent, time steady, parallel base flow $\Uvp(y) = (\Up$, $0)$, where $\Up(y) = \cosh(H^{1/2}y)/\cosh(H^{1/2})$, with only $H=10$ investigated. As the governing equations include a friction term of the form $-H\uvp$, they satisfy extended Galilean invariance \citep{Pope2000turbulent}, and thereby at finite $H$, wall-driven flows are identical to pressure-driven ones, when viewed in the appropriate frame of reference \citep{Camobreco2021stability}.
Due to this, a constant mass flux condition is \textit{not} enforced for the full nonlinear flow, as this would be akin to the inclusion of both a constant pressure gradient condition
and a constant flow rate condition.
The perturbation definitions, Eq.~(\ref{eq:def_pert}), are substituted into the Q2D equivalent of the Navier--Stokes equations, and only terms linear in the perturbation are retained.
Taking twice the curl of the result, applying the divergence-free constraint on the perturbation, and assuming plane wave solutions with streamwise variation $\exp(\ii\alpha x)$ provides the linearized perturbation evolution equation
\begin{equation} \label{eq:ltg_forward}
\pde{\vhp}{t} = \mathcal{A}^{-1}\left[-\ii\alpha \Up \mathcal{A} + \ii\alpha \Up'' + \frac{1}{\mathit{Re}}\mathcal{A}^2 - \frac{H}{\mathit{Re}}\mathcal{A} \right]\vhp = \mathcal{L}\vhp,
\end{equation}
where $\mathcal{L}$ is the linear evolution operator, $\mathcal{A} = D^2-\alpha^2$ and $D$ represents $\partial_y$.
\subsection*{Methods and validation: computing linear transient growth optimals}
The linearly optimized initial condition maximizing growth in the functional
$G=|| \uvph(t=\tau) || / || \uvph(t=0) ||$ is sought, for a prescribed target time $\tau$, and wavenumber $\alpha$.
$G$ represents the gain in perturbation kinetic energy under the norm $|| \uvph || = \int \uvph \bcdot \uvph \,\mathrm{d}\Omega$ \citep{Barkley2008direct}, over computational domain $\Omega$.
The initial condition is computed with two independent solvers; the MATLAB solver is detailed here; see \citep{Camobreco2020role} for the primitive variable solver and see Table \ref{tab:ltg_comp} for validation of the independent methods.
The optimal is found by seeking the stagnation points of functional $G(\vhp)$ under the constraints that $\vhp$ satisfies (\ref{eq:ltg_forward}), mass is conserved and that the perturbation energy is normalized to unity. The Lagrange multiplier for $\vhp$ is the adjoint velocity perturbation $\xhp$ introduced in \citep{Schmid2001stability} and satisfies the adjoint evolution equation
\begin{equation} \label{eq:ltg_adjoint}
\pde{\xhp}{t} = \mathcal{A}^{-1}\left[\ii\alpha \Up \mathcal{A} + 2\ii\alpha \Up' D + \frac{1}{\mathit{Re}}\mathcal{A}^2 - \frac{H}{\mathit{Re}}\mathcal{A} \right]\xhp = \mathcal{L}^\ddag\xhp,
\end{equation}
derived from Eq.~(\ref{eq:ltg_forward}), where $\mathcal{L}^\ddag$ is the adjoint evolution operator.
To solve this problem numerically, the domain $y \in [-1,1]$ is discretized with $\Nc+1$ Chebyshev nodes \citep{Trefethen2000spectral,Weideman2001differentiation}, and $D$
approximated by derivative matrices incorporating boundary conditions \citep{Trefethen2000spectral}.
A third-order forward Adams--Bashforth scheme \citep{Hairer1993solving} integrates Eq.~(\ref{eq:ltg_forward}) from $t=0$ to $t=\tau$, and with `initial' condition $\xhp(\tau)=\vhp(\tau)$, integrates Eq.~(\ref{eq:ltg_adjoint}) from $t=\tau$ back to $t=0$.
After normalizing to $||\vhp(0)||=1$ the next iteration proceeds.
Boundary conditions are $\vhp = D\vhp = \xhp = D\xhp = 0$ at all walls.
The $j$'th eigenvalue $\lambda_{\mathrm{G},j}$ of the operator representing the sequential action of direct and adjoint evolution is determined with a Krylov subspace scheme \citep{Barkley2008direct, Blackburn2008convective}.
With eigenvalues sorted in ascending order by largest real component, the optimized growth $G = \lambda_{\mathrm{G},1}$ with corresponding eigenvector $\vtpG(t=0)$.
The iterative scheme is initialized with random noise.
To calculate the nonlinear evolution of the linear optimals, the initial condition is seeded onto the base flow as $\uvp(t=0) = \Up + \uvphG$,
where $\uvphG = \chi\Rez[(\ii\partial_y\vtpG/\alpha,\vtpG)\exp(\ii\alpha x)]$, and where $\chi$ allows the initial perturbation energy, quoted as $\Ezero(t=0) = \int \uvphG^2(t=0) \mathrm{d} \Omega\,/\!\int \Up^2 \mathrm{d} \Omega$, to be varied.
$\Ezero$ is adjusted as appropriate, until an accuracy of at least 4 significant figures is attained in the delineation energy $\ELD$.
\subsection*{Methods: computing the leading direct and adjoint eigenmodes}
Eqs.~(\ref{eq:ltg_forward}) and (\ref{eq:ltg_adjoint}) can be rewritten as eigenvalue problems by assuming exponential time dependence of the form $\exp(-\ii\omega t)$ for the direct eigenmodes, and $\exp(+\ii\omega^\ddag t)$ for the adjoint eigenmodes.
The discretized direct eigenvalue problem $-\ii\omega\vtp = \bm{L}\vtp$ is solved in MATLAB via \texttt{eigs}($\ii\bm{L}$), while the discretized adjoint eigenvalue problem $\ii\omega^\ddag\xtp = \bm{L}^\ddag\xtp$ is solved in MATLAB via \texttt{eigs}($-\ii\bm{L}^\ddag$).
The first direct (\emph{resp.} adjoint) eigenvalue, sorted in descending (\emph{resp.} ascending) order by their imaginary part, becomes the leading direct (\emph{resp.} adjoint) mode. The wavenumber minimizing the decay rate of the leading direct eigenmode $\amax$ is obtained by varying $\alpha$.
Linear growth from an initial condition composed of an adjoint eigenmode can be computed in two ways. First, growth can be measured via evolution of Eq.~(\ref{eq:ltg_forward}), with the leading adjoint mode as initial condition. Alternately, growth can be measured by first computing the amplitudes of the adjoint mode in the direct eigenvector basis $\gamma=\bm{E}^{-1}\xtp$, where $\bm{E}$ is a matrix with the direct eigenvectors as columns. These amplitudes $\gamma$ have known time dependence (given by the direct eigenvalues), and the amplitude spectrum at a later time $t$ in the forward system can be computed as $\gamma(t) = \gamma(0)\exp(-\ii\omega_\mathrm{c} t)$, where $\omega_\mathrm{c}$ is a column vector of the eigenvalues of the forward system. The adjoint mode at time $t$ is reconstructed directly as $\xtp(t) = \bm{E}\gamma(t)$ and the norm $\lVert\xtp(t)\rVert/\lVert\xtp(0)\rVert$ computed.
\subsection*{Methods: weakly nonlinear analysis}
To verify the importance of the leading modal instability in generating the edge state, its weakly nonlinear interactions are computed, and compared to full DNS results at times on the edge state.
The $\kappa=1$ harmonic extracted from DNS (to which all amplitudes contribute) is directly compared to the linearly computed modal instability (only the normalized amplitude of exponent unity contributes).
The weakly nonlinear interaction of the leading eigenmode with itself is compared to the $\kappa=2$ harmonic extracted from DNS, and the weakly nonlinear interaction of the leading eigenmode with its complex conjugate to the $\kappa=0$ mode extracted from DNS. Again, the DNS has contributions from all amplitudes, while the weakly nonlinear equivalents (at identical harmonic of $n=0$ or $n=2$) only have contributions from the normalized amplitude of exponent two.
Note that at $\rrc=0.9$ the energy of the edge state is approximately four orders of magnitude below the energy at which turbulence is observed, supporting the validity of weakly nonlinear analysis.
To perform weakly nonlinear analysis, the amplitude dependence of the plane-wave mode $\vhpn(y) =\vhp(y)e^{\ii\alpha n x}$ is expanded as
\begin{equation}
\vhpn = \sum_{m=0}^{\infty}\epsilon^{|n|+2m}\tilde{A}^{|n|}|\tilde{A}|^{2m}\vhpnnm,
\end{equation}
where $\vhpnnm$ denotes a perturbation ($n$ is the harmonic, $|n|+2m$ the amplitude), in line with \citep{Hagan2013weakly}, and $\tilde{A} = A/\epsilon$ is the normalized amplitude.
Nonlinear interaction between the linear mode $\woo$ and itself excites a second harmonic $\wtt$.
Nonlinear interaction between the linear mode $\woo$ and its complex conjugate $\wno$ generates a modification to the base flow $\uot$ \citep{Hagan2013weakly}.
The equations governing the $n$'th harmonic of the base flow and perturbation are \citep{Hagan2013weakly}
\begin{equation}\label{eq:L_b_p4}
D_0^2\uotm - H\uotm = \hat{g}_{0,2m},
\end{equation}
\begin{equation}\label{eq:L_n_p4}
[(D_n^2 - \ii\lambda n)D_n^2 - H D_n^2 - \ii\alpha n(\uoo''-\uoo D_n^2)]\vhpn = \hat{h}_{n,|n|+2m}^w,
\end{equation}
respectively, where $D_n = \partial_y \vect{e}_y + \ii n\alpha \vect{e}_x$ and where
\begin{equation}
\hat{g}_{0,2m}=\ii\sum_{m\neq 0}^{\infty}(\alpha m)^{-1}\vhpm^*D_0^2\vhpm,
\end{equation}
\begin{equation}\label{eq:h_eq_p4}
\hat{h}_{n,|n|+2m}^w = n\sum_{m\neq 0}^{\infty}m^{-1}(\vhpnm D_m^2D\vhpm - D(\vhpm)D_{n-m}^2\vhpnm),
\end{equation}
are from the curl of the nonlinear term. Here $*$ denotes complex conjugation.
If the right hand sides of Eqs.~(\ref{eq:L_b_p4}) and (\ref{eq:L_n_p4}) are set to zero, taking $m=0$, $n=1$, and assuming time dependence for perturbations of the form $e^{-\ii\lambda t}$, an equation governing the laminar base flow, and the forward linear evolution equation for the perturbation, are respectively recovered.
Note that the laminar base flow $\Up(y)$ is equivalent to $\uoo$. Equations (\ref{eq:L_b_p4}) through (\ref{eq:h_eq_p4}) are discretized into matrix operators and solved as follows.
First, solution of the SM82-modified Orr--Sommerfeld eigenvalue problem,
\begin{equation}\label{eq:lin_HP_p4}
[\bm{A}_1^{-1}\bm{M}_1(\uoo) - \lambda \bm{I}]\wo = 0,
\end{equation}
provides the leading eigenvalue $\lambda_1$ with corresponding eigenvector $\woo$. Note that the weakly nonlinear problem is formulated slightly differently (e.g.~neutral conditions satisfy $\Rez(\lambda_1)=0$), and so the leading eigenmode is recomputed via Eq.~(\ref{eq:lin_HP_p4}), rather than as discussed in the section `Methods: computing the leading direct and adjoint eigenmodes'. The following are solved for the weakly nonlinear interactions,
\begin{equation}
\uot = -2\alpha^{-1}(\bm{D}^2 - H\bm{I})^{-1}\Imz(\woo^*\bm{F}_0 \bm{D}^2\woo),
\end{equation}
\begin{equation}
\wtt = (\bm{M}_2 - 2\ii\Imz(\lambda_1)\bm{A}_2)^{-1}[2\bm{F}_2\bm{D}(\bm{D}(\woo\bm{D}\woo) - 2(\bm{D}\woo)^2)],
\end{equation}
where
\begin{equation}
\bm{A}_n = \bm{D}^2 - n^2\alpha^2\bm{I}, \, \bm{N}_n(\uoo) = \ii\alpha(\uoo''\bm{I}-\uoo\bm{A}_n), \, M_n = \bm{F}_n[\bm{A}_n^2 - H\bm{A}_n + \mathit{Re}\bm{N}_n],
\end{equation}
with boundary condition matrix $\bm{F}_n$ as given in \citep{Hagan2013capacitance, Hagan2013weakly}.
The weakly nonlinear analysis has been previously validated in \citep{Camobreco2020transition}, when establishing the subcritical nature of the bifurcation.
The linear $\woo$ and weakly nonlinear $\uot$, $\wtt$ results are compared to the corresponding Fourier components from DNS on the edge state.
Fourier components were obtained by projecting the $\Np=19$ DNS results to $\Np=50$, and computing the Fourier coefficients, $c_\kappa = \lvert (1/\Nf)\sum_{n=0}^{n=\Nf-1}\hat{f}(x_n) e^{-2\pi \ii \kappa n/\Nf}\rvert$, along 4000 $y$-slices, each with $\Nf=1000$, where $\hat{f}$ is the variable of interest, e.g.~$\vhp$.
All except the $j$'th and $(\Nf-j)$'th Fourier coefficients were set to zero $c_{\kappa,\neg j}=0$, and the inverse discrete Fourier transform $\hat{f}_{|\kappa| = j}= \sum_{\kappa=0}^{\kappa=\Nf-1}c_{\kappa,j} e^{2\pi \ii \kappa n/\Nf}$ computed, isolating the $j$'th mode in the physical domain.
\subsection*{Methods: Fourier spectra and edge states}
To support the classification of initial conditions realizing turbulence, energy spectra are computed at select instants in time, as the discretized streamwise direction (of the in-house solver) is periodic.
All Fourier spectra presented in both the main work and supporting information are therefore based on perturbation quantities.
Fourier coefficient magnitudes $c_\kappa = \lvert (1/\Nf)\sum_{n=0}^{n=\Nf-1}[\uhp^2(x_n)+\vhp^2(x_n)] e^{-2\pi i \kappa n/\Nf}\rvert$ were computed with the discrete Fourier transform in MATLAB, where $x_n$ represents the $n$'th $x$-location linearly spaced between $x_0=0$ and $x_{\Nf}=2\pi/\alpha$.
A mean Fourier coefficient $\meanfoco$ is obtained by averaging the coefficients obtained at 21 $y$-values.
Although $\Nf=10000$ was chosen, only the first 70 to 80 Fourier modes (and their conjugates) are well resolved by the streamwise discretization.
A time-averaged mean Fourier coefficient is further obtained by averaging approximately 20 data sets in time (sparsely recorded over a window of approximately $10^4$ time units).
As both transient turbulent episodes and indefinitely sustained turbulence are observed (depending both on $\mathit{Re}$ and $\alpha$), two criteria are introduced to define observations of possible edge states preceding the departure to turbulence.
If the turbulent state is sustained, and if the edge state is able to maintain a near constant energy for an arbitrarily long time (as $\Ezero \rightarrow \ELD$), the classical definition \citep{Duguet2009localized, Beneitez2019edge, Vavaliaris2020optimal} suffices.
Two energy bounds, one above and one below the edge state energy, are specified.
If an initial condition crosses the upper bounding energy (from below), it is considered to reach the turbulent basin of attraction, while if it crosses the lower bounding energy (from above) it is considered to have relaminarized.
However, if the delineating energy does not yield a clear edge state (i.e.~a perturbation energy which does not fluctuate about a constant for a reasonable length of time) the alternate definition introduced in Ref.~\citep{Camobreco2020role} is adopted.
In such cases, if the energy time history attains a secondary local maximum after the initial linear transient growth, the initial condition is considered to relaminarize.
Instead, if there is a secondary inflection point after the linear transient growth, the initial condition is classed as reaching the turbulent basin.
To investigate the possible edge states of the system, and particularly which conditions are necessary to sustain turbulence, three general types of initial condition are investigated.
Different initial conditions are obtained by varying the domain length (via $\alpha$) and the target time to achieve optimal growth.
The first type, hereafter case 0, simulates conventional linear transient growth optimals, with both $\alpha$ and $\tau$ optimized to yield the largest growth over all initial conditions, wavenumbers and target times.
The second, case 1, has the wavenumber set to that achieving minimal decay rate of the leading modal instability (henceforth $\amax$) at the chosen $\mathit{Re}$, with $\tau$ optimized for maximal growth at this fixed wavenumber.
The third, case 2, again sets the wavenumber to $\amax$, but is time optimized at $\alpha=2\amax$, so that the optimal initial condition has two repetitions within the full $\amax$ based domain.
Thus, case 2 initially has an effective wavenumber closer to $\alphaOpt$, but if a period doubling occurs, will have an effective wavenumber of $\amax$.
Note that these three cases (0 through 2) are only investigated at $\rrc = \mathit{Re}/\ReyCrit = 0.9$, as $\rrc = 0.9$ has been shown to sustain turbulence that was generated from a supercritically evolved modal instability \citep{Camobreco2020transition}.
For reference, at $H=10$, $\ReyCrit = 79123.2$ and at $\rrc = 0.9$, $\alphaOpt = 1.49$ and $\amax = 0.979651$.
\subsection*{Methods and validation: in-house and Dedalus solvers}
While an in-house spectral element solver performed the majority of the simulations presented, the Dedalus solver \cite{Burns2020dedalus} was also employed to independently verify the edge state behaviour and delineation energies, as well as to perform some auxiliary computations. The reader is referred to Ref.~\cite{Camobreco2020role} for an extensive description and validation of the in-house solver. In Dedalus, the governing equations solved in primitive variables are identical to those of the in-house solver, with similarly identical boundary conditions (excepting a slightly different handling of the pressure boundary conditions). Time integration was performed with a 4th order forward Runge--Kutta scheme (RK443) with timestep $\Delta t = 10^{-3}$. 1000 Chebyshev polynomials were employed in the spanwise direction, with up to 256 Fourier modes in the streamwise direction to determine $\ELD$ (depending on $\rrc$) and 864 to continue turbulent evolution. Both streamwise and spanwise directions were $3/2$ de-aliased. All nonlinear evolutions performed with Dedalus were initiated with a perturbation of the leading adjoint mode rescaled to finite energy. See Table \ref{tab:ded_comp} for validation of the delineation energy to 4 significant figures between solvers. Explicitly, if the initial condition was a linear transient optimal, the in-house solver was used, if the initial condition was a leading adjoint mode, Dedalus was used. Unless specified otherwise, the domain length in Dedalus was set to $2\pi/\amax$, with a single repetition of the adjoint initial condition present. Note that the initial conditions loaded into Dedalus were generated with MATLAB, which was also used to determine $\amax$, as discussed in the `Methods: computing the leading direct and adjoint eigenmodes' section.
\subsection*{Results: determining the conditions required to trigger and/or sustain turbulence}
\subsubsection*{Additional data supporting the claim that only linear optimals of wavenumber close to the leading eigenmode were found to trigger transition to turbulence}
The effect of varying the wavenumber with linear transient growth optimals ($T=\tau/\tauOpt=1$) is considered first, see \fig~\ref{fig:delin_ex} and
Table \ref{tab:delin_comp}.
Case 2 exhibits the largest $\ELD$, and so is furthest from the turbulent attractor, and of least interest.
Case 2 represents a compromise between maintaining the largest linear transient growth while still having access to $\amax$, via period doubling.
Although period doubling was often observed in an isolated exponential boundary layer \citep{Camobreco2020role}, it was prohibited by the extra constriction from the upper wall in a full duct flow; \fig\ \ref{fig:delin_ex}(c), right column, indirectly implies that period doubling did not occur.
Compared to case 2, case 0 generated approximately 6\% more linear transient growth, yet reduced $\ELD$ by a factor of more than one half.
However, the maximal linear transient growth still does not yield the lowest $\ELD$.
This, in itself, already provides an indication that searching for the most explosively growing initial conditions is not optimal when attempting to trigger transitions to turbulence.
Since in Q2D flows $\Gmax$ is typically 100 times lower than in 3D flows \citep{Butler1992optimal,Camobreco2020transition}, this may indicate that linear transient growth is less relevant, although by no means unimportant, to Q2D transitions to turbulence.
Case 1, by contrast, simultaneously generated the least linear transient growth and yielded the lowest $\ELD$, of the three cases investigated at $\rrc=0.9$, with $\ELD$ over 20 times smaller than for case 0.
Case 1 selected the wavenumber minimizing the decay rate of the leading modal instability.
After the initial linear transient growth, the optimal adjusts so as to transfer most of its energy to the leading eigenmode.
This minimizes its decay rate, and keeps $G$ larger for larger $\tau$ \citep{Trefethen1993hydrodynamic}.
Furthermore, the Fourier spectra at select instants in time in the right column of \fig\ \ref{fig:delin_ex} show that while cases 0 and 2 are unable to sustain turbulence, case 1 can.
Hence, targeting the leading eigenmode (by selecting $\amax$, or a wavenumber in the immediate vicinity) is more important than solely optimizing transient growth.
\subsubsection*{Additional data supporting the claim that a turbulent state is indeed reached in case 1}
Turbulence is identified with two key measures, the number of energized modes \citep{Grossmann2000onset}, and the formation of an inertial sub-range, identified by a ($y$ averaged) perturbation energy spectrum $\meanfoco$ with ($-5/3$) power law dependence \citep{Pope2000turbulent}.
Considering the Fourier spectra of \fig\ \ref{fig:delin_ex}\tix{b}, as it focuses on the initial transition, while on the edge state ($1.92\times 10^3<t<9.23\times 10^3$), only 10-15 modes have non-floor energy levels.
Upon departing the edge state there is a steady increase in the number of energized modes, with 30 or so being energized by $t=1.008\times10^4$.
Once a turbulent state is achieved around $t=1.100\times10^4$, all resolvable modes are appreciably energized. Furthermore, for $t \geq 1.100\times10^4$, even without a time average, perturbation energy spectra computed from instantaneous velocity fields for case 1 scale with $\kappa^{-5/3}$ quite well.
However, for cases 0 and 2, the $\kappa^{-5/3}$ scaling is not maintained.
In the case 0 example, \fig\ \ref{fig:delin_ex}\tix{a}, recorded times $1.87 \times10^3\lesssim t\lesssim 2.38 \times10^3$ fit such a $\kappa^{-5/3}$ trend, indicating that a turbulent episode was triggered.
Soon thereafter, the energy in each mode decays.
At even larger times, only the base flow modulation ($\kappa=0$) has appreciable energy, although it too slowly decays, with the laminar base flow reinstated.
Similarly, case 2 only triggers a single turbulent episode, with relaminarization following the rapid decay of all but the zeroth (slowly decaying) mode.
Overall, case 1, being both at $\amax$ and targeting an instability that energized the leading eigenmode (large $\tau$), was the only combination able to sustain turbulence.
While either on the edge state and before a turbulent transition, for case 1, or as the turbulent state relaminarizes, for case 0 in particular, the energy contained in each Fourier mode scales exponentially with wavenumber.
This is highlighted in \fig\ \ref{fig:delin_ex}\tix{b}, where for $1.92\times10^3 < t < 7.48\times10^3$, the energy contained in each mode varies as $\exp(-3\kappa/2)$ for the case 1 edge state.
The $\kappa$ coefficient (with magnitude less than $3/2$ for $t<6.48\times10^3$) is time dependent as the flow relaminarizes for case 0, \fig\ \ref{fig:delin_ex}\tix{a}, but the energy variation is still approximately exponential.
Simulations at $r_c=0.9$ and $\amax$ (case 1) for linear optimals computed at several values of $T$ are gathered on \fig~\ref{fig:zin_intermittent}(c).
Regardless of $T$, the turbulent state stochastically collapses (to a secondary stable state), and then is reinstated, without a clearly identifiable pattern or frequency.
Hence, sustained subcritical turbulence is achieved at $\rrc=0.9$, but is intermittent in time.
\subsubsection*{Additional data supporting the claims that sustained subcritical turbulence was only found at criticalities $\boldsymbol{r_c>0.8}$}
The effect of varying the criticality $\rrc=\mathit{Re}/\ReyCrit$ and the target time $\tau = T\tauOpt$ at fixed $\amax$ is shown on \figs~\ref{fig:fo_time_hist} and \ref{fig:edge_comp} and in Table \ref{tab:edge_comp}.
Only case 1 is considered hereafter, since neither case 0 nor case 2 were able to sustain turbulence. Six Reynolds numbers, corresponding to $\rrc=0.3$, $0.4$, $0.6$, $0.7$, $0.8$ and $0.9$ were investigated in detail, see \fig~\ref{fig:edge_comp}, with time histories of the perturbation energy in each Fourier component depicted in \fig~\ref{fig:fo_time_hist}. Assessments of turbulent sustainment were for linear transient optimals with $T=8$ (or $T=6$ at $\rrc=0.3$) as initial condition. Note that time-resolved Fourier data is gathered roughly at the point of departure from the edge.
At $\rrc=0.3$, \fig~\ref{fig:fo_time_hist}(a) or \ref{fig:edge_comp}(a), a small amount of nonlinear growth occurs after departing the edge state.
However, this is insufficient to trigger turbulence, or even to slightly excite modes with $\kappa \gtrsim 60$.
A secondary stable finite amplitude state is instead maintained.
At $\rrc=0.4$, \fig\ \ref{fig:fo_time_hist}(b) or \ref{fig:edge_comp}(b), nonlinear growth is still insufficient to trigger turbulence, with a secondary stable finite amplitude state again attained.
However, there is some energization of all resolved modes.
Thus, for this case, the lack of the formation of an inertial sub-range, see \fig~\ref{fig:R3R6_comp}(c), was used to rule out turbulence at $\rrc=0.4$.
Although insufficient for turbulence, $\rrc=0.4$ was sufficient for the almost spontaneous introduction of a broadband oscillation of the energy in all Fourier modes, around $t\approx 4\times10^3$.
At $\rrc=0.6$, \fig\ \ref{fig:fo_time_hist}(c) or \ref{fig:edge_comp}(c), a single turbulent episode is triggered.
However, the turbulent episode is very brief, with saturation again to a secondary stable state, one very similar to those observed at $\rrc=0.3$ and $\rrc=0.4$.
Note that fluctuations in the lower, smaller wavenumber modes appear to die out earlier, with increasingly higher wavenumber modes only damped at increasingly larger times.
At $\rrc=0.7$, \fig\ \ref{fig:edge_comp}(d), saturation again occurs to a secondary stable state, after a single turbulent episode, in a very similar manner to $\rrc=0.6$.
By comparison, at $\rrc=0.8$, \fig\ \ref{fig:edge_comp}(e), the first signs of turbulence being indefinitely sustained are observed, at least for the computed length of the simulation.
Finally, at $\rrc=0.9$, \fig\ \ref{fig:fo_time_hist}(d) or \ref{fig:edge_comp}(f), turbulence is clearly triggered and indefinitely sustained, as discussed in the section `Additional data supporting the claim that a turbulent state is indeed reached in case 1'.
Note that the length of time required to simulate the turbulent phases of the flow (at reduced $\Delta t$) limit the ability to gather a continuous data set covering the initial transition, a turbulent phase, a sporadic collapse of turbulence, and then a return to turbulence.
Note that although Ref.~\citep{Avila2011onset} manage to obtain a threshold Reynolds number accurate to $\approx 0.5$\% (in their case $\mathit{Re} > 2040 \pm 10$ are sufficient to sustain subcritical turbulence for hydrodynamic pipe flow), this accuracy was attained only with time horizons of the order of $10^7$, accessible only to an exceedingly long experimental pipe setup. The DNS time horizons of Ref.~\citep{Avila2011onset} reached $\approx 10^4$, with a similar time horizon of $\approx 4\times10^4$ as the cutoff in this work. Hence, without experimental verification (required to obtain the thousand-fold increase in observation time), only the approximate threshold level of criticality of $\rrc \gtrsim 0.8$ can currently be identified here.
\subsubsection*{Additional data supporting either the transient or the intermittent nature of the turbulence observed, depending on $\boldsymbol{\rrc}$}
Simulations at $\rrc=0.3$, $0.4$, $0.6$ and $0.9$ (case 1), with $\Ezero>\ELD$ and $\Ezero<\ELD$, are presented in a state space in \fig~\ref{fig:zin_intermittent}(a,b), with an event indicating the sporadic collapse and reinstatement of turbulence highlighted at $\rrc=0.9$.
The state space in \fig~\ref{fig:zin_intermittent}(a,b) further reinforces the different behaviours observed at various $\rrc$ (results at $T=6$ or $T=8$). At $\rrc=0.3$, nonlinear energy growth is limited, and the region of the state space where turbulence is attained is never reached.
Notably, this region of the state space is reached at $\rrc=0.4$.
However, as discussed earlier, it is unlikely that $\rrc=0.4$ ever truly triggers turbulence, in spite of bearing some of the hallmarks of turbulence at higher wavenumbers, recalling \fig~\ref{fig:fo_time_hist}.
At $\rrc=0.6$, turbulence is transiently achieved. Like $\rrc=0.4$ and $\rrc=0.3$, $\rrc=0.6$ saturates to a secondary stable finite amplitude state, as indicated by the counter-clockwise spiralling trajectory depicted in the state space.
Finally, at $\rrc=0.9$, clear disorder is present in the trajectory, once sustained turbulence is reached. However, a clear collapse of the turbulent state can also be observed on this trajectory (heading toward the bottom left corner), before the trajectory turns around (toward the top right), and again ends up buried in the turbulent portion of the state space.
The effect of $\rrc$, and particularly the criticality, is further considered in \fig\ \ref{fig:zin_intermittent}\tix{d}.
As a baseline, the results at $\rrc=0.6$ are included on the same plot, to provide a reference for the energy level of the secondary stable finite amplitude state (although this varies with $\rrc$).
Comparing this to both $\rrc=0.9$ and $\rrc = 1.1$, although both sustain turbulence at a similar energy level, neither is fully developed.
Even the supercritical $\rrc=1.1$ cases depict sporadic, transient collapses of the turbulent state, with varying degrees of extremity.
Note that at $\rrc=1.1$, both a nonmodal perturbation (the choice of initial energy not greatly relevant) and white noise initial seed were separately evolved.
Furthermore, note that the $\rrc=1.1$ white noise initial condition data is an extension of that from Ref.~\citep{Camobreco2020transition}.
\subsection*{Results: additional data supporting the independence of the edge state from the initial conditions considered}
The energy-time histories for various $\rrc$ are displayed in \fig\ \ref{fig:edge_comp}, for all $T$ tested, with the key results collated in Tables \ref{tab:varyT_comp} and \ref{tab:edge_comp}.
\Fig\ \ref{fig:edge_comp} depicts initial conditions for all $T$ tested with $\Ezero$ just above and just below $\ELD$. Although both the initial energies and initial mode structures differ with $T$, all curves collapse to the same edge state (at each $\rrc$), with a common edge state energy $\EE$.
The edge state energy also allows for an alternate means of determining the efficiency of an initial condition.
The $T=1$ case generates a great deal of inconsequential linear transient growth, before falling back to the edge state, whereas the overshoot for $T=2$ through $8$ are much smaller.
The transient growth and delineation energies for $\rrc$ from $0.3$ to $0.9$ and $T = \tau/\tauOpt$ from $1$ to $8$ are collated in Table \ref{tab:varyT_comp}.
The delineation energy increases by about an order of magnitude for each of the $\rrc$ reductions shown in Table \ref{tab:varyT_comp}.
However, for each $\rrc$, $\ELD$ converges with increasing $T$.
Thus, increasing $T$ yields a more efficient initial condition to trigger a turbulent transition, with the initial condition converging toward the leading adjoint mode, which is again confirmed as the most efficient means of triggering turbulence in this Q2D setup.
This is further supported by the comparison of the values of $\ELD$ determined directly with the leading adjoint mode to those obtained with linear transient growth optimals at large $T$ (Table \ref{tab:ded_comp}).
\subsection*{Results: additional data supporting the claims that a secondary stable finite amplitude state is observed which has the hallmarks of the leading eigenmode}
The saturation to a secondary stable finite amplitude state is observable in the energy-time histories in \figs~\ref{fig:fo_time_hist}(a-c) and \ref{fig:edge_comp}(a-d). To support this classification, Fourier spectra computed from snapshots at distinct times up to $t\approx3.5\times10^4$ are presented in \fig~\ref{fig:R3R6_comp}(a, c \& e), and DNS extractions at large times isolating individual Fourier components in \fig~\ref{fig:R3R6_comp}(b, d \& f). In addition, flow fields at $\rrc=0.3$, $0.4$ and $0.6$ (post relaminarization) are depicted in \fig~\ref{fig:DNS_galactic}.
The neutral stability of the finite amplitude state at $\rrc=0.3$ and $\rrc=0.4$, post nonlinear growth, and at $\rrc=0.6$, after the turbulent episode, are supported by the Fourier spectra depicted in \fig\ \ref{fig:R3R6_comp}.
At $\rrc = 0.3$, in \fig\ \ref{fig:R3R6_comp}\tix{a}, the perturbation energy exhibits an exponential variation with $\kappa$ for all $t$, both when on the edge, and once having saturated to the secondary stable state.
This stable state is characterized by an $\exp(-0.156\kappa)$ dependence, which is maintained from $t=6.48\times10^3$ until at least $t=34.75\times10^3$ (its computed extent, or apparently indefinitely).
Note that for the secondary stable state at $\rrc=0.3$, only $10 \lesssim \kappa \lesssim 60$ follow this trend. Lower wavenumbers, particularly $\kappa=0$, $1$ and $2$, clearly contain a significant fraction of the total energy, supporting the conclusion that the secondary stable state is itself based around the weakly nonlinear edge state (with the leading three modes cumulatively containing 94.3\% of the total energy). By comparison, higher wavenumbers ($\kappa \gtrsim 60$) may be exhibiting chaotic dynamics, or may not be well resolved.
At $\rrc = 0.4$, the story is similar, as shown in \fig\ \ref{fig:R3R6_comp}\tix{c}, although the trend for the finite amplitude state is $\exp(-0.098\kappa)$. Note that the possibility of turbulence being observed at $\rrc=0.4$, hinted at by the raising of the energy floor in \fig\ \ref{fig:fo_time_hist}(b), can be ruled out by \fig\ \ref{fig:R3R6_comp}\tix{c}.
Finally, at $\rrc = 0.6$, \fig\ \ref{fig:R3R6_comp}\tix{e}, the flow is briefly turbulent (around $t=5.76\times10^3$). A secondary stable state, characterized by an $\exp(-0.055\kappa)$ perturbation energy dependence on wavenumber, is maintained from $t=10.33\times10^3$ to at least $t=35.05\times10^3$ (again with no sign of decay). Similarly, once having saturated to the secondary stable state, the first three modes cumulatively contain 93.2\% of the total perturbation.
To further investigate the leading three harmonics, \fig~\ref{fig:R3R6_comp}(b, d \& f) depict Fourier components extracted from DNS isolating harmonics $\kappa=0$, $1$ and $2$, at times when the secondary stable state appears to have saturated. Extractions corresponding to the $\kappa=0$ and $\kappa=1$ harmonics appear to vary little in time, particularly at lower $\rrc$. Only $\kappa=2$ (and higher $\kappa$; not shown) exhibit some time variations. It is unclear whether these fluctuations would also settle with additional time evolution. For the sake of comparison, the results of weakly nonlinear analysis are included at each $\rrc$; note that these are \textit{not} supposed to match the DNS extractions, as they did on the edge state (see main text). However, the $\kappa=1$ Fourier components extracted from DNS do still bear a strong resemblance to the TS wave.
Example flow fields for the secondary stable states (all Fourier modes) are provided in \fig\ \ref{fig:DNS_galactic}, for each $\rrc$.
There are clear similarities between the flow structures at all three $\rrc$ (0.3, 0.4 and 0.6), regardless of whether the secondary stable state forms after a brief turbulent episode or not.
In addition, the secondary stable states exhibit clear similarities to their 2D hydrodynamic equivalent \citep{Jimenez1990transition}.
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
\footnotesize{(a)} & \hspace{4mm} \footnotesize{Case 0; $\alpha = \alphaOpt$, $\tau=\tauOpt$ (unconstrained $\alpha$)} & & \\
\makecell{\rotatebox{90}{{$E$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3a.eps}} &
\makecell{\rotatebox{90}{{$\meanfoco$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3b.eps}} \\
\footnotesize{(b)} & \hspace{4mm} \footnotesize{Case 1; $\alpha = \amax$, $\tau = \tauOpt$ (at $\alpha = \amax$)} & & \\
\makecell{\rotatebox{90}{{$E$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3c.eps}} &
\makecell{\rotatebox{90}{{$\meanfoco$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3d.eps}} \\
\footnotesize{(c)} & \hspace{4mm} \footnotesize{Case 2; $\alpha = \amax$, $\tau = \tauOpt$ (at $\alpha = 2\amax$)} & & \\
\makecell{ \rotatebox{90}{{$E$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3e.eps}} &
\makecell{ \rotatebox{90}{{$\meanfoco$}}} & \makecell{\includegraphics[width=0.452\textwidth]{Fig3f.eps}} \\
& \hspace{41mm} {$t$} & & \hspace{41mm} {$\kappa$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{Nonlinear evolution of linear optimals at $\rrc = 0.9$, for the three domain/initial condition combinations of interest. Left column: determination of delineation energy. Right column: Fourier spectra at select instants in time (see legends) to indicate whether a flow (with $\Ezero>\ELD$) sustains turbulence. Dashed black lines denote $\kappa^{-5/3}$ trends; dash-dotted black lines denote $\exp(-3\kappa/2)$ trends.}
\label{fig:delin_ex}
\end{figure}
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
\footnotesize{(a)} & \footnotesize{\hspace{7mm} $\rrc = 0.3$}
& \footnotesize{(b)} & \footnotesize{\hspace{7mm} $\rrc = 0.4$}\\
\makecell{\rotatebox{90}{{$\meanfoco$}}} &
\makecell{\includegraphics[width=0.452\textwidth]{Fig35a.eps}} &
\makecell{\rotatebox{90}{{$\meanfoco$}}} &
\makecell{\includegraphics[width=0.452\textwidth]{Fig35b.eps}} \\
& \hspace{41mm} \footnotesize{$t$} & & \hspace{41mm} \footnotesize{$t$} \\
\footnotesize{(c)} & \footnotesize{\hspace{7mm} $\rrc = 0.6$}
& \footnotesize{(d)} & \footnotesize{\hspace{7mm} $\rrc = 0.9$}\\
\makecell{\rotatebox{90}{{$\meanfoco$}}} &
\makecell{\includegraphics[width=0.452\textwidth]{Fig35c.eps}} &
\makecell{\rotatebox{90}{{$\meanfoco$}}} &
\makecell{\includegraphics[width=0.452\textwidth]{Fig35d.eps}} \\
& \hspace{41mm} {$t$} & & \hspace{41mm} {$t$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{Time histories of $y$-averaged Fourier coefficients of perturbation energy for various $\rrc$, with $\alpha = \amax$ and $T=8$ ($T=6$ for $\rrc=0.3$). At $\rrc=0.3$ turbulence is not triggered. At $\rrc=0.4$ it is unclear whether turbulence is triggered. At $\rrc=0.6$ a transient turbulent episode is triggered. At $\rrc=0.9$, turbulence is triggered and indefinitely sustained. Symbols at each recorded time instant are included on streamwise modes 0 through 10. Thereafter, only lines are plotted, and only for every 2nd mode, up to the 200th, and then every 6th, up to the 476th.}
\label{fig:fo_time_hist}
\end{figure}
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
\footnotesize{(a)} & \footnotesize{\hspace{7mm} $\rrc=0.3$} & \footnotesize{(b)} & \footnotesize{\hspace{7mm} $\rrc=0.4$} \\
\makecell{ \rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig9d.eps}} &
\makecell{ \rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig9c.eps}} \\
\footnotesize{(c)} & \footnotesize{\hspace{7mm} $\rrc=0.6$} & \footnotesize{(d)} & \footnotesize{\hspace{7mm} $\rrc=0.7$} \\
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig9b.eps}} &
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig15a.eps}} \\
\footnotesize{(e)} & \footnotesize{\hspace{7mm} $\rrc=0.8$} & \footnotesize{(f)} & \footnotesize{\hspace{7mm} $\rrc=0.9$} \\
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig15b.eps}} &
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig9a.eps}} \\
& \hspace{41mm} \footnotesize{$t$} & & \hspace{41mm} \footnotesize{$t$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{DNS initialized with linear transient optimals for various $\rrc$ of $0.3$ through $0.9$ and $T$ of $1$ through $8$. $\Ezero$ is selected to be just above and below $\ELD$. Regardless of $T$ (i.e.~regardless of the initial condition), the edge state energy $\EE$ (dot-dashed line) is the same for each $\rrc$. However, larger $T$ have smaller $\Ezero$, and overshoot $\EE$ less. Inset figures at $\rrc=0.7$ and $\rrc=0.8$ highlight a single transient turbulent episode and possible indefinite sustainment of turbulence, respectively.}
\label{fig:edge_comp}
\end{figure}
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
\footnotesize{(a)} & & \footnotesize{(b)} & \\
\makecell{\rotatebox{90}{{$\int \vhp^2 \mathrm{d}\Omega$}}}
& \makecell{\includegraphics[width=0.446\textwidth]{Fig14a.eps}} &
\makecell{\rotatebox{90}{{$\int \vhp^2 \mathrm{d}\Omega$}}}
& \makecell{\includegraphics[width=0.446\textwidth]{Fig14b.eps}} \\
& \hspace{36mm} \footnotesize{$\int \uhp^2 \mathrm{d}\Omega$} & & \hspace{36mm} \footnotesize{$\int \uhp^2 \mathrm{d}\Omega$} \\
\footnotesize{(c)} & & \footnotesize{(d)} & \\
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig13a.eps}} &
\makecell{\rotatebox{90}{{$E$}}}
& \makecell{\includegraphics[width=0.452\textwidth]{Fig13b.eps}} \\
& \hspace{41mm} {$t$} & & \hspace{41mm} {$t$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{\tix{a} State space representation of the largest $T$, smallest $\ELD$ cases for each $\rrc$, with $\Ezero$ just above and just below $\ELD$. The initial conditions are marked with filled black circles. \tix{b} Zoom in on the fate of cases with $\Ezero > \ELD$. \tix{c} DNS initialized with linear transient optimals with $T=\tau/\tauOpt$ of 1 through 8 at $\rrc = 0.9$; data identical to \fig~\ref{fig:edge_comp}\tix{f} except zoomed in. \tix{d} DNS initialized with linear transient optimals with $T=1$, varying $\rrc$; data identical to \figs~\ref{fig:delin_ex}\tix{b} and \ref{fig:edge_comp}\tix{f} except zoomed in, excepting the white noise case, from Ref.~\cite{Camobreco2020transition}.}
\label{fig:zin_intermittent}
\end{figure}
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
\footnotesize{(a)} & \footnotesize{\hspace{7mm} $\rrc=0.3$} & \footnotesize{(b)} & \footnotesize{\hspace{7mm} $\kappa=0$} \\
\makecell{\rotatebox{90}{{$\meanfoco$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig10a.eps}} &
\makecell{\rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig17a.eps}} \\
& \hspace{39mm} \footnotesize{$\kappa$} & & \hspace{31mm} \footnotesize{$\hat{u}_{\perp,\kappa = 0}$, $\uot$} \\
\footnotesize{(c)} & \footnotesize{\hspace{7mm} $\rrc=0.4$} & \footnotesize{(d)} & \footnotesize{\hspace{7mm} $\kappa=1$} \\
\makecell{\rotatebox{90}{{$\meanfoco$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig10c.eps}} &
\makecell{\rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig17b.eps}} \\
& \hspace{39mm} \footnotesize{$\kappa$} & & \hspace{31mm} \footnotesize{$\hat{v}_{\perp,|\kappa| = 1}$, $\woo$} \\
\footnotesize{(e)} & \footnotesize{\hspace{7mm} $\rrc=0.6$} & \footnotesize{(f)} & \footnotesize{\hspace{7mm} $\kappa=2$} \\
\makecell{\rotatebox{90}{{$\meanfoco$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig10b.eps}} &
\makecell{\rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.42\textwidth]{Fig17c.eps}} \\
& \hspace{39mm} {$\kappa$} & & \hspace{31mm} {$\hat{v}_{\perp,|\kappa| = 2}$, $\wtt$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{Fourier spectra at select instants in time (left column) and comparison between weakly nonlinear (WNL) results and DNS extractions at large times (right column; normalized and phase shifted as described in main text). At large times, the Fourier spectra settle as a stable finite amplitude state forms. \tix{a} $\rrc = 0.3$. The dash-dotted black line denotes an $\exp(-0.156\kappa)$ trend. \tix{c} $\rrc = 0.4$. The dash-dotted black line denotes an $\exp(-0.098\kappa)$ trend. \tix{e} $\rrc = 0.6$. The dashed black line denotes a $\kappa^{-5/3}$ trend, and dash-dotted black line an $\exp(-0.055\kappa)$ trend. \tix{b} $\kappa=0$, streamwise velocity perturbation. \tix{d} $\kappa=1$, spanwise velocity perturbation. \tix{f} $\kappa=2$, spanwise velocity perturbation. }
\label{fig:R3R6_comp}
\end{figure}
\begin{figure}
\centering
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ llll }
& \hspace{8mm} \makecell{\includegraphics[width=0.28\textwidth]{vleg.eps}} &
& \hspace{8mm} \makecell{\includegraphics[width=0.28\textwidth]{uleg.eps}} \\
\footnotesize{(a)} & \footnotesize{\hspace{5mm} $\rrc=0.3$, $v_\mathrm{m}=\max(\vhp)=6.415\times10^{-2}$} & & \footnotesize{\hspace{5mm} $u_\mathrm{m}=\max(\uhp)=2.081\times10^{-1}$} \\
\makecell{\vspace{2mm} \rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11a.eps}} &
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11b.eps}} \\
\footnotesize{(b)} & \footnotesize{\hspace{5mm} $\rrc=0.4$, $v_\mathrm{m}=\max(\vhp)=1.115\times10^{-1}$} & & \footnotesize{\hspace{5mm} $u_\mathrm{m}=\max(\uhp)=3.082\times10^{-1}$} \\
\makecell{\vspace{2mm} \rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11c.eps}} &
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11d.eps}} \\
\footnotesize{(c)} & \footnotesize{\hspace{5mm} $\rrc=0.6$, $v_\mathrm{m}=\max(\vhp)=1.646\times10^{-1}$} & & \footnotesize{\hspace{5mm} $u_\mathrm{m}=\max(\uhp)=3.825\times10^{-1}$} \\
\makecell{\vspace{2mm} \rotatebox{90}{{$y$}}}
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11e.eps}} &
& \makecell{\includegraphics[width=0.46\textwidth]{Fig11f.eps}} \\
& \hspace{41mm} {$x$} & & \hspace{41mm} {$x$} \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\caption{Snapshots of the spanwise (left column) and streamwise (right column) velocity fields representing the secondary stable states obtained from DNS. \tix{a} $\rrc=0.3$ at $t=1.492\times10^4$. \tix{b} $\rrc=0.4$ at $t=1.099\times10^4$. \tix{c} $\rrc = 0.6$ at $t=1.762\times10^4$ (post turbulent episode).}
\label{fig:DNS_galactic}
\end{figure}
\begin{table}
\begin{center}
\caption{Validation of the transient growth $\boldsymbol{\Gtau(\tau)}$ achievable for various cases (differing $\boldsymbol{\alpha}$, $\boldsymbol{\tau}$) at $\boldsymbol{\rrc = 0.9}$. The MATLAB solver had $\boldsymbol{\Nc = 80}$ Chebyshev points, timestep $\boldsymbol{\Delta t=4\times10^{-5}}$ and 20 forward-backward iterations. The discretization for the linear transient growth solver is detailed in text, with $\boldsymbol{\Delta t=1.25\times 10^{-3}}$, and a convergence tolerance of $\boldsymbol{10^{-7}}$ between iterations. Percent errors are relative to MATLAB results.}
\begin{tabular}{ cccc }
\hline
Case & MATLAB & Linear primitive & $|$\% Error$|$ \\
\hline
$0$ & $97.55253053$ & $97.55254271$ & $1.25\times10^{-5}$ \\
$1$ & $78.14835970$ & $78.14836801$ & $1.06\times10^{-5}$ \\
$2$ & $91.63907657$ & $91.63914267$ & $7.21\times10^{-5}$ \\
\hline
\end{tabular}
\label{tab:ltg_comp}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Estimates of the delineation energy, comparing rescaled linear transient growth optimals evolved nonlinearly in the in-house solver to rescaled leading adjoint eigenvectors nonlinearly evolved in Dedalus. Excepting $\boldsymbol{\rrc=0.3}$, both computations yield identical $\boldsymbol{\ELD}$ to 4 significant figures. The discrepancy at $\boldsymbol{\rrc=0.3}$ is likely due to $\boldsymbol{T=6}$ not being a large enough time horizon for linear optimization; although an equivalent at $\boldsymbol{T=8}$ was difficult to extract from the in-house solver at $\boldsymbol{\rrc=0.3}$. For the in-house solver, $\boldsymbol{\Delta t=1.25\times10^{-3}}$, with 912 data points in $\boldsymbol{y}$ and 228 in $\boldsymbol{x}$. For Dedalus, $\boldsymbol{\Delta t=10^{-3}}$, with 1000 internal Chebyshev polynomials in $\boldsymbol{y}$ and $\boldsymbol{\Nx}$ Fourier modes, the latter varied for each $\boldsymbol{\rrc}$. The requirement of increasing $\boldsymbol{\Nx}$ with decreasing $\boldsymbol{\rrc}$ is likely due to increasing initial energies, which active higher order nonlinearities earlier.}
\begin{tabular}{ccc}
\hline
$\rrc$ & \makecell{$\ELD$ via rescaled linear \\ transient optimal; in-house} & \makecell{$\ELD$ via rescaled leading \\ adjoint mode; Dedalus} \\
\hline
$0.9$ & $1.473\times10^{-6}$ ($T=8$) & $1.473\times10^{-6}$ ($\Nx=32$) \\
$0.8$ & $4.327\times10^{-6}$ ($T=8$) & $4.327\times10^{-6}$ ($\Nx=32$) \\
$0.7$ & $1.012\times10^{-5}$ ($T=8$) & $1.012\times10^{-5}$ ($\Nx=32$) \\
$0.6$ & $2.297\times10^{-5}$ ($T=8$) & $2.297\times10^{-5}$ ($\Nx=32$) \\
$0.4$ & $1.717\times10^{-4}$ ($T=8$) & $1.717\times10^{-4}$ ($\Nx=64$) \\
$0.3$ & $7.379\times10^{-4}$ ($T=6$) & $7.374\times10^{-4}$ ($\Nx=256$) \\
\hline
\end{tabular}
\label{tab:ded_comp}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Key results for the three cases investigated in this work, ordered by increasing $\boldsymbol{\tau}$, or equally, reducing $\boldsymbol{\ELD}$. Case 1 yields the smallest $\boldsymbol{\ELD}$, in spite of the smallest linear transient growth, by virtue of energizing the leading modal instability (i.e.~selecting $\boldsymbol{\alpha=\amax}$ directly), while still time optimizing for maximum growth. Focusing on Case 1 in the lower portion of the table, except constrained to the leading adjoint mode (rather than a linear transient growth optimal), the wavenumber is further varied to determine the overall minimum $\boldsymbol{\ELD}$. This occurs at relatively low maximum growth, and at a wavenumber somewhere between $\boldsymbol{\amax}$ (minimum decay rate over all $\boldsymbol{\alpha}$), and $\boldsymbol{\alphaOptAdj}$ (maximum growth over all $\boldsymbol{\alpha}$, once constrained to the leading adjoint mode). Computations of $\boldsymbol{\ELD}$ with Dedalus employed $\boldsymbol{\Nx=32}$ as validated with $\boldsymbol{\Nx=64}$ (not shown). Both at $\boldsymbol{\Nx=32}$ and an increased resolution of $\boldsymbol{\Nx=384}$, $\boldsymbol{\alphaOptAdj}$, like $\boldsymbol{\alphaOpt}$, was unable to sustain turbulence (also not shown). However, at $\boldsymbol{\alpha=1.01}$ (minimum $\boldsymbol{\ELD}$ overall), turbulence appears to be indefinitely sustained, i.e. $\boldsymbol{\alphaOptAdj}$ appears to be too far from $\boldsymbol{\amax}$, whereas $\boldsymbol{\alpha=1.01}$ is not.}
\begin{tabular}{ ccccccc }
\hline
Case & $T = t/\tauOpt$ & $\alpha$ & \hspace{2mm} $\tau$ \hspace{2mm} & $\max(E(t\leq T))/\Gmax$ & $\ELD$ \\
\hline
$2$ & $1$ & $0.979651$ ($\amax$) & $19.4$ & $0.939$ & $2.547\times10^{-4}$ \\
$0$ & $1$ & $1.49$ ($\alphaOpt$) & $23.4$ & $1$ & $6.347\times10^{-5}$ \\
$1$ & $1$ & $0.979651$ ($\amax$) & $31.0$ & $0.801$ & $3.0577\times10^{-6}$ \\
\hline
$1$ & $\rightarrow\infty$ & $0.979651$ ($\amax$) & $-$ & $0.443$ & $1.473\times10^{-6}$ \\
$1$ & $\rightarrow\infty$ & $1.00$ & $-$ & $0.451$ & $1.263\times10^{-6}$ \\
$1$ & $\rightarrow\infty$ & $1.01$ & $-$ & $0.455$ & $1.240\times10^{-6}$ \\
$1$ & $\rightarrow\infty$ & $1.02$ & $-$ & $0.459$ & $1.253\times10^{-6}$ \\
$1$ & $\rightarrow\infty$ & $1.1546$ ($\alphaOptAdj$) & $-$ & $0.482$ & $3.467\times10^{-6}$ \\
\end{tabular}
\label{tab:delin_comp}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Convergence to a minimum overall $\boldsymbol{\ELD}$ with increasing $\boldsymbol{T}$ for each $\boldsymbol{\rrc}$. This is in spite of the reduced linear transient growth, where $\boldsymbol{E_\mathrm{G} = \max[E(t \leq T\tauOpt)]/E(t=0)}$ represents the maximum growth at any $\boldsymbol{t \leq T\tauOpt}$, relative to $\boldsymbol{G(1\tauOpt)}$. Note that $\boldsymbol{\rrc = 0.3}$ has $\boldsymbol{G(8\tauOpt) <1}$, and so does not provide an equivalent optimal in the primitive variable solver. $\boldsymbol{\ELD}$ is computed to 4 significant figures.}
\begin{tabular}{ ccccccccc }
\hline
& \multicolumn{2}{c}{$\rrc=0.3$} & \multicolumn{2}{c}{$\rrc=0.4$} & \multicolumn{2}{c}{$\rrc=0.6$} & \multicolumn{2}{c}{$\rrc=0.9$} \\
\hline
T & $E_\mathrm{G}/G$ & $\ELD$ & $E_\mathrm{G}/G$ & $\ELD$ & $E_\mathrm{G}/G$ & $\ELD$ & $E_\mathrm{G}/G$ & $\ELD$\\
\hline
$1$ & $1$ & $2.630\times10^{-3}$ & $1$ & $3.142\times10^{-4}$ & $1$ & $4.328\times10^{-5}$ & $1$ & $3.058\times10^{-6}$ \\
$2$ & $0.3700$ & $1.535\times10^{-3}$ & $0.3653$ & $2.970\times10^{-4}$ & $0.3940$ & $3.382\times10^{-5}$ & $0.4127$ & $1.954\times10^{-6}$ \\
$3$ & $0.5463$ & $7.984\times10^{-4}$ & $0.5532$ & $1.792\times10^{-4}$ & $0.5518$ & $2.351\times10^{-5}$ & $0.5461$ & $1.495\times10^{-6}$ \\
$4$ & $0.5799$ & $7.453\times10^{-4}$ & $0.5726$ & $1.727\times10^{-4}$ & $0.5618$ & $2.304\times10^{-5}$ & $0.5520$ & $1.475\times10^{-6}$ \\
$6$ & $0.5857$ & $7.379\times10^{-4}$ & $0.5759$ & $1.718\times10^{-4}$ & $0.5635$ & $2.298\times10^{-5}$ & $0.5530$ & $1.473\times10^{-6}$ \\
$8$ & $-$ & $-$ & $0.5761$ & $1.717\times10^{-4}$ & $0.5636$ & $2.297\times10^{-5}$ & $0.5530$ & $1.473\times10^{-6}$ \\
\hline
\end{tabular}
\label{tab:varyT_comp}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{For various $\boldsymbol{\rrc}$ (largest $\boldsymbol{T}$ cases), the delineation energy $\boldsymbol{\ELD}$, energy of the edge state $\boldsymbol{\EE}$ (the amplitude yielding a balance in the rate of linear decay to the rate of weakly nonlinear growth) and the maximum energy overall $\boldsymbol{\Emax}$ are tabulated, as are their ratios. Whether a transition to turbulence was observed, and if transition occurred, whether the turbulence was indefinitely sustained, are also indicated.}
\begin{tabular}{ cccccccc }
\hline
$\rrc$ & $\ELD$ & $\EE$ & $\EE/\ELD$ & $\Emax$ & $\Emax/\EE$ & Transition (Y/N) & Sustained Turbulence (Y/N) \\
\hline
$0.3$ & $7.379\times10^{-4}$ & $1.25\times10^{-2}$ & $16.9$ & $0.1136$ & $9.09$ & N & - \\
$0.4$ & $1.717\times10^{-4}$ & $3.15\times10^{-3}$ & $18.3$ & $0.3243$ & $1.03\times10^{2}$ & N & - \\
$0.6$ & $2.297\times10^{-5}$ & $7.18\times10^{-4}$ & $31.3$ & $0.6060$ & $8.44\times10^{2}$ & Y & N \\
$0.7$ & $1.012\times10^{-5}$ & $3.88\times10^{-4}$ & $38.3$ & $0.6752$ & $1.74\times10^{3}$ & Y & N \\
$0.8$ & $4.327\times10^{-6}$ & $1.99\times10^{-4}$ & $46.0$ & $0.7385$ & $3.71\times10^{3}$ & Y & Y \\
$0.9$ & $1.473\times10^{-6}$ & $8.03\times10^{-5}$ & $54.5$ & $0.6582$ & $8.20\times10^{3}$ & Y & Y \\
\hline
\end{tabular}
\label{tab:edge_comp}
\end{center}
\end{table}
\FloatBarrier
\section*{Results}
\subsection*{Evidence of subcritical turbulence}
The starting point is to determine whether turbulence in quasi-2D shear flows exists in the subcritical range. In other words: can turbulence exist at Reynolds numbers below the critical value $\ReyCrit$. In the subcritical regime,
turbulence originates from perturbations that are sufficiently intense to activate nonlinear amplification mechanisms that infinitesimal ones cannot. However, unlike 3D flows, seeding the subcritical laminar shear flow with noise, even at high level, does not ignite turbulence \cite{Camobreco2020transition}. Thus, the initial perturbation must be more carefully chosen. Since at this stage, our purpose is to find turbulence, but not necessarily the most efficient perturbation to trigger it,
we choose a type of perturbation that experiences some growth through the system dynamics, though not exponential growth. The idea is that even though these perturbations would eventually decay if they remained infinitesimal, they could experience sufficient transient growth to activate nonlinearities if given a small but finite initial energy.
The laminar state is then seeded with the perturbation of optimal transient growth and its evolution is simulated until the flow either returns to its initial laminar state or breaks down into a turbulent one. The prodecure is repeated, varying the initial perturbation energy.
To implement this strategy, we chose a configuration we have previously studied in detail, in which we found sustained turbulence in the supercritical regime (\emph{i.e.} for $Re>Re_c$)
\cite{Camobreco2020transition}. It consists of a quasi-2D incompressible flow (of viscosity $\nu$) in a duct of rectangular cross-section driven by the motion of its side walls, distant by $2L$, at a constant velocity $U_0$ (Fig.~\ref{fig:schematic}). The quasi-2D plane is imposed, for example, by a transverse magnetic field \cite{Sommeria1982why} perpendicular to the moving walls. It includes the direction of wall motion and the spanwise direction normal to the moving walls. Quasi-2D flows of this type obey the 2D Navier--Stokes equations augmented with a linear friction term that accounts for friction in the viscous boundary layers near the static walls, which is the only physical manifestation of the third dimension in these 2D equations.\\
\begin{figure}[]
\begin{center}
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ l }
{(a)} \\
\makecell{ \\ \vspace{6mm} \rotatebox{90}{{$\Euv$}}} \makecell{\includegraphics[width=0.4\textwidth]{Fig1aT-eps-converted-to.pdf}} \\
\hspace{40mm} $t$\\
{(b)} \\
\makecell{ \\ \vspace{6mm} \rotatebox{90}{{$\meanfoco$}}}
\makecell{\includegraphics[width=0.4\textwidth]{Fig1bT-eps-converted-to.pdf}} \\
\hspace{40mm} $\kappa$\\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\end{center}
\caption{(a) Nonlinear evolution of linear optimals of wavenumber $\amax$ at $\rrc = 0.9$ and several levels of initial energy: for $\Ezero<\ELD=3.0577\times10^{-6}$ (blue), the flow visits the edge state but relaminarizes, whereas it departs the edge state and becomes turbulent for $\Ezero>\ELD$ (red). Once turbulent, the flow sporadically relaminarizes on multiple occasions (see inset), but otherwise remains turbulent indefinitely. (b) Fourier spectra at select instants in time showing that a flow with $\Ezero=3.0577\times10^{-6}>\ELD$ sustains turbulence. While on the edge, the Fourier spectra follow an $\exp(-3\kappa/2)$ trend (dash-dotted black line). Once turbulent, a time-average of the spectra (purple data set) follows a $\kappa^{-5/3}$ trend (dashed black line).}
\label{fig:delin_ex}
\end{figure}
\begin{figure}[]
\begin{center}
\begin{tabular}{ c }
\hspace{1mm} \makecell{\includegraphics[width=0.55\columnwidth]{ColorbarGSUD-eps-converted-to.pdf}} $|\hat{\omega}_{z,|\kappa| \geq 10}|$\\
\makecell{\rotatebox{90}{{$y$}}}
\makecell{\includegraphics[width=0.94\columnwidth]{Fig3-eps-converted-to.pdf}} \\
\hspace{4mm} $x$\\
\end{tabular}
\end{center}
\caption{Streamwise high-pass filtered snapshot of vorticity perturbation $|\hat{\omega}_{z,|\kappa| \geq 10}|$ from DNS at $\rrc = 0.9$ and $\amax$, after the transition to turbulence, at $t=1.1\times10^4$.}
\label{fig:DNS_snap4}
\end{figure}
Fig.~\ref{fig:delin_ex}(a) depicts the most interesting set of simulations for which the initial perturbation has been calculated to maximize
the transient growth of energy over all possible target times, but whose streamwise wavenumber $\alpha$ was chosen to match that of the least damped TS wave $\alpha_{\rm max}$, obtained for the leading eigenmode from linear stability analysis \cite{Camobreco2020transition}. The Reynolds number $Re=U_0 L/\nu$ is fixed at $0.9Re_c$. A turbulent state is indeed reached for any normalized initial energy
$E_0>\ELD$, where $E_0=E/\EB$, $\EB$ is the energy of the laminar base flow, and the delineation energy $\ELD$, discussed in the following section, lies within $3.0576\times10^{-6}<\ELD<3.0577\times10^{-6}$. Evidence of a turbulent state is found in the turbulent spectra $\meanfoco(\kappa)$ of Fig.~\ref{fig:delin_ex}(b), which show how the perturbation's kinetic energy is distributed between flow components corresponding to different streamwise wavenumbers $\kappa$: while non-turbulent states contain energy in a few of the lower wavenumber modes and others hit the noise floor (around $10^{-13}$), all modes are energized in the turbulent cases \citep{Grossmann2000onset}, with an extended inertial range in which $\meanfoco(\kappa)\sim \kappa^{-5/3}$ \cite{Pope2000turbulent}. These two features are typical of turbulent flows. A visual manifestation of the erratic and multiscale nature of the flow is seen in the snapshot of vorticity in Fig.~\ref{fig:DNS_snap4}.\\
These simulations establish the existence of subcritical turbulence in quasi-2D flows. In addition, they reveal
two remarkable features of the turbulence state so attained. First, it is intermittent, with the flow exhibiting sporadic excursions to a non-turbulent state different from the laminar one, but invariably returning to turbulence, as highlighted in Fig.~\ref{fig:schematic}(b) and the inset of Fig.~\ref{fig:delin_ex}(a). This was also observed in the supercritical regime of the same problem \cite{Camobreco2020transition}. It is reminiscent of
the spatial intermittency
in various 3D shear flows, some of which were subject to Coriolis, buoyancy or Lorentz forces \cite{Cros2002spatiotemporal, Barkley2007mean, Moxey2010distinct, Brethouwer2012turbulent, Khapko2014complexity}, and which exists at any finite Reynolds number and in domains of any length.
The duration or spatial extent of the laminar patches was observed by Refs. \cite{Hof2006finite,Avila2013nature} to reduce superexponentially with $\mathit{Re}$.\\
Second, although an energy threshold where the flow
does not immediately relaminarize exists for $\mathit{Re}$ down to $0.3\ReyCrit$, transition to an indefinitely sustained turbulent state was only found for $\mathit{Re} \gtrsim 0.8\ReyCrit$ (see SI). Hence, the level of criticality required to sustain turbulence is much higher than in 3D flows. In the range $0.4\lesssim \rrc \lesssim 0.8$, a single turbulent episode with finite lifetime proportional to $\mathit{Re}$ was observed (see SI).\\ %
\subsection*{Nonlinear Tollmien--Schlichting waves are the tipping point between laminar and turbulent states}
Having established that subcritical turbulence exists, we now turn to the question of which pathway is followed from the laminar base flow to the turbulent state. To answer this question, we seek the `edge state', \emph{i.e.}~the state acting as a tipping point from which the flow can either revert to its original laminar state, or break down into turbulence \cite{Skufca2006Edge}.
The edge state is reached for the initial value of the perturbation energy $\ELD$, which separates perturbations triggering turbulence from those decaying. $\ELD$ is thus called the \emph{delineation energy}. The closer the initial energy to $\ELD$, the longer the edge state survives. $\ELD$ is found iteratively with a bisection method
\cite{Itano2001dynamics},
and the edge state corresponding to the energy plateau visible on Fig.~\ref{fig:delin_ex}(a) is depicted in Fig.~\ref{fig:comp_DNS_WNL}(a). It consists of a travelling wave of very similar topology to the infinitesimal TS wave. While TS waves play little role in 3D shear flows, where they are superseded by fully 3D mechanisms, the similarity observed here suggests they may play a role in the quasi-2D transition to turbulence.
To investigate this possibility, we calculated a weakly nonlinear flow state, keeping only the base flow, the leading eigenmode and its complex conjugate (both of wavenumber $\kappa$), as well as the modulation of the base flow (wavenumber 0) and the second harmonic ($2\kappa$) resulting from their nonlinear interactions \cite{Hagan2013weakly,Camobreco2020transition}. This enabled us to compare the fully nonlinear flow state (with all possible modes) obtained from DNS, with its asymptotic approximation to third order in the perturbation amplitude and in which nonlinearities only arise out of combinations of the leading TS wave.
\begin{figure}[h]
\begin{center}
\begin{tabular}{ l }
{(a)} \\
\hspace{1mm} \makecell{\includegraphics[width=0.6\columnwidth]{ColorbarRB-eps-converted-to.pdf}} $\vhp,\woo$ \\
\makecell{\rotatebox{90}{{$y$}}}
\makecell{\includegraphics[width=0.94\columnwidth]{Fig4a0-eps-converted-to.pdf}} \\
\hspace{43mm} $x$\\
{(b)} \\
\hspace{20mm} \color{mybd}{$\uot$} \hspace{31mm} \color{mygd}{$\wtt$} \\
\makecell{\rotatebox{90}{{$y$}}}
\makecell{\includegraphics[width=0.94\columnwidth]{Fig3T-eps-converted-to.pdf}} \\
\hspace{37mm} \color{myrd}{$\woo$} \\
\end{tabular}
\end{center}
\caption{(a) Contours of the spanwise velocity field $\mathbf u\cdot\mathbf e_y$ from DNS at $\rrc = 0.9$ and $\amax$, representing the edge state (at $t=7.48\times10^3$), a coherent travelling wave. The least damped TS wave $\woo$ obtained from linear stability analysis is overlayed (black lines; same contour levels), and has remarkably similar appearance to $\vhp$. (b) Comparisons between the modes accounted for in the weakly nonlinear analysis ($\woo$, $\wtt$ and $\uot$) (dashed black lines), and corresponding Fourier components from DNS with $\Ezero = 3.0577\times10^{-6}>\ELD$ at different times (colored lines; see legend). DNS results are at $\kappa=0$, $1$ and $2$, respectively (streamwise, spanwise, spanwise velocity components, respectively). The solution remains in the neighbourhood of the edge state for $1.92\times10^3 <t<7.48\times10^3$, while $t=10.08\times10^3$ is during the departure from the edge state toward turbulence.}
\label{fig:comp_DNS_WNL}
\end{figure}
Fig.~\ref{fig:comp_DNS_WNL}(b) shows the velocity profiles from each of the modes accounted for in the weakly nonlinear analysis, plotted alongside each of the corresponding Fourier components of the same wavelength extracted from the full DNS evolved from the linear optimal with $\alpha=\amax$ with initial energy close to $\ELD$. Both
the leading eigenmode ($\kappa=1$) and its nonlinear interaction ($\kappa=2$) match their DNS counterpart to high precision in the early stage of evolution. The modulated base flows ($\kappa=0$) exhibit small differences at this stage,
that vanish as the influence of our particular choice of initial condition does. Additionally, the cumulated kinetic energy of all three components forming the weakly nonlinear approximation represents over $93.7$\%
of the total energy in the DNS while on the edge (for times between $0.192\leq t/10^{4} \leq 0.748$). These results prove that the build-up of the edge state originates almost exclusively from the dynamics of the TS wave. Even after its breakdown to full turbulence, these leading three Fourier modes still contain $\approx 60$\% of the total kinetic energy. As such, this transition mechanism differs radically from its counterpart in 3D flows, where a bypass transition involving rapidly growing streamwise structures takes place at much lower levels of criticality \cite{Zammert2019transition}, and where the TS waves are too damped to play any role.
\subsection*{Fast transient growth is not the pathway to quasi-2D turbulence}
With the pathway to the edge state and then to turbulence now clarified, the question remains as to its robustness against our choice of initial condition. A corollary question is that of the role played by optimal transients that initiated our DNS. To answer this question, DNS were performed with the initial conditions chosen as the modes optimizing growth at increasingly large times $T = \tau/\tauOpt$ (where $\tauOpt$ is the time of optimal growth for a given
wavenumber $\alpha$, \emph{i.e.} $\tauOpt\simeq31.0$ for $\alpha=\alpha_{\rm \max}$ at $\rrc=0.9$).
The corresponding delineation energy provides a measure of how easily these transients ignite turbulence and therefore of their role in doing so.
First, simulations show that all optimals for a given $\alpha$ evolve into the same edge state (see SI), which therefore appears independent of initial conditions among those tested.
Second, as $T$ increases, the profiles of spanwise velocity of the initial condition optimizing growth with $\alpha=\amax$, shown in Fig.~\ref{fig:int_opt_comp}(b), converge toward the leading adjoint mode, depicted in Fig.~\ref{fig:int_opt_comp}(a), which by construction optimally energizes the TS wave (see SI).
Once the time of optimal growth $T$ is large enough for the optimal initial condition to have converged toward the leading adjoint mode, both the corresponding transient growth at the target optimization time and the delineation energy drop, as shown in Fig.~\ref{fig:int_opt_comp}(c), and by $T=8$, the delineation energy of the optimal matches that of the leading adjoint to $\sim \pm 0.01\%$. In other words, initial conditions leading to lower transient growth turn out to be \emph{more} efficient at triggering turbulence. This approach was repeated with modes optimizing transient growth across streamwise wavenumbers (see SI); all yielded higher delineation energy than initial conditions most efficiently energizing the leading eigenmode. This establishes that the leading adjoint eigenmode is a more efficient initial condition to reach turbulence than any initial condition producing optimal transient growth. Consequently, transient growth does not greatly favour the transition, unlike in 3D shear flows where the transient growth associated with the lift-up mechanism is an essential part of the transition process \cite{Reddy1998stability,Pringle2012minimal}.
Lastly, the same procedures applied at lower levels of criticality (see SI) produced the same reduction in delineation energy with increasing $T$ (increasing resemblance to the adjoint), and also exhibited edge states independent of $T$. For $0.3 \lesssim \rrc \lesssim 0.8$, after departure from the edge state, and in some cases after experiencing a finite turbulent episode, a secondary stable state was observed \cite{Jimenez1990transition,Falkovich2018turbulence}, which was equally independent of initial condition.
\section*{Materials and Methods}
\subsection*{Physical model}
All calculations are performed on
a rectangular incompressible duct flow, with walls in the $(x,z)$ plane moving at a constant velocity $U_0\mathbf{e}_x$. The base flow is streamwise invariant and periodic boundary conditions are imposed in the streamwise direction ($x$).
The flow is assumed quasi-2D in the sense that all quantities are invariant along the $z$ direction except in thin boundary layers near the fixed walls in $(x,y)$ planes. Such flows are well described by shallow-water equations, where the 2D Navier--Stokes equations for the velocity and pressure averaged along the $z$ direction are supplemented by a linear Rayleigh friction term accounting for the friction incurred by these layers on the $z$-averaged flow. Depending on the expression of the friction parameter $H$, these equations represent flows in high, imposed, transverse magnetic field $B\mathbf{e}_z$ \cite{Sommeria1982why}, the flow in a thin film \cite{Buhler1996instabilities}, or a flow in strong background rotation (with the addition of the Coriolis force) \cite{Pedlosky1987geophysical}.
With length, velocity, time and pressure non-dimensionalized by $L$, $U_0$, $L/U_0$ and $\rho U_0^2$, respectively, these equations are written:
\begin{equation}\label{eq:Q2Dc}
\bnp \cdot \uvp = 0,
\end{equation}
\begin{equation} \label{eq:Q2Dm}
\pde{\uvp}{t} + (\uvp \cdot \bnp) \uvp = - \bnp \pp + \frac{1}{\mathit{Re}}\bnp^2 \uvp - \frac{H}{\mathit{Re}} \uvp,
\end{equation}
where $\uvp=(\up$, $\vp)$, $\bnp = (\partial_x,\partial_y)$, $\rho$ is the fluid's density, $2L$ is the distance between the moving walls, and with non-dimensional boundary conditions $\uvp(y=\pm1) = (1,0)$.
\subsection*{Linear stability, linear optimal growth and weakly nonlinear stability}
The methods to find the leading eigenmode, the perturbation optimizing growth at a prescribed time $T$ and the weakly nonlinear interactions are derived and described in detail in Ref. \cite{Camobreco2020role}. We cursorily recall their principle here for completion. More details on their implementation and validation for the present work are given in the SI.\\
The leading eigenmode and perturbation maximizing transient growth are found by decomposing all fields into the laminar base flow and an
infinitesimal perturbation $(\uvp,\pp)=(\Uvp+\uvph,\php)$. The equations governing infinitesimal perturbations follow from the linearisation of (\ref{eq:Q2Dc}-\ref{eq:Q2Dm}) about the base laminar solution $(\Uvp,0)$. The leading eigenmode is found by decomposing the perturbation into normal modes $(\mathbf {\tilde{u}_\perp}, \tilde{p}_\perp ) \exp(\ii[\alpha x-\omega t])$ of wavenumber $\alpha$ and complex frequency $\omega$. Under this form, the perturbation equation becomes an eigenvalue problem for $\omega$. The leading eigenmode is the least damped by the linear dynamics and corresponds to the eigenvalue of highest imaginary part.\\
The initial condition maximizing transient growth at time $\tau$ is found by optimizing the energy growth functional $G=||\uvph(t=\tau) || / || \uvph(t=0)||$, for a prescribed target time $\tau$, and wavenumber $\alpha$, under the constraints of satisfying the perturbation equation and that the initial energy be normalized to unity. $G$ represents the gain in perturbation kinetic energy under the norm $|| \uvph || = \int \uvph \cdot \uvph \,\mathrm{d}\Omega$ \citep{Barkley2008direct}, where $\Omega$ is the fluid domain.
The optimization is performed on the linearized rather the full nonlinear equation, as it is significantly more efficient and we verified that the two hardly differ for this problem \cite{Camobreco2020role}.
\begin{figure}[h]
\begin{center}
\addtolength{\extrarowheight}{-10pt}
\addtolength{\tabcolsep}{-2pt}
\begin{tabular}{ l }
{(a)} \\
\makecell{\rotatebox{90}{$y$}}
\makecell{\includegraphics[width=0.4\textwidth]{Fig5a-eps-converted-to.pdf}}
\makecell{\rotatebox{90}{\color{white}{$10^6\ELD$}}} \\
\hspace{40mm} $x$ \\
{(b)} \\
\makecell{\rotatebox{90}{$y$}}
\makecell{\includegraphics[width=0.4\textwidth]{Fig4TaH-eps-converted-to.pdf}}
\makecell{\rotatebox{90}{\color{white}{$10^6\ELD$}}} \\
{(c)} \\
\makecell{\vspace{2mm} \rotatebox{90}{\color{mybl}{$E(T)$}, \color{mybd}{$\max(E(t))$}}}
\makecell{\includegraphics[width=0.4\textwidth]{Fig4Tb-eps-converted-to.pdf}}
\makecell{\vspace{4mm} \rotatebox{90}{\color{myrd}{$10^6\ELD$}}} \\
\hspace{37mm} $T = \tau/\tauOpt$ \\
\end{tabular}
\addtolength{\tabcolsep}{+2pt}
\addtolength{\extrarowheight}{+10pt}
\end{center}
\caption{(a) Contours of the spanwise velocity field $\mathbf u\cdot\mathbf e_y$ representing the leading adjoint mode at $\rrc = 0.9$ and $\amax$, which optimally energizes the least damped TS wave as $\tau\rightarrow \infty$; contour levels as for Fig.~\ref{fig:comp_DNS_WNL}(a).
(b) Comparison between the initial condition yielding the linear transient optimal at $\tau=T\tauOpt$ (solid lines) and the leading adjoint mode (dashed lines), for various $T$ (horizontally shifted for clarity).
(c) The corresponding delineation energy obtained via bisection from DNS, and the energy growth attained from linear analysis, for each $T$. In spite of reduced transient growth, $\ELD$ decreases as $T$ increases. Once the initial condition resembles the leading adjoint mode, i.e.~$T\gtrsim4$, the maximum growth attained at any $\tau\leq T$, and the delineation energy, plateau.}
\label{fig:int_opt_comp}
\end{figure}
The initial condition is computed in two independent ways: either by a matrix method with Chebyshev discretization in the spanwise direction \cite{Camobreco2020transition}, or through a classical adjoint-based formulation where the optimal initial condition is found by iterating from a random initial condition through a linear evolution from $t=0$ to $t=\tau$ and then backwards in time through the adjoint of the linear evolution from $t=\tau$ to $t=0$ \cite{Camobreco2020role}.\\
The weakly nonlinear equations are a more precise version of the perturbation equations obtained by truncating the governing equations to the third order in the perturbation amplitude, compared to the linearized version used to calculate the leading eigenmode and the perturbation maximising transient growth. They are found by expanding the harmonics of the perturbation in powers of its amplitude. For example, for a leading eigenmode of amplitude $\epsilon$, the n$^\mathrm{th}$ harmonic of the spanwise velocity component is written as:
\begin{equation}\label{eq:amp_exp}
\vhpn = \sum_{m=0}^{\infty}\epsilon^{|n|+2m}\tilde{A}^{|n|}|\tilde{A}|^{2m}\vhpnnm,
\end{equation}
where $\vhpnnm$ denotes a perturbation ($n$ refers to the harmonic, $|n|+2m$ to the amplitude's order) and $\tilde{A} = A/\epsilon$ is the normalized amplitude. Nonlinear interaction between the linear mode $\woo$ and itself excites a second harmonic $\wtt$. Nonlinear interaction between the linear mode $\woo$ and its complex conjugate $\wno$ generates a modification to the base flow $\uot$. The full equations governing the n$^\mathrm{th}$ harmonic of the base flow and perturbation follow from inserting this decomposition into Eqs. (\ref{eq:Q2Dc}-\ref{eq:Q2Dm}); they are expressed in full in Ref. \citep{Camobreco2020transition} and the general method is detailed in Ref. \citep{Hagan2013weakly}.\\
The full nonlinear evolution of the flow is obtained by solving the system (\ref{eq:Q2Dc}-\ref{eq:Q2Dm}) numerically. The initial condition is the laminar base flow and a perturbation computed via linear transient growth optimization. The amplitude of the perturbation is normalized to set the perturbation energy $E$ to the value required for each of the cases presented.
\subsection*{Numerical implementation}
The time evolution of the velocity and pressure through the full system (\ref{eq:Q2Dc}-\ref{eq:Q2Dm}), the linearized version of this system, and through their adjoint equations, is calculated numerically using a primitive variable spectral element solver. The primitive variable solver, incorporating the friction term, has been previously introduced and validated, in this and similar configurations \citep{Camobreco2020role, Camobreco2020transition, Cassels2019from3D, Hussam2012optimal}. The time integration scheme is of the third order backward, with operator splitting, while high-order Neumann pressure boundary conditions are imposed on the lateral walls to maintain third order time accuracy \citep{Karniadakis1991high}. The $x$-$y$ plane is discretized with quadrilateral elements, within which Gauss--Legendre--Lobatto nodes corresponding to polynomial order $\Np$ are distributed. The mesh design is based on the resolution testing of Refs. \cite{Camobreco2020role,Camobreco2020transition}. 12 spectral elements are equispaced in the streamwise direction, and 48 in the wall-normal direction (geometrically biased toward both lateral walls, with bias ratio 0.7). A polynomial order of $\Np=13$ was sufficient for computing linear optimals, and increased to $\Np=19$ for nonlinear evolution \cite{Camobreco2020role}. A timestep of $\Delta t = 1.25\times10^{-3}$, following Refs. \cite{Camobreco2020role,Camobreco2020transition}, was also used (although $\Delta t$ must be greatly reduced once turbulence is triggered). In addition, the Dedalus solver \cite{Burns2020dedalus} was employed to independently verify the edge state behaviour and delineation energies (see SI).\\
\section*{Discussion}
This study establishes that subcritical turbulence exists in quasi-2D shear flows and that it can be reached directly from a quasi-2D laminar state, without a transition though a state of 3D turbulence. The quasi-2D transition mechanism bears important similarities with its 3D cousin: it is ignited by a perturbation of finite amplitude and first reaches an edge state that is seemingly independent of this initial perturbation. The edge state subsequently breaks down into a turbulent state if the initial perturbation energy exceeds the delineation energy $\ELD$ for that particular form of the perturbation. As in the 3D problem, the turbulence so attained is not fully developed: for instance, localized turbulence appears first in 3D pipe flows
\cite{Wygnanski1973p1,Wygnanski1975p2}.
Departure from fully established turbulence in quasi-2D shear flows expresses as
time intermittency, with sporadic excursions to a non-turbulent state that is different from the base laminar flow.
Thus, as suggested by Ref. \cite{Avila2011onset} for 3D flows, observing laminar to turbulent transition near the threshold for sustained turbulence, where sporadic relaminarization may occur,
requires an experimental setup. \\
On the other hand, the subcritical transition in quasi-2D shear flows exhibits specificities that distinguish it sharply from the 3D one. Chiefly, the 3D lift-up mechanism that underpins the transition in 3D can be ignited at very low criticality. So low in fact, that at this level, the Tollmien--Schlichting waves are strongly suppressed by the linear dynamics despite being the least damped perturbation of infinitesimal amplitude.
Because of this, they are never observed and do not play a role in the 3D transition.
In quasi-2D flows by contrast, the 3D mechanism is absent and our study shows that the dynamics are dominated by the TS waves, with the edge state resulting directly from their weakly nonlinear evolution. The edge even shares their topology, but for a small nonlinear modulation. This leading role played by the TS wave may also explain why the transition in quasi-2D flows is only weakly subcritical, compared to its 3D counterpart: at lower levels of criticality, TS waves are so strongly linearly damped that their nonlinear growth is stifled.\\
This new transition mechanism reopens a great many of the questions that had found an answer in the 3D case: for example, how does the intermittency or localization of the turbulence evolve into the supercritical regime, for example following the mechanism outlined by Ref. \cite{Mellibovsky2015mechanism}? Does the transition to the fully turbulent state obey a second order phase transition of the universality class of directed percolations as for other shear flows \cite{Lemoult2016directed}? Conversely, much remains to be discovered on the subcritical response to finite amplitude perturbations: how does the delineation energy vary with criticality, especially considering the relatively short subcritical range in which turbulence can be sustained? Can this subcritical response be manipulated to prevent or promote turbulence (for example, to enhance heat transfer in the heat exchangers of plasma fusion reactors)? On the one hand these questions call for expensive numerical simulations, but also for experiments with well controlled perturbations, since this first evidence of subcritical transitions in quasi-2D shear flows is currently purely numerical.
\acknow{C.J.C.\ receives support from the Australian Government Research Training Program (RTP). This research was supported by the Australian Government via the Australian Research Council (Discovery Grant DP180102647), the National Computational Infrastructure (NCI) and Pawsey Supercomputing Centre (PSC), by Monash University via the MonARCH HPC cluster and by the Royal Society under the International Exchange Scheme between the UK and Australia (Grant E170034).}
\showacknow{}
|
2,869,038,154,889 | arxiv | \section{Introduction}\label{introduction}
Panel data allows researchers to better account for individual heterogeneity and estimate richer models than cross-sectional data. Traditionally, the literature on panel data models focused on fully parametric models. Popular textbooks written by \citet{arellano_2003}, \citet{hsiao_2003}, and \citet{baltagi_2013} give excellent overviews of such models. However, parametric models may be misspecified, and more flexible panel data models may be needed.
Semiparametric models, such as partially linear or varying coefficient models, serve as an attractive alternative to fully parametric models. While being more flexible than parametric models, they are more tractable than fully nonparametric models and alleviate the curse of dimensionality. \citet{ai_li_2008}, \citet{su_ullah_2011}, \citet{rodriguez-poo_soberon_2017}, and \citet{parmeter_racine_2018} provide excellent surveys of recent developments in semiparametric panel data models.
While there is a growing literature on estimation of semiparametric panel data models, the literature on specification testing in this setting remains scarce. There are three possible reasons for this. First, in the presence of fixed effects, one needs to transform data to eliminate them before the model can be estimated. Estimating the transformed model in itself can be challenging when kernel methods are used.
Second, the asymptotic theory for consistent specification tests in semiparametric panel data models can be challenging. For instance, \citet{henderson_et_al_2008} propose kernel-based specification tests both for parametric and semiparametric fixed effects panel data models and suggest using the bootstrap to obtain critical values, but they do not derive asymptotic properties of their tests. In turn, \citet{lin_et_al_2014} develop an asymptotic theory for kernel-based specification tests for panel data models with fixed effects, but they only consider parametric models.
Third, it has long been known in the literature on consistent specification tests that asymptotic approximations often do not work well in finite samples even with cross-sectional data (see, e.g., \citet{li_wang_1998}). The bootstrap is typically used to improve the finite sample performance of consistent specification tests, but it may be computationally costly.
In this paper I rely on the results on series estimation of fixed effects panel data models from \citet{baltagi_li_2002} and \citet{an_et_al_2016} and develop a consistent Lagrange Multiplier (LM) type specification test for semiparametric panel data models. My test overcomes all three challenges described above.
First, the use of series methods leads to a model that is linear in parameters. As a result, transforming the data, e.g. applying the within transformation or taking first differences, to eliminate fixed effects is straightforward. Thus, the test is simple to implement.
Second, as in cross-sectional models in \citet{korolev_2019}, the projection property of series estimators allows me to develop a degrees of freedom correction. Intuitively, when series methods are used, the restricted residuals are orthogonal to the series terms included in the restricted model. This means that even under the alternative, only a subset of moment conditions, rather than all of them, can be violated, which in turn affects the normalization of the test statistic. This degrees of freedom correction has two important consequences.
From the theoretical point of view, it leads to a tractable asymptotic theory for the test and allows me to obtain refined asymptotic results. In my asymptotic analysis, I decompose the test statistic into the leading term and the remainder. By relying on the projection nature of series estimators, I can directly account for the estimation variance, so that only bias enters the remainder term. Because of this, I only need to control the rate at which bias goes to zero to bound the remainder term, while variance can remain large. As a result, I can derive the asymptotic distribution of the test statistic under fairly weak rate conditions.
From the practical point of view, the degrees of freedom correction substantially improves the finite sample performance of the test. While I propose a wild bootstrap procedure and establish its asymptotic validity, I show using simulations that the asymptotic version of the proposed test with the degrees of freedom correction performs almost as well as its wild bootstrap version. Hence, the degrees of freedom correction serves as a computationally attractive analytical way to obtain a test with good small sample behavior.
The remainder of the paper is organized as follows. Section~\ref{model_test} introduces the model and describes how to construct the series-based specification test for semiparametric fixed effects models. Section~\ref{asymptotics} develops the asymptotic theory for the proposed test. Section~\ref{simulations} studies the behavior of the proposed test in simulations. Section~\ref{empirical_example} applies my test to the data from \citet{cornwell_rupert_1988} and \citet{baltagi_khanti-akom_1990}. Section~\ref{conclusion} concludes.
\ref{appendix_tables_figures} collects all tables and figures. \ref{appendix_proofs} contains proofs of my results.
\section{The Model and Proposed Test}\label{model_test}
I consider a general nonparametric panel data model with fixed effects:
\begin{align}\label{np_model}
Y_{it} = g(X_{it}) + u_{it} = g(X_{it}) + \mu_i + \varepsilon_{it}, \quad E[\varepsilon_{it} | X_{i}, \mu_i ] = 0,
\end{align}
where $X_i = (X_{i1},...,X_{iT})'$, $t = 1, ..., T$, and $i = 1, ..., n$. $\mu_i$ denotes the fixed effect, which captures unobserved heterogeneity and may be correlated with the regressors $X_i$. In my asymptotic analysis, I will assume that $T$ is fixed while $n$ grows to infinity.
The goal of this paper is to test that the true model is semiparametric, i.e. that
\begin{align}\label{generic_h0}
H_0^{SP}: P_{X} \left(g(X_{it}) = f(X_{it}, \theta_0, h_0) \right) = 1 \text{ for some } \theta_0 \in \Theta, h_0 \in \mathcal{H},
\end{align}
where $f: \mathcal{X} \times \Theta \times \mathcal{H} \to \mathbb{R}$ is a known function, $\theta \in \Theta \subset \mathbb{R}^{d}$ is a finite-dimensional parameter, and $h \in \mathcal{H} = \mathcal{H}_1 \times ... \times \mathcal{H}_q$ is a vector of unknown functions. For instance, if the semiparametric model is partially linear, then $f(X_{it},\theta,h) = X_{1it}' \theta + h(X_{2it})$, where $X_{it} = (X_{1it}',X_{2it}')'$. Many other semiparametric models can also be written in this form.
The global alternative is
\begin{align}\label{generic_h1}
H_1: P_{X} \left(g(X_{it}) \neq f(X_{it}, \theta, h) \right) > 0 \text{ for all } \theta \in \Theta, h \in \mathcal{H}
\end{align}
\subsection{Series Estimators}
As in \citet{korolev_2019}, I use series methods to replace unknown functions with their finite series expansions. Namely, for any variable $z$, let $Q^{a_n}(z) = (q_1(z), ..., q_{a_n}(z))'$ be an $a_n$-dimensional vector of approximating functions of $z$, where the number of series terms $a_n$ is allowed to grow with the sample size $n$. Then an unknown function $g(z)$ can be approximated as $g(z) \approx \sum_{j=1}^{a_n}{q_j(z) \gamma_j} = Q^{a_n}(z)' \gamma$. I replace all unknown functions in $f(X_{it}, \theta, h)$ with their finite series expansions and write the semiparametric model in a series form as
\begin{align}\label{sp_model_series}
Y_{it} = W_{it}' \beta_1 + R_{it} + \mu_i + \varepsilon_{it},
\end{align}
where $W_{it} := W^{m_n}(X_{it}) := (W_{1}(X_{it}), ..., W_{m_n}(X_{it}))'$ are appropriate regressors or basis functions, such as power series or splines, $m_n$ is the number of parameters in the semiparametric null model, $R_{it} = f(X_{it}, \theta, h) - W_{it}' \beta_1$ is the approximation error.
\begin{comment}
For any variable $z$, let $Q^{a_n}(z) = (q_1(z), ..., q_{a_n}(z))'$ be an $a_n$-dimensional vector of approximating functions of $z$, where the number of series terms $a_n$ is allowed to grow with the sample size $n$. Possible choices of series functions include:
\begin{enumerate}[(a)]
\item
Power series. For univariate $z$, they are given by:
\begin{align}\label{series}
Q^{a_n}(z) = (1, z, ..., z^{a_n-1})'
\end{align}
\item
Splines. Let $s$ be a positive scalar giving the order of the spline, and let $t_{1}, ..., t_{a_n - s - 1}$ denote knots. Then for univariate $z$, splines are given by:
\begin{align}\label{splines}
Q^{a_n}(z) = (1, z, ..., z^s, \mathbbm{1}\{ z > t_1 \} (z-t_1)^s, ..., \mathbbm{1}\{ z > t_{a_n - s -1} \} (z - t_{a_n - s -1})^s)
\end{align}
\end{enumerate}
Multivariate power series or splines can be formed from products of univariate ones.
\end{comment}
\subsection{Test Statistic}
To construct a specification test, I include additional series terms, $Z_{it} := Z^{r_n}(X_{it}) := (Z_{1}(X_{it}), ..., Z_{r_n}(X_{it}))'$, that capture possible deviations from the null hypothesis:
\begin{align}\label{np_model_series}
Y_{it} = W_{it}' \beta_1 + Z_{it}' \beta_2 + R_{it} + \mu_i + \varepsilon_{it} = P_{it}' \beta + R_{it} + \mu_i + \varepsilon_{it},
\end{align}
where $P_{it} := P^{k_n}(X_{it}) := (W_{it}', Z_{it}')'$, $k_n = m_n + r_n$ is the total number of parameters, and $\beta = (\beta_1', \beta_2')'$. For instance, in the partially linear model example above, the additional series terms can include nonlinear terms in $X_{1it}$ and interactions between $X_{1it}$ and $X_{2it}$.
Due to the presence of fixed effects $\mu_i$ that may be correlated with $X_{it}$, it is problematic to estimate or test this model directly. Instead, I use first differencing or the within transformation to get rid of fixed effects. The model becomes
\[
\hat{Y}_{it} = \hat{W}_{it}' \beta_1 + \hat{Z}_{it}' \beta_2 + \hat{R}_{it} + \hat{\varepsilon}_{it} = \hat{P}_{it}' \beta + \hat{R}_{it} + \hat{\varepsilon}_{it},,
\]
where in the former case for any variable $A_{it}$, $\hat{A}_{it} = A_{it} - A_{i,t-1}$, and in the latter case $\hat{A}_{it} = A_{it} - \frac{1}{T}\sum_{s=1}^{T}{A_{is}}$. The specification test reduces to testing the hypothesis $\beta_2 = 0$.
For any variable $A_{it}$, let $A_i = (A_{i1},...,A_{iT})'$ and $A = (A_1', ..., A_n')'$. The restricted estimate of $\beta_1$ is obtained from the regression of $\hat{Y}$ on $\hat{W}$ and is given by
\[
\tilde{\beta}_1 = (\hat{W}' \hat{W})^{-1} \hat{W}' \hat{Y},
\]
and the restricted residuals are
\[
\tilde{e} = \hat{Y} - \hat{W} (\hat{W}' \hat{W})^{-1} \hat{W}' \hat{Y} = M_{\hat{W}} \hat{Y},
\]
where $M_{\hat{W}} = I - \hat{W} (\hat{W}' \hat{W})^{-1} \hat{W}'$. If the null is true, it can be shown that
\[
\tilde{e} = M_{\hat{W}} \hat{\varepsilon} + M_{\hat{W}} \hat{R}
\]
The test will be based on the moment condition $E[\hat{P}_i' \hat{\varepsilon}_i] = 0$. The sample analog of this moment condition is $\sum_{i=1}^{n}{\hat{P}_i' \tilde{e}_i}/n$. Note that $\sum_{i=1}^{n}{\hat{W}_i' \tilde{e}_i}/n = 0$, so the test is essentially based on $\sum_{i=1}^{n}{\hat{Z}_i' \tilde{e}_i}/n$.
Let $\tilde{Z} = M_{\hat{W}} \hat{Z}$. The LM type test statistic is given by:
\begin{align}\label{xi_hc}
\xi_{HC} &= \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right)
\end{align}
Alternatively, in the homoskedastic case, it can be simplified as follows:
\begin{align}\label{xi}
\xi = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right),
\end{align}
where $\tilde{\Sigma}_T = \frac{1}{n} \sum_{i=1}^{n}{\tilde{e}_i \tilde{e}_i'}$.
These two test statistics resemble the parametric LM test statistic. However, the number of restrictions $r_n$ is allowed to grow to infinity. Thus, in order to obtain convergence in distribution, a normalization is needed. The normalized test statistics are given by
\begin{align}\label{t_test_statistics}
t_{HC} = \frac{\xi_{HC} - r_n}{\sqrt{2 r_n}} \quad \text{and} \quad t = \frac{\xi - r_n}{\sqrt{2 r_n}}
\end{align}
I will show in the next section that under appropriate conditions, the normalized test statistics are asymptotically standard normal.
\section{Asymptotic Theory}\label{asymptotics}
In this section, I develop the asymptotic theory for the proposed specification test. I analyze its behavior under the null hypothesis and under a fixed alternative.
\subsection{Behavior of the Test Statistic under $H_0$}
This section derives the asymptotic distribution of the test statistic when the semiparametric model is correctly specified. I start with my assumptions. First, I impose some regularity conditions on the data generating process.
\begin{assumption}\label{dgp}
$(Y_{it}, X_{it}')' \in \mathbb{R}^{1+d_x}, d_x \in \mathbb{N}, i = 1, ..., n$ are independent across individuals, i.e. $(Y_i', X_i')'$ are i.i.d. random draws of the random variables $(Y_1', X_1')'$, and the support of $X_1$, $\mathcal{X}$, is a compact subset of $\mathbb{R}^{d_x}$.
\end{assumption}
\begin{assumption}\label{errors_unified}
Let $\varepsilon_i = Y_i - E[Y_i | X_i]$. The following two conditions hold:
\begin{enumerate}[(a)]
\item
$\Sigma(x) = E[\varepsilon_{i} \varepsilon_{i}' | X_i = x]$ is bounded.
\item
$E[\varepsilon_{it}^4 | X_i]$ is bounded.
\end{enumerate}
\end{assumption}
The following assumption deals with the behavior of the approximating series functions. From now on, let $\| A \| = [tr(A'A)]^{1/2}$ be the Euclidian norm of a matrix $A$. Let $x \in R^{d_x}$ be a realization of the random variable $X_{it}$.
\begin{assumption}\label{series_norms_eigenvalues}
For each $m$, $r$, and $k$ there are matrices $B_1$, and $B_2$ such that, for $\bar{W}^{m}(x) = B_1 \hat{W}^{m}(x)$, $\bar{Z}^{r}(x) = B_2 \hat{Z}^{r}(x)$, and $\bar{P}^{k}(x) = (\bar{W}^{m}(x)', \bar{Z}^{r}(x)')$,
\begin{enumerate}[(a)]
\item
There exists a sequence of constants $\zeta(\cdot)$ that satisfies the conditions $\sup_{x \in \mathcal{X}} \| \bar{W}^{m}(x) \| \leq \zeta(m)$, $\sup_{x \in \mathcal{X}} \| \bar{Z}^{r}(x) \| \leq \zeta(r)$, and $\sup_{x \in \mathcal{X}} \| \bar{P}^{k}(x) \| \leq \zeta(k)$.
\item
The smallest eigenvalue of $E[\bar{P}^{k}(X_{it}) \bar{P}^{k}(X_{it})']$ is bounded away from zero uniformly in $k$.
\end{enumerate}
\end{assumption}
\begin{assumption}\label{series_approx}
Suppose that $H_0$ holds. There exist $\alpha > 0$ and $\beta_1 \in \mathbb{R}^{m_n}$ such that
\[
\sup_{x \in \mathcal{X}}{| f(x,\theta_0,h_0) - W^{m_n}(x)' \beta_1 |} = O(m_n^{-\alpha})
\]
\end{assumption}
$\beta_1$ in this assumption can be defined in various ways. One natural definition is projection: $\beta_1 = E[\hat{W}_{it} \hat{W}_{it}']^{-1} E[\hat{W}_{it} \hat{f}(X_{i},\theta_0,h_0)]$, where $\hat{f}(\cdot)$ is an appropriate transformation of $f(\cdot)$.
\begin{theorem}\label{asy_distr_t_r_n_hc}
Assume that Assumptions \ref{dgp}, \ref{errors_unified}, \ref{series_norms_eigenvalues}, and \ref{series_approx} are satisfied, and the following rate conditions hold:
\begin{align}
\label{rate_cond_r_1_hc} (m_n/n + m_n^{-2\alpha}) \zeta(r_n)^2 r_n^{1/2} &\to 0 \\
\label{rate_cond_r_2_hc} \zeta(r_n) r_n / n^{1/2} &\to 0 \\
\label{rate_cond_r_3_hc} \zeta(k_n) m_n^{1/2} k_n^{1/2}/n^{1/2} &\to 0 \\
\label{rate_cond_r_4_hc} n m_n^{-2\alpha}/ r_n^{1/2} &\to 0 \\
\label{rate_cond_r_5_hc} \zeta(r_n)^2/n^{1/2} &\to 0
\end{align}
Also assume that $\| \hat{\Omega} - \tilde{\Omega} \| = o_p(r_n^{-1/2})$, where
\[
\tilde{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{e}_i \tilde{e}_i' \hat{Z}_i} \text{ and } \hat{\Omega} = n^{-1}\sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i}.
\]
Then under $H_0$
\begin{align}\label{eqn_t_r_n_HC1}
t_{HC} = \frac{\xi_{HC} - r_n}{\sqrt{2 r_n}} \overset{d}{\to} N(0,1),
\end{align}
where $\xi_{HC}$ is as in Equation~\ref{xi_hc}.
If, in addition to the assumptions above, $\Sigma(x) \equiv \Sigma$ and $\| \hat{\Omega} - \tilde{\Omega} \| = o_p(r_n^{-1/2})$, where
\[
\tilde{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{\Sigma}_T \hat{Z}_i} \text{ and } \hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i},
\]
then
\[
t = \frac{\xi - r_n}{\sqrt{2 r_n}} \overset{d}{\to} N(0,1),
\]
where $\xi$ is as in Equation~\ref{xi}.
\end{theorem}
The normalization I use, $r_n$, differs from the normalization used in most series-based specification tests for parametric models with cross-sectional data, which use the total number of parameters in the nonparametric model $k_n$ (see equations (2.1) and (2.2) in \citet{hong_white_1995} and Lemma 6.2 in \citet{donald_et_al_2003}). This difference can be viewed as a degrees of freedom correction.
The fact that I am dealing with semiparametric, as opposed to parametric, models requires me to modify the key step of my proof, going from the transformed semiparametric regression residuals $\tilde{e}$ to the transformed true errors $\hat{\varepsilon}$. My approach relies on the projection property of series estimators to eliminate the estimation variance and hence only needs to deal with the approximation bias. Specifically, it uses the equality $\tilde{e} = M_W \hat{\varepsilon} + M_W \hat{R}$, applies a central limit theorem for $U$-statistics to the quadratic form in $M_W \hat{\varepsilon}$, and bounds the remainder terms by requiring the approximation error $R$ to be small.
The conventional approach does not impose any special structure on the model residuals and uses the equality $\tilde{e} = \hat{\varepsilon} + (\hat{g} - \tilde{g})$. In parametric models, $\hat{g} - \tilde{g} = \hat{X}' (\beta - \hat{\beta})$, and $\hat{\beta}$ is $\sqrt{n}$-consistent. This makes it possible to apply a central limit theorem for $U$-statistics to the quadratic form in $\hat{\varepsilon}$ and bound the remainder terms that depend on $\hat{X}' (\beta - \hat{\beta})$. However, in semiparametric models this approach needs to deal with both the bias and variance of semiparametric estimators. Specifically, $\hat{g} - \tilde{g} = \hat{R} + \hat{W}'(\beta_1 - \tilde{\beta}_1)$, where $\hat{R}$ can be viewed as the bias term and $\hat{W}'(\beta_1 - \tilde{\beta}_1)$ as the variance term. Thus, in order for $(\hat{g} - \tilde{g})$ to be small, both bias and variance need to vanish sufficiently fast, and the resulting rate conditions turn out to be very restrictive. To see this, it is useful to look at the rates that would be permissible with and without the degrees of freedom correction.
Usually $\zeta(k) = O(k^{1/2})$ for splines and $\zeta(k) = O(k)$ for power series. It can be shown that if splines are used, the rates $k_n = O(n^{2/7})$, $r_n = O(n^{2/7})$, $m_n = O(n^{1/4})$ are permissible if $\alpha \geq 4$. If power series are used, the rates $k_n = O(n^{2/9})$, $r_n = O(n^{2/9})$, $m_n = O(n^{1/5})$ are permissible if $\alpha \geq 5$.
Without the degrees of freedom correction, in order for the test to be asymptotically valid, $m_n$ typically has to be of the order $o(k_n^{1/2})$. Hence, $k_n = O(n^{2/7})$ would require $m_n = o(n^{1/7})$ and $\alpha \geq 7$ if splines are used. If power series are used, $k_n = O(n^{2/9})$ would require $m_n = o(n^{1/9})$ and $\alpha \geq 9$.
\subsection{A Wild Bootstrap Procedure}\label{wild_bootstrap}
In this section I propose a wild bootstrap procedure that can be used to obtain critical values for my test and establish its asymptotic validity. I will compare the small sample behavior of the asymptotic and bootstrap versions of the test in simulations.
Because I am interested in approximating the asymptotic distribution of the test under the null hypothesis, the bootstrap data generating process should satisfy the null. Moreover, because my test is robust to heteroskedasticity, the bootstrap data generating process should be able to accommodate heteroskedastic errors. Finally, because the errors in panel data models may be correlated over time (but not across units), the bootstrap procedure should take this into account. The wild bootstrap can satisfy both these requirements.
The bootstrap procedure will be based on the residuals based on the transformed data $\tilde{e}_{it} = \hat{Y}_{it} - \hat{W}_{it}' \tilde{\beta}_1$. I require the bootstrap errors to satisfy the following two requirements:
\[
\text{(i) } E^*[\hat{\varepsilon}_i^*] = 0, \quad \text{(ii) } E^*[\hat{\varepsilon}_i^{*} \hat{\varepsilon}_i^{* \prime}] = \tilde{e}_i \tilde{e}_i',
\]
where $E^*[\cdot] = E[\cdot | \mathcal{Z}_{n,T}]$ is the expectation conditional on the data $\mathcal{Z}_{n,T} = \{ (Y_{it}, X_{it}')' \}_{i=1,t=1}^{n,T}$. To satisfy these requirements, I let $\hat{\varepsilon}_i^* = V_i^* \tilde{e}_i$, where $V_i^*$ is a two-point distribution. Note that I use the same $V_i^*$ for all time periods for a given $i$. By doing so, I maintain the intertemporal correlation of the transformed errors and residuals in the original sample.
Various choices of $V_i^*$ are possible. One popular option is Mammen's two point distribution, originally introduced in \citet{mammen_1993}:
\[
V_i^* =
\begin{cases}
(1-\sqrt{5})/2 & \quad \text{with probability } (\sqrt{5}+1)/(2 \sqrt{5}),\\
(1+\sqrt{5})/2 & \quad \text{with probability } (\sqrt{5}-1)/(2 \sqrt{5}).
\end{cases}
\]
Another possible choice is the Rademacher distribution, as suggested in \citet{davidson_flachaire_2008}:
\[
V_i^* =
\begin{cases}
-1 & \quad \text{with probability } \frac{1}{2}, \\
1 & \quad \text{with probability } \frac{1}{2}.
\end{cases}
\]
The wild bootstrap procedure then works as follows:
\begin{enumerate}
\item
Obtain the estimates $\tilde{\beta}_1$ and residuals $\tilde{e}_i$ from the restricted model $\hat{Y}_{it} = \hat{W}_{it}' \beta_1 + \hat{e}_{it}$.
\item
Generate the wild bootstrap error $\hat{\varepsilon}_i^* = V_i^* \tilde{e}_i$.
\item
Obtain $\hat{Y}_{it}^* = \hat{W}_{it}' \tilde{\beta}_1 + \hat{\varepsilon}_{it}^*$. Then estimate the restricted model and obtain the restricted bootstrap residuals $\tilde{e}_{it}^*$ using the bootstrap sample $\{ (\hat{Y}_{it}^*, \hat{W}_{it}')' \}_{i=1,t=1}^{n,T}$.
\item
Use $\tilde{e}_{it}^*$ in place of $\tilde{e}_{it}$ to compute the bootstrap test statistic $t_{HC,r_n}^*$ or $t_{r_n}^*$.
\item
Repeat steps 2--4 $B$ times (e.g. $B=399$) and obtain the empirical distribution of the $B$ test statistics $t_{r_n}^*$ or $t_{HC,r_n}^*$. Use this empirical distribution to compute the bootstrap critical values of the bootstrap $p$-values.
\end{enumerate}
Then the following is true.
\begin{theorem}\label{asy_distr_t_r_n_hc_boot}
Assume that Assumptions of Theorem~\ref{asy_distr_t_r_n_hc} hold. Let $\mathcal{Z}_{n,T} = \{ (Y_{it}, X_{it}')' \}_{i=1,t=1}^{n,T}$. Then
\[
F_{HC,n}^*(t) \to \Phi(t) \text{ in probability},
\]
for all $t$, as $n \to \infty$, where $F_{HC,n}^*(t)$ is the bootstrap distribution of $t_{HC,r_n}^*|\mathcal{Z}_{n,T}$ and $\Phi(\cdot)$ is the standard normal CDF.
\end{theorem}
A similar result can be obtained for the homoskedastic test statistic $t_{r_n}^*$. It is omitted for brevity.
\subsection{Behavior of the Test Statistic under a Fixed Alternative}
This section discusses the behavior of the test statistic under a fixed alternative. First, a cautionary note is in order. Note that the null hypothesis concerns the model
\[
Y_{it} = g(X_{it}) + \mu_i + \varepsilon_{it},
\]
while the semiparametric series estimation method is based on the transformed model
\[
\hat{Y}_{it} = \hat{g}(X_{i}) + \hat{\varepsilon}_{it},
\]
where the model is transformed by taking the first differences or using the within transformation.\footnote{Because of this, $\hat{g}(\cdot)$ may depend on all elements of $X_i$, not just $X_{it}$.} In particular, the model is estimated based on the series form
\[
\hat{Y}_{it} = \hat{W}_{it}' \beta_1 + e_{it}
\]
Because of this, my test will only be able to detect specification errors that are present in the transformed model. In other words, the null hypothesis essentially becomes $H_0: P(\hat{g}(X_i) = \hat{f}(X_i,\theta_0,h_0)) = 1$ for some $\theta_0$ and $h_0$. Because most of the time researchers work with transformed models when they deal with fixed effects, I believe this is a reasonable hypothesis to test. Other specification tests for fixed effects panel data models, e.g. in \citet{lin_et_al_2014}, are also usually based on the transformed residuals. As long as the transformation used does not eliminate the specification error in the original model, the test will be consistent for the original model.
\begin{assumption}\label{assumption_din_1}
(\citet{donald_et_al_2003}, Assumption 1)
Assume that $E[\hat{P}_{it} \hat{P}_{it}']$ is finite for all $k$, and for any $a(x)$ with $E[a(X_{i})^2]<\infty$ there are $k \times 1$ vectors $\gamma^{k}$ such that, as $k \to \infty$,
\[
E[(a(X_{i}) - \hat{P}_{it}' \gamma^{k})^2] \to 0
\]
\end{assumption}
Lemma~\ref{lemma_din_1} in the appendix shows that when this assumption is satisfied, the conditional moment restriction $E[\varepsilon_i | X_i] = 0$ is equivalent to a growing number of unconditional moment restrictions. The class of functions $a(x)$, for which the equivalence between the conditional and unconditional restrictions holds, consists of functions that can be approximated (in the mean squared sense) using series as the number of series terms grows. While it is difficult to give a necessary and sufficient primitive condition that would describe this class of functions, the test will likely be consistent against continuous and smooth alternatives, while it may not be consistent against alternatives that exhibit jumps.
This is a population result in the sense that it does not involve the sample size $n$. In order to use this result in practice, I require the number of series terms used to construct the test statistic, $k_n$, to grow with the sample size. By doing so, I ensure that the unconditional moment restriction $E[\hat{P}_i' \hat{\varepsilon}_i] = 0$, on which the test is based, is equivalent to the conditional moment restriction $E[\hat{\varepsilon}_i | X_i]$. Thus, the test will be consistent against a wide class of alternatives satisfying Assumption~\ref{assumption_din_1}.
In order to analyze the behavior of the test under a fixed alternative, I introduce some notation first. The true model is nonparamertic:
\[
Y_{it} = g(X_{it}) + \mu_i + \varepsilon_{it}, \quad E[\varepsilon_i | X_i] = 0
\]
An alternative way to write this model is
\[
Y_{it} = f(X_{it},\theta^*,h^*) + \mu_i + \varepsilon_{it}^*,
\]
where $\theta^*$ and $h^*$ are pseudo-true parameter values and $\varepsilon_{it}^* = \varepsilon_{it} + (g(X_{it}) - f(X_{it},\theta^*,h^*)) = \varepsilon_{ti} + d(X_{it})$ is a composite error term. The pseudo-true parameter values minimize
\[
E[(g(X_{it}) - f(X_{it},\theta,h))^2]
\]
over a suitable parameter space.
Note that the model can be written as
\[
Y_{it} = W_{it}' \beta_1^{*} + \mu_i + \varepsilon_{it}^* + R_{it}^*,
\]
where $R_{it}^* = (f(X_{it},\theta^*,h^*) - W_{it}' \beta_1^{*})$. After transforming the data, the model becomes
\[
\hat{Y}_{it} = \hat{W}_{it}' \beta_1^* + \hat{\varepsilon}_{it}^* + \hat{R}_{it}^*
\]
The pseudo-true parameter value $\beta_1^*$ solves the moment condition $E[\hat{W}_{it} (\hat{Y}_{it} - \hat{W}_{it}' \beta_1^{*})] = 0$, and the semiparametric estimator $\tilde{\beta}_1$ solves its sample analog $\hat{W}' (\hat{Y} - \hat{W} \tilde{\beta}_1)/n = 0$.
The following theorem provides the divergence rate of the test statistic under the fixed alternative.
\begin{theorem}\label{global_alternative_t_r}
Let $\Omega^* = E[\hat{Z}_i' \hat{\varepsilon}_i \hat{\varepsilon}_i' \hat{Z}_i]$. In the heteroskedastic case, let
\[
\hat{\Omega} = n^{-1}\sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i},
\]
and in the homoskedastic case let
\[
\hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i}
\]
Suppose that there exists $\beta_1^*$ such that $\sup_{x \in \mathcal{X}}{| f(x,\theta^*,h^*) - W^{m_n}(x)' \beta_1^* |} \to 0$, $\| \hat{\Omega} - \Omega^* \| \overset{p}{\to} 0$, the smallest eigenvalue of $\Omega^*$ is bounded away from zero, $m_n \to \infty$, $r_n \to \infty$, $r_n/n \to 0$, $E[\hat{\varepsilon}_i^{*\prime} T_i] \Omega^{*-1} E[T_i' \hat{\varepsilon}_i^*] \to \Delta$, where $\Delta$ is a constant. Then under homoskedasticity
\[
\frac{\sqrt{r_n}}{n} \frac{\xi - r_n}{\sqrt{2 r_n}} \overset{p}{\to} \Delta/\sqrt{2},
\]
and under heteroskedasticity
\[
\frac{\sqrt{r_n}}{n} \frac{\xi_{HC} - r_n}{\sqrt{2 r_n}} \overset{p}{\to} \Delta/\sqrt{2}
\]
\end{theorem}
\section{Simulations}\label{simulations}
In this section, I study the finite sample performance of the proposed test using simulations. I have several goals: first, to illustrate the importance of the degrees of freedom correction; second, to study the sensitivity of the test to the choice of basis functions and tuning parameters; third, to compare the asymptotic version of my test with its bootstrap version; finally, to study the effect of the sample size on the test behavior.
The setup I use resembles the one in \citet{korolev_2019} but includes fixed effects:
\[
Y_{it} = \mu_i + X_{1it} \beta + g(X_{2it}) + \varepsilon_{it}
\]
Here $\varepsilon_{it}$ are independent across individuals $i$ and time $t$, while $\alpha_i$ are fixed effects that are correlated with both the regressors and error terms for individual $i$. More specifically,
\[
\mu_i = \nu_i + \mu_{X,i},
\]
where $\nu_i \sim \text{i.i.d. } N(0,2.25)$ and $\mu_{X,i} = \sum_{t=1}^{T}{(0.6 X_{1it}+0.4 X_{2it})}$. In this setting, estimating the model $Y_{it} = 2 X_{1it} + g(X_{2it}) + e_{it}$, where $e_{it} = \mu_i + \varepsilon_{it}$, would result in inconsistent estimates, so it is crucial to account for the panel nature of the data and for the presence of fixed effects. To achieve this, I use the within transformation.\footnote{I have tried using first differencing instead of the within transformation and obtained similar results.} After that, I estimate the model and compute the proposed test statistic.
I test the following null hypothesis:
\begin{align*}
H_0^{SP}: P \left(E[Y_{it} | \mu_i, X_{it}] = \mu_i + X_{1it} \beta + g(X_{2it}) \right) = 1 \text{ for some } \beta, g(X_{2it})
\end{align*}
against the alternative
\begin{align*}
H_1: P \left(E[Y_{it} | \mu_i, X_{it}] \neq \mu_i + X_{1it} \beta + g(X_{2it}) \right) > 0 \text{ for all } \beta, g(X_{2it})
\end{align*}
I use two data generating processes:
\begin{enumerate}
\item
Semiparametric partially linear, which corresponds to $H_0^{SP}$:
\begin{align}\label{DGP_SP}
\begin{split}
Y_{it} &= \mu_i + 2 X_{1it} + g(X_{2it}) + \varepsilon_{it} \\
g(X_{2it}) &= 3 + 2(\exp(X_{2it})-2 \ln(X_{2it}+3))
\end{split}
\end{align}
\item
Nonparametric, which corresponds to $H_1$:
\begin{align}\label{DGP_NP}
\begin{split}
Y_{it} &= \mu_i + 2 X_{1it} + g(X_{2it}) + h(X_{1it},X_{2it}) + \varepsilon_{it} \\
h(X_{1it},X_{2it}) &= h_1(X_{1it}) h_2(X_{2it}) \\
h_1(X_{1it}) &= 1.25 \cos(X_{1it}-2), \quad h_2(X_{2it}) = \sin(0.75 X_{2it})
\end{split}
\end{align}
\end{enumerate}
These two DGPs are very similar to the ones used in \citet{korolev_2019}, but include fixed effects. For more details on the two DGPs, see \citet{korolev_2019}. I consider four setups: Setup 1 with $(n = 250, T = 2)$, Setup 2 with $(n = 250, T = 4)$, Setup 3 with $(n = 500, T = 2)$, Setup 4 with $(n = 500, T = 4)$. I separately consider two settings: with homoskedastic errors and with heteroskedastic errors.
To implement the test, I use both power series and cubic splines as basis functions due to their popularity. Instead of studying the behavior of my test for a given (arbitrary) number of series terms, I vary the number of terms in univariate series expansions to investigate how the behavior of the test changes as a result. The total number of parameters $k_n$ ranges from 15 to 39 in Setups 1 and 2 and to 52 in Setups 3 and 4.\footnote{For more details, see the online supplement to \citet{korolev_2019}.}
\subsection{Homoskedastic Errors}\label{simulations_homoskedastic_errors}
First, I investigate the performance of the test when the errors are homoskedastic. The errors are normally distributed and independent across both $i$ and $t$: $\varepsilon_{it} \sim \text{i.i.d. } N(0,4)$. I consider tests based both on the LM type test statistic
\[
\xi = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right) \overset{a}{\sim} \chi^2(\tau_n)
\]
and on the normalized statistic $t_{\tau_n} = \frac{\xi - \tau_n}{\sqrt{2 \tau_n}} \overset{a}{\sim} N(0,1)$.
I start by looking at the simulated size of the test at the nominal 5\% level. Figures~\ref{fig_simulated_size_set1}, \ref{fig_simulated_size_set2}, \ref{fig_simulated_size_set3}, and \ref{fig_simulated_size_set4} plot the simulated size as a function of the number of series terms in univariate series expansions $a_n$ (including the constant term) for the four setups I consider. The upper panels of these figures use the LM type statistic $\xi$ , while the bottom panels use the normalized test statistic $t$. The left panels use power series and the right panels use splines. I consider four versions of the test: the asymptotic version with $\tau_n = r_n$ (red solid lines), the asymptotic version with $\tau_n = k_n$ (magenta solid lines), the wild bootstrap version with the Rademacher distribution (cyan dash-dotted lines), and the wild bootstrap with Mammen's distribution (blue dashed lines).
As we can see, the asymptotic test without the degrees of freedom correction (i.e. with $\tau_n = k_n$) is severely undersized. In turn, the asymptotic test with the degrees of freedom correction (i.e. with $\tau_n = r_n$) based on the $t$ statistic is slightly oversized, while the asymptotic test based on the $\xi$ statistic controls size very well. Depending on the setup, its performance is either very close to, or even better than, that of the wild bootstrap tests. We can also see that the performance of the test is fairly robust to the choice of basis functions and tuning parameters.
Next, I turn to the test power. Figures~\ref{fig_simulated_power_set1}, \ref{fig_simulated_power_set2}, \ref{fig_simulated_power_set3}, and \ref{fig_simulated_power_set4} plot the simulated power of the nominal 5\% level test as a function of the number of series terms in univariate series expansions $a_n$. Given that the asymptotic test without the degrees of freedom correction is undersized, it is not surprising that it also has very low power. In turn, the power of the asymptotic version of the test with the degrees of freedom correction is very similar to the power of the wild bootstrap tests. As could be expected, the power increases as the sample size (the number of units $n$ or the number of periods $T$) increases. Finally, the power decreases as the number of series terms grows. This is due to the fact that the alternative is smooth and can be captured by the first few series terms. I will turn to a data-driven method to choose tuning parameters later.
\subsection{Heteroskedastic Errors}\label{simulations_heteroskedastic_errors}
In this section I investigate the performance of the test when the errors are heteroskedastic. The errors are normally distributed and independent across both $i$ and $t$, but not identically distributed: $\varepsilon_{it} \sim \text{i.n.i.d. } N(0,1+1.75 \exp(0.75(X_{1it}+X_{2it})))$. I consider tests based both on the heteroskedasticity-robust LM type test statistic
\[
\xi_{HC} = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right)
\]
and on the normalized statistic $t_{\tau_n,HC} = \frac{\xi_{HC} - \tau_n}{\sqrt{2 \tau_n}} \overset{a}{\sim} N(0,1)$.
First, I look at the simulated size of the test at the nominal 5\% level. Figures~\ref{fig_simulated_size_set1_hc}, \ref{fig_simulated_size_set2_hc}, \ref{fig_simulated_size_set3_hc}, and \ref{fig_simulated_size_set4_hc} plot the simulated size as a function of the number of series terms in univariate series expansions $a_n$. We can see that the asymptotic test without the degrees of freedom correction is again severely undersized. The asymptotic test with the degrees of freedom correction based on the $\xi_{HC}$ statistic is also undersized, though its size becomes closer to the nominal level as the sample size grows. In turn, the simulated size of the asymptotic test with the degrees of freedom correction based on the $t_{HC}$ statistic is pretty close to the nominal level. In fact, in Setups 1 and 2, when splines are used, it controls size even better that the wild bootstrap tests.
Next, I turn to the test power. Figures~\ref{fig_simulated_power_set1_hc}, \ref{fig_simulated_power_set2_hc}, \ref{fig_simulated_power_set3_hc}, and \ref{fig_simulated_power_set4_hc} plot the simulated power of the nominal 5\% level test as a function of the number of series terms in univariate series expansions $a_n$. The asymptotic test without the degrees of freedom correction has low power in all setups. The asymptotic test with the degrees of freedom correction based on the $\xi_{HC}$ test statistic is less powerful than the wild bootstrap tests, but the power loss decreases as the sample size grows. Finally, the power of the asymptotic test with the degrees of freedom correction based on the $t_{HC}$ statistic is fairly close to the power of the bootstrap tests, especially with larger sample sizes.
To summarize, even though the performance of the asymptotic test with the degrees of freedom correction deteriorates when the errors are heteroskedastic, as opposed to homoskedastic, it nevertheless comes close to the wild bootstrap tests in most setups. With larger sample sizes, its performance is almost indistinguishable from that of the bootstrap tests.
\subsection{Data-Driven Methods for Tuning Parameters Choice}\label{simulations_data_driven}
In the simulations presented above, the alternative was smooth and the power of the test declined as the number of series terms increased. However, this is not always the case. There exist alternatives that are orthogonal to the first few series terms, and in order to detect such alternatives, one needs to include higher order series terms. In this section, I investigate the finite sample performance of a data-driven method to select tuning parameters.
In order to simplify the problem, I abstract away from the task of selecting the number of series terms under the null and consider a linear univariate null model:
\[
Y_{it} = \mu_i + 2 X_{1it} + \varepsilon_{it}
\]
The smooth alternative is given by
\[
Y_{it} = \mu_i + 2 X_{1it} + \cos(X_{1it} - 2) + \varepsilon_{it}
\]
I also consider an alternative that is orthogonal to the first four power terms in $X_1$. A data-driven test should be able to adapt to a wide class of alternatives and choose tuning parameters appropriately.
I use a modified version of the approach proposed in \citet{guay_guerre_2006}. I use the $\xi$ test statistic and pick the value of $r_n$ that maximizes
\[
\xi(r_n) - r_n - \gamma_n \sqrt{2(r_n - r_{n,min})},
\]
where $\gamma_n = c \sqrt{2 \ln{\Card{r_n}}}$, $c$ is a constant that satisfies $c \geq 1 + \varepsilon$ for some $\varepsilon > 0$, $\Card{r_n}$ is the cardinality of the set of possible numbers of restrictions, and $r_{n,min}$ is the lowest possible number of restrictions across different choices of $r_n$. The notation $\xi(r_n)$ emphasizes the dependence of the test statistic $\xi = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right)$ on the number $r_n$ of elements in $\tilde{Z}$. Intuitively, $r_n$ is the center term of $\xi(r_n)$, while $\gamma_n \sqrt{2(r_n - r_{n,min})}$ is the penalty term that rewards simpler alternatives. In my analysis, I set $c=5$.
Table~\ref{Tbl_data_driven} presents the results. I report the simulated size, power against the standard alternative, and power against the orthogonal alternative for the data driven test and the test with the fixed number of series term equal to $a_n = 4$ and $a_n = 9$ (including the constant term). The former choice of $a_n$ is typically optimal under the regular alternative but has no power against the orthogonal alternative. The latter choice of $a_n$ typically leads to good power against the orthogonal alternative but results in the loss of power against the well-behaved alternative.
As we can see, the data-driven test is slightly oversized in the first three setups and is slightly undersized in the last setup, but overall its size is close to the nominal level. Moreover, it has excellent power against the standard alternative and pretty good power against the orthogonal alternative. Even though a more careful investigation of data-driven specification tests for panel data models is beyond the scope of this paper, my simulations suggest that the proposed procedure performs well in finite samples.
\section{Empirical Example}\label{empirical_example}
In this section, I apply my test to the PSID data\footnote{Available at \url{http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=4338&itemId=1118672321&resourceId=13452}.} that was used in \citet{cornwell_rupert_1988} and \citet{baltagi_khanti-akom_1990}. The dataset contains 7 years of observations on 595 heads of household between the ages of 18 and 65 in 1976 with a positive reported wage in some private, non-farm employment for all 7 years. Among other models, the authors estimated the following wage equation with fixed effects:
\begin{align}\label{quadratic_model}
\begin{split}
&LWAGE_{it} = \alpha_1 WKS_{it} + \delta_1 EXP_{it} + \lambda_1 EXP_{it}^2 + D_{it}' \gamma_1 + \mu_{i} + \varepsilon_{it} \\
&E[\varepsilon_{i} | WKS_{i}, EXP_{i}, D_{i}, \mu_i] = 0
\end{split}
\end{align}
where $LWAGE_{it}$ is the natural logarithm of the wage of individual $i$ in year $t$, $WKS$ is weeks worked, $EXP$ is experience, and $D_{it}$ includes the following dummy variables: occupation ($OCC= 1$ if the individual has blue-collar occupation), industry ($IND= 1$ if the individual works in a manufacturing industry), residence ($SOUTH = 1$, $SMSA = 1$ if the individual resides in the south, or in a standard metropolitan statistical area), marital status ($MS = 1$ if the individual is married), union coverage ($UNION = 1$ if the individual's wage is set by a union contract).
While my test is fairly general and applies to semiparametric as well as parametric models, I focus on a parametric model because parametric panel data models are prevalent in applications. I test this parametric model against the alternative which is fully nonparametric in weeks worked and experience but is parametric in the dummy variables:
\[
LWAGE_{it} = g(WKS_{it},EXP_{it}) + D_{it}' \gamma + \mu_{i} + \varepsilon_{it},
\]
where $D_{it}$ includes the six dummy variables listed above. Due to the number of dummy variables, considering the alternative which is fully nonparametric appears implausible, as it would essentially require me to split the dataset into $2^{6} = 64$ bins and estimate it within each bin separately.
In order to implement the test, I need to select the basis functions and the number of series terms. I use both power series and splines and utilize a data-driven procedure to select the number of series terms. Note that the number of terms under the null is fixed because the null model is parametric. Thus, I only need to choose the number of series terms under the alternative. Following the approach discussed in Section~\ref{simulations_data_driven}, I vary the number of series terms in univariate series expansions in $WKS$ and $EXP$ from 3 to 8 (not including the constant) and pick the value of $r_n$ that maximizes
\[
\xi_{HC}(r_n) - r_n - \gamma_n \sqrt{2(r_n - r_{n,min})}
\]
I find that the optimal number of terms is equal to 3 (not including the constant term), i.e. that a cubic polynomial should be used. In this case, power series and splines coincide as there are no knots yet. The resulting number of restrictions is $r_n = 12$. The upper panel of Table~\ref{Tbl_testing} reports the heteroskedastic test statistic $\xi_{HC}$ as well as the standardized statistic $t_{HC}$. As we can see, the null hypothesis that the model is correctly specified is not rejected at the 5\% level, but it is rejected at the 10\% level.
Next, I repeat this exercise for the specification that drops the quadratic term in experience. I use the same nonparametric alternative as before. Because the null model has one regressor less than before, I am testing $r_n = 13$ restrictions. The middle panel of Table~\ref{Tbl_testing} reports the results. All for types of the test reject the null hypothesis at any conventional confidence level.
Finally, I estimate a semiparametric model that is nonparametric in experience but is parametric in the remaining variables:
\begin{align}\label{semiparametric_model}
LWAGE_{it} = \alpha_2 WKS_{it} + g_2(EXP_{it}) + D_{it}' \gamma_2 + \mu_{i} + \varepsilon_{it}
\end{align}
I estimate this semiparametric model using power series with up to cubic terms. The semiparametric model leads to $r_n = 11$ restrictions. Figure~\ref{figure_exp_effects} plots the estimated effects of experience for the linear, quadratic, and emiparametric models. As we can see, the semiparametric model appears to be pretty similar to the quadratic model, and the linear model is not too far off. However, specification testing draws a somewhat different picture. As we can see from the bottom panel of Table~\ref{Tbl_testing}, while the linear model is overwhelmingly rejected and the quadratic model is rejected at the 10\% level, there is no evidence against the semiparametric model.
Because in this paper I develop a specification test and not a model selection procedure, one should be careful with applying my test to several models sequentially. However, it appears that there is substantial evidence against the linear model, while there is little evidence agains the quadratic specification employed by \citet{cornwell_rupert_1988} and \citet{baltagi_khanti-akom_1990}. If the researcher worries about the borderline results and wants to be on the safe side, it may be plausible to use a more flexible semiparametric model that is fully nonparametric in experience.
\section{Conclusion}\label{conclusion}
In this paper, I develop a Lagrange Multiplier type specification test for semiparametric panel data models with fixed effects. The test achieves consistency by turning a conditional moment restriction into a growing number of unconditional moment restrictions. Unlike in the traditional parametric Lagrange Multiplier test, both the number of parameters and the number of restrictions are allowed to grow with the sample size. I develop an asymptotic theory that explicitly takes this into account and prove that the normalized test statistic converges in distribution to the standard normal.
My test has several attractive features. First, fixed effects panel data models typically require researchers to transform their data, by taking first differences or applying the within transformation. This makes semiparametric estimation and specification testing that involves kernel methods problematic, as it is difficult to impose the additive structure on kernel estimators. In contrast, with series methods, the transformed model remains linear in parameters, and the proposed test is very simple to implement.
Second, the projection property of series estimators allows me to develop a degrees of freedom correction, which explicitly accounts for the variance of semiparametric estimators. Thus, I only need to control the bias, and my rate conditions are relatively mild. Moreover, the degrees of freedom correction results in good performance of the test in simulations.
In future research, I plan to extend the proposed test to semiparametric dynamic panel data models. The presence of endogenous variables calls for the use of instrumental variables. Estimation of such models, with endogeneity only in the parametric part, has been studied in \citet{baltagi_li_2002} and \citet{an_et_al_2016}. A possible concern for specification testing in these models is that nonparametric instrumental variables models are subject to the ill-posed inverse problem, so the unrestricted nonparametric model may not be identified. It remains to be seen whether this identification problem poses a challenge for specification testing in dynamic panel data models.
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\section{Tables and Figures}\label{appendix_tables_figures}
\onehalfspacing
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, $n=250$, $T = 2$}\label{fig_simulated_size_set1}
\includegraphics[scale=0.5]{LM_size_set1_panel_series} \includegraphics[scale=0.5]{LM_size_set1_panel_splines}
\includegraphics[scale=0.5]{t_size_set1_panel_series} \includegraphics[scale=0.5]{t_size_set1_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, $n=250$, $T = 4$}\label{fig_simulated_size_set2}
\includegraphics[scale=0.5]{LM_size_set2_panel_series} \includegraphics[scale=0.5]{LM_size_set2_panel_splines}
\includegraphics[scale=0.5]{t_size_set2_panel_series} \includegraphics[scale=0.5]{t_size_set2_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, $n=500$, $T = 2$}\label{fig_simulated_size_set3}
\includegraphics[scale=0.5]{LM_size_set3_panel_series} \includegraphics[scale=0.5]{LM_size_set3_panel_splines}
\includegraphics[scale=0.5]{t_size_set3_panel_series} \includegraphics[scale=0.5]{t_size_set3_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, $n=500$, $T = 4$}\label{fig_simulated_size_set4}
\includegraphics[scale=0.5]{LM_size_set4_panel_series} \includegraphics[scale=0.5]{LM_size_set4_panel_splines}
\includegraphics[scale=0.5]{t_size_set4_panel_series} \includegraphics[scale=0.5]{t_size_set4_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, $n=250$, $T = 2$}\label{fig_simulated_power_set1}
\includegraphics[scale=0.5]{LM_power_set1_panel_series} \includegraphics[scale=0.5]{LM_power_set1_panel_splines}
\includegraphics[scale=0.5]{t_power_set1_panel_series} \includegraphics[scale=0.5]{t_power_set1_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, $n=250$, $T = 4$}\label{fig_simulated_power_set2}
\includegraphics[scale=0.5]{LM_power_set2_panel_series} \includegraphics[scale=0.5]{LM_power_set2_panel_splines}
\includegraphics[scale=0.5]{t_power_set2_panel_series} \includegraphics[scale=0.5]{t_power_set2_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, $n=500$, $T = 2$}\label{fig_simulated_power_set3}
\includegraphics[scale=0.5]{LM_power_set3_panel_series} \includegraphics[scale=0.5]{LM_power_set3_panel_splines}
\includegraphics[scale=0.5]{t_power_set3_panel_series} \includegraphics[scale=0.5]{t_power_set3_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, $n=500$, $T = 4$}\label{fig_simulated_power_set4}
\includegraphics[scale=0.5]{LM_power_set4_panel_series} \includegraphics[scale=0.5]{LM_power_set4_panel_splines}
\includegraphics[scale=0.5]{t_power_set4_panel_series} \includegraphics[scale=0.5]{t_power_set4_panel_splines}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi$ test statistic from Equation~\ref{xi}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, Heteroskedastic Errors, $n=250$, $T = 2$}\label{fig_simulated_size_set1_hc}
\includegraphics[scale=0.5]{LM_size_set1_panel_series_hc} \includegraphics[scale=0.5]{LM_size_set1_panel_splines_hc}
\includegraphics[scale=0.5]{t_size_set1_panel_series_hc} \includegraphics[scale=0.5]{t_size_set1_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, Heteroskedastic Errors, $n=250$, $T = 4$}\label{fig_simulated_size_set2_hc}
\includegraphics[scale=0.5]{LM_size_set2_panel_series_hc} \includegraphics[scale=0.5]{LM_size_set2_panel_splines_hc}
\includegraphics[scale=0.5]{t_size_set2_panel_series_hc} \includegraphics[scale=0.5]{t_size_set2_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, Heteroskedastic Errors, $n=500$, $T = 2$}\label{fig_simulated_size_set3_hc}
\includegraphics[scale=0.5]{LM_size_set3_panel_series_hc} \includegraphics[scale=0.5]{LM_size_set3_panel_splines_hc}
\includegraphics[scale=0.5]{t_size_set3_panel_series_hc} \includegraphics[scale=0.5]{t_size_set3_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Size of the Test, Heteroskedastic Errors, $n=500$, $T = 4$}\label{fig_simulated_size_set4_hc}
\includegraphics[scale=0.5]{LM_size_set4_panel_series_hc} \includegraphics[scale=0.5]{LM_size_set4_panel_splines_hc}
\includegraphics[scale=0.5]{t_size_set4_panel_series_hc} \includegraphics[scale=0.5]{t_size_set4_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated size of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, Heteroskedastic Errors, $n=250$, $T = 2$}\label{fig_simulated_power_set1_hc}
\includegraphics[scale=0.5]{LM_power_set1_panel_series_hc} \includegraphics[scale=0.5]{LM_power_set1_panel_splines_hc}
\includegraphics[scale=0.5]{t_power_set1_panel_series_hc} \includegraphics[scale=0.5]{t_power_set1_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, Heteroskedastic Errors, $n=250$, $T = 4$}\label{fig_simulated_power_set2_hc}
\includegraphics[scale=0.5]{LM_power_set2_panel_series_hc} \includegraphics[scale=0.5]{LM_power_set2_panel_splines_hc}
\includegraphics[scale=0.5]{t_power_set2_panel_series_hc} \includegraphics[scale=0.5]{t_power_set2_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, Heteroskedastic Errors, $n=500$, $T = 2$}\label{fig_simulated_power_set3_hc}
\includegraphics[scale=0.5]{LM_power_set3_panel_series_hc} \includegraphics[scale=0.5]{LM_power_set3_panel_splines_hc}
\includegraphics[scale=0.5]{t_power_set3_panel_series_hc} \includegraphics[scale=0.5]{t_power_set3_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Simulated Power of the Test, Heteroskedastic Errors, $n=500$, $T = 4$}\label{fig_simulated_power_set4_hc}
\includegraphics[scale=0.5]{LM_power_set4_panel_series_hc} \includegraphics[scale=0.5]{LM_power_set4_panel_splines_hc}
\includegraphics[scale=0.5]{t_power_set4_panel_series_hc} \includegraphics[scale=0.5]{t_power_set4_panel_splines_hc}
\end{center}
\footnotesize{\vspace{-0.4cm}This figure plots the simulated power of the nominal 5\% test against the number of series terms in univariate series expansions, $a_n$ (including the constant). The left panel uses power series. The right panel uses splines. The upper panel uses the $\xi_{HC}$ test statistic from Equation~\ref{xi_hc}. The lower panel uses the $t$ test statistic from Equation~\ref{t_test_statistics}.
The red solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = r_n$. The magenta solid line corresponds to the test that uses the asymptotic critical values and normalization $\tau_n = k_n$. The cyan dash-dotted line corresponds to the test that uses the wild bootstrap critical values based on Rademacher distribution. The blue dashed line corresponds to the test that uses the wild bootstrap critical values based on Mammen's distribution. The results are based on $M=1,000$ simulations and $B=399$ bootstrap iterations.}
\end{figure}
\begin{figure}[H]
\begin{center}
\caption{Experience Effects from Different Methods}\label{figure_exp_effects}
\includegraphics[scale=0.8]{exp_effects_final}
\end{center}
\begin{singlespace}\footnotesize{This figure plots the estimated effects of experience from different models. The dash-dotted line shows the estimated experience effect from the linear model. The dashed line shows the estimated experience effect from the quadratic model. The solid line shows the estimated experience effect from the semiparametric model. The $x$ axis plots experience. The $y$ axis plots the logarithm of wage.}
\end{singlespace}
\end{figure}
\begin{table}[h]
\begin{center}
\caption{Simulated Size and Power of Data-Driven Test}\label{Tbl_data_driven}
\begin{tabular}{l | c c c | c c c}
& \multicolumn{3}{c}{Power Series} & \multicolumn{3}{c}{Splines} \\
& Data-Driven & $a_n = 4$ & $a_n = 9$ & Data-Driven & $a_n = 4$ & $a_n = 9$ \\ \hline\hline
& \multicolumn{6}{c}{$n = 250, T = 2$} \\
Size & 0.055 & 0.046 & 0.049 & 0.054 & 0.046 & 0.047 \\
Power, regular & 0.369 & 0.363 & 0.196 & 0.369 & 0.363 & 0.201 \\
Power, orthogonal & 0.105 & 0.051 & 0.224 & 0.096 & 0.051 & 0.225 \\ \hline\hline
& \multicolumn{6}{c}{$n = 250, T = 4$} \\
Size & 0.064 & 0.061 & 0.041 & 0.063 & 0.061 & 0.044 \\
Power, regular & 0.825 & 0.825 & 0.622 & 0.825 & 0.825 & 0.621 \\
Power, orthogonal & 0.447 & 0.062 & 0.654 & 0.440 & 0.062 & 0.644 \\\hline\hline
& \multicolumn{6}{c}{$n = 500, T = 2$} \\
Size & 0.058 & 0.054 & 0.055 & 0.056 & 0.054 & 0.054 \\
Power, regular & 0.673 & 0.671 & 0.444 & 0.673 & 0.671 & 0.448 \\
Power, orthogonal & 0.238 & 0.048 & 0.421 & 0.241 & 0.048 & 0.426 \\ \hline\hline
& \multicolumn{6}{c}{$n = 500, T = 4$} \\
Size & 0.041 & 0.037 & 0.052 & 0.041 & 0.037 & 0.051 \\
Power, regular & 0.990 & 0.990 & 0.935 & 0.990 & 0.990 & 0.932 \\
Power, orthogonal & 0.856 & 0.033 & 0.933 & 0.843 & 0.033 & 0.939
\end{tabular}
\end{center}
\begin{singlespace}\footnotesize{The table reports the simulated size, power against the regular alternative, and power against the orthogonal alternative of the test based on the test statistic $\xi$. The left panel uses power series and the right panel uses splines. The column called ``Data-Driven'' uses the data-driven value of the tuning parameter $r_n$ as described in Section~\ref{simulations_data_driven}. Other columns use the fixed number of series terms $a_n$ equal to 4 or 9 (including the constant). The results are based on $M=1,000$ simulations.}
\end{singlespace}
\end{table}
\begin{table}[h]
\begin{center}
\caption{Specification Testing Results}\label{Tbl_testing}
\begin{tabular}{l | c | c | c}
& Test Statistic & 5\% Critical Value & 10\% Critical Value \\ \hline\hline
\multicolumn{4}{c}{Quadratic Model} \\
$\xi_{HC}$ & 19.835 & 21.026 & 18.549 \\
$t_{HC}$ & 1.599 & 1.645 & 1.282 \\ \hline\hline
\multicolumn{4}{c}{Linear Model} \\
$\xi_{HC}$ & 42.318 & 22.362 & 19.812 \\
$t_{HC}$ & 5.750 & 1.645 & 1.282 \\ \hline\hline
\multicolumn{4}{c}{Semiparametric Model} \\
$\xi_{HC}$ & 15.957 & 19.675 & 17.275 \\
$t_{HC}$ & 1.057 & 1.645 & 1.282
\end{tabular}
\end{center}
\begin{singlespace}\footnotesize{The table reports the values of the test statistics $\xi_{HC} \overset{a}{\sim} \chi^2(r_n)$ and $t_{HC} \overset{a}{\sim} N(0,1)$ for the quadratic specification from Equation~\ref{quadratic_model}, linear specification, and semiparametric specification from Equation~\ref{semiparametric_model}. The number of restrictions is $r_n = 12$, $r_n = 13$, and $r_n = 11$ respectively. The corresponding critical values are shown together with the test statistics.}
\end{singlespace}
\end{table}
\clearpage
\section{Proofs}\label{appendix_proofs}
\subsection{Proof of Theorem~\ref{asy_distr_t_r_n_hc}}
The homoskedastic and heteroskedastic test statistics have different expressions, and it is convenient to introduce some notation that allows me to write them in a similar form. Denote $\hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i}$ in the heteroskedastic case or $\hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i}$ in the homoskedastic case. Then both test statistics can be written as
\[
\xi = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( n \hat{\Omega} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right) = \tilde{e}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \tilde{e} = (\hat{\varepsilon} + \hat{R})' M_{\hat{W}} \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' M_{\hat{W}} (\hat{\varepsilon} + \hat{R})
\]
Because $\tilde{Z} = M_{\hat{W}} \hat{Z}$ and $M_{\hat{W}} \tilde{Z} = M_{\hat{W}} M_{\hat{W}} \hat{Z} = M_{\hat{W}} \hat{Z} = \tilde{Z}$, the test statistic can be rewritten as
\[
\xi = (\hat{\varepsilon} + \hat{R})' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' (\hat{\varepsilon} + \hat{R})
\]
The proof consists of several steps.
Step 1. Decompose the test statistic and bound the remainder terms.
\[
(\hat{\varepsilon} + \hat{R})' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' (\hat{\varepsilon} + \hat{R}) =
\hat{\varepsilon}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \hat{\varepsilon} + \hat{R}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \hat{R}
+ 2 \hat{R}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \hat{\varepsilon}
\]
By Lemma~\ref{lemma_PP}, the smallest and largest eigenvalues of $\tilde{Z} \tilde{Z}'/(nT)$ converge to one. Because $\tilde{Z}' \tilde{Z}/(nT)$ and $\tilde{Z} \tilde{Z}'/(nT)$ have the same nonzero eigenvalues, $\lambda_{\max}(\tilde{Z} \tilde{Z}'/(nT))$ converges in probability to 1. Moreover, the eigenvalues of $\hat{\Omega}$ are bounded below and above. Thus, by Assumption~\ref{series_approx},
\[
\hat{R}' \tilde{Z} (n \hat{\Omega})^{-1} \tilde{Z}' \hat{R} \leq T C \hat{R}' ((nT)^{-1} \tilde{Z} \tilde{Z}') \hat{R} \leq C \hat{R}' \hat{R} = O_p(n m_n^{-2\alpha}),
\]
where $T$ is absorbed by $C$ because the length of the panel is fixed.
Next,
\[
\Big{|} \hat{R}' \tilde{Z} (n \hat{\Omega})^{-1} \tilde{Z}' \hat{\varepsilon} \Big{|} \leq \Big{|} T C \lambda_{\max}(\tilde{Z} \tilde{Z}'/(nT)) \hat{R}' \hat{\varepsilon} \Big{|} \leq \Big{|} C \hat{R}' \hat{\varepsilon} \Big{|} = O_p(n^{1/2} m_n^{-\alpha})
\]
Thus,
\[
(\hat{\varepsilon} + \hat{R})' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' (\hat{\varepsilon} + \hat{R}) = \hat{\varepsilon}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \hat{\varepsilon} + O_p(n m_n^{-2\alpha}) + O_p(n^{1/2} m_n^{-\alpha})
\]
Step 2. Further decompose the leading term and bound the new remainder terms.
\[
\hat{\varepsilon}' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' \hat{\varepsilon} = \hat{\varepsilon}' \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} - 2 \hat{\varepsilon}' P_{\hat{W}} \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} + \hat{\varepsilon}' P_{\hat{W}} \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' P_{\hat{W}} \hat{\varepsilon}
\]
Let $\chi_i = \hat{Z}_i' \hat{\varepsilon}_i$, $\Omega = E[\hat{Z}_i' \hat{\varepsilon}_i \hat{\varepsilon}_i' \hat{Z}_i] = E[\chi_i \chi_i']$. Note that
\begin{align*}
&E[(n^{-1} \hat{\varepsilon}' \hat{Z}) \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})]/n =E[\chi_i' \Omega^{-1} \chi_i] = E[tr(\chi_i' \Omega^{-1} \chi_i)]/n \\
&= E[tr(\Omega^{-1} \chi_i \chi_i')]/n = tr(\Omega^{-1} E[\chi_i \chi_i'])/n = tr(I_{r_n}) /n= r_n/n
\end{align*}
Thus, by Markov's inequality,
\[
\| \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}) \| \leq C \sqrt{(n^{-1} \hat{\varepsilon}' \hat{Z}) \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})} = O_p(\sqrt{r_n/n})
\]
Because the eigenvalues of $\Omega$ are bounded below and above w.p.a. 1, it is also true that $\| n^{-1} \hat{Z}' \hat{\varepsilon} \| = O_p(\sqrt{r_n/n})$. Similarly, $\| n^{-1} \hat{W}' \hat{\varepsilon} \| = O_p(\sqrt{m_n/n})$. Using this result and the inequality $\|A B \|^2 \leq \| A \|^2 \| B \|^2$, get
\begin{align*}
&\Big{\|} \hat{\varepsilon}' P_{\hat{W}} \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} \Big{\|} = \Big{\|} \hat{\varepsilon}' \hat{W} (\hat{W}'\hat{W})^{-1} \hat{W}' \hat{Z} (n \hat{\Omega})^{-1} \hat{T}' \hat{\varepsilon} \Big{\|} \\
&= \Big{\|} (nT^2) \left( \hat{\varepsilon}' \hat{W}/(nT) \right) \left( \hat{W}'\hat{W}/(nT) \right)^{-1} \left( \hat{W}' \hat{T}/(nT) \right) \hat{\Omega}^{-1} \left( \hat{T}' \hat{\varepsilon}/(nT) \right) \Big{\|} \\
&\leq C n \Big{\|} \left( \hat{\varepsilon}' \hat{W}/(nT) \right) \left( \hat{W}' \hat{Z}/(nT) \right) \left( \hat{Z}' \hat{\varepsilon}/(nT) \right) \Big{\|} \\
&\leq C n \Big{\|} \hat{\varepsilon}' \hat{W}/(nT) \Big{\|} \; \Big{\|} \hat{W}' \hat{Z}/(nT) \Big{\|} \; \Big{\|} \hat{Z}' \hat{\varepsilon}/(nT) \Big{\|} \\
&= n O_p(\sqrt{m_n/n}) O_p(\zeta(k_n) \sqrt{k_n/n}) O_p(\sqrt{r_n/n}) = O_p(\zeta(k_n) \sqrt{m_n k_n r_n/n})
\end{align*}
In turn,
\begin{align*}
&\Big{\|} \hat{\varepsilon}' P_{\hat{W}} \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' P_{\hat{W}} \hat{\varepsilon} \Big{\|} = \Big{\|} \hat{\varepsilon} \hat{W} (\hat{W}'\hat{W})^{-1} \hat{W}' \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{W} (\hat{W}'\hat{W})^{-1} \hat{W}' \hat{\varepsilon} \Big{\|} \\
&= \Big{\|} (nT^2) \left( \hat{\varepsilon}' \hat{W}/(nT) \right) \left( \hat{W}' \hat{W}/(nT) \right)^{-1} \left( \hat{W}' \hat{Z}/(nT) \right) \hat{\Omega}^{-1} \\
&\left( \hat{Z}' \hat{W}/(nT) \right) \left( \hat{W}' \hat{W}/(nT) \right)^{-1} \left( \hat{W}' \hat{\varepsilon}/(nT) \right) \Big{\|} \\
&\leq C n \Big{\|} \left( \hat{\varepsilon}' \hat{W}/(nT) \right) \left( \hat{W}' \hat{Z}/(nT) \right) \left( \hat{Z}' \hat{W}/(nT) \right) \left( \hat{W}' \hat{\varepsilon}/(nT) \right) \Big{\|} \\
&\leq C n \Big{\|} \hat{\varepsilon}' \hat{W}/(nT) \Big{\|} \; \Big{\|} \hat{W}' \hat{Z}/(nT) \Big{\|} \; \Big{\|} \hat{Z}' \hat{W}/(nT) \Big{\|} \; \Big{\|} \hat{W}' \hat{\varepsilon}/(nT) \Big{\|} \\
&= n O_p(\sqrt{m_n/n}) O_p(\zeta(k_n) \sqrt{k_n/n}) O_p(\zeta(k_n) \sqrt{k_n/n}) O_p(\sqrt{m_n/n}) = O_p(\zeta(k_n)^2 m_n k_n/n)
\end{align*}
Thus,
\begin{align}\label{eqn_remainders}
\begin{split}
&(\hat{\varepsilon} + \hat{R})' \tilde{Z} \left( n \hat{\Omega} \right)^{-1} \tilde{Z}' (\hat{\varepsilon} + \hat{R}) = \hat{\varepsilon}' \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} + O_p(n m_n^{-2\alpha}) + O_p(n^{1/2} m_n^{-\alpha}) \\
&+ O_p(\zeta(k_n) \sqrt{m_n k_n r_n/n}) + O_p(\zeta(k_n)^2 m_n k_n/n) = \hat{\varepsilon}' \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} + o_p(\sqrt{r_n})
\end{split}
\end{align}
Step 3. Deal with the leading term.
As shown in Lemma~\ref{diff_r_n_small},
\begin{align}\label{eqn_t_r_3_hc}
\frac{n (n^{-1} \hat{\varepsilon}' \hat{Z}) \hat{\Omega}^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}) - n (n^{-1} \hat{\varepsilon}' \hat{Z}) \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})}{\sqrt{r_n}} \overset{p}{\to} 0
\end{align}
Note that
\[
\hat{\varepsilon}' \hat{Z} (n \Omega)^{-1} \hat{Z}' \hat{\varepsilon} = n^{-1} \sum_{i=1}^{n}{\sum_{j = 1}^{n}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j}} = n^{-1} \sum_{i=1}^{n}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_i' \hat{\varepsilon}_i} + 2 n^{-1} \sum_{i=1}^{n-1}{\sum_{j > i}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j}}
\]
Thus,
\begin{align}\label{eqn_t}
t_{*} = \frac{\xi_{*} - r_n}{\sqrt{2 r_n}} = \frac{n^{-1} \sum_{i=1}^{n}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_i' \hat{\varepsilon}_i} - r_n}{\sqrt{2 r_n}} + \frac{\sqrt{2} \sum_{i=1}^{n-1}{\sum_{j > i}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j}}}{\sqrt{n^2 r_n}} = t_1 + t_2,
\end{align}
where
\begin{align*}
t_1 &= \frac{n^{-1} \sum_{i=1}^{n}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_i' \hat{\varepsilon}_i} - r_n}{\sqrt{2 r_n}} \\
t_2 &= \frac{\sqrt{2} \sum_{i=1}^{n-1}{\sum_{j > i}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j}}}{\sqrt{n^2 r_n}}
\end{align*}
Note that
\[
E[t_1] = \frac{E[\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_i' \hat{\varepsilon}_i] - r_n}{\sqrt{2 r_n}} = \frac{E[\chi_i' \Omega^{-1} \chi_i] - r_n}{\sqrt{2 r_n}} = 0,
\]
because
\[
E[\chi_i' \Omega^{-1} \chi_i] = r_n
\]
Next,
\begin{align*}
Var(t_1) &\leq E[(\chi_i' \Omega^{-1} \chi_i)^2]/(2n r_n) \leq C E[\| \chi_i \|^4]/(n r_n) \leq C E[\| \hat{\varepsilon}_i \|^4 \| \hat{Z}_i \|^4]/(n r_n) \\
&\leq C E[\| \hat{Z}_i \|^4]/(n r_n) \leq C \zeta(r_n)^2 r_n/(2n r_n) = C \zeta(r_n)^2/n \to 0,
\end{align*}
so, by Markov's inequality, $t_1 \overset{p}{\to} 0$.
Next, note that $t_2 = \sum_{i=1}^{n-1}{\sum_{j > i}{H_n(\chi_i,\chi_j)}}$, where
\[
H_n(\chi_i,\chi_j) = \sqrt{\frac{2}{n^2 r_n}} \chi_i' \Omega^{-1} \chi_j = \sqrt{\frac{2}{n^2 r_n}} \hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j
\]
Then
\begin{align*}
G_n(u,v) &= E[H_n(\chi_1,u) H_n(\chi_1,v)] = \frac{2}{n^2 r_n} E[\chi_1' \Omega^{-1} u \chi_1' \Omega^{-1} v] \\
&= \frac{2}{n^2 r_n} E[u' \Omega^{-1} \chi_1 \chi_1' \Omega^{-1} v] = \frac{2}{n^2 r_n} u' \Omega^{-1} v = \sqrt{\frac{2}{n^2 r_n}} H_n(u,v)
\end{align*}
Note that $E[H_n(\chi_1,\chi_2)|\chi_1] = \chi_1' \Omega E[\chi_2] = 0$ and that
\begin{align*}
E[H_n(\chi_1,\chi_2)^2] &= \frac{2}{n^2 r_n} E[\chi_1' \Omega^{-1} \chi_2 \chi_1' \Omega^{-1} \chi_2] = \frac{2}{n^2 r_n} E[\chi_1' \Omega^{-1} \chi_2 \chi_2' \Omega^{-1} \chi_1] = \frac{2}{n^2 r_n} E[\chi_1' \Omega^{-1} \chi_1] \\
&= \frac{2}{n^2 r_n} E[tr(\chi_1' \Omega^{-1} \chi_1)] = \frac{2}{n^2 r_n} E[tr(\chi_1 \chi_1' \Omega^{-1})] = \frac{2}{n^2}
\end{align*}
Thus, we have:
\[
\frac{E[G_n(\chi_1,\chi_2)^2]}{\{E[H_n(\chi_1,\chi_2)^2]\}^2} = \frac{(2/n^2 r_n) (2/n^2)}{(4/n^2)} = \frac{1}{r_n} \to 0
\]
and, using the Cauchy-Schwartz inequality,
\begin{align*}
\frac{n^{-1} E[H_n(\chi_1,\chi_2)^4]}{\{E[H_n(\chi_1,\chi_2)^2]\}^2} &= \frac{(4/n^5 r_n^2) E[(\chi_1' \Omega^{-1} \chi_2)^4]}{(4/n^4)} \leq \frac{E[(\chi_1' \Omega^{-1} \chi_1)^2 (\chi_2' \Omega^{-1} \chi_2)^2]}{n r_n^2} \\
&= \frac{E[(\chi_1' \Omega^{-1} \chi_1)^2]^2}{n r_n^2} = \left[ \frac{E[(\chi_1' \Omega^{-1} \chi_1)^2]}{r_n \sqrt{n}} \right]^2 \to 0
\end{align*}
Thus, the conditions of Theorem 1 in \citet{hall_1984} hold, and
\begin{align}\label{eqn_t2}
t_2 = \sqrt{\frac{2}{n^2 r_n}} \sum_{i=1}^{n-1}{\sum_{j > i}{\hat{\varepsilon}_i' \hat{Z}_i \Omega^{-1} \hat{Z}_j' \hat{\varepsilon}_j}} \overset{d}{\to} N(0,1)
\end{align}
The result of the theorem now follows from equations~\ref{eqn_remainders}, \ref{eqn_t_r_3_hc}, \ref{eqn_t}, and~\ref{eqn_t2}. \qed
\subsection{Proof of Theorem~\ref{asy_distr_t_r_n_hc_boot}}
The wild bootstrap test statistic is given by
\[
\xi_{HC}^* = \left( \sum_{i=1}^{n}{\tilde{e}_i^{* \prime} \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i^* \tilde{e}_i^{* \prime} \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i^*} \right) = \tilde{e}^{* \prime} \tilde{Z} \left( n \hat{\Omega}^* \right)^{-1} \tilde{Z}' \tilde{e}^*,
\]
where $\hat{\Omega}^* = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i^* \tilde{e}_i^{* \prime} \tilde{Z}_i}$.
Note that the bootstrap data is generated as
\[
\hat{Y}^* = \hat{W} \tilde{\beta}_1 + \hat{\varepsilon}^*
\]
The bootstrap residuals are given by $\tilde{e}^* = M_{\hat{W}} \hat{Y}^* = M_{\hat{W}} \hat{\varepsilon}^*$. Thus, the bootstrap test statistic can be rewritten as
\[
\xi_{HC}^* = \hat{\varepsilon}^{* \prime} M_{\hat{W}} \tilde{Z} \left( n \hat{\Omega}^* \right)^{-1} \tilde{Z}' M_{\hat{W}} \hat{\varepsilon}^*
\]
The rest of the proof is very similar to the proof of Theorem~\ref{asy_distr_t_r_n_hc}, so I only provide a sketch here. First, one can show that
\[
\frac{\hat{\varepsilon}^{* \prime} M_{\hat{W}} \tilde{Z} \left( n \hat{\Omega}^* \right)^{-1} \tilde{Z}' M_{\hat{W}} \hat{\varepsilon}^* - \hat{\varepsilon}^{* \prime} Z \left( n \Omega^* \right)^{-1} Z \hat{\varepsilon}^*}{\sqrt{r_n}} \overset{p}{\to} 0,
\]
where $\Omega^* = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{e}_i \tilde{e}_i' \hat{Z}_i}$. Next, one can deal with the leading term $\hat{\varepsilon}^{* \prime} Z \left( n \Omega^* \right)^{-1} Z \hat{\varepsilon}^*$ as in Step 3 of the proof of Theorem~\ref{asy_distr_t_r_n_hc}, but now conditional on the data $\mathcal{Z}_{n,T}$. The result of the theorem follows.
\subsection{Proof of Theorem~\ref{global_alternative_t_r}}
Recall that $\hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i}$ in the homoskedastic case and $\hat{\Omega} = n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i}$ in the heteroskedastic case. Next, note that
\[
\sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} = \tilde{Z}' \tilde{e}
\]
where $\tilde{Z} = (\tilde{Z}_1',...,\tilde{Z}_n')'$ is $nT \times r_n$, $\tilde{e} = (\tilde{e}_1',...,\tilde{e}_n')'$ is $nT \times 1$.
Then under homoskedasticity
\[
\xi = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right) = n \left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right),
\]
while under heteroskedasticity
\[
\xi_{HC} = \left( \sum_{i=1}^{n}{\tilde{e}_i' \tilde{Z}_i} \right) \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i} \right)^{-1} \left( \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i} \right) = n \left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right)
\]
Also note that
\[
\frac{\sqrt{r_n}}{n} \frac{n \left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right) - r_n}{\sqrt{2 r_n}} = \frac{1}{\sqrt{2}} \left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right) + T_2,
\]
where $T_2 = - r_n/(n \sqrt{2}) \to 0$.
Hence, it suffices to show that $\left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right) \overset{p}{\to} \Delta$.
Next, note that due to the projection nature of the series estimators, $\tilde{e} = M_{\hat{W}} \hat{Y} = M_{\hat{W}} \hat{\varepsilon}^* + M_{\hat{W}} \hat{R}^*$. Hence,
\begin{align*}
&\left( n^{-1} \tilde{e}' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' \tilde{e} \right) = \left( n^{-1} (M_{\hat{W}} \hat{\varepsilon}^* + M_{\hat{W}} \hat{R}^*) ' \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' (M_{\hat{W}} \hat{\varepsilon}^* + M_{\hat{W}} \hat{R}^*) \right) \\
&= \left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \tilde{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \tilde{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right) \\
&= \left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right)
\end{align*}
Thus, it suffices to show that
\[
n \left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right) \overset{p}{\to} \Delta
\]
Next,
\begin{align*}
&n \left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right) = \left( n^{-1} \hat{\varepsilon}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{\varepsilon}^* \right) \\
&+ 2 \left( n^{-1} \hat{R}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{\varepsilon}^* \right) + \left( n^{-1} \hat{R}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{R}^* \right)
\end{align*}
Similarly to the proof of Theorem~\ref{asy_distr_t_r_n_hc}, but using the fact that $\sup_{x \in \mathcal{X}}{R^*(x)} = o(1)$ instead of $\sup_{x \in \mathcal{X}}{R(x)} = O(m_n^{-\alpha})$,
\begin{align*}
\left( n^{-1} \hat{R}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{\varepsilon}^* \right) \leq C \hat{R}^{* \prime} \hat{\varepsilon}^*/(n \tilde{\sigma}^2) = O_p(n^{-1/2}) o_p(1) = o_p(1)
\end{align*}
and
\[
\left( n^{-1} \hat{R}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{R}^* \right) \leq C \hat{R}^{* \prime} \hat{R}^*/(n \tilde{\sigma}^2) = o_p(1)
\]
Thus,
\[
n \left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right) = \left( n^{-1} \hat{\varepsilon}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{\varepsilon}^* \right) + o_p(1)
\]
Next, given that, as shown in the proof of Theorem~\ref{asy_distr_t_r_n_hc}, $M_{\hat{W}} \hat{Z} = \hat{Z} + o_p(1)$ and the eigenvalues of $\hat{\Omega}$ are bounded above and below w.p.a. 1,
\[
\left( n^{-1} \hat{\varepsilon}^{* \prime} M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} \hat{\varepsilon}^* \right) = \left( n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' \hat{\varepsilon}^* \right) + o_p(1)
\]
Next,
\begin{align*}
&\Big{|} (n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) (\hat{\Omega}^{-1} - \Omega^{*-1}) (n^{-1} \hat{Z}' \hat{\varepsilon}^{*}) \Big{|} \leq \Big{|} (n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) (\Omega^{*-1} (\hat{\Omega} - \Omega^*) \hat{\Omega}^{*-1} (\hat{\Omega} - \Omega^*) \Omega^{*-1}) (n^{-1} \hat{Z}' \hat{\varepsilon}^{*}) \Big{|} \\
&+\Big{|} (n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) (\Omega^{*-1} (\hat{\Omega} - \Omega^*) \Omega^{*-1}) (n^{-1} \hat{Z}' \hat{\varepsilon}^{*}) \Big{|} \leq \| \Omega^{*-1} n^{-1} \hat{Z}' \hat{\varepsilon}^{*} \|^2 (\| \hat{\Omega} - \Omega^* \| + C \| \hat{\Omega} - \Omega^* \|^2) = o_p(1)
\end{align*}
Thus, $(n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) \hat{\Omega}^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^*) = (n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^{*}) + o_p(1)$.
To complete the proof, note that $Var(\hat{Z}_i' \hat{\varepsilon}_i^*) \leq \Omega^*$, because $\Omega^* = E[\hat{Z}_i' \hat{\varepsilon}_i \hat{\varepsilon}_i' \hat{Z}_i]$. Then
\begin{align*}
&E[(n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])' \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])] \\
&\leq E[(n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])' Var(\hat{Z}_i' \hat{\varepsilon}_i^*)^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])] \\
&= E[tr\left(Var(\hat{Z}_i' \hat{\varepsilon}_i^*)^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*]) (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])'\right)] =tr(I_{r_n})/n = r_n/n \to 0
\end{align*}
Thus,
\begin{align*}
&\Big{|} (n^{-1} \hat{\varepsilon}^{* \prime} \hat{Z}) \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^{*}) - E[\hat{\varepsilon}_i^{*\prime} \hat{Z}_i] \Omega^{*-1} E[\hat{Z}_i' \hat{\varepsilon}_i^*] \Big{|} \\
&\leq \Big{|} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])' \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*]) \Big{|} + 2 \Big{|} E[\hat{\varepsilon}_i^{* \prime} \hat{Z}_i] \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*]) \Big{|} \\
&\leq o_p(1) + 2 \sqrt{E[\hat{\varepsilon}_i^* \hat{Z}_i'] \Omega^{*-1} E[\hat{Z}_i \hat{\varepsilon}_i^*]} \sqrt{(n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])' \Omega^{*-1} (n^{-1} \hat{Z}' \hat{\varepsilon}^* - E[\hat{Z}_i' \hat{\varepsilon}_i^*])} \\
&= o_p(1) + 2 \sqrt{\Delta} o_p(1) = o_p(1)
\end{align*}
Combining the results above, $\left( n^{-1} (\hat{\varepsilon}^* + \hat{R}^*)' M_{\hat{W}} \hat{Z} \right) \hat{\Omega}^{-1} \left( n^{-1} \hat{Z}' M_{\hat{W}} (\hat{\varepsilon}^* + \hat{R}^*) \right) \overset{p}{\to} \Delta$. \qed
\subsection{Auxiliary Lemmas}
\begin{lemma}[\citet{donald_et_al_2003}, Lemma A.2]\label{series_normalization}
If Assumption \ref{series_norms_eigenvalues} is satisfied then it can be assumed without loss of generality that $\hat{P}^{k}(x) = \bar{P}^{k}(x)$ and that $E[\bar{P}^{k}(X_{it}) \bar{P}^{k}(X_{it})'] = I_{k}$.
\end{lemma}
\begin{remark}
This normalization is common in the literature on series estimation, when all elements of $P^{k}(X_i)$ are used to estimate the model. In my setting, $P^{k}(X_i)$ is partitioned into $W^{m}(X_i)$, used in estimation, and $T^{r}(X_i)$, used in testing. The normalization implies that $W_i$ and $T_i$ are orthogonal to each other. This can be justified as follows. Suppose that $W_i$ and $T_i$ are not orthogonal. Then one can take all elements of $(W_i', T_i')'$ and apply the Gram-Schmidt process to them. Because the orthogonalization process is sequential, it will yield the vector $(W_i^{0},T_i^{0})'$ such that $W_i^{0}$ spans the same space as $W_i$, $T_i^{0}$ spans the same space as $T_i$, and $W_i^{0}$ and $T_i^{0}$ are orthogonal. Thus, the normalization is indeed without loss of generality.
\end{remark}
\begin{lemma}[\citet{baltagi_li_2002}, Theorem 2.2]\label{series_f_rates}
Let $f(x) = f(x,\theta_0,h_0)$, $f_i = f(X_i)$, $\tilde{f}(x)= f(x, \tilde{\theta}, \tilde{h}) = W^{m_n}(x)' \tilde{\beta}_0$, and $\tilde{f}_i = \tilde{f}(X_i) = W_i' \tilde{\beta}_0$. Under Assumptions \ref{dgp}, \ref{series_norms_eigenvalues}, and \ref{series_approx}, the following is true:
\[
\frac{1}{n}\sum_{i=1}^{n}{(\tilde{f}_i - f_i)^2} = O_p(m_n/n + m_n^{-2\alpha})
\]
\end{lemma}
\begin{lemma}\label{lemma_din_1}
Suppose that Assumption~\ref{assumption_din_1} is satisfied and $E[\hat{\varepsilon}_{i} \hat{\varepsilon}_{i}']$ is finite. If $E[\hat{\varepsilon}_{i} | \hat{X}_i] = 0$ then $E[\hat{P}_{i}' \hat{\varepsilon}_{i}] = 0$ for all $k$. Furthermore, if $E[\hat{\varepsilon}_{i} | X_i] \neq 0$ then $E[\hat{P}_{i}' \hat{\varepsilon}_{i}] \neq 0$ for all $k$ large enough.
\end{lemma}
\begin{proof}
The proof is similar to the proof of Lemma 2.1 in \citet{donald_et_al_2003} and is thus omitted.
\end{proof}
\begin{lemma}\label{lemma_PP}
If Assumptions of Theorem~\ref{asy_distr_t_r_n_hc} hold, then $\| \hat{P}'\hat{P}/(nT)- I_{k_n}\| = O_p(\zeta(k_n) \sqrt{k_n/n})$, $\| \hat{W}'\hat{W}/(nT) - I_{m_n}\| = O_p(\zeta(m_n) \sqrt{m_n/n})$ and $\| \hat{Z}'\hat{Z}/(nT) - I_{r_n}\| = O_p(\zeta(r_n) \sqrt{r_n/n})$. Moreover, $\| \hat{W}'\hat{Z}/(nT) \| = O_p(\zeta(k_n) \sqrt{k_n/n})$.
\end{lemma}
\begin{proof}
By Lemma A.1 in \citet{baltagi_li_2002}, $\| \hat{P}'\hat{P}/(nT)- I_{k_n}\| = O_p(\zeta(k_n) \sqrt{k_n/n})$. Similarly, it can be shown that $\| \hat{W}'\hat{W}/(nT) - I_{m_n}\| = O_p(\zeta(m_n) \sqrt{m_n/n})$ and $\| \hat{Z}'\hat{Z}/(nT) - I_{r_n}\| = O_p(\zeta(r_n) \sqrt{r_n/n})$. Moreover, note that
\[
\hat{P}'\hat{P}/(nT) - I_{k_n} = \begin{pmatrix}
\hat{W}'\hat{W}/(nT) & \hat{W}'\hat{Z}/(nT) \\
\hat{Z}'\hat{W}/(nT) & \hat{Z}'\hat{Z}/(nT)
\end{pmatrix} -
\begin{pmatrix}
I_{m_n} & \textbf{0}_{m_n \times r_n} \\
\textbf{0}_{r_n \times m_n} & I_{r_n}
\end{pmatrix}
\]
Hence, $\| \hat{W}'\hat{Z}/(nT) \| = O_p(\zeta(k_n) \sqrt{k_n/n})$.
\end{proof}
\begin{lemma}\label{omegas_T_hc}
In the homoskedastic case, let $\tilde{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{\Sigma}_T \hat{Z}_i}$, $\breve{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \Sigma_T \hat{Z}_i}$.
In the heteroskedastic case, let $\tilde{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{e}_i \tilde{e}_i' \hat{Z}_i}$, $\breve{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \hat{\varepsilon}_i \hat{\varepsilon}_i' \hat{Z}_i}$, and $\bar{\Omega} = n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \Sigma_i \hat{Z}_i}$, where $\Sigma_i = E[\hat{\varepsilon}_i \hat{\varepsilon}_i' | X_i]$.
Let $\Omega = E[\hat{Z}_i \Sigma_i \hat{Z}_i']$.
Suppose that Assumptions \ref{errors_unified}(ii), \ref{series_norms_eigenvalues}, and \ref{series_approx} are satisfied. Then
\begin{align*}
\| \tilde{\Omega} - \breve{\Omega} \| &= O_p \left( \zeta(r_n)^2 (m_n/n + m_n^{-2\alpha}) \right) \\
\| \breve{\Omega} - \bar{\Omega} \| &= O_p(\zeta(r_n) r_n^{1/2}/n^{1/2}) \\
\| \bar{\Omega} - \Omega \| &= O_p(\zeta(r_n) r_n^{1/2}/n^{1/2})
\end{align*}
If Assumption~\ref{errors_unified}(i) is also satisfied then $1/C \leq \lambda_{\min}(\Omega) \leq \lambda_{\max}(\Omega) \leq C$, and if $\zeta(r_n)^2 (m_n/n + m_n^{-2\alpha}) \to 0$ and $\zeta(r_n) r_n^{1/2}/n^{1/2} \to 0$, then w.p.a. 1, $1/C \leq \lambda_{\min}(\tilde{\Omega}) \leq \lambda_{\max}(\tilde{\Omega}) \leq C$ and $1/C \leq \lambda_{\min}(\bar{\Omega}) \leq \lambda_{\max}(\bar{\Omega}) \leq C$.
Moreover, if $\| \hat{\Omega} - \tilde{\Omega} \| = o_p(1)$, then $1/C \leq \lambda_{\min}(\hat{\Omega}) \leq \lambda_{\max}(\hat{\Omega}) \leq C$.
\end{lemma}
\begin{proof}[Sketch of the Proof of Lemma~\ref{omegas_T_hc}]
Note that in the homoskedastic case, all matrices involved in the lemma can be written as
\begin{align*}
\hat{\Omega} &= n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{\Sigma}_T \tilde{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{\tilde{\sigma}_{ts} n^{-1} \sum_{i=1}^{n}{ \tilde{Z}_{it} \tilde{Z}_{is}'}}} \\
\tilde{\Omega} &= n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{\Sigma}_T \hat{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{\tilde{\sigma}_{ts} n^{-1} \sum_{i=1}^{n}{ \hat{Z}_{it} \hat{Z}_{is}'}}} \\
\breve{\Omega} &= n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \Sigma_T \hat{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{\sigma_{ts} n^{-1} \sum_{i=1}^{n}{ \hat{Z}_{it} \hat{Z}_{is}'}}}
\end{align*}
In the heteroskedastic case, all matrices involved in the lemma can be written as
\begin{align*}
\hat{\Omega} &= n^{-1} \sum_{i=1}^{n}{\tilde{Z}_i' \tilde{e}_i \tilde{e}_i' \tilde{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{n^{-1} \sum_{i=1}^{n}{\tilde{e}_{it} \tilde{e}_{is} \tilde{Z}_{it} \tilde{Z}_{is}'}}} \\
\tilde{\Omega} &= n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \tilde{e}_i \tilde{e}_i' \hat{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{n^{-1} \sum_{i=1}^{n}{\tilde{e}_{it} \tilde{e}_{is} \hat{Z}_{it} \hat{Z}_{is}'}}} \\
\breve{\Omega} &= n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \hat{\varepsilon}_i \hat{\varepsilon}_i' \hat{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{n^{-1} \sum_{i=1}^{n}{\hat{\varepsilon}_{it} \hat{\varepsilon}_{is} \hat{Z}_{it} \hat{Z}_{is}'}}} \\
\bar{\Omega} &= n^{-1} \sum_{i=1}^{n}{\hat{Z}_i' \Sigma_i \hat{Z}_i} = \sum_{t=1}^{T}{\sum_{s=1}^{T}{n^{-1} \sum_{i=1}^{n}{\sigma_{ts} \hat{Z}_{it} \hat{Z}_{is}'}}}
\end{align*}
Because $T$ is finite, Lemma~\ref{omegas_T_hc} can be proved by applying the results from Lemmas A.5 and A.6 in \citet{korolev_2019} element by element.
\end{proof}
\begin{lemma}\label{diff_r_n_small}
If Assumptions of Theorem~\ref{asy_distr_t_r_n_hc} hold, then
\[
\frac{n (n^{-1} \hat{\varepsilon}' \hat{Z}) \hat{\Omega}^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}) - n (n^{-1} \hat{\varepsilon}' \hat{Z}) \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})}{\sqrt{r_n}} \overset{p}{\to} 0
\]
\end{lemma}
\begin{proof}
Note that $\hat{\varepsilon}' \hat{Z} (n \hat{\Omega})^{-1} \hat{Z}' \hat{\varepsilon} = n (n^{-1} \hat{\varepsilon}' \hat{Z}) \hat{\Omega}^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})$. Then
\begin{align*}
&\Bigg{|} \frac{n (n^{-1} \hat{\varepsilon}' \hat{Z}) \hat{\Omega}^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})}{\sqrt{2 r_n}} - \frac{n (n^{-1} \hat{\varepsilon}' \hat{Z}) \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon})}{\sqrt{2r_n}} \Bigg{|} = \Bigg{|} \frac{n (n^{-1} \hat{\varepsilon}' \hat{Z}) (\hat{\Omega}^{-1} - \Omega^{-1}) (n^{-1} \hat{Z}' \hat{\varepsilon})}{\sqrt{2r_n}} \Bigg{|} \\
&\leq \frac{n \| \Omega^{-1} n^{-1} \hat{Z}' \hat{\varepsilon} \|^2 (\| \hat{\Omega} - \Omega \| + C \| \hat{\Omega} - \Omega \|^2)}{\sqrt{2r_n}}
\end{align*}
As shown above, $\| \Omega^{-1} (n^{-1} \hat{Z}' \hat{\varepsilon}) \| = O_p(\sqrt{r_n/n})$.
Then
\begin{align*}
\frac{n \| \Omega^{-1} n^{-1} \hat{Z}' \hat{\varepsilon} \|^2 (\| \tilde{\Omega} - \Omega \| + C \| \tilde{\Omega} - \Omega \|^2)}{\sqrt{2r_n}} = \frac{n O_p(r_n/n) o_p(1/\sqrt{r_n})}{\sqrt{2r_n}} = \frac{o_p(\sqrt{r_n})}{\sqrt{2r_n}} = o_p(1),
\end{align*}
provided that $\|\hat{\Omega} - \Omega \| = o_p(1/\sqrt{r_n})$, which holds under the rate conditions~\ref{rate_cond_r_1_hc}--\ref{rate_cond_r_5_hc} and the high level assumption that $\| \hat{\Omega} - \tilde{\Omega} \| = o_p(r_n^{-1/2})$.
\end{proof}
\clearpage
|
2,869,038,154,890 | arxiv | \section{Introduction}
From the early work of \cite{biermann32r} onwards research on convection has remained a major
challenge in stellar astrophysics, in particular as convection turned out to be one of the most important
mechanisms of energy transport and mixing in stars. As is described, for instance, in \cite{canuto09b}, when
we compare the spatial scales of viscous processes derived from the results of \cite{chapman54b} on
fully ionised gases with the spatial scales of convective flow observed at stellar surfaces \citep{kupka17b},
stellar convection is characterised by very high Reynolds numbers. Stellar convective flows are thus
highly turbulent, even though the direct detection of turbulence is difficult due to the nature and resolution
of observational methods available to us (cf.\ \citealt{kupka17b}).
To model this class of flows poses serious challenges for stellar structure and evolution models
(for introductions and reviews see, e.g., \citealt{cox04b}, \citealt{canuto09b}, \citealt{kupka17b}, \citealt{kupka20b}).
Due to the extreme range of scales in space and time numerical, hydrodynamical simulations
cannot be used directly in stellar evolution calculations (cf.\ the estimates given in \citealt{kupka17b}).
Consequently, turbulent convection has to be modelled in a framework affordable for direct
coupling into one-dimensional stellar models of stellar evolution. The turbulent convection models (TCM)
used in this approach differ widely in computational costs, physical completeness, and general principles
considered in their derivation, from completely phenomenological to more systematic approaches based
on turbulence theory (see \citealt{kupka17b} for an overview).
One methodological way to derive TCM equations which are suitable for stellar evolution calculations
is the Reynolds stress approach. The splitting of variables in turbulent flow into a mean and
a fluctuating component was first introduced by \cite{reynolds1894b}, followed by the suggestion
of \cite{keller25b} to consider this Reynolds splitting for a moment expansion approach that
was first completed by \cite{chou45b}. Dynamical variables such as velocity $\vec v$,
density $\rho$, or entropy $s$, for example, can be subject to such splitting:
\begin{align*}
\vec v=\overline{\vec v}+\vec v\, ', \,\,\,\, \rho=\overline{\rho}+\rho', \,\,\,\, s=\overline{s}+s', \,\,\dots
\end{align*}
Strictly speaking these are {\em ensemble averages} over different initial conditions. In practice,
the variables are also subject to spatial averaging, in one-dimensional stellar models typically over
the $\theta$ and $\phi$ directions, to which the overbar in the above notation refers to whereas the component
with a prime refers to the fluctuating part of each quantity.
Due to their immediate physical meaning the higher order combinations of the fluctuating parts
which appear in such Reynolds stress models of turbulent convection are also model predictions
of direct astrophysical interest. The second order moment of velocity fluctuations,
characterising the {\em turbulent kinetic energy (TKE)} of the convective flow, is directly related to the highly
efficient chemical mixing induced by convection. In stars with nuclear burning in convective
cores this has a direct impact on the luminosity and the lifetime of
the nuclear burning phase. Similarly, the second order moment of velocity and entropy
fluctuations, related to the convective flux, determines the energy transported by convection.
Computing the convective flux allows predicting the temperature gradient in convective regions. Recently,
the temperature gradient in core boundary layers of an intermediate-mass main-sequence star was
probed using asteroseismology \citep{michielsen2021}, an observation that can directly be compared
to results from a TCM.
Presently, the most commonly used theory to describe convection in stellar
structure and evolution models is still the mixing length theory
\citep[][ MLT]{bv58b}. However, MLT is not able
to describe the convective boundary in a physically accurate way. Observations
have shown that chemical mixing beyond the boundary of convectively
unstable regions, commonly known as \textit{overshooting}, is
required \citep[see, for example,][]{maeder1981,bressan1981,pietrinferni2004}. In stellar
models using MLT parametrised ad~hoc mixing beyond the boundary is
introduced to achieve this. Likewise, the temperature structure of an overshooting region
cannot be predicted by MLT. These examples highlight the need for more physical theories of
convection, like TCM, being included in stellar structure and evolution models.
A large number of TCM have been developed
\citep{xiong1997,canuto92b,canuto93b,canuto98b,li2001,li2007,kuhfuss86b,kuhfuss1987}
which differ in the set of variables used and the set of approximations and assumptions made (see \citealt{canuto93b}
and \citealt{kupka17b} for comparisons and a review). Among other physical effects the dissipation
of TKE requires a careful discussion in the context of TCM. Acting as a sink term
for TKE in overshooting layers the dissipation rate has a direct impact on the extent of convectively mixed regions. Assuming a
Kolmogorov spectrum of turbulence the dissipation rate of TKE can conveniently be computed by a local
expression involving a dissipation length scale with a single constant parameter. This expression is, however,
inapplicable in non-local situations, encountered in layers adjacent to convectively unstable zones. To
treat the dissipation of TKE in non-local convection models a physically more complete description of the
dissipation rate is required \citep{zeman77b,canuto98b}.
We begin this paper by discussing local and non-local descriptions of the dissipation of TKE in Sect.~\ref{sect_diss}.
From the dissipation rate in non-local convection theories we derive a model to account for the
dissipation of TKE by buoyancy waves in overshooting layers in Sect.~\ref{Sect_GARSTEC}. In Sect.~\ref{sect_discussion}
we then discuss implications of the improved dissipation model when applied to stellar models. For the computation of
the stellar models we use the TCM derived by \cite{kuhfuss1987} implemented into the GARching STellar Evolution
Code \citep[GARSTEC,][]{weiss2008}. The key assumptions and approximations of the \cite{kuhfuss1987} model are
reviewed in Appendix~\ref{secKuh}. Using the local expression for the dissipation rate of the TKE we find an excessive
overshooting extent beyond convective cores. When including the dissipation by buoyancy waves this overshooting
is limited to a physically more reasonable range. This allows us to predict the convective core sizes and temperature
structures of stars with different masses. We present our conclusions in Sect.~\ref{sect_conclusions}. A detailed discussion
of the results obtained from the improved TCM can be found in \cite{paper2} (Paper~{\sc II} in the following).
\section{On the dissipation rate $\epsilon$ of turbulent kinetic energy}
\label{sect_diss}
The necessity to account for the dissipation rate of turbulent kinetic energy,
$\epsilon$, in models of convection stems from the fact that it is not a negligibly
small quantity. Indeed, the expression from which $\epsilon$ is computed is
proportional to the kinematic viscosity $\nu$. The latter is small in stars compared
to the radiative diffusivity $\chi$ which results in the small values of the Prandtl
number ${\rm Pr}=\nu/\chi$ typical for stars. Energy conservation requires
$\epsilon$ to remain finite and non-negligible even in the limit of small viscosity
(see \citealt{canuto97c}). Neglecting compressibility (for its modelling cf.\ \citealt{canuto97b})
we can compute $\epsilon$ from
\begin{equation} \label{eq_eps_spectrum}
\epsilon = 2\,\nu \overline{\left(\frac{\partial u_i}{\partial x_i}\right)^2} = 2\,\nu\,\Omega = 2\,\nu\,\int k^2E(k){\rm d}k
\end{equation}
in case of a locally isotropic, homogeneous flow. Here, $u_i$ is the \mbox{$i$-th} velocity component,
$x_i$ is the \mbox{$i$-th} component of location, and $E(k)$ is the spectrum of turbulent kinetic
energy as a function of wavenumber $k$.\footnote{Concerning notation the convention of
summation over equally named indices is assumed.}
Although convection is neither isotropic nor homogeneous on those large scales on
which its contribution to energy transport is maximal, Eq.~(\ref{eq_eps_spectrum}) is
a sufficient approximation to explain some basic properties of turbulent
convection.\footnote{In real world systems the spectra
of turbulent kinetic energy, $E(k)$, usually depend on location ${\bf r}$ and in the most
general sense an averaging over directions in $k$-space would have to be performed,
i.e., $E(k)$ becomes a two-point correlation function $E({\bf k},{\bf r})$ and would
also have to account for density fluctuations.}
In a quasi-stationary state where the amount of kinetic energy injected into the system
per unit of time equals $\epsilon$, the enstrophy $\Omega$ of the flow increases, if $\nu$ decreases.
The latter follows from the vorticity $\omega$ through
$2\,\Omega = \overline{\omega^2}$. Thus, $\epsilon$ is constrained by energy
conservation and quantifies the amount of kinetic energy converted to thermal one.
If for a flow both the first and second Kolmogorov hypotheses hold
\citep{pope00b}, then there exists a range of length scales $\ell = \pi/k$
for which $\epsilon$ is independent of $\nu$ and independent of the details
of the large scale input of kinetic energy into the flow. This region
is known as the {\em Kolmogorov inertial range}. In that region $\epsilon$
is solely described by the exchange of energy between larger and
smaller scales. If this exchange peaks between neighbouring scales
(see \citealt{lesieur08b}), which is assumed to hold for turbulent flows
except for corrections due to intermittency (see also \citealt{pope00b}),
it can be modelled as a flux in $k$-space. This is one of the basic inputs
for the turbulence model of \citet{canuto96d} used in \citet{canuto98b} to
justify the mathematical form and the constants involved in closure relations
derived for their one-point closure Reynolds stress model of convection (see their Eq.~(9c)).
One important consequence for Eq.~(\ref{eq_eps_spectrum}) is the following
one: if an inertial range exists, it can be shown to require
\begin{equation} \label{eq_Kolomogorov_spectrum}
E(k)={\rm Ko}\,\epsilon^{2/3}\,k^{-5/3}
\end{equation}
to hold, i.e., a {\em Kolomogorov spectrum} to exist. Here, ${\rm Ko}$ is
the {\em Kolomogorov constant} which also turns out to equal $5/3$ in
the model of \citet{canuto96d} and \citet{canuto98b}, just as the {\em power
law index} for $k$ in the spectrum Eq.~(\ref{eq_Kolomogorov_spectrum}).
Recalling Eq.~(\ref{eq_eps_spectrum}) this demands that the contributions
of small scales $k$ to $\epsilon$ {\rm increase} with $k^{1/3}$ and a region
where the Kolmogorov inertial range no longer holds, just around the dissipation scale
$k_{d}=\pi/{\ell}_d$, would have to be characterised more accurately than through
Eq.~(\ref{eq_Kolomogorov_spectrum}) for a direct computation of $\epsilon$ from
the spectral energy distribution $E(k)$ and Eq.~(\ref{eq_eps_spectrum}). Within
a one-point closure model and thus in any of the prescriptions used in astrophysics
to compute the convective flux inside a stellar structure code, this is not feasible
and a different approach is required to compute $\epsilon$.
\subsection{Computation in local models: spectra and local limits}
One way around computing the spectrum $\epsilon(k)$ is to just compute its integral
value $\epsilon$ from a model of $E(k)$ as follows. Assume that
$\nu$ is negligibly small. In the limit of vanishing $\nu$ the latter can ensure that
its product with $\int k^2\, E(k)\,{\rm d}k$ remains finite even though $\Omega$ might
increase indefinitely for arbitrarily large $k$. Hence, as in the derivation of
Eq.~(5b) of \citet{canuto98b} and as also in their Section~6.4, assume that
$E(k)$ is given by Eq.~(\ref{eq_Kolomogorov_spectrum}) from a certain
value $k_0$ onwards, i.e., the entire energy spectrum is given by a Kolmogorov
spectrum with an energy cutoff for $k < k_0$. Thus, $E(k)=0$ for $k < k_0$
and $E(k) \sim k^{-5/3}$ for arbitrarily large $k$ with $k \geqslant k_0$.
In this case, it is easy to first obtain $K$, the turbulent kinetic energy (TKE),
from integration of $E(k)$ over all wavenumbers:
\begin{equation} \label{eq_Kint}
K = \int_0^{\infty} E(k)\,{\rm d}k = \int_{k_0}^{\infty} E(k)\,{\rm d}k, \quad {\rm if}\,\, E(k)=0\,\, {\rm for}\,\, k < k_0,
\end{equation}
and with Eq.~(\ref{eq_Kolomogorov_spectrum}) we obtain from Eq.~(\ref{eq_Kint}) that
\begin{equation}
K = {\rm Ko}\,\epsilon^{2/3}\left.\frac{k^{-2/3}}{-2/3}\right|_{k_0}^{\infty} =
\frac{3\,{\rm Ko}}{2} \epsilon^{2/3} k_0^{-2/3}.
\end{equation}
For $\ell_0 = \pi/k_0$ this can be quickly rearranged to yield
\begin{equation} \label{eq_MLT_epsilon}
\epsilon = \pi \left(\frac{2}{3\,{\rm Ko}}\right)^{3/2} \frac{K^{3/2}}{\ell_0} = c_{\epsilon} \frac{K^{3/2}}{\ell_0}=c_{\epsilon}\frac{K^{3/2}}{\Lambda},
\end{equation}
as shown in \citet{canuto98b}.\footnote{Note there is a typo in their Eq.~(5c) which should have the constant ${\rm Ko}$
in the denominator.}
This is also the standard ``local'' or ``mixing length'' prescription for the computation of $\epsilon$.
It assumes maximum separation of the energy carrying scales around $\ell_0$ and the
Kolmogorov dissipation scale $\ell_d$ (assumed to be negligibly small,
and not be confused with $\Lambda$).
Moreover, it assumes validity of the inertial range as if all the energy input were at one
length scale only, i.e., at $\ell_0$, here set to be equal to $\Lambda$.
All other scales for which $\ell \gtrsim \ell_0$ behave as if they were unaffected
by the very small scales (scale separation) and also by the details of the energy input.
Thus, a perfect energy cascade is assumed. Mixing length theory (MLT) {\em in addition}
replaces $E(k)$ with a $\delta$-function peaked at $l_0$ such that its integral yields
Eq.~(\ref{eq_MLT_epsilon}). It is thus a ``one-eddy approximation'' where all the energy
transport due to convection occurs on the critical (mixing) length scale $\Lambda$
which has to be computed for each layer. Either way, the challenge of computing $\epsilon$ turns
into the challenge of prescribing $\Lambda$.
Evidently, this cannot be an accurate model, since at least a range of scales spanning
easily an order of magnitude (consider different granule sizes as an example) is expected
to contribute to energy transport at convective stellar surfaces such as those of our Sun.
Thus, Eq.~(\ref{eq_MLT_epsilon}) can at most be an estimate of order $O(1)$.
For some flows such as a shear flow in a pipe (Poiseuille flow), for which the mixing length
formalism to compute the turbulent viscosity had originally been proposed by L.~Prandtl
(see \citealt{pope00b}, for example), this length can be fairly well constrained from geometrical
arguments. Not surprisingly this is the application for which this prescription is most reliable.
For compressible convection on the other hand this length is much more difficult to constrain
and the standard choice is to assume that
\begin{equation} \label{eq_MLT_alpha}
\Lambda = \alpha\,H_p
\end{equation}
where $\alpha$ is the MLT-parameter or mixing length parameter and $H_p$
is the local pressure scale height in the convective zone. This situation has motivated
\citet{canuto91b,canuto92c} to suggest a new convection model in which
$\epsilon$ is computed directly from Eq.~(\ref{eq_Kint}). That removes the
uncertainties introduced by the one-eddy approximation, but a scale length
$\Lambda$ is still introduced in this model. It compares the geometric size of
flow features which transport most of the energy with the length scales dominated
by dissipation. This has permitted easy implemention into existing stellar evolution
codes based on MLT. The same approach was used by \citet{canuto96b}.
But that concept collapses if an overshooting zone has to be
modelled. In such a region, located just underneath or above a convectively unstable
zone, the convective flow is fundamentally non-local: the only way to sustain
a non-vanishing solution is transport of kinetic and potential energy from the
adjacent convective zone (cf.\ Sect.~10 in \citealt{canuto98b}). For such a region
there is no reason to assume that the prescription of Eq.~(\ref{eq_MLT_alpha})
with an $\alpha$ {\em independent of vertical location} can still hold.
Thus, even if other equations in a convection model are treated non-locally,
the continued use of Eq.~(\ref{eq_MLT_alpha}) with Eq.~(\ref{eq_MLT_epsilon})
along with a constant $\alpha$ even just within a single object may lead to inconsistent
or unphysical results, a fact long acknowledged in the atmospheric sciences
by much more advanced modelling (see, for instance, \citealt{zeman77b}).
As we show below, this is exactly the problem one encounters when using the
3-equation Kuhfu{\ss} theory \citep{kuhfuss1987}, and it motivated the present work:
how to proceed and improve the computation of $\epsilon$ in such a case?
\subsection{Computation in non-local models: the dissipation rate equation} \label{subsecdissip}
A common starting point for non-local models of convection is the
dynamical equation for turbulent kinetic energy:
\begin{eqnarray}
\lefteqn{\partial_t K + \partial_z \left(\frac{1}{2}\overline{q^2 w} + \overline{p w}\right) =
g \alpha_{\rm v} \overline{w\theta} - \epsilon} \nonumber\\
& & {} + \partial_z \left(\nu \partial_z K\right) + \frac{1}{2}C_{ii}, \label{eq_Kfull}
\end{eqnarray}
as given in \citet{canuto93b}, for example. In the Boussinesq case though, the
corrections due to compressibility given by the term $C_{ii}$ are zero. For the case
of a low Prandtl number and if there are no contributions by a mean shear or rotation,
we obtain \citep{canuto92b}
\begin{eqnarray}
\lefteqn{\partial_t K + \partial_z \left(\frac{1}{2}\overline{q^2 w} + \overline{p w}\right) =
g \alpha_{\rm v} \overline{w\theta} - \epsilon} \label{eq_Keq}
\end{eqnarray}
which within the Boussinesq approximation is an {\rm exact} equation, though yet
unclosed. Here, $\partial_t$ and $\partial_z$
are partial derivatives with respect to time $t$ and vertical (radial) coordinate $z$. This is
a prognostic equation for the second order moment $K=\overline{q^2}/{2}$
with $\overline{q^2}=\overline{w^2+v_\theta^2+v_\phi^2}$ derived directly from the Boussinesq
approximation of the Navier-Stokes equations through ensemble averaging. The non-local transport
includes the flux of kinetic energy (in the Boussinesq approximation given by
$F_{\rm kin}=\rho\,\overline{q^2\,w}/{2}$ with $w$ as the fluctuating component of vertical velocity)
and of pressure fluctuations, $\overline{p\,w}$. This is to be balanced by local production,
$g \alpha_{\rm v} \overline{w\theta}$, and the local sink given by $-\epsilon$. Through
the cross-correlation $\overline{w\theta}$ the production is readily linked to the
convective (enthalpy) flux $F_{\rm conv} = c_p\,\rho\,\overline{w\theta}$. The latter
is exact in the Boussinesq approximation and can be generalised to a compressible flow.
The quantities $g$, $\alpha_{\rm v}$, $c_p$, and $\rho$ are the local (vertical)
gravitational acceleration, the volume expansion coefficient, the specific heat at constant
pressure, and mass density.
To solve Eq.~(\ref{eq_Keq}) we need to know $\epsilon$.
The exact evolution equation for $\epsilon$ was first derived by \citet{davidov61b}.
In their Sect.~3, \citet{hanjalic72b} emphasised\footnote{In the literature the model
discussed here is known as $K-\epsilon$ model or ``Imperial College model'' since
there the model had been developed by \citet{hanjalic72b}.}
why it is difficult to close this equation. But in the same paper they also point out
how to proceed to derive a new equation which models the transport of $\epsilon$.
One term (diffusional transport due to pressure fluctuations) is argued to be small on
general grounds compared to other contributions while others are modelled such that
the ensuing closure constants can be determined in the case of simple flows directly
from experiments: decaying turbulence behind a grid and a constant-stress layer
adjacent to a wall. Their model equation for $\epsilon$ eventually reads
\begin{equation} \label{eq_epsilon_HL}
\partial_t \epsilon + D_{\rm f}(\epsilon) =
c_1 \epsilon K^{-1} P - c_2 \epsilon^2 K^{-1}
+ \partial_z (\nu\partial_z \epsilon),
\end{equation}
where $P$ means production of dissipation (due to shear or buoyancy or both).
The term $\partial_z (\nu\partial_z \epsilon)$ is only relevant at moderate or low
Reynolds numbers and can always be neglected for small Prandtl numbers as is
the case for stars. The term $D_{\rm f}(\epsilon)$ was suggested to be parametrised as
\begin{equation} \label{eq_diffeps}
D_{\rm f}(\epsilon) \equiv \partial_z(\overline{\epsilon w})
\approx -\frac{1}{2}\partial_z
\left[(\nu_{\rm t})\partial_z \epsilon \right].
\end{equation}
where $\nu_t$ requires a model for turbulent viscosity such as\footnote{Note that this definition is different
from \cite{canuto98b}, Eq.~(24c), which appears to have a typo.}
$\nu_t = C_{\mu}\, K^2/\epsilon$ with a closure constant $C_{\mu}$. Although this term is mainly relevant for
moderate to low Reynolds numbers, it must be kept and modelled, since this is just what
we also encounter in the case of overshooting zones. This is in contrast with terms only
relevant for moderate to large Prandtl numbers (i.e., only in a non-stellar case) or which are small
independently of the parameter space considered: those we can safely neglect for our applications.
We emphasise that contrary to Eq.~(\ref{eq_Keq}) all contributions to Eq.~(\ref{eq_epsilon_HL})
contain closure approximations. Hence, Eq.~(\ref{eq_epsilon_HL}) is essentially a model for
$\partial_t \epsilon$ and not an exact evolution equation.
Eq.~(\ref{eq_epsilon_HL}) was reconsidered by \citet{canuto94b} and
\citet{canuto98b}, who also suggested the additional
contribution to Eq.~(\ref{eq_epsilon_HL}) introduced in \citet{zeman77b}:
\begin{eqnarray} \label{eq_epsilon}
\partial_t \epsilon + D_{\rm f}(\epsilon) & = &
c_1 \epsilon K^{-1} g \alpha_{\rm v} \overline{w\theta} - c_2 \epsilon^2 K^{-1}
+ c_3 \epsilon \tilde{N}
+ \partial_z (\nu\partial_z \epsilon), \nonumber \\
\tilde{N} & \equiv & \sqrt{g \alpha_{\rm v} |\beta|}.
\end{eqnarray}
Here, $\beta=-((\partial T/\partial z)-(\partial T/\partial z)_{\rm ad})$ is the superadiabatic gradient.
In addition to $c_1=1.44$ and $c_2=1.92$, which is close to the middle of the typical range of values in
earlier work \citep{tennekes72b,hanjalic76b}, \citet{canuto98b} suggested $C_{\mu}=0.08$ from
their turbulence model \citep{canuto96d}, which they obtained using Eq.~(\ref{eq_Kolomogorov_spectrum}).
Before quantifying the new term $c_3\, \epsilon\, \tilde{N}$ more closely, the physical origin
of the contributions to Eq.~(\ref{eq_epsilon}) requires some explanation. The first
term on the right hand side provides a closure for the production of dissipation by buoyancy
\citep{hanjalic72b}. The second term was discussed already in detail by \citet{hanjalic72b}
and represents a closure for the combined effects of the exact terms describing the generation
of vorticity fluctuations through self-stretching in turbulent flows and the decay of turbulence due
to viscosity. For the exact term of diffusion of $\epsilon$ by velocity fluctuations, $D_{\rm f}(\epsilon)$,
both a down-gradient closure \citep{hanjalic72b} and a direct closure based on the flux of turbulent
kinetic energy \citep{canuto92b} have been proposed. The viscous diffusion term
$\partial_z (\nu\partial_z \epsilon)$ is also part of the exact expression for diffusional transport and
is suggested to be kept when modelling flows in the regime of low to moderately high Reynolds
numbers, especially in the case of moderate to high Prandtl numbers (see \citealt{hanjalic76b}).
For buoyancy driven flows Eq.~(\ref{eq_epsilon_HL}) requires several changes in comparison
with \citet{hanjalic72b,hanjalic76b}. We refer the reader to the work by \citet{zeman76b} and
\citet{zeman77b} which eventually allowed the derivation of Eq.~(\ref{eq_epsilon}). What follows
from their and similar considerations is that, irrespectively of the detailed physical nature of
increased local dissipation in the overshooting zone, a separately parametrised loss term that involves
the superadiabatic temperature gradient $\beta$, or actually, the Brunt-V\"ais\"al\"a frequency, $\tilde{N}$,
is needed. With hindsight gravity waves are expected to play the most important role as a source of $\epsilon$.
As argued by \citet{zeman77b}, this involves a characteristic length scale which can be computed
from the ratio of flow velocity $w^2$ and $\tilde{N}$.
It can also be viewed as the distance which eddies of a certain size that penetrate into the
stable layer with a certain lapse rate can travel until their potential energy is fully converted
into kinetic energy. It turns out that this yields the same expression as the parametrisation
of dissipation by internal gravity waves: their contributions may differ in magnitude, but
their functional form remains the same.
Hence, \citet{canuto94b} suggested that this term should indeed be added to the standard
form of Eq.~(\ref{eq_epsilon_HL}). As they pointed out, this contribution also allows
to maintain stationarity in homogeneous, stratified turbulence as confirmed by
data from direct numerical simulations of shear turbulence by \citet{holt92b}. Thus,
\citet{canuto94b} suggested $c_3=0.3$ for stably stratified layers and $c_3=0$
elsewhere to complete Eq.~(\ref{eq_epsilon}). \citet{canuto98b} followed that proposal.
Clearly though, among all the parametrisations which appear in Eq.~(\ref{eq_epsilon}),
$c_3\, \epsilon\, \tilde{N}$ remains the most uncertain one, but yet it is also crucial. Its
choice requires to be tested carefully. Otherwise, the width of convective overshooting
may turn out to be sensitive to the detailed calibration of its
parameters. In Sect.~\ref{secEq11} we discuss more recent suggestions
to further improve the physical content of Eq.~(\ref{eq_epsilon}).
\section{A new model for the dissipation rate in non-local convection models in GARSTEC} \label{Sect_GARSTEC}
\subsection{The problem: overshooting zones of convective cores growing unlimitedly during main-sequence stellar evolution}
The Garching Stellar Evolution Code (GARSTEC) (see \citealt{weiss2008}) offers several models
to compute the contributions of convection to energy transport and mixing in stellar evolution
calculations (including those of \citealt{bv58b}, \citealt{canuto91b}, \citealt{kuhfuss1987}). In particular,
the model of \citet{kuhfuss1987} has been implemented (\citealt{Flaskamp02b}, \citealt{flaskamp2003})
in GARSTEC both in its 1-equation version, i.e., with an additional differential equation for
turbulent kinetic energy, $K$, and in its full, 3-equation version
(\citealt{kuhfuss1987}; for a brief discussion of this model see App.~\ref{secKuh}).
The latter features differential equations for the TKE, the squared fluctuations of entropy,
$\Phi$, and for the turbulent flux of entropy fluctuations, $\Pi$.
Those three equations are essentially equivalent to the dynamical equations for the TKE,
the squared fluctuations of temperature $\overline{\theta^2}$, and for the cross correlation
between velocity and temperature fluctuations, denoted here by
$J=\overline{w\theta}$. The latter can be derived
from the phyically more complete model of \citet{canuto98b} by
assuming (i) an isotropic velocity distribution, (ii)
a local prescription to compute the distribution of the dissipation
rate $\epsilon$, (iii) the diffusion approximation
for the non-local fluxes, and (iv) some minor simplifications in the closures used in the dynamical
equations.\footnote{Entropy gradients in turn are numerically easier to compute than the small differences
between temperature and adiabatic temperature gradients during stellar evolution calculations.}
As a variant, the 3-equation model may be used with local limit
expressions for the non-local transport terms for $\overline{\theta^2}$ as well as $J$.
As a theoretical analysis shows (see \citealt{kupka20b} and references therein)
only a full 3-equation model can feature a countergradient or ``Deardorff'' layer where
$J$ is positive, while the superadiabatic gradient $\beta$ is
negative. Only in such a model both quantities
can change their sign independently (the key to a positive convective flux in
a countergradient stratification is the non-local transport of $\overline{\theta^2}$
as originally shown by \citealt{deardorff61b} and \citealt{deardorff66b}). However,
in both the fully non-local and the local limit of the 3-equation model variant as described above,
overshooting gradually mixes the entire star in a stellar evolution calculation for a 5 solar mass (B-type)
main-sequence star. In Fig.~\ref{figenergyorig} we show the profile of the TKE as a function of fractional
mass in this calculation. It can be seen that the energy extends substantially beyond the Schwarzschild
boundary, reaching very close to the surface of the star. Due to the high efficiency of convective
mixing the whole star would become essentially homogeneous which is
unrealistic, because the star would evolve from the hydrogen to the
helium main-sequence, i.e.\ to the left in the colour-magnitude
diagram, contrary to all observations \citep[see][ Chap.~23.1]{kippenhahn2012}.
This problem was originally identified in the PhD thesis of \citet{flaskamp2003}.
To solve this problem \citet{flaskamp2003} suggested to give up the assumption of isotropy of
TKE of the model of \citet{kuhfuss1987} in the overshooting (OV) zone and let the ratio
of vertical to horizontal kinetic energy tend to zero. This limits the mixing efficiency in the outer layers
of the OV zone, located above the stellar convective core, and avoids its unphysical growth throughout
main-sequence evolution. If this simulation were plausible, also a more realistic model for the
anisotropy of the convective velocity field, derived, for instance, from the stationary limit of
Eq.~(19d) of \citet{canuto98b}, should solve this problem. Both variants of this approach
are discussed below in Sect.~\ref{sect_flowaniso}.
\begin{figure}
\centering
\includegraphics{figures/energy_unlimited.pdf}
\caption{TKE as a function of the fractional mass for the original Kuhfu\ss~model. The formal Schwarzschild boundary, define by $\nabla_{\rm rad}=\nabla_{\rm ad}$, is indicated by a dashed black line.}
\label{figenergyorig}
\end{figure}
\subsection{A comparison with a fully non-local Reynolds stress model}
\begin{figure*
\centering
\includegraphics[width=0.4\textwidth]{figures/C2001-TOMs/Teff8000lgg444solarXYZ_FcFtot_bw.pdf}
\includegraphics[width=0.4\textwidth]{figures/DGA-TOMs/DGA_Teff8000lgg444solarXYZ_FcFtot_bw.pdf}\\
\includegraphics[width=0.4\textwidth]{figures/C2001-TOMs/Teff8000lgg444solarXYZ_wrms_bw.pdf}
\includegraphics[width=0.4\textwidth]{figures/DGA-TOMs/DGA_Teff8000lgg444solarXYZ_wrms_bw.pdf}\\
\includegraphics[width=0.4\textwidth]{figures/C2001-TOMs/Teff8000lgg444solarXYZ_eps_vs_epsMLT_bw.pdf}
\includegraphics[width=0.4\textwidth]{figures/DGA-TOMs/DGA_Teff8000lgg444solarXYZ_eps_vs_epsMLT_bw.pdf}
\caption{\textit{Left panels}: convective flux in units of total flux, root mean square vertical velocity in units of km/s,
and dissipation rate $\epsilon$ from Eq.~(\ref{eq_epsilon}) relative to a value computed from
Eq.~(\ref{eq_MLT_epsilon}) and~(\ref{eq_MLT_alpha}) with $\alpha$ as given in the figure legend.
\textit{Right panels}: same quantities as left panels, however, the downgradient approximation is used to compute third order
moments instead of the full model used in \citet{kupka02b}. The results are for one of the A-star envelope
models discussed in
\citet{kupka02b}.
}
\label{fig1}
\end{figure*}
A progressive growth of the overshooting zone with time is not observed in 3D radiation hydrodynamical
simulations of overshooting in DA white dwarfs \citep{kupka18b} either. Since the extension of the different zones
in that case (Schwarzschild unstable convective zone with $J > 0$ and $\beta > 0$, countergradient region
with $J > 0$ and $\beta < 0$, plume dominated region with $J < 0$ and $\beta < 0$, and wave dominated
region with $J \approx 0$ and $\beta < 0$) compare quite well with results from the non-local Reynolds stress
model of \citet{canuto98b} solved in \citet{montgomery04b} for the same type of stars, the latter can
provide a guideline for the behaviour of variables such as $\epsilon$ as a function of depth. The overall
structure of the OV zones and the behaviour of the convection related variables described in \citet{montgomery04b}
is very similar to that one which had already been found for A-type main-sequence stars in \citet{kupka02b}
which in turn had been compared to earlier 2D RHD simulations of \citet{freytag96b}.
We hence use the Reynolds stress convection model calculations of \citet{kupka02b}
in Fig.~\ref{fig1} to illustrate the convective flux, the root mean square vertical velocity, and the
dissipation rate as a function of depth. The left panels show results for the full third order moment model
while the right panels shows results computed using the downgradient approximation. For $T_{\rm eff} = 8000\,\rm K$ and $\log g$ slightly below
the main sequence (see \citealt{kupka02b} for further details) we find two convective
zones, an upper one due to ionisation of neutral hydrogen and a lower one caused by double-ionisation of
helium. They are connected by an overshooting region at a radius of $\sim 931$~Mm and there is another
overshooting region underneath the lower convective zone at $\sim 926.5$~Mm.
For this setting we compare the computation of dissipation rates from the full equation of
\citet{canuto98b} with the standard mixing length prescription for a range of bulk convective and overshooting layers.
Clearly, the dissipation rate $\epsilon$ becomes much larger than the value computed from the MLT prescription
as soon as the plume region of the OV zones (with $J < 0$ and $\beta < 0$) is reached, and which can be determined
from the behaviour of the convective flux. At the bottom of the lower overshooting zone, $\epsilon$ becomes
even order(s) of magnitudes larger than the oversimplified MLT prescription would predict.
Note that if the downgradient (diffusion) approximation is used to compute third
order moments such as $\overline{q^2w}$ in the model of \citet{canuto98b} (the non-local fluxes of
$K$, $J$, $\overline{\theta^2}$, and $\overline{w^2}$), a smaller
overshooting is obtained in comparison with the complete third order moment model used in \citet{kupka02b}.
Hence, the two convection zones become separated at $T_{\rm eff} = 8000\,\rm K$ which allows observing
this behaviour of $\epsilon$ even between the two convective zones. At lower $T_{\rm eff}$, for example
at 7500~K, convection and overshooting are stronger also for the downgradient approximation of third order
moments and the same behaviour is recovered as for the physically more complete third order moment model
already for $T_{\rm eff} = 8000\,\rm K$. For that latter model the two convective zones become more tightly
coupled and the increase of $\epsilon$ compared to the MLT prescription is eventually restricted to the lower
overshooting zone only, for instance, for models with $T_{\rm eff} = 7200\,\rm K$.
We hence can draw the following conclusions from solutions of the Reynolds stress model of
\citet{canuto98b} for convective envelopes of A-type stars:
irrespective of the various situations described above,
deep inside the plume-dominated region characterised by $J < 0$ and $\beta < 0$ the MLT prescription
to compute $\epsilon$ begins to fail by entirely missing out the drastic increase in dissipation in that region.
However, the proper computation of $\epsilon$ is essential to determine
the extent of the mixed region, since it drains kinetic energy
from the overshooting flow. From Eq.~(\ref{eq_MLT_epsilon}) one can immediately conclude that
{\em underestimating} $\epsilon$ in the MLT framework can be easily caused by {\em overestimating}
the mixing length $\Lambda$ or $\ell_0$.
\subsection{Reducing the mixing length in the OV zone} \label{sect_reduced_ML}
There is also a physical argument why the mixing length must be limited and
even gradually shrink in the OV zone on top of a stellar convective core. Taking
$\Lambda$ to be about a pressure scale height at the convective core boundary
results in a very large length scale. This is essentially the size of the convective core
itself. The claim that such a large structure penetrates into the radiative zone makes
no sense, both from the viewpoint of available potential energy and from the viewpoint
of the typical size of a convective structure. We note here that existing numerical
simulations of convective cores are actually for extremely different physical
parameter regimes, featuring mostly ${\rm Pr} \gtrsim 1$ or even ${\rm Pr} \gg 1$
(see, for instance, \citealt{Rogers13b}, \citealt{Rogers15b}, \citealt{Edelmann19b}).
They are unable to reproduce the very small levels of superadiabaticity
($\beta > 0$, but $|\beta/(\partial T/\partial r)_{\rm ad}| \ll 1$) at realistic stellar
luminosities. This inevitably leads to excessive numerical heat diffusion and
unrealistically small effective Peclet numbers (see \citealt{kupka17b} for a discussion).
Numerical simulations of convective cores are hence likely also subject to the convective
conundrum problem reported for the Sun (cf.\ \citealt{gizon12b}, \citealt{hanasoge16b}). Probably,
they are not as reliable for guiding us as numerical simulations are in the case of convective overshooting
near stellar surfaces (cf.\ \citealt{freytag96b}, \citealt{tremblay15b}, \citealt{kupka18b}, and many others).
We return to the problem of comparing results on convective cores from stellar evolution models
with 3D hydrodynamical simulations of convective cores in Sect.~\ref{sect_discussion_modsim}.
In the following, we thus use a different chain of arguments to derive an improved estimate of $\Lambda$.
As a very first step, one could let $\Lambda$ decay to zero within the OV zone,
either linearly or exponentially, from the value it has at the boundary of the convective zone.
This ad~hoc ``fix'' has been implemented into GARSTEC. The exponential decay model was chosen
and {\em indeed} this easily stops the growth of the overshooting zone as a function of stellar evolution
time. The so enhanced dissipation rate introduced can be seen in Fig.~\ref{figdissexp} at the outer edge
of the convective region. The model including the exponential decay has a central hydrogen abundance of
0.6. The stellar model computed with the original Kuhfu\ss~model was chosen to have the same maximum TKE in
the convection zone to make the dissipation rates comparable.
\begin{figure}
\includegraphics{figures/dissipation_ratecomparison_5.pdf}
\caption{Dissipation rate as a function of fractional mass for the original Kuhfu\ss~model and the
Kuhfu\ss~model including an ad hoc exponential decay of the dissipation length,
shown with a grey dotted and a blue continuous line, respectively. The ad~hoc exponential decay of
the dissipation length leads to an increased dissipation rate at the beginning of the overshooting zone,
indicated by the local maximum beyond the Schwarzschild boundary,
followed by a sharp drop due to the rapid decay of TKE.
The models have been chosen to have the same
maximum TKE.}
\label{figdissexp}
\end{figure}
Physically plausible extensions of the OV zone can be obtained from a ``reduction factor'', which forces
an e-folding extent of the ``decay'' of the mixing length of 2\% to 6\% of the mass of the Schwarzschild-unstable region.
In a $5\,M_\odot$ main-sequence star this limits the OV zone to contain about 12\% to 29\% in terms of the Schwarzschild
core mass. The relative extent of the overshooting region in terms of the Schwarzschild core mass remains mostly constant
along the main-sequence. For an e-folding extent of 4\% the overshooting region contains about 5\% of the stellar mass at
the beginning of the main-sequence while it is shrinking to about 2\% of the total mass at the end of the main-sequence.
The procedure introduces a free parameter, but it is sufficient as a proof of concept: a physically more complete model of
$\epsilon$ constrains the OV contrary to earlier, alternative explanations that require unphysical parameter values to do so
(such as $\overline{w^2}/ K \rightarrow 0$
which is at variance with \citealt{kupka18b}, see Sect.~\ref{sect_flowaniso} below).
\subsection{Boundary conditions and regularity constraints}
As a prerequisite to derive an improved estimate for $\Lambda$ we first discuss its
asymptotic behaviour in the centre of a convective core.
Regularity properties of non-local models of convection at the centre of
stellar cores are a rather delicate issue which has been analysed in \citet{roxburgh07b}.
Under the assumption that non-zero convective motions
can also occur at the centre of a convective core, for the second order moments they demonstrated
that $\overline{w^2}$, $K$, and $\overline{\theta^2}$ are all positive and have an even order
expansion in $r$ just like the gas pressure $P$. Moreover, from their Eq.~(11), the horizontal
component of TKE has to balance the vertical component in the sense that
$\overline{v_r^2} = \overline{v_{\theta}^2} = \overline{v_{\phi}^2}$ for the velocity components
in spherical polar coordinates $(r,\theta,\phi)$. Hence, $\overline{w^2}/K = 2/3$ and the flow
is isotropic. Clearly, also $\epsilon$ has to be positive in this case.
Thus, if the relation $\epsilon = c_{\epsilon} K^{3/2} / \Lambda$ is used, a positive
$\Lambda$ guarantees positivity of $\epsilon$. An appropriate prescription which ensures
this property is to use the curvature of the pressure profile to define a local scale height,
since its gradient vanishes at the centre.
This has been worked out in \citet{roxburgh07c} where the scale height at the centre is defined from
$H_c^2 = -P / (\partial^2 P/\partial r^2) = 2 r H_p = 3 P_c / (2\pi G \rho_c^2)$ and the subscript
$c$ denotes the value of the local scale height, $H_c$, of pressure, $P_c$, and density, $\rho_c$,
at the centre (and $G$ is, of course, the gravitational constant). For the centre, $\Lambda = \alpha H_c$
and in general $\Lambda = \alpha \min(H_p,H_c)$. \citet{roxburgh07c} suggest a smooth interpolation
between the limit at the core centre and the expression for $H_p \gg
H_c$.
For reasons of regularity and energy conservation, $F_{\rm conv} \rightarrow 0$ at the centre
in that case, which is fulfilled by the above prescription of the mixing length. An
expansion in odd powers of $r$ is found for the Reynolds stress equation
for $J$ and thus for $F_{\rm conv}$ \citep{roxburgh07b}. This implies non-trivial constraints
on closures for the third order moments. \citet{roxburgh07b} demonstrate that the downgradient
closure forces the core centre to be convectively neutral ($J \propto r^3$ instead of $J \propto r$)
while other closures have to be modified to ensure regularity of the solution.
In GARSTEC, the \cite{wuchterl1995} prescription for $\Lambda$
is used by default. This requires a different approach at the core centre, as it assumes
$\Lambda \rightarrow 0$ for $r \rightarrow 0$. Thus, in GARSTEC, it is ensured by power series
expansions that the convective variables are not forced to zero while the temperature gradient
at the centre is the adiabatic one. As a result, the convective quantities become small in the
central region (see Paper~{\sc II}). Are differences between these two rather different prescriptions
for the convective variables in the centre of a convective core relevant for applications? Fortunately,
it turns out that they remain constrained to less than the innermost 10\% of the stellar core. In either
case, the stellar core is predicted to be fully mixed and has a temperature gradient close to the
adiabatic one. For this study, we hence prefer to stay within the standard setup used for GARSTEC,
i.e., the prescription for $\Lambda$ proposed by \cite{wuchterl1995}.
\subsection{Some input from the dissipation rate equation}
Can we carry over some of the physics contained in Eq.~(\ref{eq_epsilon_HL}) or Eq.~(\ref{eq_epsilon})
into a local model for $\epsilon$ which avoids the solution of an additional differential equation?
If we model the non-local transport of TKE in Eq.~(\ref{eq_Keq}) by a downgradient approximation,
the closure $\overline{w\epsilon} = (3/2)\tau^{-1}\,F_{\rm kin}$ relates $\overline{w\epsilon}$ to $\partial_z \overline{w^2}$ in Eq.~(\ref{eq_epsilon}).
The same behaviour is found for a direct downgradient closure for $\overline{w\epsilon}$ (i.e., computing it from $\partial_z \epsilon$)
as for example Eq.~(\ref{eq_diffeps}). Let us hence assume a local
approximation for $D_f(\epsilon)$, the non-local flux of $\epsilon$,
which replaces the derivatives of the outer divergence operator and the gradient
operator in Eq.~(\ref{eq_epsilon}) by a product of reciprocal length scales, $1 / {\ell}^2$.
Inspecting Eq.~(\ref{eq_epsilon}), for the sake of simplicity, it appears desirable to model as many
contributions as possible by expressions of type $\epsilon^2/K \propto \epsilon / \tau$. Instead
of a diffusion length scale ($\ell$) we hence use the characteristic transport time scale $\tau = 2K/\epsilon$
to approximate $D_f(\epsilon) \propto -\alpha_{\epsilon} \epsilon / \tau$.
The same can be done also in the case of Eq.~(\ref{eq_epsilon_HL}).
If we furthermore assume
the local limit of Eq.~(\ref{eq_Keq}), $P = P_{\rm b} = \epsilon$, i.e.\ production of TKE by
buoyancy equals its dissipation, and if we also assume $c_3 = 0$, we obtain the following
approximation for both Eq.~(\ref{eq_epsilon_HL}) and Eq.~(\ref{eq_epsilon}):
\begin{equation} \label{eq_eps_alpha}
-\alpha_{\epsilon} \epsilon / \tau = 2\, c_1 g \alpha J / \tau - 2\, c_2 \epsilon / \tau.
\end{equation}
To remain consistent with $g \alpha J = \epsilon$ we have to require that
$\alpha_{\epsilon} = 2 c_2 - 2 c_1$ if $\epsilon$ itself is computed from
Eq.~(\ref{eq_MLT_epsilon})--(\ref{eq_MLT_alpha}). In this case we obtain a completely
local model for the computation of $\epsilon$.
We can use Eq.~(\ref{eq_eps_alpha}) to understand some implications from the different
physical contributions which its physically more complete counterpart, Eq.~(\ref{eq_epsilon_HL}),
would instead account for. To this end let us relax the requirement $P_{\rm b} = \epsilon$ in Eq.~(\ref{eq_Keq})
somewhat. In this case, whether the 1-equation or the 3-equation version of the \citet{kuhfuss1987} model
is used (cf.\ Appendix~\ref{secKuh}), due to the non-locality of the flux of kinetic energy
in Eq.~(\ref{eq_Keq}), $\partial_z(\overline{q^2 w}/2) \neq 0$, there is always a point where $J=0$
(cf.\ Chap.~5 in \citealt{kupka20b}). At such a point, $\alpha_{\epsilon}=2 c_2$ is required from
Eq.~(\ref{eq_eps_alpha}) for a non-vanishing dissipation rate $\epsilon$. Right next to
such a point, where $\epsilon > 0$ with $J<0$, a value of $\alpha_{\epsilon} > 2 c_2$ would be
required whereas $\alpha_{\epsilon} < 2 c_2$ where $J>0$. So $\alpha_{\epsilon}$ would have
to be a function that has to be fine-tuned to obtain consistent results from Eq.~(\ref{eq_eps_alpha})
in the vicinity of $J=0$. Moreover, because of the downgradient closure for $\overline{w\epsilon}$
also constraints on $\overline{w^2}/K$ would be imposed.
Such constraints appear unphysical: Eq.~(\ref{eq_eps_alpha}) does not
provide a good starting point for a local model capable to capture at least the main gist of
either Eq.~(\ref{eq_epsilon_HL}) or Eq.~(\ref{eq_epsilon}). To proceed we need a physically more
complete model for $\epsilon$, i.e., we either have to abandon the mixing length prescription altogether
or we need a more complete model equation than Eq.~(\ref{eq_epsilon_HL}) to start from. Let us hence
first have a look at Eq.~(\ref{eq_epsilon}), i.e., we no longer impose $c_3=0$ everywhere. The sibling
of Eq.~(\ref{eq_eps_alpha}) which accounts for the production of dissipation by gravity waves in stably
stratified fluid then reads:
\begin{equation} \label{eq_eps_alpha_NBV}
-\alpha_{\epsilon} \epsilon / \tau = 2\, c_1 g \alpha J / \tau - 2\, c_2 \epsilon / \tau + c_3 \epsilon \tilde{N}.
\end{equation}
If we were to combine this equation with the 1-equation model of \citet{kuhfuss86b}, $\beta$ and $J$ change
sign at the same point so the perfect balancing constraint between $D_f(\epsilon)$ and $-2\, c_2 \epsilon / \tau$
reappears. In the region where $J < 0$, more freedom of how $D_f(\epsilon)$ behaves is permitted.
This changes once we switch to the 3-equation model of \citet{kuhfuss1987}: since $\beta$ and $J$ then
change sign at different locations, $\alpha_{\epsilon}$ is no longer forced by $c_2$ at any point.
In the end, the $c_3 \epsilon \tilde{N}$ contribution decouples both $D_f(\epsilon)$ and $\overline{w^2}/K$
from peculiar constraints required to be fulfilled at where $\beta=0$ or where $J=0$.
On the other hand, now there is an efficient local source for $\epsilon$ also where $\beta < 0$. This
is particularly important for the 3-equation model which through its countergradient layer permits
much larger enthalpy (and hence also TKE) fluxes in this region: considering that property it is
understandable that
the 3-equation model can be prone to large overshooting,
unless the latter is limited by efficient dissipation. And this is just what gravity waves can provide.
\subsection{Deriving a local model for $\epsilon$ with enhanced dissipation}
\label{Sec_newdiss}
For the sake of physical completeness it would be preferable
to switch to Eq.~(\ref{eq_epsilon}) and give up the local model
Eq.~(\ref{eq_MLT_epsilon})--(\ref{eq_MLT_alpha}) altogether. However, as a first step into
that direction we can aim at modifying the computation of $\Lambda$ for the stably stratified layers
by guiding the necessary physical input through Eq.~(\ref{eq_epsilon}) and
in particular through its local approximation,
Eq.~(\ref{eq_eps_alpha_NBV}).
In a local framework we cannot accurately account for $D_f(\epsilon)$. Hence, we first
express $\tau$ in terms of $\Lambda$ in the local limit,
\begin{equation}
\epsilon = \frac{2 K}{\tau} = c_{\epsilon} \frac{K^{3/2}}{\Lambda},
\end{equation}
from which we obtain that
\begin{equation} \label{eq_tau_Lambda}
\tau = \frac{2}{c_{\epsilon}} \frac{\Lambda}{K^{1/2}}.
\end{equation}
To proceed we can now rewrite $c_3 \epsilon \tilde{N}$ as follows:
\begin{equation} \label{eq_c3}
c_3 \epsilon \tilde{N} = c_3 \frac{\epsilon}{\tau_{b}} = 2\,c_3 \frac{K}{\tau\,\tau_{\rm b}}.
\end{equation}
Following the analysis in the previous subsection we now compare Eq.~(\ref{eq_c3}) with
\begin{equation} \label{eq_c2}
-c_2 \frac{\epsilon^2}{K} = -2\,c_2 \frac{\epsilon}{\tau}.
\end{equation}
In the stationary, local limit and assuming that we can absorb the contribution
from $\alpha_{\epsilon} \epsilon / \tau + 2\, c_1 g \alpha J / \tau$ into $-2\,c_2 \epsilon /\tau$
for sufficiently small $J$ and $\overline{w\epsilon}$ we obtain from
Eqs.~(\ref{eq_eps_alpha_NBV}), (\ref{eq_c3}), and (\ref{eq_c2}) that
\begin{equation} \label{eq_c3_c2_scale}
\frac{c_3/\tau_{\rm b}}{2\,c_2/\tau} = \frac{c_3}{2\,c_2}\frac{\tau}{\tau_{\rm b}} \approx 0.078125 \frac{\tau}{\tau_{\rm b}} = \frac{25}{320} \frac{\tau}{\tau_{\rm b}} \approx 1,
\end{equation}
where the numerical value is obtained from setting $c_2=1.92$ and $c_3=0.3$.
Contributions absorbed into the $-2\,c_2 \epsilon / \tau$ term could be accounted for
by small change of $c_2$. As inspection of the full Reynolds stress models solved in
\citet{kupka02b} demonstrates this is well justified since the two terms compared in
Eq.~(\ref{eq_c3_c2_scale}) completely dominate where $J < 0$.
This motivates the idea to also scale $\Lambda$, which according to Eq.~(\ref{eq_tau_Lambda})
is proportional to $\tau$, by a contribution $\propto \frac{25}{320} \frac{\tau}{\tau_{\rm b}}$.
In GARSTEC the mixing length required for the turbulent convection model
of \citet{kuhfuss1987} is computed following the prescription of \cite{wuchterl1995},
\begin{equation} \label{eq_lambda_GARSTEC}
\frac{1}{\Lambda} = \frac{1}{\alpha\,H_p} + \frac{1}{\beta_{\rm s}\,r},
\end{equation}
where $\beta_{\rm s}$ is a factor chosen to be $1$ in convectively unstable layers, where
$\beta = -(dT/dr-(dT/dr)_{\rm ad}) > 0$ and thus $\nabla-\nabla_{\rm ad} > 0$,
and $\beta_{\rm s}$ is possibly less than $1$
elsewhere. We now account for the effect of enhanced dissipation by gravity waves through reducing
$\beta_{\rm s}$ to values less than 1.
To this end we can interpolate between the two asymptotic cases
$\tilde{N} \rightarrow 0$ and $\tilde{N} = \tau_{\rm b}^{-1} \gg \tau^{-1}$ through
\begin{equation} \label{eq_scale}
\beta_{\rm s} = (1 + \lambda_{\rm s}\,\tilde{N})^{-1} \quad \mbox{\rm for} \quad M_r > M_{\rm schw}
\end{equation}
where $M_{\rm schw}$ is the mass of the convectively unstable core and thus identifies the mass shell
for which $\nabla = \nabla_{\rm ad}$ and $\lambda_s$ is a model parameter. Comparisons with solutions of the non-local Reynolds
stress model of \citet{canuto98b} for A-type stars \citep{kupka02b} show that $\tau_{\rm b} \approx 0.1\,\tau$
where $F_{\rm conv}$ reaches its negative minimum. This range of values for $\tau_{\rm b}$ is what we also
expect from Eq.~(\ref{eq_c3_c2_scale}) for a moderate variation of $c_2$.
\begin{figure*
\centering
\includegraphics[width=0.90\textwidth]{figures/time_scale_ratios_mid-Astar_DGA.pdf}
\caption{Ratio of $\tau/\tau_{\rm b}$ as a function of convective stability from a solution of
the non-local Reynolds stress model as presented in \citet{kupka02b} assuming the
downgradient approximation for third order moments. The time scale $\tau_{\rm b}$ is computed
from $\tilde{N}^{-1}$ where the absolute value of $\beta$ is taken. Sign changes are hence
indicated by spikes. Both the overshooting zones below and above the lower and the upper
convectively unstable zone show the same increase of $\tau/\tau_{\rm b}$ from 0 to more than 10
(the finite grid resolution prevents $\tau/\tau_{\rm b}$ from becoming actually zero).
}
\label{fig2}
\end{figure*}
The results of \citet{kupka02b} can hence provide a rough guideline for the choice of $\lambda_{\rm s}$
and imply that $\Lambda$ is rapidly reduced by an order of magnitude {\em already within the countergradient region}
from the value it has at the Schwarzschild stability boundary (see Fig.~\ref{fig2}). This value is then maintained
throughout the remainder of the countergradient region and the entire region where $F_{\rm conv} < 0$, in agreement
with the $\tau\,\tilde{N} = O(1)$ suggested by \citet{canuto11d} in his Eq.~(5h). The preceding arguments and
the analysis in the previous subsection show how this relation is connected with the full Eq.~(\ref{eq_epsilon})
and how this result can be implemented into a physically motivated reduction factor for
the mixing length through Eq.~(\ref{eq_lambda_GARSTEC}) and~(\ref{eq_scale}). Since the rough constancy
of $\tau/\tau_{\rm b}$ (or the ``dominance'' of the term $c_3 \epsilon \tilde{N}$ in Eq.~(\ref{eq_epsilon}))
also causes the linear decay of the root mean square velocity as a function of distance in
the results of \citet{kupka02b} and \citet{montgomery04b}, and because the latter has also been recovered
from 3D radiation hydrodynamical simulations \citep{kupka18b} for just those layers, the entire procedure
is at least indirectly supported by this physically much more complete modelling. Similar results are not
yet available for convective cores, however.
In spite of its simplicity the disadvantage of Eq.~(\ref{eq_scale}) is the fact
that $\lambda_{\rm s}$ is a {\em dimensional} parameter. It hence has to be determined separately
for each stellar evolution model by numerical experiments which yield the value it has to have
for a sufficient reduction of $\Lambda$ by an order of magnitude. For stars of different
mass this may have to be changed, and for later stages of stellar
evolution it will be even less convenient.
What we need here is an estimate for $\tau$. Without solving Eq.~(\ref{eq_epsilon}) this is akin
to a hen and egg problem, since in the end this would require just the quantity $\Lambda$ we are
up to compute: $\lambda_{\rm s} = (25/320) \,\tau$ with $\tau$ computed from Eq.~(\ref{eq_tau_Lambda}).
We could simplify this by setting $\tau = (2/c_{\epsilon}) (\alpha H_p K^{-1/2})$ or $\tau = (2/c_{\epsilon}) (r K^{-1/2})$,
as this formula is to be used only for $r>0$ and $H_p < \infty$ anyway. However, this has the
disadvantage that near the outer edge of the overshooting zone where $K \rightarrow 0$ one
obtains $\tau \rightarrow \infty$. From standard calculus applied to Eq.~(\ref{eq_lambda_GARSTEC})
we then obtain that $\Lambda \approx \alpha H_p$ right there which is exactly {\em not} what
we want.
But we can rewrite Eq.~(\ref{eq_scale}) into
\begin{equation} \label{eq_scale2}
\beta_{\rm s} = (1 + c_4 \Lambda K^{-1/2} \tilde{N})^{-1} \quad \mbox{\rm for} \quad M_r > M_{\rm schw}
\end{equation}
with
\begin{equation} \label{eq_c4}
c_4 = \frac{c_3}{2\,c_2} \frac{2}{c_\epsilon} \approx
\frac{25}{320} \frac{2}{c_\epsilon} \approx \frac{5}{32 c_\epsilon} =
0.19659 \approx 0.2,
\end{equation}
for which we have used $c_{\epsilon} = \pi (2/(3\,\rm Ko))^{3/2} \approx 0.7948 \approx 0.8$ with $\rm Ko=5/3$
from \citet{canuto98b}\footnote{If we used the value of $c_{\epsilon} \approx 2.18$ suggested in \citet{kuhfuss1987} we would instead obtain that
$c_4 \approx 0.07$. However, in the product $c_\epsilon K^{3/2}/\Lambda$ the constant $c_{\epsilon}$ to some extent cancels out, hence, the
overshooting distance is only weakly depending on this parameter. We discuss this further in Appendix~B of Paper~{\sc II}.}.
This is achieved by realising that $\lambda_{\rm s}\,\tilde{N} = c_4 \Lambda K^{-1/2} \,\tilde{N}
= ((2 c_3) / (2 c_2 c_{\epsilon}) \Lambda K^{-1/2} \,\tilde{N} = (c_3 / (2 c_2)) \tau_{\rm b}^{-1} (2/c_{\epsilon}) \Lambda K^{-1/2}
= ((c_3/\tau_{\rm b})/(2 c_2 / \tau))\cdot (2 \Lambda K^{-1/2} / (\tau c_{\epsilon})) = (c_3 / (2 c_2)) \cdot (\tau / \tau_{\rm b}))$
which is just Eq.~(\ref{eq_c3_c2_scale}) and where we have used Eq.~(\ref{eq_tau_Lambda}) for the last step.
Eq.~(\ref{eq_scale2}) is equivalent to Eq.~(\ref{eq_scale}) and also interpolates between the two asymptotic
cases, the transition between locally stable to unstable stratification ($\tilde{N} \rightarrow 0$) as well as
the overshooting region far away from the convective zone, where flow motions are dominated by waves
($\tilde{N} = \tau_{\rm b}^{-1} \gg \tau^{-1}$).
Eq.~(\ref{eq_lambda_GARSTEC}) combined with Eq.~(\ref{eq_scale2})--(\ref{eq_c4})
can be rewritten into a quadratic equation for $\Lambda$ for which the positive branch can be taken
or which can be solved implicitly, for instance, by an iterative scheme (the former will be done in Paper~{\sc II}).
In principle, the parameter $c_4$ could be adjusted to achieve the goal of $\tau_{\rm b} \approx 0.1\,\tau$
or rather $\Lambda(\min(F_{\rm conv})) \approx 0.1 \Lambda(M_r=M_{\rm schw})$ which mimics the result
discussed in Fig.~\ref{fig2} and in the previous paragraphs. However, we prefer to assume sufficient
generality of Eq.~(\ref{eq_epsilon}) and its parameters and therefore use them without further adjustments.
Some numerical experiments on the effects of varying $c_4$ can be found in Appendix~B of Paper~{\sc II}.
In the next section we show that this procedure also leads to a finite overshooting layer
which does not (notably) grow during stellar evolution.
\section{Discussion: Kuhfu{\ss} 3-equation model with enhanced dissipation} \label{sect_discussion}
\subsection{Flow anisotropy instead of enhanced dissipation} \label{sect_flowaniso}
A very important difference between the \cite{kuhfuss1987} and the
\cite{canuto98b} model is the set of convective variables
considered. In addition to the TKE \cite{canuto98b} also solve for
the vertical TKE. This means that the ratio of $\overline{w^2}/K$ is
not fixed a priori but is an outcome of the theory. \cite{kuhfuss1987}
on the other hand assumes full isotropy in the whole convection zone
which translates to a fixed ratio of
$\overline{w^2}/K=2/3$. Furthermore, the Kuhfu\ss~model uses an
isotropic estimate of the radial velocity
$v_\text{radial}=\sqrt{2/3\omega}$ in the non-local terms. Hence,
these terms are potentially overestimated by overestimating the ratio
of vertical to total kinetic energy. This could result in an
unreasonably large overshooting zone. The treatment of the flow
anisotropy is especially problematic at convective boundaries where
the flow turns over. In the convective boundary layers the motions
change from being predominantly radial to becoming predominantly
horizontal. This means that the ratio of vertical to total kinetic
energy should drop from the isotropic value to smaller
values.
To study the impact of anisotropy we mimic the change of the flow
pattern by introducing an artificial anisotropy factor
$\xi^2=\overline{w^2}/K$. This anisotropy factor is set to a value of
$\xi=\sqrt{2/3}$ in the bulk of the convection zone and then linearly
decreases to a value of zero from the Schwarzschild boundary
outwards. This is most probably not a very physical behaviour but just
meant for illustrative purposes. The profile of this artificial
anisotropy factor is shown in the upper panel of
Fig.~\ref{figadhocanis}. The profile of the TKE computed with this
anisotropy factor is shown in the lower panel of the same figure. The
black dashed line indicates the Schwarzschild boundary. It can be seen
that an overshooting zone beyond the Schwarzschild boundary emerges,
which has, however, a clearly limited extent. As intended, a limitation of the
anisotropy could solve the problems observed with the original version of the 3-equation model.
The description requires another free parameter which is the slope of the linear function.
The slope parameter directly controls the overshooting distance which is very similar to other
ad hoc descriptions of convective overshooting. Also, the functional form of $\xi$ has not been
determined by physical arguments but has been chosen arbitrarily.
\begin{figure}
\centering
\includegraphics{figures/adhoc_anisotropy.pdf}
\caption{Artificial anisotropy factor $\xi$ and TKE as a function of fractional mass in the upper and lower panel, respectively. The black dashed line indicates the Schwarzschild boundary.}
\label{figadhocanis}
\end{figure}
This unfavourable situation should be avoided by a physically motivated estimate for the anisotropy factor. This requires to compute the vertical kinetic energy. To obtain an estimate of the distribution of the turbulent kinetic energy in the \cite{kuhfuss1987} model we start from the fourth equation of the \cite{canuto98b} model:
\begin{align}
\dpar{}{t}\frac{1}{2}\overline{w^2}+D_f\left(\frac{1}{2}\overline{w^2}\right)=-\frac{1}{\tau_{pv}}\left(\overline{w^2}-\frac{2}{3}K\right)+\frac{1}{3}(1+2\beta_5)g\alpha J-\frac{1}{3}\epsilon\label{eqvertkin}
\end{align}
which solves for the vertical turbulent kinetic energy
$\overline{w^2}$. Not solving for $\overline{w^2}$ implies that also $D_f\left(\frac{1}{2}\overline{w^2}\right)$ is unknown. A reasonable way to compute this quantity from the \cite{kuhfuss1987} model is again to assume an isotropic distribution of the fluxes: $D_f\left(\frac{1}{2}\overline{w^2}\right)=\frac{1}{3}D_f(K)$. By rearranging and neglecting the time-dependence in Eq.~(\ref{eqvertkin}) we can define an anisotropy factor:
\begin{align}
\frac{\overline{w^2}}{K}=\frac{2}{3}-\frac{\tau_{pv}}{K}\left(\frac{1}{3}D_f(K)-\frac{1}{3}(1+2\beta_5)g\alpha J+\frac{1}{3}\epsilon\right)\label{eqanis}
\end{align}
All quantities in Eq.~(\ref{eqanis}) can be computed within the Kuhfu\ss~3-equation model.
We have computed the anisotropy factor according to Eq.~(\ref{eqanis}) for a stellar model which used the original version
of the Kuhfu\ss~3-equation model. The result is shown in Fig.~\ref{figaniso}. In the bulk of the convection zone within the Schwarzschild boundary the estimated anisotropy
points towards a radially dominated flow. Directly beyond the Schwarzschild boundary the estimated anisotropy factor drops below
the isotropic value of 2/3. This can be attributed to the negative convective flux in the overshooting zone which according to
Eq.~(\ref{eqanis}) reduces the ratio of vertical to total kinetic energy. Further out in mass coordinate the estimated anisotropy
increases again slightly above a value of 2/3 and remains to a good approximation constant over the region in which positive
kinetic energy is observed (see Fig.~\ref{figenergyorig}).
Introducing this anisotropy factor into the Kuhfu\ss~3-equation model would not substantially reduce the estimate of the radial velocity.
On the contrary, over large parts of the model the value of the radial velocity would be even larger than the current estimate as
we find an anisotropy factor above the isotropic value of 2/3. To finally settle the question of the flow anisotropy in Reynolds stress models
one also has to solve the respective equation for the vertical kinetic energy (Eq.~(\ref{eqvertkin}) shown here, as taken
from the \citealt{canuto98b} model) self-consistently
coupled to the non-local convection model.
However, since such a more realistic anisotropy factor cannot resolve the problem of
excessive mixing found in the original Kuhfu\ss~3-equation model and because its implementation
as an additional differential equation increases the complexity of the model, we first perform
a thorough analysis of the improved 3-equation model in Paper~{\sc II} and postpone the extension
of this new model to future work.
\begin{figure}
\centering
\includegraphics{figures/anisotropy400.pdf}
\caption{Estimate of the anisotropy factor according to Eq.~(\ref{eqanis}) for a 3-equation model without limited dissipation length-scale $\Lambda$.
The profile of the turbulent kinetic energy of this model is shown in Fig.~\ref{figenergyorig}.
}
\label{figaniso}
\end{figure}
\subsection{Dissipation in the Kuhfu\ss~1- and 3-equation model}
We have implemented the enhanced dissipation mechanism, developed in Sect.~\ref{Sec_newdiss}, into GARSTEC. For the details of the implementation we here refer to Paper~II. With this implementation we solve the stellar structure equations and the convective equations (\ref{eqKuh1}) - (\ref{eqKuh3}) self-consistently.
{We note that for consistency and to simplify the comparison between
the 1-equation and the 3-equation model, we set $c_{\epsilon}=C_D$
(see Appendix~\ref{secKuh}), whence it follows that $c_4 \approx 0.072$ in those
calculations.
As an example we show here the TKE in a $5\,M_\odot$ main-sequence star in Fig.~\ref{figenergyextended}. The Schwarzschild boundary is indicated with a black dashed line. In this model the convective energy extends slightly beyond the Schwarzschild boundary which means that an overshooting zone emerges consistently from the solution of the model equations. However, in contrast to Fig.~\ref{figenergyorig} the energy does no longer extend throughout the whole star but has a clearly limited extent as one would expect for this kind of star in this evolutionary phase.
This shows already that the enhanced dissipation mechanism proposed
above is able to solve the problems observed in the original version
of the 3-equation Kuhfu\ss~convection model. The detailed structure
and the behaviour of stellar models with different initial masses will
be discussed in Paper~II.
\begin{figure}
\centering
\includegraphics{figures/convective_energy_5.pdf}
\caption{Convective energy as a function of the fractional mass for the Kuhfu\ss~model
including the improved dissipation mechanism. The Schwarzschild boundary is
indicated by a dashed black line.}
\label{figenergyextended}
\end{figure}
The results obtained from the different versions of the Kuhfu\ss~model
can be interpreted by studying the individual terms of the TKE
equation (Eq.~\ref{eqKuh1}) in more detail. In
Fig.~\ref{figtermcompare} we show the three terms of the TKE
equation---buoyant driving, dissipation and non-local flux---with a corresponding
red, black, and blue line respectively for the 1-equation model (panel
a), the original 3-equation model (panel b) and the improved
3-equation model (panel c).
Stellar models applying the non-local 1-equation theory posses a clearly bounded convective region with a reasonable extent.
However, this is achieved by suppressing the countergradient layer and artificially coupling the sign of
the convective flux to that one of the superadiabatic gradient.
When using the 3-equation model in its original version this welcome property vanishes and the stellar models become fully convective.
As discussed in Appendix~\ref{secKuh} the 3-equation model does not approximate the convective flux by a local model but rather solves
an additional differential equation for it. This reduces the coupling of the different convective variables. Intuitively one would expect this
model to be physically more complete than the 1-equation model and to yield physically improved models (see the discussion in Sect.~5 of \citealt{kupka20b}).
However, the stellar models computed with the 3-equation model look physically unreasonable,
as the existence of fully convective B-stars with $5\,M_{\odot}$ is
excluded from the lack of stars hotter than the hydrogen main-sequence.
This rises the question why a seemingly physically more complete model leads to worse results.
It can be illustrated by comparing the TKE terms in the 1- and original 3-equation models shown in panels a) and b) in Fig.~\ref{figtermcompare}.
In the 1-equation model the buoyant driving term which is proportional to the convective flux shows negative values in the overshooting zone,
which is expected due to the buoyant braking in the stable layers. The buoyant term even exceeds the actual dissipation term in magnitude.
This means that in the 1-equation model it is not the dissipation term but rather the buoyant driving term which acts as the main sink term
in the overshooting zone. When applying the 3-equation model the buoyant term is still negative in the overshooting zone. The values are, however,
much smaller in magnitude compared to the 1-equation model. The dissipation and non-local flux term have about the same magnitude
in the overshooting zone as obtained with the 1-equation model, because their functional form did not change. Considering that it was
the buoyant driving term which was acting as the main sink term, the 3-equation model in its original form is lacking a sink term in the overshooting zone.
This naturally explains the excessive overshooting distance found for this model.
To understand how the dissipation by buoyancy waves can mitigate this problem it is worth to recall the approximation for the convective flux in the 1-equation model. \cite{kuhfuss1987} has approximated this to be $\Pi\propto(\nabla-\nabla_\text{ad})$. As the convective flux is the major sink term in the overshooting zone in the 1-equation model one possibility is to introduce a dissipation term which has the same dependence, $\epsilon\propto(\nabla-\nabla_\text{ad})$. A process with this dependence would be, for example, the dissipation by buoyancy waves as proposed above. We have demonstrated that the enhanced dissipation by buoyancy waves reduces the overshooting distance again to a more reasonable extent for the TKE (see Fig.~\ref{figenergyextended}). The related terms of the TKE equation are shown in Fig.~\ref{figtermcompare} in panel c). In the overshooting zone the magnitude of the dissipation term is now substantially larger than the negative buoyancy term such that it acts as the dominant sink term. Also the shape of the dissipation profile has changed compared to the original 3-equation case. The transition from finite to zero values looks smoother for the improved 3-equation model because the temperature gradient which has readjusted differs in comparison with the 1-equation model.
This comparison shows why the original version of the 3-equation model results in fully convective stars. The fact that a sink term is missing
points again at the importance of a dissipation term which is proportional to $(\nabla-\nabla_\text{ad})$. On a first glance, a negative convective flux
with larger magnitude in the overshooting zone could also increase the sink term in the TKE equation. But the following line of arguments shows that this
hypothesis leads to unplausibly large non-local fluxes.
\begin{figure}
\centering
\includegraphics{figures/terms_comparison.pdf}
\caption{Comparison of the different terms in the TKE equation (Eq.~\ref{eqKuh1}) in the Kuhfu\ss~ 1-equation (panel a), original 3-equation (panel b) and improved 3-equation (panel c) model. The buoyant driving term, the dissipation term and the non-local flux term are shown with a red, black, and blue line here.}
\label{figtermcompare}
\end{figure}
Here, we consider Eq.~(\ref{eqKuh1})--(\ref{eqKuh3}).\footnote{We point out that exactly the same sequence of arguments applies to the equivalent three
equations for the turbulent kinetic energy $K=\overline{q^2}/2$, the squared fluctuation of the difference between local temperature and its Reynolds
average, $\overline{\theta^2}$, and the cross correlation between velocity and temperature fluctuations, $J=\overline{w \theta}$, as they appear in the model
of \citet{canuto98b} and discussed in \citet{kupka20b}.}
Let us assume that $J$ becomes larger, or, equivalently, $\Pi$ in Eq.~(\ref{eqKuh1})--(\ref{eqKuh3}) becomes larger in magnitude in the region
where it is negative. Then, the buoyant driving term shown in panel~b of Fig.~\ref{figtermcompare} changes towards more negative values. This permits the source,
the divergence of the flux of kinetic energy, to become smaller. However, in that case the buoyant driving term (containing $\Pi$) also becomes larger in
Eq.~(\ref{eqKuh3}), which predicts the magnitude of entropy fluctuations.
Since the vertical velocities have to become smaller, when the non-local flux of kinetic energy becomes smaller (and we assume a constant
anisotropy in this thought experiment), the squared fluctuations of entropy, $\Phi$, or of temperature, $\overline{\theta^2}$, have to become {\em larger} instead.
But for $\Pi < 0$ in the region we consider here, both $-\Pi/\tau_{\rm rad}$ and $(2\nabla_\mathrm{ad}T / H_p) \Phi$ act as sources which are boosted in Eq.~(\ref{eqKuh2}).
Unless we would consider a large rate of change in the non-local transport of convective flux and entropy fluctuations, the only way to obtain
an equilibrium solution in this model is to increase velocities and thus also the flux of kinetic energy. This is exactly the solution observed in
panel~b of Fig.~\ref{figtermcompare} with its excessively extended overshooting. The closure used in \citet{canuto93b} and \citet{canuto98b}, which also accounts
for buoyancy contributions to the correlation between fluctuations of temperature and the pressure gradient (the $-\Pi/\tau_{\rm rad}$ term in Eq.~(\ref{eqKuh2}))
does not change this argument. But a scenario that builds up large fluctuations of entropy in the overshooting region, where radiative cooling should efficiently
smooth them while it has to suppress high velocities, appears
unphysical. Thus, this alternative can be excluded.
Since extensive overshooting, which eventually mixes the entire B-star, is ruled out by observations, we are left with flow anisotropy or enhanced dissipation
due to the generation of waves as physical mechanisms to limit overshooting in the 3-equation framework. Because extreme levels of flow anisotropy are neither found
in solar observations nor in numerical simulations of overshooting in white dwarfs \citep{kupka18b}, nor in solutions of the model of \citet{canuto98b}
for A-stars \citep{kupka02b} or white dwarfs \citep{montgomery04b}, there is hardly evidence for this idea. On the contrary, the enhanced energy dissipation rate is
contained in the full model of \citet{canuto98b} which yields at least some qualitative agreement with numerical simulations of several scenarios of stellar overshooting
(see \citealt{kupka02b} and \citealt{montgomery04b} and compare with \citealt{kupka18b} for the latter). This makes the improved computation of the dissipation rate
of kinetic energy the most plausible improvement of the 3-equation model to remove the deficiency the model has had in its original version proposed by \citet{kuhfuss1987}.
\subsection{Comparing the Kuhfu\ss~3-equation model with overshooting models and numerical simulations} \label{sect_discussion_modsim}
\citet{viallet2015} have reviewed several models suitable for parametrization of
overshooting above stellar convective cores. One of them is the model proposed by
\citet{freytag96b} based on 2D hydrodynamical simulations of thin convective zones
which appear in the atmosphere and upper envelope of stars. The simulations had to be restricted
to low Peclet (Pe) numbers where highly efficient radiative diffusion competes with convective
energy transport. The simple exponential decay law for velocity as a function of distance from
the convection zone has been particularly attractive for stellar evolution modelling and
the model is available in most actively used stellar evolution codes including GARSTEC.
As the velocity scales with the pressure scale height, this model requires an additional
cut-off to prevent diverging overshoot from very small convective cores as found in
stars with less than two solar masses. We will discuss this issue in detail in Paper~{\sc II}.
Additionally, \citet{kupka18b} have pointed out that within the countergradient and plume
dominated regions of convective overshooting zones exponential decay rates for velocity
work only within a limited spatial range \citep[see also][]{montgomery04b}.
\citet{cunningham19b} argued for different decay rates for the plume dominated
and the wave dominated regime. Such distinctions are, however, not made in
applications of that model. We refer to Paper~{\sc II} to a detailed comparsion
of convective core sizes between the Kuhfu\ss~3-equation model and the exponential
overshooting model, and here just emphasize that the energy loss of turbulent flows
due to waves is readily built into the improved Kuhfu\ss~3-equation model.
Another model, suitable for a higher Pe regime, where penetrative
convection due to plumes occurs, is the one originally suggested by \citet{zahn91b}.
When applied to convective cores his model had to rely on invoking Roxburgh's integral
constraint \citep{roxburgh89b} for self-consistent predictions which effectively turns it
into a model similar in complexity to the 1-equation model by \citet{kuhfuss1987}. We recall
here that the 3-equation model with enhanced dissipation has a built-in dependence on Pe
by accounting for radiative losses in its dynamical equations. A detailed discussion
on the role of Pe in the 1-equation and 3-equation models can also be found in
Paper~{\sc II}. The latter model is also not subject to the simplifications made in
\citet{roxburgh89b} concerning the treatment of the dissipation rate $\epsilon$.
Finally, for the very high Pe regime of convective entrainment \citet{viallet2015}
considered a model based on estimates relying on the variation of the inverse buoyancy
time scale in the stably stratified layer next to a convective zone and the kinetic energy
available at the boundary of the convective zone. The Kuhfu\ss~3-equation model can
also deal with this case since it is the regime in which heat conduction is negligibly small.
Hence, instead of relying on physically different models which have not been designed to be
compatible among each other, the new 3-equation model can deal with the different regimes
discussed in \citet{viallet2015} within a single formalism and without the necessity of fine
tuning for these different cases.
Comparing the predictions of the new model with those concluded from 3D hydrodynamical simulations
of convection is more difficult: as already mentioned in Sect.~\ref{sect_reduced_ML} they currently have
to be restricted to a different parameter range. In \citet{kupka20b} it is explained why the effective (numerical)
heat conductivity in the simulations has to be higher than the physical one which leads to values of Pe several orders of
magnitudes smaller than those found in stars. As the numerical diffusion of momentum and heat in high Pe
simulations have to remain comparable to each other, we have to expect differences in flow structures
and overshooting distances when compared to the actual, stellar parameter range (see, for instance,
\citealt{scheel17b} and \citealt{kp19b}). Nevertheless, it is a very important finding for the veracity
of the enhanced Kuhfu\ss~3-equation model that the 3D simulation results concerning convective cores
by \citet{browning04b}, \citet{gilet2013}, \citet{Rogers13b}, \citet{Augustson16b}, and \citet{edelmann2019},
among others, and the related simulations of convective shells by \citet{meakin2007}, all show that convective
zones excited by nuclear burning are subject to convective entrainment and penetration, respectively, depending
on the specific setup, and in each case gravity waves are excited which extend throughout the radiative stellar
envelope. This supports the theoretical analysis of \citet{linden75b} and \citet{zeman77b} for the equivalent
scenario in meteorology which lead to the non-local dissipation rate equation proposed in \citet{canuto94b}
and generalized to applications in stellar convection by \citet{canuto98b}, see Eq.~(\ref{eq_epsilon}), which is
the starting point for our investigations we detail in this paper.
\section{Conclusions}
\label{sect_conclusions}
The original model by \citet{kuhfuss1987} was shown by \citet{flaskamp2003} to lead to
convective overshooting zones on top of convective cores that fully
mix the entire object on a fraction of its main sequence life time. We verified that
the ad hoc cure to reduce the ratio of vertical to total TKE to zero
no longer works once realistic models for that quantity are used.
From a physical point of view the ad hoc cure is hence
ruled out as an explanation for this deficiency of the model by \citet{kuhfuss1987}.
In this paper a physically motivated modification of the mixing length
has hence been suggested which takes into account that the dissipation rate
of TKE has been underestimated by the original 3-equation model of \citet{kuhfuss1987}.
In Paper~{\sc ii} we present more detailed tests of the improved 3-equation model
proposed in this paper based on stellar evolution tracks for A- and B-type main sequence
stars of different masses.
One conclusion from these analyses appears to be that the minimum physics
to obtain realistic models of overshooting layers require to account for non-locality
of the fluxes of kinetic energy and potential temperature (as intended by \citealt{kuhfuss1987})
and in addition to account for the variation of the anisotropy of turbulent kinetic energy
as a function of local stability and non-local transport. If the latter is done in a realistic
way, it becomes also clear that a physically more complete model of the dissipation
rate of TKE is needed. All these features are already provided by the
model of \citet{canuto98b} which in its most simple form accounts for non-locality
with the downgradient approximation (as in the model of \citealt{kuhfuss1987}). The
present simplification is an attempt to carry over the most important features of
the more complete model by \citet{canuto98b} into the \citet{kuhfuss1987} model
which is already coded within GARSTEC.
Switching to more complex non-local convection models in a stellar evolution
code is not an easy task. This requires that the model and its implementation also account
for the following:
\begin{enumerate}
\item Realistic, mathematically self-consistent boundary conditions. This is taken
care of in the current implementation of the \citet{kuhfuss1987} model in GARSTEC.
\item A fully implicit, relaxation based numerical solver for the resulting set of equations.
This is fulfilled by GARSTEC as well. Adding further differential equations always
means some non-trivial work on this side.
\item A stable, monotonic interpolation scheme for the equation of state.
Again this is fulfilled in GARSTEC \citep{weiss2008}.
If this is not fulfilled, $\beta$ cannot be computed correctly and
any closure depending on its sign becomes uncertain, since oscillations may be
fed into its computation.
\item A robust formulation of the dynamical equations which avoids cancellation errors
introduced through a nearly perfectly adiabatic stratification. This is
realised in the implementation of the \citet{kuhfuss1987} model in GARSTEC indicated by
the smoothness of the equation terms in Fig.~\ref{figtermcompare}. This can be attributed
to the fact that the implementation uses Eq.~(\ref{eq3eqnabla}) to compute the temperature
gradient instead of numerical derivatives.
\end{enumerate}
Naturally, as discussed in \citet{ireland18b}, in \citet{augustson19b}, and in \citet{korre21b},
among others, rotation and magnetic fields influence convection and convective overshooting.
A path towards including rotation in non-local convection models has been investigated, e.g.,
by \citet{canuto98c} and by \citet{canuto11b}, but such extensions have to be left
for future work: the present model is only a first step beyond MLT-like models.
If the modified mixing length Eq.~(\ref{eq_lambda_GARSTEC}) and~(\ref{eq_scale})
and even more so Eq.~(\ref{eq_lambda_GARSTEC}) with Eq.~(\ref{eq_scale2})--(\ref{eq_c4})
turns out to produce stable, physically meaningfully evolving overshooting zones
with GARSTEC, further tests of this approach are highly warranting. These may
also motivate the implementation of fully non-local Reynolds stress models
at the complexity level of \citet{canuto98b} which completely avoid the introduction
of a mixing length with all its shortcomings.
\section{The Kuhfu{\ss}\ convection model} \label{secKuh}
In this appendix we summarise the turbulent convection model developed by \cite{kuhfuss1987} who
derived dynamical equations for three of the second order moments to model turbulent convection
in the stellar interior: the turbulent kinetic energy, the turbulent convective flux, and the squared entropy
fluctuations. The (specific) turbulent kinetic energy (TKE) is denoted by $K$ in the main text. Here,
we summarise those equations as used inside GARSTEC. They model entropy fluctuations instead
of temperature fluctuations. To avoid confusion with other models and their implementation
here we stick to the notation of \cite{kuhfuss1987}: TKE
is denoted by $\omega$. The radial component of the turbulent convective flux is written as
$\Pi$ and is computed from entropy fluctuations, consistent with choosing the squared entropy
fluctuations $\Phi$ as the third dynamical variable of the system. Hence, the Reynolds splitting
is performed for
\begin{align*}
\vec v=\overline{\vec v}+\vec v\, ', \,\,\,\, \rho=\overline{\rho}+\rho', \,\,\,\, s=\overline{s}+s', \,\,\dots
\end{align*}
and the second order moments are computed from
\begin{align*}
\omega=\overline{\vec v\, '^2/2},\,\,\,\, \Pi=\overline{\vec v\, '\cdot s'}_r,\,\,\,\, {\rm and}\,\,\,\, \Phi=\overline{s'^2/2}.
\end{align*}
As for any TCM a number of assumptions and approximations is required to obtain closed
systems of equations that can actually be applied in stellar structure and evolution models.
In the following we will briefly review the key assumptions of the Kuhfu\ss\ model.
By using only the total TKE $\omega$ the \cite{kuhfuss1987} model is not able to account for a variable distribution
of the kinetic energy in radial and horizontal directions. Instead the distribution of kinetic energy in radial and horizontal directions is assumed
to be isotropic at all radii, such that one third of the energy is attributed to each spatial direction. The Kuhfu\ss~ model further neglects turbulent
pressure fluctuations. As pointed out by \cite{viallet2013} pressure fluctuations play an important role for convection in envelopes, hence the
Kuhfu\ss~ model is probably not suited to model envelope convection. Finally \cite{kuhfuss1987} also made use of the Boussinesq approximation.
In the current implementation suggested by \cite{flaskamp2003} we also neglect effects due to the chemical composition, e.g. composition gradients.
\subsection{Viscous dissipation}
In the Kuhfu{\ss}\ model most terms containing the molecular viscosity are neglected because they are of minor importance compared to competing terms.
Only the viscous dissipation term for the kinetic energy is considered to be non-negligible. \cite{kuhfuss1987} models the dissipation of the kinetic energy
with a Kolmogorov-type term \citep{kolmogorov1968,kolmogorov1962}:
\begin{align}
\epsilon=C_D\frac{\omega^{3/2}}{\Lambda},
\label{eqdiss}
\end{align}
where $C_D$ is a parameter. \cite{kuhfuss1987} suggests a value of $C_D = 8/3\cdot\sqrt{2/3}$ to be compatible
with MLT in the local limit of his model.
In the Kolmogorov picture kinetic energy is dissipated thanks to a cascade through which energy is transferred to smaller and smaller spatial scales.
The rate at which this dissipation happens is dominated by the largest scales at which energy is fed into the cascade. In Eq.~(\ref{eqdiss})
the length-scale $\Lambda$ refers to this largest scale of the turbulent cascade. As in the mixing length theory the length-scale is parametrised
using the pressure scale height $H_p$ and an adjustable parameter $\alpha$: $\Lambda=\alpha H_p$. Problems with this parametrisation
are discussed in the main text.
\subsection{Radiative dissipation}
Convective elements lose energy through radiation. This is considered in the energy conservation equation by including radiative fluxes
as sink terms. In the Kuhfu{\ss}\ equations the radiative losses finally appear as dissipation terms:
\begin{align*}
\epsilon_{\mathrm{rad},\Pi}=\frac{1}{\tau_\mathrm{rad}}\Pi\,,\,\,\,\,\,\epsilon_{\mathrm{rad},\Phi}=\frac{2}{\tau_\mathrm{rad}}\Phi,
\end{align*}
where \cite{kuhfuss1987} models radiative dissipation by introduing the radiative time-scale $\tau_\text{rad}$, which he defines as:
\begin{align*}
\tau_\text{rad}=\frac{c_p\kappa\rho^2\Lambda^2}{4\sigma T^3\gamma_R^2}.
\end{align*}
Here, $\gamma_R$ is a parameter which \cite{kuhfuss1987} sets to $2\sqrt{3}$ , again to recover the MLT model in the local limit.
Furthermore, $c_p$ refers to the specific heat capacity at constant pressure, $\kappa$ to Rosseland opacity and $\sigma$ to the
Stefan-Boltzmann-constant. The variables $T$ and $\rho$ are temperature and density, as usual in stellar structure models.
\subsection{Higher order moments}
The Navier-Stokes equations contain non-linear advection terms. When constructing the equations for the second order moments
these advection terms give rise to third order moments (TOMs). These higher order moments are the source of the non-local behaviour
of the convection model. They can be cast into the form:
\begin{align*}
\mathcal{F}_a & = \frac{1}{\overline{\rho}}\operatorname{div}(j_a) \,\,\,\, {\rm with} \,\,\,\, j_a = \overline{\rho}\,\overline{\vec v\, ' a},
\end{align*}
where $a$ is a second order quantity. The closure of these TOMs is one of the main challenges of any TCM.
\cite{kuhfuss1987} closes the system of equations at second order and describes each TOM using the so-called down-gradient
approximation \citep[e.g.,][]{daly1970,launder1975,xiong78b,li2007}. In the down-gradient approximation the fluxes $j_a$ are modelled
following Fick's law:
\begin{align}
\vec j_a & = -D_a\nabla \overline{a}, \\\label{eqfluxomega}
D_a & = \alpha_a\overline{\rho}\Lambda \sqrt{\omega}.
\end{align}
This approximation is applied for the TOMs appearing in the equations for $\omega$, $\Pi$, and $\Phi$ with $a=\vec v\, '^2/2$,
$\vec v\, ' s'$, or $s'^2/2$. The parameters $\alpha_a$ control the impact of the non-local terms. \cite{kuhfuss1987} suggests
a default value of $\alpha_\omega\approx0.25$. The values for the parameters $\alpha_{\Pi,\Phi}$ are calibrated to MLT in a local version
of the Kuhfu{\ss}\ theory. However, no values for the non-local case are provided.
Alternatively, one could compute the TOMs by deriving equations for them in the same way as for the second order moments.
This has been shown in \cite{canuto92b,canuto93b}, \cite{canuto98b}, or \cite{xiong1997}, for example, and introduces fourth order moments
which again have to be closed.
\subsection{Final model equations}
The above listed approximations are implemented in the derivation of the Kuhfu{\ss}~model. The final set of partial differential equations reads:
\begin{align}
\text{d}_t\omega&=\frac{\nabla_\mathrm{ad}T}{H_p}\Pi-\frac{C_D}{\Lambda}\omega^{3/2}-\mathcal{F}_\omega,\label{eqKuh1}\\
\text{d}_t\Pi&=\frac{2\nabla_\mathrm{ad}T}{H_p}\Phi+\frac{2c_p}{3H_p}(\nabla-\nabla_\mathrm{ad})\omega-\mathcal{F}_\Pi-\frac{1}{\tau_\text{rad}}\Pi,\label{eqKuh2}\\
\text{d}_t\Phi&=\frac{c_p}{H_p}(\nabla-\nabla_\mathrm{ad})\Pi-\mathcal{F}_\Phi-\frac{2}{\tau_\text{rad}}\Phi,\label{eqKuh3}
\end{align}
where $\nabla$ and $\nabla_\mathrm{ad}$ refer to the model and adiabatic temperature gradient, respectively. The substantial derivative
is defined as $\mathrm{d}_t=\partial_t+\overline{\vec v}\cdot\nabla$. For more details about the derivation we refer to the original work by \cite{kuhfuss1987}
and \cite{flaskamp2003}.
Using the convective flux from the convection model one can compute the temperature gradient of the stellar model self-consistently from
\begin{align}
\nabla=\nabla_\text{rad}-\frac{H_p\rho}{k_\text{rad}}\Pi,
\label{eq3eqnabla}
\end{align}
with
\begin{align*}
k_\text{rad}=\frac{4acT^3}{3\kappa\rho}\,\,.
\end{align*}
where $a$ and $c$ denote the radiation constant and the speed of light respectively. Here, we neglect the kinetic energy flux $\vec j_\omega$, which is assumed to be small compared to the convective flux. Equation~(\ref{eq3eqnabla})
couples the convection model to the stellar structure equations. The self-consistent computation of the temperature gradient allows to study its behaviour
in the overshooting region. This is an advantage over ad hoc descriptions of overshooting in which the temperature gradient is set manually.
\section{Alternatives to improve Eq.~(\ref{eq_epsilon})} \label{secEq11}
Eq.~(\ref{eq_epsilon}) is heavily parametrised. \citet{canuto09b} hence discussed
a number of simplified models used in geophysics for the computation of $\epsilon$.
They are based on modified mixing lengths which account for physical processes relevant
to dissipation. However, those models are not directly applicable to stellar convection:
some of them consider a solid wall as a boundary and none of them has been designed
for the extreme density contrast of deep stellar envelopes or the peculiarities of convective
cores in massive stars.
As the closures used to derive Eq.~(\ref{eq_epsilon_HL}), which were modelled on the
basis of turbulent channel flows and freely decaying turbulence, may not be universal,
they should ideally be obtained from a more general framework. This approach has been taken
in \citet{canuto10b} who derived a dynamical equation for the TKE dissipation rate $\epsilon$
using the general turbulence model of \citet{canuto96d}. That requires the spectrum
of the source driving turbulence to be known. For shear-driven flows power law spectra for the
TKE and the Reynolds stress spectrum can readily be specified. Note that these concern scales
$k < k_0$, i.e., below the maximum of the TKE spectrum $E(k)$. In addition, energy conservation
is invoked which allows computing the non-local contribution to $\epsilon$ from the flux of turbulent
kinetic energy. That closure was already used in \citet{canuto92b} (Eq.~(37f)) and
the non-local character it introduces into Eq.~(\ref{eq_epsilon}) was discussed in
Sect.~11 of \citet{canuto93b}. It was tested in \citet{kupka07e} who found it to be one of
the most robust ones among all the closures suggested for the Reynolds
stress models of \citet{canuto92b}, \citet{canuto93b}, \citet{canuto98b}, \citet{canuto01b}, and
in \citet{canuto09b}. It specifies that $\overline{w\epsilon} = (3/2)\tau^{-1}\,F_{\rm kin}$ with
$F_{\rm kin} = \rho\,\overline{q^2 w}/2$. In practice, the accuracy of this closure is degraded, if
$\overline{q^2 w}$ can only be computed from a downgradient approximation, but even in
this case it justifies that $D_{\rm f}(\epsilon)$ can be evaluated from
$D_{\rm f}(K)$ which is required anyway. Hence, \citet{canuto10b} use
the (exact) dynamical equation for the turbulent kinetic energy and a closure for $F_{\rm kin}$ to
compute $D_{\rm f}(\epsilon)$. The equivalents of $c_1$ and
$c_2$ of Eq.~(\ref{eq_epsilon_HL}) are obtained from within the model, too. The resulting
dissipation rate equation passes the same tests as the original Eq.~(\ref{eq_epsilon_HL})
for turbulent channel flow and also two tests concerning the shear dominated planetary
boundary of the Earth atmosphere. Unfortunately, this procedure is currently
not feasible for the case of convection in stars, since this would require accurate knowledge
of the turbulent kinetic energy spectrum over a large range of scales and as a function of depth
throughout the star (see also the discussions in \citealt{gizon12b} and Fig.~5 in \citet{hanasoge16b}
on difficulties in modelling the turbulent kinetic energy spectrum for the Sun).
The dissipation rate equation Eq.~(\ref{eq_epsilon_HL}) has hence remained part of
the Reynolds stress model of \citet{canuto11b}, whether for dealing with double-diffusive
convection \citep{canuto11c} or overshooting \citep{canuto11d}. The
latter paper provides a detailed
discussion of the computation of $\epsilon$, which considers Eq.~(\ref{eq_epsilon_HL})
and $\overline{w\epsilon} = (3/2)\tau^{-1}\,F_{\rm kin}$ for non-local contributions. The role of
gravity waves as a source of dissipation in the overshooting zone is emphasised, too. From earlier
work of \citet{kumar99b}, it is concluded in \citet{canuto11c} that $\epsilon \approx 10^{-3}\,{\rm cm}^2\,{\rm s}^{-3}$.
However, as also pointed out in \citet{canuto11d}, it is unclear how this result could be
applied to overshooting zones other than the solar tachocline. Thus, in his Eq.~(5h),
\citet{canuto11d} suggests to use $\tau\,\tilde{N} = O(1)$ to compute $\tau$ and hence
via $\tau = 2\,K/\epsilon$ the dissipation rate $\epsilon$ in the overshooting region.
This, however, is consistent with the claim that the term $c_3\, \epsilon\, \tilde{N}$,
neglected in the explicit form of the $\epsilon$-equation in \citet{canuto11b,canuto11c,canuto11d},
actually dominates in the overshooting region.
Recalling \citet{kupka02b} and \citet{montgomery04b}, who had found the term
$c_3\, \epsilon\, \tilde{N}$ to dominate the solution of Eq.~(\ref{eq_epsilon})
in their applications of the Reynolds stress model of \citet{canuto98b} to overshooting in
envelopes of A-stars and white dwarfs and taking into account the confirmation of their
results for the case of a DA white dwarf by 3D radiation hydrodynamical simulations in \citet{kupka18b},
Eq.~(\ref{eq_epsilon}) is still the physically most complete model for the computation
of $\epsilon$ available at the moment. It is thus used to guide the considerations in Sect.~\ref{Sect_GARSTEC}.
Note that the dynamical equation for $\epsilon$ which is discussed here does not account for physical
effects due to compressibility. \citet{canuto97b} has presented several different models to extend
Eq.~(\ref{eq_epsilon}) beyond its {\em solenoidal} (incompressible) component $\epsilon_s$ and
account for a dilation (compressible) contribution $\epsilon_d$ (cf.\ Sect.~14 in that paper). For current
modelling in stellar structure and evolution theory such extensions appear yet too advanced: the very first step
is to give up the MLT approach to compute $\epsilon$ as specified by Eq.~(\ref{eq_MLT_epsilon})--(\ref{eq_MLT_alpha}).
|
2,869,038,154,891 | arxiv | \section{Introduction}
In network inference applications, it is important to detect community structure, i.e., cluster vertices into potential blocks. However, it can be prohibitively expensive to observe the entire graph in many cases, especially for large graphs. Thus it becomes essential to identify vertices that have the most impact on block structure and only check whether there are edges between them to save significant resources but still recover the block structure.
Many classical methods only consider the adjacency or Laplacian matrices for community detection~\cite{Fortunato2016}. By contrast, vertex covariates can also be taken into consideration for the inference. These covariate-aware methods rely on either variational methods~\cite{Choi2012,Roy2019,Sweet2015} or spectral approaches~\cite{Binkiewicz2017,Huang2018,Mele2019,Mu2022}. However, none of them focus on the problem of clustering vertices for partially observed graphs. To address this issue, existing methods propose different types of random and adaptive sampling strategies to minimize the information loss from the data reduction~\cite{Yun2014,Purohit2017}.
We propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel (SBM) when we only have limited resources to check whether there are edges between certain selected vertices. The innovation of our approach is the application of Chernoff information. To our knowledge, this is the first time that it has been applied to network sampling problems. Motivated by the Chernoff analysis, we not only propose a dynamic network sampling scheme to optimize block recovery, but also provide the framework and justification for using Chernoff information in subsequent inference for graphs.
The structure of this article is summarized as follows. Section~\ref{sec:2} reviews relevant models for random graphs and the basic idea of spectral methods. Section~\ref{sec:3} introduces the notion of Chernoff analysis for analytically measuring the performance of block recovery. Section~\ref{sec:4} includes our dynamic network sampling scheme and theoretical results. Section~\ref{sec:5} provides simulations and real data experiments to measure the algorithms' performance in terms of actual block recovery results. Section~\ref{sec:6} discusses the findings and presents some open questions for further investigation. Appendix provides technical details for our theoretical results.
\section{Models and Spectral Methods}
\label{sec:2}
In this work, we are interested in the inference task of block recovery (community detection). To model the block structure in edge-independent random graphs, we focus on the SBM and the generalized random dot product graph (GRDPG).
\begin{definition}[Generalized Random Dot Product Graph~\cite{Rubin-Delanchy2017}]
\label{def:GRDPG}
Let $ \mathbf{I}_{d_+ d_-} = \mathbf{I}_{d_+} \bigoplus \left(-\mathbf{I}_{d_-} \right) $ with $ d_+ \geq 1 $ and $ d_- \geq 0 $. Let $ F $ be a $ d $-dimensional inner product distirbution with $ d = d_+ + d_- $ on $ \mathcal{X} \subset \mathbb{R}^d $ satisfying $ \mathbf{x}^\top \mathbf{I}_{d_+ d_-} \mathbf{y} \in [0, 1] $ for all $ \mathbf{x}, \mathbf{y} \in \mathcal{X} $. Let $ \mathbf{A} $ be an adjacency matrix and $ \mathbf{X} = [\mathbf{X}_1, \cdots, \mathbf{X}_n]^\top \in \mathbb{R}^{n \times d} $ where $ \mathbf{X}_i \sim F $, i.i.d. for all $ i \in \{ 1, \cdots, n \} $. Then we say $ (\mathbf{A}, \mathbf{X}) \sim \text{GRDPG}(n, F, d_+, d_-) $ if for any $ i, j \in \{ 1, \cdots, n \} $
\begin{equation}
\mathbf{A}_{ij} \sim \text{Bernoulli}(\mathbf{P}_{ij}) \qquad \text{where} \qquad \mathbf{P}_{ij} = \mathbf{X}_{i}^\top \mathbf{I}_{d_+ d_-} \mathbf{X}_j.
\end{equation}
\end{definition}
\begin{definition}[$ K $-block Stochastic Blockmodel Graph~\cite{Holland1983}]
\label{def:SBM}
The $ K $-block stochastic blockmodel (SBM) graph is an edge-independent random graph with each vertex belonging to one of $ K $ blocks. It can be parametrized by a block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $ summing to unity. Let $ \mathbf{A} $ be an adjacency matrix and $ \bm{\tau} $ be a vector of block assignments with $ \tau_i = k $ if vertex $ i $ is in block $ k $ (occuring with probability $ \pi_k $). We say $ (\mathbf{A}, \bm{\tau}) \sim \text{SBM}(n, \mathbf{B}, \bm{\pi}) $ if for any $ i, j \in \{ 1, \cdots, n \} $
\begin{equation}
\mathbf{A}_{ij} \sim \text{Bernoulli}(\mathbf{P}_{ij}) \qquad \text{where} \qquad \mathbf{P}_{ij} = \mathbf{B}_{\tau_i \tau_j}.
\end{equation}
\end{definition}
\begin{remark}
\label{remark:GRDPG-SBM}
The SBM is a special case of the GRDPG model. Let $ (\mathbf{A}, \bm{\tau}) \sim \text{SBM}(n, \mathbf{B}, \bm{\pi}) $ as in Definition~\ref{def:SBM} where $ \mathbf{B} \in (0, 1)^{K \times K} $ with $ d_+ $ strictly positive eigenvalues and $ d_- $ strictly negative eigenvalues. To represent this SBM in the GRDPG model, we can choose $ \bm{\nu}_1, \cdots, \bm{\nu}_K \in \mathbb{R}^d $ where $ d = d_+ + d_- $ such that $ \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_\ell = \mathbf{B}_{k \ell} $ for all $ k, \ell \in \{ 1, \cdots, K \} $. For example, we can take $ \bm{\nu} = \mathbf{U}_B |\mathbf{S}_B|^{1/2} $ where $ \mathbf{B} = \mathbf{U}_B \mathbf{S}_B \mathbf{U}_B^\top $ is the spectral decomposition of $ \mathbf{B} $ after re-ordering. Then we have the latent position of vertex $ i $ as $ \mathbf{X}_i = \bm{\nu}_k $ if $ \tau_i = k $.
\end{remark}
The parameters of the models can be estimated via spectral methods~\cite{Von2007}, which have been widely used in random graph models for community detection~\cite{Lyzinski2014,Lyzinski2016,McSherry2001,Rohe2011}. Two particular spectral embedding methods, adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE), are popular since they enjoy nice propertices including consistency~\cite{Sussman2012} and asymptotic normality~\cite{Athreya2016,Tang2018}.
\begin{definition}[Adjacency Spectral Embedding]
Let $ \mathbf{A} \in \{0, 1 \}^{n \times n} $ be an adjacency matrix with eigendecomposition $ \mathbf{A} = \sum_{i=1}^{n} \lambda_i \mathbf{u}_i \mathbf{u}_i^\top $ where $ |\lambda_1| \geq \cdots \geq |\lambda_n| $ are the magnitude-ordered eigenvalues and $ \mathbf{u}_1, \cdots, \mathbf{u}_n $ are the corresponding orthonormal eigenvectors. Given the embedding dimension $ d < n $, the adjacency spectral embedding (ASE) of $ \mathbf{A} $ into $ \mathbb{R}^d $ is the $ n \times d $ matrix $ \mathbf{\widehat{X}} = \mathbf{U}_A |\mathbf{S}_A|^{1/2} $ where $ \mathbf{S}_A = \text{diag}(\lambda_1, \cdots, \lambda_d) $ and $ \mathbf{U}_A = [\mathbf{u}_1 | \cdots | \mathbf{u}_d] $.
\end{definition}
\begin{remark}
\label{remark:dhat}
There are different methods for choosing the embedding dimension~\cite{Hastie2009,Jolliffe2016}; we adopt the simple and efficient profile likelihood method~\cite{Zhu2006} to automatically identify ``elbow", which is the cut-off between the signal dimensions and the noise dimensions in scree plot.
\end{remark}
\section{Chernoff Analysis}
\label{sec:3}
To analytically measure the performance of algorithms for block recovery, we consider the notion of Chernoff information among other possible metrics. Chernoff information enjoys the advantages of being independent of the clustering procedure, i.e., it can be derived no matter which clustering methods are used, and it is intrinsically relating to the Bayes risk~\cite{Tang2018,Athreya2017,Karrer2011}.
\begin{definition}[Chernoff Information~\cite{Chernoff1952,Chernoff1956}]
Let $ F_1 $ and $ F_2 $ be two continuous multivariate distributions on $ \mathbb{R}^d $ with density functions $ f_1 $ and $ f_2 $. The Chernoff information is defined as
\begin{equation}
\begin{split}
C(F_1, F_2) & = - \log \left[\inf_{t \in (0,1)} \int_{\mathbb{R}^d} f_1^t(\mathbf{x}) f_2^{1-t}(\mathbf{x}) d\mathbf{x} \right] \\
& = \sup_{t \in (0, 1)} \left[- \log \int_{\mathbb{R}^d} f_1^t(\mathbf{x}) f_2^{1-t}(\mathbf{x}) d\mathbf{x} \right].
\end{split}
\end{equation}
\end{definition}
\begin{remark}
Consider the special case where we take $ F_1 = \mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1) $ and $ F_2 = \mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2) $; then the corresponding Chernoff information is
\begin{equation}
C(F_1, F_2) = \sup_{t \in (0, 1)} \left[ \frac{1}{2} t (1-t) (\bm{\mu}_1 - \bm{\mu}_2)^\top \bm{\Sigma}_t^{-1} (\bm{\mu}_1 - \bm{\mu}_2) + \frac{1}{2} \log \frac{\lvert \bm{\Sigma}_t \rvert}{\lvert \bm{\Sigma}_1 \rvert^t \lvert \bm{\Sigma}_2 \rvert^{1-t}} \right],
\end{equation}
where $ \bm{\Sigma}_t = t \bm{\Sigma}_1 + (1-t) \bm{\Sigma}_2 $.
\end{remark}
The comparsion of block recovery via Chernoff information is based on the statistical information between the limiting distributions of the blocks and smaller statistical information implies less information to discriminate between different blocks of the SBM. To that end, we also review the limiting results of ASE for SBM, essential for investigating Chernoff information.
\begin{theorem}[CLT of ASE for SBM~\cite{Rubin-Delanchy2017}]
\label{thm:CLT-ASE-SBM}
Let $ (\mathbf{A}^{(n)}, \mathbf{X}^{(n)}) \sim \text{GRDPG}(n, F, d_+, d_-) $ be a sequence of adjacency matrices and associated latent positions of a $ d $-dimensional GRDPG as in Definition~\ref{def:GRDPG} from an inner product distribution $ F $ where $ F $ is a mixture of $ K $ point masses in $ \mathbb{R}^d $, i.e.,
\begin{equation}
F = \sum_{k=1}^{K} \pi_k \delta_{\bm{\nu}_k} \qquad \text{with} \qquad \forall k, \; \pi_k > 0 \quad \text{and} \quad \sum_{k=1}^{K} \pi_k = 1,
\end{equation}
where $ \delta_{\bm{\nu}_k} $ is the Dirac delta measure at $ \nu_k $. Let $ \Phi(\mathbf{z}, \bm{\Sigma}) $ denote the cumulative distribution function (CDF) of a multivariate Gaussian distribution with mean $ \bm{0} $ and covariance matrix $ \bm{\Sigma} $, evaluated at $ \mathbf{z} \in \mathbb{R}^d $. Let $ \mathbf{\widehat{X}}^{(n)} $ be the ASE of $ \mathbf{A}^{(n)} $ with $ \mathbf{\widehat{X}}^{(n)}_i $ as the $ i $-th row (same for $ \mathbf{X}^{(n)}_i $). Then there exists a sequence of matrices $ \mathbf{M}_n \in \mathbb{R}^{d \times d} $ satisfying $ \mathbf{M}_n \mathbf{I}_{d_+ d_-} \mathbf{M}_n^\top = \mathbf{I}_{d_+ d_-} $ such that for all $ \mathbf{z} \in \mathbb{R}^d $ and fixed index i,
\begin{equation}
\mathbb{P} \left\{ \sqrt{n} \left(\mathbf{M}_n \mathbf{\widehat{X}}^{(n)}_i - \mathbf{X}^{(n)}_i \right) \leq \mathbf{z} \; \big| \; \mathbf{X}^{(n)}_i = \bm{\nu}_k \right\} \to \Phi(\mathbf{z}, \bm{\Sigma}_k),
\end{equation}
where for $ \bm{\nu} \sim F $
\begin{equation}
\label{eq:Sigmax}
\bm{\Sigma}_k = \bm{\Sigma}(\bm{\nu}_k) = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \mathbb{E} \left[ \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \bm{\nu} \bm{\nu}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-},
\end{equation}
with
\begin{equation}
\label{eq:Delta}
\bm{\Delta} = \mathbb{E} \left[ \bm{\nu} \bm{\nu}^\top \right].
\end{equation}
\end{theorem}
For a $ K $-block SBM, let $ \mathbf{B} \in (0, 1)^{K \times K} $ be the block connectivity probability matrix and $ \bm{\pi} \in (0, 1)^K $ be the vector of block assignment probabilities. Given an $ n $ vertex instantiation of the SBM parameterized by $ \mathbf{B} $ and $ \bm{\pi} $, for sufficiently large $ n $, the large sample optimal error rate for estimating the block assignments using ASE can be measured via Chernoff information as~\cite{Tang2018,Athreya2017}
\begin{equation}
\label{eq:rho}
\rho = \min_{k \neq l} \sup_{t \in (0, 1)} \left[ \frac{1}{2} n t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) + \frac{1}{2} \log \frac{\lvert \bm{\Sigma}_{k \ell}(t) \rvert}{\lvert \bm{\Sigma}_k \rvert^t \lvert \bm{\Sigma}_\ell \rvert^{1-t}} \right],
\end{equation}
where $ \bm{\Sigma}_{k\ell}(t) = t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell $, $ \bm{\Sigma}_k = \bm{\Sigma}(\bm{\nu}_k) $ and $ \bm{\Sigma}_\ell = \bm{\Sigma}(\bm{\nu}_\ell) $ are defined as in Eq.~\eqref{eq:Sigmax}. Also note that as $ n \to \infty $, the logarithm term in Eq.~\eqref{eq:rho} will be dominated by the other term. Then we have the approximate Chernoff information as
\begin{equation}
\label{eq:rhoapprox}
\rho \approx \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}),
\end{equation}
where
\begin{equation}
\label{eq:C_kl}
C_{k ,\ell}(\mathbf{B}, \bm{\pi}) =\sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \right].
\end{equation}
We also introduce the following two notions, which will be used when we describe our dynamic network sampling scheme.
\begin{definition}[Chernoff-active Blocks]
For $K$-block SBM parametrized by the block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and the vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. The Chernoff-active blocks $ (k^*, \ell^*) $ are defined as
\begin{equation}
(k^*, \ell^*) = \arg \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}),
\end{equation}
where $ C_{k ,\ell}(\mathbf{B}, \bm{\pi}) $ is defined as in Eq.~\eqref{eq:rhoapprox}.
\end{definition}
\begin{definition}[Chernoff Superiority]
For $K$-block SBMs, given two block connectivity probability matrices $ \mathbf{B}, \mathbf{B}^\prime \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. Let $ \rho_B $ and $ \rho_{B^\prime} $ denote the Chernoff information obtained as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively. We say that $ \mathbf{B} $ is Chernoff superior to $ \mathbf{B}^\prime $, denoted as $ \mathbf{B} \succ \mathbf{B}^\prime $, if $ \rho_B > \rho_{B^\prime} $.
\end{definition}
\begin{remark}
If $ \mathbf{B} $ is Chernoff superior to $ \mathbf{B}^\prime $, then we can have a better block recovery from $ \mathbf{B} $ than $ \mathbf{B}^\prime $. In addition, Chernoff superiority is transitive, which is straightforward from the definition.
\end{remark}
\section{Dynamic Network Sampling}
\label{sec:4}
Consider the $ K $-block SBM parametrized by the block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and the vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $ with $ K > 2 $. Given initial sampling parameter $ p_0 \in (0, 1) $, initial sampling is uniformly at random, i.e.,
\begin{equation}
\label{eq:B0}
\mathbf{B}_0 = p_0 \mathbf{B}.
\end{equation}
\begin{theorem}
\label{thm:Chernoff-Superiority}
For $K$-block SBMs, given two block connectivity probability matrices $ \mathbf{B}, p\mathbf{B} \in (0, 1)^{K \times K} $ with $ p \in (0, 1) $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $, we have $ \mathbf{B} \succ p \mathbf{B} $.
\end{theorem}
The proof of Theorem~\ref{thm:Chernoff-Superiority} can be found in Appendix. As an illustration, consider a 4-block SBM parametrized by block connectivity probability matrix $ \mathbf{B} $ as
\begin{equation}
\label{eq:exampleB}
\mathbf{B} =
\begin{bmatrix}
0.04 & 0.08 & 0.10 & 0.18 \\
0.08 & 0.16 & 0.20 & 0.36 \\
0.10 & 0.20 & 0.25 & 0.45 \\
0.18 & 0.36 & 0.45 & 0.81
\end{bmatrix}.
\end{equation}
Figure~\ref{fig:rho0} shows Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and $ p \mathbf{B} $ for $ p \in (0, 1) $. In addition, Figure~\ref{fig:rho0a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho0b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. As suggested by Theorem~\ref{thm:Chernoff-Superiority}, for any $ p \in (0, 1) $ we have $\rho_{B} > \rho_{pB} $ and thus $ \mathbf{B} \succ p \mathbf{B} $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho0a}]{
[width=0.45\textwidth]{figures/rho0a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho0b}]{
[width=0.45\textwidth]{figures/rho0b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and $ p \mathbf{B} $ for $ p \in (0, 1) $.}
\label{fig:rho0}
\end{figure}
Now given dynamic network sampling parameter $ p_1 \in (0, 1-p_0) $, the baseline sampling scheme can proceed uniformly at random again, i.e.,
\begin{equation}
\label{eq:B1}
\mathbf{B}_1 = \mathbf{B}_0 + p_1 \mathbf{B} = (p_0 + p_1) \mathbf{B}.
\end{equation}
\begin{corollary}
\label{cor:Chernoff-Superiority}
For $K$-block SBMs, given block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. We have $ \mathbf{B} \succ \mathbf{B}_1 \succ \mathbf{B}_0 $ where $ \mathbf{B}_0 $ is defined as in Eq.~\eqref{eq:B0} with $ p_0 \in (0, 1) $ and $ \mathbf{B}_1 $ is defined as in Eq.~\eqref{eq:B1} with $ p_1 \in (0, 1-p_0) $.
\end{corollary}
The proof of Corollary~\ref{cor:Chernoff-Superiority} can be found in Appendix. This corollay implies that we can have a better block recovery from $ \mathbf{B}_1 $ than $ \mathbf{B}_0 $.
\begin{assumption}
\label{cond:1}
The Chernoff-active blocks after initial sampling is unique, i.e., there exists an unique pair $ \left(k_0^*, \ell_0^* \right) \in \{(k, \ell) \; | \; 1 \leq k < \ell \leq K \} $ such that
\begin{equation}
\left(k_0^*, \ell_0^* \right) = \arg \min_{k \neq l} C_{k ,\ell}(\mathbf{B}_0, \bm{\pi}),
\end{equation}
where $ \mathbf{B}_0 $ is defined as in Eq.~\eqref{eq:B0} and $ \bm{\pi} $ is the vector of block assignment probabilities.
\end{assumption}
To improve this baseline sampling scheme, we concentrate on the Chernoff-active blocks $ \left(k_0^*, \ell_0^* \right) $ after initial sampling assuming Assumption~\ref{cond:1} holds. Instead of sampling from the entire block connectivity probability matrix $ \mathbf{B} $ like the baseline sampling scheme as in Eq.~\eqref{eq:B1}, we only sample the entries associated with the Chernoff-active blocks. As a competitor to $ \mathbf{B}_1 $, our Chernoff-optimal dynamic network sampling scheme is then given by
\begin{equation}
\label{eq:B1tilde}
\widetilde{\mathbf{B}}_1 = \mathbf{B}_0 + \frac{p_1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*},
\end{equation}
where $ \circ $ denotes Hadamard product, $ \pi_{k_0^*} $ and $ \pi_{\ell_0^*} $ denote the block assignment probabilities for block $ k_0^* $ and $ \ell_0^* $ respectively, and $ \mathbf{1}_* $ is the $ K \times K $ binary matrix with 0's everywhere except for 1's associated with the Chernoff-active blocks $ \left(k_0^*, \ell_0^* \right) $, i.e., for any $ i, j \in \{1, \cdots, K \} $
\begin{equation}
\mathbf{1}_{k_0^*, \ell_0^*}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in \left\{ \left(k_0^*, k_0^* \right), \; \left(k_0^*, \ell_0^* \right), \; \left(\ell_0^*, k_0^* \right), \; \left(\ell_0^*, \ell_0^* \right) \right\} \\
0 & \text{otherwise}
\end{cases}
.
\end{equation}
Note that the multiplier $ \frac{1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2} $ on $ p_1 \mathbf{B} \circ \mathbf{1}_* $ assures that we sample the same number of potential edges with $ \widetilde{\mathbf{B}}_1 $ as we do with $ \mathbf{B}_1 $ in the baseline sampling scheme. In addition, to avoid over-sampling with respect to $ \mathbf{B} $, i.e., to ensure $ \widetilde{\mathbf{B}}_1[i, j] \leq \mathbf{B}[i, j] $ for any $ i, j \in \{1, \cdots, K \} $, we require
\begin{equation}
\label{eq:p1max}
p_1 \leq p_1^{\text{max}} = \left( 1 - p_0 \right) \left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2.
\end{equation}
\begin{assumption}
\label{cond:Chernoff-Superiority2}
For $K$-block SBMs, given a block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. Let $ p_1^* \in (0, p_1^{\text{max}}] $ be the smallest positive $ p_1 \leq p_1^{\text{max}} $ such that
\begin{equation}
\arg \min_{k \neq l} C_{k ,\ell}(\widetilde{\mathbf{B}}_1, \bm{\pi})
\end{equation}
is not unique where $ p_1^{\text{max}} $ is defined as in Eq.~\eqref{eq:p1max} and $ \widetilde{\mathbf{B}}_1 $ is defined as in Eq.~\eqref{eq:B1tilde}. If the arg min is always unique, let $ p_1^* = p_1^{\text{max}} $.
\end{assumption}
For any $ p_1 \in (0, p_1^*)$, we can have a better block recovery from $ \widetilde{\mathbf{B}}_1 $ than $ \mathbf{B}_1 $, i.e., our Chernoff-optimal dynamic network sampling sheme is better than the baseline sampling scheme in terms of block recovery.
As an illustaration, consider the 4-block SBM with initial sampling parameter $ p_0 = 0.01 $ and block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. Figure~\ref{fig:rho1} shows the Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1 $ as in Eq.~\eqref{eq:B1tilde} with dynamic network sampling parameter $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}. In addition, Figure~\ref{fig:rho1a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho1b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. Note that for any $ p_1 \in (0, p_1^*) $ we have $\rho_{B} > \rho_{\widetilde{B}_1} > \rho_{B_1} > \rho_{B_0} $ and thus $ \mathbf{B} \succ \widetilde{\mathbf{B}}_1 \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho1a}]{
[width=0.45\textwidth]{figures/rho1a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho1b}]{
[width=0.45\textwidth]{figures/rho1b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1 $ as in Eq.~\eqref{eq:B1tilde} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}.}
\label{fig:rho1}
\end{figure}
As described earlier, it may be the case that $ p_1^* < p_1^{\text{max}} $ at which point Chernoff-active blocks change to $ (k_1^*, \ell_1^*) $. This potential non-uniquess of the Chernoff argmin is a consequence of our dynamic network sampling scheme. In the case of $ p_1 > p_1^* $, our Chernoff-optimal dynamic network sampling scheme is adopted as
\begin{equation}
\label{eq:B1tildestar}
\widetilde{\mathbf{B}}_1^* = \mathbf{B}_0 + \left(p_1 - p_1^* \right) \mathbf{B} + \frac{p_1^*}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*},
\end{equation}
Similarly, the multiplier $ \frac{1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2} $ on $ p_1^* \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*} $ assures that we sample the same number of potential edges with $ \widetilde{\mathbf{B}}_1^* $ as we do with $ \mathbf{B}_1 $ in the baseline sampling scheme. In addition, to avoid over-sampling with respect to $ \mathbf{B} $, i.e., $ \widetilde{\mathbf{B}}_1^*[i, j] \leq \mathbf{B}[i, j] $ for any $ i, j \in \{1, \cdots, K \} $, we require
\begin{equation}
\label{eq:p11max}
p_1 \leq p_{11}^{\text{max}} = 1 - p_0 - \frac{p_1^*}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } + p_1^*.
\end{equation}
For any $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $, we can have a better block recovery from $ \widetilde{\mathbf{B}}_1^* $ than $ \mathbf{B}_1 $, i.e., our Chernoff-optimal dynamic network sampling sheme is again better than the baseline sampling scheme in terms of block recovery.
As an illustration, consider a 4-block SBM with initial sampling parameter $ p_0 = 0.01 $ and block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. Figure~\ref{fig:rho2} shows the Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} with dynamic network sampling parameter $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2} and $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. In addition, Figure~\ref{fig:rho2a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho2b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. Note that for any $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ we have $\rho_{B} > \rho_{\widetilde{B}_1^*} > \rho_{B_1} > \rho_{B_0} $ and thus $ \mathbf{B} \succ \widetilde{\mathbf{B}}_1^* \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho2a}]{
[width=0.45\textwidth]{figures/rho2a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho2b}]{
[width=0.45\textwidth]{figures/rho2b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2} and $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}.}
\label{fig:rho2}
\end{figure}
We summarize the uniform dynamic sampling scheme (baseline) as Algorithm~\ref{algo:1} and our Chernoff-optimal dynamic network sampling scheme as Algorithm~\ref{algo:2}. Recall given potential edge set $ E $ and initial sampling parameter $ p_0 \in (0, 1) $, we have the initial edge set $ E_0 \subset E $ with $ \lvert E_0 \rvert = p_0 \lvert E \rvert $. The goal is to dynamically sample new edges from the potential edge set so that we can have a better block recovery given limited resources.
\begin{algorithm}
\label{algo:1}
\SetAlgoNoLine
\KwIn{Number of vertices $ n $; potential edge set $ E = \{(i, j) \; | \; i, j \in \{1, \cdots, n \} \} $; initial edge set $ E_0 \subset E $; dynamic network sampling parameter $ p_1 \in \left(0, 1- \frac{\lvert E_0 \rvert}{\lvert E \rvert} \right) $}
Construct dynamic edge set as
\begin{equation*}
E_1 = \left\{(i ,j) \; | \; (i ,j) \in E \setminus E_0 \right\} \qquad \text{with} \qquad \lvert E_1 \rvert = p_1 \lvert E \rvert.
\end{equation*} \\
Construct dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \bigcup E_1 \;\; \text{or} \;\; (j, i) \in E_0 \bigcup E_1 \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Estimate dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using Gaussian mixture modeling (GMM) to estimate the block assignments as $ \bm{\widehat{\tau}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via Bayesian Information Criterion (BIC).
\KwOut{Block assignments $ \bm{\widehat{\tau}} $.}
\caption{Uniform dynamic network sampling scheme (baseline)}
\end{algorithm}
\begin{algorithm}
\label{algo:2}
\SetAlgoNoLine
\KwIn{Number of vertices $ n $; potential edge set $ E = \{(i, j) \; | \; i, j \in \{1, \cdots, n \} \} $; initial edge set $ E_0 \subset E $; dynamic network sampling parameter $ p_1 \in \left(0, 1- \frac{\lvert E_0 \rvert}{\lvert E \rvert} \right) $}
Construct dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \;\; \text{or} \;\; (j, i) \in E_0 \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Estimate dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using GMM to estimate the initial block assignments as $ \bm{\widehat{\xi}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via BIC. \\
Estimate the dynamic block assignment probability vector as $ \bm{\widehat{\pi}} \in (0, 1)^K $ where for $ k \in \{1, \cdots, K \} $
\begin{equation*}
\widehat{\pi}_k = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1} \{\bm{\widehat{\xi}}_i = k \}.
\end{equation*} \\
Estimate the dynamic block connectivity probability matrix as
\begin{equation*}
\mathbf{\widehat{B}} = \bm{\widehat{\mu}} \mathbf{I}_{d_+ d_-} \bm{\widehat{\mu}}^\top \in [0,1]^{\widehat{K} \times \widehat{K}},
\end{equation*}
where $ \bm{\widehat{\mu}} \in \mathbb{R}^{\widehat{K} \times \widehat{d}} $ is the estimated means of all clusters. \\
Find the Chernoff-active blocks as
\begin{equation*}
\left(k^*, \ell^* \right) = \arg \min_{k \neq l} C_{k ,\ell} \left(\mathbf{\widehat{B}}, \bm{\widehat{\pi}} \right).
\end{equation*} \\
Construct dynamic edge set as
\begin{equation*}
\begin{split}
E_1 \subseteq E_* \qquad & \text{with} \qquad \lvert E_1 \rvert = \min \left\{p_1 \lvert E \rvert \left(\widehat{\pi}_{k^*} + \widehat{\pi}_{\ell^*}\right)^2, \lvert E_* \rvert \right\}, \\
E_{11} \subset E \setminus \left(E_0 \bigcup E_1 \right) \qquad & \text{with} \qquad \lvert E_{11} \rvert = p_1 \lvert E \rvert - \lvert E_1 \rvert,
\end{split}
\end{equation*}
where
\begin{equation*}
E_* = \left\{(i ,j) \; | \; (i ,j) \in E \setminus E_0 \; \text{and} \; \widehat{\xi}_i, \widehat{\xi}_j \in \{k^*, \ell^* \} \right\}.
\end{equation*} \\
Update dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \bigcup E_1 \bigcup E_{11} \;\; \text{or} \;\; (j, i) \in E_0 \bigcup E_1 \bigcup E_{11} \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Update dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of updated $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using GMM to estimate the block assignments as $ \bm{\widehat{\tau}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via BIC.
\KwOut{Block assignments $ \bm{\widehat{\tau}} $.}
\caption{Chernoff-optimal dynamic network sampling scheme}
\end{algorithm}
\section{Experiments}
\label{sec:5}
\subsection{Simulations}
In addition to Chernoff analysis, we also evalute our Chernoff-optimal dynamic network sampling sheme via simulations. In particular, consider the 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and dynamic network sampling parameter $ p_1 \in (0, p_{11}^{\text{max}}] $ where $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. We fix initial sampling parameter $ p_0 = 0.01 $. For each $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}, we simulate 50 adjacency matrices with $ n = 12000 $ vertices from $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1} and $ \widetilde{\mathbf{B}}_1 $ as in Eq.~\eqref{eq:B1tilde} respectively. For each $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $, we simulate 50 adjacency matrices with $ n = 12000 $ vertices from $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1} and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} respectively. In addition, Figure~\ref{fig:sim0a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $, i.e., 3000 vertices in each block, and Figure~\ref{fig:sim0b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $, i.e., 1500 vertices in two of the blocks and 4500 vertices in the other two blocks. We then apply ASE $ \circ $ GMM (Step 3 and 4 in Algorithm~\ref{algo:1}) to recover block assignments and adopt adjusted Rand index (ARI) to measure the performance. Figure~\ref{fig:sim0} shows ARI (\texttt{mean$ \pm $stderr}) associated with $ \mathbf{B}_1 $ for $ p_1 \in (0, p_{11}^{\text{max}}] $, $ \widetilde{\mathbf{B}}_1 $ for $ p_1 \in (0, p_1^*) $, and $ \widetilde{\mathbf{B}}_1^* $ for $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where the dashed lines denote $ p_1^* $. Note that we can have a better block recovery from $ \widetilde{\mathbf{B}}_1 $ and $ \widetilde{\mathbf{B}}_1^* $ than $ \mathbf{B}_1 $, which argee with our results from Chernoff analysis.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:sim0a}]{
[width=0.45\textwidth]{figures/sim0a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:sim0b}]{
[width=0.45\textwidth]{figures/sim0b.png}
}
\caption{Simulations for 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in (0, p_{11}^{\text{max}}] $ where $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. The dashed lines denote $ p_1^* $ which is defined as in Assumption~\ref{cond:Chernoff-Superiority2}.}
\label{fig:sim0}
\end{figure}
Now we compare the performance of Algorithms~\ref{algo:1} and~\ref{algo:2} by actual block recovery results. In particular, we start with the 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. We consider dynamic network sampling parameter $ p_1 \in (0, 1-p_0) $ where $ p_0 $ is the initial sampling parameter. For each $ p_1 $, we simulate 50 adjacency matrices with $ n = 4000 $ vertices and retrieve associated potential edge sets. We fix initial sampling parameter $ p_0 = 0.15 $ and randomly sample initial edge sets. We then apply both algorithms to estimate the block assignments and adopt ARI to measure the performance. Figure~\ref{fig:sim1} shows ARI (\texttt{mean$ \pm $stderr}) of two algorithms for $ p_1 \in (0, 0.85) $ where Figure~\ref{fig:sim1a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $, i.e., 1000 vertices in each block, and Figure~\ref{fig:sim1b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $, i.e., 500 vertices in two of the blocks and 1500 vertices in the other two blocks. Note that both algorithms tend to have a better performance as $ p_1 $ increases, i.e., as we sample more edges, and Algorithm~\ref{algo:2} can always recover more accurate block structure than Algorithm~\ref{algo:1}.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:sim1a}]{
[width=0.45\textwidth]{figures/sim1a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:sim1b}]{
[width=0.45\textwidth]{figures/sim1b.png}
}
\caption{Simulations for 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} with initial sampling parameter $ p_0 = 0.15 $ and dynamic network sampling parameter $ p_1 \in (0, 0.85) $.}
\label{fig:sim1}
\end{figure}
\subsection{Real Data}
We also evaluate the performance of Algorithms~\ref{algo:1} and~\ref{algo:2} for real application. We conduct real data experiments on a diffusion MRI connectome dataset~\cite{Priebe2019}. There are 114 graphs (connectomes) estimated by the NDMG pipeline~\cite{Kiar2018} in this dataset. Each vertex in these graphs (the number of vertices $ n $ varies from 23728 to 42022) has a \{Left,~Right\} hemisphere label and a \{Gray,~White\} tissue label. We consider the potential 4 blocks as \{LG,~LW,~RG,~RW\} where L and R denote the Left and Right hemisphere label, G and W denote the Gray and White tissue label. Here we consider initial sampling parameter $ p_0 = 0.25 $ and dynamic network sampling parameter $ p_1 = 0.25 $. Let $ \Delta = \text{ARI(Algo2)} - \text{ARI(Algo1)} $ where ARI(Algo1) and ARI(Algo2) denotes the ARI when we apply Algorithms~\ref{algo:1} and~\ref{algo:2} respectively. The following hypothesis testing yields \texttt{p-value=0.0184}.
\begin{equation}
H_0: \; \text{median}(\Delta) \leq 0 \qquad \text{v.s.} \qquad H_A: \; \text{median}(\Delta) > 0.
\end{equation}
Furthermore, we test our algorithms on a Microsoft bing entity dataset~\cite{Agterberg2020}. There are 2 graphs in this dataset where each has 13535 vertices. We treat block assignments estimated from the complete graph as ground truth. We consider initial sampling parameter $ p_0 \in \left\{0.2, \; 0.3 \right\} $ and dynamic network sampling parameter $ p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $. For each $ p_1 $, we sample 100 times and compare the overall performance of Algorithm~\ref{algo:1} and~\ref{algo:2}. Figure~\ref{fig:real2} shows the results where ARI is reported as \texttt{mean($\pm$stderr)}.
\begin{figure}[h!]
\subfigure[$ p_0 = 0.2, \; p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $ \label{fig:real2a}]{
[width=0.45\textwidth]{figures/real2a.png}
}
\hfil
\subfigure[$ p_0 = 0.3, \; p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $ \label{fig:real2b}]{
[width=0.45\textwidth]{figures/real2b.png}
}
\caption{Algorithms' comparative performance on Microsoft bing entity data via ARI with different initial sampling parameter $ p_0 $ and dynamic network sampling parameter $ p_1 $.}
\label{fig:real2}
\end{figure}
We also conduct real data experiments with 2 social network datasets.
\begin{itemize}
\item LastFM asia social network data set~\cite{Leskovec2014,Rozemberczki2020}: Vertices (the number of vertices $ n = 7624 $) represent LastFM users from asian countries and edges (the number of edges $ e = 27806 $) represent mutual follower relationships. We treat 18 different location of users, which are derived from the country field for each user, as the potential block.
\item Facebook large page-page network data set~\cite{Leskovec2014,Rozemberczki2019}: Vertices (the number of vertices $ n = 22470 $) represent official Facebook pages and edges (the number of edges $ e = 171002 $) represent mutual likes. We treat 4 page types \{Politician,~Governmental~Organization,~Television~Show,~Company\}, which are defined by Facebook, as the potential block.
\end{itemize}
We consider initial sampling parameter $ p_0 \in \left\{0.15, \; 0.35 \right\} $ and dynamic network sampling parameter $ p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $. For each $ p_1 $, we sample 100 times and compare the overall performance of Algorithm~\ref{algo:1} and~\ref{algo:2}. Figure~\ref{fig:real3} shows the results where ARI is reported as \texttt{mean($\pm$stderr)}.
\begin{figure}[h!]
\subfigure[LastFM: $ p_0 = 0.15, \; p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $ \label{fig:real3a}]{
[width=0.45\textwidth]{figures/real3a.png}
}
\hfil
\subfigure[Facebook: $ p_0 = 0.35, \; p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $ \label{fig:real3b}]{
[width=0.45\textwidth]{figures/real3b.png}
}
\caption{Algorithms' comparative performance on social network data via ARI with different initial sampling parameter $ p_0 $ and dynamic network sampling parameter $ p_1 $.}
\label{fig:real3}
\end{figure}
\section{Discussion}
\label{sec:6}
We propose a dynamic network sampling scheme to optimize block recovery for SBM. Theoretically, we provide justification of our proposed Chernoff-optimal dynamic sampling scheme via the Chernoff information. Practically, we evaluate the performance, in terms of block recovery (community detection), of our method on several real datasets including diffusion MRI connectome dataset, Microsoft bing entity graph transitions dataset and social network datasets. Both theoretically and practically results suggest that our method can identify vertices that have the most impact on block structure and only check whether there are edges between them to save significant resources but still recover the block structure.
As the Chernoff-optimal dynamic sampling scheme depends on the initial clustering results to identify Chernoff-active blocks and construct dynamic edge set. Thus the performance could be impacted if the initial clustering is not very ideal. One of the future direction is to design certain strategy to reduce this dependency such that the proposed scheme is more robust.
\section*{Appendix}
\begin{proof}[Proof of Theorem~\ref{thm:Chernoff-Superiority}.]
Let $ \mathbf{B} = \mathbf{U} \mathbf{S} \mathbf{U}^\top $ be the spectral decomposition of $ \mathbf{B} $ and $ \mathbf{B}^\prime = p \mathbf{B} $ with $ p \in (0, 1) $. Then we have
\begin{equation}
\label{eq:Bprime}
\mathbf{B}^\prime = \mathbf{U}^\prime \mathbf{S} \left(\mathbf{U}^\prime\right)^\top \qquad \text{where} \qquad \mathbf{U}^\prime = \sqrt{p} \mathbf{U}.
\end{equation}
By Remark~\ref{remark:GRDPG-SBM}, to represent these two SBMs parametrized by two block connectivity matrices $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively (with the same block assignment probability vector $ \bm{\pi} $) in the GRDPG models, we can take
\begin{equation}
\label{eq:nunuprime}
\begin{split}
\bm{\nu} & = \begin{bmatrix}
\bm{\nu}_1 & \cdots & \bm{\nu}_K
\end{bmatrix}^\top = \mathbf{U} |\mathbf{S}|^{1/2} \in \mathbb{R}^{K \times d}, \\
\bm{\nu}^\prime & = \begin{bmatrix}
\bm{\nu}_1^\prime & \cdots & \bm{\nu}_K^\prime
\end{bmatrix}^\top = \mathbf{U}^\prime |\mathbf{S}|^{1/2} = \sqrt{p} \mathbf{U} |\mathbf{S}|^{1/2} = \sqrt{p} \bm{\nu} \in \mathbb{R}^{K \times d}.
\end{split}
\end{equation}
Then for any $ k \in \{1, \cdots, K \} $, we have $ \bm{\nu}_k^\prime = \sqrt{p} \bm{\nu}_k \in \mathbb{R}^{d} $. By Theorem~\ref{thm:CLT-ASE-SBM}, we have
\begin{equation}
\begin{split}
\bm{\Delta} & = \sum_{k=1}^{K} \pi_k \bm{\nu}_k \bm{\nu}_k^\top \in \mathbb{R}^{d \times d}, \\
\bm{\Delta}^\prime & = \sum_{k=1}^{K} \pi_k \bm{\nu}_k^\prime \left(\bm{\nu}_k^\prime\right)^\top = p \sum_{k=1}^{K} \pi_k \bm{\nu}_k \bm{\nu}_k^\top = p \bm{\Delta} \in \mathbb{R}^{d \times d}.
\end{split}
\end{equation}
Note that $ \mathbf{B} $ and $ \mathbf{B}^\prime $ have the same eigenvalues, thus we have $ \mathbf{I}_{d_+ d_-} = \mathbf{I}_{d_+ d_-}^\prime $. See also Lemma 2~\cite{Gallagher2019}. Then for $ k \in \{1, \cdots, K \} $, we have
\begin{equation}
\begin{split}
\bm{\Sigma}_k & = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \mathbb{E} \left[ \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \bm{\nu} \bm{\nu}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[\sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \in \mathbb{R}^{d \times d}, \\[1em]
\bm{\Sigma}_k^{\prime} & = \frac{1}{p^2} \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[p^2 \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-p \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[p \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& \qquad + \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[(1-p) \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = p \bm{\Sigma}_k + \mathbf{V}^\top \mathbf{D}_k(p) \mathbf{V} \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\mathbf{V} & = \bm{\nu} \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \in \mathbb{R}^{K \times d}, \\
\mathbf{D}_k(p) & = (1-p) \text{diag} \left(\pi_1 \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_1, \cdots, \pi_K \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_K \right) \in (0, 1)^{K \times K}.
\end{split}
\end{equation}
Recall that by Remark~\ref{remark:GRDPG-SBM}, we have $ \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_\ell = \mathbf{B}_{k \ell} \in (0, 1) $ for all $ k, \ell \in \{ 1, \cdots, K \} $. Then we have $ \mathbf{D}_k(p) $ is positive-definite for any $ k \in \{1, \cdots, K \} $ and $ p \in (0, 1) $. For $ k, \ell \in \{1, \cdots, K \} $ and $ t \in (0, 1) $, let $ \bm{\Sigma}_{k\ell}(t) $ and $ \bm{\Sigma}_{k\ell}^{\prime}(t) $ denote the matrics as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively, i.e.,
\begin{equation}
\begin{split}
\bm{\Sigma}_{k\ell}(t) & = t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell \in \mathbb{R}^{d \times d}, \\[1em]
\bm{\Sigma}_{k\ell}^{\prime}(t) & = t \bm{\Sigma}_k^{\prime} + (1-t) \bm{\Sigma}_\ell^{\prime} \\
& = t \left[p \bm{\Sigma}_k + \mathbf{V}^\top \mathbf{D}_k(p) \mathbf{V} \right] + (1-t) \left[p \bm{\Sigma}_\ell + \mathbf{V}^\top \mathbf{D}_\ell(p) \mathbf{V} \right] \\
& = p \left[t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell \right] + \mathbf{V}^\top \left[t \mathbf{D}_k(p) + (1-t) \mathbf{D}_\ell(p) \right] \mathbf{V} \\
& = p \bm{\Sigma}_{k\ell}(t) + \mathbf{V}^\top \mathbf{D}_{k \ell}(p, t) \mathbf{V} \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\mathbf{D}_{k \ell}(p, t) = t \mathbf{D}_k(p) + (1-t) \mathbf{D}_\ell(p) \in \mathbb{R}_+^{K \times K}.
\end{equation}
Recall that $ \mathbf{D}_k(p) $ and $ \mathbf{D}_\ell(p) $ are both positive-definite for any $ k, \ell \in \{1, \cdots, K \} $ and $ p \in (0, 1) $, thus $ \mathbf{D}_{k \ell}(p, t) $ is also positive-definite for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $. Now by the Sherman-Morrison-Woodbury formula~\cite{Horn2012}, we have
\begin{equation}
\begin{split}
\left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} & = \left[p \bm{\Sigma}_{k\ell}(t) + \mathbf{V}^\top \mathbf{D}_{k \ell}(p, t) \mathbf{V} \right]^{-1} \\
& = \frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \left[\mathbf{D}_{k \ell}^{-1}(p, t) + \frac{1}{p} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \right]^{-1} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \\
& = \frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \mathbf{M}_{k \ell}^{-1}(p, t)\mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\mathbf{M}_{k \ell}(p, t) = \mathbf{D}_{k \ell}^{-1}(p, t) + \frac{1}{p} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \in \mathbb{R}^{K \times K}.
\end{equation}
Recall that for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $, $ \mathbf{D}_{k \ell}(p, t) $ and $ \bm{\Sigma}_{k\ell}(t) $ are both positive-definite, thus $ \mathbf{M}_{k \ell}(p, t) $ is also positive-definite. Then for any $ k, \ell \in \{1, \cdots, K \} $ and $ p,t \in (0, 1) $, we have
\begin{equation}
\label{eq:nuSigma}
\begin{split}
(\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime) & = p (\bm{\nu}_k - \bm{\nu}_\ell)^\top \\
& \quad \left[\frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \right] \\
& \quad (\bm{\nu}_k - \bm{\nu}_\ell) \\
& = (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \\
& \quad - \frac{1}{p} \mathbf{x}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{x} \\
& = (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) - h_{k \ell}(p, t),
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\mathbf{x} & = \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \in \mathbb{R}^K, \\
h_{k \ell}(p, t) & = \frac{1}{p} \mathbf{x}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{x}.
\end{split}
\end{equation}
Recall that for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $, $ \mathbf{M}_{k \ell}(p, t) $ is positive-definite, thus we have $ h_{k \ell}(p, t) > 0 $. Together with Eq.~\eqref{eq:nuSigma}, we have
\begin{equation}
t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) > t (1-t) (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime).
\end{equation}
Thus for any $ k, \ell \in \{1, \cdots, K \} $, we have
\begin{equation}
\begin{split}
C_{k ,\ell}(\mathbf{B}, \bm{\pi}) & =\sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \right], \\
& > \sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime) \right] \\
& = C_{k ,\ell}(\mathbf{B}^\prime, \bm{\pi}).
\end{split}
\end{equation}
Let $ \rho_B $ and $ \rho_{B^\prime} $ denote the Chernoff information obtained as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively (with the same block assignment probability vector $ \bm{\pi} $). Then we have
\begin{equation}
\rho_{B} \approx \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}) > \min_{k \neq l} C_{k ,\ell}(\mathbf{B}^\prime, \bm{\pi}) \approx \rho_{B^\prime}.
\end{equation}
Thus we have $ \mathbf{B} \succ \mathbf{B}^\prime = p \mathbf{B} $ for $ p \in (0, 1) $.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:Chernoff-Superiority}.]
By Eq.~\eqref{eq:B0} and Eq.~\eqref{eq:B1}, we have
\begin{equation}
\begin{split}
\mathbf{B}_0 & = \frac{p_0}{p_0+p_1} \mathbf{B}_1, \\
\mathbf{B}_1 & = (p_0 + p_1) \mathbf{B}.
\end{split}
\end{equation}
Recall that $ p_0 \in (0, 1) $ and $ p_1 \in (0, 1-p_0) $. Then by Theorem~\ref{thm:Chernoff-Superiority}, we have $ \mathbf{B} \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\end{proof}
\begin{backmatter}
\section*{Funding
Cong Mu's work is partially supported by the Johns Hopkins Mathematical Institute for Data Science (MINDS) Data Science Fellowship.
\section*{Abbreviations
\textbf{SBM}: Stochastic Blockmodel \\
\textbf{GRDPG}: Generalized Random Dot Product Graph \\
\textbf{ASE}: Adjacency Spectral Embedding \\
\textbf{LSE}: Laplacian Spectral Embedding \\
\textbf{GMM}: Gaussian Mixture Modeling \\
\textbf{BIC}: Bayesian Information Criterion \\
\textbf{ARI}: Adjusted Rand Index \\
\textbf{stderr}: Standard Error \\
\textbf{NDMG}: NeuroData’s Magnetic Resonance Imaging to Graphs
\section*{Availability of data and materials
Social network datasets are available at \href{http://snap.stanford.edu/data/}{http://snap.stanford.edu/data/}.
\section*{Authors' information
\bibliographystyle{bmc-mathphys}
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2,869,038,154,892 | arxiv | \section{Introduction}
The pandemic spreading of the Coronavirus infection 2019 (COVID-19)~\cite{oms,who,hopkins} is forcing billion of people to live in isolation. The related economical degrowth is producing dramatic conditions for workers, trade and industry.
In all Countries the political decisions aim to reduce the spreading and to achieve an almost stable configuration of coexistence with the disease, where a small number of new infected individuals per day is sustainable. In this new condition, the containment effort is usually reduced in favor of a gradual reopening of the social life and of the various economical sectors: the so called phase 2 (Ph2).
In the Ph2 the spread usually restarts and the evaluation of the regrowth of the infection diffusion is a complex problem: microscopic models require a coupled dynamics of the stakeholders, implying a strong model dependence and a large number of free parameters~\cite{napoco1,napoco2,napoco3,napoco4,pluc, epid1}.
For example, the asymptomatic population has been estimated about $\le 50$ \cite{imp}, $\simeq 10$ \cite{istat}, $\simeq 3 -4 $ \cite{lancet, noib} times the symptomatic one and
the simulation in the Italian report on the effects of the reopening on the National Health System is based on a stochastic epidemic model including the age dependence, the demographic structure, the heterogeneity of social contacts in different meeting places (home, school, work, public transportation, cultural activity, shop, bank, post office) and many work sub-sectors (public health, manufacturing, building, trade,~...)~\cite{gov}.
On the other hand, complementary approaches, which outline the Covid-19 evolution in Ph2 in a model independent way, on the basis of macroscopic growth laws (with few parameters) are a useful tool for monitoring the regrowth of the spreading by collecting data after few days from the end of the lockdown or, in general, of the restarting phase.
In this paper we propose a method, based on macroscopic variables~\cite{noi1,noi2,noia,noib} and with no explicit reference to the underlying dynamics, which analyzes the quantitative consequences of the impairment of the constraints.
The starting point is the observation that the Covid-19 spreading, after an initial exponential increase and a subsequent small slowdown (which follows the Gompertz law (GL)~\cite{gompertz} or other non linear trends), reaches a saturation, stable phase, described by the GL or by a logistic equation (LL)~\cite{logistic}, after which the Ph2 starts.
The GL, initially applied to human mortality tables (i.e. aging), also describes tumor growth, kinetics of enzymatic reactions, oxygenation of hemoglobin, intensity of photosynthesis as a function of CO2 concentration, drug dose-response curve, dynamics of growth, (e.g., bacteria, normal
eukaryotic organisms). The LL~\cite{logistic} has been used in population dynamics, in economics, in material science and in many other sectors.
The previous macroscopic growth laws, GL and LL, depend on two parameters, related to the initial exponential trend and to the maximum number of infected individuals, $N_\infty$, called carrying capacity.
It is well known that the carrying capacity changes according to some ``external'' conditions in many biological, economical and social systems~\cite{review2}.
In tumor growth it is related to a multi-stage evolution~\cite{wehldon}. In population dynamics, new technologies affect how resources are consumed, and since the carrying capacity depends on the availability of that resource, its value changes~\cite{pop}.
Therefore a simple method of monitoring the Ph2 is to understand how the carrying capacity (CC) increases due to the reduction of the social isolation and to the restarting of the economical activities. As discussed, this modification is difficult to predict, but different scenarios of regrowth (i.e. with different time dependence of CC in the Ph2, for example) are analyzed in the next sections.
By monitoring the initial data in the new phase one outlines the behavior of the spreading for longer time to evaluate the possible effects of new mobility constraints or new total lockdown.
If the infection regrows exponentially, the (re)lockdown and/or other containment efforts have to be decided as soon as possible. On the other hand, a small change of the specific rate in the Ph2, parameterized by a slight modification of the CC, should require less urgent political choices.
\section{\label{sec:1} Theory and calculations}
\subsection{Macroscopic growth law with time dependent carrying capacity}
The macroscopic growth laws for a population $N(t)$ are solutions of a general differential equation that can be written as
\be
\frac{1}{N(t)} \frac{dN(t)}{dt} = f[N(t)]
\ee
where $f(N)$ is the specific growth rate and its $N$ dependence describes the feedback effects during the time evolution. If $f(N)=$~constant, the growth follows an exponential pattern.
In particular, the Gompertz and the logistic equations are
\be\label{eq:G}
\frac{1}{N(t)}\frac{dN(t)}{dt} = - k_g\;\ln \frac{N(t)}{N_\infty^g} \qquad \text{Gompertz}\;,
\ee
\be\label{eq:L}
\frac{1}{N(t)}\frac{dN(t)}{dt} = k_l \left(1- \frac{N(t)}{N_{\infty}^l}\right)
\qquad
\text{logistic},
\ee
where $k_g \ln(N_\infty^g)$ and $k_l$ are respectively the initial exponential rates and the other terms determine their slowdown. In both cases the steady state condition, $dN/dt =0$ is reached when $N$ is equal to the carrying capacity $N_\infty$.
As shown in refs. \cite{noib,noia} the coronavirus spreading has, in general, three phases: an initial exponential behavior, followed by a Gompertz one and a final logistic phase, due to lockdown.
In many dynamical systems the previous, simple, GL or LL solutions give a good quantitative understanding of the growth. However the CC can be modified by effects not included in eqs.~(\ref{eq:G},~\ref{eq:L}). For example, the invention and diffusion of technologies lift the growth limit.
For Covid-19 infection, in the Ph2 phase there is a fast increase of the human mobility and aggregation which, considering the large number of asymptomatic individuals, modifies the CC.
Moreover, one has to take into account that other pathological features of the virus could be different in the new phase (viral load, external temperature, ...) also.
Therefore one introduces an extension to the widely-used macroscopic model to allow for a time dependent carrying capacity and a change in the parameter which characterize the exponential initial phase, due to
the different infectious features of the Covid-19. In other terms, eqs.~(\ref{eq:G},~\ref{eq:L}) are now
respectively coupled with different values of the constant $k_g$ or $k_l$ and a differential equation for the evolution of the CC, i.e. (g=Gompertz, l=logistic)
\be\label{eq:CC}
\frac{d N_\infty^{g,l}}{dt} = \beta^{g,l}(t)
\ee
where $\beta^{g,l}(t)$ are the rates: $\beta=0$, $\beta=$constant, $\beta \simeq t^n$, $\beta \simeq c\;\exp(b\;t)$ give respectively constant, linear, power law and exponential time dependence of the CC.
\subsection{Covid-19 spreading in phase 2 - formulation}
The application of the previous differential equations to the spreading of Covid-19 in the Ph2 in different Countries requires: a) the time, $t^\star$, of the beginning of the change of the isolation conditions and/or of the restarting phase;
b) a stable phase of the infection diffusion for $t < t^\star$: the effects of the political decision of reducing the constraints start (or should start) when the disease shows a clear slowdown (see below).
Therefore for $t \le t^\star$ the total number of infected individuals is described by eqs.~(\ref{eq:G},\ref{eq:L}) with constant $N_\infty^{g,l}$ fitted by the available data, and for $t \ge t^\star$ one has to solve the system of coupled differential equations~(\ref{eq:G}-\ref{eq:CC}) where the CC is a function of time, with the initial condition that $N_\infty^{g,l}(t^\star)= N_\infty^{g,l}$.
A simple example is useful to outline the strategy. If a time $t^*$ the spread is stable, then $N(t^*) \simeq N_{\infty}^{(g,l)}$ and the specific rate is very small. Let us assume tha for $t>t^*$ there is a fast
rate of the spreading which follows the GL in the new phase with a new CC, i.e.
\be
\frac{1}{N(t)}\frac{dN(t)}{dt} = - k_g\;\ln \frac{N(t)}{N_\infty^{(g2)}} \phantom{....} for \phantom{....} t > t^*\;,
\ee
where $N_\infty^{(g2)} > N_\infty^g$ is the carrying capacity in the new phase and $N(t^*)$ is the initial value of the regrowth. If $N_\infty^{(g2)}= \gamma N_\infty^g$, with constant $\gamma$, the Gompertz equation for $t>t^*$ is given by
\be
\frac{1}{N(t)}\frac{dN(t)}{dt} = - k_g\;\ln \frac{N(t)}{N_\infty^{(g2)}}= - k_g\;\ln \frac{N(t)}{\gamma N_\infty^g}
\ee
that is
\be
\frac{1}{N(t)}\frac{dN(t)}{dt} = + k_g\;\ln \gamma - k_g\;\ln \frac{N(t)}{\gamma N_\infty^g}
\ee
and if $\ln \gamma >> \ln [N(t^*)/N_\infty^g]$ a new exponential phase of the spreading starts for $t > t^*$.
The condition $t>t^\star$ has to be better clarified. The instantaneous change of the CC is unphysical since there is a time interval to observe a possible increase of the spreading due to the Covid-19 incubation time, $\Delta$. Therefore in the time interval $t^\star < t < t^\star + \Delta $ the growth behavior still follows the initial phase, with a fixed CC.
The study of the incubation time is crucial to define the delay (after $t^\star$) of a possible regrowth. This aspect is discussed in the next section
and to clarify the proposed method let us assume that a logistic trend up to $t^\star=60$ days, with a CC, $N_\infty^l= 2883$, is modified at the day $t^\star=60 + \Delta$, with $\Delta=5$, by an increase of the CC by a constant factor ($1.02$, $1.1$, $1.20$). Fig.~\ref{fig:4} shows the cumulative number of detected infected individuals.
The previous examples are for illustrative purposes and in the next sections we apply the proposed approach to Singapore, France, Spain and Italy, including the effects of the delay $\Delta$.
\begin{figure}
\includegraphics[width=\columnwidth]{logisticmap}
\caption{Variation of a logistic growth due to a sudden change in the CC: $N_\infty^{l\,\star} = k\;N_\infty^l$, with $k=1.02$ (orange), $k=1.1$ (red) and $k=1.2$ (purple).}
\label{fig:4}
\end{figure}
\subsection{Covid-19 incubation time}
The definition of the incubation time, or the time from infection to illness onset, is necessary to inform choices of quarantine periods, active monitoring, surveillance, control and modeling. COVID-19 emerged just recently, and the presence of a high rate of asymptomatic individuals, does not currently allow a precise estimation of incubation time.
Different studies, especially at the beginning of the pandemic, tried to define the incubation period, obtaining a mean time varying between 4.0 and 6.4 days~\cite{baker,guan,li}.
This value of incubation time is similar to other Coronaviruses, such as MERS-CoV and SARS-CoV, and generally accepted as a reliable estimate.
However, $95\%$ confidence intervals are large, varying from 2.4 days to 15.5 days~\cite{baker}. This strong variability is related to an uncertainty of the most probable date of exposure and onset of symptoms and this is the main reason why the WHO recommended an isolation time of 14 days after exposure to avoid more spreading of the infection~\cite{lei}.
In our study, knowledge of the incubation time is necessary to model possible consequence of a re-opening. As a matter of fact, reduction of social isolation will increase the CC, and our attention should still be at its highest levels for at least two entire incubation periods, to promptly recognize any warning signal and apply the right control measures.
Therefore the incubation time $\Delta \simeq 8 \pm 6 $ days can be considered and $\Delta=6$ will be used in the next sections. Let us recall that an increase of Covid-19 mortality in Ph2 should be observed after a longer time interval. In Italy, for example, the correlation between the rate of infected people per day and the corresponding mortality rate shows a delay of about 8 days (see figs.~\ref{fig:ConfDay} and \ref{fig:DDay}). Therefore an increase of mortality could be expected after 14-22 days from $t^\star$.
\begin{figure}
\includegraphics[width=\columnwidth]{ConfirmedXday}
\caption{Italy - Confirmed daily rate.}
\label{fig:ConfDay}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{DeathsXday}
\caption{Italy - Mortality daily rate.}
\label{fig:DDay}
\end{figure}
\section{\label{sec:4} Results and Discussion}
The analysis of the regrowth phase has been done with three possible trends:
the new phases is described by a LL/GL with carrying capacity
\begin{itemize}
\item[a)]
$N_\infty^{(2)} = \gamma\; N_\infty^{(1)}$,
\item[b)] $N_\infty^{(2)} = N_\infty^{(1)} + \gamma\;\left(t-t_0\right)$,
\item[c)] $N_\infty^{(2)} = N_\infty^{(1)} \; e^{\gamma\;\left(t-t_0\right)}$.
\end{itemize}
In the next sections, $G^{a,b,c}$ and $L^{a,b,c}$ indicate the fits and the time evolution with the GL and LL, respectively, in the corresponding case $a,b,c$.
\subsection{Singapore: an early case of regrowth}
In Singapore, after reaching a stable phase, a new strong growth of the infection spreading has been observed, due to the immigration of workers from neighboring Countries.
This effect can be described in terms of a modified CC with respect to the saturation phase. The data of $N(t)$ before the restart of the infection can be fitted either by Gompert (red line) or by logistic (orange line), as shown in fig.~\ref{fig:Singapore}. The initial day of the PH2, $t^\star$, corresponds to about $t=32$ and $t^*+\Delta =38$ (March 2).
\begin{figure}
\includegraphics[width=\columnwidth]{Singapore}
\caption{Singapore before the restarting of the infection. GL (red) and Logistic (orange) fit are plotted. Time zero corresponds to the initial day - 23/01.}
\label{fig:Singapore}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{Singapore2}
\caption{Singapore: GL (red) and Logistic (orange) fit with an exponential grow for $N_\infty^{g,l}(t)$ are plotted. Time zero corresponds to the initial day - 23/01.}
\label{fig:Singapore2}
\end{figure}
By applying the method discussed in the previous section, the entire data sample can be fitted by assuming that the CC in the new phase, $N_\infty^{g,l}(t)$,
has an exponential growth, as shown in fig.~\ref{fig:Singapore2}.
The indication coming from the previous analysis is that the increase of the spreading rate, observed immediately after the starting of the new phase,
is so strong to require a sudden (re)lockdown of the Country.
\section{ Phase 2 in Italy: possible scenarios}
In the first phase, the Italian data followed a GL.
Recently, the Ph2 phase started in early September, and
different regrowth scenarios will be outlined by assuming
an increase of the CC.
For the previous trends (a,~b,~c), figures~\ref{fig:ItalyC}, \ref{fig:ItalyCday} and~\ref{fig:ItalyCtot} show respectively the comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from $t^* + \Delta=$ September the 1st to the final day, of the daily number of confirmed infected individuals in the same period, and of the daily number of confirmed infected individuals from the initial day 22/Feb to the final day, respectively.
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyC}
\caption{Italy: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from September the 1st to the final day in figure.}
\label{fig:ItalyC}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyCday}
\caption{Italy: comparison of the growth laws with the data of the daily number of confirmed infected individuals from September the 1st to the final day in figure.}
\label{fig:ItalyCday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyCtot}
\caption{Italy: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the initial day 22/Feb to the final day in figure.}
\label{fig:ItalyCtot}
\end{figure}
Figures~\ref{fig:ItalyD}, \ref{fig:ItalyDday} and~\ref{fig:ItalyDtot} depict the analogous comparisons for the cumulative number of deaths, for the daily number of deaths and for the daily number of deaths from the initial day 22/Feb to the final day, respectively.
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyD}
\caption{Italy: comparison of the growth laws with the data of the cumulative number of deaths from September the 1st to the final day in figure.}
\label{fig:ItalyD}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyDday}
\caption{Italy: comparison of the growth laws with the data of the daily number of deaths from September the 1st to the final day in figure.}
\label{fig:ItalyDday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{ItalyDtot}
\caption{Italy: comparison of the growth laws with the data of the cumulative number of deaths from the initial day 22/Feb to the final day in figure.}
\label{fig:ItalyDtot}
\end{figure}
The effects of the mobility constraints decided by the Italian government can be monitored by looking at the different predicted trends.
\section{\label{sec:FRANCE} France}
France is in a strong spread of the virus, started in August 2020, with an almost total lockdown.
As in the previous case, figures~\ref{fig:FranceC}, \ref{fig:FranceCday} and~\ref{fig:FranceCtot} are devoted to the comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the 15th of August to the final day, of the daily number of confirmed infected individuals in the same period and of the daily number of confirmed infected individuals from the initial day 22/Feb to the final day, respectively.
\begin{figure}
\includegraphics[width=\columnwidth]{FranceC}
\caption{France: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the 15th of August to the final day in figure.}
\label{fig:FranceC}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{FranceCday}
\caption{France: comparison of the growth laws with the data of the daily number of confirmed infected individuals from the 15th of August to the final day in figure.}
\label{fig:FranceCday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{FranceCtot}
\caption{France: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the initial day 22/Feb to the final day in figure.}
\label{fig:FranceCtot}
\end{figure}
In figures~\ref{fig:FranceD}, \ref{fig:FranceDday} and~\ref{fig:FranceDtot} the number of deaths for the same time periods are reported.
\begin{figure}
\includegraphics[width=\columnwidth]{FranceD}
\caption{France: comparison of the growth laws with the data of the cumulative number of deaths from the 15th of August to the final day in figure.}
\label{fig:FranceD}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{FranceDday}
\caption{France: comparison of the growth laws with the data of the daily number of deaths from the 15th of August to the final day in figure.}
\label{fig:FranceDday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{FranceDtot}
\caption{France: comparison of the growth laws with the data of the cumulative number of deaths from the initial day 22/Feb to the final day in figure.}
\label{fig:FranceDtot}
\end{figure}
\section{\label{sec:SPAIN} Spain}
Figures~\ref{fig:SpainC}, \ref{fig:SpainCday} and~\ref{fig:SpainCtot} compare the growth laws with the data of the cumulative number of confirmed infected individuals from the 15th of August to the final day, of the daily number of confirmed infected individuals in the same period, and of the daily number of confirmed infected individuals from the initial day 22/Feb to the final day, respectively.
\begin{figure}
\includegraphics[width=\columnwidth]{SpainC}
\caption{Spain: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the 15th of August to the final day in figure.}
\label{fig:SpainC}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SpainCday}
\caption{Spain: comparison of the growth laws with the data of the daily number of confirmed infected individuals from the 15th of August to the final day in figure.}
\label{fig:SpainCday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SpainCtot}
\caption{Spain: comparison of the growth laws with the data of the cumulative number of confirmed infected individuals from the initial day 22/Feb to the final day in figure.}
\label{fig:SpainCtot}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SpainD}
\caption{Spain: comparison of the growth laws with the data of the cumulative number of deaths from the 15th of August to the final day in figure.}
\label{fig:SpainD}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SpainDday}
\caption{Spain: comparison of the growth laws with the data of the daily number of deaths from the 15th of August to the final day in figure.}
\label{fig:SpainDday}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SpainDtot}
\caption{Spain: comparison of the growth laws with the data of the cumulative number of deaths from the initial day 22/Feb to the final day in figure.}
\label{fig:SpainDtot}
\end{figure}
Figures~\ref{fig:SpainD}, \ref{fig:SpainDday} and~\ref{fig:SpainDtot} report the growth laws results versus data of the cumulative number of deaths from the 15th of August to the final day, of the daily number of deaths in the same period, and of the daily number of deaths from the initial day 22/Feb to the final day, respectively.
\section{\label{sec:cc}Comments and Conclusions}
The purpose of this paper is a demonstration-of-concept: one takes simple growth models, considers the available data and shows different scenarios of the future trend of the spreading.
The method applies a time dependent carrying capacity since the reduction of containment efforts change this crucial parameters of the macroscopic growth laws.
Different time behaviors of the CC outline various trends in the new phase.
Therefore a comparison of data, collected in a short time interval, with the plots obtained by the various ansatz for the CC can help to decide if the social isolation conditions have to be strengthened or weakened. Moreover, a large variation of the CC signals an increase of the pressure on the National Health systems.
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2,869,038,154,893 | arxiv | \section*{Abstract}
We model a social-encounter network where linked nodes match for reproduction in a manner depending probabilistically on each node's attractiveness.
The developed model reveals that increasing either the network's mean degree or the ``choosiness'' exercised during pair-formation increases the strength of positive assortative mating.
That is, we note that attractiveness is correlated among mated nodes.
Their total number also increases with mean degree and selectivity during pair-formation.
By iterating over model mapping of parents onto offspring across generations, we study the evolution of attractiveness.
Selection mediated by exclusion from reproduction increases mean attractiveness, but is rapidly balanced by skew in the offspring distribution of highly attractive mated pairs.
\section*{Introduction}
Most animals assort positively for mating \cite{Jiang_2013}; that is, values of a phenotypic or genotypic trait correlate positively across a population's mated pairs \cite{Alpern_1999,Shine_2001}.
The strength of assortment varies among taxonomic groups and categorical traits.
But phenotypic similarity between paired females and males, often with respect to body size or visual signals, occurs far more often that does negative assortment or random mating \cite{Jiang_2013,Janetos_1980}.
Humans are not an exception \cite{Berscheid_1971}.
Mate choice in humans produces partner similarity with respect to several traits, including age, social attitudes, height and attractiveness \cite{Zietsch_2011}. Our study focuses on attractiveness, which we invoke as a surrogate for any genetically variant, continuous trait correlating positively across pairs.
Assortative mating is sometimes adaptive. Under disruptive selection, individuals may adaptively avoid producing lower-fitness intermediates by assorting positively for reproduction \cite{Coyne_2004}. However, in many cases assortative mating arises as a consequence of some other ecological process. For example, if a phenotypic trait covaries spatially or temporally with habitat in both sexes, the population's spatio-temporal structure can induce assortative mating in the absence of selection on mate choice \cite{Jiang_2013}. Importantly, trait similarity within mating pairs may drive the evolution of that trait, independently of the reason for assortative mating \cite{Lynch_1998,Bolnick_2012}.
Several disciplines, including population genetics and social psychology, have explored relationships involving individual pair-bonding preferences, assortative mating, and the evolutionary consequences of homogamy \cite{Kalick_1986,Kondrashov_1998,McPherson_2001,Bearhop_2005}. Certain models include the realism of stochasticity in encounters between potential mates, and in both pair formation and dissociation \cite{Jia_2015,ZHJWW_2014}. However, most available models for assortative mating assume fully-connected social structure. That is, every female may encounter every male in the same population. Realistically however, any individual has social contact with a limited number of potential mates, which can be characterized by the degree distribution of a bipartite network (females and males) \cite{Corviello_2012,Jia_2014}. The network topology has a profound impact on many properties of social networks.\cite{BA_1999,WS_1998,JK_2013,SSSK_2013,KSSK_2015}
In this case, it can govern many aspects of a system such as the number of mated nodes in a population and the strength of assortative mating across those mated nodes \cite{Jia_2015,ZHJWW_2014}.
Our analysis extends this line of inquiry.
We assume assortative mating with respect to attractiveness, and vary the degree of ``choosiness'' or selectivity exercised during pair-formation.
We show how the strength of assortative mating increases with an encounter network's average degree.
We also find, contrary to intuition, that the number of mated nodes in a large population increases as this population becomes more selective during pairing, provided that the system has sufficient time to complete interactions.
Finally, we assume that attractiveness is a heritable trait, and show how the population-level distribution of attractiveness evolves under assortative mating.
\section*{Methods}
Throughout this work, we assume that each discrete generation has the same size, and that the sex ratio is unity. Selection acts on individuals through access to reproduction. All successfully paired individuals have the same mean number of offspring, independently of their attractiveness; any remaining individuals leave no offspring. This assumption lets us focus on how network topology and attractiveness interactively affect breeding inclusion \emph{vs} exclusion.
\subsection*{The Encounter Network}
We construct a bipartite graph with $2N$ nodes divided equally between subsets $A$ and $B$.
Each node is assigned links according to a degree distribution $P(k)$; a node's links connect it to nodes of the other subset.
The network's average degree is then $\langle k \rangle=\Sigma kP(k)$.
Each node $A_i \in A$ has a weight $a_i \in [0,1]$ which is a continuous random variable that represents the node's attractiveness.
In this work, we initialize these variables as uniform random variables on $[0,1]$
Nodes in subset $B$ are assigned their weights in exactly the same way.
We then link nodes stochastically based on the degree distribution and denote the set of all links as $L$.
The links enable nodes for interaction and eventually pairing.
\subsection*{Pair-formation dynamics}
To begin, consider the pairing dynamics described in \cite{Jia_2014} (which we modify below).
If nodes $A_i$ and $B_j$ are linked, we denote that link as $l_{i,j}=\{A_i,B_j\} \in L$ and has an associated weight defined by its endpoints as
\begin{equation}
w_{i,j}=a_ib_j.
\label{linkstr}
\end{equation}
In the model, all links are initially in the \emph{potential} state.
There are two other states of a link, temporary and permanent.
Only three transitions are possible, from potential to the temporary state and from temporary to either the potential or permanent state; the permanent state being an absorbing state.
The general flow of the pair-formation dynamics goes as follows:
\begin{enumerate} \itemsep 0pt
\item[1. ] A random link $l_{i,j}=\{A_i,B_j\}$ is chosen from set $L$.
\item[2. ] A uniform random number $r \in U(0,1)$ is generated and if $r<(w_{i,j})^{\beta}$ (where the exponent $\beta \geq 0$ controls the strength of selectivity (see next section for details)), the pairing condition is met and one of two transitions occur.
\item[3a.] If $l_{i,j}$ is in the potential state, it transitions to a temporary state and every other link in a temporary state with one of its endpoints being $A_i$ or $B_j$ returns to the potential state.
\item[3b.] If $l_{i,j}$ is in the temporary state, it transitions to the permanent state.
Its endpoints, to which we will refer to as a mated pair, are then placed in the set $M$, and all of their links are removed from the graph.
\item[4. ] The simulation time is increased by $\dfrac{-\ln(q)}{|L|}$ where $q \in U(0,1)$.
\item[5. ] This repeats until subsets $A$ and $B$ are empty or contain only isolated nodes.
\end{enumerate}
Iteration of the process results in $M$ containing all mated pairs with all other nodes discarded.
Given the pair-formation dynamics, an individual must pair before too many of its links are removed from the network, since each such removal decreases the chances to mate.
Note that the pair-formation dynamics implies that the average attractiveness in subsets $A$ and $B$ prior to pair-formation can differ from the average attractiveness among individuals that become mated pairs. Below we refer to this difference as the selection differential. First, we specify how pair-formation might depend on ``choosiness,'' or partner selectivity.
\subsection*{Matching Selectivity}
The criterion for meeting the pairing condition (step 2 of the pair-formation dynamics) has been generalized compared to \cite{Jia_2014} in which $\beta =1$.
As a convenience, we refer to $\beta$ as selectivity; as $\beta$ increases, the randomly selected link is less likely to meet the pairing condition when sampled.
Of course, if $\beta = 0$, every sampled link meets the pairing condition, and $\beta = 1$ corresponds to the original step 2 above.
One can mathematically show that the introduction of $\beta$ is similar to transforming the initial population to a new distribution.
Supplementary Information presents a derivation of the continuous random variable $Y$, with realizations $w_{i,j}^{\beta} = \left( a_i ~b_j\right)^{\beta}$.
Increased selectivity should extend the time elapsing before all nodes in subsets $A$ and $B$ are removed or isolated.
Ecologically, the time available for pair-bonding may be constrained, in which case greater selectivity might exclude more individuals from breeding \cite{Janetos_1980}.
Our pair-formation dynamics does not take into account this constraint, but selectivity still can affect the likelihood a node of given degree forming a pair.
It is worth noting that nodes in $A$ and $B$ are assigned attractiveness and connectivity by the same random process and are drawn from the same distributions.
It follows that sets $A$ and $B$ have statistically equivilent properties of attractiveness and interconnectivity.
\subsection*{Computational Procedures}
To model a network of individuals and their encounter links, we use an Erd\H{o}s-R\'{e}nyi graph \cite{Erdos_1959}.
The degree distribution will approximate a truncated Poisson probability function with mean $\langle k \rangle$.
Hence nodes of high degree occur rarely.
The graph is constructed by selecting two random nodes, one from subset $A$ and the other from subset $B$.
These nodes become linked so long as the selected nodes do not already have a link connecting them.
The created link is then added to $L$.
This process is continued until $|L| = N\langle k \rangle$.
\subsubsection*{Rejection-free simulation}
During pair-formation, changes occur, that is the system's history is updated, only when the pairing condition is met. Advancing the simulation so that we skip the events in which the pairing condition is not met offers a computational advantage, as when $\beta$ increases the frequency of failure to meet the pairing condition increases. Therefore, we model pairing as the sequence of events generated by $|L|$ independent Poisson processes employing a rejection-free scheme \cite{BKL_1975,Gillespie_1976,Gilmer_1976}.
First, we construct the probability distribution for choosing which link will next meet the pairing condition.
The probability that any given link will be chosen in step 1 of the pair formation dynamics is $\dfrac{1}{|L|}$.
It follows that the probability that a given link $l_{i,j}$ will meet the pairing condition stated above is $\dfrac{(w_{i,j})^{\beta}}{|L|}$.
By excluding the outcomes where the pairing condition is not met, the probability distribution for which link will meet the pairing condition is as follows.
\begin{equation}
P_{l_{i,j}}=\dfrac{ w_{i,j}^{\beta}}{\sum\limits_{l_{i',j'} \in L} w_{i',j'}^{\beta}}
\label{gillspie}
\end{equation}
\noindent
This distribution remains static until the network structure changes.
The transition from a potential to temporary state does not impact the transition rates of any link in the system.
Only a transition to a permanent state causes the distribution to be updated.
For a rejection-free scheme, we modify the following in the pair-formation dynamics. In step 1, a link is selected by generating a uniformly distributed random number $r\in U(0,1)$ and mapping it into the inverse of the cumulative distribution function associated with Eq. \ref{gillspie} to identify the selected link $l_{i,j}$, which by definition meets the pairing condition of step 2.
Step 4 is also modified so that the time between events of meeting the pairing conditions is the random variable $\Delta T$ defined by the following equation.
\begin{equation}
\Delta T=\dfrac{-\ln(q)}{\sum\limits_{l_{i',j'} \in L} w_{i',j'}^{\beta}}
\label{gill_time}
\end{equation}
where $q$ is a uniformly distributed random number on $(0,1)$.
Our implementation serves primarily to accelerate simulation. It also helps to explain the limiting case of $\beta \rightarrow \infty$.
Let $w_{max}$ be the maximum link weight in the population and $n_{max}\geq 1$ be the number of links with such weight.
After dividing the numerator and denominator in Eq. \ref{gillspie} by $w^{\beta}_{max}$ in the limit of $\beta \rightarrow \infty$ the denominator $\sum_{l_{i',j'}\in L} (w_{i',j'}/w_{max})^{\beta}$ tends to $n_{max}$ because all terms corresponding to links with less than the maximum weight will be reduced to $0$.
Likewise, the numerator $(w_{i,j}/w_{max})^{\beta}$ will go to either $1$ if $w_{i,j}=w_{max}$ or $0$ otherwise.
Hence, we will have a constant probability of $1/n_{max}$ to select a link with the maximum weight.
This choice becomes deterministic if there is only one such link. Note that when $w_{max}<1$, using this limit causes the time between each pairing conditions to become infinite. In such a case, we cannot use Eq. \ref{gill_time} to analyze the time series unless we normalize the link weights $w_{i,j}^{\beta} \rightarrow \left( \frac{w_{i,j}}{w_{max}} \right)^{\beta}$.
This produces a very trivial time series as the denominator summation in Eq. \ref{gill_time} will tend to $n_{max}$ and become independent of which link is chosen.
\subsection*{Reproduction and Offspring Attractiveness}
Once pair-formation has ended, we use the mated pairs in set $M$ to produce the next generation of nodes.
$A^{(g)}$ and $B^{(g)}$ represent the subsets of nodes in generation $g$, \emph{prior to} that generation's pair-formation.
Initially by definition, we have $A^{(0)}\equiv A$ and $B^{(0)}\equiv B$.
We define $a^{(g)}$ and $b^{(g)}$ as attractiveness values before pair-formation in generation $g$ with their initial values being $a^{(0)}\equiv a$ and $b^{(0)}=b$.
Pair-formation in generation $g$ constructs the set $M^{(g)}$ with its attractiveness-pairs $\{{\hat{a}}^{(g)}, {\hat{b}}^{(g)}\}$.
Thereafter, the mated pairs in set $M^{(g)}$ produce the next generation's $A^{(g+1)}$ and $B^{(g+1)}$.
We assume that any offspring of a given pair in $M^{(g)}$ has attractiveness $x$ sampled from a truncated normal density with moments conditioned on the parents' attractiveness levels.
That is, offspring attractiveness has the conditional probability density:
\begin{equation}
f(x|\mu ,\sigma )=\dfrac{2 e^{-\dfrac{1}{2}\left( \dfrac{x-\mu}{\sigma} \right)^2} }{\sigma \left( \erf{ \left( \dfrac{\mu}{\sigma \sqrt{2}} \right)} + \erf{ \left( \dfrac{1-\mu}{\sigma \sqrt{2}} \right) } \right) } ; x\in [0,1]
\end{equation}
\noindent
Reproduction proceeds for $N$ steps as follows.
\begin{enumerate} \itemsep 0pt
\item Randomly choose a mated pair $\{A_i^{(g)}, B_j^{(g)}\}$ in set $M^{(g)}$.
\item Generate a node in each of the subsets $A^{(g+1)}$ and $B^{(g+1)}$; for each node generated, sample its attractiveness from $f(x|\mu ,\sigma )$ with $\mu =\dfrac{{\hat{a}_i}^{(g)} + {\hat{b}_j}^{(g)}}{2}$ and $\sigma =G\vert \mu - {\hat{a}_i}^{(g)}\vert$ where $G$ is the offspring variance with $G > 0$.
\end{enumerate}
\noindent
After all nodes have been generated, we assign links in a similar manner as their parents (using the same distribution and parameters), but completely independently.
This process also preserves the statistically similar properities that a given generation has in terms of attractiveness and interconnectivity between sets $A^{(g)}$ and $B^{(g)}$.
Several assumptions introduced here require elaboration.
We restrict attractiveness to the unit interval, whereas quantitative genetic models commonly treat them as unbounded phenotypic traits \cite{Slatkin_1976}.
This can be overcome by transforming the attractiveness distribution to one which is boundless.
This transformation must be in line with the probabilities of meeting the pairing condition in the intended system.
Bounds on attractiveness induce an obvious complication under assortative mating.
Given a mated pair $\{{\hat{a}}^{(g)}, {\hat{b}}^{(g)}\}$, both attractiveness values can be close to unity.
The upper bound on $x$ implies that their offspring will likely be less attractive than the midparent $\mu$, since $f(x|\mu ,\sigma )$ has negative skew. (Figure \ref{xpdf}) Similarly, the converse is also true when the mated pair has $\mu$ close to zero. The offspring distribution near the parental-phenotype boundaries does not strongly affect our analysis of attractiveness evolution, since we focus on population change at interior values.
We assume that the mean and the (approximate) variance of the offspring trait distribution depend on the parents' phenotypes.
Most quantitative genetic models assume a reproductive variance independent of the parents' trait values \cite{Slatkin_1976}.
Ultimately, we anticipate that phenotypic variability among a given pair's offspring will increase with the difference between parental trait values.
\subsection*{Selection: Differential and Response}
In our model, the selection differential $S_g$ is the difference between mean attractiveness among individuals of generation $g$ that pair for reproduction and mean attractiveness among members of the same generation at birth.
$S_g \neq 0$ implies that average attractiveness differs between individuals who attract a mate and those that do not.
We have:
\begin{equation}
S_g = \frac{\sum_i {\hat{a}_i}^{(g)} + \sum_j {\hat{b}_j}^{(g)} }{2 \vert M^{(g)}\vert } - \frac{\sum_i {a_i}^{(g)} + \sum_j {b_j}^{(g)}}{2N}
\end{equation}
\noindent
Our model's response to selection $R_g$ is the difference between mean attractiveness among individuals of generation $(g + 1)$ at birth and mean attractiveness among their parents at birth.
We have:
\begin{equation}
R_g = \left[ \sum_i {a_i}^{(g+1)} + \sum_j {b_j}^{(g+1)} - \sum_i {a_i}^{(g)} - \sum_j {b_j}^{(g)} \right]/2N
\end{equation}
\noindent
By definition, the population evolves when $R_g \neq 0$.
\section*{Results}
\subsection*{Assortative Mating: Selectivity and Degree}
First, we consider the distribution of mated pairs $\{\hat{a}, \hat{b}\}$ formed during simulation, as selectivity $\beta$ and average number of links per node $\langle k \rangle$ are varied.
Each simulation included $2N = 2 \times 10^4$ nodes.
Using the same initial conditions (uniformly distributed attractiveness values), we conducted 20 simulations and averaged results.
Figure \ref{gen_1_dists} shows the relative frequencies of attractiveness values for mated pairs.
Fixing $\langle k \rangle$, increasing selectivity promotes the strength of assortative mating.
Fixing $\beta$ while increasing the network's average degree increases the strength of assortment.
For $\beta < 1$, the effect of greater average degree is relatively small, since encounters so readily lead to pairing.
Indeed, the combination of low $\beta$ and small $\langle k \rangle$ (upper left panel) generates mated pairs with attractiveness close to uniformly distributed over all levels.
High selectivity and large degree (lower right panel) nearly eliminate bonding of mated pairs with a large difference in attractiveness, and matching is highly assortative.
These results on pair-formation accord with intuition, and motivate us to apply the model to the evolution of attractiveness.
Note that in each panel of Figure \ref{gen_1_dists}, the most frequent mated pairs involve two highly attractive individuals.
As described above, these matches tend to form earlier during the pair-formation process; highly attractive individuals are unlikely to be excluded from reproduction.
\subsection*{Number of Mated Pairs Formed}
The total number of mated pairs formed, $\vert M \vert = n$, increases in a decelerating manner as the mean node degree increases (Figure \ref{max_matching}a).
Not surprisingly, increasing the total number of feasible encounters results in fewer individuals excluded from reproduction.
Fixing $\langle k \rangle$, we find that greater selectivity results in an increased number of mated pairs (Figure \ref{max_matching}b).
The effect of selectivity is strongest in networks of low average degree $(\langle k \rangle = 3, 4)$.
Interestingly, the ratio of $n$ to the maximal number of mated pairs that \emph{could} form (in the same network) is minimal where the effect of selectivity on $n$ is maximal (Figure \ref{max_matching}c, \ref{max_matching}d).
That is, where network topology results in the greatest proportional exclusion of individuals from mating, the increase in pairing due to greater selectivity attains a maximum.
Averaged over attractiveness, increased selectivity during pair-formation reduces the likelihood that an individual will be excluded from mating.
To explain this observation, Figure S1 shows that nodes of low degree, averaged over attractiveness, have a greater probability of becoming apart of a mated pair as $\beta \rightarrow \infty$.
This is because increasing selectivity increases the strength of assortative mating, and that mated pairs of mutually high-attractiveness form earliest for any $(\langle k \rangle,~\beta)$-combination.
Figure S1 suggests that greater selectivity causes nodes with low degree, but high attractiveness to have an increased chance of becoming a mated pair, decreasing the number of links removed when this happens.
Fewer nodes of intermediate (or low) attractiveness then need be excluded from mating due to more links being available, therefore $n$ increases.
\subsection*{Time to Match}
In the model, simulations do not have an upper bound on how long individuals are allowed to match.
In general, real life situations have an upper bound on how long interactions are allowed to take place.
Figure \ref{time_b} shows the number of matches as a function of time for various $<k>$ at constant $\beta$.
Figure \ref{time_k} shows similar plots, but for various $\beta$ at constant $<k>$.
We observe that for a given average node degree, the time required to match increases by orders of magnitude as selectivity increases.
This is expected as increased selectivity in general increases the time between successful pairing conditions.
As shown above with sufficient time higher selectivities eventually catch up to lower selectivity in terms of the number of matches.
In addition there is an overhead time before any system can begin producing mated pairs which exists in all realizations.
This is due to the courtship mechanism where all links need to meet the pairing condition at least twice.
Because there is a low chance to randomly select a given link, it on average takes a significant amount of time for the first mated pair to form.
\section*{Attractiveness Evolution}
To address phenotypic evolution, we first fix selectivity $\beta$, and vary the network's mean degree.
Figure S2 shows how the bivariate distribution of mated pairs changes from an initial to post third round of pair-formation, with $\beta = 1$.
The distribution of mated pair attractiveness levels becomes more condensed after each generation in each case.
The dispersion of the distribution looks similar for various $\langle k \rangle$, but is statistically different.
Since assortative mating increases with $\langle k \rangle$, the distribution's small dependence on $\langle k \rangle$ suggests that the distribution of phenotypes at birth rapidly changes from generation to generation.
Figure S2 might suggest bias towards phenotypic values near the center of the distribution.
But the result merely reflects the below average chance that individuals of high attractiveness will be excluded from reproduction (directional selection), and the negative skew of the offspring distribution of high-attractiveness parents - so that their offspring are of intermediate attractiveness.
Fixing $\langle k \rangle$, we show effects of increasing selectivity in Figure \ref{gen_dists_beta}.
Again, the bivariate distribution of mated pairs begins to converge quickly.
For $\beta < 1$, the distribution's radial symmetry (after only three generations) suggests a statistical loss of assortative mating.
However, for $\beta > 1$, the distribution regains positive assortment, and the most common mated pairs are between two highly attractive nodes - a consequence of the underlying evolution of the trait to a higher mean attractiveness.
The mean of the distribution also increases with increasing $\beta$.
We briefly note that increasing $G$, which increases the variance of the offspring-trait distribution for any mated pair $\{\hat(a), \hat(b)\}$, increases the dispersion of the bivariate distribution of mated pairs (Figure S3).
Iterating the selective pair-formation process and subsequent reproduction will drive the population to statistical equilibrium; that is, $R_g \rightarrow 0$ as $g \rightarrow \infty$.
At or near equilibrium, the expected increase in mean attractiveness per generation balances the decrease due to the negative skew in the offspring-production distribution $f(x|\mu ,\sigma )$.
Figure \ref{fit_dists} shows a temporal series of univariate attractiveness distributions for several different selectivity values.
Mean attractiveness increases with $\beta$; recall Figure \ref{gen_dists_beta}.
The variance of the distribution appears independent of selectivity, since dispersion depends strongly on $G$.
To find the long term behavior, we look at 100 generations for various parameter values.
The mean and variance of these distributions can be seen in Figure S5.
Both metrics reach a statistical equilibrium after a sufficient number of generations.
\subsection*{Offspring Number of Matches}
The number of matches is examined as a function of generation.
Figure S4 shows an interesting decline in the number of matches.
Because all other factors known to change the number of matches are constant, this indicates that the number of matches is dependent on the attractiveness distribution of the population.
This can be caused by the input distributions for the matching process approaching a similar value.
As mentioned above, selectivity can also mathematically be thought as modifying the input distribution for the matching process.
For $\beta =0$, the input distribution is a singularity at one, which corresponds to all individuals possessing the same attractiveness.
For later generations, the distributions are narrowing.
This approach to distributions where all nodes possess the same attractiveness is likely causing this decrease as there are too few low degree, high attractiveness nodes to increase the number of matches.
\section*{Discussion}
The first set of results presented here addresses variation in assortative mating with respect to attractiveness.
Increasing either the encounter network's mean degree or pair-formation selectivity increases assortment across mated pairs.
Associated with these intuitive results, we find that the earliest and most frequent mated pairs involve two highly attractive individuals.
Our results indicate that increasing mean degree increases the number of mated pairs formed within a generation.
Counter-intuitively, we find that increased selectivity increase the number of mated pairs.
Hence, fewer individuals are excluded from reproduction when the pair-formation dynamics exhibit greater selectivity.
The latter effect is associated, statistically, with nodes of low degree.
The second set of results investigate evolution of mean attractiveness when mated pairs assort positively.
Selection mediated by exclusion from breeding can increase mean attractiveness.
Reproductive variance led to production of enough moderate phenotypes to balance any selective advantage of high attractiveness.
For $\beta < 1$ (weak selectivity) evolution over several generations reduced assortative mating.
However, for $\beta > 1$ (strong selectivity) the evolving population retained positive assortment by attractiveness.
\section*{Acknowledgments}
This research was supported in part
by the Office of Naval Research Grant No. N00014-15-1-2640, and
the Army Research Laboratory under Cooperative Agreement Number W911NF-09-2-0053 (the ARL Network Science CTA).
T. Jia is also supported by the Natural Science Foundation of China (No. 6160309).
B. K. Szymanski is also supported by the European Commission under the 7th Framework Programme, Agreement Number 316097, and by the Polish National Science Centre, the decision no. DEC-2013/09/B/ST6/02317.
The views and conclusions contained in this document are those of the authors and should not be interpreted as
representing the official policies, either expressed or implied, of the Army Research Laboratory or the US Government.
\section*{Author Contributions}
B.K.S. conceived the research;
S.D., T.J., T.C., G.K., and B.K.S. designed the research;
S.D. and T.J. implemented and performed numerical experiments and simulations;
S.D., T.J., T.C., G.K., and B.K.S. analyzed data and discussed results;
S.D., T.J., T.C., G.K., and B.K.S. wrote, reviewed, and revised the manuscript.
\section*{Additional Information}
Competing financial interests: The authors declare no competing financial interests.
|
2,869,038,154,894 | arxiv | \section{Introduction}
Sergey Fomin and Andrei Zelevinsky have introduced and studied cluster algebras in a series of four
articles~\cite{FZ1,FZ2,BFZ3,FZ4} (one of which is coauthored by Arkady Berenstein) in order
to study Lusztig's canonical basis and total positivity. Cluster algebras are commutative algebras which are
constructed by generators and relations. The generators are called cluster variables and they are grouped
into several overlapping sets, so-called clusters. A combinatorial mutation process relates the clusters and
provides the defining relations of the algebra. The rules for this mutation process are encoded
in a rectangular exchange matrix usually denoted by the symbol $\tilde{B}$.
Surprinsingly, Fomin-Zelevinsky's Laurent phenomenon~\cite[Thm.~3.1]{FZ1} asserts that
every cluster variable can be expressed as a Laurent polynomial in an arbitrarily chosen cluster.
The second main theorem of cluster theory is the classification of cluster algebras with only finitely
many cluster variables by finite type root systems, see Fomin-Zelevinsky~\cite{FZ2}.
The connections of cluster algebras to other areas of mathematics are manifold.
A major contribution is Caldero-Chapoton's map~\cite{CC} which relates cluster algebras and
representation theory of quivers. Another contribution is the construction of cluster algebras
from oriented surfaces which relates cluster algebras and differential geometry,
see Fomin-Shapiro-Thurston~\cite{FST1} and Fomin-Thurston~\cite{FT2}.
Arkady Berenstein and Andrei Zelevinsky~\cite{BZ} have introduced the concept of quantum cluster algebras.
Quantum cluster algebras are $q$-deformations which specialise to the ordinary
cluster algebras in the classical limit $q=1$. Such quantisations play an important role in cluster theory:
on the one hand, quantisations are essential when trying to link cluster algebras to Lusztig's canonical bases,
see for example~\cite{Lu,Le,La1,La2,GLS,HL}.
On the other hand, Goodearl-Yakimov~\cite{GY} use quantisations to approximate cluster algebras
by their upper bounds.
The latter result is particularly important since it enables us to study cluster algebras as unions of
Laurent polynomial rings.
In general, the notion of $q$-deformation turns former commutative structures into
non-commutative ones. In the case of quantum cluster algebras, this yields $q$-commutativity between
variables within the same quantum cluster which is stored in an additional matrix usually denoted by the
symbol $\Lambda$. In order to keep the $q$-commutativity intact under mutation,
Berenstein and Zelevinsky require some compatibility relation between
the matrices $\tilde{B}$ and $\Lambda$.
The very same compatibility condition also parametrises compatible Poisson structures for
cluster algebras, see Gekhtman-Shapiro-Vainshtein~\cite{GSV}.
Unfortunately, not every cluster algebra admits a quantisation,
because not every exchange matrix admits a compatible $\Lambda$.
But in the case where there exists such a quantisation,
Berenstein-Zelevinsky have shown that $\tilde{B}$ is of full rank.
This paper has several aims. Firstly, we reinterpret what it means for quadratic exchange matrices
to be of full rank via Pfaffians and perfect matchings.
Secondly, we show the converse of the above statement
(which was posted as a question during Zelevinsky's lecture at a workshop):
assuming $\tilde{B}$ is of full rank, there always exists a quantisation.
This result we show by using concise linear algebra arguments.
It should be noted that Gekhtman-Shapiro-Vainshtein in~\cite[Thm.~4.5]{GSV} prove a similar statement
in the language of Poisson structures.
Thirdly, when a quantisation exists, it is not necessarily unique.
This ambiguity we make more precise by relating all such quantisations
via matrices we construct from a given $\tilde{B}$ using particular minors.
\section{Berenstein-Zelevinsky's quantum cluster algebras}
\subsection{Notation}
Let $m,n$ be integers with $1 \leq n \leq m$ and
$A$ an $m\times n$ matrix with integer entries.
For $n<m$ we use the notation $[n,m] = \{n+1, \ldots, m\}$ and in the case $n=1$ the
shorthand $[m]=[1, m]$.
For a subset $J \subseteq [m]$ we denote by $A_J$ the submatrix of $A$
with rows indexed by $J$ and all columns. By $q$ we denote throughout the paper a formal
indeterminate.
\subsection{The definition of quantum cluster algebras}
Let $\tilde{B}=(b_{i,j})$ be an $m\times n$ matrix
with integer entries. For further use we write
$\tilde{B}=\left[\begin{smallmatrix}B\\C\\\end{smallmatrix}\right]$
in block form with an $n\times n$ matrix $B$ and an $(m-n)\times n$ matrix $C$.
The matrix $B$ is called the \emph{principal part} of $\tilde{B}$.
We call indices $i\in [n]$ \emph{mutable} and the indices $j\in [n+1,m]$ \emph{frozen}.
We say that the principal part $B$ is \emph{skew-symmetrisable} if there exists a diagonal $n\times n$ matrix
$D=\textrm{diag}(d_1,d_2,\ldots,d_n)$ with positive integer diagonal entries such that the matrix
$DB$ is skew-symmetric, i.\,e. $d_ib_{i,j}=-d_jb_{j,i}$ for all $1\leq i,j\leq n$.
The matrix $D$ is then called a \emph{skew-symmetriser} for $B$ and $\tilde{B}$ is called an
\emph{exchange matrix}. Note that skew-symmetrising from the right yields identical restraints and
$b_{i,j}\neq 0$ if and only if $b_{j,i}\neq 0$.
The skew-symmetriser is essentially unique by the following discussion:
Consider the unoriented simple graph $\Delta(B)$ with
vertex set $\{1,2,\ldots ,n\}$ such that there is an edge between two vertices $i$ and $j$ if and only if $b_{ij}\neq 0$.
We say that the principal part $B$ is \emph{connected} if $\Delta(B)$ is connected.
Note that the connectedness of the principal part is mutation invariant, i.\,e.
if $B$ is connected, then the principal part of $\mu_k(\tilde{B})$ is connected for all $1\leq k\leq n$ as well.
Assume now that $B$ is connected.
Suppose there exist two diagonal $n\times n$-matrices
$D$ and $D'$ with positive integer diagonal entries such that both $DB$ and $D'B$ are skew-symmetric.
Then there exists a rational number $\lambda$ with $D=\lambda D'$,
as for all indices $i,j$ with $b_{ij}\neq 0$ the equality $d_i/d_j=d'_i/d'_j$ holds true.
We refer to the smallest such $D$ as the \emph{fundamental skew-symmetriser}.
If $B$ is not connected, then every skew-symmetriser $D$ is a $\mathbb{N}^{+}$-linear combination
of the fundamental skew-symmetrisers of the connected components of $B$.
This concludes the discussion of the first datum to construct quantum cluster algebras.
The next piece of data is the notion of compatible matrix pairs.
From now on, assume that $\tilde{B}=\left[\begin{smallmatrix}B\\C\\\end{smallmatrix}\right]$
is a not necessarily connected matrix with skew-symmetrisable principal part $B$.
A skew-symmetric $m\times m$ integer matrix $\Lambda = (\lambda_{i,j})$ is called \emph{compatible}
if there exists a diagonal $n\times n$ matrix $D'=\textrm{diag}(d'_1,d'_2,\ldots,d'_n)$ with
positive integers $d'_1,d'_2,\ldots,d'_n$ such that
\begin{equation}\label{eq:comp}
\tilde{B}^T\Lambda =\left[\begin{matrix}D'&0\end{matrix}\right]
\end{equation}
as a $n\times n$ plus $n\times (m-n)$ block matrix.
In this case we call $(\tilde{B}, \Lambda)$ a \emph{compatible} pair.
To any $m\times n$ matrix $\tilde{B}$ there need not exist a compatible $\Lambda$.
As a necessary condition Berenstein-Zelevinsky~\cite[Prop.~3.3]{BZ} note that if a matrix $\tilde{B}$
belongs to a compatible pair $(\tilde{B}, \Lambda)$, then its principal part $B$ is skew-symmetrisable,
$D'$ itself is a skew-symmetriser and $\tilde{B}$ itself is of full rank, i.\,e. $\textrm{rank}(\tilde{B})=n$.
Let us fix a compatible pair $(\tilde{B},\Lambda)$.
We are now ready to complete the necessary data to define quantum cluster algebras.
First of all, let $\{e_i\colon 1\leq 1\leq m\}$ be the standard basis of $\mathbb{Q}^m$.
With respect to this standard basis the skew-symmetric matrix $\Lambda$ defines a
skew-symmetric bilinear form $\beta\colon\mathbb{Q}^m\times \mathbb{Q}^m\to \mathbb{Q}$.
The \emph{based quantum torus} $\mathcal{T}_{\Lambda}$ associated with $\Lambda$
is the $\mathbb{Z}[q^{\pm 1}]$-algebra with $\mathbb{Z}[q^{\pm 1}]$-basis $\{X^a\colon a\in\mathbb{Z}^m\}$
where we define the multiplication of basis elements by the formula
$X^aX^b=q^{\beta(a,b)}X^{a+b}$ for all elements $a,b\in\mathbb{Z}^m$.
It is an associative algebra with unit $1=X^0$ and every basis element $X^a$ has an inverse $(X^a)^{-1}=X^{-a}$.
The based quantum torus is commutative if and only if $\Lambda$ is the zero matrix,
in which case $\mathcal{T}_{\Lambda}$ is a Laurent polynomial algebra.
In general, it is an Ore domain, see~\cite[Appendix]{BZ} for further details.
We embed $\mathcal{T}_{\Lambda}\subseteq \mathcal{F}$ into an ambient skew field.
Although $\mathcal{T}_{\Lambda}$ is not commutative in general, the relation $X^aX^b=q^{2\beta(a,b)}X^bX^a$
holds for all elements $a,b\in\mathbb{Z}^m$.
Because of this relation we say that the basis elements are $q$-\emph{commutative}.
Put $X_i=X^{e_i}$ for all $i\in[m]$. The definition implies $X_iX_j=q^{\lambda_{i,j}}X_jX_i$ for all $i,j\in[m]$.
Then we may write
$\mathcal{T}_{\Lambda}=\mathbb{Z}[q^{\pm 1}][X_1^{\pm 1},X_2^{\pm 1},\ldots,X_m^{\pm 1}]$
and the basis vectors satisfy the relation
\begin{align*}
X^{a}=q^{\sum_{i>j}\lambda_{i,j}a_ia_j}X_1^{a_1}X_2^{a_2}\cdot\ldots\cdot X_m^{a_m}
\end{align*}
for all $a=(a_1,a_2,\ldots,a_m)\in\mathbb{Z}^m$.
We call a sequence of pairwise $q$-commutative and algebraically independent elements
such as $X=(X_1,X_2,\ldots,X_m)$ in $\mathcal{F}$ an \emph{extended quantum cluster},
the elements $X_1,X_2,\ldots,X_n$ of an extended quantum cluster \emph{quantum cluster variables},
the elements $X_{n+1},X_{n+2},\ldots,X_m$ \emph{frozen variables} and
the triple $(\tilde{B},X,\Lambda)$ a \emph{quantum seed}.
Let $k$ be a mutable index.
Define mutation map $\mu_k\colon (\tilde{B},X,\Lambda) \mapsto (\tilde{B}',X',\Lambda')$ as follows:
\begin{enumerate}[($M_1$)]
\item The matrix $\tilde{B}' = \mu_k(\tilde{B})$ the well-known
mutation of skew-symmetrisable matrices.
\item The matrix $\Lambda' = (\lambda_{i,j}')$ is the $m\times m$ matrix with entries
$\lambda_{i,j}'=\lambda_{i,j}$ except for
\[
\begin{aligned}
\lambda_{i,k}' &= -\lambda_{i,k} + \sum_{r\neq k} \lambda_{i,r} \max(0,-b_{r,k}) \text{ for all } i \in [m]\backslash \{k\},\\
\lambda_{k,j}' &= -\lambda_{k,j} - \sum_{r\neq k} \lambda_{j,r} \max(0,-b_{r,k}) \text{ for all } j \in [m]\backslash \{k\}.
\end{aligned}
\]
\item To obtain the quantum cluster $X'$ we replace the element $X_k$ with the element
\[
X'_k=X^{-e_k+\sum \limits_{i=1}^{m} \max(0,b_{i,k})e_i}+X^{-e_k+\sum \limits_{i=1}^{m} \max(0,-b_{i,k})e_i}\in\mathcal{F}.
\]
\end{enumerate}
The variables $X'=(X_1',X_2',\ldots,X_m')$ are pairwise $q$-commutative: for all $j\in[m]$ with $j\neq k$ the integers
\begin{align*}
&\beta\left(-e_k+\sum_{i=1}^m\max(0,b_{i,k})e_i,e_j\right)=-\lambda_{k,j}+\sum_{i=1}^m\max(0,b_{i,k})\lambda_{i,j}\\
&\beta\left(-e_k+\sum_{i=1}^m\max(0,-b_{i,k})e_i,e_j\right)=-\lambda_{k,j}+\sum_{i=1}^m\max(0,-b_{i,k})\lambda_{i,j}
\end{align*}
are equal, because their difference is equal to the sum $\sum_{i=1}^m b_{i,k}\lambda_{i,j}$
which is the zero entry indexed by $(k,j)$ in the matrix $\tilde{B}^T\Lambda$.
So the variable $X_k'$ $q$-commutes with all $X_j$.
Hence the variables $X'=(X'_1,X'_2,\ldots,X'_m)$ generate again a based quantum torus
whose $q$-commutativity relations are given by the skew-symmetric matrix $\Lambda'$.
Moreover, the pair $(\tilde{B}',\Lambda')$ is compatible by~\cite[Prop. 3.4]{BZ} so that
the matrix $\tilde{B}'$ has a skew-symmetrisable principle part.
We conclude that the mutation $\mu_k(\tilde{B}',X',\Lambda')=(\tilde{B}',X',\Lambda')$ is again an extended quantum seed.
Note that the mutation map is involutive, i.\,e. $(\mu_k\circ\mu_k)(\tilde{B},X,\Lambda)=(\tilde{B},X,\Lambda)$.
Here we see the importance of the compatibility condition.
A main property of classical cluster algebras are the binomial exchange relations.
For the quantised version we require pairwise $q$-commutativity for the quantum cluster variables in a single cluster.
This implies that a monomial $X_1^{a_1}X_2^{a_2}\cdot\ldots\cdot X_m^{a_m}$ with $a\in\mathbb{Z}^m$
remains (up to a power of $q$) a monomial under reordering the quantum cluster variables.
We call two quantum seeds $(\tilde{B},X,\Lambda)$ and $(\tilde{B}',X',\Lambda')$ \emph{mutation equivalent}
if one can relate them by a sequence of mutations.
This defines an equivalence relation on quantum seeds, denoted by $ (\tilde{B},X,\Lambda) \sim (\tilde{B}',X',\Lambda')$.
The \emph{quantum cluster algebra} $\mathcal{A}_q(\tilde{B},X,\Lambda)$ associated to a given quantum seed
$(\tilde{B},X,\Lambda)$ is the $\mathbb{Z}[q^{\pm 1}]$-subalgebra of $\mathcal{F}$ generated by the set
\[
\chi(\tilde{B},X,\Lambda) = \left\{ X_{i}^{\pm 1}\;|\; i \in [n+1,m] \, \right\} \cup
\bigcup_{ (\tilde{B}',X',\Lambda') \sim (\tilde{B},X,\Lambda)} \left \{X_i' \;|\; i\in [n] \,\right \}.
\]
The specialisation at $q=1$ identifies the quantum cluster algebra $\mathcal{A}_q(\tilde{B},X,\Lambda)$ with
the classical cluster algebra $\mathcal{A}(\tilde{B},X)$. Generally, the definitions of classical and
quantum cluster algebras admit additional analogies.
One such analogy is the quantum Laurent phenomenon,
as proven in~\cite[Cor.~5.2]{BZ}: $\mathcal{A}_q(\tilde{B},X,\Lambda) \subseteq \mathcal{T}_\Lambda$.
Remarkably, $\mathcal{A}_q(\tilde{B},X,\Lambda)$ and $\mathcal{A}(\tilde{B},X)$ also possess
the same exchange graph by~\cite[Thm.~6.1]{BZ}.
In particular, quantum cluster algebras of finite type are also classified by Dynkin diagrams.
\section{The quantisation space}
\subsection{Remarks on skew-symmetric matrices of full rank}
How can we decide whether an exchange matrix $\tilde{B}$ has full rank?
Let us consider the case $n=m$.
Multiplication with a skew-symmetriser $D$ does not change the rank,
so without loss of generality we may assume that $\tilde{B}=B$ is skew-symmetric.
In this case, $B=B(Q)$ is the signed adjacency matrix of some quiver $Q$ with $n$ vertices.
First of all, if $n$ is odd, then $B$ can not be of full rank, because $\det(B)=(-1)^n\det(B)$ implies $\det(B)=0$.
Especially, no (coefficient-free) cluster algebra attached to a quiver $Q$ with an odd number of vertices admits a quantisation.
Now suppose that $n=m$ is even.
In this case, a theorem of Cayley \cite{C} asserts that there exists a polynomial $\operatorname{Pf}(B)$
in the entries of $B$ such that $\det(B)=\operatorname{Pf}(B)^2$. The polynomial is called the \emph{Pfaffian}.
For example, if $n=4$, then $\operatorname{Pf}(B)=b_{12}b_{34}-b_{13}b_{24}+b_{14}b_{23}$.
For general $n$ we have
\[
\operatorname{Pf}(B)=\sum \operatorname{sgn}(i_1,\ldots,i_{n/2},j_1,\ldots,j_{n/2}) b_{i_1j_1}b_{i_2j_2}\cdots b_{i_{n/2}j_{n/2}}
\]
where the sum is taken over all $(n-1)(n-3)\cdots1$ possibilities of writing the set $\{1,2,\ldots,n\}$ as
a union $\{i_1,j_2\}\cup\{i_2,j_2\}\cup\ldots \cup\{i_{n/2},j_{n/2}\}$ of $\frac n2$ sets of cardinality $2$ and
$\operatorname{sgn}(i_1,\ldots,i_{n/2},j_1,\ldots,j_{n/2})\in\{\pm 1\}$ is the sign of the permutation
$\sigma\in S_n$ with $\sigma(2k-1)=i_k$ and $\sigma(2k)=j_k$ for all $k\in\{1,2,\ldots,\frac{n}{2}\}$,
cf. Knuth~\cite[Equation (0.1)]{K}. It is easy to see that the sum is well-defined.
Note that in the sum above a summand vanishes unless
$\{i_1,j_2\},\{i_2,j_2\},\ldots,\{i_{n/2},j_{n/2}\}$ is a \emph{perfect matching}
of the underlying undirected graph of $Q$.
For example, let $Q=\overrightarrow{A_n}$ be an orientation of a Dynkin diagram of type $A_n$ with an even number $n$.
Then $Q$ admits exactly one perfect matching $\{1,2\},\{3,4\},\ldots,\{n-1,n\}$. Hence, $\det(B(Q))=\pm 1$ so that $B(Q)$ is regular.
The same is true for all quivers $Q$ of type $\overrightarrow{E_6}$ or $\overrightarrow{E_{8}}$.
On the other hand, there does not exist a perfect matching for a Dynkin diagram of type $D_n$.
Hence, $\det(B(Q))=0$ for all quivers $Q$ of type $D_n$.
To summarize, a (coefficient-free) cluster algebra of finite type has a quantization if and only if
it is of Dynkin type $A_n$ with even $n$ or of type $E_6$ or $E_8$.
(These are precisely the Dynkin diagrams for which the stable category
$\underline{\operatorname{CM}}(R)$ of Cohen-Macaulay modules of the corresonding
hypersurface singularity $R$ of dimension $1$ does not have an indecomposable rigid object,
see Burban-Iyama-Keller-Reiten~\cite[Theorem 1.3]{BIKR}.)
\subsection{Existence of quantisation}\label{subsec:existence}
Suppose that $\operatorname{rank}(\tilde{B})=n$.
In this subsection we prove that the cluster algebra $\mathcal{A}(\tilde{\mathbf{x}},\tilde{B})$ admits a quantisation.
The $n$ column vectors of $\tilde{B}$ are linearly independent elements in $\mathbb{Q}^m$.
We extend them to a basis of $\mathbb{Q}^m$ by adding $(m-n)$ appropriate column vectors.
Hence, there is an invertible $m \times n$ plus $(m-n)\times m$ block matrix
$\left[\begin{matrix}\tilde{B}&\tilde{E}\end{matrix}\right]\in\operatorname{GL}_m(\mathbb{Q})$
which we denote by $M$. We also write $\tilde{E}$ itself in block form as
$\tilde{E}=\left[\begin{smallmatrix}E\\F\\\end{smallmatrix}\right]$
with an $n\times (m-n)$ matrix $E$ and an $(m-n)\times (m-n)$ matrix $F$.
Of course, the choice for the basis completion is not canonical.
In particular, one can choose standard basis vectors for columns of $\tilde{E}$, making it sparse.
After these preparations we are ready to state the theorem about the existence of a quantization.
\begin{Theorem}\label{thm:existence}\label{Quantizations}
Let $D$ be a skew-symmetriser of $B$.
There exists a skew-symmetric $m\times m$-matrix $\Lambda$ with integer coefficients
and a multiple $D'=\lambda D$ with $\lambda\in\mathbb{Q}^{+}$ such that
$\tilde{B}^T\Lambda =\left[\begin{matrix}D'&0\end{matrix}\right]$.
\end{Theorem}
\begin{proof}
Put
\[
\Lambda_0=M^{-T}\left[
\begin{matrix}
DB&DE\\
-E^TD&0\\
\end{matrix}\right]M^{-1}\in\operatorname{Mat}_{m\times m}(\mathbb{Q})
\]
and let $\Lambda$ be the smallest multiple of $\Lambda_0$ which lies in
$\operatorname{Mat}_{m\times m}(\mathbb{Z})$.
The matrix $\Lambda$ is skew-symmetric by construction and the relation $M^TM^{-T}=I_{m,m}$ implies
$\tilde{B}^TM^{-T}=\left[\begin{matrix}I_{n,n}&0_{n,m-n}\end{matrix}\right]$. Thus,
\[
\tilde{B}^T\Lambda_0=\left[\begin{matrix}DB&DE\\\end{matrix}\right]M^{-1}=
D\left[\begin{matrix}B&E\\\end{matrix}\right]M^{-1}=D\left[\begin{matrix}I_{n,n}&0_{n,m-n}\\\end{matrix}\right]=
\left[\begin{matrix}D&0\end{matrix}\right].
\]
Scaling the equation yields $\tilde{B}^T\Lambda=\left[\begin{matrix}D'&0\end{matrix}\right]$
for some multiple $D'$ of $D$.
\end{proof}
Together with Berenstein-Zelevinsky's initial result this means that a cluster algebra
$\mathcal{A}(\tilde{B})$ admits a quantisation if and only if $\tilde{B}$ has full rank.
Since the rank of the exchange matrix is mutation invariant,
one can use any seed to check whether a cluster algebra admits a quantisation.
Zelevinsky~\cite{Z} suggests to reformulate the statement in terms of bilinear forms.
With respect to the standard basis, the matrix $\Lambda$ defines a skew-symmetric bilinear form.
Let us change the basis. The column vectors $b_1,b_2,\ldots,b_n$ of $\tilde{B}$ are linearly independent over $\mathbb{Q}$.
Let $V'=\operatorname{span}_{\mathbb{Q}}(b_1,b_2,\ldots,b_n)$ be the column space of $\tilde{B}$.
The column vectors $\tilde{e}_{n+1},\tilde{e}_{n+2},\ldots,\tilde{e}_m$ of $\tilde{E}$ extend to a basis of $V=\mathbb{Q}^m$.
Let $V''=\operatorname{span}_{\mathbb{Q}}(\tilde{e}_{n+1},\tilde{e}_{n+2},\ldots,\tilde{e}_{m})$.
The compatibility condition $\tilde{B}^T\Lambda =\left[\begin{matrix}D'&0\end{matrix}\right]$ says
that for any given $D'$, the skew-symmetric bilinear form $V\times V\to \mathbb{Q}$ is completely determined on $V'\times V$,
hence also on $V\times V'$.
Such a bilinear form can be chosen freely on $V''\times V''$ giving a $\frac{1}{2}(m-n-1)(m-n-2)$-dimensional solution space.
In particular, the quantisation is essentially unique (i.\,e. unique up to a scalar) when there are only $0$ or $1$ frozen vertices.
\section{A minor generating set}\label{sec:minor}
In the previous section we observed that any full-rank
skew-symmetrisable matrix $\tilde{B}$ admits a quantisation.
In the construction yielding Theorem~\ref{thm:existence},
we chose some $m \times (m-n)$
integer matrix $\tilde{E}$ which completed a basis for
$\mathbb{Q}^m$. This choice we now reformulate by giving
a generating set of integer matrices for the equation
\begin{equation}\label{eq:homog}
\tilde{B}^T\Lambda = \left[\begin{matrix}0 &0\end{matrix}\right].
\end{equation}
As previously remarked, this ambiguity does not occur for
$0$ or $1$ frozen vertices, hence we may start
with the case $m=n+2$ in Section~\ref{subsec:two}.
From this result we construct
such a generating set for arbitrary $m$ with $|m-n|>2$ in the subsequent section.
The construction below holds in more generality than what is naturally required in our setting.
Thus we now consider an arbitrary integer matrix $A$ of dimension $m\times n$ instead of $\tilde{B}$
and obtain the generating set for equation~\eqref{eq:homog} as a consequence.
\subsection{Minor blocks} \label{subsec:two}
In this subsection we assume $m = n+2$.
For distinct $i,j \in [m]$ define a reduced indexing set
$R(i,j)$ as the $n$-element subset of $[m]$ in which $i$ and $j$ do not occur.
To an arbitrary $m \times n$ integer matrix $A = (a_{i,j})$ we associate
the skew-symmetric $m\times m$-integer
matrix $M = M(A) = \left( m_{i,j} \right)$ with entries
\begin{equation} \label{eq:defm}
m_{i,j} = \begin{cases}
(-1)^{i+j} \cdot \det ( A_{R(i,j)} ), & i < j,\\
0, & i = j, \\
(-1)^{i+j+1} \cdot \det ( A_{R(i,j)} ), & j < i.
\end{cases}
\end{equation}
Then we first observe the following property of $M$, which carries some similarity to the well-known Pl\"ucker relations.
\begin{Lemma}\label{prop:sol}
For $A$ an $m \times n$ integer matrix, we obtain
\[
A^T \cdot M = [ 0 \; 0 ].
\]
\end{Lemma}
\begin{proof}
By definition, we have
\[
\left[ A^T \cdot M \right]_{i,j} = \sum_{k=1}^m a_{k,i} m_{k,j} \notag \\
= \sum_{k\in [m]\backslash\{j\}} a_{k,i} m_{k,j}.
\]
Now let $A_j$ be the matrix we obtain from $A$ by removing the $j$-th row and
$A_j^i$ the matrix that results from attaching the $i$-th column of
$A_j$ to itself on the right. Then $\det(A_j^i) = 0$ and we observe that
using the Laplace expansion along the last column, we obtain the right-hand side
of the above equation up to sign change. The claim follows.
\end{proof}
\begin{Example}
Let $\alpha, a,b,c$ and $d$ be positive integers. Then consider the quiver $Q$ given by:
\begin{center}
\begin{tikzpicture}[scale=0.76]
\node[rounded corners, draw] at (0,0) (v1) {$1$};
\node[rounded corners, draw] at (4,0) (v2) {$2$};
\node[rounded corners, fill=black!20] at (0,-4) (v3) {$3$};
\node[rounded corners, fill=black!20] at (4,-4) (v4) {$4$};
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v1) edge node [above] {$\alpha$} (v2);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v3) edge node [left=1pt] {$a$} (v1);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v3) edge node [left=20pt, below=3pt] {$b$} (v2);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v4) edge node [right=20pt, below=6pt] {$c$} (v1);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v4) edge node [right=1pt] {$d$} (v2);
\end{tikzpicture}
\end{center}
Thus the matrices $\tilde{B}$ and $M$ are
\[
\tilde{B} = \begin{bmatrix}
0 & \alpha\\
-\alpha & 0 \\
a & b \\
c & d
\end{bmatrix}, \qquad
M = \begin{bmatrix}
0 & - ad + bc & -\alpha d & \alpha b \\
ad - bc & 0 & \alpha c & -\alpha a\\
\alpha d & -\alpha c & 0 & -\alpha^2 \\
-\alpha b & \alpha a & \alpha^2 & 0
\end{bmatrix},
\]
and we immediately see the result of the previous lemma, namely $\tilde{B}^T \cdot M = [ 0 \; 0].$
\end{Example}
\subsection{Composition of minor blocks} \label{subsec:many}
In this section let $n+2<m$ and as before, let
$A\in \mat{m}{n}{Z}$ be some rectangular integer matrix.
Choose a subset $F\subset [m]$ of cardinality $n$ and
obtain a partition of the indexing set $[m]$ of the rows of $A$ as $[m] = F \sqcup R$.
Note that $|R| = m-n$.
For distinct $i,j \in R$ set the \emph{extended indexing set associated to $i, j$} to be
\[
E(i,j) := F\cup \{ i,j\}.
\]
By Lemma~\ref{prop:sol} (after a reordering of rows) and slightly abusing the notation, there exists an $ (n+2) \times (n+2) $ matrix
$M_{E(i,j)}= \left( m_{r, s} \right) $ such that
\begin{equation}
A_{E(i,j)}^T \cdot M_{E(i,j)} = [0 \; 0 ]. \label{eq:sol2}
\end{equation}
Now let $\mathfrak{M}_{E(i,j)} = \mathfrak{M}_{E(i,j)}(A) = \left( \mathfrak{m}_{r, s} \right) $ be the
\emph{enhanced solution matrix associated to $i, j$},
the $ m \times m $ integer matrix we obtain from $M_{E(i,j)}$ by
filling the entries labeled by $E(i,j) \times E(i,j)$ with $M_{E(i,j)}$
consecutively and setting all other entries to zero.
\begin{Example}
Consider the quiver $Q$ with associated exchange matrix $\tilde{B}$ as below:
\begin{center}
\begin{tikzpicture}[scale=0.76]
\node at (-1,-1) {$Q:$};
\node[rounded corners, draw] at (2,0) (v1) {$1$};
\node[rounded corners, draw] at (6,0) (v2) {$2$};
\node[rounded corners, fill=black!20] at (0,-2) (v3) {$3$};
\node[rounded corners, fill=black!20] at (4,-2) (v5-227) {$4$};
\node[rounded corners, fill=black!20] at (8,-2) (v5-228) {$5$};
\node at (12,-1) {and $\qquad \tilde{B} = \begin{bmatrix} 0 & \alpha \\ -\alpha & 0 \\ a & 0 \\ b & 0 \\ 0 & c \end{bmatrix} $.};
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v1) edge node [below] {$\alpha$} (v2);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v3) edge node [below] {$a$} (v1);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v5-227) edge node [below] {$b$} (v1);
\path[->,thick,shorten <=2pt,shorten >=2pt,>=stealth'] (v5-228) edge node [below] {$c$} (v2);
\end{tikzpicture}
\end{center}
We choose $F=\{1,2\}$, assuming $\alpha \neq 0$ and get the following matrices $M_{E(i,j)}$
and their enhanced solution matrices for distinct $i,j \in \{3,4,5\}$:
\begin{align*}
M_{E(3, 4)} &= \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & \alpha b & -\alpha a \\
0 & -\alpha b & 0 & -\alpha^2 \\
0 & \alpha a & \alpha^2 & 0
\end{bmatrix}, &&\mathfrak{M}_{E(3, 4)} = \left[ \begin{array}{cccc>{\columncolor{black!15}}c}
0 & 0 & 0 & 0 & 0\\
0 & 0 & \alpha b & -\alpha a & 0\\
0 & -\alpha b & 0 & -\alpha^2 & 0\\
0 & \alpha a & \alpha^2 & 0 & 0\\
\rowcolor{black!15}
0 & 0 & 0 & 0 & 0
\end{array} \right], \\
M_{E(3, 5)} &= \begin{bmatrix}
0 & -ac & -\alpha c & 0 \\
ac & 0 & 0 & -\alpha a \\
\alpha c & 0 & 0 & -\alpha^2 \\
0 & \alpha a & \alpha^2 & 0
\end{bmatrix}, && \mathfrak{M}_{E(3, 5)} = \left[ \begin{array}{ccc>{\columncolor{black!15}}cc}
0 & -ac & -\alpha c & 0 & 0 \\
ac & 0 & 0 & 0 & -\alpha a \\
\alpha c & 0 & 0 & 0 & -\alpha^2 \\
\rowcolor{black!15}
0 & 0 & 0 & 0 & 0\\
0 & \alpha a & \alpha^2 & 0 & 0
\end{array} \right], \\
M_{E(4, 5)} &= \begin{bmatrix}
0 & -bc & -\alpha c & 0 \\
bc & 0 & 0 & -\alpha b \\
\alpha c & 0 & 0 & -\alpha^2 \\
0 & \alpha b & \alpha^2 & 0
\end{bmatrix}, && \mathfrak{M}_{E(4, 5)} = \left[ \begin{array}{cc>{\columncolor{black!15}}ccc}
0 & -bc & 0 & -\alpha c & 0 \\
bc & 0 & 0 & 0 & -\alpha b \\
\rowcolor{black!15}
0 & 0 & 0 & 0 & 0\\
\alpha c & 0 & 0 & 0 & -\alpha^2 \\
0 & \alpha b & 0 & \alpha^2 & 0
\end{array} \right].
\end{align*}
Here we highlighted the added $0$-rows/columns in gray.
We observe by considering the lower right $3 \times 3$ matrices of
$\mathfrak{M}_{E(3, 4)}, \mathfrak{M}_{E(3, 5)}, \mathfrak{M}_{E(4, 5)}$ that these matrices
are linearly independent.
This we generalise in the theorem below.
\end{Example}
\begin{Theorem}\label{prop:gensol}
Let $A\in \mathbb{Z}^{m \times n}$ as above.
Then for distinct $i,j \in R$ we have
\[
A^T \cdot \mathfrak{M}_{E(i,j)} = 0.
\]
Furthermore, if $A$ is of full rank and $F$ is chosen such that
the submatrix $A_F$ yields the rank,
then the matrices $\mathfrak{M}_{E(i,j)}$ are linearly independent.
\end{Theorem}
\begin{proof}
By construction, for $s \in R\backslash\{i,j\}$ the $s$-th column of
$ \mathfrak{M}_{E(i,j)}$ contains nothing but zeros. Hence for arbitrary
$r\in [m]$, we have
\begin{equation}
\left[ A^T \cdot \mathfrak{M}_{E(i,j)} \right]_{r,s} = 0. \label{eq:proofconstr}
\end{equation}
Now let $s \in E(i,j)$. Then
\[
\sum_{k=1}^m a_{k,r} \mathfrak{m}_{k,s} = \sum_{k \in E(i,j)} a_{k,r} m_{k,s}=0,
\]
by Lemma~\ref{prop:sol}.
Without loss of generality, assume $i < j$ and
$F=[n]$. Then by assumption on the rank,
$\beta := (-1)^{i+j} \det \left( A_{[n]} \right) \neq 0$
and by construction, $\mathfrak{M}_{E(i,j)}$ is of the form as in Figure~\ref{fig:thm}.
Then $\pm \beta$ is the only entry of the submatrix of $A$ indexed by $[n+1,m] \times [n+1,m]$.
This immediately provides the linear independence.
\end{proof}
\begin{figure}[!ht] \label{fig:thm}
\[
\begin{blockarray}{cc|ccccccc}
& 1\, \cdots \, n& n+1 & \cdots & i & \cdots & j & \cdots & m\\
\begin{block}{c[c|ccccccc]}
1 & & & & & & & & \\
\vdots & \Big{$\ast$} & & & & \Big{$\ast$} & & & \\
n & & & & & & & & \\\cline{1-9}
n+1 & & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\vdots & & \vdots & \ddots& & & & & \vdots \\
i & & 0 & & 0 & &\beta & & 0 \\
\vdots & \Big{$\ast$} & \vdots & & & \ddots & & & \vdots\\
j & & 0 & & -\beta & & 0 & & 0 \\
\vdots & & \vdots & & & & & \ddots & \vdots \\
m & & 0 & \cdots & 0 & \cdots & 0 & \cdots & 0 \\
\end{block}
\end{blockarray}.
\]
\caption{An example of the form of enhanced solution matrices}
\end{figure}
As an immediate consequence we obtain that there are at least
$\binom{m-n}{2}$ many $m\times m $ integer matrices
$M$ satisfying
\[
A^T \cdot M = [ 0 \; 0 ].
\]
Together with the final remark from Subsection~\ref{subsec:existence},
we can thus conclude that the above constructed matrices
form a basis of the homogeneous equation~\eqref{eq:homog}.
\section{Conclusion}
When does a quantisation for a given cluster algebra $\mathcal{A}(\tilde{B})$ exist and how unique is it?
The answer we have seen above: it depends on the rank of $\tilde{B}$ and the number of frozen indices.
If the rank of $\tilde{B}$ is small then no quantisation exists.
On the other hand, if $\tilde{B}$ is of full rank, we distinguish two cases.
If there is no or only one frozen index, the quantisation is essentially unique.
Otherwise, we remarked at the end of Section~\ref{subsec:existence}
that the solution space of matrices satisfying the
compatibility equation~\eqref{eq:comp} to a fixed skew-symmetriser
is a vector space over the rational numbers of dimension
$\left( \begin{smallmatrix} m-n \\ 2 \end{smallmatrix} \right)$.
In particular, this space is not empty and it contains at least one solution $\Lambda_0$.
Since Theorem~\ref{prop:gensol} with $A=\tilde{B}$ together with
an appropriate indexing set $F$ yields a
linearly independent set of $\left( \begin{smallmatrix} m-n \\ 2 \end{smallmatrix} \right)$
solutions to the homogeneous compatibility equation,
all other solutions $\Lambda$ can be constructed as the sum of $\Lambda_0$ and
a linear combination of all $\mathfrak{M}_{E(i,j)}$ for $i,j\in [m]$.
In the special case where the principal part of $\tilde{B}$ is already invertible,
quantisations of full subquivers with all mutable and two frozen vertices
yield a basis of the homogeneous solution space.
To construct quantum seeds, it is necessary to have integer solutions $\Lambda$
for the compatibility equation. Both $\Lambda$ from Theorem~\ref{thm:existence}
and the enhanced solution matrices $\mathfrak{M}_{E(i,j)}$ have integer entries.
However, they do not generate the semigroup of all integer quantisations in general.
What came as a surprise to us is the simple structure of the matrices $\mathfrak{M}_{E(i,j)}$.
Their computation only depends on $(n+2)\times(n+2)$ minors of $\tilde{B}$,
which can be realised with little effort.
The authors used SAGE in their investigations of the problem and
the first author makes a complementary website available at~\cite{G}.
There, one can follow the construction of the matrices above in detail,
compute a general solution to~\eqref{eq:comp} and a generating set of matrices to~\eqref{eq:homog}.
|
2,869,038,154,895 | arxiv | \section{Introduction} Since Maxwell and Boltzmann predictions in 1860-70, it is well-known that molecules of a gas move erratically with a Gaussian velocity distribution, as experimentally verified later \cite{Miller55}. This allows thermodynamic and transport properties of molecular gases to be described. However, these kinetic theory results do not hold if the particle interactions are dissipative or depend on their velocities (e.g., in relativistic plasmas \cite{Landau}). A well-known example of dissipative gas is the granular gas (see \cite{GranularGas,GranularGasD} for a recent collection of papers). Since collisions between granular particles are inelastic, a continuous input of energy (by vibrating a piston or the container) is required to reach a nonequilibrium steady state. In this regime, granular matter sometimes seems to behave like an molecular gas in which particles follow erratic motions, but several experiments have displayed striking different properties: Instability of the homogeneous density state leading to cluster formation \cite{Gollub97,FalconMug,Falcon01}, non-Gaussian nature of the velocity distribution \cite{Rouyer00}, anomalous scaling of the pressure \cite{Falcon01,McNamara03}. These effects have been also numerically simulated and some of them have been theoretically understood \cite{Goldhirsch03}.
In this paper, we report a 3D experiment of a dilute granular medium fluidized by sinusoidal vibrations in a low gravity environment. The motivation for low gravity is to achieve an experimental situation in which inelastic collisions are the only interaction mechanism, and where only one ``input'' variable (the inverse vibration frequency) has the dimension of time \cite{FalconMug}. This eliminates possible resonances between the time of flight of a particle under gravity and the period of vibration. The aim is to observe new phenomena which result from the inelasticity of the collisions, thus absent in molecular gases. We first study the scaling of the collision frequency with a container wall with respect to the vibration velocity, $V$, and the particle number, $N$. We also measure the time lag distribution between successive collisions with the wall and the impulse distribution. We show in particular that two measurements display significant differences from the behaviours observed in molecular gases: the scalings of the collision frequency and of the particle impulse distribution with $N$. The scaling of the granular temperature with $V$ has been extensively investigated \cite{GranularGas}, but there exists only one 2D experiment for the scaling with $N$ \cite{Warr95}. We emphasize that velocity distributions in granular gases have been measured so far only for nearly 2D geometries. In the 3D case, it is much easier to measure the distribution of impact velocities at a boundary as done here. This measurement involves a similar information content and can be easily compared to molecular dynamics simulations \cite{Aumaitre2}.
\begin{figure}[h]
\centerline{
\epsfysize=50mm
\epsffile{fig01.eps} }
\caption{Typical time recording of the force sensor [impulse response $I(t)$] during 10 periods of vibration showing 106 collisions. Inset: zoom of this signal during 1.3 ms showing two detected collision peaks ($\circ$) and the typical damping time of the oscillatory response of the sensor. The parameters of vibration are: $N=12$, $f=40$ Hz, $A=1.96$ mm (not listed in Table\ \ref{tab01}).}
\label{Fig01}
\end{figure}
\section{Experimental setup} A fixed transparent Lexan tube, $D=12.7$ mm in inner diameter and $L=10$ mm in height, is filled with $N$ steel spheres, $d = 2$ mm in diameter. Experiments have been performed with $N=12$, 24, 36 and 48 respectively corresponding to $n=$ 0.3, 0.6, 0.9 and 1.2 particle layers at rest (packing fraction from 0.04 to 0.18). A piezoelectric force sensor (PCB 200B02), 12.7 mm in diameter, is fixed at the top of the cell in order to record the particle collisions with the upper wall. A piston made of duralumin, 12 mm in diameter, is driven sinusoidally at the bottom of the cell by an electromagnetic shaker. The frequency $f$ is in the range 40 to 91 Hz and the maximal displacement amplitude $A$ is varied from 0.4 to 2 mm. The vibration parameters during the time line are listed in Table\ \ref{tab01}. Vibration amplitudes are measured by piezoelectric accelerometers (PCB 356A08) screwed in the shaft in a triaxial way. Typical output sensitivities in the vibration direction and in the perpendicular directions are, respectively, 0.1 and 1 V/$g$, where $g=9.81$ m/s$^2$ is the acceleration of gravity. Typical force sensor characteristics are a 11.4 mV/N output sensitivity, a 70 kHz resonant frequency, and a 10 $\mu$s rise time. Low gravity environment (about $\pm 5\times 10^{-2} g$) is repetitively achieved by flying with the specially modified {\em Airbus A300 Zero-G} aircraft through a series of parabolic trajectories which result in low gravity periods, each of 20 s. An absolute acceleration sensor allows the detection of the low gravity phases and the automatic increment of the vibration parameters after each parabola. The output signals of force, respectively accelerations, are stored on a computer on 16 bits at a 2 MHz sampling rate, respectively, on 12 bits at 10 kHz.
\begin{table}[ht]
\begin{center}
\begin{tabular}{ccccccc}
$N$ & $A$ & $f$ & $V$ & $\Gamma$ & $N_w$ & Symbols \\
& (mm) & (Hz) & (m/s) & & & in Fig.\ \ref{Fig03} \\
\hline
12 &0.92 & 40 & 0.23 & 5.9 & 2591 & $\times$\\
12 &0.65 & 59.7 & 0.24 & 9.3 & 2605 & $\circ$\\
12 &0.88 & 80 & 0.44 & 22.7 & 5756 & $\bullet$\\
12 &0.64 & 90.9 & 0.37 & 21.4 & 4617 & $+$\\
24 &0.96 & 40 & 0.24 & 6.2 & 5097 & $\ast$\\
24 &0.67 & 59.7 & 0.25 & 9.6 & 4078 & $\Diamond$\\
24 &0.89 & 80 & 0.44 & 22.8 & 8362 & $\bigtriangledown$\\
36 &0.44 & 40& 0.11 & 2.8 & 2538 & $\bigtriangleup$\\
36 &0.67 & 59.7& 0.25 & 9.7 & 6496 & $penta.$\\
36 &0.89 & 80& 0.44 & 22.8 & 9744 & $\circ$\\
36 &0.69 & 90.9& 0.39 & 22.9 & 9741 & $\times$\\
48 &0.42 & 40& 0.11 & 2.7 & 2728 & $\times$ \\
48 &0.69 & 59.7& 0.26 & 9.9 & 8650 & $hexa.$\\
48 &0.89 & 80& 0.45 & 22.9& 10906 & $\circ$ \\
48 &0.73 & 90.9& 0.41 & 24.2 & 12512 & $\Box$
\end{tabular}
\end{center}
\caption{Vibration parameters during each parabola. $V=2\pi Af$ and $\Gamma=4\pi^{2}Af^{2}/g$ are respectively the maximal piston velocity and the dimensionless acceleration. The number of collisions $N_w$ on the sensor is detected during $\theta = 16$ s of low gravity to avoid transient states.}
\label{tab01}
\end{table}
\section{Detection of collisions} A typical time recording of the force sensor shows a succession of peaks corresponding to particle collisions, as displayed in Fig.\ \ref{Fig01} for 10 periods of vibration. Bursts of peaks roughly occur in phase with the vibration but the number of peaks in each burst and their amplitude are random (see Fig.\ \ref{Fig01}). A peak corresponds to the collision of a single sphere, which leads to an almost constant impact duration from 5 to 6 $\mu$s for our range of particle velocities $v$ (assumed of the order of $V$). Indeed, the Hertz's law of contact between a sphere of radius $R$ and a plane made of same material, leads to a duration of collision $\tau = Y R / v^{1/5}$, where $Y=6.9\times 10^{-3}$ $($s$/$m$)^{4/5}$ for steel \cite{Falcon98}. The signal recorded by the sensor corresponds to an impulse response, $I(t)$. Each peak due to a collision is thus followed by an oscillatory tail at the sensor resonance frequency (roughly 100 kHz) damped over 500 $\mu$s (see inset of Fig.\ \ref{Fig01}). A thresholding technique is applied to the signal in order to detect the collisions. We have to discard a time interval of 100 $\mu$s around each detected peak in order to avoid counting the first maxima of each oscillatory tail as additional collisions.
Thus an additional weak collision occuring in the oscillatory tail due to the previous one may be missed by our detection process. However, the discarded time interval is small compared to the mean time lag (a few ms) between successive collisions if their statistics is assumed Poissonian (see below). Consequently, the probability of possibly discarded collisions is small.
\section{Collision frequency scaling} The number of collisions $N_w$ with the top wall (i.e., the sensor) is obtained by the previous thresholding technique, for each parameter listed in Table\ \ref{tab01}, during $\theta=16$ s of low gravity to avoid possible transient states. For a fixed number of particles, $N$, Fig.\ \ref{Fig02} shows that the collision frequency, $\nu_w = N_w/\theta$, is proportional to the maximal piston velocity, $V$, for $0.1 \le V \le 0.5$ m/s. As also shown in Fig.\ \ref{Fig02}, $\nu_w \propto VN^{\alpha},$ with $\alpha = 0.6 \pm 0.1$ for our range of $N$. This result strongly differs from the kinetic theory of molecular gases for which $\nu_w$ varies linearly with $N$.
It cannot be explained either in the very dilute limit (Knudsen regime). Indeed, assuming that each particle mostly collides with the boundaries of the container and does not interact with others, leads to $\nu_w \propto VN / [2(L-d)]$.
Therefore particles do interact significantly with each other through inelastic collisions. We will show below that this anomalous scaling is a consequence of the dissipative nature of collisions. Note also that this scaling law with $N$ has been recently recovered in 2D numerical simulations \cite{Aumaitre2}. It thus appears to be a robust and generic behaviour of granular gases as we will explain below.
\begin{figure}[h]
\centerline{
\epsfysize=60mm
\epsffile{fig02.eps}}
\caption{Collision frequency rescaled by the number of particles, $\nu_w / N^{0.6}$, as a function of $V$ for $N=$12 ($\Box$) and ($\blacksquare$); 24($\Diamond$); 36($\bigtriangledown$); 48($\circ$). $\blacksquare$-marks are from a previous set of experiments at fixed $N=12$ for 15 different velocities which are not listed in Table\ \ref{tab01}. Solid line corresponds to the fit $\nu_w / N^{0.6} = V/l_0$ where $l_0\simeq5.9$ mm.}
\label{Fig02}
\end{figure}
\section{Time lag distribution} The probability density functions (PDF) of the time lag $\Delta t$ between two successive collisions with the top wall is displayed in Fig.\ \ref{Fig03} for 4 different values of $N$ and various parameters of vibration. These PDFs are found to decrease exponentially with $\Delta t$ and to scale like $V$ for our range of $V$. This exponential distribution for the time lag statistics is the expected one for Poissonian statistics. As already shown for the data of Fig.\ \ref{Fig02}, these PDFs can be collapsed by the $N^{0.6}$ rescaling. We also observe in Fig.\ \ref{Fig03} that even the largest values of $\Delta t V$ are smaller than $L$. In our range of $N$, the Knudsen number, $K = l / L$, is in the range $0.1 - 1$, where $l$ is the mean free path, $l = \Omega/(N \pi d^2)$. We are thus in a transition regime from a Knudsen regime to a kinetic regime. It corresponds to a crossover between the very dilute regime for which each particle mostly collides with the boundaries ($l$ of order $L$ independent of $N$), to the kinetic regime ($l$ inversely proportional to $N$). Finally, if the amplitude of vibration, $A$, is not negligible with respect to $L$ (i.e., $A/L \ge 0.17$), the time lag distributions are no longer exponential (not shown here).
\begin{figure}[ht]
\centerline{
\epsfysize=78.95mm
\epsffile{fig03.eps} }
\caption{Probability density functions of the time lag $\Delta t$ between two successive collisions rescaled by $V$ ($V = 2\pi Af$), for $N=12$, 24, 36 and 48 particles, and for different vibration parameters (see symbols in Table\ \ref{tab01}).}
\label{Fig03}
\end{figure}
\begin{figure}[hb]
\centerline{
\epsfysize=78.95mm
\epsffile{fig04.eps} }
\caption{Probability density functions of the impulse $I$ of the impacts on the sensor rescaled by $V$, for different vibration parameters. Symbols are the same as Fig.\ \ref{Fig03}.}
\label{Fig04}
\end{figure}
\section{Impulse distribution} The PDFs of the maxima $I$ of the impacts recorded in Fig.\ \ref{Fig01} are displayed in Fig.\ \ref{Fig04} for 4 different values of $N$ and various parameters of vibration. Note that the low impulse events are not resolved because of noise. Indeed, we expect that the PDFs vanish for $I=0$. We first observe that they scale like $V$, for our range of $V$ (see Fig.\ \ref{Fig04}). Second, they display exponential tails with a slope increasing with the particle number $N$.
Third, the PDF for different values of $N$ can be roughly collapsed when $I$ is scaled like $V / N^{\beta}$ with $\beta \approx 0.8 \pm 0.2$. This shows that the mean particle velocity $\overline v$ near the wall scales like,
$\overline v \propto V / N^{\beta}$, which gives for the granular temperature near the wall, $T_w \propto V^2 / N^{2 \beta}$ (see below). Finally, we observe that the shape of the distributions of $I$ also differ from kinetic theory of molecular gas. They display an exponential tail instead of the Gaussian one. Note however, that it has been observed many times that a universal shape of the bulk velocity distribution do not exist for granular gases. In particular, a significant effect of side walls has been reported \cite{Rouyer00}.
\section{Discussion and concluding remarks} The experimental results obtained on the collision frequency and the distribution of impulse at the wall are related and can be used to extract information on various quantities of interest. Indeed, keeping only quantities that depends on $V$ and $N$, we have $\nu_w \propto \rho_w \overline {v} \propto \rho_w \overline {I}$ where $\rho_w$ is the particle density at the wall. Thus, we get $\rho_w \propto N^{\alpha + \beta}$ and $T_w \propto V^2 N^{- 2\beta}$. The density of particles close to the wall opposite to the piston increases faster than $N$ ($\alpha + \beta = 1.4 \pm 0.3$) because the density gradient becomes larger when $N$ increases. Indeed, it is well known that the granular temperature decreases away from the piston because of inelastic collisions. Thus, the density has to increase in order to keep the pressure $P$ constant (in zero gravity environment). We also have from the state equation for a dilute gas, $P \propto \rho_w T_w \propto V^2 N^{\alpha - \beta}$.
The dependence on $N$ of the collision frequency can be understood as follows: we have $\nu_w \propto \rho_w \overline {v}$. In a molecular gas, $\rho_w \propto N$ and $\overline {v}$ is fixed by the thermostat and does not depend on $N$, thus we have $\nu_w \propto N$. In a granular gas, $\overline {v}$ or the total kinetic energy $E$ are determined from the balance between the injected power by the vibrating piston and the dissipated one by inelastic collisions. For a dilute gas with a restitution coefficient $r$ very close to 1 such that the density is roughly homogeneous ($\rho_w \propto N$), $E$ does not depend on $N$, and thus $\overline {v} \propto 1/\sqrt{N}$. Thus, we get $\nu_w \propto \sqrt{N}$ \cite{Aumaitre2}. In the present experiment, both the scaling of the density and the one of the mean velocity differ from this limit case, because gradients of density and granular temperature cannot be neglected. However, the prediction in the limit $r \simeq 1$ gives a good approximation to the observed scaling of the collision frequency. This shows that a granular gas driven by a vibrating piston strongly differs from a molecular gas in contact with a thermostat even in the limit $r \simeq 1$.
\acknowledgments
We thank P.~Chainais and S.~McNamara for discussions. This work has been supported by the European Space Agency and the Centre National d'\'Etudes Spatiales. The flight has been provided by Novespace. {\em Airbus A300 Z\'ero-G} aircraft is a program of CNES and ESA. We gratefully acknowledge the Novespace team for his kind technical assistance.
|
2,869,038,154,896 | arxiv | \chapter{Back Reaction}
\label{ch:BR}
In this section we make some general comments on how one can include the effect of back reaction for an infalling observer during gravitational collapse. In this section we do not completely solve the equations of motion for the included back reaction, we merely set up the situation and make some comments about it.
To incorporate back reaction into gravitational collapse, one must consider the entire Hamiltonian, as in Chapter \ref{ch:entropy}, as well as the interaction Hamiltonian between the domain wall and the induced radiation.
Thus the total Hamiltonian is given by
\begin{align}
H=&H_{Wall}+H_{Rad}+H_{Int}\nonumber\\
=&4\pi \sigma R^2 [ \sqrt{1+R_\tau^2} - 2\pi G\sigma R] +\sum_{modes}\left[\sqrt{1+R_{\tau}^2}\frac{\Pi_b}{2m}+\frac{|R_{\tau}|}{2B}Kb^2\right]+T_{\mu\nu}S^{\mu\nu}
\end{align}
where $H_{Int}=T_{\mu\nu}S^{\mu\nu}$ is the interaction Hamiltonian and $T_{\mu\nu}$ and $S^{\mu\nu}$ are the energy-momentum tensors for the radiation and the domain wall, respectively, which are given by
\begin{align}
T_{\mu\nu}=&\int d^4x\left[\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-\partial_{\mu}\Phi\partial_{\nu}\Phi\right]\nonumber\\
=&2\pi^2\int dt\int drr^2\left[\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-\partial_{\mu}\Phi\partial_{\nu}\Phi\right]
\label{T_rad}
\end{align}
and
\begin{equation}
S^{\mu\nu}\sqrt{-g}=\sigma\int d^3\xi\gamma^{ab}\partial_aX^{\mu}\partial_bX^{\nu}\delta^{(4)}(X^{\sigma}-X^{\sigma}(\xi^a)).
\label{S_shell}
\end{equation}
From the expansion of the scalar field in Eq.(\ref{Phi_Exp}), we can see that the stress-energy tensor for the scalar field takes on the form
\begin{equation}
(T_{\mu\nu})=\left[ \begin{array}{cccc}
T_{00} & T_{01} & 0& 0 \\ T_{10} & T_{11} &0 &0 \\ 0 & 0 & T_{22} & 0 \\ 0 & 0 & 0 & T_{33}
\end{array} \right].
\end{equation}
While from Eq.(\ref{S_shell}) we see that the stress-energy tensor for the domain wall takes the form,
\begin{equation}
(S^{\mu\nu})=\left[ \begin{array}{cccc}
S^{00} & 0 & 0& 0 \\ 0 & S^{11} &0 &0 \\ 0 & 0 & S^{22} & 0 \\ 0 & 0 & 0 & 0
\end{array} \right].
\end{equation}
Hence we see that the interaction Hamiltonian doesn't contain any off-diagonal terms.
\section{Stress-Energy Tensor}
Here we develop the stress-energy tensor for the radiation and the domain wall, respectively. First we will discuss the stress-energy tensor for the induced radiation. Second we will discuss the stress-energy tensor for the domain wall.
\subsection{Radiation Stress-Energy Tensor}
Here we examine the stress-energy tensor for the radiation. From the discussion in Chapter \ref{ch:radiation} we can write the stress-energy tensor as
\begin{align}
T_{\mu\nu}=&4\pi\int d\tau\Big{[}\frac{1}{2}g_{\mu\nu}\Big{(}-\int_0^{R(\tau)} drr^2\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2-\int_{R(\tau)}^{\infty} drr^2\frac{B}{\sqrt{B+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}\nonumber\\
&+\int_0^{R(\tau)} drr^2\sqrt{1+R_{\tau}^2}(\partial_r\Phi)^2+\int_{R(\tau)}^{\infty} drr^2\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{)}\nonumber\\
&-\int_0^{\infty} drr^2\partial_{\mu}\Phi\partial_{\nu}\Phi\Big{]}
\end{align}
Now using the metric we can write the individual terms, which are given as
\begin{align}
T_{00}=&4\pi\int d\tau\Big{[}-\frac{1}{2}\Big{(}-\int_0^{R(\tau)} drr^2\left(1-\frac{R_s}{r}\right)\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2\nonumber\\
&-\int_{R(\tau)}^{\infty} drr^2\frac{B}{\sqrt{B+R_{\tau}^2}}(\partial_{\tau}\Phi)^2+\int_0^{R(\tau)} drr^2\left(1-\frac{R_s}{r}\right)\sqrt{1+R_{\tau}^2}(\partial_r\Phi)^2\nonumber\\
&+\int_{R(\tau)}^{\infty} drr^2\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)^2(\partial_r\Phi)^2\Big{)}-\int_0^{\infty} drr^2(\partial_{\tau}\Phi)^2\Big{]},
\label{T00}
\end{align}
\begin{align}
T_{01}=-4\pi\int d\tau\int drr^2\partial_{\tau}\Phi\partial_r\Phi,
\label{T01}
\end{align}
\begin{equation}
T_{10}=-4\pi\int d\tau\int drr^2\partial_r\Phi\partial_{\tau}\Phi,
\label{T10}
\end{equation}
\begin{align}
T_{11}=&4\pi\int d\tau\Big{[}\frac{1}{2}\Big{(}-\int_0^{R(\tau)} drr^2\frac{1}{\sqrt{1+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}\nonumber\\
&-\int_{R(\tau)}^{\infty} drr^2\frac{B}{\sqrt{B+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{(1-R_s/r)^2}+\int_0^{R(\tau)} drr^2\sqrt{1+R_{\tau}^2}\frac{(\partial_r\Phi)^2}{1-R_s/r}\nonumber\\
&+\int_{R(\tau)}^{\infty} drr^2\frac{\sqrt{B+R_{\tau}^2}}{B}(\partial_r\Phi)^2\Big{)}-\int_0^{\infty} drr^2(\partial_{r}\Phi)^2\Big{]},
\label{T11}
\end{align}
\begin{align}
T_{22}=&4\pi\int d\tau\Big{[}\frac{1}{2}\Big{(}-\int_0^{R(\tau)} drr^4\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2-\int_{R(\tau)}^{\infty} drr^4\frac{B}{\sqrt{B+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}\nonumber\\
&+\int_0^{R(\tau)} drr^4\sqrt{1+R_{\tau}^2}(\partial_r\Phi)^2+\int_{R(\tau)}^{\infty} drr^4\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{)}\Big{]},
\label{T22}
\end{align}
and
\begin{align}
T_{33}=&\frac{8}{3}\pi\int d\tau\Big{[}\frac{1}{2}\Big{(}-\int_0^{R(\tau)} drr^4\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2-\int_{R(\tau)}^{\infty} drr^4\frac{B}{\sqrt{B+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}\nonumber\\
&+\int_0^{R(\tau)} drr^4\sqrt{1+R_{\tau}^2}(\partial_r\Phi)^2+\int_{R(\tau)}^{\infty} drr^4\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{)}\Big{]}.
\label{T33}
\end{align}
Here note that from Eqs.(\ref{T22}) and (\ref{T33}) show that $T_{33}=(2/3)T_{22}$.
For a full analysis of the stress-energy tensor we will look in the near the horizon limit. Ideally we would like to extend this analysis to the near singularity limit as well. However, we are working in Schwarzschild coordinates, which we cannot extend to the near singularity limit due to the fact that the observer is being constantly accelerated (see Chapter \ref{ch:Classical}). We will then exam the behavior of the stress-energy tensor near the horizon, i.e. in the region $R\sim R_s$.
Of interest is the behavior of the stress-energy tensor near the horizon. To investigate the effect of the radiation we will change the limit of integration from $R(\tau)$ to $R_s$, allowing us to find the dominate terms in this regime. From Eq.(\ref{T00}) we see that in this limit and with the expansion in modes we have
\begin{eqnarray*}
T_{00}&=&4\pi\int d\tau\Big{[}-\frac{1}{2}\Big{(}-\int_0^{R_s} drr^2\left(1-\frac{R_s}{r}\right)\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2\\
&&+\int_{R_s}^{\infty} drr^2\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)^2(\partial_r\Phi)^2\Big{)}-\int_0^{\infty} drr^2(\partial_{\tau}\Phi)^2\Big{]}\\
&=&\int d\tau\left[\frac{1}{2}\frac{1}{\sqrt{1+R_{\tau}^2}}\dot{a}_k\tilde{{\bf A}}_{kk'}\dot{a}_{k'}-\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf C}}_{kk'}a_{k'}-\frac{1}{2}\dot{a}_k\tilde{{\bf D}}_{kk'}\dot{a}_{k'}\right],
\end{eqnarray*}
from Eq.(\ref{T01})
\begin{equation*}
T_{01}=-\int d\tau\dot{a}_k\tilde{{\bf E}}_{kk'}a_{k'},
\end{equation*}
from Eq.(\ref{T10})
\begin{equation*}
T_{10}=-\int d\tau a_k\tilde{{\bf E}}^{-1}_{kk'}\dot{a}_{k'},
\end{equation*}
from Eq.(\ref{T11})
\begin{eqnarray*}
T_{11}&=&4\pi\int d\tau\Big{[}\frac{1}{2}\Big{(}-\int_0^{R_s} drr^2\frac{1}{\sqrt{1+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}\\
&&+\int_{R_s}^{\infty} drr^2\frac{\sqrt{B+R_{\tau}^2}}{B}(\partial_r\Phi)^2\Big{)}-\int_0^{\infty} drr^2(\partial_{r}\Phi)^2\Big{]}\\
&=&\int d\tau\left[-\frac{1}{2}\frac{1}{\sqrt{1+R_{\tau}^2}}\dot{a}_k\tilde{{\bf F}}_{kk'}\dot{a}_{k'}+\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf G}}_{kk'}a_{k'}-\frac{1}{2}a_k\tilde{{\bf H}}_{kk'}a_{k'}\right],
\end{eqnarray*}
and from Eq.(\ref{T22}) we have
\begin{eqnarray*}
T_{22}&=&4\pi\int d\tau\left[\frac{1}{2}\Big{(}-\int_0^{R_s} drr^4\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2+\int_{R_s}^{\infty} drr^4\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{)}\right]\\
&=&\int d\tau\left[-\frac{1}{2}\frac{1}{\sqrt{1+R_{\tau}^2}}\dot{a}_k\tilde{{\bf J}}_{kk'}\dot{a}_{k'}+\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf K}}_{kk'}a_{k'}\right],
\end{eqnarray*}
where the matrices are defined by
\begin{align}
\tilde{{\bf A}}_{kk'}=&4\pi\int_0^{R_s}drr^2\left(1-\frac{R_s}{r}\right)f_kf_{k'},\\
\tilde{{\bf C}}_{kk'}=&4\pi\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)^2f'_kf'_{k'},\\
\tilde{{\bf D}}_{kk'}=&8\pi\int_0^{\infty}drr^2f'_kf_{k'},\\
\tilde{{\bf E}}_{kk'}=&4\pi\int_0^{\infty}drr^2f_kf'_{k'},\\
\tilde{{\bf F}}_{kk'}=&4\pi\int_0^{R_s}drr^2\left(1-\frac{R_s}{r}\right)^{-1}f_kf_{k'},\\
\tilde{{\bf G}}_{kk'}=&4\pi\int_{R_s}^{\infty}drr^2f'_kf'_{k'},\\
\tilde{{\bf H}}_{kk'}=&8\pi\int_0^{\infty}drr^2f'_kf'_{k'},\\
\tilde{{\bf J}}_{kk'}=&4\pi\int_0^{R_s}drr^4f_kf_{k'},\\
\tilde{{\bf K}}_{kk'}=&4\pi\int_{R_s}^{\infty}drr^4\left(1-\frac{R_s}{r}\right)f'_kf'_{k'}.
\label{Matrices}
\end{align}
To further investigate the problem, for a moment let us assume that the basis functions are planewaves. This is a valid approximation in the asymptotic regime, however, this will give us some insight into the problem here. For the basis functions as plane waves we have
\begin{equation*}
f_k=e^{ikr},
\end{equation*}
however since we are requiring real basis functions then we will take the real part of this. Therefore we can write
\begin{equation*}
Re\left(\int_0^{\infty}drr^2f_k'f_{k'}\right)=Re\left(\int_0^{\infty}drr^2e^{i(k+k')r}\right).
\end{equation*}
Performing the integral over $r$ we then have
\begin{equation*}
Re\left(\int_0^{\infty}drr^2f_k'f_{k'}\right)=\delta(k+k')
\end{equation*}
which is finite. Hence the other terms in Eqs.(\ref{T00}), (\ref{T11}), (\ref{T22}) and (\ref{T33}) are dominate due to the divergences of these terms. Therefore we can ignore these extra terms, so we then have
\begin{eqnarray*}
T_{00}&=&\int d\tau\left[\frac{1}{2}\sqrt{1+R_{\tau}^2}\dot{a}_k(\tilde{{\bf A}}^{-1})_{kk'}\dot{a}_{k'}-\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf C}}_{kk'}a_{k'}\right],\\
T_{01}&=&-\int d\tau\dot{a}_k\tilde{{\bf E}}_{kk'}a_{k'},\\
T_{10}&=&-\int d\tau a_k\tilde{{\bf E}}^{-1}_{kk'}\dot{a}_{k'},\\
T_{11}&=&\int d\tau\left[-\frac{1}{2}\sqrt{1+R_{\tau}^2}\dot{a}_k(\tilde{{\bf F}}^{-1})_{kk'}\dot{a}_{k'}+\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf G}}_{kk'}a_{k'}\right],\\
T_{22}&=&\int d\tau\left[-\frac{1}{2}\sqrt{1+R_{\tau}^2}\dot{a}_k(\tilde{{\bf J}}^{-1})_{kk'}\dot{a}_{k'}+\frac{1}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}a_k\tilde{{\bf K}}_{kk'}a_{k'}\right],
\end{eqnarray*}
which is of the same form as $H_{rad}$. We can see that the matrices in Eq.(\ref{Matrices}) are just multiples of the matrices in Eq.(\ref{AC}). Therefore we can see that the eigenvalues of Eq.(\ref{Matrices}) are multiples of those of Eq.(\ref{AC}), hence we can simultaneously diagonalize the matrices as we did in Chapter \ref{ch:radiation}. Finally we can write
\begin{align}
T_{00}=&\int d\tau\left[-\frac{1}{2}\frac{\sqrt{1+R_{\tau}^2}}{nm}\frac{\partial^2}{\partial b^2}-\frac{nK}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}b^2\right],\label{T_00n}\\
T_{01}=&-\int d\tau\dot{a}_k\tilde{{\bf E}}_{kk'}a_{k'},\label{T_01n}\\
T_{10}=&-\int d\tau a_k\tilde{{\bf E}}^{-1}_{kk'}\dot{a}_{k'},\label{T_10n}\\
T_{11}=&\int d\tau\left[\frac{1}{2}\frac{n\sqrt{1+R_{\tau}^2}}{m}\frac{\partial^2}{\partial b^2}+\frac{K}{2n}\frac{\sqrt{B+R_{\tau}^2}}{B}b^2\right],\label{T_11n}\\
T_{22}=&\int d\tau\left[\frac{1}{2}\frac{\sqrt{1+R_{\tau}^2}}{m}\frac{\partial^2}{\partial b^2}+\frac{K}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}b^2\right]\label{T_22n},
\end{align}
where as in Chapter \ref{ch:radiation}, $m$ and $K$ are eigenvalues, $b$ are the eigenmodes and $n$ is a constant multiple. Here we note that $T_{00}$, $T_{11}$ and $T_{22}$ have the structure of a Harmonic oscillator.
We now calculate the expectation value of the stress-energy tensor. To do this we consider
\begin{equation}
\langle T_{\mu\nu}\rangle=\langle0|T_{\mu\nu}|0\rangle
\end{equation}
where $|0\rangle$ is the vacuum state. Since the components of the stress-energy tensor have the structure of a harmonic oscillator we take that the vacuum state is the ground state of the harmonic oscillator (see Chapter \ref{ch:radiation}). The ground state of the harmonic oscillator is given by Eq.(\ref{HO_basis}), thus the expectation value is
\begin{equation}
\langle T_{\mu\nu}\rangle=\int db\left(\frac{m\omega_0}{\pi}\right)^{1/2}e^{-m\omega_0b^2/2}T_{\mu\nu}e^{-m\omega_0b^2/2}.
\end{equation}
From the structure of Eqs.(\ref{T_00n})-(\ref{T_22n}), we can see that there is a kinetic term and a potential term which all have the same dependence on the eigenmode $b$. So, using Eq.(\ref{HO_basis}) we can write
\begin{eqnarray*}
\text{Kinetic Term}&=&\int db\left(e^{-m\omega_0b^2/2}\frac{\partial^2}{\partial b^2}e^{-m\omega_0b^2/2}\right)\\
&=&-\frac{\sqrt{m\omega_0\pi}}{4},
\end{eqnarray*}
and
\begin{eqnarray*}
\text{Potential Term}&=&\int db\left(e^{-m\omega_0b^2/2}b^2e^{-m\omega_0b^2/2}\right)\\
&=&\frac{\sqrt{\pi}}{4(m\omega_0)^{3/2}}
\end{eqnarray*}
where we used the fact that there are an infinite number of eigenmodes $b$ (hence the integrals are from zero to infinity). The individual components are then,
\begin{align}
\langle T_{00}\rangle=&\int d\tau\left[\frac{1}{2}\frac{\sqrt{1+R_{\tau}^2}}{nm}\left(\frac{\sqrt{m\omega_0}}{4}\right)-\frac{nK}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}\left(\frac{\sqrt{\pi}}{4(m\omega_0)^{3/2}}\right)\right],\label{T_00nh}\\
\langle T_{11}\rangle=&\int d\tau\left[-\frac{1}{2}\frac{n\sqrt{1+R_{\tau}^2}}{m}\left(\frac{\sqrt{m\omega_0}}{4}\right)-\frac{K}{2n}\frac{\sqrt{B+R_{\tau}^2}}{B}\left(\frac{\sqrt{\pi}}{4(m\omega_0)^{3/2}}\right)\right],\label{T_11nh}\\
\langle T_{22}\rangle=&\int d\tau\left[-\frac{1}{2}\frac{\sqrt{1+R_{\tau}^2}}{m}\left(\frac{\sqrt{m\omega_0}}{4}\right)-\frac{K}{2}\frac{\sqrt{B+R_{\tau}^2}}{B}\left(\frac{\sqrt{\pi}}{4(m\omega_0)^{3/2}}\right)\right]\label{T_22nh}.
\end{align}
Investigating Eqs.(\ref{T_00nh})-(\ref{T_22nh}) we can see that as $R\rightarrow R_s$ the potential term diverges such as in the Hamiltonian of the induced radiation. Therefore we can conclude that the components of the stress-energy tensor are divergent, however, not due to the usual reasons. Typically this divergence of the stress-energy tensor is associated with the divergence in the frequency $\omega$ (see for example Refs.\cite{Davies,Unruh,Birrell}). To get around this divergence, one usually either applies a cut-off for the allowed frequency or applies a renormalization technique that makes the stress-energy tensor finite. Here we can see that this process is not needed since the divergence is not due to the frequency (since we never specify the basis functions), however the divergence is due to the metric itself.
The divergence in Eqs.(\ref{T_00nh})-(\ref{T_22nh}) in the regime $R\sim R_s$ is due to the $B$ term in the potential term. As stated earlier, Chapters \ref{ch:Classical} and \ref{ch:radiation}, this is due to the fact that we are using Schwarzschild coordinates. The Schwarzschild observer is in an accelerated reference frame, which causes the divergence. Therefore, as we saw in Chapter \ref{ch:radiation}, to study the question of backreaction the more appropriate observer to use would be a truly free-falling observer such as an Eddington-Finkelstein observer. However, we will not investigate such an observer here.
\subsection{Stress-Energy of the domain wall}
From Eq.(\ref{S_shell}) we can write the determinant of the induced metric as
\begin{align}
\sqrt{-\gamma}&=R^2\sin\theta\sqrt{B-\frac{\dot{R}^2}{B}}\nonumber\\
&=R^2\sin\theta\sqrt{B-\frac{BR_{\tau}^2}{B+R_{\tau}^2}}
\end{align}
where we used Eq.(\ref{dtdtau}). The stress-energy tensor for the domain can then be written as
\begin{equation}
S^{\mu\nu}=-\sigma\gamma^{ab}
\end{equation}
where $\gamma^{ab}$ is again the induced metric on the surface of the domain wall. This is expected from Eq.(\ref{Stress-Energy}) in the case of the domain wall ($\sigma=\eta$).
\section{Quantum Hamiltonian}
Here we wish to find an appropriate way to take into account the fact that the mass of the domain wall is changing, due to the fact that the radiation is taking mass away from the system. To do this we will follow a technique that was first introduced in Ref.\cite{Vach}.
Eq.(\ref{Mass_tau}) tells us that the mass of the domain wall is approximately the Hamiltonian, therefore we can write the Schwarzschild radius as
\begin{equation}
R_s\rightarrow2GH_{Wall}.
\label{RsH}
\end{equation}
The factor $B$ in the Hamiltonian for the radiation contains the energy of the wall via the Schwarzschild radius. So we then have
\begin{equation}
B=1-\frac{2GH_{Wall}}{R}.
\label{BH}
\end{equation}
In the near horizon limit Eq.(\ref{Mass_tau}) can be written as
\begin{equation}
M = 4\pi \sigma R_s^2 [ \sqrt{1+R_\tau^2} - 2\pi G\sigma R_s] =H_{wall}.
\end{equation}
Assuming that the velocity at the horizon is small and dropping the constant terms we can write this as
\begin{equation}
H_{wall}=2\pi \sigma R_s^2R_{\tau}^2.
\end{equation}
Using Eq.(\ref{RsH}) we can then rewrite this as
\begin{align}
H_{wall}&=\frac{1}{8\pi\sigma G^2R_{\tau}^2}\\
&=\left(\frac{\Pi_R^2}{16\pi\sigma G^2}\right)^{1/3}
\end{align}
where in the second line we used Eq.(\ref{Pi_tau}).
The total Hamiltonian in terms of a single mode then becomes
\begin{align}
H=&\left(\frac{\Pi_R^2}{8\pi\sigma G^2}\right)^{1/3}+\left(1+\left(\frac{1}{1024\pi\sigma G^2\Pi_R}\right)^{2/3}\right)\frac{\Pi^2_b}{2m}\nonumber\\
&+\frac{RKb^2}{2\left((16\pi\sigma G^2)^{1/3}R-2G(\Pi^2_R)^{1/3}\right)(\Pi_R^2)^{2/3}}+T_{\mu\nu}S^{\mu\nu}.
\label{Ham_tot_BR}
\end{align}
Here we note an unusual property of Eq.(\ref{Ham_tot_BR}), the appearance of the fractional derivatives. In general, fractional derivatives are non-local, that is, one cannot say that the fractional derivative at a point $x$ of a function $f$ depends only on the graph of $f$ very near $x$, see for example Ref.\cite{Wiki}. Therefore it is expected that the theory of fractional derivatives involves some sort of boundary conditions, involving information further out. The most general definition of the fractional derivative is
\begin{equation}
_aD^q_t=\begin{cases} \frac{d^q}{dx^q}, & Re(q)>0\\
1, & Re(a)=0\\
\int_a^t(dx)^{-q}, & Re(a)<0. \end{cases}
\end{equation}
Here the first case is defined as
\begin{equation}
\frac{d^q}{dx^q}x^k=\frac{k!}{(k-a)!}x^{k-q}=\frac{\Gamma(k+1)}{\Gamma(k-q+1)}x^{k-q},
\end{equation}
while the third case is defined as
\begin{equation}
\int_a^t(dx)^{|q|}f(x)=\frac{1}{\Gamma(q)}\int_0^x(x-t)^{|q|-1}f(t)dt.
\end{equation}
Hence Eq.(\ref{Ham_tot_BR}) is an differential-integral equation.
The study of the behavior backreaction is therefore very complicated. However, the interesting thing to point out here is that, similarly to the investigation of the quantum mechanical effects studied in Chapter \ref{ch:quantum}, the presence of the non-locality again emerges. However, in Chapter \ref{ch:quantum} the non-locality was only present when investigating the near classical singularity regime. Here, the non-local effect is even present in the near horizon regime.
\section{Discussion}
In this chapter we investigated the stress-energy tensor for the radiation given off during collapse as well as investigated a way to include the loss of mass during this collapse. We found some interesting properties of these two quantities.
First for the stress-energy tensor for the radiation, we found that the expectation value for the stress-energy tensor is in fact infinite. However, this is not due to the usual difficulties. Generally when one investigates the stress-energy tensor, the infinities arise from the basis function. Traditionally one assumes a plane-wave basis function for the radiation, and the divergence is therefore due to the frequency of the basis function. To avoid these infinities, one usually institutes a cut-off frequency. Here, we do not have this problem. This is due to the fact that we never actually specify our basis functions, hence we do not have the problem of infinities in the basis function. The divergence in this case is due to the presence of the $B^{-1}$ in the potential term. As $R\rightarrow R_s$, $B\rightarrow0$ which causes the divergence. As stated in Chapters \ref{ch:Classical} and \ref{ch:radiation} this is due to the fact that we are using Schwarzschild coordinates, where the observer is being accelerated.
To include the loss of mass into the Hamiltonian of the system, we used the technique originally developed in Ref.\cite{Vach}. Here one uses the approximation that the Hamiltonian of the domain wall is approximately the mass of the domain wall. Therefore one can replace the mass in the Schwarzschild radius by the Hamiltonian of the wall. Using this, we can then rewrite the total Hamiltonian as in Eq.(\ref{Ham_tot_BR}). The interesting thing here is that the Hamiltonian is now in terms of fraction, not whole or partial, derivatives. By definition, fractional derivatives are not strictly local quantities and will either give a differential or integral equation depending on the sign of the fractional derivative. As in Chapter \ref{ch:quantum}, we recover the non-locality of the quantum effects during gravitational collapse. Unlike in Chapter \ref{ch:quantum}, these effects are now manifest even near the horizon.
\chapter{Classical Treatment}
\label{ch:Classical}
In this chapter we wish to study the classical equations of motion of the collapsing spherically symmetric domain wall. To do so, we will consider the cases for two different foliations of space-time. Here we consider the collapse of the spherically symmetric domain wall from the point of view of an asymptotic observer (one who is at rest with respect to the collapse) and from the point of view of an infalling observer (one who is riding along with the shell). This will be done by considering Eq.(\ref{Mass_T}) and Eq.(\ref{Mass_tau}).
A naive approach to obtaining the dynamics for the spherical domain wall is to insert Eq.(\ref{Met_out}) and Eq.(\ref{Met_in}), as well as Eq.(\ref{Mass_tau}) into the original action Eq.(\ref{gen_action}). Upon doing so it is known that this approach does not give the correct dynamics for gravitating systems. Therefore we will take an alternative approach for finding the action. We will find the action that does in fact lead to the correct mass conservation law. The form of the action can be deduced from Eq.(\ref{Mass_T}) (Eq.(\ref{Mass_tau})).
\section{Asymptotic Observer}
First we will consider the equations of motion from the view point of the asymptotic observer. The asymptotic observer is any observer stationary with respect to the collapsing domain wall, typically taken to be located at infinity. Here we summarize the work originally done in Ref.\cite{Stojkovic}.
From Eq.(\ref{Mass_T}) we find the effective action for the spherically symmetric domain wall to be
\begin{equation}
S_{eff}=-4\pi\sigma\int dTR^2\left[\sqrt{1-R_T^2}-2\pi\sigma GR\right].
\label{Seff_T}
\end{equation}
Using Eq.(\ref{dTdt}) we can write Eq.(\ref{Seff_T}) in terms of the asymptotic observer time $t$ as
\begin{equation}
S_{eff}=-4\pi\sigma\int dtR^2\left[\sqrt{B-\frac{\dot{R}^2}{B}}-2\pi\sigma GR\sqrt{B-\frac{(1-B)}{B}\dot{R}^2}\right].
\label{Seff_t}
\end{equation}
From Eq.(\ref{Seff_t}) the effective Lagrangian for the system is
\begin{equation}
L_{eff}=-4\pi\sigma R^2\left[\sqrt{B-\frac{\dot{R}^2}{B}}-2\pi\sigma GR\sqrt{B-\frac{(1-B)}{B}\dot{R}^2}\right].
\label{Leff_t}
\end{equation}
The generalized momentum $\Pi_R$ can be derived from Eq.(\ref{Leff_t}) in the usual manner, this is given by
\begin{equation}
\Pi_R=\frac{4\pi\sigma R^2\dot{R}}{\sqrt{B}}\left[\frac{1}{\sqrt{B^2-\dot{R}^2}}-\frac{2\pi\sigma GR(1-B)}{\sqrt{B^2-(1-B)\dot{R}^2}}\right].
\label{Pi_R}
\end{equation}
Therefore from the Lagrangian, Eq.(\ref{Leff_t}), and the generalized momentum, Eq.(\ref{Pi_R}), the Hamiltonian can then be written as
\begin{equation}
H=4\pi\sigma B^{3/2}R^2\left[\frac{1}{\sqrt{B^2-\dot{R}^2}}-\frac{2\pi\sigma GR(1-B)}{\sqrt{B^2-(1-B)\dot{R}^2}}\right].
\label{Ham_Rdot}
\end{equation}
For later convenience, we wish to find the Hamiltonian as a function of $(R,\Pi_R)$. To do so, we need to eliminate $\dot{R}$ in favor of $\Pi_R$ using Eq.(\ref{Pi_R}). This can be done, in principle, but is very messy (the solution will involve solutions of a quartic polynomial). However, we will be interested in what is happening as the shell approaches the horizon, i.e. when $R$ is close to $R_s$ and hence $B\rightarrow0$, since this is the most interesting region of study. In this limit one can see that the denominators of the two terms in Eq.(\ref{Pi_R}) are equal. So we can rewrite Eq.(\ref{Pi_R}) as
\begin{equation}
\Pi_R\approx\frac{4\pi\mu R^2\dot{R}}{\sqrt{B}\sqrt{B^2-\dot{R}^2}}
\label{P_R Hor}
\end{equation}
where
\begin{equation}
\mu\equiv\sigma(1-2\pi\sigma GR_s).
\end{equation}
In the region $R\sim R_s$ the Hamiltonian, Eq.(\ref{Ham_Rdot}), is then approximately given by
\begin{eqnarray}
H&\approx&\frac{4\pi\mu B^{3/2}R^2}{\sqrt{B^2-\dot{R}^2}}\label{Ham Hor}\\
&=&\sqrt{(B\Pi_R)^2+B(4\pi\mu R^2)^2}.
\label{Ham Hor Pi_R}
\end{eqnarray}
Here we note that the Hamiltonian, written in the form of Eq.(\ref{Ham Hor Pi_R}), has the form of the energy of a relativistic particle with a position dependent mass.
Since the mass is a constant of motion, the Hamiltonian is a conserved quantity, so from Eq.(\ref{Ham Hor}) we can write
\begin{equation}
h=\frac{B^{3/2}R^2}{\sqrt{B^2-\dot{R}^2}}
\label{h}
\end{equation}
where $h\equiv H/4\pi\mu$ is a constant.
Solving Eq.(\ref{h}) for $\dot{R}$ we obtain
\begin{equation}
\dot{R}=\pm B\sqrt{1-\frac{BR^4}{h^2}}.
\label{dotRgen}
\end{equation}
In the region $R\sim R_s$, this takes the form
\begin{equation}
\dot{R}\approx\pm B\left(1-\frac{1}{2}\frac{BR^4}{h^2}\right).
\end{equation}
Since in the region $R\sim R_s$, $B\rightarrow0$, the dynamics for the collapsing spherically symmetric domain wall in this region can be obtained by solving the expression
\begin{equation}
\dot{R}=\pm B.
\label{Rdot}
\end{equation}
To leading order in $R-R_s$, the solution is
\begin{equation}
R(t)\approx R_s+(R_0-R_s)e^{\pm t/R_s}
\label{R(t)}
\end{equation}
where $R_0$ is the radius of the shell at $t=0$. Since we are interested in the collapsing shell, we take the negative sign in the exponential term, Eq.(\ref{R(t)}).
Here we wish to make some comments on Eq.(\ref{R(t)}). By virtue of the negative sign in the exponential, this then implies that, from the classical point of view, the asymptotic observer never sees the formation of the horizon of the black hole, since Eq.(\ref{R(t)}) equals $R_s$ only as $t\rightarrow\infty$. This is in agreement with the fact that it takes an infinite amount of time for a photon to reach the horizon of a pre-existing black hole, as seen by an asymptotic observer (see for example Ref.\cite{Wheeler}). Therefore, Eq.(\ref{R(t)}) makes sense from this point of view.
In Figure \ref{Rt} we plot the position of the domain wall for the asymptotic observer. Here we see the asymptotic behavior of the time dependence of the position of the domain wall. As shown in Eq.(\ref{R(t)}), the domain wall asymptotes to the Schwarzschild radius, taking an infinite amount of time for the domain wall to reach the Schwarzschild radius.
\begin{figure}[htbp]
\includegraphics{Rt.eps}
\caption{Here we plot the the solution in Eq.(\ref{R(t)}).}
\label{Rt}
\end{figure}
Figure \ref{R_dot} shows the corresponding velocity of the domain wall as seen by the asymptotic observer. Here we see that the velocity of the domain wall asymptotes to zero as the domain wall collapses toward the Schwarzschild radius, as given in Eq.(\ref{Rdot}).
\begin{figure}[htbp]
\includegraphics{R_dot.eps}
\caption{Here we plot the the solution in Eq.(\ref{Rdot}).}
\label{R_dot}
\end{figure}
\section{Infalling Observer}
Now we turn our attention to the infalling observer case, where the conserved mass is given by Eq.(\ref{Mass_tau}). Here we point out the misnomer in the name infalling. The infalling observe here is not to be confused with the traditional view point of an infalling observer, one who is traveling along a geodesic, or a freely falling observer. The observer in this case is infalling from the fact that the observer is attached to the domain wall and is infalling with the wall. Therefore, eventhough the observer is in a locally Minkowski reference frame, the overall reference frame is still Schwarzschild, since at any point in time the observer is in a Schwarzschild reference frame. Here we summarize the work originally done in Ref.\cite{GreenStoj}.
The effective action consistent with Eq.(\ref{Mass_tau}) is
\begin{equation}
S_{eff}=-4\pi\sigma\int d\tau R^2\left[\sqrt{1+R_{\tau}^2}-R_{\tau}\sinh^{-1}(R_{\tau})-2\pi\sigma GR\right].
\label{Seff_tau}
\end{equation}
Therefore the effective Lagrangian expressed in terms of the infalling observer's time $\tau$ is given by
\begin{equation}
L_{eff}=-4\pi\sigma R^2\left[\sqrt{1+R_{\tau}^2}-R_{\tau}\sinh^{-1}(R_{\tau})-2\pi\sigma GR\right].
\label{Leff_tau}
\end{equation}
From Eq.(\ref{Leff_tau}) the generalized momentum $\tilde{\Pi}_R$ is derived to be
\begin{equation}
\tilde{\Pi}_R=4\pi\sigma R^2\sinh^{-1}(R_{\tau}).
\label{Pi_tau}
\end{equation}
From Eq.(\ref{Leff_tau}) and Eq.(\ref{Pi_tau}), the Hamiltonian in terms of $R_{\tau}$ is given by
\begin{equation}
H=4\pi\sigma R^2\left[\sqrt{1+R_{\tau}^2}-2\pi\sigma GR\right]
\label{Ham_Rtau}
\end{equation}
which is just Eq.(\ref{Mass_tau}) as expected.
From Eq.(\ref{Ham_Rtau}) we can calculate $R_{\tau}$
\begin{equation}
R_{\tau}=\pm\sqrt{\left(\frac{\tilde{h}}{R^2}+2\pi\sigma GR\right)^2-1}
\label{dRdtau}
\end{equation}
where $\tilde{h}\equiv H/4\pi\sigma$. In general, Eq.(\ref{dRdtau}) cannot be solved analytically, at least not in very nice way. However, we can take some special cases to investigate the behavior of the solution to find the time dependence.
As a first case we consider the zeroth order behavior near the horizon, i.e. $R\sim R_s$. In this region we note that we can write Eq.(\ref{dRdtau}) as
\begin{equation}
R_{\tau}=\pm\sqrt{\left(\frac{\tilde{h}}{R_s^2}+2\pi\sigma GR_s\right)^2-1}
\label{dRdtau0}
\end{equation}
hence $R_{\tau}$ is constant. Integrating Eq.(\ref{dRdtau0}) gives
\begin{equation}
R(\tau)=\tilde{R}_0-\tau\sqrt{\left(\frac{\tilde{h}}{R_s^2}+2\pi\sigma GR_s\right)^2-1}
\label{R_tau_0}
\end{equation}
where $\tilde{R}_0$ is the radius of the shell at $\tau=0$.
As a second case we consider the case that $\tilde{h}/R^2>>2\pi\sigma GR>>1$. Therefore we can write Eq.(\ref{dRdtau}) as
\begin{equation}
R_{\tau}=-\frac{\tilde{h}}{R^2}.
\label{dRdtauL}
\end{equation}
Integrating Eq.(\ref{dRdtauL}) we then have the solution
\begin{equation}
R(\tau)=\left(\tilde{R}_0^3-3\tilde{h}\tau\right)^{1/3}.
\label{RtauL}
\end{equation}
Eq.(\ref{RtauL}) then gives that the time for an infalling observer to reach $R_s$ is
\begin{equation}
\tau=\frac{R_0^3-R_s^3}{3\tilde{h}}.
\end{equation}
Here we make some comments on Eq.(\ref{R_tau_0}) and Eq.(\ref{RtauL}). These solutions imply that the infalling observer will reach $R_s$ in a finite amount of his/her proper time. This result is expected from classical general relativity, since the observer is in a locally flat Minkowski reference frame. Therefore, there is no difficulty for the observer once he/she reaches the horizon, the horizon is just another locally flat point in space according to this observer.
\begin{figure}[htbp]
\includegraphics{RtauFull.eps}
\caption{Here we plot the numerical solution to Eq.(\ref{dRdtau}).}
\label{RtauFull}
\end{figure}
For consistency, in Figure \ref{RtauFull} we plot the numerical solution of Eq.(\ref{dRdtau}) for the parameters $\tilde{h}=1/2$ and $\sigma=0.1R_s$. Figure \ref{RtauFull} shows that the observer does in fact reach $R_s$ in a finite amount of his proper time. Figure \ref{RtauBoth} compares the special cases discussed above with that of the numerical solution. Here the blue curve is the full solution of Eq.(\ref{dRdtau}), the green curve is the solution to Eq.(\ref{dRdtau0}) and the red curve is the solution to Eq.(\ref{dRdtauL}).
\begin{figure}[htbp]
\includegraphics{RtauBoth.eps}
\caption{Here we plot a comparison of the different approximations for the solution of Eq.(\ref{dRdtau}). The blue curve is the solution to Eq.(\ref{dRdtau}), the green curve is the solution to Eq.(\ref{dRdtau0}) and the red curve is the solution to Eq.(\ref{dRdtauL}).}
\label{RtauBoth}
\end{figure}
In Figure \ref{veloc_tau} we plot the numerical solution for the velocity of the domain wall as seen by the infalling observer, from Eq.(\ref{dRdtau}). Since the domain wall crosses its own Schwarzschild radius at a time of $\tau=1.66$, Figure \ref{veloc_tau} shows that the velocity is infact approximately constant as $R\rightarrow R_s$. After the domain wall crosses the Schwarzschild radius, the velocity then increases and diverges as $R\rightarrow0$ (the classical singularity).
\begin{figure}[htbp]
\includegraphics{veloc_tau.eps}
\caption{Here we plot the corresponding numerical solution for Eq.(\ref{dRdtau}). Here we see that as $R\rightarrow R_s$ ($\tau=1.66$), the velocity of the domain wall is approximately constant. However, after the domain wall passes the Schwarzschild radius the velocity diverges as $R\rightarrow0$.}
\label{veloc_tau}
\end{figure}
In Figure \ref{accel_tau} we plot the numerical solution for the acceleration of the domain wall as seen by the infalling observer, from Eq.(\ref{dRdtau}). Since the domain wall crosses its own Schwarzschild radius at a time of $\tau=1.66$, Figure \ref{accel_tau} shows that the acceleration is increasing almost linearly as $R\rightarrow R_s$. After the domain wall crosses the Schwarzschild radius, the acceleration then increases and diverges as $R\rightarrow0$ (the classical singularity). Therefore, we can conclude that even though the observer is attached to the domain wall, he/she is not a truly free-falling observer (as stated at the beginning of this Section). The infalling Schwarzschild observer is an accelerated observer during the entire duration of the collapse.
\begin{figure}[htbp]
\includegraphics{accel_tau.eps}
\caption{Here we plot the corresponding numerical solution for the acceleration associated with Eq.(\ref{dRdtau}). Here we see that as $R\rightarrow R_s$ ($\tau=1.66$), the acceleration of the domain wall increases almost linearly. However, after the domain wall passes the Schwarzschild radius the acceleration diverges as $R\rightarrow0$.}
\label{accel_tau}
\end{figure}
\section{Comparing Asymptotic versus Infalling}
Here we wish to investigate the discrepancy of the observation between the infalling and asymptotic observers.
To understand this discrepancy we turn to Eq.(\ref{dtdtau}). As discussed in the previous section, in the limit $R\rightarrow R_s$, Eq.(\ref{dRdtau}) is approximately constant. This means that we can then write
\begin{equation}
\Delta t\approx \frac{1}{B}\sqrt{B+c}\Delta\tau.
\end{equation}
Now, since $B\rightarrow0$ in this limit and the proper time taken to reach the Schwarzschild radius is finite, we can then see that
\begin{equation}
\lim_{R\rightarrow R_s}\Delta t\rightarrow\infty.
\label{grav_red}
\end{equation}
This is a fairly crude approximating, thus in Figure \ref{t_dot} we plot $dt/dtau$ versus $R/R_s$. Figure \ref{t_dot} shows that as $R\rightarrow R_s$, $dt/d\tau$ does indeed diverge as given in Eq.(\ref{grav_red}). Therefore Eq.(\ref{grav_red}) can be thought of as the gravitational red-shift, which is the source of the discrepancy between the observations between the two observers. In Figure \ref{Rt_vs_Rtau} we show the the position of the domain wall as a function of time for both the asymptotic and infalling observers. Figure \ref{Rt_vs_Rtau} shows that initially the two observers are in agreement on where the domain wall is compared to the Schwarzschild radius. However, as the time increases (both asymptotic, $t$, and infalling, $\tau$) the discrepancy becomes more apparent. According to the asymptotic observer, the domain wall asymptotes to the Schwarzschild radius, while according to the infalling observer this happens in a finite amount of time (as stated earlier).
\begin{figure}[htbp]
\includegraphics{tdot.eps}
\caption{Here we plot $dt/d\tau$ versus $R/R_s$. Here we can see that as $R\rightarrow R_s$, $dt/d\tau\rightarrow\infty$ as given in Eq.(\ref{grav_red}).}
\label{t_dot}
\end{figure}
\begin{figure}[htbp]
\includegraphics{Rt_vs_Rtau.eps}
\caption{Here we plot a comparison of the position of the domain wall relative to the Schwarzschild radius for both of the two different observers. Here the position of the domain wall as seen by the infalling observer is given in blue, while that of the asymptotic observer is given in green.}
\label{Rt_vs_Rtau}
\end{figure}
\section{Discussion}
In this section we investigated the classical equations of motion for the collapsing spherically symmetric domain wall. As discussed in the first section, the asymptotic observer sees the domain wall collapse to the Schwarzschild radius $R_s$ only as $t\rightarrow\infty$. This is not an unreasonable result since in classical General Relativity an asymptotic observer never sees a photon cross the Schwarzschild radius since it take an infinite amount of time $t$ to reach the horizon. Therefore one can easily believe the result here, since the time taken is due to the gravitational redshift of the photon. As the shell approaches $R_s$ the redshift will increase until it becomes infinite by the time it reaches the horizon.
In the second section, the infalling observer will see the shell collapse to the horizon in a finite amount of time. As discussed earlier, this is because the infalling observer is always in a locally flat Minkowski frame. Thus, the horizon is not a significant point for the observer, therefore there is no problem for him/her to pass right through and not even know it. This result is again expected from classical General Relativity.
\chapter{Entropy}
\label{ch:entropy}
In 1972 Bekenstein argued that a black hole of mass $M$ has an entropy proportional to its surface area, see Ref.\cite{Bekenstein}. Further calculations by Gibbons and Hawking showed that the entropy of a black hole is always a constant, despite the type of metric which is used, see Ref.\cite{GibbonsHawking}. They showed that the expression for the entropy is given by
\begin{equation}
S_{BH}=\frac{A_{hor}}{4}=\pi R_s^2
\label{SBH}
\end{equation}
where $A_{hor}$ is the surface area of the event horizon and $R_s$ is the Schwarzschild radius for a black hole which contains only mass.
The typical method for calculating the entropy of the black hole is to first calculate the temperature of the black hole using the so-called Bogolyubov method. Here, one considers that the system starts in an asymptotically flat metric (typically Minkowski), then the system evolves to a new asymptotically flat metric (in the case of just mass, the typically final metric is that of Schwarzschild). One then matches the coefficients between the two asymptotically flat spaces at the beginning and end of the gravitational collapse. The mismatch of these two vacua gives the number of particles produced during the collapse. What happens in between is then beyond the scope of the Bogolyubov method, since the method is generally independent of time. Therefore the time-evolution of the thermodynamics properties of the collapse cannot be investigated in the context of the Bogoyubov method.
Here we will investigate the time-evolution of a spherically symmetric infinitely shell of collapsing matter in the context of the Functional Schr\"odinger formalism. Since the Functional Schr\"odinger formalism depends on the observer's degrees of freedom, one can introduce the ``observer" time into the quantum mechanical processes, with the use of the Wheeler-de Witt equation, in the form of the Schr\"odinger equation, see Chapter \ref{ch:formal}. To study the case of gravitational collapse, one can then choose the classical Hamiltonian of the collapsing object, then employ the standard quantization condition. The wavefunctional is then dependent on the observer time chosen, hence one can view the quantum mechanical processes of a given system under any foliation of space-time that one chooses. The benefit of using the Functional Schr\"odinger formalism is that, in principle, one can solve the time-dependent wavefunctional equation exactly, as discussed in the previous chapters. Therefore the Functional Schr\"odinger formalism goes beyond the approximations of the Bogolyubov method, since the system is allowed to evolve over time, which allows one to investigate the intermediate regime during the collapse. Since the wavefunctional contains all the information of the system, one can, in principle, study the time evolution of the thermodynamical processes of the system. Of current interest is the time-evolution of the entropy of a collapsing gravitational object. We will do so from the view point of a stationary asymptotic observer, since this is the more relevant question. In this chapter we summarize the work originally done in Ref.\cite{EG_ent}.
\section{Partition Function}
To study the entropy of the system, we will first develop the partition function for the system. In order to study the time-evolution of the entropy we shall employ the so-called Liouville-von Neumann approach, which was developed to study equilibrium and non-equilibrium quantum processes (see Ref.\cite{Thermal}). The Liouville-von Neumann approach is a canonical method which unifies the Liouville-von Neumann equation and the Functional Schr\"odinger equation. This approach utilizes the invariant operator approach developed by Lewis and Riesenfeld (see Ref.\cite{Lewis}, Chapter \ref{ch:number} and Appendix \ref{ch:Invariant}), which allows one to exactly solve time-indepedent and time-dependent quantum systems. The Liouville-von Neumann approach has been employed for several different situations ranging from Condensed matter physics to Cosmology, see for example see Ref.\cite{Kim}. The basic assumption of the Liouville-von Neuman approach is that non-equilibrium processes are consequences of underlying microscopic processes which are well described by quantum theory. The details about the collapse will depend on the particular foliation of space-time used to study the system. From the point of view of an infalling observer, in order to calculate the backreaction and local effect around the event horizon it is important to choose a state that is non-singular at the horizon. In this region, the vacuum of choice is the Unruh vacuum (see Refs.\cite{Davies,Unruh}). However, discussed above, we are interested in the view point of the asymptotic observer.
Using the Liouville-von Neumann approach, and following the procedure used in Ref.\cite{Kim}, we can write the partition function as
\begin{equation}
Z=\textrm{Tr}\left[e^{-\beta I}\right]
\label{Z}
\end{equation}
where $I$ is any operator which satisfies the equation
\begin{equation}
\frac{dI}{dt}=\frac{\partial I}{\partial t}-i\left[I,H\right]=0
\label{Heisenberg}
\end{equation}
and $\beta$ is a free parameter. Here we note that Eq.(\ref{Heisenberg}) is just the Heisenberg equation of motion for the operator $I$, see Ref.\cite{Sakurai}, where the total time derivative of the operator is zero. In the case that the total derivative is equal to zero in the Heisenberg, this case is known as the Liouville-von Neumann equation, see Ref.\cite{Lewis}.
From Ref.\cite{Pedrosa}, we can write the invariant operator $I$ as
\begin{equation}
I=\frac{1}{2}\left[\sqrt{\frac{b}{\rho}}+\left(\pi_b\rho-m\rho_{\eta}b\right)^2\right].
\label{I}
\end{equation}
Here we note that the invariant operator $I$ is time dependent since $\rho$ is time dependent (see Eq.(\ref{gen_rho})). Using Eq.(\ref{I}) we can therefore write the partition function, Eq.(\ref{Z}), as
\begin{equation}
Z=\textrm{Tr}\exp\left[-\beta\frac{1}{2}\left[\sqrt{\frac{b}{\rho}}+\left(\pi_b\rho-m\rho_{\eta}b\right)^2\right]\right].
\end{equation}
In this form we can see that the partition function is time dependent since the invariant operator $I$ is time dependent by virtue of Eq.(\ref{I}).
We note that we can rewrite the invariant operator in a more suggestive manner by writing Eq.(\ref{I}) as
\begin{eqnarray}
I&=&\left(\frac{1}{\sqrt{2}}\right)^2\left[\left(\frac{b}{\rho}\right)^{1/4}-i\left(\pi_b\rho-m\rho_{\eta}b\right)\right]\left[\left(\frac{b}{\rho}\right)^{1/4}+i\left(\pi_b\rho-m\rho_{\eta}b\right)\right]\nonumber\\
&\equiv&n(t)+\frac{1}{2}
\end{eqnarray}
where
\begin{equation}
n(t)=a^{\dagger}(t)a(t)
\label{n_t}
\end{equation}
and
\begin{equation}
a(t)\equiv\frac{1}{\sqrt{2}}\left[\left(\frac{b}{\rho}\right)^{1/4}+i\left(\pi_b\rho-m\rho_{\eta}b\right)\right].
\end{equation}
Here $n(t)$ is the time dependent number of states. Hence, the invariant operator $I$ takes on the form of a time-dependent harmonic oscillator Hamiltonian, where the number operator is time dependent.
For a physical meaning of the partition function, we need to act the invariant operator on a quantum state. In the Heisenberg picture, the quantum states span a particular Hilbert space. A convienient basis in this Hilbert space is the so-called Fock space representation, see Ref.\cite{Birrell}. This basis is an eigenstate of the Number operator, Eq.(\ref{n_t}). Thus at a particular time $t$, one has in the Fock space representation
\begin{equation}
n(t)\big{|}n,t\rangle=n\big{|}n,t\rangle.
\end{equation}
Thus, in this space we can then write the partition function as
\begin{eqnarray}
Z&=&\textrm{Tr}\exp\left[-\beta\omega_0\left(n+\frac{1}{2}\right)\right]\nonumber\\
&=&\frac{1}{2\sinh\left(\frac{\beta\omega_0}{2}\right)}.
\label{part func}
\end{eqnarray}
At first glance, one would be tempted to say that the partition function in Eq.(\ref{part func}) is not time-dependent since the partition function now only depends on the initial frequency of the induced scalar field. However, recall that $\beta$ is free parameter which we can choose. Here we discuss our choice in the free parameter $\beta$.
In Refs.\cite{Stojkovic,Greenwood} one can define the occupation number for a frequency $\bar{\omega}$, Eq.(\ref{OccNum}). Then by fitting the number of particles created as the usual Planck distribution Eq.(\ref{OccPlanck}), one can then in principle fit the temperature of the radiation. Here, we then choose to define $\beta$ as
\begin{equation}
\beta=\frac{\partial\ln\left(1+1/N\right)}{\partial\bar{\omega}}.
\label{beta}
\end{equation}
This implies that all of the time dependence of the system is encoded into the temperature of the system.
Therefore we can see that Eq.(\ref{part func}) is just the standard entropy for a time-independent harmonic oscillator, however, the temperature here is time-dependent. Thus we recover the time-dependence of the partition function. Since the partition function is time-dependent, therefore the entropy is also time-dependent.
\section{Entropy}
In terms of the partition function, the thermodynamic definition of entropy is given by, see for example Ref.\cite{Landau},
\begin{equation}
S=\ln Z-\beta\frac{\partial\ln Z}{\partial\beta}.
\end{equation}
Using Eq.(\ref{part func}), we can then write the entropy of the system as
\begin{equation}
S=-\ln\left(1-e^{-\beta\omega_0}\right)+\beta\frac{e^{-\beta\omega_0}}{1-e^{-\beta\omega_0}}.
\end{equation}
Therefore, this is again just the entropy of the usual time-independent harmonic oscillator. From Eq.(\ref{beta}) it follows that the temperature is time-dependent.
To be able to calculate the entropy of the domain wall we will consider the entropy of the entire system, i.e. the domain wall and radiation, and the radiation alone. We will assume that the total entropy is a linear equation in the entropy of the domain wall and the entropy of the radiation. Thus we will write the total entropy as
\begin{equation}
S_{SR}=S_S+S_R,
\end{equation}
where the subscripts $SR$ stands for domain wall and radiation, $S$ for just domain wall and $R$ radiation only, respectively. Then by subtracting these two quantities one can then determine the entropy of the domain wall
\begin{equation}
S_S=S_{SR}-S_R.
\label{S_shell}
\end{equation}
In Chapter \ref{ch:radiation} we considered the wavefunction and occupation number of the radiation only system. To proceed further, we must now consider the wavefunction and occupation number for the entire system, $SR$.
\subsection{Entire System}
To find the wavefunction and occupation number for the entire system, we first note that from Eq.(\ref{Ham_Rdot}) we can approximate the Hamiltonian of the domain wall as
\begin{equation}
H_{wall}\approx-B\Pi_R.
\label{HW_t}
\end{equation}
Then using Eq.(\ref{HW_t}) and Eq.(\ref{Pre-Rad_Schrod_t}) we can write the Hamiltonian of the entire system as
\begin{equation}
H=H_{wall}+H_b=-B\Pi_R+B\frac{\Pi_b^2}{2m}+\frac{K}{2}b^2
\label{tot Ham}
\end{equation}
where $\Pi_R$ is given in Chapter \ref{ch:Classical} and $\Pi_b$ is given by
\begin{equation}
\Pi_b=-i\frac{\partial}{\partial b}
\end{equation}
The wavefunction for the entire system is then a function of $b$, $R$, and $t$, which we can write as
\begin{equation}
\Psi=\Psi(b,R,t).
\label{psi_b_R_t}
\end{equation}
Substituting Eq.(\ref{tot Ham}) into Eq.(\ref{FSE}), we can then write the Functional Schr\"odinger equation as
\begin{equation}
iB\frac{\partial\Psi}{\partial R}-\frac{B}{2m}\frac{\partial^2\Psi}{\partial b^2}+\frac{K}{2}b^2\Psi=i\frac{\partial\Psi}{\partial t}.
\label{Schrod1}
\end{equation}
To solve Eq.(\ref{Schrod1}) we will use the semiclassical case, i.e. we will use the classical background for the collapsing shell. Since the distance of the shell only depends on the time, see Eq.(\ref{Rdot}), we can then write
\begin{equation*}
iB\frac{dt}{dR}\frac{\partial\Psi}{\partial t}-\frac{B}{2m}\frac{\partial^2\Psi}{\partial b^2}+\frac{K}{2}b^2\Psi=i\frac{\partial\Psi}{\partial t}.
\label{Schrod2}
\end{equation*}
Hence, we are eliminating the $R$ dependence from Eq.(\ref{psi_b_R_t}), so $\Psi(b,R,t)\rightarrow\Psi(b,t)$. Rewriting gives
\begin{equation}
-\frac{B}{2m}\frac{\partial^2\Psi}{\partial b^2}+\frac{K}{2}b^2\Psi=i\frac{\partial\Psi}{\partial t}\left(1-B\frac{dt}{dR}\right).
\label{Schrod3}
\end{equation}
Making use of Eq.(\ref{Rdot}), i.e. $dt/dR=-B$, this becomes
\begin{equation}
-\frac{B}{2m}\frac{\partial^2\Psi}{\partial b^2}+\frac{K}{2}b^2\Psi=2i\frac{\partial\Psi}{\partial t}.
\label{ent Schrod}
\end{equation}
We now rewrite Eq.(\ref{ent Schrod}) in the standard form
\begin{equation}
\left[-\frac{1}{2m}\frac{\partial^2}{\partial b^2}+\frac{m}{2}\omega^2(\tilde{\eta})b^2\right]\psi(b,\tilde{\eta})=i\frac{\partial\psi(b,\tilde{\eta})}{\partial \tilde{\eta}}
\label{Ent Schrod}
\end{equation}
where
\begin{equation}
\tilde{\eta}=\frac{1}{2}\int_0^tdt'\left(1-\frac{R_s}{R}\right)
\label{tilde_eta}
\end{equation}
and
\begin{equation}
\omega^2(\tilde{\eta})=\frac{K}{m}\frac{1}{1-R_s/R}\equiv\frac{\omega_0^2}{1-R_s/R}.
\end{equation}
Here we have chosen to set $\tilde{\eta}(t=0)=0$.
The solution to Eq.(\ref{Ent Schrod}) is given by Eq.(\ref{PedWave}), as discussed in Chapter \ref{ch:radiation}. We can then find the occupation number $N$ for the entire system, Eq.(\ref{OccNum}).
Here we will make some quick comments regarding the occupation number. We can see that from Eqs.(\ref{Ent Schrod}) and (\ref{Rad_Schrod_t}), the Schr\"odinger equations for the entire system and radiation only are of the same form. Hence one would expect that there is no difference between the occupation number for the entire system and the radiation only. However, the time parameters $\tilde{\eta}$ and $\eta$, given in Eqs.(\ref{tilde_eta}) and (\ref{eta_t}), are different. Hence the occupation numbers of the two systems will evolve differently, which leads to different temperatures in each of the two systems. Therefore, the entropy of each system will be different.
\section{Analysis}
First we consider the entropy of the entire system. In Figure \ref{EntEntire} we plot the entropy of the entire system as a function of dimensionless time $t/R_s$. Figure \ref{EntEntire} shows that the system starts with an initial entropy of zero. This is expected since initially there is only one degree of freedom, meaning that $S=\ln(1)=0$. Here we have normalized the initial entropy of the shell to be zero. To justify this normalization, consider a solar mass black hole. Under the usual Bekenstein-Hawking entropy, the order of magnitude estimate of the entropy of a solar mass black hole is $S_{BH}\approx10^{75}$. Now consider that the shell is actually made up of protons. The initial entropy of the shell then is approximately $S_{S,0}\approx10^{57}$. Comparing the entropy of the final black hole versus the initial entropy of the shell, the entropy of the final black hole is much much greater than that of the initial entropy of the shell, thus the initial entropy of the shell only contributes a negligible amount of entropy to the entropy of the final black hole. Thus our normalization of the initial entropy of the shell to zero is justified. As $t/R_s$ increases, initially the entropy increases rapidly, then settles down to increase approximately linearly. Due to the linear increase, we see that as $t/R_s$ goes to infinity, the entropy will then diverge. This is again expected since as the asymptotic time goes to infinity, the number of particles that are produced diverges (see Ref.\cite{Stojkovic}). This is a consequence of the fact that we keep the background fixed (i.e. $R_s$ is a constant). In reality, $R_s$ should decrease over time since the radiation is taking away mass and energy from the system. Therefore as $t/R_s$ goes to infinity, the entropy of the entire system as measured by the asymptotic observer diverges as $R\rightarrow R_s$.
\begin{figure}[htbp]
\includegraphics{EntEntire.eps}
\caption{We plot the entropy of the entire system as a function of asymptotic observer time $t$.}
\label{EntEntire}
\end{figure}
This is consistent with the results found in Refs.\cite{Saida}. Here the authors consider the time-dependent non-equilibrium evolution of a black hole as well as the incorporation of the given off radiation. Here one can see that the entropy of the system diverges as the time goes to infinity.
The results of Figure \ref{EntEntire} are consistent with the generalized second law of black hole thermodynamics. The generalized second law states that, see for example Ref.\cite{Jacobson} and references there in
\begin{equation}
\delta(S_{out}+A/4)\geq0
\label{GSL}
\end{equation}
where here, $S_{out}=S_R$, $A/4=S_S$ and $S_{out}+A/4=S_{SR}$, respectively. Eq.(\ref{GSL}) simply states that the total entropy of the system must constantly be increasing as in agreement with thermodynamics entropy Ref.\cite{Landau}. As stated above, a realistic model for gravitational collapse will have that the Schwarzschild radius $R_s$ will decrease over time, since the domain wall is losing mass. Eq.(\ref{GSL}) allows for this result as long as the entropy increase of the radiation compensates for the loss in entropy of the collapsing domain wall.
Now we consider the radiation only. Considering just the particles which are created, i.e. the radiation, during the collapse, we can then plot the entropy as a function or the rescaled asymptotic time $t/R_s$, see Figure \ref{EntR}. Figure \ref{EntR} shows initially the entropy of the system is zero. Again, this is expected since initially the domain wall is in vacuum, meaning that there are no particles produced. Therefore the only degree of freedom is that of the domain wall, this then gives that the initial entropy must be zero. As the asymptotic observer time increases, initially there is rapid increase in the entropy, but again, the entropy then increases linearly as the asymptotic observer time increases. As in the case of the entire system, as the time measured by the asymptotic observer goes to infinity, the entropy of the particles created during the time of collapse diverges. This is expected since the number of particles which are created during the time of collapse diverges as $R\rightarrow R_s$, hence as the domain wall approaches the horizon the number of particles created during the collapse diverges. This result again is in agreement with the generalized second law of black hole thermodynamics, Eq.(\ref{GSL}).
\begin{figure}[htbp]
\includegraphics{EntR.eps}
\caption{We plot the entropy of the particles created during the collapse as a function of asymptotic time $t$.}
\label{EntR}
\end{figure}
In Figure \ref{EntB} we plot the entropy as a function of the rescaled asymptotic observer time $t/R_s$ of both the entire system and the particles created during the time of collapse. Figure \ref{EntB} shows that except for the initial increase in the entropy, for later asymptotic observer time, the slopes of the entropy versus time are approximately equal. Therefore, one can expect that the entropy of the domain wall is approximately constant for late times.
\begin{figure}[htbp]
\includegraphics{EntB.eps}
\caption{We plot the entropy as a function of asymptotic observer time $t$ for both the entire system and the particles created during the time of collapse.}
\label{EntB}
\end{figure}
As stated earlier, what is of interest is the entropy of the collapsing domain wall, since this will collapse to form a black hole. To find the entropy of the domain wall, we can take the entropy of the entire system and subtract off the entropy of the particles produced (since these are the only relevant objects which contribute to the entropy), see Eq.(\ref{S_shell}). The result is then given in Figure \ref{EntShell}. Figure \ref{EntShell} shows that initially the entropy of the domain wall is zero. As stated above, this is expected since initially there is only one degree of freedom. As asymptotic time increases, the entropy of the domain wall rapidly increases. However, for late times, the entropy of the domain wall goes to a constant. As stated above, this is expected since the late time entropies for entire system and for the particles created during collapse are approximately parallel. However, as discussed earlier, one would expect that in a realistic model the entropy of the domain wall should in fact decrease over time since $R_s$ is decreasing because the domain wall is losing mass. The entropy here, however, is constant since we are assuming that the mass is approximately the Hamiltonian of the system, which is a constant of motion, see Chapters \ref{ch:Classical} and \ref{ch:model}. This means that since we are holding the mass of the domain wall constant, we need to keep adding energy to the system to counter act the loss of mass from the Hawking radiation. Therefore one can expect that the entropy of the domain wall must be a constant for late times.
In reality, radiation takes mass away from the system, so the entropy of the domain wall will go to zero as $R_s$ goes to zero. This means that after the black hole disappears, all the entropy will go into the entropy of the radiation, which is in agreement with the generalized second law of black hole thermodynamics.
\begin{figure}[htbp]
\includegraphics{EntShell.eps}
\caption{We plot the entropy of the shell as a function of asymptotic observer time $t$.}
\label{EntShell}
\end{figure}
From Figure \ref{EntShell}, we see that our numerical value for the late time entropy of the domain wall is
\begin{equation*}
S\approx0.7 R_s^2.
\end{equation*}
Comparing with Eq.(\ref{SBH}), we can view this discrepancy as a shift in the Schwarzschild radius $R_s$. In order to get the theoretical value for the entropy, Eq.(\ref{SBH}), we see that we would require $R_s\rightarrow2.11R_s$. This is an understandable numerical error, which implies that our numerical solution is of the same order as the Hawking-Bekenstein entropy.
Another interesting thing to note is that Figure \ref{EntShell} tells us that change in entropy occurs for early times, then gets frozen as time increases. From the plot we see that the change in entropy occurs during the time range $0\leq t/R_s<7.5$. At first sight this seems to be an arbitrary value for the entropy of the domain wall to stop increasing. However, from Eq.(\ref{R(t)}) one can see that this time is not an arbitrary value.
To see this, let us first consider Eq.(\ref{R(t)}) and make the requirement that $R_0=nR_s$, where $n$ is some integer. Then we can write Eq.(\ref{R(t)}) as
\begin{equation*}
R(t)=R_s\left(1+(n-1)e^{-t/R_s}\right).
\end{equation*}
For illustration purposes let's restrict the value of $n$ to be $n\leq10$, which is a restriction that the domain wall starts off at a position ten times it's Schwarzschild radius. In Figure \ref{Position_n} we plot $R/R_s$ versus $t/R_s$ for various values of $n$. For each value of $n$ chosen, we see that the value $R/R_s\approx1$ occurs for $t/R_s\approx7$. In the case of $n=10$, we see that $R=1.005R_s$, while the value is less than that for smaller values of $n$. Hence, the time $t/R_s=7.5$ seems to be a universal time when the domain wall is almost to the Schwarzschild radius. From Eq.(\ref{Rdot}) we see that by this time we have
\begin{equation*}
\dot{R}=-B\approx0.
\end{equation*}
Hence in this time limit, the velocity of the domain wall is approximately zero, meaning that as far as the asymptotic observer is concerned the domain wall has stopped moving and there are no more dynamics. This can be seen in Figure \ref{velocity_n}, where we plot the corresponding velocities for the same values of $n$. Figure \ref{velocity_n} also shows that the time $t/R_s=7.5$ corresponds to a universal time of when the different velocities go approximately to zero. Recall from Chapter \ref{ch:Classical} that it takes an infinite amount of time for the domain wall to reach the horizon, so from $t/R_s=7.5$ to infinity the entropy is constant since all the dynamics are essentially done and the shell is approximately stationary for the observer. Hence the volume of the spherically symmetric domain wall becomes essentially constant by the time $t/R_s=7.5$.
\begin{figure}[htbp]
\includegraphics{Position_n.eps}
\caption{We plot $R/R_s$ versus $t/R_s$ for various values of $n$. Here the blue curve corresponds to $n=2$, the green curve corresponds to $n=5$ and the red curve corresponds to $n=10$.}
\label{Position_n}
\end{figure}
Second, we can show that the entire system and the induced radiation come into thermal equilibrium at this time. In Figure \ref{beta} we plot $\beta$ versus $t/R_s$ for the entire system (continuous curve) and the induced radiation (dashed curve). Figure \ref{beta} shows that for the time $t/R_s\approx7.5$ the values of the two $\beta$'s become approximately equal, meaning that the entire system and the induced radiation are now at the same temperature. Therefore the system is now in thermal equilibrium, meaning that there is no more change in entropy of the domain wall as $t/R_s$ increases. Further more, the fluctuations (departure from thermality) in $\beta$ become very small at this time, as discussed in Refs.\cite{Stojkovic,Greenwood}.
\begin{figure}[htbp]
\includegraphics{velocity_n.eps}
\caption{We plot $\dot{R}/R_s$ versus $t/R_s$ for various values of $n$. Here the blue curve corresponds to $n=2$, the green curve corresponds to $n=5$ and the red curve corresponds to $n=10$.}
\label{velocity_n}
\end{figure}
Finally we can evaluate the the chemical potential for both the entire system and for the induced radiation. From definition we can write the chemical potential as
\begin{equation}
\mu=\frac{\partial S}{\partial N}.
\end{equation}
In Figure \ref{ChemPot} we plot the chemical potential for both the entire system and for the induced radiation. We can see that as $t/R_s$ increases the chemical potential of the entire system and the induced radiation goes to zero. This means that the dispersion of particles goes to zero and the system goes into equilibrium.
\begin{figure}[htbp]
\includegraphics{ChemPot.eps}
\caption{We plot $\mu$ versus $t/R_s$. The solid line corresponds to the entire system while the dashed line corresponds to the induced radiation only. Here we see that as $t/R_s$ increases, the chemical potential for each goes to zero.}
\label{ChemPot}
\end{figure}
During the dynamical process, the entropy increases almost linearly. If one applies a best-fit line, we see that the entropy oscillates about the best line. These oscillations may be attributed to several different circumstances. First, the oscillations may be caused by the non-thermal property of the radiation (see Ref.\cite{Stojkovic}). Secondly, these oscillations may be a manifestation of the error associated with the numerical calculations. Lastly, the oscillations may be an artifact of expanding the calculations beyond the region of validity, since we are using the near horizon approximation. Hence for values large compared to $R_s$, we cannot completely trust our result.
\section{Discussion}
Here we have shown that the entropy of the collapsing domain wall and the entropy of the radiation given off during the time of collapse are in agreement with the generalized second law of black hole thermodynamics. The results of Figure \ref{EntEntire} are clearly in agreement with Eq.(\ref{GSL}). The results of Figure \ref{EntShell} are in agreement with the results of Hawking and Gibbons, Eq.(\ref{SBH}), that the entropy of the black hole is in fact finite and proportional to the area of the event horizon.
Note, here we do not discuss or explain the origin of Eq.(\ref{SBH}), we merely verify that our model gives the correct result. The origin of Eq.(\ref{SBH}) is still not understood, however, many attempts have been made to make sense of this result (see for example Refs.\cite{Strominger,tHooft,Susskind}). However, the answer to this question may lie in understanding the entanglement nature between the particles inside and outside of the event horizon, see for example Refs.\cite{Sorkin,Bombelli,Frolov}.
\chapter{Gauss Codazzi}
\label{ch:GC}
\section{The Gauss-Codazzi Formalism}
Here we wish to solve Einstein's equations in the presence of stress-energy sources confined to three-dimensional time-like hypersurfaces for a general metric. Following the methods used by Ipser and Sikivie, Ref.\cite{Ipser}, we shall use the Gauss-Codazzi formalism.
The Gauss-Codazzi equations relate the four-dimensional geometry of the overall global space-time to their projection onto a three-dimensional hypersurface embedded within the original four-dimensional space-time. This is done by investigating the intrinsic and extrinsic curvature of the three-dimensional time-like hypersurface. The Gauss-Codazzi formalism allows one to find the equations of motion for a collapsing domain wall in a very systematic way. To find the equations of motion, one needs to specify the metric (and associated energy-momentum tensor) only.
In this chapter we wish to develop the Gauss-Codazzi formalism for a general metric where the only initial requirement is that the coefficients of the metric depend on position and time only. We will then arrive a final equation which depends on the coefficient (and derivatives of), as well as its associated energy-momentum tensor, which will allow us to find the equations of motion for the collapsing domain wall once the metric is completely specified. After we develop the general equations, we will compare our result with that found in the literature for two different specified metrics: the Schwarzschild and Reissner-Nordstr\"om metrics, respectively. The Schwarzschild and Reisner-Norstr\"om metric coefficients both depend on position only, hence these are an example of a special case of the general method we are working with here.
\subsection{The Equations}
Here we follow the technologies developed in Ref.\cite{Ipser}. Let $S$ denote a three-dimensional time-like hypersurface containing stress-energy and let $\xi^a$ be its unit spacelike normal ($\xi_a\xi^a=1$). The three-metric intrinsic to the hypersurface $S$ is
\begin{equation}
h_{ab}=g_{ag}-\xi_a\xi_b
\end{equation}
where $g_{ab}$ is the four-metric of the space-time. Here $h_{ab}$ is known as the projected tensor for the hypersurface $S$, see Ref.\cite{Carroll}. This is due to the fact that, when acting $h_{ab}$ on a vector $v^a$, it will project it tangent to the hypersurface, hence orthogonal to $\xi^a$,
\begin{align*}
(h_{ab}v^a)\xi^b&=g_{ab}v^a\xi^b-\xi_a\xi_bv^a\xi^b\\
&=v^a\xi_a-v^a\xi_a\\
&=0.
\end{align*}
Let $\nabla_a$ denote the covariant derivative associated with $g_{ab}$ and let
\begin{equation}
D_a=h_a{}^b\nabla_b,
\label{D_a}
\end{equation}
hence $D_a$ is the covariant derivative on the induced three-dimensional hypersurface. The extrinsic curvature of $S$, denoted by $\pi_{ab}$, is defind by
\begin{equation}
\pi_{ab}\equiv D_a\xi_b=\pi_{ba}.
\label{pi}
\end{equation}
The extrinsic curvature depends on how the hypersurface is embedded in the full four-dimensional space-time. The extrinsic curvature is used to differentiate different topologies. For example, intrinsic geometry of a cylinder and a torus can be flat, however, we know the exterior geometry of each is different. This different topology is given in the extrinsic curvature, which will tell us that we are actually on a torus or a cylinder.
The contracted forms of the first and second Gauss-Codazzi equations are then given by
\begin{eqnarray}
^3R+\pi_{ab}\pi^{ab}-\pi^2&=&-2G_{ab}\xi^a\xi^b\label{Gauss}\\
h_{ab}D_c\pi^{ab}-D_a\pi&=&G_{bc}h^b{}_a\xi^c\label{Codazzi}.
\end{eqnarray}
Here $^3R$ is the Ricci scalar curvature of the three-geometry $h_{ab}$ of $S$, $\pi$ is the trace of the extrinsic curvature, and $G_a{}^b$ is the Einstein tensor in four-dimensional space-time.
Here we will be working with infinitely thin domain walls. The stress-energy tensor $T_{ab}$ of four-dimensional space-time then is assumed to have a $\delta$-function singularity on $S$. This in turn implies that the extrinsic curvature has a jump discontinuity across $S$, since the extrinsic curvature is analogous to the gradient of the Newtonian gravitational potential. Therefore we can introduce
\begin{equation}
\gamma_{ab}\equiv\pi_{+ab}-\pi_{-ab}
\end{equation}
which is the difference between the exterior and interior extrinsic curvatures, and
\begin{equation}
S_{ab}\equiv\int dlT_{ab},
\end{equation}
where $l$ is the proper distance through $S$ in the direction of the normal $\xi^a$, and where the subscripts $\pm$ refer to values just off the surface on the side determined by the direction of $\pm\xi^a$. Hence the direction for, say $+\xi^a$ will be in the direction of the exterior geometry of the domain wall, while $-\xi^a$ will denote the direction of the interior geometry of the domain wall. As we shall discuss below, these geometries will be different for the case of the spherically symmetric domain wall. Using Einstein's and the Gauss-Codazzi equations, one can show that (see Ref.\cite{Wheeler})
\begin{equation}
S_{ab}=-\frac{1}{8\pi G_N}\left(\gamma_{ab}-h_{ab}\gamma_c{}^c\right).
\label{action}
\end{equation}
We can also introduce the ``average" extrinsic curvature
\begin{equation}
\tilde{\pi}_{ab}=\frac{1}{2}\left(\pi_{+ab}+\pi_{-ab}\right)
\label{tilde_pi}
\end{equation}
which will be important later.
\subsection{The Surface Stress-Energy Tensor}
Here we restrict ourselves to sources for which the stress energy tensor is given by, see Ref.\cite{Ipser}
\begin{equation}
S^{ab}=\sigma u^au^b-\eta\left(h^{ab}+u^au^b\right)
\label{Stress-Energy}
\end{equation}
which is the material sources consisting of a perfect fluid. In Eq.(\ref{Stress-Energy}) $u^a$ is the four-velocity of any observer whose world line lies within $S$ and who sees no energy flux in his local frame, and where $\sigma$ is the energy per unit area and $\eta$ is the tension measured by the observer. For a dust wall it is well known that $\eta=0$, while for a domain wall $\eta=\sigma$. For a domain wall Eq.(\ref{Stress-Energy}) reduces to
\begin{equation}
S^{ab}=-\sigma h^{ab}.
\end{equation}
We also note that the four-velocity $u^a$ is a time-like unit vector orthogonal to the space-like unit normal $\xi^a$, i.e.,
\begin{equation*}
u_au^a=-1, \hspace{2mm} \xi_au^a=0, \hspace{2mm} \xi_a\xi^a=+1.
\end{equation*}
\subsection{Attractive Energy}
Here we derive equations for an observer who is hovering just above the surface $S$ on either side. Let the vector field $u^a$ be extended off $S$ in a smooth fashion. The acceleration
\begin{eqnarray}
u^a\nabla_au^b&=&(h^b{}_c+\xi^b\zeta_c)u^a\nabla_au^c\nonumber\\
&=&h^b{}_cu^a\nabla_au^c-\xi^bu^au^c\pi_{ab}
\end{eqnarray}
has a jump discontinuity across $S$ since the extrinsic curvature has such a discontinuity. The perpendicular components of the accelerations of observers hovering just off $S$ on either side satisfy
\begin{align}
\xi_bu^a\nabla_au^b\Big{|}_++\xi_bu^a\nabla_au^b\Big{|}_-=&-2u^au^b\tilde{\pi}_{ab}\nonumber\\
=&-2\frac{\eta}{\sigma}(h^{ab}+u^au^b)\tilde{\pi}_{ab}-2\frac{1}{\sigma}S^{ab}\tilde{\pi}_{ab}
\label{perp1}
\end{align}
and
\begin{eqnarray}
\xi_bu^a\nabla_au^b\Big{|}_+-\xi_bu^a\nabla_au^b\Big{|}_-&=&-u^au^b\gamma_{ab}\nonumber\\
&=&4\pi G_n(\sigma-2\eta).
\label{perp2}
\end{eqnarray}
Here we comment on the precense of the second term on the right hand side of Eq.(\ref{perp1}). This term takes into account the contributions to the energy-tensor $T_{ab}$ which are present in the vacuo on opposite sides of $S$. For example, if there is only mass present, then $T_{ab}$ vanishes off the shell, hence the second term is zero. In the case of charge present, then $T_{ab}$ does not vanish, then the contribution to $T_{ab}$ outside can be taken from the Maxwell tensor.
\section{Spherical Walls}
In this section we shall obtain the asymptotically flat solutions to Einstein's equations for spherically symmetric domain walls with an arbitrary metric. Here we will consider two cases. First we will consider the case where the metric coefficients only depend on the radial position of the domain wall. Second, we will consider the case where the metric coefficients depend on both the radial position of the domain wall and the time.
\subsection{Radial dependence only}
For a spherical shell of stress-energy, let the unit normal $\xi_+$ point in the outward radial direction. It is well known that asymptotic flatness and spherical symmetry requires that the interior geometry is flat (Birkhoff's theorem). For the external geometry we will choose an arbitrary metric. First we shall consider the case where the coefficients only depend on position. Hence,
\begin{align}
(ds^2)_+=&-e^{v(r)}dt^2+e^{u(r)}dr^2+r^2d\Omega^2\nonumber\\
=&-A(r)dt^2+B(r)^2dr^2+r^2d\Omega^2 \hspace{2mm} \text{for $r>R(t)$}
\label{out_metric}
\end{align}
and
\begin{equation}
(ds^2)_-=-dT^2+dr^2+r^2d\Omega^2 \hspace{2mm} \text{for $r<R(t)$}
\label{in_metric}
\end{equation}
where
\begin{equation}
d\Omega^2=d\theta^2+\sin^2\theta d\phi^2.
\end{equation}
Here the equation of the wall is
\begin{equation}
r=R(t).
\label{wallEq}
\end{equation}
One finds for the components of $u^a$ and $\xi^a$ $(a=t$ or $T,r,\theta,\phi$, in that order$)$
\begin{align}
(u^a{}_+)&=(\beta A(r)^{-1},R_{\tau},0,0), \hspace{2mm} (u^a{}_-)=(\alpha,R_{\tau},0,0),\nonumber\\
(\xi^a{}_+)&=(R_{\tau}A^{-1},\beta(AB)^{-1},0,0), \hspace{2mm} (\xi^a{}_-)=(R_{\tau},\alpha,0,0).
\label{u_xi}
\end{align}
Here $R_{\tau}=dR/d\tau$, where $\tau$ is the propertime of an observer moving with four-velocity $u^a$ at the wall, and
\begin{align}
\alpha\equiv&T_{\tau}=\sqrt{1+R_{\tau}^2},\\
\beta\equiv&At_{\tau}=\sqrt{A(r)+A(r)B(r)R_{\tau}^2}.
\label{beta1}
\end{align}
However, here we should comment that the condition that $\xi$ is of unit normal, this then implies the condition that
\begin{equation}
B(r)=\frac{1}{A(r)}.
\end{equation}
Therefore we can rewrite Eq.(\ref{beta1}) as
\begin{align}
\alpha\equiv&T_{\tau}=\sqrt{1+R_{\tau}^2},\label{alpha}\\
\beta\equiv&=\sqrt{A(r)+R_{\tau}^2}.
\label{beta2}
\end{align}
These expressions and the definitions Eqs.(\ref{D_a}), (\ref{pi}) and (\ref{tilde_pi}) imply that
\begin{equation}
(h^{ab}+u^au^b)\tilde{\pi}_{ab}=(\xi^r{}_++\xi^r{}_-)\frac{1}{R},
\end{equation}
and
\begin{align}
\xi_bu^a\nabla_au^b\Big{|}_+&=\frac{1}{\beta}\left[R_{\tau\tau}+\frac{A'}{2}\right]\nonumber\\
\xi_bu^a\nabla_au^b\Big{|}_-&=\frac{1}{\alpha}R_{\tau\tau}
\label{acceleration}
\end{align}
where
\begin{equation}
A'=\frac{dA(r)}{dr}\Big{|}_{r=R(t)}.
\label{A_prime}
\end{equation}
Substituting into Eqs.(\ref{perp1}) and (\ref{perp2}) then yields the equations of motion
\begin{align}
(\alpha+\beta)R_{\tau\tau}=&-2\frac{\eta}{\sigma}\frac{\alpha\beta(\alpha+\beta)}{R}-\frac{\alpha A'}{2}-2\frac{\alpha\beta}{\sigma}S^{ab}\tilde{\pi}_{ab}\label{plus}\\
(\alpha-\beta)R_{\tau\tau}=&4\pi\alpha\beta G(\sigma-2\eta)-\frac{\alpha A'}{2}.\label{minus}
\end{align}
Taking the ratio of Eqs.(\ref{plus}) and (\ref{minus}) allows us to eliminate $R_{\tau\tau}$ from the expression, so we then find
\begin{equation}
\sigma(\sigma-2\eta)+\frac{(1-A)\eta}{2\pi(\alpha+\beta)GR}-\frac{A'\sigma}{4\pi(\alpha+\beta)G}+\frac{(1-A)S^{ab}\tilde{\pi}_{ab}}{4\pi(\alpha+\beta)^2G}=0
\label{EOM}
\end{equation}
Here we make some general comments on Eqs.(\ref{plus}) and (\ref{minus}). First, in the absence of stress-energy outside the domain wall, $R_{\tau\tau}$ is always negative provided $\eta\geq0$. Hence a spherical domain wall with, say only mass, with $\eta\geq0$ will always collapse to a black hole, regardless of its size. Second, in the presence of stress-energy outside the domain wall, $R_{\tau\tau}$ is always positive provided that the source term is small compared to the other terms. However, if the source term is large compared to the other terms, $R_{\tau\tau}$ can become positive at some point. This means that the collapsing object will turn around and begin to expand.
Eq.(\ref{EOM}) allows us to find the equations of motion for a specific geometry, provided that the coefficient $A=A(r)$, i.e. is only a function of position. In the next section we demonstrate the findings in Eq.(\ref{EOM}) for two specific cases found in the literature. This will allow us to demonstrate ease of the general form of the equations of motion.
\subsection{Radial and Time dependence}
In this section we will write the exterior metric, Eq.(\ref{out_metric}), as
\begin{align}
(ds^2)_+=&-e^{v(r,t)}dt^2+e^{u(r,t)}dr^2+r^2d\Omega^2\nonumber\\
=&-A(r,t)dt^2+B(r,t)^2dr^2+r^2d\Omega^2 \hspace{2mm} \text{for $r>R(t)$}
\label{out_metric_t}
\end{align}
where we will maintain that the interior metric is still given by the Minkowski line element. We will again take that the equation of the wall is given by Eq.(\ref{wallEq}), this then gives that the components of $u^a$ and $\xi^a$ are unchanged in form from Eq.(\ref{u_xi}). Note however that one does have to make the change from $A(r)$ and $B(r)$ to $A(r,t)$ and $B(r,t)$, respectively. As in the case of radial dependence only, the condition that $\xi^a$ is a normalized space-like vector, we again have the condition that
\begin{equation}
B(r,t)=\frac{1}{A(r,t)}.
\end{equation}
Therefore we define $\alpha$ and $\beta$ in the same manner as in the case with only radial dependence, using the suitable substitution.
We can then find that the acceleration outside and inside the domain wall are given by
\begin{align}
\xi_bu^a\nabla_au^b\Big{|}_+=&\frac{1}{\beta}\left[R_{\tau\tau}+\frac{A'}{2}\right]\nonumber\\
&+\frac{\dot{A}\dot{R}}{2A^3\beta}\left[A^2+2\dot{R}^2(A-\beta)-3A\beta\right]\nonumber\\
\xi_bu^a\nabla_au^b\Big{|}_-=&\frac{1}{\alpha}R_{\tau\tau}
\label{acceleration_t}
\end{align}
where $A'$ is given in Eq.(\ref{A_prime}) and
\begin{equation}
\dot{A}=\frac{dA}{dt}.
\end{equation}
Comparing the acceleration outside the domain wall for the radial and time dependent metric coefficient, Eq.(\ref{acceleration_t}), to that of the acceleration outside the domain wall for the radially dependent metric coefficient, Eq.(\ref{acceleration}), we see that the acceleration outside the domain in the new case is just the acceleration in the radial case modified by an additional term which depends on $t$-derivatives of the metric coefficient. This is not an unexpected result.
Substituting Eq.(\ref{acceleration_t}) into Eqs.(\ref{perp1}) and (\ref{perp2}) then yields the equations of motion
\begin{align}
(\alpha+\beta)R_{\tau\tau}=&-2\frac{\eta}{\sigma}\frac{\alpha\beta(\alpha+\beta)}{R}-\frac{\alpha A'}{2}-2\frac{\alpha\beta}{\sigma}S^{ab}\tilde{\pi}_{ab}\nonumber\\
&-\frac{\dot{A}\dot{R}\alpha}{2A^3}\left[A^2+2\dot{R}^2(A-\beta)-3A\beta\right]\label{plus2}\\
(\alpha-\beta)R_{\tau\tau}=&4\pi\alpha\beta G_N(\sigma-2\eta)-\frac{\alpha A'}{2}\nonumber\\
&-\frac{\dot{R}\dot{A}\alpha}{2A^3}\left[A^2+2\dot{R}^2(A-\beta)-3A\beta\right].\label{minus2}
\end{align}
Taking the ratio of Eqs.(\ref{plus2}) and (\ref{minus2}) allows us to eliminate $R_{\tau\tau}$ from the expression, so we then find
\begin{align}
0=&\sigma(\sigma-2\eta)+\frac{(1-A)\eta}{2\pi(\alpha+\beta)G_NR}-\frac{A'\sigma}{4\pi(\alpha+\beta)G_N}\nonumber\\
&+\frac{(1-A)S^{ab}\tilde{\pi}_{ab}}{4\pi(\alpha+\beta)^2G_N}\nonumber\\
&-\frac{\dot{A}\dot{R}\sigma}{4\pi G_NA^3(\alpha+\beta)}\left[A^2+2\dot{R}^2(A-\beta)-3A\beta\right]
\label{EOM_t}
\end{align}
Here we make some general comments on Eqs.(\ref{plus2}) and (\ref{minus2}). First, we again see that the first three terms in Eq.(\ref{plus2}) and the first two terms in Eq.(\ref{minus2}) are identical to the radially dependent metric coefficients only, where the last term comes from the time dependence of the metric coefficients. Second, it is not as obvious in this case the behavior of the domain wall. In the case of gravitational collapse, $\dot{R}<0$, making the last term positive.
\section{Examples}
In this section we present some examples using the equation of motion in Eq.(\ref{EOM}). First, we will investigate the case of a massive domain wall. We will show that Eq.(\ref{EOM}) automatically leads to the equation of motion arrived at by Ipser and Sikivie, see Ref.\cite{Ipser}. Second, we will investigate the case of a massive-charged domain wall. We will show that Eq.(\ref{EOM}) automatically leads to the equation of motion arrived at by L\'opez, see Ref.\cite{Lopez}.
Here we note that the usual procedure for determining the metric coefficients is to consider the asymptotic region of space-time (see for example Ref.\cite{Weinberg}). Here one writes the Ricci tensor, which gives the equations of motion for the the metric coefficients. Then using the asymptotic requirements of the space-time, one integrates the equations of motion for the metric coefficients and fixes the integration constant. As stated above, we will just start with the metric coefficients to find the conserved quantities for the collapsing domain wall.
\subsection{Massive Domain Wall}
It is well known that asymptotic flatness and spherical symmetry require the exterior geometry to be Schwarzschild. Therefore we can write Eq.(\ref{out_metric}), the exterior metric, as
\begin{equation}
(ds^2)_+=-\left(1-\frac{2GM}{r}\right)dt^2+\left(1-\frac{2GM}{r}\right)^{-1}dr^2+r^2d\Omega^2.
\end{equation}
Comparing the Schwarzschild metric with Eq.(\ref{out_metric}), one can then identify
\begin{equation}
A(r)=1-\frac{2GM}{r}.
\end{equation}
Since the domain wall only contains mass, the stress-energy is only present on the domain wall. Hence $T_{ab}$ vanishes outside of the domain wall. Therefore using Eq.(\ref{EOM}) we can immediately write
\begin{equation}
\sigma(\sigma-2\eta)-\frac{2GM}{R^2}\frac{(\sigma-2\eta)}{4\pi(\alpha+\beta)G}=0,
\end{equation}
or rearranging the terms we have
\begin{align}
M=&\frac{1}{2}(\alpha+\beta)4\pi\sigma R^2\nonumber\\
=&\frac{1}{2}\left[\sqrt{1+R_{\tau}^2}+\sqrt{1-\frac{2GM}{R}+R_{\tau}^2}\right]4\pi\sigma R^2
\end{align}
where in the second line we use the definition of $\alpha$ and $\beta$, Eqs.(\ref{alpha}) and (\ref{beta2}) respectively. This is identical to Eq.(3.8) in Ref.\cite{Ipser}, for the case of the massive domain wall.
\subsection{Massive-Charged Domain Wall}
Since the domain wall is charged, and spherically symmetric, the geometry outside the domain wall is given by the Reissner-Nordstr\"om solution to Einstein equations. Therefore we can write Eq.(\ref{out_metric}), the exterior metric, as
\begin{align}
(ds^2)_+=&-\left(1-\frac{2GM}{r}+\frac{Q^2}{r^2}\right)dt^2+\left(1-\frac{2GM}{r}+\frac{Q^2}{r^2}\right)^{-1}dr^2+r^2d\Omega^2.
\end{align}
Comparing the Reissner-Nordstr\"om metric with Eq.(\ref{out_metric}), one can then identify
\begin{equation}
A(r)=1-\frac{2GM}{r}+\frac{Q^2}{r^2}.
\end{equation}
In this case the domain wall contains both mass and charge, thus the stress-energy outside of the shell is taken from Maxwell's tensor, since the inside portion of the spherically symmetric domain wall will not feel the influence of the charge. The only nonvanishing components outside the domain wall are
\begin{equation}
T_0{}^0=T_1{}^1=-T_2{}^2=-T_3{}^3=-\frac{Q^2}{8\pi r^4}.
\end{equation}
By taking the difference of Eq.(\ref{Gauss}) on opposite sides of $S$, one finds
\begin{equation}
-\frac{2}{\sigma}S^{ab}\tilde{\pi}_{ab}=\frac{Q^2}{4\pi\sigma R^4}.
\end{equation}
Therefore using Eq.(\ref{EOM}) we can write
\begin{equation}
\left[(\sigma-2\eta)+\frac{Q^2}{4\pi(\alpha+\beta)R^3}\right]\left[\sigma-\frac{(GM-\frac{Q^2}{2R})}{2\pi(\alpha+\beta)R^2}\right]=0.
\end{equation}
Although this is an algebraic equation of second degree in $\sigma$, only one of the two roots holds
\begin{equation}
\sigma=\frac{(GM-\frac{Q^2}{2R})}{2\pi(\alpha+\beta)R^2},
\end{equation}
which, using Eqs.(\ref{alpha}) and (\ref{beta2}) can be put in the form
\begin{equation}
\alpha-\beta=4\pi\sigma GR.
\end{equation}
Therefore, solving for the mass yields
\begin{equation}
M=\frac{Q^2}{2GR}+4\pi\sigma R^2\left[\sqrt{1+R_{\tau}^2}-2\pi\sigma GR\right]
\end{equation}
which is identical to Eq.(61) in Ref.\cite{Lopez}, for the case of the massive-charged domain wall.
\chapter{Invariant Method and the Schr\"odinger Equation}
\label{ch:Invariant}
In this section we breifly review the invariant operator method developed by Lewis and Reisenfeld in Ref.\cite{Lewis} as a solution to the time-dependent Schr\"odinger Equation.
Consider a system whose Hamiltonian operator $H(t)$ is an explicit function of time, and assume the existence of another explicitly time-dependent non-trivial Hermitian operator $I(t)$, which is invariant. To say that $I(t)$ is invariant means that $I(t)$ satisfies the Liouville-von Neumann equation
\begin{equation}
\frac{dI}{dt}=\frac{\partial I}{\partial t}+\frac{1}{i}\left[I,H\right]=0
\label{dIdt}
\end{equation}
and since $I(t)$ is Hermitian we have
\begin{equation}
I^{\dagger}=I.
\end{equation}
Here we will consider the analysis for a state vector $\big{|}\psi\rangle$, however, in general this also works for a wavefunction since $\psi(x)=\langle x\big{|}\psi\rangle$. We can then write the Schr\"odinger equation as
\begin{equation}
H(t)\big{|}\psi\rangle=i\frac{\partial}{\partial t}\big{|}\psi\rangle.
\end{equation}
By operating with the left-hand side of Eq.(\ref{dIdt}) on the state vector and using the Schr\"odinger equation, we obtain the relation
\begin{equation}
i\frac{\partial}{\partial t}\left(I\big{|}\psi\rangle\right)=H\left(I\big{|}\psi\rangle\right),
\end{equation}
which implies that the action of the invariant operator on a Schr\"odinger state vector produces another solution of the Schr\"odinger equation. In general, this result is valid for any invariant, even if the invariant involves the operation of time differentiation. However, for our purposes, we shall consider invariants which do not involve time differentiation. This choice allows one to derive simple and explicit rules for choosing the phases of the eigenstates of $I(t)$ such that these states themselves satisfy the Schr\"odginer equation.
Assume that the invariant is one of a complete set of commuting observables, so that there is a complete set of eigenstates of $I$. Denote the eigenvalues of $I$ by $\lambda$, and the orthonormal eigenstates associated with a given $\lambda$ by $\big|\lambda,\kappa\rangle$, where $\kappa$ represents all of the quantum numbers other than $\lambda$ that are necessary for specifying the eigenstates:
\begin{align}
I(t)\big|\lambda,\kappa\rangle=&\lambda\big|\lambda,\kappa\rangle\label{actionI}\\
\langle\lambda',\kappa'\big|\lambda,\kappa\rangle=&\delta_{\lambda'\lambda}\delta_{\kappa'\kappa}.
\end{align}
Since the invariant is Hermitian, the eigenvalues $\lambda$ are real. They are also time-independent as we shall now see. By differentiating Eq.(\ref{actionI}) with respect to time, we obtain
\begin{equation}
\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle+I\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle=\frac{\partial\lambda}{\partial t}\big|\lambda,\kappa\rangle+\lambda\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle.
\label{dIdt_full}
\end{equation}
Using Eq.(\ref{dIdt}) we can write
\begin{equation}
i\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle+IH\big|\lambda,\kappa\rangle-\lambda H\big|\lambda,\kappa\rangle=0.
\label{dIdt_action}
\end{equation}
The scalar product of Eq.(\ref{dIdt_action}) with a state $\big|\lambda',\kappa'\rangle$ is
\begin{equation}
i\langle\lambda',\kappa'\big{|}\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle+(\lambda'-\lambda)\langle\lambda',\kappa'\big{|}H\big|\lambda,\kappa\rangle=0
\label{eigen}
\end{equation}
which then implies
\begin{equation}
\langle\lambda',\kappa'\big{|}\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle=0.
\end{equation}
Taking the scalar product of Eq.(\ref{dIdt_full}) with $\big|\lambda,\kappa\rangle$, we obtain
\begin{equation}
\frac{\partial\lambda}{\partial t}=\langle\lambda,\kappa\big{|}\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle=0.
\label{dlambdadt}
\end{equation}
Since the eigenvalues are time-independent, it is clear that the eigenstates must be time-dependent.
To investigate the connection between the eigenstates of $I$ and the solutions so the Schr\"odinger equation, we first write the equation of motion of $\big|\lambda,\kappa\rangle$ starting from Eq.(\ref{dIdt_full}) and using Eq.(\ref{dlambdadt}):
\begin{equation}
(\lambda-I)\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle=\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle.
\end{equation}
By taking the scalar production with $\big|\lambda',\kappa'\rangle$ and using Eq.(\ref{eigen}) to eliminate
\begin{equation*}
\langle\lambda',\kappa'\big{|}\frac{\partial I}{\partial t}\big|\lambda,\kappa\rangle
\end{equation*}
we get
\begin{equation}
i(\lambda-\lambda')\langle\lambda',\kappa'\big{|}\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle=(\lambda-\lambda')\langle\lambda',\kappa'\big{|}H\big|\lambda,\kappa\rangle.
\label{eigen_relation}
\end{equation}
From this, for $\lambda'\not=\lambda$, we infer
\begin{equation}
i\langle\lambda',\kappa'\big{|}\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle=\langle\lambda',\kappa'\big{|}H\big|\lambda,\kappa\rangle.
\label{partial_H}
\end{equation}
Eq.(\ref{eigen_relation}) does not imply
\begin{equation*}
i\langle\lambda',\kappa'\big{|}\frac{\partial}{\partial t}\big|\lambda,\kappa\rangle=\langle\lambda',\kappa'\big{|}H\big|\lambda,\kappa\rangle.
\end{equation*}
If Eq.(\ref{partial_H}) held for $\lambda'=\lambda$ as well as for $\lambda'\not=\lambda$, then we would immediately deduce that $\big|\lambda,\kappa\rangle$ satisfies the Schr\"odinger equation, that is $\big|\lambda,\kappa\rangle$ is a special case of $\big{|}\psi\rangle$.
Note that the phase of $\big|\lambda,\kappa\rangle$ has not been fixed by our definitions. We are still free to multiply $\big|\lambda,\kappa\rangle$ by an arbitrary time-dependent phase factor. Thus, we can define a new set of eigenvectors of $I(t)$ related to our initial set by a time-dependent gauge transformation
\begin{equation}
\big|\lambda,\kappa\rangle_{\alpha}=e^{i\alpha_{\lambda\kappa}(t)}\big|\lambda,\kappa\rangle,
\end{equation}
where the $\alpha_{\lambda\kappa}(t)$ are arbitrary real functions of time. Because $I(t)$ is assumed not to contain time-derivative operators, the $\big|\lambda,\kappa\rangle_{\alpha}$ are orthonormal eigenstates of $I(t)$ just as are the $\big|\lambda,\kappa\rangle$. For $\lambda'\not=\lambda$, Eq.(\ref{eigen_relation}) also holds for matrix elements taken with respect to the new eigenstates. Each of the new eigenstates will statisfy the Schr\"odinger if we choose the phases $\alpha_{\lambda\kappa}(t)$ such that Eq.(\ref{eigen_relation}) holds for $\lambda'=\lambda$. This requirement is equivalent to the following first-order differential equation for the $\alpha_{\lambda\kappa}(t)$:
\begin{equation}
\delta_{\kappa\kappa'}\frac{d\alpha_{\lambda\kappa}}{dt}=\langle\lambda,\kappa'\big{|}i\frac{\partial}{\partial t}-H\big|\lambda,\kappa\rangle.
\end{equation}
Since each of the new set of eigenstates of $I(t)$, $\big|\lambda,\kappa\rangle_{\alpha}$, satisfies the Schr\"odinger equation, the general solution is
\begin{equation}
\big{|}t\rangle=\sum_{\lambda,\kappa}c_{\lambda\kappa}e^{i\alpha_{\lambda\kappa}(t)}\big|\lambda,\kappa;t\rangle,
\label{t_state}
\end{equation}
where the $c_{\lambda\kappa}$ are time-independent coefficients. All of the state vectors with which we have dealt so far are time-dependent, while in Eq.(\ref{t_state}) we modified the notation to indicate the dependence on time explicitly. The Schro\"odinger state vector is now denoted by $\big{|}t\rangle$ and the eigenstates of the invariant by $\big|\lambda,\kappa;t\rangle$.
Assume that in the remote past the Hamiltonian $H(t)$ is a constant operator $H(-\infty)$ having a complete, orthonormal set of time-independent eigenstates $\big{|}n;i\rangle$, $n$ being a label for all relevant quantum numbers and $i$ standing for ``initial state." Similiarly, assume that in the remote future, the Hamiltonian is a constant operator $H(\infty)$ and it possesses time-independent eigenstates $\big{|}m;f\rangle$, $m$ labeling the quantum numbers and $f$ standing for ``final state." The explicit time variation of $H(t)$ for intermediate times is arbitrary except for piecewise continuity; in particular, we do not exclude the possibility of variations rapid enough to render an analysis in terms of quasistationary states of $H(t)$ impossible.
We want to calculate the transition amplitude $T(n\rightarrow m)$ connecting an initial state $\big{|}n;i\rangle$ to a final state $\big{|}m;f\rangle$. Thus we consider the case in which the Schr\"odinger state vector $\big{|}-\infty\rangle$ corresponds to an eigenstate $\big{|}n;i\rangle$. The superposition coefficients of Eq.(\ref{t_state}) for this problem are given by
\begin{equation}
c_{\lambda\kappa}=e^{-i\alpha_{\lambda\kappa}(-\infty)}\langle\lambda,\kappa;-\infty\big|n;i\rangle
\end{equation}
from which we obtain
\begin{equation}
\big|t\rangle=\sum_{\lambda,\kappa}\exp\left(i\left[\alpha_{\lambda\kappa}(t)-\alpha_{\lambda\kappa}(-\infty)\right]\right)\big|\lambda,\kappa;t\rangle\langle\lambda,\kappa;-\infty\big|n;i\rangle.
\end{equation}
The transition amplitude is therefore given by
\begin{align}
T(n\rightarrow m)&=\langle m;f\big|\infty\rangle\nonumber\\
&=\sum_{\lambda,\kappa}\exp\left(i\left[\alpha_{\lambda\kappa}(\infty)-\alpha_{\lambda\kappa}(-\infty)\right]\right)\langle m;f\big|\lambda,\kappa;\infty\rangle\langle\lambda,\kappa;-\infty\big|n;i\rangle.
\label{trans_amp}
\end{align}
The properties of $I(t)$ apply equally well to any operator that is an invariant corresponding to a given $H(t)$. In general, for a system of $f$ degrees of freedom, there is an infinite family of such invariants, the members of which are functions of a set of $f$ independent invariants. Two such invariants will, in general, have different eigenstates, different time derivatives, and different commutators with the Hamiltonian. However, one must arrive at the same physical results no matter what invariant we use and, therefore, the choice of which particular invariant to use may be made on the basis of mathematical convenience. Here we demonstrate that the physical result is independent of the choice of invariant, we give a direct proof that a transition amplitude, such as in Eq.(\ref{trans_amp}) is indeed independent of our choice of invariant.
Suppose that we have two complete orthonormal sets of states, $\big{|}v;t\rangle$ and $\big{|}w;t\rangle$, all of which satisfy the time-dependent Schr\"odinger equation; and suppose that the states $\big|v;t\rangle$ are eigenstates of one set of operators, whose eigenvalues are labeled by $v$, and that the states $\big|w;t\rangle$ are eigenstates of a different set of operators, whose eigenvalues are labeled by $w$. The transition amplitude $T(n\rightarrow m)$ can be expressed as
\begin{equation}
T(n\rightarrow m)=\sum_v\langle m;f\big|v;\infty\rangle\langle v;-\infty\big|n;i\rangle
\label{trans_amp1}
\end{equation}
or as
\begin{equation}
T(n\rightarrow m)=\sum_w\langle m;f\big|w;\infty\rangle\langle w;-\infty\big|n;i\rangle.
\label{trans_amp2}
\end{equation}
We want to show directly that these two expressions are the same. The completeness of the states $\big|w;t\rangle$ requires
\begin{equation}
\big|v;t\rangle=\sum_w\big|w;t\rangle\langle w;t\big|v;t\rangle.
\end{equation}
Operating on this equation with $(i(\partial/\partial t)-H)$, and using the facts that all of the states satisfy the Schr\"odinger equation and that the states $\big|w;t\rangle$ are orthogonal, we obtain
\begin{equation}
\frac{\partial}{\partial t}\langle w;t\big|v;t\rangle=0.
\label{diff_t_state}
\end{equation}
Thus the quantity $\langle w;t\big|v;t\rangle$ is independent of time. We now use the completeness of the state $\big|v;t\rangle$ and $\big|w;t\rangle$, Eq.(\ref{diff_t_state}), and the orthonormality of the states $\big|w;t\rangle$ to rewrite Eq.(\ref{trans_amp1}) as
\begin{align}
T(n\rightarrow m)=&\sum_{v,w,w'}\langle m;f\big|w;\infty\rangle\langle w;\infty\big|v;\infty\rangle\langle v;-\infty\big|w';-\infty\rangle\langle w';-\infty\big|n;i\rangle\nonumber\\
=&\sum_{v,w,w'}\langle m;f\big|w;\infty\rangle\langle w;-\infty\big|v;-\infty\rangle\langle v;-\infty\big|w';-\infty\rangle\langle w';-\infty\big|n;i\rangle\nonumber\\
=&\sum_w\langle m;f\big|w;\infty\rangle\langle w;-\infty\big|n;i\rangle.
\end{align}
Thus, Eqs.(\ref{trans_amp1}) and (\ref{trans_amp2}) are the same, as asserted.
Suppose for simplicity that the eigenstates of $I$ are nondegenerate, so that the eigenvalue of $I$ is the only quantum number required for describing the system. When this is so, as it is in our discussion of the time-dependent harmonic oscillator, then it is particularly convenient to choose an invariant having the property that it becomes time-independent as $t\rightarrow-\infty$ so that the commutator $[I(-\infty),H(-\infty)]$ vanishes. Then the normalized eigenvectors of $H(-\infty)$ and $I(-\infty)$ are identical to within constant phase factors. Consequently, we may choose the initial state $|n;i\rangle$ simply to be a eigenstate of $I(-\infty)$, say $|\lambda;-\infty\rangle$. Eq.(\ref{trans_amp}) then reduces to
\begin{equation}
T(n\rightarrow m)=\exp\left(i\left[\alpha_n(\infty)-\alpha_n(-\infty)\right]\right)\langle m;f|\lambda_n;\infty\rangle,
\end{equation}
and the transition probability is given by
\begin{align}
P_{nm}&=\left|T(n\rightarrow m)\right|^2\nonumber\\
&=\left|\langle m;f|\lambda_n;\infty\rangle\right|^2.
\label{P_mn}
\end{align}
As $t\rightarrow\infty$, the invariant operator $I(t)$ in general remains time-dependent and does not commute with the Hamiltonian. Therefore, the state $|\lambda_m;\infty\rangle$ in Eq.(\ref{P_mn}) is a superposition of eigenstates of $H(\infty)$; this is another expression of the fact that energy is not conserved in our system.
From the structure of Eq.(\ref{trans_amp}), it is apparent that we may express the transition amplitude as a matrix element of an $S$ matrix by writing
\begin{equation*}
S=\sum_{\lambda,\kappa}e^{i\alpha_{\lambda\kappa}(\infty)}|\lambda,\kappa;\infty\rangle\langle\lambda,\kappa;-\infty|e^{-\alpha_{\lambda\kappa(-\infty)}},
\end{equation*}
\begin{equation}
T(n\rightarrow m)=\langle m;f|S|n;i\rangle.
\end{equation}
It is easily verified that this operator is unitary:
\begin{equation}
S^{\dagger}S=SS^{\dagger}=1.
\end{equation}
In the special case that the Hamiltonian operators in the remote past and distant future are identical, $H(-\infty)=H(\infty)$, so that the initial and final states are the same set, we may define an elastic scattering operator $R$ in the standard fashion:
\begin{equation}
S=1+2\pi iR.
\end{equation}
The operator $R$ describes the nondiagonal transitions just as $S$ does, but subtracts a noninteracting part from the diagonal amplitudes so that $\langle n|R|n\rangle$ represents a ``forward reaction amplitude" from the state $|n\rangle$ to the same state. The unitarity of the $S$ matrix implies
\begin{equation}
\sum_m\left|\langle m|R|n\rangle\right|^2=\frac{1}{\pi}Im(\langle n|R|n\rangle),
\end{equation}
which is a statement of the optical theorem: the total reaction probability is proportional to the imaginary part of the forward reaction amplitude.
\chapter*{Dedication}
\addcontentsline{toc}{chapter}{Dedication}
This body of work is dedicated to my father, Richard D. Greenwood Sr., and my grandfather, Albert Kneaskern. My grandfather for his faith in me beyond anyone else I have ever known, even before I had matured enough to realize the implication of his faith. For my father who taught me the true nature of hard work and devotion. Together they have embodied everything that I have ever aspired to in life. Yours memories shall forever live in me, may you both rest in peace.
\newpage
\chapter*{Acknowledgements}
\addcontentsline{toc}{chapter}{Acknowledgements}
I would like to thank my committee members and the faculty and staff in the physics department at the University at Buffalo.
I would like to thank first and foremost my advisor Dr.~Dejan Stojkovic for taking me under his wing and giving me an opportunity to blossom as both an individual and a physicist. His insight and understanding of the material is something that I myself only hope to achieve as a physicist. I would like to thank Dr.~Ulrich Baur for being my surrogate adviser for a year; his mentoring during that time was invaluable to me for both his insight and his willingness to explore subjects outside his specialization. I would like to thank Dr.~Doreen Wackeroth for her help, kindness, wisdom and tolerance of me in her classes (sorry for being such a trouble maker during your lectures!). I would also like to thank Dr.~Francis Gasparini for his compassion and console during some very difficult times during my duration at the University at Buffalo. His understanding and advise where inspirational to me during these trying times and having faith in me to teach both summer and regular semester courses.
I would also like to thank my family for their constant support and devotion over the years before and during this body of work. Your support was and is very comforting to me. A very special thank you to my friends that have become an intricate part of my life during the torture which is graduate school. I especially want to thank my friends Tyler Glembo, Andr\'as Sablauer and Kenneth Smith, for without them I would have never made it this far in both my schooling and my life.
\newpage
\tableofcontents
\listoffigures
\addcontentsline{toc}{chapter}{List of Figures}
\newpage
\chapter*{Abstract}
\addcontentsline{toc}{chapter}{Abstract}
In this thesis we investigate quantum mechanical effects to various aspects of gravitational collapse. These quantum mechanical effects are implemented in the context of the Functional Schr\"odinger formalism. The Functional Schr\"odinger formalism allows us to investigate the time-dependent evolutions of the quantum mechanical effects, which is beyond the scope of the usual methods used to investigate the quantum mechanical corrections of gravitational collapse. Utilizing the time-dependent nature of the Functional Schr\"odinger formalism, we study the quantization of a spherically symmetric domain wall from the view point of an asymptotic and infalling observer, in the absence of radiation. To build a more realistic picture, we then study the time-dependent nature of the induced radiation during the collapse using a semi-classical approach. Using the domain wall and the induced radiation, we then study the time-dependent evolution of the entropy of the domain wall. Finally we make some remarks about the possible inclusion of backreaction into the system.
\newpage
\pagenumbering{arabic}
\chapter{Introduction}
\label{ch:intro}
This thesis is based on work done in a series of nine papers, which have been published in several different journals, see Refs.\cite{Stojkovic,GreenStoj,Greenwood,EG_ent,WangGreenStoj,EG_RNrad,GreenHalHao,GreenHalPolStoj}. The work here is varied and takes several different aspects into account, however, for this thesis we concentrate on only a subset of these papers. The subset of interest here are those papers which include gravitational collapse of a massive shell only. Even though this is only a subset of the possible parameters that a black hole can have, this subset displays most of the interesting features that illustrate the core of our work.
\newpage
\include{formal}
\newpage
\include{number}
\newpage
\include{model}
\newpage
\include{Classical}
\newpage
\include{quantum}
\newpage
\include{Radiation}
\newpage
\include{Entropy}
\newpage
\include{BR}
\newpage
\chapter{Conclusion}
\label{ch:Conclusion}
In this thesis we have investigated quantum mechanical effects of gravitational collapse by utilizing the time-dependent nature of the Functional Schr\"odinger formalism. As stressed throughout this thesis, the Functional Schr\"odinger formalism allows us to investigate the intermediate regimes that the standard methods cannot. Therefore we can obtain a better understanding of what is happening during the evolution of the collapse, at least in the context of the Functional Schr\"odinger formalism. As we have seen in the previous chapters, the effects in this intermediate regime are robust and give good insight into the process of collapse.
This thesis is not meant to be an exhaustive list of the different types of gravitational collapse. Here we solely concentrated on a massive domain wall, while ignoring all other observable quantities (such as charge and angular momentum). However, one can ``easily" incorporate these observable into the system as well. For example, one can repeat the steps above for the case of a massive-charged domain wall (i.e. Reissner-Nordstr\"om). This has been done for the classical and quantum solutions in Ref.\cite{WangGreenStoj} and for the semi-classical radiation in Ref.\cite{EG_RNrad}. The analysis can also be repeated for different topologies, other than spherically symmetric domain walls, as well as for different asymptotic space-times (such as de Sitter or anti-de Sitter). In Ref.\cite{GreenHalHao} the classical and quantum solutions are studied for a $(3+1)$-dimensional BTZ black string in AdS space. It is well know that a $(3+1)$-dimensional BTZ black string has the topology of a cylinder and is asymptotic to AdS space-time, due to the negative cosmological constant.
It is also important to note here that the Functional Schr\"odinger formalism is not restricted to gravitational collapse. One could also apply the formalism to expanding systems as well, which are essentially collapsing systems in reverse. In this case, one can investigate an expanding de Sitter or anti-de Sitter universe and consider the radiation and entropy during the evolution of expansion. For subsequent work on the radiation given off during expansion, see ``Time dependent fluctuations and particle production in cosmological de Sitter and anti-de Sitter spaces," by E. Greenwood, D. Dai and D. Stojkovic (submitted for publication in Phys.~Rev.~{\bf D}). In the case of de Sitter expansion, which is represented by the Freedman-Robertson-Walker metric, the horizon is the largest comoving distance which light emitted now can reach the observer at any time in the future. It is expected that that de Sitter space can produce thermal radiation as well (for some counter arguments see Refs.\cite{Sekiwa,Volovik}). In the case of anti-de Sitter expansion, unlike de Sitter expansion, the space-time does not contain an event horizon. Therefore, one would not expect thermal radiation with a constant temperature. However, due to the time-dependent metric, particle production is still expected. Here it is expected that after a short time of expansion, the universe starts recollapsing and ends up forming a black hole, see for example Refs.\cite{Coleman_deLuccia,AbbottColeman}. The Functional Schr\"odinger formalism can be applied to these situations as well to help shed light on these questions.
\newpage
\chapter{Radiation}
\label{ch:radiation}
In this chapter we wish to investigate one of the thermodynamic properties of gravitational collapse. Two of the most important thermodynamic properties of a black hole are the temperature (discussed in this chapter) and the entropy (discussed in Chapter \ref{ch:entropy}). As mentioned in Chapter \ref{ch:formal}, the benefit of the Functional Schr\"odinger equation is that this will allow us to evolve the system over time. This is in contrast with the usual method used to evaluate the thermodynamic properties of a black hole.
The most widely used method of determining thermodynamic properties of a black hole is the so-called Bogolyubov method. The method here is as follows. One considers an initial asymptotically flat space-time, usually Minkowski, at the beginning of the gravitational collapse. The system is then allowed to evolve to a final asymptotically flat space-time, Schwarzschild in the context of a shell of matter only, with no knowledge of what happens in between. Then by matching the coefficients between these two space-times, the mismatch of these two vacua gives the number of produced particles. As mentioned, what happens in between the initial vacua and the final vacua is beyond the scope of the Bogolyubov method.
Since the Functional Schr\"odinger equation allows one to find the time dependent wavefunction for the system, one can, in principle, ask the question of what happens during the evolution of the collapse. In this chapter we will investigate the time evolution of the radiation, in the form of the occupation developed in Chapter \ref{ch:number}, during the time of gravitational collapse. The occupation number will then allow us to fit the temperature of the radiation, and compare with that of the pre-formed black hole.
As discussed in Chapter \ref{ch:model}, we can consider the radiation given off during the collapse of the domain wall by considering a massless scalar field $\Phi$ that is coupled to the gravitational field. The action of the scalar field can then be written as in Eq.(\ref{gen_action})
\begin{equation}
S=\int d^4x\frac{1}{2}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi.
\label{rad_act}
\end{equation}
We decompose the (spherically symmetric) scalar field into a complete set of real basis functions denoted by $\{f_k(r)\}$
\begin{equation}
\Phi=\sum_ka_k(t)f_k(r).
\label{mode exp}
\end{equation}
The exact form of the function $f_k(r)$ will not be important for us. We will, however, be interested in the wavefunction for the mode coefficients $\{a_k(t)\}$. Note, again here we use ``$t$" to be anytime coordinate of interest to us, not necessarily the asymptotic time.
From Eq.(\ref{Met_in}) and Eq.(\ref{Met_out}) we can see that the action for both the different foliations of space-time will consist of two parts, one from Eq.(\ref{Met_in}) and the second from Eq.(\ref{Met_out}). In both foliations, the asymptotic observer and the infalling observer, we will be interested in the region $R\sim R_s$, therefore we will explicitly write out Eq.(\ref{rad_act}) then take the limit to find the dominating contributions.
From this action we can then find the Hamiltonian of the system using the usual methods. Substituting the Hamiltonian into Eq.(\ref{FSE}) we can find the time-dependent wavefunction of the system. As discussed in Chapter \ref{ch:number}, the time-dependent wavefunction allows us to find the occupation number $N$, which will allow us to fit the temperature of the radiation (which we will describe below).
In 1975 Hawking showed that for a pre-existing static black hole, the black hole will radiate its mass away, see Ref.\cite{1975Hawking}. The radiation that is given off has a finite temperature, as viewed by an asymptotic observer, which is known as the Hawking temperature. Therefore, it will be instructive for us to compare our late time result with Hawking's original calculation.
For the infalling observer, the calculation is also instructive to give us an idea of what the region is like for this observer. Some unanswered questions are: since the temperature as measured by the asymptotic observer is finite, what is the temperature at the horizon for the local observer? If the temperature is infinite at the horizon, as one would expect since the temperature for the asymptotic observer is finite, will the infalling observer burn up before he reaches the horizon? To answer this question, we will use two different foliations of space time, that of Schwarzschild and Eddington-Finkelstein, respectively.
\section{Asymptotic Observer}
Here we consider the radiation as measured by the asymptotic observer. For this section we summarize the work originally done in Ref.\cite{Stojkovic}.
As stated above, the action for the scalar field can be written in two parts
\begin{equation}
S=S_{in}+S_{out}
\label{action_two}
\end{equation}
where
\begin{align}
S_{in}&=2\pi\int dt\int_0^{R(t)}drr^2\left[-\frac{(\partial_t\Phi)^2}{\dot{T}}+\dot{T}(\partial_r\Phi)^2\right]\label{S_t_in}\\
S_{out}&=2\pi\int dt\int_{R(t)}^{\infty}drr^2\left[-\frac{(\partial_t\Phi)^2}{1-R_s/r}+\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\right]
\label{S_t_out}
\end{align}
where $\dot{T}$ is given in Eq.(\ref{dTdt}), which with Eq.(\ref{dotRgen}), gives
\begin{equation}
\dot{T}=B\sqrt{1+(1-B)\frac{R^4}{h^2}}.
\end{equation}
As mentioned above, we are interested in the $R\sim R_s$ behavior of the action. As $R\rightarrow R_s$, we see that $\dot{T}\sim B\rightarrow0$. Therefore the kinetic term in Eq.(\ref{S_t_in}) diverges as $(R-R_s)^{-1}$ in this limit, while the kinetic term in Eq.(\ref{S_t_out}) diverges logarithmically. Therefore the divergence of the kinetic term in Eq.(\ref{S_t_in}) dominates over that of the divergence of the kinetic term in Eq.(\ref{S_t_out}). The gradient term in Eq.(\ref{S_t_in}) vanishes in this limit, while the gradient term in Eq.(\ref{S_t_out}) becomes finite. Thus the gradient term in Eq.(\ref{S_t_out}) is dominant over that of the gradient term in Eq.(\ref{S_t_in}) in this limit. Hence the action can be written as
\begin{equation}
S\sim2\pi\int dt\left[-\frac{1}{B}\int_0^{R_s}drr^2(\partial_t\Phi)^2+\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\right]
\label{Rad_act}
\end{equation}
where we have changed the limits of integration to $R_s$ since this is the region of interest. This approximation is valid provided the contribution from $r\in(R_s,R(t))$ to the integrals remains subdominant, and also the time variation introduced by the true integration limit can be ignored.
Now, using Eq.(\ref{mode exp}) we write the action in Eq.(\ref{Rad_act}) as
\begin{equation}
S=\int dt\left[-\frac{1}{2B}\dot{a}_k(t){\bf M}_{kk'}\dot{a}_{k'}(t)+\frac{1}{2}a_k(t){\bf N}_{kk'}a_{k'}(t)\right]
\end{equation}
where ${\bf M}$ and ${\bf N}$ are matrices that are independent of $R(t)$ and are given by
\begin{align}
{\bf M}_{kk'}&=4\pi\int_0^{R_s}drr^2f_k(r)f_{k'}(r)\label{M}\\
{\bf N}_{kk'}&=4\pi\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)f'_k(r)f'_{k'}(r).
\label{N}
\end{align}
Using the standard quantization procedure and Eq.(\ref{FSE}), the wavefunction $\psi(a_k,t)$ satisfies
\begin{equation}
\left[\left(1-\frac{R_s}{R}\right)\frac{1}{2}\Pi_k({\bf M}^{-1})_{kk'}\Pi_{k'}+\frac{1}{2}a_k(t){\bf N}_{kk'}a_{k'}(t)\right]\psi=i\frac{\partial\psi}{\partial t}
\end{equation}
where
\begin{equation}
\Pi_k=-i\frac{\partial}{\partial a_k(t)}
\end{equation}
is the momentum operator conjugate to $a_k(t)$.
The problem of radiation from the collapsing domain wall is equivalent to the problem of an infinite set of uncoupled harmonic oscillators whose masses go to infinity with time. We can see from Eq.(\ref{M}) and Eq.(\ref{N}) that the matrices ${\bf M}$ and ${\bf N}$ are hermitian. Therefore, it is possible to do a principal axis transformation to simultaneously diagonalize ${\bf M}$ and ${\bf N}$ (see Sec. 6-2 of Ref. \cite{Goldstein}) for example). Then for a single eigenmode, the Schr\"odinger equation takes the form
\begin{equation}
\left[-\left(1-\frac{R_s}{R}\right)\frac{1}{2m}\frac{\partial^2}{\partial b^2}+\frac{1}{2}Kb^2\right]\psi(b,t)=i\frac{\partial\psi(b,t)}{\partial t}
\label{Pre-Rad_Schrod_t}
\end{equation}
where $m$ and $K$ denote eigenvalues of ${\bf M}$ and ${\bf N}$, and $b$ is the eigenmode.
Dividing Eq.(\ref{Pre-Rad_Schrod_t}) through by $B$, we can write in the standard form
\begin{equation}
\left[-\frac{1}{2m}\frac{\partial^2}{\partial b^2}+\frac{m}{2}\omega(\eta)^2b^2\right]\psi(b,\eta)=i\frac{\partial\psi(b,\eta)}{\partial\eta}
\label{Rad_Schrod_t}
\end{equation}
where
\begin{equation}
\eta=\int_0^tdt\left(1-\frac{R_s}{R}\right)
\label{eta_t}
\end{equation}
and
\begin{equation}
\omega^2(\eta)=\frac{K}{m}\frac{1}{1-R_s/R}\equiv\frac{\omega_0^2}{1-R_s/R}
\label{omega_t}
\end{equation}
where we have chosen to set $\eta(t=0)=0$.
From Eq.(\ref{R(t)}) we can see that the classical late time behavior of the shell is given by $1-R_s/R\sim\exp(-t/R_s)$. For early times, the behavior depends on how the spherical domain wall was created and we are free to choose a behavior for $R(t)$ that is convenient for calculations and interpretation. The most convenient case to use is a static beginning. This can be obtained if we artificially take the collapse to stop at some time, $t_f$. Eventually we can then take $t_f\rightarrow\infty$, as given by Chapter \ref{ch:Classical}. We will then chose $R$ to be
\begin{equation}
1-\frac{R_s}{R}=\begin{cases} 1 & t\in(-\infty,0)\\
e^{-t/R_s}, &t\in(0,t_f)\\
e^{-t_f/R_s}, &t\in(t_f,\infty). \end{cases}
\label{background_t}
\end{equation}
With the choice of initial static space-time of the domain wall, the initial vacuum state for the modes is the simple harmonic oscillator ground state,
\begin{equation}
\psi(b,\eta=0)=\left(\frac{m\omega_0}{\pi}\right)^{1/4}e^{-m\omega_0b^2/2}.
\label{HO_basis}
\end{equation}
The exact solution for late times is given by Eq.(\ref{PedWave}) with initial conditions given by Eq.(\ref{IC}).
As discussed in Chapter \ref{ch:number} an observer with a detector will interpret the wavefunction of a given mode $b$ at late times in terms of simple harmonic oscillator states at the final frequency
\begin{equation}
\bar{\omega}=\omega_0e^{t_f/2R_s}
\label{LateOmega_t}
\end{equation}
where we have made use of Eq.(\ref{background_t}).
The number of quanta in eigenmode $b$ can be evaluated from Eq.(\ref{OccNum}). By calculating $\dot{N}$ it can be checked that $N$ remains constant for $t<0$ and also $t>t_f$. Hence all the particle production occurs for $0< t<t_f$ and is a consequence of the gravitational collapse.
Now we can take the limit $t_f\rightarrow\infty$. In this limit, $\rho$ remains finite but $\rho_{\eta}\rightarrow-\infty$ as $t>t_f\rightarrow\infty$, provided $\omega_0\not=0$ (see Appendix \ref{ch:rho_t} for details). However, we are interested in the behavior of $N$ for fixed frequency, $\bar{\omega}$. From Eq.(\ref{LateOmega_t}) in this limit implies $\omega_0\rightarrow0$. From the discussion in Appendix \ref{ch:rho_t}, we also know that $\rho\rightarrow\infty$ as $\omega_0\rightarrow0$. Hence we find
\begin{equation}
N(t,\bar{\omega})\sim\frac{\bar{\omega}\rho^2}{\sqrt{2}}\sim\frac{e^{t/(2R_s)}}{\sqrt{2}}, \hspace{2mm} t>t_f\rightarrow\infty.
\label{LateOccNum_t}
\end{equation}
Therefore the occupation number at any frequency diverges in the infinite time limit when backreaction is not taken into account. In Figure \ref{Nt_vs_t} we have plotted the occupation number $N$ versus $t/R_s$ for various values of $\bar{\omega}R_s$. Figure \ref{Nt_vs_t} confirms the late time behavior of the time dependence of the occupation number, Eq.(\ref{LateOccNum_t}).
\begin{figure}[htbp]
\includegraphics{Nt_vs_t.eps}
\caption{$N$ versus $t/R_s$ for various fixed values of $\bar{\omega}R_s$. The curves are lower for higher $\bar{\omega}R_s$.}
\label{Nt_vs_t}
\end{figure}
We have also numerically evaluated the spectrum of mode occupation numbers at any finite time and show the results in Figure \ref{Nt_vs_w} for several different values of $t/R_s$. Figure \ref{Nt_vs_w} shows that as the asymptotic observer's time increases, the occupation number of larger values of $\bar{\omega}R_s$ increases. This is consistent with Eq.(\ref{omega_t}), since as $t\rightarrow\infty$, $\bar{\omega}\rightarrow\infty$.
\begin{figure}[htbp]
\includegraphics{Nt_vs_w.eps}
\caption{$N$ versus $\bar{\omega}R_s$ for various fixed values of $t/R_s$. The occupation number at any frequency grows as $t/R_s$ increases.}
\label{Nt_vs_w}
\end{figure}
To find the temperature of the radiation, we compare the curve in Figure \ref{Nt_vs_w} with the occupation numbers for the Planck distribution
\begin{equation}
N_P(\omega)=\frac{1}{e^{\beta\omega}-1}
\label{OccPlanck}
\end{equation}
where $\beta$ is the inverse temperature. We can see that the spectrum of occupation numbers is non-thermal. As an example, there is no singularity in $N$ at $\omega=0$ at finite time. However, as $t\rightarrow\infty$, the peak at $\omega=0$ does diverge and the distribution becomes more and more thermal for these times. There are also oscillations in $N$.
We now wish to fit the temperature of the radiation. From Eq.(\ref{OccPlanck}), we can find the inverse temperature to be
\begin{equation}
\beta=\frac{\ln(1+1/N_P)}{\bar{\omega}R_s}=T^{-1}.
\end{equation}
In Figure \ref{LnNt_vs_w} we plot $\ln(1+1/N)$ versus $\bar{\omega}R_s$ for various values of $t/R_s$. Here we see that for smaller values of $t/R_s$ the spectrum for $\beta$ is non-thermal. For example, Figure \ref{LnNt_vs_w} shows a thermal-like distribution for only small values of $\bar{\omega}R_s$ for $t/R_s=2$, while the larger values or $\bar{\omega}R_s$ are not yet thermally induced. If one fitted the slope of $\beta$ for this particular time, the only relevant region is that between $0<\bar{\omega}R_s<200$. We can also see that the fluctuations of $\beta$ are large. However, as $t/R_s$ increases more and more values of $\bar{\omega}R_s$ are thermally induced, hence one can fit a larger region. The fluctuations for larger values of $t/R_s$ become much smaller, until they become almost completely non-existent. Another feature which occurs as $t/R_s$ increases is that the slope of $\beta$ goes to zero, which would imply that the temperature of the radiation in fact goes to infinity, not to a finite number as predicted by Hawking.
\begin{figure}[htbp]
\includegraphics{LnNt_vs_w.eps}
\caption{$\ln(1+1/N)$ versus $\bar{\omega}R_s$ for various values of $t/R_s$.}
\label{LnNt_vs_w}
\end{figure}
However, from Eq.(\ref{Rad_Schrod_t}) we see that the time derivative of the wavefunction on the right-hand side is with respect to $\eta$, not with respect to $t$, and $\omega$ is the mode frequency with respect to $\eta$ as well. Eq.(\ref{eta_t}) tells us that the frequency in $t$ is $(1-R_s/R)$ times the frequency in $\eta$, so at the final time $t_f$, this implies
\begin{equation}
\omega^{(t)}=e^{-t_f/R_s}\bar{\omega}
\end{equation}
where the superscript $(t)$ on $\omega$ refers to the fact that this frequency is with respect to time $t$.
However, since we are interested in the temperature in time $t$, we must also rescale the temperature in the same manner as the frequency. So the temperature seen by the asymptotic observer is
\begin{equation}
T=e^{-t_f/R_s}\beta^{-1}(t_f).
\end{equation}
Fitting a thermal spectrum to the collapsed spectrum of Figure \ref{Nt_vs_w}, as shown in Figure \ref{LnNt_vs_w8}, we obtain
\begin{equation}
T\approx\frac{0.19}{R_s}=2.4T_H
\label{HawkTemp}
\end{equation}
where $T_H=1/4\pi R_s$ is the Hawking temperature. Since there is ambiguity in fitting the non-thermal spectrum by a thermal distribution, we can only say that the constant temperature, $T$, and the Hawking temperature are of comparable magnitude.
\begin{figure}[htbp]
\includegraphics{LnNt_vs_w8.eps}
\caption{$\ln(1+1/N)$ versus $\bar{\omega}R_s$ for $t/R_s=8$. The dashed line shows $\ln(1+1/N_P)$ versus $\bar{\omega}R_s$ where $N_P$ is a Planck distribution. The slope gives $\beta^{-1}$ and the temperature is given in Eq.(\ref{HawkTemp}).}
\label{LnNt_vs_w8}
\end{figure}
\section{Infalling Observer}
Here we consider the radiation as measured by the infalling observer. To do so we will consider two different foliations of space time. As seen in Chapter \ref{ch:Classical}, the acceleration of the Schwarzschild observer becomes divergent as the observer crosses the horizon. Therefore it is important to switch to another observer whose acceleration is no longer divergent upon crossing the horizon. For this observer, we will work in Eddington-Finkelstein coordinates. In this section we summarize the work originally done in Ref.\cite{Greenwood}.
First we will consider the Schwarzschild observer, after which we will consider the Eddington-Finkelstein observer.
\subsection{Schwarzschild Coordinates}
The action can again be written in two parts, see Eq.(\ref{action_two}), where
\begin{align}
S_{in}=&2\pi\int d\tau\int_0^{R(\tau)}drr^2\left[-\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2+\sqrt{1+R_{\tau}^2}(\partial_r\Phi)^2\right]\\
S_{out}=&2\pi\int d\tau\int_{R(\tau)}^{\infty}drr^2\left[-\frac{B}{\sqrt{B+R_{\tau}^2}}\frac{(\partial_{\tau}\Phi)^2}{1-R_s/r}+\frac{\sqrt{B+R_{\tau}^2}}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\right].
\end{align}
The most interesting things happen when the shell approaches the Schwarzschild radius. From Eq.(\ref{dRdtau}) we see that $R_{\tau}$ is constant in the limit when $R\rightarrow R_s$. Therefore the kinetic term for $S_{in}$ is roughly constant. The kinetic term in $S_{out}$ goes to zero as $R\rightarrow R_s$, so the $S_{in}$ kinetic term is dominant. Similarly the potential term in $S_{in}$ goes to a constant while the potential term in $S_{out}$ becomes very large, so the potential term in $S_{out}$ dominates. Therefore in the region $R\sim R_s$ we can write the action as
\begin{equation}
S\approx\int d\tau\left[-\int_0^{R_s}drr^2\frac{1}{\sqrt{1+R_{\tau}^2}}(\partial_{\tau}\Phi)^2+\int_{R_s}^{\infty}drr^2\frac{|R_{\tau}|}{B}\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\right]
\label{rad_act_tau}
\end{equation}
where we have changed the limits of integration from $R(\tau)$ to $R_s$ since this is the region of interest.
Using the expansion in modes, Eq.(\ref{mode exp}), we can write the action as
\begin{equation}
S=\int d\tau\left[-\frac{1}{2}\frac{1}{\sqrt{1+R_{\tau}^2}}\dot{a}_k(\tau){\bf A}_{kk'}\dot{a}_{k'}(\tau)+\frac{|R_{\tau}|}{2B}a_k(\tau){\bf C}_{kk'}a_{k'}(\tau)\right]
\end{equation}
where $\dot{a}=da/d\tau$, and ${\bf A}$ and ${\bf C}$ are matrices that are independent of $R(\tau)$ and are given by
\begin{align}
{\bf A}_{kk'}&=4\pi\int_0^{R_s}drr^2f_k(r)f_{k'}(r)\\
{\bf C}_{kk'}&=4\pi\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)f'_k(r)f'_{k'}(r).
\label{AC}
\end{align}
From the action Eq.(\ref{rad_act_tau}) we can find the Hamiltonian, and according to the standard quantization procedure, the wave function $\psi(a_k,\tau)$ must satisfy the Functional Schr\"odinger equation. We can write the Schr\"odinger equation as
\begin{equation}
i\frac{\partial\psi}{\partial\tau}=\left[\frac{1}{2}\sqrt{1+R_{\tau}^2}\Pi_k({\bf A}^{-1})_{kk'}\Pi_{k'}+\frac{|R_{\tau}|}{2B}a_k(\tau){\bf C}_{kk'}a_{k'}(\tau)\right]\psi
\end{equation}
where
\begin{equation}
\Pi_k=-i\frac{\partial}{\partial a_k(\tau)}
\end{equation}
is the momentum operator conjugate to $a_k(\tau)$.
Again, the problem of radiation from the collapsing domain wall for the infalling observer is equivalent to the problem of an infinite set of uncoupled harmonic oscillators with time dependent mass and frequency. Following the principal axis transformation used in the section above, the single eigenmode Schr\"odinger equation take the form
\begin{equation}
\left[-\frac{1}{2m}\sqrt{1+R_{\tau}^2}\frac{\partial^2}{\partial b^2}+\frac{|R_{\tau}|}{2B}Kb^2\right]\psi(b,\tau)=i\frac{\partial\psi(b,\tau)}{\partial\tau}
\label{Schrod_rad_tau}
\end{equation}
where $m$ and $K$ denote eigenvalues of ${\bf A}$ and ${\bf C}$, and $b$ is the eigenmode.
Re-writing Eq.(\ref{Schrod_rad_tau}) in the standard form we obtain
\begin{equation}
\left[-\frac{1}{2m}\frac{\partial^2}{\partial b^2}+\frac{m}{2}\omega^2(\eta)b^2\right]\psi(b,\eta)=i\frac{\partial\psi(b,\eta)}{\partial\eta}
\label{Schrod_Rad_tau}
\end{equation}
where
\begin{equation}
\omega^2(\eta)=\frac{K}{m}\frac{|R_{\tau}|}{B\sqrt{1+R_{\tau}^2}}\equiv\omega_0^2\frac{|R_{\tau}|}{B\sqrt{1+R_{\tau}^2}}
\end{equation}
and
\begin{equation}
\eta=\int d\tau'\sqrt{1+R_{\tau}^2}
\label{eta_tau}
\end{equation}
where we defined $\omega_0^2\equiv K/m$.
To proceed further, we will use the classical background of the collapsing domain wall Eq.(\ref{R_tau_0}). The initial vacuum state for the modes is the simple harmonic oscillator ground state,
\begin{equation}
\psi(b,\eta=0)=\left(\frac{m\omega_0}{\pi}\right)^{1/4}e^{-m\omega_0b^2/2}.
\end{equation}
The exact solution for late times is given by Eq.(\ref{PedWave}) with initial conditions given by Eq.(\ref{IC}).
As discussed in Chapter \ref{ch:number} an observer with a detector will interpret the wavefunction of a given mode $b$ at late times in terms of simple harmonic oscillator states at the final frequency $\bar{\omega}$.
The number of quanta in eigenmode $b$ can be evaluated from Eq.(\ref{OccNum}). By calculating $N_{\tau}$ it can be checked that $N$ remains constant for $\tau<0$ and also $\tau>\tau_f$. Hence all the particle production occurs for $0< \tau<\tau_f$ and is a consequence of the gravitational collapse.
Now we can take the limit $\tau_f\rightarrow\tau_c$. In this limit, $\rho$ remains finite but $\rho_{\eta}\rightarrow-\infty$ as $\tau>\tau_f\rightarrow\tau_c$, provided $\omega_0\not=0$ (see Appendix \ref{ch:rho(tau)} for details). However, we are interested in the behavior of $N$ for fixed frequency, $\bar{\omega}$. From the discussion in Appendix \ref{ch:rho(tau)}, we also know that $\rho\rightarrow\infty$ as $\omega_0\rightarrow0$. Therefore the occupation number at any frequency diverges in the infinite time limit when backreaction is not taken into account.
In Figure \ref{Ntau_vs_tau} we plot the occupation number of produced particles as a function of time (for several fixed frequencies $\bar{\omega}R_s$). The amount of proper time needed for the shell to reach $R_s$ can be obtained by integrating Eq.(\ref{dRdtau}). For $\sigma=0.01R_s^{-3}$ this critical proper time is $\tau_c=7/3R_s$. Figure \ref{Ntau_vs_tau} shows that, as the infalling observer approaches $R_s$, the occupation number increases and diverges exactly at $R_s$. The same conclusion as found by analyzing the occupation number $N$ as a function of $\rho$ and $\rho_{\tau}$ (see Appendix \ref{ch:rho(tau)}). This is in agreement with what one would expect in the absence of backreaction. Hawking showed, see Ref.\cite{1975Hawking}, that the flux of particles at late times diverges for a fixed background, i.e. fixed mass of the object. Here, from Eq.(\ref{Mass_tau}), we are treating the mass of the domain wall as a constant of motion. This means that we keep adding energy to the domain wall during the time of collapse, despite the the loss of mass due to the radiation. For the asymptotic observer it takes an infinite amount of his time for the domain wall to collapse to $R_s$, see Chapter \ref{ch:Classical} for discussion of this. However, this infinite time interval corresponds to a finite amount of time for the infalling observer's time. Thus, one may conclude that the infalling observer has to encounter the infinite number of particles produced during this finite amount of time before he reaches $R_s$.
\begin{figure}[htbp]
\includegraphics{Ntauvstau.eps}
\caption{The occupation number $N$ as a function of proper time $\tau/R_s$ for various fixed values of particle frequencies $\bar{\omega}R_s$. The curves are lower for higher values of $\bar{\omega}R_s$. The occupation number diverges as the infalling observer approaches $R_s$, which happens as $\tau\rightarrow\tau_c$.}
\label{Ntau_vs_tau}
\end{figure}
We have also numerically evaluated the spectrum of mode occupation numbers at any finite time and show the results in Figure \ref{Ntau_vs_w} for several different values of $\tau/R_s$. Figure \ref{Ntau_vs_w} shows that as the infalling observer time increases, the occupation number of larger values of $\bar{\omega}R_s$ increases.
\begin{figure}[htbp]
\includegraphics{Ntauvsomega.eps}
\caption{The occupation number $N$ as a function of frequency $\bar{\omega}R_s$ for various fixed values of proper time $\tau/R_s$. The occupation number increases for larger values of $\tau/R_s$ as $\tau\rightarrow\tau_c$.}
\label{Ntau_vs_w}
\end{figure}
To find the temperature of the radiation, we again compare the curve in Figure \ref{Ntau_vs_w} with the occupation numbers for the Planck distribution, which is given by Eq.(\ref{OccPlanck}), where $\beta$ is again the inverse temperature. We can see that the spectrum of occupation numbers is non-thermal. As an example, there is no singularity in $N$ at $\omega=0$ at finite time. However, as $\tau\rightarrow\tau_c$, the peak at $\omega=0$ does diverge and the distribution becomes more and more thermal for these times. There are also oscillations in $N$, which are not present in the Planck distribution.
We now wish to fit the temperature of the radiation, however, from Eq.(\ref{Schrod_Rad_tau}) we see that the time derivative of the wavefunction on the right-hand side is with respect to $\eta$, not with respect to $\tau$, and $\omega$ is the mode frequency with respect to $\eta$ as well. Eq.(\ref{eta_tau}) tells us that the frequency in $\tau$ is $\sqrt{1+R_{\tau}^2}$ times the frequency in $\eta$. However, recall from Chapter \ref{ch:Classical} Eq.(\ref{dRdtau}) tells us that as $R\rightarrow R_s$, $R_{\tau}$ is in fact a constant. Therefore, $\eta$ and $\tau$, for the case of the infalling observer, only differ by a constant amount. Hence, without loss of generality, we can ignore this shift by a constant amount, since the general features of the temperature will be the same. From Eq.(\ref{OccPlanck}), we can find the inverse temperature to be
\begin{equation}
\beta=\frac{\ln(1+1/N_P)}{\bar{\omega}R_s}=T^{-1}.
\end{equation}
In Figure \ref{LnNtau_vs_w} we fit a thermal spectrum to the collapsed spectrum of Figure \ref{Ntau_vs_w}. Several important features of the Hawking-like radiation can be taken from from this plot. First, the non-thermal features of the radiation are apparent. However, the departure from thermality (the fluctuations) are larger for earlier times, hence larger frequencies. This observation was first argued in Ref.\cite{Stojkovic}. Second, as $\tau\rightarrow\tau_c$ and the infalling observer approaches $R_s$, the radiation becomes more and more thermal even at large frequencies. Third, at $\tau=\tau_c$, i.e. $R=R_s$, the radiation becomes purely thermal. At this point, the black hole is formed and the radiation becomes thermal, as known from various studies of quantum radiation from a pre-existing black hole. Finally, it is apparent that the slope of $\ln(1+1/N)$ versus $\bar{\omega}R_s$ is decreasing as the infalling observer approaches $R_s$. Exactly at $R_s$, the slope of the curve is zero, indicating that the temperature of the radiation is infinite. This is not surprising since, as it is well known, the asymptotic observer in the nearly flat asymptotic region will register Hawking radiation with a finite temperature (see previous section). When the temperature is blue-shifted back to $R_s$, it clearly diverges.
\begin{figure}[htbp]
\includegraphics{LnNtauvsomega.eps}
\caption{Plot of $\ln(1+1/N)$ as a function of frequency $\bar{\omega}R_s$ for various fixed values of proper time $\tau/R_s$. The slope of the best fit line is $\beta$, which is the inverse temperature. The non-thermal features disappear and the temperature diverges as the Schwarzschild radius is approached, i.e. $\tau\rightarrow\tau_c$.}
\label{LnNtau_vs_w}
\end{figure}
\subsection{Infalling Eddington-Finkelstein Coordinates}
Now we consider the collapse from the point of view of an infalling Eddington-Finkelstein observer. This is a different space-time foliation than that in Schwarzschild coordinates, and we expect crucially different results. In particular, since the metric is not divergent at the horizon, we do not expect infinite temperature there.
For this purpose, we define the ingoing null coordinate $v$ as
\begin{equation}
v=t+r^*
\end{equation}
where $r^*$ is the tortoise coordinate. We can then rewrite Eq.(\ref{Met_out}) as
\begin{equation}
ds^2=-\left(1-\frac{R_s}{r}\right)dv^2+2dvdr+r^2d\Omega^2, \hspace{2mm} r>R(v).
\label{out_Met}
\end{equation}
where the trajectory of the collapsing wall is $r=R(v)$.
The interior metric is the same as in Eq.(\ref{Met_in}). The interior time coordinate, $T$, is related to the ingoing null coordinate, $v$, via the proper time on the shell, $\tau$. The relations are
\begin{equation}
\frac{dT}{d\tau}=\sqrt{1+\left(\frac{dR}{d\tau}\right)^2}
\label{dTdtau}
\end{equation}
and
\begin{equation}
\frac{dv}{d\tau}=\frac{1}{B}\left(\frac{dR}{d\tau}-\sqrt{B+\left(\frac{dR}{d\tau}\right)^2}\right)
\label{dvdtau}
\end{equation}
where
\begin{equation}
B\equiv 1-\frac{R_s}{R}.
\end{equation}
Consider again a massless scalar field $\Phi$ which propagates in the background of the collapsing shell. The action for the scalar field is
\begin{equation}
S=\int d^4x\sqrt{-g}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi,
\label{action}
\end{equation}
where $g^{\mu\nu}$ is the background metric given by Eqs.(\ref{Met_in}) and (\ref{out_Met}). Decomposing the (spherically symmetric) scalar field into a complete set of real basis functions denoted by $\{f_k(r)\}$
\begin{equation}
\Phi=\sum_ka_k(v)f_k(r)
\label{mode_ex}
\end{equation}
we can find a complete set of independent eigenmodes $\{b_k\}$ for which the Hamiltonian is a sum of terms.
Since the metric inside and outside of the shell have different forms, we again split the action into two parts
\begin{equation}
S_{in}=2\pi \int dT\int_0^{R(v)}drr^2\left[-(\partial_T\Phi)^2+(\partial_r\Phi)^2\right],
\label{S_in}
\end{equation}
\begin{align}
S_{out}=2\pi\int dv\int_{R(v)}^{\infty}drr^2&\Big{[}\partial_v\Phi\partial_r\Phi+\partial_r\Phi\partial_v\Phi\nonumber\\
&+\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{]}.\label{S_out}
\end{align}
We are again interested in the near horizon behavior of the radiation, i.e. as $R\rightarrow R_s$. In this limit we can write Eq.(\ref{dvdtau}) as
\begin{equation}
\frac{dv}{d\tau}\approx-\frac{1}{2R_{\tau}}
\end{equation}
where $R_{\tau}=dR/d\tau$. Then with the help of Eq.(\ref{dTdtau}) we can write Eq.(\ref{S_in}) as
\begin{align}
S_{in}=2\pi\int dv\int_0^{R(v)}drr^2&\Big{[}-\frac{1}{2}\frac{1}{\sqrt{R_v/2(R_v/2+1)}}(\partial_v\Phi)^2\nonumber\\
&+2\sqrt{R_v/2(R_v/2+1)}(\partial_r\Phi)^2\Big{]}\label{S_in_v}
\end{align}
where $R_v=dR/dv$. Obviously, the action is not singular as $R(v) \rightarrow R_s$, unlike the Schwarzschild case. From Eqs.(\ref{S_out}) and (\ref{S_in_v}) we can write the total action as
\begin{align}
S\approx&2\pi\int dv\Big{[}-\int_0^{R_s}drr^2\frac{1}{2}\frac{1}{\sqrt{R_v/2(R_v/2+1)}}(\partial_v\Phi)^2\nonumber\\
&+\int_{R_s}^{\infty}drr^2\partial_v\Phi\partial_r\Phi+\int_{R_s}^{\infty}drr^2\partial_r\Phi\partial_v\Phi\nonumber\\
&+\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)(\partial_r\Phi)^2\Big{]}
\end{align}
where we have changed the limits of integration from $R(v)$ to $R_s$ since this is the region of interest.
Now using the expansion in modes Eq.(\ref{mode_ex}), we can rewrite the action as
\begin{align}
S\approx\int dv&\Big{[}-\frac{1}{2}\frac{1}{\sqrt{R_v/2(R_v/2+1)}}\dot{a}_k{\bf A}_{kk'}\dot{a}_{k'}\nonumber\\
&+\frac{1}{2}\dot{a}_k{\bf Y}_{kk'}a_{k'}+\frac{1}{2}a_k{\bf Y}_{kk'}^{-1}\dot{a}_{k'}+\frac{1}{2}a_k{\bf C}_{kk'}a_{k'}\Big{]}
\end{align}
where $\dot{a}=da/dv$, and ${\bf A}$, ${\bf Y}$ and ${\bf C}$ are matrices that are independent of $R(v)$ and are given by
\begin{align}
{\bf A}_{kk'}=2\pi\int_0^{R_s}drr^2f_k(r)f_{k'}(r),\\
{\bf Y}_{kk'}=4\pi\int_{R_s}^{\infty}drr^2f_k(r)f'_{k'}(r),\\
{\bf C}_{kk'}=8\pi\int_{R_s}^{\infty}drr^2\left(1-\frac{R_s}{r}\right)f'_k(r)f'_{k'}(r).
\end{align}
However if we take that the matrices are symmetric and real, we can see that ${\bf Y}={\bf Y}^{-1}$, so we can write the action as
\begin{align}
S\approx\int dv&\Big{[}-\frac{1}{2}\frac{1}{\sqrt{R_v/2(R_v/2+1)}}\dot{a}_k{\bf A}_{kk'}\dot{a}_{k'}\nonumber\\
&+\frac{1}{2}{\bf Y}_{kk'}\left(\dot{a}_ka_{k'}+a_k\dot{a}_{k'}\right)+\frac{1}{2}a_k{\bf C}_{kk'}a_{k'}\Big{]}.\label{Action}
\end{align}
From the action Eq.(\ref{Action}) we can find the Hamiltonian, and according to the standard quantization procedure, the wave function $\psi(a_k,v)$ must satisfy
\begin{equation}
i\frac{\partial\psi}{\partial v}=H\psi,
\end{equation}
or
\begin{align}
i\frac{\partial\psi}{\partial v}=&\Big{[}\frac{1}{2}\sqrt{R_v/2(R_v/2+1)}\Pi_k({\bf A^{-1}})_{kk'}\Pi_{k'}\nonumber\\
&+\frac{1}{2}a_k\left(\sqrt{R_v/2(R_v/2+1)}{\bf Y}^2_{kk'}({\bf A^{-1}})_{kk'}+{\bf C}_{kk'}\right)a_{k'}\nonumber\\
&+\frac{1}{2}\sqrt{R_v/2(R_v/2+1)}\Pi_k{\bf Y}_{kk'}({\bf A^{-1}})_{kk'}a_{k'}\Big{]}\psi
\end{align}
where
\begin{equation}
\Pi_k=-i\frac{\partial}{\partial a_k}
\end{equation}
is the momentum operator conjugate to $a_k$. Using the momentum operator conjugate to $a_k$, we can rewrite the Schr\"odinger equation as
\begin{align}
i\frac{\partial\psi}{\partial v}=&\Big{[}\frac{1}{2}\sqrt{R_v/2(R_v/2+1)}\Pi_k({\bf A^{-1}})_{kk'}\Pi_{k'}\nonumber\\
&+\frac{1}{2}a_k\left(\sqrt{R_v/2(R_v/2+1)}{\bf Y}^2_{kk'}({\bf A^{-1}})_{kk'}+{\bf C}_{kk'}\right)a_{k'}\nonumber\\
&-i\frac{1}{2}\sqrt{R_v/2(R_v/2+1)}{\bf Y}_{kk'}({\bf A^{-1}})_{kk'}\delta_{kk'}\Big{]}\psi
\end{align}
where $\delta_{kk'}$ is the Kronecker delta function.
So the problem of radiation from the collapsing domain wall for the infalling observer is equivalent to the problem of solving an infinite set of decoupled damped harmonic oscillators with time-dependent frequency. Since ${\bf A}$, ${\bf Y}$ and ${\bf C}$ are symmetric and real, it is possible to simultaneously diagonalize them using the principle axis transformation. Then for a single eigenmode, the Schr\"odinger equation takes the form
\begin{align}
i\frac{\partial\psi}{\partial v}=&\Big{[}-\frac{1}{2m}\sqrt{R_v/2(R_v/2+1)}\frac{\partial^2}{\partial b^2}\nonumber\\
&+\frac{1}{2}\left(\sqrt{R_v/2(R_v/2+1)}\frac{y^2}{m}+K\right)b^2\nonumber\\
&-i\frac{y}{2m}\sqrt{R_v/2(R_v/2+1)}\Big{]}\psi
\label{schrod}
\end{align}
where $m$, $y$ and $K$ denote eigenvalues of ${\bf A}$, ${\bf Y}$ and ${\bf C}$, and $b$ is the eigenmode.
Re-writing Eq.(\ref{schrod}) in the standard form we obtain
\begin{equation}
\left[-\frac{1}{2m}\frac{\partial^2}{\partial b^2}+\frac{m}{2}\omega^2(\eta)-i\frac{y}{2m}\right]\psi(b,\eta)=i\frac{\partial\psi(b,\eta)}{\partial\eta}
\label{Schrod}
\end{equation}
where
\begin{align}
\omega^2(\eta)&=\frac{y^2}{m^2}+\frac{K}{m}\frac{1}{\sqrt{R_v/2(R_v/2+1)}}\nonumber\\
&\equiv\frac{y^2}{m^2}+\frac{\omega_0^2}{\sqrt{R_v/2(R_v/2+1)}}
\end{align}
and
\begin{equation}
\eta=\int dv'\sqrt{R_v/2(R_v/2+1)}
\end{equation}
where we defined $\omega_0^2\equiv K/m$. To find solutions to equation Eq.(\ref{Schrod}) we use the ansatz
\begin{equation}
\psi(b,\eta)=e^{-y\eta/2m}\phi(b,\eta).
\label{ansatz}
\end{equation}
This leads to the equation for $\phi(b,\eta)$
\begin{equation}
-\frac{1}{2m}\frac{\partial^2\phi}{\partial b^2}+\frac{m\omega^2}{2}b^2\phi=i\frac{\partial\phi}{\partial\eta}.
\end{equation}
The exact solution for late times is given by Eq.(\ref{PedWave}) with initial conditions given by Eq.(\ref{IC}).
Then Eq.(\ref{ansatz}) gives
\begin{equation}
\psi=e^{-y\eta/2m}\phi(b,\eta)
\label{psi}
\end{equation}
where $\phi$ given in Eq.(\ref{PedWave}).
As discussed in Chapter \ref{ch:number} an observer with a detector will interpret the wavefunction of a given mode $b$ at late times in terms of simple harmonic oscillator states at the final frequency $\bar{\omega}$.
In Fig.~\ref{N_versus_v} we plot $N$ versus $v/R_s$ for various fixed values of $\bar{\omega}R_s$. We can see that the occupation number at any frequency increases as $v/R_s$ decreases. Thus more particles are created as the shell reaches and crosses the horizon. However, the number of created particles does not diverge as $R(v) \rightarrow R_s$.
We then numerically evaluate the spectrum of mode occupation numbers at any finite time and show the results in Fig.~\ref{N_versus_w} for several values of $v/R_s$. The first sign of non-thermality is the fact that the occupation number is non-divergent at $\bar{\omega}=0$, as opposed to the thermal Planck distribution in Eq.(\ref{OccPlanck}).
\begin{figure}[htbp]
\includegraphics{N_versus_v.eps}
\caption{Here we plot $N$ versus $v/R_s$ for various fixed values of $\bar{\omega} R_s$. The curves are lower for higher values of $\bar{\omega} R_s$. }
\label{N_versus_v}
\end{figure}
\begin{figure}[htbp]
\includegraphics{N_versus_w.eps}
\caption{Here we plot $N$ versus $\bar{\omega} R_s$ for various fixed values of $v/R_s$. The occupation number at any frequency grows as the collapse progresses (i.e. $v/R_s$ decreases) but in never diverges.}
\label{N_versus_w}
\end{figure}
In Fig.~\ref{LnN_versus_w} we plot $\ln(1+1/N)$ versus $\bar{\omega}R_s$ for various values of $v/R_s$. As $v/R_s$ decreases (as the shell is collapsing), the curves decrease. A thermal spectrum should gives us a straight line, however, we see that is not the case here. The best one can do is to fit the low frequency part of the spectrum and get the temperature in that regime. In our case we get $T=(0.17R_s)^{-1}$. Unlike the case of Schwarzschild coordinates, where the spectrum becomes thermal in the whole frequency range, in Eddington-Finkelstein coordinates the spectrum never becomes thermal in the high frequency range. Another feature is apparent in Fig.~\ref{LnN_versus_w}. As the collapse progresses, the fluctuations in the spectrum become more violent. This is indicative of the shell approaching the actual singularity at $R=0$ which is the region of strong gravitational fields.
\begin{figure}[htbp]
\includegraphics{LnN_versus_w.eps}
\caption{Here we plot $\ln(1+1/N)$ versus $\bar{\omega} R_s$ for various fixed values of $v/R_s$. The curves are lower and display more fluctuations as $v/R_s$ decreases.}
\label{LnN_versus_w}
\end{figure}
\section{Discussion}
In this chapter we investigated the Hawking-like radiation produced during the time of gravitational collapse for both the asymptotic observer and the infalling observer. The occupation number of the radiation was then used to fit the temperature of the radiation as the shell approaches $R_s$. When considering Schwarzschild coordinates, in both cases the resulting analysis lead to the same conclusions: First, that the spectrum of the occupation display non-thermal characteristics during the time of collapse. This non-thermality is seen by a non-divergent occupation number when $\bar{\omega}=0$ and in oscillations about thermality. Second, the spectrum becomes more and more thermal as the domain wall approaches $R_s$, corresponding to large $\bar{\omega}$ values. Finally, the spectrum becomes purely thermal when the domain wall reaches $R_s$. When consider Eddington-Finkelstein coordinates for the infalling observer, we find that the spectrum never becomes thermal in the high frequency range.
In the case of the asymptotic observer, upon fitting the temperature, we find that the temperature of the radiation is on the order of the Hawking temperature. This value is not exactly the Hawking temperature for two reasons. First, when fitting the temperature we use a best fit approximation for the slope of $\beta$. However, there is ambiguity for choosing the best fit approximation, thus the true slope of $\beta$ may be different from the one chosen. Second, we are fitting the temperature numerically. There is always an inherent approximation used we using numerical methods, therefore our calculation is inherently ambiguous.
In the case of the infalling observer, upon fitting the temperature in Schwarzschild coordinates, we find that when the shell reaches $R_s$ the temperature of the radiation becomes divergent. This would seem to imply that the local temperature measured by the observer is then infinite, meaning that the observer will burn up before he/she reaches $R_s$. However, this is not necessarily the case. It has been argued in Ref.\cite{Fulling}, where a simple $1+1$ model was studied, that the local vacuum polarization will cancel out the divergent temperature energy density due to the radiation. Therefore, the true local value of the stress-energy tensor is small in the region $R\sim R_s$. A simple reason for this divergent temperature is that the Schwarzschild observer is actually an accelerated observer, so to truly investigate the local temperature one must consider a truly freely falling observer, i.e. the Eddington-Finkelstein coordinates. In this case, upon fitting the temperature, we find that the local temperature is in fact finite.
\chapter{Formalism}
\label{ch:formal}
The most important part of our research is the formalism used to study the quantum mechanical effects of gravitational collapse. To study these quantum effects of gravitational collapse we will institute the Functional Schr\"odinger equation. In this part of the thesis we will first derive the Functional Schr\"odinger equation, which will be the primary equation used to study the quantum effects of gravitational collapse.
The main purpose of the Functional Sch\"odinger equation is to introduce the ``observer" time into the Wheeler-de Witt equation. It is well known in General Relativity that for different foliations of space-time, different physical observations occur. For example, one can consider the equations of motion for a black hole. Let us consider an object which is falling into a black hole from different points of view. If one chooses the time as observed by an asymptotic observer as the desired foliation of space-time, upon solving the equations of motion of the object, one finds that it takes an infinite amount of time for the object to fall into the black (even if the object is a photon). However, if one choses the time as observed by a freely falling observer (one that is falling into the black hole along a geodesic) as the desired foliation of space-time, upon solving the equations of motion of the object in this case, one finds that it takes a finite amount of time for the object to fall into the black hole. Therefore, one can see that it is important and instructive to consider different foliations of the space-time to learn different aspects of the system of gravitational collapse. The Functional Schr\"odinger equation allows for one to specify the particular foliation of space-time that is of interest and study the system from that view point.
A second purpose of the Functional Schr\"odinger equation is to allow one to investigate the time evolution of the system. Typically this is not done in the study of gravitational collapse. The preferred method of study is to consider an initial static asymptotically flat space-time, let the system evolve (with no knowledge of the evolution), then consider a final different static asymptotically flat space-time. Then by comparing these two different space-times, one can in principle have some understanding of what the evolution was like between these two events. The Functional Schr\"odinger equation will in principle allow us to study the total time evolution of the system, not just the static asymptotically flat regions of space-time.
\section{Functional Schr\"odinger Equation}
In this section we will derive the Functional Schr\"odinger equation.
The Wheeler-de Witt equation for a closed universe is given by, see Ref.\cite{DeWitt},
\begin{equation}
H\Psi=0
\label{WdW}
\end{equation}
where $H$ is the total Hamiltonian and $\Psi$ is the total wavefunction for all the ingredients of the system, including the observer's degrees of freedom denoted by ${\cal{O}}$. Eq.(\ref{WdW}) is a consequence of the idea that there is no ``God" time, or no preferred time, or no super-observer time. Therefore Eq.(\ref{WdW}) is written in a gauge independent fashion, since there is no preferred observer to observe the system.
In general we can write the wavefunction in Eq.(\ref{WdW}) as
\begin{equation}
\Psi=\Psi\left(X^{\alpha},g_{\mu\nu},\Phi,{\cal{O}}\right).
\label{wPsi}
\end{equation}
Here $X^{\alpha}=X^{\alpha}(\zeta^a)$ describes the location of the wall as a function of the internal wall world volume coordinates $\zeta^a$, $g_{\mu\nu}$ is the metric, and $\Phi$ is a scalar field. The Roman indices go over the internal domain wall world volume coordinates and the Greek indices go over space-time coordinates. Note that the wavefunctional in Eq.(\ref{wPsi}) is a functional of the fields but not the space-time coordinates. In general, the total Hamiltonian is a linear combination of the Hamiltonian of the system itself and that of the Hamiltonian of the observer. Therefore we will separate the Hamiltonian into two parts, one for the system and the other for the observer, which can be written as
\begin{equation}
H=H_{sys}+H_{obs}.
\label{commutation}
\end{equation}
Any (weak) interaction terms between the observer and the wall-metric-scalar system are included in $H_{sys}$. The observer is assumed to not significantly affect the evolution of the system and vice versa. In mathematical language this means that we are assuming that the Hamiltonian for the system and the observer commute with each other
\begin{equation}
\left[H_{sys},H_{obs}\right]=0.
\end{equation}
The total wavefunction Eq.(\ref{wPsi}) can be written as a sum over eigenstates
\begin{equation}
\Psi=\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)
\label{totPsi}
\end{equation}
where $k$ labels the eigenstates and $c_k$ are complex coefficients.
To solve the full Wheeler-de Witt equation is very difficult since it involves all the degrees of freedom, both that of the system and the observer. Here we shall utilize the frequently employed strategy of truncating the field degrees of freedom to a finite subset, hence we will be consider with the minsuperspace version of the Wheeler-de Witt equation. As long as we keep all the relevant degrees of freedom that are of interest, this is a useful truncation. Since we are only considering a subset of the total degrees of freedom, this is now considered an ``open" system. In an ``open" system, one can then define an appropriate ``observer" time in which one chooses to make measurements. Therefore we can write the Schr\"odinger equation for the observation as
\begin{equation}
i\frac{\partial\Psi^k_{obs}}{\partial t}\equiv H_{obs}\Psi_{obs}^k.
\label{ObsSchrod}
\end{equation}
This is convenient, however, we wish to make observations on the system not on the observer. To transform this to observation on the system we will make use Eq.(\ref{WdW}).
To introduce the observer time on observations of the system, we use Eq.(\ref{WdW}), Eq.(\ref{totPsi}) and Eq.(\ref{ObsSchrod}), therefore we can write
\begin{align}
H\Psi&=\left(H_{sys}+H_{obs}\right)\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)\nonumber\\
&=H_{sys}\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)+H_{obs}\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)\nonumber\\
&=H_{sys}\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)+\sum_kc_k\Psi^k_{sys}(sys,t)H_{obs}\Psi_{obs}^k({\cal{O}},t)\nonumber\\
&=H_{sys}\sum_kc_k\Psi^k_{sys}(sys,t)\Psi_{obs}^k({\cal{O}},t)+i\sum_kc_k\Psi^k_{sys}(sys,t)\partial_t\Psi_{obs}^k({\cal{O}},t)
\label{der}
\end{align}
where we made use of Eq.(\ref{commutation}) and in the last line we used Eq.(\ref{ObsSchrod}).
Now consider the integral of the last term in Eq.(\ref{der}), we have
\begin{align}
\int_{t_i}^{t_f}dt\sum_kc_k\Psi_{sys}^k(sys,t)\partial_t\Psi_{obs}^k({\cal{O}},t)=&\sum_kc_k\Psi_{sys}^k(sys,t)\Psi_{obs}^k({\cal{O}},t)\Big{|}_{t_i}^{t_f}\nonumber\\
&-\int_{t_i}^{t_f}dt\sum_k\left(\partial_t\Psi_{sys}^k(sys,t)\right)\Psi_{obs}^k({\cal{O}},t)\nonumber\\
=&\Psi\Big{|}_{t_i}^{t_f}-\int_{t_i}^{t_f}dt\sum_k\left(\partial_t\Psi_{sys}^k(sys,t)\right)\Psi_{obs}^k({\cal{O}},t).
\label{by_parts}
\end{align}
However, by virtue of the Wheeler-de Witt equation, the total wavefunction $\Psi$ is time-independent. Therefore the first term on the right hand side in Eq.(\ref{by_parts}) is zero. Shrinking the integral we then have,
\begin{equation}
\sum_kc_k\Psi_{sys}^k(sys,t)\partial_t\Psi_{obs}^k({\cal{O}},t)=-\sum_kc_k\left(\partial_t\Psi_{sys}^k(sys,t)\right)\Psi_{obs}^k({\cal{O}},t).
\label{equiv}
\end{equation}
Substituting Eq.(\ref{equiv}) into Eq.(\ref{der}) we can then write,
\begin{equation}
H_{sys}\sum_kc_k\Psi_{sys}^k(sys,t)\Psi_{obs}^k({\cal{O}},t)=i\sum_kc_k\left(\partial_t\Psi_{sys}^k(sys,t)\right)\Psi_{obs}^k({\cal{O}},t)
\end{equation}
or interms of one $k$ value, we then arrive at the Functional Schr\"odinger equation
\begin{equation}
H_{sys}\Psi_{sys}^k=i\frac{\partial\Psi_{sys}^k}{\partial t}.
\end{equation}
For convenience, from now on we will denote the system wavefunction simply by $\Psi$ and drop the superscript $k$ and the subscript ``sys". Similarly $H$ will now denote $H_{sys}$, and the Schr\"odinger equation reads
\begin{equation}
H\Psi=i\frac{\partial\Psi}{\partial t}.
\label{FSE}
\end{equation}
\section{Discussion}
Here we have derived the Functional Schr\"odinger equation. As discussed above, the purpose of the Functional Schr\"odinger equation is to introduce the ``observer" time into the Wheeler-de Witt equation, Eq.(\ref{WdW}). This will allow us to be able to use the classical Hamiltonian of the system of gravitational collapse, then study the evolution of the system from the view point of any observer of our choosing. This has two benefits: First the formalism allows us to choose the ``observer" we wish to study. As discussed earlier, different ``observers" will observer different phenomena which are of interest. Secondly, the formalism will allow us to evolve the system quantum mechanically over time. One of the benefits of this approach is that we can, in principle, observe thermodynamic properties of the system in a time-dependent fashion, which we will discuss in Chapters \ref{ch:radiation} and \ref{ch:entropy}. As we will discuss, this is something which is beyond the scope of the usual methods used to study the thermodynamic properties of the system.
\chapter{Model}
\label{ch:model}
To study a concrete realization of black hole formation we consider a spherically symmetric Nambu-Goto domain wall (representing a shell of matter) that is collapsing. To include the possibility of (spherically symmetric) radiation, as discussed in the previous chapter (Chapter \ref{ch:number}), we consider a massless scalar field, $\Phi$, that is coupled to the gravitational field but not directly to the domain wall. The action for the system is then given as
\begin{equation}
S=\int d^4x\sqrt{-g}\left[-\frac{1}{16\pi G}{\cal{R}}+\frac{1}{2}(\partial_{\mu}\Phi)^2\right]-\sigma\int d^3\zeta\sqrt{-\gamma}+S_{obs}.
\label{gen_action}
\end{equation}
The first term is the Einstein-Hilbert action for the gravitational field, the second is the scalar field action, the third is the domain wall action in terms of the wall world volume coordinates, $\zeta^a$ ($a=0,1,2$), the wall tension $\sigma$, and the induced world volume metric
\begin{equation}
\gamma_{ab}=g_{\mu\nu}\partial_aX^{\mu}\partial_bX^{\nu}.
\end{equation}
As stated in Chapter \ref{ch:formal}, the coordinates $X^{\mu}=X^{\mu}(\zeta^a)$ describe the location of the wall. The term $S_{obs}$ in Eq.(\ref{gen_action}) denotes the action for the observer.
As discussed earlier, a general treatment of full Wheeler-de Witt equation, Eq.(\ref{WdW}), is very difficult. So, we shall use the frequently employed strategy of truncating the field degrees of freedom to a finite set, typically including only the relevant degrees of freedom. In other words, we will consider the minisuperspace version of the Wheeler-de Witt equation. As long as we keep all the relevant degrees of freedom, this is a useful truncation. Since we are considering spherically symmetric domain walls, we will assume spherical symmetry for all the fields. Thus, the wall is described by the radial degree of freedom $R(t)$ only.
The metric for the wall is then taken to be the solution to Einstein equations for a spherical domain wall. In Ref.\cite{Ipser} the metric, as follows from the spherical symmetry, outside the wall is given by
\begin{equation}
ds^2=-\left(1-\frac{R_s}{r}\right)dt^2+\left(1-\frac{R_s}{r}\right)^{-1}dr^2+r^2d\Omega^2, \hspace{2mm} r>R(t)
\label{Met_out}
\end{equation}
where $R_s=2GM$ is the Schwarzschild radius in terms of the mass $M$ of the wall, and
\begin{equation}
d\Omega^2=d\theta^2+\sin^2\theta d\phi^2.
\end{equation}
By Birkhoff's theorem, in the interior of the spherical domain wall the line element of the metric is flat, i.e. Minkowski, which is given by
\begin{equation}
ds^2=-dT^2+dr^2+r^2d\Omega^2, \hspace{2mm} r<R(t).
\label{Met_in}
\end{equation}
Here $T$ is the interior time coordinate, not to be confused with temperature. The interior time coordinate is related to the asymptotic observer time coordinate $t$ via the proper time $\tau$ of the domain wall. By matching the coordinates for the interior and exterior at the wall, in analogy with the Isreal junction condition (see Ref.\cite{Israel}), and assuming that the wall is infinitely thin, we have the relations
\begin{equation}
\frac{dT}{d\tau}=\sqrt{1+\left(\frac{dR}{d\tau}\right)^2}
\label{dTdtau}
\end{equation}
and
\begin{equation}
\frac{dt}{d\tau}=\frac{1}{B}\sqrt{B+\left(\frac{dR}{d\tau}\right)^2}
\label{dtdtau}
\end{equation}
where
\begin{equation}
B\equiv1-\frac{R_s}{R}.
\label{B}
\end{equation}
By taking the ratio of Eq.(\ref{dTdtau}) and Eq.(\ref{dtdtau}), the relationship between the interior time $T$ and the asymptotic time $t$ is given by
\begin{equation}
\frac{dT}{dt}=\frac{\sqrt{1+R_{\tau}^2}B}{\sqrt{B+R_{\tau}^2}}=\sqrt{B-\frac{(1-B)}{B}\dot{R}^2}
\label{dTdt}
\end{equation}
where $R_{\tau}=dR/d\tau$ and $\dot{R}=dR/dt$.
Since we are restricting the system to fields with spherical symmetry only, we need not include other metric degrees of freedom. Thus, the scalar field can also be truncated to be the spherically symmetric modes
\begin{equation}
\Phi=\Phi(r,t).
\end{equation}
In Ref.\cite{Ipser}, Ipser and Sikivie integrated the equations of motion for the spherically symmetric domain wall. They found that the mass is actually a constant of motion and is given by, see Appendix \ref{ch:GC} for a sketch of the method used,
\begin{equation}
M=\frac{1}{2}4\pi\sigma R^2\left[\sqrt{1+R_{\tau}^2}+\sqrt{B+R_{\tau}^2}\right]
\label{mass}
\end{equation}
where it is assumed that max($R$)$>1/4\pi\sigma G$. This assumption is just used to ensure that one does not start off inside of the collapsing spherical domain wall.
By virtue of Eq.(\ref{B}), Eq.(\ref{mass}) is implicit since $R_s=2GM$. Solving for $M$ explicitly in terms of $R_{\tau}$ gives
\begin{equation}
M=4\pi\sigma R^2\left[\sqrt{1+R_{\tau}^2}-2\pi\sigma GR\right].
\label{Mass_tau}
\end{equation}
However, making use of Eq.(\ref{dTdt}) we can solve for $M$ in terms of $R_T=dR/dT$
\begin{equation}
M=4\pi\sigma R^2\left[\frac{1}{\sqrt{1-R_T^2}}-2\pi\sigma GR\right].
\label{Mass_T}
\end{equation}
Before we proceed we wish to discuss the physical relevance of Eq.(\ref{Mass_tau}). First consider the case where $R_{\tau}=0$, i.e. for a static domain wall. The first term in the square bracket is just the total rest mass of the shell. When the shell is moving, i.e. $R_{\tau}\not=0$, the first term in the square bracket takes the kinetic energy of the domain wall into account. The last term in the square bracket is the self-gravity, or the binding energy of the domain wall. Therefore we can identify Eq.(\ref{Mass_tau}) (Eq.(\ref{Mass_T})) as the total energy of the system, hence the Hamiltonian of the system. Thus, we will refer to Eq.(\ref{Mass_tau}) (Eq.(\ref{Mass_T})) as the Hamiltonian.
\section{Discussion}
Here we developed the classical Hamiltonian for a massive spherically symmetric domain wall undergoing gravitational collapse. As stated in the last paragraph, Eq.(\ref{Mass_tau}) is the conserved mass of the system, however, it can be interpreted as the Hamiltonian. For the remainder of the text, we will use Eq.(\ref{Mass_tau}) (Eq.(\ref{Mass_T})) as the Hamiltonian for the system.
\chapter{Occupation Number}
\label{ch:number}
Throughout the text we will be interested in the number of particles created during the gravitational collapse of our object, i.e. the radiation. Therefore we will derive the occupation number of the particles created during the time of collapse in this chapter for future convenience. Throughout this text we will be interested in systems with spherical symmetry, since this is the simplest case to consider. Here we note that due to the spherical symmetry, gravitational radiation is excluded from the system, thus the radiation which we will consider will be from the excitation of particles due to the time-dependent nature of the gravitational metric.
To consider the radiation we will consider a quantum scalar field $\phi$ in the background of the gravitational collapsing object, which is given by
\begin{equation}
S=\int d^4x\frac{1}{2}\sqrt{-g}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi
\label{ScalarAct}
\end{equation}
where $S$ is the action of the scalar field. The reason we are considering only a scalar field is that this is the simplest and easiest case, which gives insight into most of the physically significant phenomena. By considering more complicated fields, one arrives at the so-called gray-body factors, see for example Ref.\cite{Birrell}, which are dependent on the type of field used. Here we derive the number of particles induced as a function of observer time ``$t$". Here ``$t$" is used for any foliation of space-time used throughout this body of work, whether the time is that of an asymptotic observer or that of an infalling observer.
In most cases we arrive at the Hamiltonian of the system from Eq.(\ref{ScalarAct}), which is of the form of a sum of uncoupled simple harmonic oscillator. To simplify the notation, we consider one eigenmode of the simple harmonic oscillator given by
\begin{equation}
H=\frac{p^2}{2m}+\frac{m}{2}\omega^2(t)x^2
\label{SHO}
\end{equation}
where $p$ is the momentum conjugate to $x$ and $x$ is the eigenmode. Using the standard quantization procedure, upon inserting Eq.(\ref{SHO}) into Eq.(\ref{FSE}), we can then write
\begin{equation}
\left[-\frac{1}{2m}\frac{\partial^2}{\partial x^2}+\frac{m}{2}\omega^2(t)x^2\right]\psi(x,t)=i\frac{\psi(x,t)}{\partial t}.
\label{SHOFS}
\end{equation}
Here Eq.(\ref{SHOFS}) can be solved exactly by utilizing the invariant operator method first developed by Lewis and Reisenfeld, see Ref.\cite{Lewis} and Appendix \ref{ch:Invariant}. Using this method, Dantas, Pedrosa and Baseia showed, see Ref.\cite{Pedrosa}, that the exact solution to Eq.(\ref{SHOFS}) at late times is given by
\begin{equation}
\psi(x,t)=e^{i\alpha(t)}\left(\frac{m}{\pi\rho^2}\right)^{1/4}\exp\left[i\frac{m}{2}\left(\frac{\rho_t}{\rho}+\frac{i}{\rho^2}\right)x^2\right]
\label{PedWave}
\end{equation}
where $\rho_t$ denotes the derivative of $\rho(t)$ with respect to $t$, and $\rho$ is given by the real solution of the non-linear auxilarly equation
\begin{equation}
\rho_{tt}+\omega^2(t)\rho=\frac{1}{\rho^3}
\label{gen_rho}
\end{equation}
with intitial conditions
\begin{equation}
\rho(t_i)=\frac{1}{\sqrt{\omega_0}}, \hspace{2mm} \rho_t(t_i)=0
\label{IC}
\end{equation}
where $t_i$ is the initial time. The time-dependent phase $\alpha$ is given by
\begin{equation}
\alpha(t)=-\frac{1}{2}\int_{t_i}^t\frac{dt'}{\rho^2(t')}.
\end{equation}
To find the occupation number of the induced radiation, consider an observer with detectors that are designed to register particles of different frequencies for the free scalar field $\Phi$ at earlier times. Such an observer will interpret the wavefunction of a given mode $x$ at late times in terms of simple harmonic oscillator states, $\{\varphi_n\}$, at final frequency $\bar{\omega}$. Here $\bar{\omega}$, is the value of the frequency evaluated at a time $t_f$ as seen by the observer. The number of quanta in eigenmode $x$ can be evaluated by decomposing the wavefunction Eq.(\ref{PedWave}) in terms of the states $\{\varphi_n\}$, and by evaluating the occupation number of that mode. To implement this, we start by writing the wavefunction for a given mode at time $t>t_f$ in terms of the simple harmonic oscillator basis at $t=0$
\begin{equation}
\psi(x,t)=\sum_nc_n(t)\varphi_n(x)
\label{Phi_Exp}
\end{equation}
where
\begin{equation}
c_n=\int dx\varphi_n^*(x)\psi(x,t)
\label{c_n}
\end{equation}
is the overlap, i.e. inner product, between the initial and final state of the wavefunction. The occupation number at eigenfrequency $\bar{\omega}$ by the time $t>t_f$, is given by the expectation value
\begin{equation}
N(t,\bar{\omega})=\sum_nn\left|c_n\right|^2.
\label{occnum}
\end{equation}
To evaluate the sum in Eq.(\ref{occnum}), we use the simple harmonic oscillator basis states but at a frequency $\bar{\omega}$ to keep track of the different $\omega$'s in the calculation. To evaluate the occupation numbers at time $t>t_f$, we need only set $\bar{\omega}=\omega(t_f)$. So the simple harmonic oscillator basis states are written as (see for example Appendix A.4 of Ref.\cite{Sakurai})
\begin{equation}
\varphi(b)=\left(\frac{m\bar{\omega}}{\pi}\right)^{1/4}\frac{e^{-m\bar{\omega}b^2/2}}{\sqrt{2^nn!}}{\cal{H}}_n\left(\sqrt{m\bar{\omega}}b\right)
\end{equation}
where ${\cal{H}}_n$ are the Hermite polynomials. Then Eq.(\ref{c_n}) and Eq.(\ref{PedWave}) together gives
\begin{eqnarray}
c_n&=&\left(\frac{1}{\pi^2\bar{\omega}\rho^2}\right)^{1/4}\frac{e^{i\alpha}}{\sqrt{2^nn!}}\int d\xi e^{-P\xi^2/2}{\cal{H}}_n(\xi)\nonumber\\
&=&\left(\frac{1}{\pi^2\bar{\omega}\rho^2}\right)^{1/4}\frac{e^{i\alpha}}{\sqrt{2^nn!}}I_n
\end{eqnarray}
where
\begin{equation}
P=1-\frac{i}{\bar{\omega}}\left(\frac{\rho_{\eta}}{\rho}+\frac{i}{\rho^2}\right).
\label{P}
\end{equation}
To find $I_n$ consider the corresponding integral over the generating function for the Hermite polynomials
\begin{eqnarray}
J(z)&=&\int d\xi e^{-P\xi^2/2}e^{-z^2+2z\xi}\nonumber\\
&=&\sqrt{\frac{2\pi}{P}}e^{-z^2(1-2/P)}.
\end{eqnarray}
Since
\begin{equation}
e^{-z^2+2z\xi}=\sum_{n=0}^{\infty}\frac{z^n}{n!}{\cal{H}}_n(\xi)
\end{equation}
we can then write
\begin{equation}
\int d\xi e^{-P\xi^2/2}{\cal{H}}_n(\xi)=\frac{d^n}{dz^n}J(z)\Big{|}_{z=0}.
\end{equation}
Therefore
\begin{equation}
I_n=\sqrt{\frac{2\pi}{P}}\left(1-\frac{2}{P}\right)^{n/2}{\cal{H}}_n(0).
\end{equation}
Since
\begin{equation}
{\cal{H}}_n(0)=(-1)^{n/2}\sqrt{2^nn!}\frac{(n-1)!!}{\sqrt{n!}}, \hspace{2mm} n=even
\end{equation}
and ${\cal{H}}_n(0)=0$ for $n=odd$, we find the coefficients $c_n$ for even values of $n$,
\begin{equation}
c_n=\frac{(-1)^{n/2}e^{i\alpha}}{(\bar{\omega}\rho^2)^{1/4}}\sqrt{\frac{2}{P}}\left(1-\frac{2}{P}\right)^{n/2}\frac{(n-1)!!}{\sqrt{n!}}.
\end{equation}
For odd $n$, $c_n=0$.
We can now find the number of particles produced during the collapse. Let
\begin{equation}
\chi=\left|1-\frac{2}{P}\right|.
\label{chi}
\end{equation}
Then using Eq.(\ref{occnum}) we have
\begin{eqnarray}
N(t,\bar{\omega})&=&\sum_{n=even}n\left|c_n\right|^2\nonumber\\
&=&\frac{2}{\sqrt{\bar{\omega}\rho^2}|P|}\chi\frac{d}{d\chi}\sum_{n=even}\frac{(n-1)!!}{n!!}\chi^n\nonumber\\
&=&\frac{2}{\sqrt{\bar{\omega}\rho^2}|P|}\chi\frac{d}{d\chi}\frac{1}{\sqrt{1-\chi^2}}\nonumber\\
&=&\frac{2}{\sqrt{\bar{\omega}\rho^2}|P|}\frac{\chi^2}{(1-\chi^2)^{3/2}}.
\label{OccNum}
\end{eqnarray}
Now inserting Eq.(\ref{P}) and Eq.(\ref{chi}) leads to
\begin{equation}
N(t,\bar{\omega})=\frac{\bar{\omega}\rho^2}{\sqrt{2}}\left[\left(1-\frac{1}{\bar{\omega}\rho^2}\right)^2+\left(\frac{\rho_t}{\bar{\omega}\rho}\right)^2\right].
\end{equation}
\section{Discussion}
Here we derived the occupation number of the radiation induced during the time of gravitational collapse. The occupation number is measured by an observer with a detector at late times $t>t_f$. As stated earlier, this was done for convenience since we will use this quantity several times during this text.
\chapter{Quantum Treatment}
\label{ch:quantum}
In this chapter we wish to study the quantum equations of motion using the Functional Schr\"odinger equation Eq.(\ref{FSE}), again for the two different foliations of space-time. The idea here is that the quantum mechanical effects will change some of the difficulties that arise from the classical solutions. The ultimate goal is that examining these quantum mechanical effects will give us insight into the quantum mechanical nature of gravitational collapse and will possibly help guide us to be able to construct the appropriate theory of quantum gravity.
Some of the difficulties that arise from the classical solutions of gravitation collapse are discussed below. This is not meant to be an exhaustive list, but give the reader of an idea of the topics that we wish to address in this text.
First we will discuss one difficulty faced by the asymptotic observer, the presence of the horizon. Why is the event horizon a difficult place for the asymptotic observer, while there is it is no problem for the infalling observer, since the horizon is nothing more than a coordinate singularity? Under the classical notion, as discussed in Chapter \ref{ch:Classical}, the presence of the coordinate singular creates an apparent gravitational time dilation, which makes the collapsing domain wall appear to stop. This effect is due to the divergence of the coordinate singularity when the observer sees the collapsing domain wall reach the Schwarzschild radius. There has been much discussion in this matter about how these quantum mechanical corrections can eliminate this effect. One such idea that has gained much attention over the past few years, we will discuss this process here. The basic idea here is that the upon quantizing the shell, the shell will now have quantum fluctuations in the position of the horizon. These fluctuations will then imply that the position of the horizon is no longer fixed, but will now be given by $R_s+\delta R_s$, where $\delta R_s$ represents the small fluctuations in the position of the horizon. These effects can then make the time as measured by the asymptotic observer finite (see for example Sec. 10.1.5 of Ref.\cite{FrolovNovikov})
\begin{equation}
\Delta t=\int_{R_s+\delta R_s}^{R_0}\frac{dr}{1-R_s/r}\sim R_s\ln\left(\frac{R_0-R_s}{\delta R_s}\right).
\end{equation}
If this were correct, we would be able to observe black hole formation (due to the collapse) and other effects in finite time. Note however, that the fluctuations can go either way. In the case of $R_s-\delta R_s$, the result becomes infinite agin.
Now we will discuss difficulties facing the infalling observer. For this observe there are really two important regimes: The regime in the region $R\sim R_s$ and the region $R\sim0$. For the region $R\sim R_s$, the concern is the exact opposite of that of the asymptotic observer. Will the quantization of the shell contradict the classical observation that the infalling observer sees the shell collapse to $R_s$ in a finite amount of time? For the region $R\sim0$, classically this represents the point of the classical singularity. Penrose and Hawking showed in Ref.\cite{PenroseHawking} that singularities are endemic in classical General Relativity. The question then arises whether these singularities are an intrinsic property of space-time or simply reflect our lack of the ultimate non-singular theory. The general belief is that quantization will rid gravitation of singularities. This is analogous to another theory where quantization got rid of the singularity, Electromagnetism. In atomic physics the singularity of the Coulomb potential, which has an identical $1/r$ behavior, was eliminated via quantization (see for example \cite{BogojevicStojkovic,Trodden,Borstnik,Shankaranarayanan}).
In this chapter we investigate these ideas. In later chapters we will investigate quantum mechanical corrects to addition aspects of gravitational collapse, those being thermodynamic.
\section{Asymptotic Observer}
First we will consider quantization of the shell from the view point of the asymptotic observer. This will be done by using Eq.(\ref{FSE}). For this section we outline the work done in Ref.\cite{Stojkovic}.
To utilize Eq.(\ref{FSE}) we need to use the Hamiltonian in Eq.(\ref{Ham Hor Pi_R}). However, we notice that Eq.(\ref{Ham Hor Pi_R}) has a squareroot in it. Therefore, we will consider the Hamiltonian squared
\begin{equation}
H^2=B\Pi_RB\Pi_R+B(4\pi\mu R^2)^2.
\label{H2}
\end{equation}
Before we proceed, we discuss the choice of ordering in the first term on the right hand side of Eq.(\ref{H2}). Since we are considering quantum mechanics, the distance $R$ and the conjugate momentum are now promoted to operators, which obey the standard commutation relations. Thus in general we would need to add terms to the squared Hamiltonian in Eq.(\ref{H2}) that depend on the commutator $[B,\Pi_R]$. However, in region of interest we find that the commutator is given by
\begin{equation*}
\left[B,\Pi_R\right]\sim\frac{1}{R_s}.
\end{equation*}
Estimating $H$ by the mass $M$ of the domain wall, as for the discussion in Chapter \ref{ch:model}, the terms due to the operater order ambiguity will be negligible provided
\begin{equation*}
M>>\frac{1}{R_s}\sim\frac{m_P^2}{M}
\end{equation*}
where $m_P$ is the Planck mass. Therefore we can ignore the ordering ambiguity and chose the ordering given in Eq.(\ref{H2}), provided that this limit is satisfied.
Now we apply the standard quantization procedure,
\begin{equation*}
\left[R,\Pi_R\right]=i.
\end{equation*}
We substitute
\begin{equation}
\Pi_R=-i\frac{\partial}{\partial R}
\label{QPR}
\end{equation}
into the squared Schr\"odinger equation
\begin{equation}
H^2\Psi=-\frac{\partial^2\Psi}{\partial t^2}.
\label{Schrod2}
\end{equation}
Inserting Eq.(\ref{H2}) into Eq.(\ref{Schrod2}) we then obtain
\begin{equation}
-B\frac{\partial}{\partial R}\left(B\frac{\partial\Psi}{\partial R}\right)+B(4\pi\mu R^2)^2\Psi=-\frac{\partial^2\Psi}{\partial t^2}.
\label{Schrod R}
\end{equation}
To find the wavefunction for the collapsing domain wall we need to solve Eq.(\ref{Schrod R}). To solve Eq.(\ref{Schrod R}) in terms of $R$ can be a formidable exercise in mathematics. However, we can simplify the matter by defining the tortoise coordinate
\begin{equation}
u=R+R_s\ln\left|\frac{R}{R_s}-1\right|.
\label{u}
\end{equation}
We can then see that Eq.(\ref{u}) then gives
\begin{equation}
B\Pi_R=-i\frac{\partial}{\partial u}
\label{BPi_R}
\end{equation}
where we used Eq.(\ref{QPR}). Using Eq.(\ref{BPi_R}) we can then rewrite Eq.(\ref{Schrod R}) as
\begin{equation}
\frac{\partial^2\Psi}{\partial t^2}-\frac{\partial^2\Psi}{\partial u^2}+B(4\pi\mu R^2)^2\Psi=0.
\label{Massive Wave}
\end{equation}
We can now identify Eq.(\ref{Massive Wave}) as just the massive wave equation in a Minkowski background with a mass term that depends on the position of the domain wall. We need to now solve Eq.(\ref{Massive Wave}) for $u$. To do so, we must first write the mass term in terms of $u$, rather than its present state of $R$. However, some care is needed since at $R=R_s$, $u$ is divergent, so we must take the appropriate branch. From Eq.(\ref{u}) we have that for the region $R\in(R_s,\infty)$ maps onto $u\in(-\infty,\infty)$ and $R\in(0,R_s)$ maps onto $u\in(0,-\infty)$.
We are interested in the situation of a collapsing shell, hence the region $R\in(R_s,\infty)$. Thus we are interested in $u$ in the region $u\in(-\infty,\infty)$. We can solve Eq.(\ref{Massive Wave}) for the entire region, however, we are mostly concerned with the effect when $R\sim R_s$. In the region $R\sim R_s$, the logarithm in Eq.(\ref{u}) dominates, so we can write
\begin{equation}
R\sim R_s+R_se^{u/R_s}.
\label{R(u)}
\end{equation}
We look for wave-packet solutions propagating toward $R_s$, or in terms of $u$, $u\rightarrow-\infty$. Thus from Eq.(\ref{R(u)}) we have
\begin{equation}
B\sim e^{u/R_s}\rightarrow0.
\end{equation}
This means that the last term in Eq.(\ref{Massive Wave}), the mass term, can be ignored in this region.
In the region $R\sim R_s$, the dynamics of the wave-packet is simply given by the free wave equation, where any function of light-cone coordinates $(u\pm t)$ is a solution. To make this explicit, we consider a Gaussian wave-packet propagating toward the Schwarzschild radius
\begin{equation}
\Psi=\frac{1}{\sqrt{2\pi}s}e^{-(u+t)^2/2s^2}
\label{wave pack t}
\end{equation}
where $s$ is some chosen width of the wave packet in the $u$ coordinate. The width of the wave-packet remains fixed in the $u$-coordinate while it shrinks in the $R$ coordinate via the relation $dR=Bdu$, as follows from Eq.(\ref{u}).
Let us consider some properties of Eq.(\ref{wave pack t}). First, we see that the wave-packet travels at the speed of light in the $u$ coordinate. This is expected since the Schr\"odinger equation takes the form of a massless wave equation in Minkowski space. Further, in the $u$ coordinate, the wave packet must travel out to $u=-\infty$ to get to the horizon, $R=R_s$. Thus we can conclude that the quantum domain wall does not collapse to $R_s$ in a finite amount of asymptotic time.
Therefore one can conclude that the quantum mechanical effect, i.e. the quantization of the domain wall, does not smear out the presence of the coordinate singularity at the Schwarzschild radius. The asymptotic will not see the formation of the horizon in a finite amount of his/her time. Hence, the quantum solution does not alter the classical result found in Chapter \ref{ch:Classical}.
\section{Infalling observer}
Now we consider quantization of the domain wall from the view point of the infalling observer. This will be done using Eq.(\ref{Mass_tau}) as the Hamiltonian of the system. As discussed earlier, here we wish to solve the Functional Schr\"odinger equation in two different regions of interest. The first region is that near the Schwarzschild radius, $R_s$. The second is the region near the classical singularity, in the region $R\sim0$. For this section we summarize the work originally done in Ref.\cite{GreenStoj}.
\subsection{Near Horizon}
The exact Hamiltonian in terms of $R_{\tau}$ is again given by Eq.(\ref{Mass_tau}). From Eq.(\ref{Mass_tau}) we again see the presence of a square-root. However, in this case it is not as easy to remedy this as it was in the case of the asymptotic observer, since upon squaring the Hamiltonian will not get rid of the square root. To simplify the analysis we will require that $R_{\tau}$ is small. This is indeed a restriction to the special motion of the wall, since in general $R_{\tau}$ can be large near $R_s$ if the shell is falling from a very large distance. However, one may always choose initial conditions in such a way that the initial position of the shell $R(\tau=0)$ is very close to $R_s$.
In the limit of small $R_{\tau}$, Eq.(\ref{Mass_tau}) simplifies to
\begin{equation}
H=4\pi\sigma R_s^2\left[1+\frac{1}{2}R_{\tau}^2-2\pi\sigma GR_s\right].
\label{Ham small Rtau}
\end{equation}
Then again in the same limit, the conjugate momentum Eq.(\ref{Pi_tau}) simplifies to
\begin{equation}
\tilde{\Pi}_R=4\pi\sigma R_s^2R_{\tau}.
\label{Pi_tau small Rtau}
\end{equation}
Ignoring the constant terms from the Hamiltonian Eq.(\ref{Ham small Rtau}) and using Eq.(\ref{Pi_tau small Rtau}) we can write the Hamiltonian as
\begin{equation}
H=\frac{\tilde{\Pi}_R}{8\pi\sigma R_s^2}.
\end{equation}
Using the standard quantization procedure, we substitute
\begin{equation}
\tilde{\Pi}_{R}=-i\frac{\partial}{\partial R}
\end{equation}
into the Schr\"odinger equation Eq.(\ref{FSE}), which yields
\begin{equation}
-\frac{1}{8\pi\sigma R_s^2}\frac{\partial^2\Psi}{\partial R^2}=i\frac{\partial\Psi}{\partial\tau}.
\label{Rs_tau SE}
\end{equation}
Investigating Eq.(\ref{Rs_tau SE}), we see that Eq.(\ref{Rs_tau SE}) is just the Schr\"odinger equation for a freely propagating ``particle" of mass $4\pi\sigma R_s^2$, as one can expect from this approximation. Since $R_s$ is only a finite distance away for an infalling observer we conclude that the wavefunction will collapse at $R_s$ in a finite amount of proper time.
For the question on if the quantization of the domain wall will cause problems for the infalling observer, we can conclude that quantum effects do not alter the classical result. Hence a collapsing shell crosses its own Schwarzschild radius in a finite proper time.
\subsection{Near the Origin}
Now we wish to investigate the quantization of the domain wall in the region of the classical singularity $R\sim0$.
The exact Hamiltonian in terms of $R_{\tau}$ is again given by Eq.(\ref{Mass_tau}), where $R_{\tau}$ is given by Eq.(\ref{dRdtau}). In the region near the classical singularity, i.e. in the limit $R\rightarrow0$, the classical expression for $R_{\tau}$ (keeping only the leading order term) becomes
\begin{equation}
R_{\tau}\approx-\frac{\tilde{h}}{R^2}
\label{RtauN0}
\end{equation}
where $\tilde{h}$ is defined in Chapter \ref{ch:Classical}. Eq.(\ref{RtauN0}) clearly shows that in this region the classical expression for $R_{\tau}$ diverges. Up to the leading term near the origin, Eq.(\ref{RtauN0}) implies that the Hamiltonian is
\begin{equation}
H=4\pi\sigma R^2R_{\tau}.
\label{Hnear0}
\end{equation}
Substituting the asymptotic behavior, Eq.(\ref{RtauN0}), in the expression for the generalized momentum Eq.(\ref{Pi_tau}) we have
\begin{equation}
\tilde{\Pi}_R=4\pi\sigma R^2\sinh^{-1}(R_{\tau}).
\label{Pi_tauN0}
\end{equation}
From Eq.(\ref{Pi_tauN0}) we see that
\begin{equation}
\lim_{R\rightarrow0}\tilde{\Pi}_R=0.
\end{equation}
This then gives
\begin{equation}
\lim_{R\rightarrow0}\frac{\tilde{\Pi}_R}{4\pi\sigma R^2}=-\infty.
\end{equation}
This implies that $R_{\tau}$, which is defined as
\begin{equation}
R_{\tau}=\sinh\left(\frac{\tilde{\Pi}_R}{4\pi\sigma R^2}\right)
\end{equation}
by virtue of Eq.(\ref{Pi_tau}), near the horizon becomes
\begin{equation}
R_{\tau}=\frac{1}{2}\exp\left(-\frac{\tilde{\Pi}_R}{4\pi\sigma R^2}\right).
\label{Rtaue0}
\end{equation}
Therefore substituting Eq.(\ref{Rtaue0}) into Eq.(\ref{Hnear0}) we have
\begin{equation}
2\pi\sigma R^2\exp\left(\frac{i}{4\pi\sigma R^2}\frac{\partial}{\partial R}\right)\Psi(R,\tau)=i\frac{\partial\Psi(R,\tau)}{\partial\tau}.
\label{SchrodN0}
\end{equation}
Let us consider some properties of Eq.(\ref{SchrodN0}). The differential operator in the exponent gives some unusual properties to the equation. First we note the presence of the $R^{-2}$ term. This implies that if we expand the exponent we can not stop the series after a finite number of terms, but instead need to include all orders of the expansion, we will make this explicit below. Thus, we need to include an infinite number of derivatives of the wavefunction $\Psi$ into the differential equation. An infinite number of derivatives of a certain function uniquely specifies the whole function. Thus, the value of (the derivative of) the function on the right hand side of Eq.(\ref{SchrodN0}) at one point depends on the values of the function at different points on the left hand side of the same equation. This is in strong contrast with ordinary local differential equations where the value of the function and certain finite number of its derivatives are related at the same point of space. This indicates that Eq.(\ref{SchrodN0}) describes physics which is not strictly local.
Here we will make the non-locality of Eq.(\ref{SchrodN0}) explicit. To illustrate we will use the expression involving the $\sinh$ term, this is not necessary, however, it will make the explanation more clear. Note that we can rewrite $\sinh$ as
\begin{equation*}
\sinh(x)=\frac{e^x-e^{-x}}{2}.
\end{equation*}
By making a change of variable we can make this more explicit. If we introduce the new variable $v=R^3$, Eq.(\ref{SchrodN0}) becomes
\begin{equation}
\pi\sigma v^{2/3}\left[\exp\left(\frac{3i}{4\pi\sigma}\right)-\exp\left(-\frac{3i}{4\pi\sigma}\right)\right]\frac{\partial}{\partial v}\Psi(v,\tau)=i\frac{\partial\Psi(v,\tau)}{\partial\tau}.
\label{Schrodv}
\end{equation}
We can then see that the the differential operator in the exponents in Eq.(\ref{Schrodv}) are just a translation operator, which shifts the argument of the wavefunction by a non-infinitesimal amount of $3i/4\pi\sigma$. Since the wavefunction is complex in general, a shift by a complex value is not a problem. Therefore Eq.(\ref{Schrodv}) can be written as
\begin{equation}
\pi\sigma v^{2/3}\left[\Psi\left(v+\frac{3i}{4\pi\sigma},\tau\right)-\Psi\left(v-\frac{3i}{4\pi\sigma},\tau\right)\right]=i\frac{\partial\Psi(v,\tau)}{\partial\tau}.
\label{ShiftSchrod}
\end{equation}
Here we make an additional change in variable and define $v'=v-3i/4\pi\sigma$. Then we can rewrite Eq.(\ref{ShiftSchrod}) as
\begin{equation}
\pi\sigma\left(v'+\frac{3i}{4\pi\sigma}\right)^{2/3}\left[\Psi\left(v'+\frac{6i}{4\pi\sigma},\tau\right)-\Psi\left(v',\tau\right)\right]=i\frac{\partial\Psi(v,\tau)}{\partial\tau}.
\end{equation}
To interpret this we rely on usual differential calculus. From calculus we have
\begin{equation}
f(x+\Delta x)-f(x)\approx f(x)+ \sum_{n=0}^{\infty}(\Delta x)^nf^{(n)}-f(x)=\sum_{n=0}^{\infty}(\Delta x)^nf^{(n)}\approx \Delta x\frac{df}{dx}.
\label{deriv}
\end{equation}
Here the last step assumes that $\Delta x$ is small so we are justified at keeping only the first term in the expansion. Now, performing the same procedure as in Eq.(\ref{deriv}) we can write
\begin{eqnarray}
\psi\left(v'+\Delta v',\tau\right)-\psi\left(v',\tau\right)&\approx&\psi(v',\tau)+\sum_{n=0}^{\infty}(\Delta v')^n\psi^{(n)}-\psi(v',\tau)\nonumber\\
&=&\sum_{n=0}^{\infty}(\Delta v')^n\psi^{(n)}
\end{eqnarray}
where
\begin{equation}
\Delta v'=-\frac{6i}{4\pi\sigma}.
\label{Deltav}
\end{equation}
However, here we cannot truncate the series after a finite number of derivatives since by virtue of Eq.(\ref{Deltav}), $\Delta v'$ is not a small shift, provided that $\sigma$ does not go to infinity. Therefore we must keep all orders of the derivative, since as in Eq.(\ref{deriv}) each derivative has higher powers of $\Delta v'$.
An interesting thing to note here is that as $\sigma\rightarrow0$, the non-local effect becomes stronger and more predominant. A possible understanding of this is as follows. Outside of the domain wall, there exists a certain Hilbert space, while on the inside there exists a second Hilbert space. The transition from the first Hilbert space to the second is not necessarily a smooth transition. When the domain wall collapses to the singularity, the effect of each of these Hilbert spaces is now taken into account. When dealing with a massive domain wall, the warping of space-time is greater, however it exists over a larger distance. This makes the transition more smooth from point to point. However, for a small domain wall, the warping is less noticeable at larger distances. Therefore, the transition is more violent the closer one gets to the classical singularity in this case, since in both cases the warping diverges.
What about the value of the wavefunction at the classical singularity? Eq.(\ref{ShiftSchrod}) shows that the wavefunction near the origin $\Psi(R\rightarrow0,\tau)$ is in fact related to the wavefunction at some distant point $\Psi(R\rightarrow(\frac{6i}{4\pi\sigma})^{1/3},\tau)$. This also implies that the wavefunction describing the collapsing domain wall is non-singular at the origin. Indeed, in the limit $R\rightarrow0$, this equation becomes
\begin{equation}
\frac{\partial\Psi(R\rightarrow0,\tau)}{\partial\tau}=0
\label{R-ind Schrod}
\end{equation}
where we used the fact that the wavefunction at some finite $R$, i.e. $\Psi(R\rightarrow(\frac{6i}{4\pi\sigma})^{1/3},\tau)$, is finite. From Eq.(\ref{R-ind Schrod}) it then follows that $\Psi(R\rightarrow0)=const$. This gives strong indication that quantization of the domain wall may indeed rid gravity of the classical singularity.
\section{Discussion}
In this chapter we quantized the collapsing domain wall and investigated the quantum corrections. In the first section we did this with respect to the asymptotic observe to see if these fluctuations changed the classical observation that the domain wall takes an infinite amount of time to reach $R_s$. Upon quantizing the domain wall, we found that in this view point the classical scenario was not changed by the quantum fluctuations.
In the second section we investigated the quantization of the domain wall from the view point of the infalling observer. Here we did this for two different points of interest, near the horizon $R\sim R_s$ and near the classical singularity $R\sim0$, respectively. In the region $R\sim R_s$, we found that upon quantizing the domain wall, the classical view point was again unchanged. The quantum fluctuations did not alter the fact that according to the infalling observer, the domain wall will collapse to $R_s$ in a finite amount of proper time. In the region $R\sim0$, we found some interesting properties of the wavefunction. First, we found that the physics of the wavefunction in this region are strongly non-local, meaning that the value of the wavefunction at $R\sim0$ depends on the value of the wavefunction some distance away from the classical singularity. This situation has been previously suggested in the context of the information loss paradox (see for example \cite{Lowe,Horowitz,Giddings}). As we pointed out in the section, what is interesting is that this non-local behavior becomes increasingly manifest in the limit that $\sigma\rightarrow0$, i.e. that the mass of the domain wall becomes very small. This may be a consequence of the non-separability of the Hilbert space between the outside of the domain wall and the singularity. In the massive domain wall scenario, the transition from the outside to the inside Hilbert space is smoother than in the case of the light domain wall. Secondly, we found that the wavefunction is in fact finite at the classical singularity, which implies that quantum fluctuations may rid gravity of the classical singularity.
\chapter{$\rho(t)$ Equation}
\label{ch:rho_t}
This work was originally completed in Ref.\cite{Stojkovic}, here we will outline the results.
In the range $t<0$, $\omega$ is a constant and the solution to Eq.(\ref{gen_rho}) is
\begin{equation}
\rho(\eta)=\frac{1}{\sqrt{\omega_0}}.
\end{equation}
In the range of interest, during the time of gravitational collapse, we do not have an analytical solution to Eq.(\ref{gen_rho}). However, we can find certain useful properties of $\rho(t)$. First note that in terms of $\eta$
\begin{equation}
\omega^2=\frac{\omega_0^2}{1-\eta/R_s}.
\end{equation}
Then after rescaling, Eq.(\ref{gen_rho}) can be written as
\begin{equation}
\frac{d^2f}{d\eta'^2}=-(\omega_0R_s)^2\left[\frac{f}{1-\eta'}-\frac{1}{f^3}\right]
\label{f_t eq}
\end{equation}
where $\eta'=\eta/R_s$, $f=\sqrt{\omega_0}\rho$. The boundary conditions are then
\begin{equation}
f(0)=1, \hspace{2mm} \frac{df(0)}{d\eta'}=0.
\end{equation}
The last term in Eq.(\ref{f_t eq}) becomes singular as $f\rightarrow0$. We can then consider a more well behaved function for $1/f^3$. For example
\begin{equation}
\frac{d^2g}{d\eta'^2}=-(\omega_0R_s)^2\left[\frac{g}{1-\eta'}-g\right]
\label{g_t eq}
\end{equation}
with boundary conditions
\begin{equation}
g(0)=1, \hspace{2mm} \frac{dg(0)}{d\eta'}=0.
\label{IC g_t}
\end{equation}
Eq.(\ref{g_t eq}) implies that $g(\eta')$ is a monotonically decreasing function as long as $g(\eta')>0$. Furthermore, it is decreasing faster than the solution for $f$ as long as $f<1$, since the $1/f^3$ in Eq.(\ref{f_t eq}) is a larger ``repulsive" force than the $g$ term in Eq.(\ref{g_t eq}). So
\begin{equation}
g(\eta')\leq f(\eta')
\end{equation}
for all $\eta'$ such that $g(\eta')>0$.
Eq.(\ref{g_t eq}) with initial conditions Eq.(\ref{IC g_t}) can be solved in terms of degenerate hypergeometric functions. The important part for us is that $g$ is positive for all $\eta'$ and, in particular, $g(1)>0$ for all the values of $\omega_0R_s$ that we have checked. Therefore $f(\eta')$ is positive, at least for a wide range of $\omega_0R_s$.
We can find some more properties of $\rho(t)$. Let $f_1=f(1)\not=0$. Then the equation for $f$ can be expanded near $\eta'=1$.
\begin{equation}
\frac{d^2f_1}{d\eta'^2}\sim-(\omega_0R_s)^2\left[\frac{f_1}{1-\eta'}-\frac{1}{f_1^3}\right].
\label{f1_t eq}
\end{equation}
This shows that
\begin{equation}
\frac{df}{d\eta'}\sim(\omega_0R_s)^2f_1\ln(1-\eta')\rightarrow-\infty
\end{equation}
as $\eta'\rightarrow1$.
Hence $\rho(\eta=R_s)$ is strictly positive and finite while $\rho_{\eta}(\eta=R_s)=-\infty$ for finite and non-zero $\omega_0$. Since $f=\sqrt{\omega_0}\rho$ and $f\rightarrow1$ for $\omega_0\rightarrow0$, we also see that $\rho\rightarrow\infty$ and $\rho_{\eta}\rightarrow0$ as $\omega_0\rightarrow0$.
In the range $t_f<t$, $\omega$ is a constant. However, the solution for $\rho$ is not constant, unlike in the range $t<0$, since the constant solution $1/\sqrt{\omega(t_f)}$ does not necessarily match up with $\rho(t_f-)$ to ensure a continuous solution. Yet it is easy to check that in this range $\dot{N}=0$ and so there is no change in the occupation numbers. So we need only find $N(t_f-,\bar{\omega})$ to determine $N(t\rightarrow\infty,\bar{\omega})$.
\chapter{$\rho(\tau)$ Equation}
\label{ch:rho(tau)}
To get an understanding of the number of particles created in the region near the horizon we need to investigate the behavior of the function $\rho(\tau)$ near the Schwarzschild radius.
Near the horizon we can then write the velocity term as
\begin{equation}
\left|R_{\tau}\right|\approx\textrm{const}\equiv A.
\end{equation}
In this limit the position of the shell is then, from Eq.(\ref{R_tau_0})
\begin{equation}
R(\tau)\approx \tilde{R}_0-A\tau
\label{A}
\end{equation}
where, as stated in Chapter \ref{ch:Classical}, $\tilde{R}_0$ is the initial position of the shell, we can write
\begin{equation}
\frac{\sqrt{1+R_{\tau}^2}}{\left|R_{\tau}\right|}\equiv C.
\label{C}
\end{equation}
Therefore the frequency becomes
\begin{equation}
\omega^2\approx\frac{\omega_0^2}{CB}.
\label{approx omega}
\end{equation}
Therefore the auxiliary equation becomes
\begin{equation*}
\rho_{\eta\eta}+\omega_0^2\frac{R_s}{C((R_0-R_s)-A\tau)}\rho=\frac{1}{\rho^3}
\end{equation*}
or using Eq.(\ref{eta_tau}) we can write this as,
\begin{equation}
\frac{1}{C^2}\frac{d^2\rho}{d\tau^2}+\omega^2_0\frac{R_s}{C((R_0-R_s)-A\tau)}\rho=\frac{1}{\rho^3}.
\end{equation}
After rescaling we can write this as
\begin{equation}
\frac{d^2f}{d\tau'^2}=-\frac{A^2\omega_0^{3/2}R_s^{3/4}C^{5/4}}{(R_0-R_s)^{11/4}}\left[\frac{f}{1-\tau'}-\frac{1}{f^3}\right]
\label{f_tau eq}
\end{equation}
where $\tau'=A\tau/(R_0-R_s)$, and $f=\sqrt{\omega_0}(R_s/C(R_0-R_s))^{1/4}\rho$. The boundary conditions are then
\begin{equation}
f(0)=\left(\frac{R_s}{C(R_0-R_s)}\right)^{1/4}, \hspace{2mm} \frac{df(0)}{d\tau'}=0.
\label{f_tau IC}
\end{equation}
The last term with the $1/f^3$ becomes singular as $f\rightarrow0$. Let us consider another equation with this term replaced by another more well behaved function. For example consider,
\begin{equation}
\frac{d^2g}{d\tau'^2}=-\frac{A^2\omega_0^{3/2}R_s^{3/4}C^{5/4}}{(R_0-R_s)^{11/4}}\left[\frac{f}{1-\tau'}-g\right]
\label{g_tau eq}
\end{equation}
where the boundary conditions in Eq.(\ref{f_tau IC}) become
\begin{equation}
g(0)=\left(\frac{R_s}{C(R_0-R_s)}\right)^{1/4}, \hspace{2mm} \frac{dg(0)}{d\tau'}=0.
\label{g_tau IC}
\end{equation}
Eq.(\ref{g_tau eq}) implies that $g(\tau')$ is a monotonically decreasing function as long as $g(\tau')>0$. It is decreasing faster than the solution for $f$ as long as $f<1$, since the $1/f^3$ term in Eq.(\ref{f_tau eq}). Therefore we have
\begin{equation}
f(\tau')\geq g(\tau')
\end{equation}
for all $\tau'$ such that $g(\tau')>0$.
The solution for $g$ is positive for all $\tau'$ and, in particular, $g(1)>0$ for all the values $A^2\omega_0^{3/2}C^{5/4}R_s^{3/2}/(R_0-R_s)^{11/4}$ that we have checked. Therefore $f(\tau')$ is positive, at least for a wide range.
Let $f_1=f(1)\not=0$. Then the Eq.(\ref{f_tau eq}) can be expanded near $\tau'=1$,
\begin{equation}
\frac{d^2f_1}{d\tau'^2}=-\frac{A^2\omega_0^{3/2}R_s^{3/4}C^{5/4}}{(R_0-R_s)^{11/4}}\left[\frac{f_1}{1-\tau'}-\frac{1}{f_1^3}\right]
\label{f1_tau eq}
\end{equation}
Integrating Eq.(\ref{f1_tau eq}) we can then write
\begin{equation}
\frac{df}{d\tau'}\sim\frac{A^2\omega_0^{3/2}R_s^{3/4}C^{5/4}}{(R_0-R_s)^{11/4}}f_1\ln(1-\tau')\rightarrow-\infty
\end{equation}
as $\tau'\rightarrow1$. Hence $\rho(\tau=(R_0-R_s)/A)$ is strictly positive and finite while $\rho_{\tau}(\tau=(R_0-R_s)/A)=-\infty$ for finite and non-zero $\omega_0$.
We are calculating the occupation number $N$ as a function of frequency $\omega$ at some fixed time. From Eq.(\ref{approx omega}) we see that, in order to keep $\omega$ fixed in time, $\omega_0\rightarrow0$ as $B\rightarrow0$. Thus, $\omega$ varies with $\omega_0$ and not with time. Since $f=(R_s/C(R_0-R_s))^{1/4}$, and $f=(R_s/C(R_0-R_s))^{1/4}$ for $\omega_0\rightarrow0$, we see that $\rho\rightarrow\infty$ and $\rho_{\tau}=0$ as $\omega_0\rightarrow0$. This implies taht the occupation number in Eq.(\ref{OccNum}) diverges as $\tau\rightarrow\tau_c$ since $B\rightarrow0$ as $\tau\rightarrow\tau_c$.
|
2,869,038,154,897 | arxiv | \section{Introduction}
In this paper we consider the question of integrability of the quantum
model related to the 1D Heisenberg chain with non-nearest,
variable range exchange interaction. It is defined by the
Hamiltonian
$$H=h_{0}\sum_{\tiny\begin{array}{c}j,k=1\\ j\neq
k\end{array}}^{N}h(j-k)P_{jk},\eqno(1)$$
where $N>2$ is an arbitrary integer, the transposition operators $\{P_{jk}\}$
form an arbitrary representation of the permutation group $S_{N}$,
in particular, they obey the relations
$$P_{jkl}=P_{ljk}=P_{klj}, \quad P_{jkl}\equiv P_{jk}P_{kl},\eqno(2)$$
for $j\not=k\not=l\not=j$, $h_0$ is a coupling constant
and $h(j-k)=\wp(j-k)$, where $\wp(x)$ is the elliptic Weierstrass function
with real period $N$ and complex period $\omega=i\kappa, \kappa\in {\bf R}$,
$\kappa $ being a free parameter. The model reduces to the Heisenberg
spin-$1\over 2$ chain
if
$$P_{jk}={1\over 2}(1+\vec\sigma_{j}\cdot\vec\sigma_{k}), \eqno(3)$$
where $\{\vec\sigma_{j}\}$ are usual Pauli matrices.
There is almost no doubt about the integrability of the above Heisenberg
chain
introduced first in \cite{Ino1}. Indeed, in [1] one of us found the Lax
representation of the Heisenberg equations of motion for (1) with (2) and
the first two nontrivial integrals of motion which can be written in compact
form
$$J=\sum_{\tiny\begin{array}{c}j,k,l=1\\ j\neq k\neq l\neq j\end{array}}^{N}
f(j-k)f(k-l)f(l-j)P_{jkl},\eqno(4)$$
where
$$f(x)={{\sigma(x+\alpha)}\over{\sigma(x)\sigma(\alpha)}}e^{-x\zeta(\alpha)}
\eqno(5)$$ and $\zeta(x), \sigma(x)$ are the Weierstrass functions
(e.g. \cite{Akh}) related to $\wp(x)$ as
$${{d\zeta(x)}\over {dx}}=-\wp(x),\quad {{d\log\sigma(x)}\over {dx}}=
\zeta(x).\eqno(6)$$
Due to the arbitrariness of the "spectral" parameter $\alpha$, (4) in fact
contains only three independent integrals of motion,
$$J=-{1\over 2}\wp'(\alpha)J_{0}+\wp(\alpha)J_{1}-{1\over 2}J_{2},$$
$$J_0=\sum_{\tiny\begin{array}{c}j,k,l=1\\ j\neq k\neq l\neq
j\end{array}}^N P_{jkl},$$
$$J_{1}=\sum_{\tiny\begin{array}{c}j,k,l=1\\ j\neq k\neq l\neq
j\end{array}}^{N}\varphi_{jkl}P_{jkl}, \quad
\varphi_{jkl}=\zeta(j-k)+\zeta(k-l)+\zeta(l-j),\eqno(7)$$
$$J_{2}=\sum_{\tiny\begin{array}{c}j,k,l=1\\ j\neq k\neq l \neq
j\end{array}}^{N}F_{jkl}P_{jkl},\eqno(8)$$
$$F_{jkl}={1\over 3}\{2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]$$
$$\times
[\wp(j-k)+\wp(k-l)+\wp(l-j)]+
\wp'(j-k)+\wp'(k-l)+\wp'(l-j)\}.\eqno(9)$$
The integral of motion $J_0$, related to the total spin for the case of
spin chain
(3), trivially commutes with any transposition $P_{jk}$ and therefore also
with $H$, $J_1$
and $J_2$. We are concerned mainly with $J_1$ and $J_2$.
It is easy to prove the identities
$$2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]\wp(j-k)+\wp'(j-k)=
2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]\times$$
$$\wp(k-l)+\wp'(k-l)=
2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]\wp(l-j)+\wp'(l-j),\eqno(10)$$
which allow one to rewrite $F_{jkl}$ in one of the following forms:
$$F_{jkl}=2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]\wp(j-k)+\wp'(j-k)=\eqno(11)$$
$$=2[\zeta(j-k)+\zeta(k-l)+\zeta(l-j)]\wp(l-j)+\wp'(l-j).\eqno(12)$$
Note that both $\varphi_{jkl}$ and $F_{jkl}$ are antisymmetric with respect
to permutations of their indices.
Unfortunately, the Lax pair formalism cannot produce higher integrals of
motion due to quantum nature of the problem. The eigenvectors of (1) with (2)
were explicitly found in \cite{Ino2} up to the solutions of the
transcendental
Bethe ansatz-like equations. In the trigonometric limit of the Weierstrass
functions $(\kappa\to\infty)$, one recovers the Haldane-Shastry model
\cite{HS},
and $J_{1}$ might be reduced (for the spin representation (3)) to the product
of the Yangian generator $\vec Y_{2}=\sum_{j=1,j\neq k}^{N}\cot{\pi\over N}
(j-k)(\vec \sigma_{j}\times\vec\sigma_{k})$ and the total spin $\vec S=
{1\over 2} \sum_{j=1}
^{N}\vec \sigma_{j}$ \cite{BGHP}. Hence in this limit the symmetry of the
model is the Yangian $Y(sl(2))$, and the mutual commutativity of $J_{1}$
and $J_{2}$ has been proved rather easily \cite{BGHP}. As for general
elliptic
case, it is highly nontrivial problem, and we would like to solve it in this
paper. If the commutativity does not take place, there would be a whole
series of the nontrivial operators $[J_{1},J_{2}]$, $[J_{1},[J_{1},J_{2}]]$
etc., commuting with the Hamiltonian (1) (as in the case of the components
of $\vec Y_{2}$ which do not commute). Till now, there is no way to include
the elliptic model (1) into the general quantum inverse scattering method
\cite{Fa}. Therefore we shall use the direct method of the evaluation of
the commutator.
\section{Commutativity of $J_1$ and $J_2$}
Let us write down the commutator of the operators (7), (8) in the form
$$[J_{1}, J_{2}]=\sum_{\tiny\begin{array}{c}j,k,l=1\\ j\neq k\neq l\neq
j\end{array}}^{N}
\sum_{\tiny\begin{array}{c}m,n,p=1\\ m\neq n\neq p\neq m\end{array}}^{N}
\varphi_{jkl}F_{mnp}[P_{jkl}, P_{mnp}],\eqno(13)$$
The commutator at the right-hand side of (13) might be nonzero if and only
if one or two
indices $(mnp)$ coincide with $(jkl)$. Consider first the coincidence
of one index (say, $m$) with one of $(jkl)$. The direct calculation
of this contribution to the commutator can be written as
$$J_{3}=9\sum_{\tiny\begin{array}{c}j,k,l,n,p=1\\{\rm
all\;different}\end{array}}^{N} (\varphi_{jnp}F_{jkl}
-\varphi_{jkl}F_{jnp})P_{jklnp},\eqno(14)$$
where $P_{jklnp}=P_{jk}P_{kl}P_{ln}P_{np}$ is symmetric with respect to all
cyclic permutations of its indices. Hence the coefficient in front of it
can be rewritten due to this symmetry, and one finds
$$J_{3}={9\over 5}\sum_{\tiny\begin{array}{c}j,k,l,n,p=1\\{\rm
all\;different}\end{array}}^{N}
\Omega_{jklnp}P_{jklnp},\eqno(15)$$
where
$$\Omega_{jklnp}=F_{jkl}(\varphi_{jnp}-\varphi_{lnp})
+F_{jnp}(\varphi_{nkl}-\varphi_{jkl})+F_{kln}(\varphi_{jkp}-\varphi_{jnp})$$
$$+F_{jkp}(\varphi_{pln}-\varphi_{kln})+F_{lnp}(\varphi_{jkl}-\varphi_{jkp}).
\eqno(16)$$
The function $\Omega_{jklnp}$ in fact depends on four arguments due to the
fact that $\varphi$ and $F$ depend only on differences of their indices.
Let us introduce the notation
$$p-j=x, p-k=y, p-l=z, n-p=v.\eqno(17)$$
Then other differences can be written as
$$n-j=v+x, n-k=v+y, n-l=v+z, j-k=y-x, j-l=z-x, k-l=z-y.\eqno(18)$$
With the use of (7), (12), (17-18), we can rewrite $\Omega$ (16) as
$$\Omega_{jklnp}=R(x,y,z,v),$$
where
$$R(x,y,z,v)= $$
$$[2(\zeta(y-x)+\zeta(z-y)+\zeta(x-z))\wp(x-z)+\wp'(x-z)][-\zeta
(v+x)+\zeta(x)+\zeta(v+z)-\zeta(z)]$$
$$+[2(\zeta(x)+\zeta(y-x)-\zeta(y))\wp(y)-\wp'(y)][\zeta
(z)+\zeta(v)-\zeta(z-y)-\zeta(v+y)]$$
$$+[2(\zeta(v)+\zeta(x)-\zeta(v+x))\wp(v+x)-\wp'(v+x)][\zeta
(v+y)-\zeta(v+z)-\zeta(y-x)+\zeta(z-x)]$$
$$+[2(-\zeta(v+z)+\zeta(v)+\zeta(z))\wp(z)+\wp'(z)][\zeta(x-z)+\zeta(z-y)-\zeta(x)+\zeta(y)]\eqno(19)$$
$$+[2(\zeta(z-y)-\zeta(v+z)+\zeta(v+y))\wp(v+y)+\wp'(v+y)]
[\zeta (y-x)-\zeta(y)-\zeta(v)+\zeta(v+x)]$$
Our goal is now to simplify this very cumbersome formula. First, let us note
that $R(x,y,z,v)$ is elliptic, i.e. double periodic function of all its
arguments.
And second, we shall use the following Laurent decomposition of $\zeta(x)$
and $\wp(x)$ near $x$=0, the only singularity point of them,
$$\wp(x)\sim x^{-2} + ax^2+O(x^4),\quad \zeta(x)\sim x^{-1}-{a\over 3}x^3+
O(x^5),\eqno(20)$$
and the differential equations for the Weierstrass $\wp$ function,
$$\wp'(x)^{2}=4\wp(x)^{3}-g_{2}\wp(x)-g_{3}, \quad \wp''(x)=6\wp(x)^{2}-
{{g_2}\over 2}, \eqno(21)$$
where $a={g_2\over 20}, g_{2}, g_{3}$ are some constants.
Consider now $R(x,y,z,v)$ as the elliptic function of $v$. It can have
simple poles at four points: $v=0, v=-x, v=-y, v=-z$ and no
other singularities on the torus ${\bf T}={\bf C}/({\bf Z}N+{\bf Z}\omega).$
It might be equal to zero if we would prove that all these poles are in fact
absent (in this case $R$ does not depend on $v$), and that $R(x,y,z,0)=0$.
Let us calculate the Laurent decomposition of $R$ near the point $v=0$.
It reads
$$R(x,y,z,v)\sim v^{-1}A(x,y,z)+B(x,y,z)+...,\eqno(22)$$
where
$$A(x,y,z)=2[\wp(x)(\zeta(y)-\zeta(z)-\zeta(y-x)+\zeta(z-x))$$
$$+\wp(z)(\zeta(x-z)+\zeta(z-y)-\zeta(x)+\zeta(y))$$
$$+\wp(y)(\zeta(x)+\zeta(z)-2\zeta(y)+\zeta(y-x)+\zeta(y-z))-\wp'(y)],\eqno(23)$$
$$B(x,y,z)=-\wp'(x)(\zeta(z)-\zeta(y)+\zeta(y-x)-\zeta(z-x))$$
$$+
\wp'(z)(\zeta(y)-\zeta(x)+\zeta(x-z)+\zeta(z-y)) $$
$$+\wp'(y)(\zeta(x)+\zeta(z)-2\zeta(y)-\zeta(z-y)+\zeta(y-x))$$
$$+2(\wp(x)-\wp(y))(\wp(z)-\wp(y))-\wp''(y). \eqno(24)$$
Consider $A(x,y,z)$ as the elliptic function of the argument $x$.
It might have poles at $x=0,x=y, x=z$. Let us calculate the first two terms
of its Laurent expansion near $x=0$:
$$A(x,y,z)\sim 2\{x^{-2}[x(\wp(z)-\wp(y))+{{x^2}\over 2}(\wp'(y)-\wp'(z))]
+\wp(z)(-x^{-1}+\zeta(y)$$
$$+\zeta(z-y)-\zeta(z))
+\wp(y)(x^{-1}+\zeta(z)-\zeta(y)+\zeta(y-z))-\wp'(y)\}\eqno(25)$$
$$=-(\wp'(y)+\wp'(z))+2(\wp(y)-\wp(z))(\zeta(z)-\zeta(y)+\zeta(y-z)).$$
Now we see that $A(x,y,z)$ has no pole at $x=0$ and $A(0,y,z)=0$ due to the
known identity (change $z$ to $-z$ in the composition formula for $\zeta$)
$${1\over 2}(\wp'(y)+\wp'(z))=(\wp(y)-\wp(z))(\zeta(z)-\zeta(y)+\zeta(y-z)).
\eqno(26).$$
An easy calculation based on (23) shows that there are no
poles of $A$ at $x=y$ and $x=z$. We conclude that
$$A(x,y,z)\equiv 0.\eqno(27)$$
Let us now simplify the expression (24) for $B(x,y,z)$. It is easy to see
that
at $x=y$ and $x=z$ there are no poles of this function.
The calculation shows that there is no pole at $x=0$ too and gives
$$B(0,y,z)={1\over 3}(\wp''(y)-\wp''(z))+(\wp'(z)-\wp'(y))(\zeta(y)-\zeta(z)+
\zeta(z-y))$$
$$-2\wp(y)(\wp(z)-\wp(y))-\wp''(y).\eqno (28)$$
By using the identity (26) and differential equations (21) one can write
$B(0,y,z)$ in the form
$$B(0,y,z)=2(\wp(y)^2-\wp(z)^2)-2\wp(y)(\wp(z)-\wp(y))$$
$$+{1\over 2}{{\wp'(z)^{2}-\wp'(y)^{2}}\over {\wp(z)-\wp(y)}}-6\wp(y)^2+
{{g_{2}}\over 2}=0.$$
Hence the elliptic function $B(x,y,z)$ has no poles and $B(0,y,z)=0$.
It results in the identity
$$B(x,y,z)\equiv 0.\eqno(29)$$
Let us summarize these steps of calculations. We proved that $R(x,y,z,v)$
has no pole at $v=0$ and $R(x,y,z,0)=0.$ But it might have poles at
$v=-x,-y,-z.$ Calculation of the asymptotics at $v\to-x$ gives
$$R(x,y,z,v)\sim {1\over{v+x}}[2(\zeta(z-y)+\zeta(x-z)+\zeta(y-x))
(\wp(y-x)-\wp(x-z))$$
$$+\wp'(y-x)+\wp'(z-x)].\eqno(30)$$
But the identity (26) shows that the right-hand side of (30) is just zero.
Similar calculations result in the absence of poles of $R(x,y,z,v)$ at
$v=-y$ and $v=-z$. Hence this function has no poles in $v$ at all and
$R(x,y,z,0)=0$. We are coming up to the identity
$$R(x,y,z,v)\equiv 0.\eqno(31)$$
It means that all contributions to the commutator $[J_{1},J_{2}]$ quartic
in permutation operators (14) disappear. Let us consider now the case of
coinciding
two pairs of indices in the sets $(jkl)$, $(mnp)$ in (13).
The corresponding contribution to the commutator consists of two parts,
$$J_{4}=9\sum_{\tiny\begin{array}{c}j,k,l,n=1\\{\rm
all\;different}\end{array}}^{N}(\varphi_{jkl}F_{jkn}-
\varphi_{jkn}F_{jkl})P_{jl}P_{kn},\eqno(32)$$
$$J_{5}=9\sum_{\tiny\begin{array}{c}j,k,l,p=1\\{\rm
all\;different}\end{array}}^{N}F_{jkp}(\varphi_{ljp}-\varphi_{klp})
P_{jkl}.\eqno(33)$$
The operator in (32) is invariant under changing indices
$(j\leftrightarrow l);(k\leftrightarrow n)$;
$(j\leftrightarrow k, l\leftrightarrow n)$. Symmetrization of the
coefficient in front of it gives,
after an easy calculation taking into account the antisymmetry of
$\varphi_{jkl}$ and $F_{jkl}$ under the
transposition of two indices, that $J_{4}\equiv 0$ for otherwise arbitrary
$\varphi_{jkl}, F_{jkl}$.
It remains to calculate $J_{5}$. Since $P_{jkl}$ is symmetric with respect
to the
cyclic permutations of $(jkl)$, (33) can be written in the form
$$J_{5}=3\sum_{\tiny\begin{array}{c}j,k,l,p=1\\{\rm
all\;different}\end{array}}^{N}T_{jklp}P_{jkl},$$
$$T_{jklp}=F_{jkp}(\varphi_{ljp}-\varphi_{klp})+F_{ljp}(\varphi_{klp}-
\varphi_{jkp})+F_{klp}(\varphi_{jkp}-\varphi_{ljp}).\eqno(34)$$
Let us introduce the notation
$$j-p=x, k-p=y, l-p=z.\eqno(35)$$
Then
$$j-k=x-y, k-l=y-z, j-l=x-z,\eqno(36)$$
and we can rewrite (34) with the use of (7), (11) as
$$T_{jklp}=\Phi(x,y,z)=\eqno(37)$$
$$[2(\zeta(x-y)+\zeta(y)-\zeta(x))\wp(x-y)+\wp'(x-y)]$$
$$\times[\zeta(z-x)+\zeta(z-y)+\zeta(x)+\zeta(y)-2\zeta(z)]$$
$$+[2(\zeta(z-x)+\zeta(x)-\zeta(z))\wp(z-x)+\wp'(z-x)]$$
$$\times[\zeta(y-z)+\zeta(y-x)+\zeta(x)+\zeta(z)-2\zeta(y)]$$
$$+[2(\zeta(y-z)+\zeta(z)-\zeta(y))\wp(y-z)+\wp'(y-z)]$$
$$\times[\zeta(x-y)+\zeta(x-z)+\zeta(y)+\zeta(z)-2\zeta(x)].$$
It is easy to see that $\Phi(x,y,z)$ is antisymmetric with respect to
permutations
of its arguments,
$$\Phi(x,y,z)=-\Phi(y,x,z)=-\Phi(x,z,y).\eqno(38)$$
The problem consists now in simplifying $\Phi(x,y,z)$ which is elliptic
function
of all its arguments. As a function of $x$, it has poles at $x=0, x=y, x=z.$
Let us calculate the first two terms of its Laurent expansion near $x=0,$
$$\Phi(x,y,z)\sim
2x^{-2}[\wp(z)-\wp(y)]+x^{-1}\{2[\zeta(z-y)-\zeta(z)+\zeta(y)
]$$
$$\times [2\wp(y-z)-\wp(y)-\wp(z)]+\wp'(y)-\wp'(z)-2\wp'(y-z)\}.$$
The coefficient at $x^{-1}$ can be drastically simplified by using the
identity
(26). Implying it two times results in
$$\Phi(x,y,z)\sim 2x^{-2}[\wp(z)-\wp(y)]+2x^{-1}[\wp'(y)-\wp'(z)].\eqno(39)$$
The first coefficients in the Laurent expansions near the points $x=y$ and
$x=z$ are
$$\Phi(x,y,z)\sim -2(x-y)^{-1}\wp'(y), \quad \Phi(x,y,z)\sim 2(x-z)^{-1}
\wp'(z). \eqno(40)$$
Let us consider now the trial function
$$\Psi(x,y,z)=2\{\wp(x-y)[\wp(y)-\wp(x)]+\wp(x-z)[\wp(x)-\wp(z)]\}.\eqno(41)$$
It is easy to see that it has poles at $x=0,x=y,x=z$ with the same residues
(39),(40) as $\Phi(x,y,z)$. Hence
$$\Phi(x,y,z)=\Psi(x,y,z)+\psi (y,z),$$
where $\psi(y,z)$ does not depend on $x$. Now, with the use of antisymmetry
of $\Phi$ (38), one finds that the only choice for $\psi$ is
$$\psi(y,z)=2\wp(y-z)[\wp(z)-\wp(y)]$$
and finally one can write the remarkable identity
$$\Phi(x,y,z)=2\{\wp(x-y)[\wp(y)-\wp(x)]$$
$$+\wp(x-z)[\wp(x)-\wp(z)]
+\wp(y-z)[\wp(z)-\wp(y)]\}.\eqno(42)$$
Now let us prove the relation
$$
\sum_{\tiny\begin{array}{c}p=1\\ p\neq
j,k,l\end{array}}^{N}\Phi(x,y,z)=0\eqno(43)
$$
for any fixed $j\neq k\neq l\neq j$.
Indeed, coming back to the notation (35-36) and using (42), one finds
$$\sum_{\tiny\begin{array}{c}p=1\\ p\neq
j,k,l\end{array}}^{N}\Phi(x,y,z)=2\{\wp(j-k)[q(k,l,j)-q(j,k,l)]
+\wp(j-l)[q(j,k,l)-q(l,j,k)]$$
$$+\wp(l-k)[q(l,j,k)-q(k,l,j)],$$
where
$$q(k,l,j)=\sum_{\tiny\begin{array}{c}p=1\\ p\neq
j,k,l\end{array}}^{N}\wp(k-p)=S(k)-\wp(k-j)-\wp(k-l),$$
$$S(k)=\sum_{\tiny\begin{array}{c}p=1\\ p\neq k\end{array}}^{N}\wp(k-p).$$
But $S(k)$ does not depend on $k$ since $\wp(k-p)$ is periodic with the
period $N$. Now it is easy to see that (43) holds for all $j,k,l$ and
the commutator $[J_{1},J_{2}]$ vanishes.
\section{Linear independence of $J_0$, $J_1$ and $J_2$}
Let us prove now that the integrals of motion $J_0$, $J_1$ and $J_2$ are
linearly
independent for $N>4$. More specifically, we prove that the operator $J_0$
is linearly
independent of $J_1$ and $J_2$ for $N\geq 3$, operators $J_1$ and $J_2$
are linearly dependent for $N=3,4$,
and operators $J_1$, $J_2$ are linearly independent for $N>4$.
To study the linear independence, we are looking for the complex numbers
$\lambda$, $\mu$, $\rho$
such that
$$\lambda J_0 +\mu J_1 + \rho J_2=0 .$$
As the coefficients in equations (7) and (8) are symmetrized with respect
to the cyclic
permutations of indices, the last relation is equivalent to
$$\lambda +\mu \varphi_{jkl} + \rho F_{jkl}=0$$
for any mutually different $j,k,l=1,\dots,N$. As $\varphi_{jkl}$ and
$F_{jkl}$ are
antisymmetric under the exchange of two indices, this is further
equivalent to
$$\lambda=0 \quad,\quad \mu \varphi_{jkl} + \rho F_{jkl}=0.$$
In particular, $J_0$ is linearly independent of $J_1$ and $J_2$.
Let us now consider the case of $N=3$. Here
$$J_1=3\varphi_{123}(J_{123}-J_{213}) \quad,\quad J_2=3
F_{123}(J_{123}-J_{213})$$
so $J_1$ and $J_2$ are linearly dependent for $N=3$. In the case of $N=4$,
we obtain
remembering that $N$ is the period of Weierstrass functions
in our considerations and their other properties
$$J_1=3\varphi_{123}(P_{123}-P_{213}+P_{124}-P_{214}+P_{134}-P_{314}+P_{234}-P_{324})
,$$
$$J_2=3F_{123}(P_{123}-P_{213}+P_{124}-P_{214}+P_{134}-P_{314}+P_{234}-P_{324})$$
with
$$\varphi_{123}=\zeta(2)-2\zeta(1) \quad,\quad
F_{123}=\frac{2}{3}[2\varphi_{123}^3-\wp'(1)]$$
and the linear dependence of $J_1$ and $J_2$ is seen for $N=4$.
Let us further on assume $N>4$ and assume that there exists $\mu$ and
$\rho$ satisfying equations
$\mu \varphi_{jkl} + \rho F_{jkl}=0$ for every possible $j,k,l$. Let us
fix $k$ and $l$ and define
a functions
$$\psi(z)=\mu a(z) + \rho b(z),$$
$$a(z)=\zeta(z-k)+\zeta(k-l)+\zeta(l-z),$$
$$c(z)= \wp'(z-k)+\wp'(k-l)+\wp'(l-z),$$
$$b(z)=\frac{1}{3}\left[c(z)+2a(z)^3\right]$$
such that our equations read $\psi(j)=0$ for
$j\in\{1,\dots,N\}\setminus\{k,l\}$.
$\psi$ is an elliptic function with periods $N$ and $\omega$. The only
possible poles are at the
points $z=k$ and $z=l$. They
are simple poles of $a$, let us calculate the behavior of $b(z)$ for
$z=k+x$, $x\to 0$:
$$a(z)=\frac{1}{x}-\zeta'(l-k)x+O(x^2)=\frac{1}{x}+\wp(l-k)x+O(x^2),$$
$$c(z)=-\frac{2}{x^3}+O(x),$$
$$b(z)=\frac{2\wp(l-k)}{x}+O(1).$$
Similar formulas hold for $z\to l$ due to the antisymmetry of $a,c,b$ with
respect to
the interchange of $k$ and $l$. Therefore $\psi$ has at most simple poles
at $k$ and
$l$. By Liouville theorem (e.g. \cite{Akh}), $\psi$ can have at most two
zeroes (modulo
periods) if it is not constant. However, there are at least $N-2>2$ zeroes
at the points
$z=j$. So $\psi(z)\equiv 0$. Looking for the behavior
at $z\to k$, we see that equation
$$\mu + 2 \wp(l-k)\rho=0$$
must be valid. The function $\wp$ can take the same value at most twice
(modulo periods) due to the
Liouville theorems. As $k\not=l$ can be chosen arbitrarily, we find two
different values among $N-1$
numbers $\wp(l-k)$ for $l-k=1,\dots,N-1$.
So necessarily $\mu=\rho=0$ and the linear independence of $J_0,J_1,J_2$
is proved for $N>4$.
Their linear independence of $H$ is trivial as different permutations
enters the definition
of $H$.
\section{Conclusions}
To summarize, we proved that the Hamiltonian (1) and operators $J_0$,
$J_{1}$ (7)
and $J_{2}$ (8) are linearly independent and generate the commutative
ring. As a byproduct,
we obtained the remarkable identities between elliptic functions (31), (42).
The proof was based on direct evaluation of $[J_{1},J_{2}]$ due to the lack
of any other methods. The model (1) with elliptic form of $h(j-k)$ is still
not immersed in the scheme of the quantum inverse scattering method.
This is highly desirable task which we postpone for further study. The
presence of the operators of higher orders in permutations commuting
with the Hamiltonian was also mentioned \cite{Ino3} but till now there is
no way
to prove their mutual commutativity.
\vspace{0.5cm}
\noindent
{\bf Acknowledgments}. The work was supported by the Ministry of Education
of the
Czech Republic under the projects LC06002 "Doppler Institute for
mathematical physics
and applied mathematics", 1P04LA213 "Collaboration of the Czech Republic
with JINR
Dubna", and by the Academy of Sciences of the Czech Republic by the NPI
research plan
AV0Z10480505.
|
2,869,038,154,898 | arxiv | \section{Introduction}
Given graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum $N$ such that whenever the edges of the complete graph on $N$ vertices $K_N$ are coloured with red and blue, there exists either a red copy of $G$ or a blue copy of $H$.
The study of Ramsey numbers has a long history, and in general it is hard to find even good upper and lower bounds on $R(G,H)$. In this paper, we are interested in the case that $G$ and $H$ are sparse graphs. In this case, if $G$ is connected and $v(G)\ge\sigma(H)$, one has the lower bound
\begin{equation}\label{eq:burr}
R(G,H)\ge \big(\chi(H)-1\big)\big(v(G)-1\big)+\sigma(H).
\end{equation}
Here $v(G)$ denotes the number of vertices of $G$, $\chi(H)$ is the chromatic number of $H$, and $\sigma(H)$ is the minimum, over all $\chi(H)$-colourings of $H$, of the smallest colour class size. This lower bound is due to Burr~\cite{Burr}, with the corresponding construction being $\chi(H)-1$ vertex-disjoint red cliques each on $v(G)-1$ vertices, plus one further red clique on $\sigma(H)-1$ vertices, and all other edges blue. When this construction gives the Ramsey number (i.e. when we have an equality in \eqref{eq:burr}), we say that $G$ is $H$\emph{-good}.
For fixed graphs $H$, the class of graphs $G$ which are $H$-good is quite well understood; see Allen, Brightwell and Skokan~\cite{ABS} and Nikiforov and Rousseau~\cite{NikiRous}. However much less is known about the case when $H$ grows with $v(G)$, or when $H=G$. Burr~\cite{Burr} conjectured that for fixed $\Delta$, every connected graph $G$ with $\Delta(G)\le\Delta$ and $v(G)$ large enough is $G$-good. This statement holds for $G=P_n$~~\cite{GG67} and $G=C_n$~\cite{BE73,R73}. However it was disproved by Graham, R\"odl and Ruci\'nski~\cite{GRR}, who showed that it fails badly for expander graphs, and again in~\cite{ABS}, where a lower bound on $R(P_n^k,P_n^k)$ better than \eqref{eq:burr} is shown for each $k\ge2$. In the latter paper, however, it is shown that Burr's conjecture is off by at most a factor (roughly) $2$ when $G$ has bounded maximum degree and sublinear bandwidth. Here the bandwidth of $G$ is the smallest $k$ such that $G$ is a subgraph of $P^k_{v(G)}$.
In~\cite{ABS}, a value for the Ramsey numbers of squares of paths, and squares of cycles on a number of vertices divisible by $3$, is conjectured. We observe that the conjectured value is wrong by one, and prove the modified conjecture.
\begin{restatable}{theorem}{thmmain}\label{thm:main}
There exists $n_0$ such that for all $n\ge n_0$ we have:
\[R(P_{3n}^2,P_{3n}^2)=R(P_{3n+1}^2,P_{3n+1}^2)=R(C_{3n}^2,C_{3n}^2)=9n-3\mbox{ and } R(P_{3n+2}^2,P_{3n+2}^2)=9n+1.\]
\end{restatable}
The lower bound part of this theorem is the following construction from~\cite{ABS}. We take disjoint vertex sets $X_1,X_2,Y_1,Y_2$ each with $2n-1$ vertices, plus $Z$ with $n-1$ vertices. We colour edges within each $X_i$ blue and within each $Y_i$ red. We colour edges in the bipartite graphs $(X_1,X_2)$ and $(X_i,Z)$ red, and in $(Y_1,Y_2)$ and $(Y_i,Z)$ blue. We colour $(X_1,X_2)$ and $(Y_1,Y_2)$ blue, and $(X_1,Y_2)$ and $(X_2,Y_1)$ red. Finally, we add a single vertex $z$, which sends blue edges to $X_1\cup X_2$ and red to $Y_1\cup Y_2$. The edges within $Z\cup\{z\}$ may be coloured arbitrarily, as illustrated in Figure~\ref{fig:construct}. A short case analysis demonstrates that this construction does not contain a monochromatic $P_{3n}^2$. Furthermore, we can add one extra vertex to each of $X_1,X_2,Y_1,Y_2$ and still have no $P_{3n+2}^2$.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\node (A) at ( 2,7) [circle, fill=blue!80,inner sep=8pt,label=left:$X_1$] {$2n-1$};
\node (B) at ( 2,4) [circle, fill=blue!80,inner sep=8pt,label=left:$X_2$] {$2n-1$};
\node (C) at ( 4,2) [circle, fill=purple!80,inner sep=5pt,label=left:$Z$] {$n-1$};
\node (D) at ( 6,7) [circle, fill=red,inner sep=8pt,label=right:$Y_1$] {$2n-1$};
\node (E) at ( 6,4) [circle, fill=red,inner sep=8pt,label=right:$Y_2$] {$2n-1$};
\node (z) at ( 4,9) [circle, fill=red,inner sep=2pt,label=left:$z$] {};
\foreach \from/\to in {A/B,A/C,B/C,A/E,B/D}
\draw [red,line width=5pt] (\from) -- (\to);
\foreach \from/\to in {D/E,D/C,E/C,A/D,B/E}
\draw [blue!80,line width=5pt] (\from) -- (\to);
\foreach \from/\to in {z/A,z/B}
\draw [blue!80,line width=2pt] (\from) -- (\to);
\foreach \from/\to in {z/D,z/E}
\draw [red,line width=2pt] (\from) -- (\to);
\foreach \from/\to in {z/C}
\draw [purple!30,line width=2pt] (\from) -- (\to);
\end{tikzpicture}
\end{center}
\caption{Lower bound construction}\label{fig:construct}
\end{figure}
In addition, we give a general upper bound on Ramsey numbers for $3$-colourable graphs with bounded maximum degree and sublinear bandwidth, which $P_{3n}^2$ demonstrates is asymptotically tight.
\begin{theorem}\label{thm:boundBW}
Given $\gamma>0$ and $\Delta$, there exist $\beta>0$ and $n_0$ such that for all $n\ge n_0$ the following holds. Suppose that $H$ is a graph with $\Delta(H)\le\Delta$, with bandwidth at most $\beta n$, and with a proper vertex $3$-colouring all of whose colour classes have at most $n$ vertices. Then $R(H,H)\le(9+\gamma)n$.
\end{theorem}
We recall from~\cite{ABS} that the bandwidth restriction in this theorem is necessary: for any given $\beta>0$, if $\Delta$ is large enough there are $n$-vertex graphs $H$ with bandwidth at most $\beta n$ and maximum degree at most $\Delta$ for which the theorem statement is false.
\medskip
Our proof method uses the stability-extremal paradigm. Using the Szemer\'edi Regularity Lemma and the Blow-up Lemma, we will argue that to find a monochromatic square of a path (or cycle, or $3$-colourable sparse graph as in Theorem~\ref{thm:boundBW}) it is enough to find in the cluster graph a monochromatic triangle factor which is `triangle connected' (which we will define later). This standard reduction leaves us looking, in a nearly complete edge-coloured graph, for a large monochromatic triangle-connected triangle factor (TCTF). The main technical work of the paper (Lemma~\ref{MainLemma}) is then to prove that a $2$-edge-coloured near complete graph on nearly $9t$ vertices will either contain a monochromatic TCTF on a little more than $3t$ vertices, or alternatively the graph must be close to the extremal example.
To prove the main lemma, we use a second partitioning method, as in~\cite{ABS}: by an iterative use of Ramsey's theorem, we partition most of the $9t$ vertices into a collection of bounded size (but quite large) monochromatic cliques. Obviously, it is easy to find a large red triangle factor in a collection of red cliques: in addition, we will see that two triangles in (or even using one edge of) the same red clique are `triangle connected' in red, and that if two red cliques are \emph{not} red triangle connected, then almost all the edges between them have to be blue. These observations were previously made in~\cite{ABS}. Where we improve compared to that paper is that we are able to deal with the interaction between cliques of different colours (whereas in~\cite{ABS} the minority colour cliques are thrown away).
\section{Notation, main lemmas and organisation}\label{sec:note}
Our graph notation is mainly standard. We will often write $|G|$ for the number of vertices in a graph $G$, and similarly $|M|$ for the number of vertices covered by a matching $M$ (i.e.\ twice the number of edges of $M$). We will often want to refer to edges (of a given colour) between two or three vertex sets. We write $(A,B)$ or $(A,B,C)$ for respectively $\{ab:a\in A,b\in B\}$ and $(A,B)\cup(A,C)\cup(B,C)$, the graph we refer to will always be clear from the context. We will work with $2$-edge-coloured graphs, and refer to the two colours as `red' and `blue'.
Given a graph $G$, we say that edges $uv$ and $uw$ of $G$ are \emph{triangle-connected} if $vw$ is an edge of $G$, we extend this to an equivalence relation on edges by transitive closure. We refer to the equivalence classes of this relation as \emph{triangle components}. We will generally want to talk about monochromatic triangle connection. Thus, if the edges of $G$ are $2$-coloured, we say that two red edges are red triangle connected if they are triangle connected in the subgraph of $G$ consisting only of red edges, we define red triangle component similarly. We also, slightly abusing notation, will say two red cliques (each with at least two vertices) are red triangle-connected if an edge (and so all edges) in one is red triangle connected to an edge (so all edges) of the other. When the colour is clear from the context (as with \emph{red} cliques) we will often just say that the two cliques are triangle connected.
A \emph{triangle factor} in a graph $G$ is a collection of vertex-disjoint triangles of $G$. It is a \emph{triangle-connected triangle factor} (TCTF) if all its edges lie in a single triangle component. Again, we will usually want to talk about monochromatic TCTFs in a $2$-edge-coloured graph $G$, and as above a red TCTF means a TCTF in the subgraph of red edges of $G$.
At this point, we are in a position to give the case analysis proving the lower bound part of Theorem~\ref{thm:main}.
\begin{proof}[Proof of Theorem~\ref{thm:main}, lower bounds]
We begin by describing the red triangle components of the lower bound construction for $P_{3n}^2$, $P_{3n+1}^2$ and $C_{3n}^2$. The edges in $Y_1$ and in $(Y_1,X_2\cup\{z\})$, form a red triangle component. Similarly, the edges in $Y_2$ and $(Y_2,X_1\cup\{z\})$ form a red triangle component. The edges $(X_1,X_2,Z)$, together with all red edges in $Z$ and all red edges from $z$ to $Z$ which lie in a red triangle, form a red triangle component. Finally, each red edge from $z$ to $Z$ which is not in a red triangle forms a triangle component. The blue components are analogous.
If the lower bound construction contains a red $P_{3n}^2$, then in particular it has a red triangle component which contains a red triangle factor with $n$ triangles. Checking each entry in the list above, observe that removing $Y_1$ from the first leaves an independent set: $X_2\cup\{z\}$ contains no red edges. But $Y_1$ contains only $2n-1$ vertices, so there cannot be a $3n$-vertex triangle factor in this component. The symmetric argument deals with the symmetric second red triangle component. For the third case, removing $Z$ leaves a bipartite graph: the only red edges are those in $(X_1,X_2)$. But $Z$ contains only $n-1$ vertices, so this component too contains no $3n$-vertex red triangle factor. Finally, trivially the single-edge components contain no red triangle factor. The argument to exclude a blue $P_{3n}^2$ is symmetric.
For the modification for $P_{3n+2}^2$, adding one vertex to each of $X_1,X_2,Y_1,Y_2$, the description of triangle components above, and the explanation that the red triangle component containing $(X_1,X_2,Z)$ does not contain $P_{3n}^2$ continues to work. Observe that $P_{3n+2}^2$ has independence number $n+1$, so removing any $2n$ vertices leaves at least one edge. This observation shows that the red component consisting of edges in $Y_1$ and $(Y_1,X_2\cup\{z\})$ does not contain a red $P_{3n+2}^2$, and the other cases are symmetric.
\end{proof}
\medskip
The main work of this paper is to prove the following stability lemma, which states that a $2$-edge-coloured nearly complete graph $G$ on almost $9t$ vertices either contains a monochromatic TCTF on a little more than $3t$ vertices, or is close to the extremal example. To state it, we need one further definition.
Given an edge-coloured graph $G$, let $A\subseteq V(G)$ and $v$ a vertex of $G$ not in $A$. For $r\in\bb{R}$, we say that $v$ is \emph{$r$-blue to $A$} if $va$ is a blue edge of $G$ for all but at most $r$ vertices $a\in A$. Similarly, given $A,B\subseteq V(G)$ disjoint, we say that \emph{$(A,B)$ is $r$-blue} if all but at most $r$ vertices in $A$ are $r$-blue to $B$ and vice versa. We define similarly \emph{$r$-red}.
We will generally use this notation with $r$ much smaller than the sets $A$ and $B$, so the reader can think of $r$-blue as meaning `almost all blue'. Our main lemma is then the following.
\begin{restatable}{lemma}{MainLem}\label{MainLemma}
There exists $\delta_0>0$ such that for any $0<h,\lambda<\delta_0$ there exist $\epsilon_0>0$ and $t_0\in\bb{N}$ such that for any $t\geq t_0$ and $0<\epsilon<\epsilon_0$ the following holds. Let $G$ be a $2$-edge-coloured graph on $(9-\epsilon)t$ vertices with minimum degree at least $(9-2\epsilon)t$. Then either $G$ contains a monochromatic TCTF on at least $3(1+\epsilon)t$ vertices or $V(G)$ can be partitioned in sets $B_1, B_2, R_1, R_2, Z, T$ such that the following hold.
\begin{enumerate}[label=(\alph*)]
\item $(2-h)t\leq \vass{B_1}, \vass{B_2}, \vass{R_1}, \vass{R_2}\leq (2+h)t$,
\item $(1-h)t\leq \vass{Z}\leq (1+h)t$,
\item all the edges in $G[B_1]$ and $G[B_2]$ are blue, and all the edges in $G[R_1]$ and $G[R_2]$ are red,
\item all the edges between the pairs $(B_1,R_1)$, $(B_2, R_2)$, $(R_1, Z)$ and $(R_2,Z)$ are blue, and those between the pairs $(B_1,R_2)$, $(B_2, R_1)$, $(B_1, Z)$ and $(B_2,Z)$ are red,
\item the pair $(B_1, B_2)$ is $\lambda t$-red, and the pair $(R_1, R_2)$ is $\lambda t$-blue, and
\item $\vass{T}\leq ht$.
\end{enumerate}
\end{restatable}
We will prove this lemma in Sections~\ref{sec:genset}--\ref{sec:finproof}.
By applying the Regularity Method in a standard way, we are able to upgrade Lemma~\ref{MainLemma} to the following superficially similar statement, in which we replace TCTF with the square of a path and cycle. We could generalise the following lemma to nearly-complete graphs easily (as in Lemma~\ref{MainLemma}), but we do not need it for the proof.
\begin{restatable}{lemma}{RegResult}\label{regularityresult} For every $\alpha>0$ there exists $\delta>0$ and $n_0\in\mathbb{N}$ such that for every $n>n_0$ the following holds. Let $N\geq (9-\delta)n$, and let $G$ be a $2$-edge-colouring of $K_N$. Then either $G$ contains both a monochromatic copy of $P_{3n+2}^2$ and of $C_{3n}^2$, or we can partition $V(G)$ into sets $X_1, X_2, Y_1, Y_2, Z$ and $R$ such that the following hold.
\begin{enumerate}[label=(\alph*)]
\item\label{conditionA} $(2-\alpha)n\leq \vass{X_1}, \vass{X_2},\vass{Y_1},\vass{Y_2}\leq (2+\alpha)n$,
\item\label{conditionB} $(1-\alpha)n\leq \vass{Z}\leq (1+\alpha)n$,
\item\label{conditionC} $\vass{R}\leq \alpha n$,
\item Vertices in the following pairs have at most $\alpha n$ red neighbours in the opposite part: $(X_1, Y_1), (X_2, Y_2), (Y_1, Y_2), (Y_1, Z)$ and $(Y_2, Z)$,
\item Vertices in the following pairs have at most $\alpha n$ blue neighbours in the opposite part: $(X_1, X_2), (X_2, Y_1), (X_1, X_2), (X_1, Z)$ and $(X_2, Z)$,
\item Vertices in $X_1$ and $X_2$ have at most $\alpha n$ red neighbours in their own part,
\item\label{conditionG} Vertices in $Y_1$ and $Y_2$ have at most $\alpha n$ blue neighbours in their own part.
\end{enumerate}
\end{restatable}
We deduce this lemma from Lemma~\ref{MainLemma} in Section~\ref{sec:reg}.
To complete the proof of Theorem~\ref{thm:main}, we need to show that a complete graph which can be partitioned as in the above Lemma~\ref{regularityresult} and which has $9n-3$ vertices necessarily contains both a monochromatic $P_{3n+1}^2$ and $C_{3n}^2$; and $9n+1$ vertices suffices for $P_{3n+2}^2$. We do this in Section~\ref{sec:exact}.
Finally, to prove Theorem~\ref{thm:boundBW} it suffices to observe that if $G$ satisfies the conditions of Lemma~\ref{MainLemma} and can be partitioned as in that lemma, then it contains a monochromatic TCTF on nearly $3t$ vertices. Together with a standard application of the Regularity Method, which we sketch in Section~\ref{sec:reg}, this completes the proof of Theorem~\ref{thm:boundBW}.
\section{Preliminary lemmas}
In this section we prove some basic Ramsey-theoretic results which we will need to prove Lemma~\ref{MainLemma}, but for which we do \emph{not} assume the conditions of Lemma~\ref{MainLemma}.
\begin{lemma}\label{lemmasteplowerbound1}
There exist $\epsilon_0>0$ and $t\in \bb{R}$ such that the following holds for any $0<\epsilon<\epsilon_0$ and $t>t_0$. Let $G$ be a graph on at least $2(1+3\epsilon)t$ vertices with minimum degree at least $\vass{G}-\epsilon t$. Any 2-edge-colouring of the edges of $G$ contains a red matching on $2(1+\epsilon)t$ vertices or a blue connected matching on $\min\left\lbrace \vass{G}-(1+2\epsilon)t, 2\vass{G}-4(1+2\epsilon)t \right\rbrace$ vertices.
\end{lemma}
\begin{proof}
Let $M$ be the largest red matching in $G$ and let $Y=V(G)\setminus M$. We may assume that $M$ has at most $2(1+\epsilon)t$ vertices. Since $M$ is maximal, every edge in $M$ has one endpoint with at most one red neighbour in $Y$. Indeed, if $xy\in M$ and both $x$ and $y$ have at least two neighbours in $Y$ we can take $x'$ in $Y$ adjacent to $x$ and $y'$ distinct from $x'$ adjacent to $y$ in $Y$, and obtain a red matching which is larger than $M$ by substituting $xy$ with $x'x$ and $y'y$.
Let $S$ be the set of vertices in $M$ with at most one red neighbour in $Y$. We can now form a blue matching $P$ (that we are going to show is connected) by greedily matching vertices in $S$ with blue neighbours in $Y$. We claim that $P$ has at least $\min\left\lbrace \vass{S}, \vass{G}-\vass{M}-2\epsilon t \right\rbrace$ edges. Indeed, since the process is greedy we stop only by finishing all the vertices of $S$ or when $S\setminus P$ is not empty, but no vertex in $S\setminus P$ has a blue neighbour in $Y\setminus P$, and this means that there are less than $2\epsilon t$ vertices not yet covered by $P$ in $Y$.
If we stopped for the first reason (if $\vass{S}<\vass{G}-\vass{M}-2\epsilon t$) we can extend $P$ to a larger blue matching $P'$: the induced graph over $Y$ contains only blue edges by maximality of $M$ and there are some edges left in $Y\setminus P$. This extension of $P$ can continue at least until all but $\epsilon t$ vertices in $Y$ are covered: we stop only when all edges in $Y$ have one vertex covered by $P'$. Therefore we have
\begin{align*}
\vass{V(P')}&\geq \overbrace{2\vass{S}}^{\text{in }P}+\overbrace{\vass{Y}-\vass{S}-\epsilon t}^{\text{in }Y}\\
&\overbrace{\geq}^{2\vass{S}\geq \vass{M}} \vass{G}-\frac{\vass{M}}{2}-\epsilon t\\
&\geq \vass{G}-(1+2\epsilon)t\,,
\end{align*}
as desired.
If on the other hand we stopped because no vertex in $S\setminus P$ has a blue neighbour in $Y\setminus P$ (but $S\setminus P$ is not empty). In particular, by definition of $S$ this means that every vertex in $S\setminus P$ has at most one neighbour in $Y\setminus P$. This can only happen if $\vass{Y \setminus P}<2\epsilon t$ and hence all but at most $2\epsilon t$ vertices of $Y$ are covered by $P$. This means that the size of $P$ is at least
\begin{align*}
\vass{P}&\geq 2 (\vass{Y}-2\epsilon t)\\
&\geq 2(\vass{G}-\vass{M}-2\epsilon t)\\
&\geq 2(\vass{G}-2(1+\epsilon)t-2\epsilon t)\\
&=2\vass{G}-4(1+2\epsilon)t\,,
\end{align*}
as desired.
In order to conclude, we must now argue that the matching $P$ (or $P'$) we obtained is blue connected. But this is the case, indeed, every edge of $P$ (or $P'$) has at least one vertex in $Y$. Indeed $\vass{Y}=\vass{G}-\vass{M}\geq 4\epsilon t$ and all edges in $Y$ are blue. By the minimum degree of $G$ each vertex of $Y$ is non-adjacent to at most $\epsilon t$ vertices of $Y$, so any pair of vertices of $Y$ has a common neighbour in $Y$, and therefore $Y$ is blue-connected.
\end{proof}
\begin{lemma}\label{lemmatwothirdsappe}
Let $G$ be a graph with minimum degree strictly greater than $\frac{2}{3}\vass{G}$. Then all the edges of $G$ are triangle connected. Moreover, there exists a TCTF on all but at most $2$ vertices of $G$.
\end{lemma}
\begin{proof}
We may notice that every three vertices of $G$ share a common neighbour by the minimum degree condition and the pigeonhole principle. This means that any couple of adjacent edges is triangle connected in a trivial way, and this property implies that connected components and triangle-connected components coincide in $G$ (because of the minimum degree condition we have that $G$ is connected and therefore every couple of edges is triangle connected).
The existence of the TCTF is given by a theorem of Corradi and Hajnal \cite{CorradiHajnal1963}.
\end{proof}
\begin{lemma}\label{lemmaconnmatchtctf}
There exist $\epsilon_0>0$ and $t\in\bb{R}$ such that the following holds for any $0<\epsilon<\epsilon_0$, any $t>0$. Let $G$ be a graph on at least $(5+100\epsilon)t$ vertices with minimum degree at least $\vass{G}-\epsilon t$. Any 2-edge-colouring of the edges of $G$ contains a red connected matching over $2(1+\epsilon)t$ vertices or a blue TCTF on $3(1+\epsilon)t$ vertices.
\end{lemma}
\begin{proof} Without loss of generality, we may assume $G$ has $(5+100\epsilon)t$ vertices.
We separate cases.
\underline{Case 1}: $G$ has a maximal red connected component $A$ that spans at least $(4+5\epsilon)t$ vertices.
Let $M$ be the largest red matching in $A$. Since $A$ is a red connected component, we may assume $\vass{M}<2(1+\epsilon)t$. Since $M$ is a maximal red matching in $A$, we know that every edge in $A\setminus M$ is blue.
Because of our assumption on the size of $A$, we have that $\vass{A\setminus M}>(2+3\epsilon)t$. We construct a matching $P$ of size $2(1+\epsilon)t$ in $A\setminus M$ greedily, which is possible by the minimum degree of $G$. By Lemma \ref{lemmatwothirdsappe}, every pair of edges in $A\setminus M$ is blue triangle connected. In particular $P$ is blue triangle connected.
We now greedily extend the edges of $P$ to blue triangles by taking vertices in $X=V(G)\setminus (P\cup M)$. We have no red edges from vertices of $X$ to vertices of $P$: if $x\in X$ is not in $A$, this is since $A$ is a red component, while if $x\in X\cap A$ then it is by maximality of $M$. We have $\vass{X}\ge (1+(k-4)\epsilon)t$, and by the minimum degree of $G$ any edge of $P$ makes a triangle with all but at most $2\epsilon t$ vertices of $X$, so the greedy extension succeeds.
\smallskip
\underline{Case 2}: $G$ has a maximal red connected component $A$ that spans at least $3(1+2\epsilon)t$ but less than $(4+5\epsilon)t$ vertices.
If $G$ has a red connected matching over $2(1+\epsilon)t$ vertices we are done, so we assume it does not. By Lemma \ref{lemmasteplowerbound1} applied to $A$, we obtain a blue connected matching $P$ in $A$ of size at least $2(1+\epsilon)t$. Now as in the previous case, we can greedily extend all the edges of $P$ to a blue triangle factor using vertices of $V(G)\setminus A$. Observe that every two blue adjacent edges in $A$ share a neighbour in $V(G)\setminus A$, therefore every blue connected component in $A$ is also blue triangle connected. In particular, $P$, and hence the blue triangle factor containing it, are triangle connected.
\smallskip
\underline{Case 3}: $G$ has two maximal red connected components $A_1$ and $A_2$ covering at least $(5+12\epsilon)t$ vertices in total, and we are not in Cases 1 or 2.
Because we are not in Cases 1 or 2, $A_1$ and $A_2$ both span less than $3(1+2\epsilon)t$ vertices and hence at least $(2+6\epsilon)t$ vertices. In addition, neither component contains a red matching on $2(1+\epsilon)t$ vertices, because otherwise we would be done. Therefore, each $A_i$ contains a blue connected matching $P_i$ on precisely $\min\big(2\vass{A_i}-4(1+2\epsilon)t,2t\big)$ vertices by Lemma~\ref{lemmasteplowerbound1}. Indeed, for the possible values of $\vass{A_i}$, we have $2\vass{A_i}-4(1+2\epsilon)t<\vass{A_i}-(1+2\epsilon)t$.
Observe that every edge between $A_1$ and $A_2$ is blue and therefore $P_1\cup P_2$ is a blue connected matching. We have $\vass{P_1},\vass{P_2}\ge 4\epsilon t$ and hence if $\vass{P_1}=2t$ we see that $P_1\cup P_2$ has at least $(1+2\epsilon)t$ edges. Similarly if $\vass{P_2}=2t$. If $\vass{P_1},\vass{P_2}<2 t$ then we have at least $\vass{A_1}+\vass{A_2}-4(1+2\epsilon)t\geq (1+4\epsilon)t$ edges, in any case we have in $P_1\cup P_2$ at least $(1+2\epsilon)t$ edges. Let $Y_i=A_i\setminus P_i$.
We extend greedily the edges of $P_1$ to a set of disjoint blue triangles $T_1$ using vertices of $Y_2$, and in the same way we greedily extend the edges of $P_2$ to a set of disjoint blue triangles $T_2$ using vertices of $Y_1$. Note that $\vass{Y_i}=4(1+2\epsilon)t-\vass{A_i}> (1+2\epsilon)t$, and therefore we are able to extend the edges of $P_1\cup P_2$, so we obtain a blue triangle factor with at least $(1+\epsilon)t$ triangles.
It now suffices to show that the triangle factor $T_1\cup T_2$ is triangle connected. Because every two blue incident edges in $A_1$ share a neighbour in $A_2$ and vice versa, we have that both $T_1$ and $T_2$ are TCTFs. Without loss of generality we assume that $\vass{P_1}\leq \vass{P_2}$. We know that $\vass{P_1}=2\vass{A_1}-4(1+2\epsilon)t>4\epsilon$. Let $xy$ be an edge in $P_2$, because every edge between $A_1$ and $A_2$ is blue, and because of the minimum degree condition we have that $x$ and $y$ share at least $\vass{P_1}-2\epsilon t$ blue neighbours in $P_1$. Because $P_1$ has a blue matching, every set in $P_1$ of size strictly bigger than $\frac{\vass{P_1}}{2}$ has an edge from $P_1$. Therefore we have that there exists $zt$ in $P_1$ such that $G[\left\lbrace x, y, z, t \right\rbrace]$ is a blue clique with $xy$ in $P_2$ and $zt$ in $P_1$. Because both $P_1$ and $P_2$ are triangle connected, we are done.
\smallskip
\underline{Case 4}: $G$ is not in any of cases 1--3, i.e.\ there is no red component of size $3(1+2\epsilon)t$ or bigger, and no two red components cover $(5+12\epsilon)t$ or more vertices.
Let $A_1, A_2, \dots{}$ be the maximal red connected components, ordered by decreasing cardinality. We have $\vass{A_1}<3(1+2\epsilon)t$ and $\vass{A_1}+\vass{A_2}<(5+12\epsilon)t$, and we can assume that $G$ does not have a red connected matching over $2(1+\epsilon)t$ vertices since otherwise we are done.
\begin{claim}
The set of blue edges of $G$ is triangle connected.
\end{claim}
\begin{claimproof}
Every blue edge in a component $A_i$ is in a blue triangle with some vertex in a different component $A_j$, so it suffices to prove that the edges between distinct components all lie in the same triangle-connected component. In particular, it is enough to show that for any $j,k\ge 2$ distinct, any $a_1a_j$ an edge between $A_1$ and $A_j$, and any $b_jb_k$ an edge between $A_j$ and $A_k$, then $a_1a_j$ and $b_jb_k$ are triangle connected. This last equivalence is due to the fact that there are at least three red components (indeed, $\vass{V(G)}-\vass{A_1\cup A_2}>(k-12)\epsilon t$).
Given $a_1,a_j,b_j,b_k$ as above, let $c$ be a common blue neighbour of $a_1, a_j, b_j$ not in $A_1\cup A_j$. This exists by minimum degree condition and by considering that $a_1, a_j, b_j$ are all in $A_1\cup A_j$ and there are at least $(k-12)\epsilon t$ vertices not in $A_1\cup A_j$. Now let us take $d$ a common blue neighbour of $c, a_j, b_j, b_k$ in $A_1$: this exists since $c,a_j,b_j,b_k$ are not in $A_1$, and using the minimum degree condition. We can now conclude since $(a_1a_jc, a_jcd, cdb_j, db_jb_k)$ is a sequence of blue triangles that proves that $a_1a_j$ and $b_jb_k$ are triangle connected.
\end{claimproof}
Because we showed that every blue edge is triangle connected, it is sufficient to find $(1+\epsilon)t$ disjoint blue triangles. We are in one of the following cases.
\underline{Case A}: Both $A_1$ and $A_2$ are larger than $2(1+20\epsilon)t$.
By Lemma \ref{lemmasteplowerbound1} we can find blue matchings $M_i\subseteq A_i$ on $2\vass{A_i}-4(1+2\epsilon)t$ vertices for $i=1,2$. Indeed, because $A_i< 3(1+2\epsilon)t$ we have $2\vass{A_i}-4(1+2\epsilon)t\leq \vass{A_i}-(1+2\epsilon)t$. We can greedily extend the matching $M_1$ to a blue triangle factor using vertices in $A_2\setminus M_2$: because $\vass{A_2\setminus M_2}=(4+8\epsilon)t-\vass{A_2}>\vass{A_1}-(2+4\epsilon)t+2\epsilon t= \frac{\vass{M_1}}{2}+2\epsilon t$ we are able to extend every edge in $M_1$ to a blue triangle. Similarly we can extend all the matching $M_2$ to a blue triangle factor using vertices in $A_1\setminus M_1$. This two triangle factors are disjoint and therefore they form a unique triangle factor that we denote with $T$. We can observe that $\vass{T}=\frac{3}{2}\vass{M_1}+\vass{M_2}=3(\vass{A_1}+\vass{A_2})-12(1+2\epsilon)t$.
Let us now denote $U_1=A_1\setminus T$, $U_2=A_2\setminus T$ and $W=V(G)\setminus (A_1\cup A_2)$. We have
\begin{align*}
\vass{U_1}&=\vass{A_1}-\vass{M_1}-\frac{\vass{M_2}}{2}\\
&=\vass{A_1}-2\vass{A_1}+4(1+2\epsilon)t+2(1+2\epsilon)t-\vass{A_2}\\
&=6(1+2\epsilon)t-(\vass{A_1}+\vass{A_2})\geq t\,.
\end{align*}
Similarly we have $\vass{U_2}\geq t$. We can also notice that $\vass{W}= (5+k\epsilon)t-(\vass{A_1}+\vass{A_2})\geq (k-12)\epsilon t$. Finally, let us observe that $\vass{U_1},\vass{U_2}> \vass{W}+4\epsilon t$ by our assumption on $|G|$. Therefore we can find a blue triangle factor on $(U_1,U_2,W)$ covering $3\vass{W}$ vertices. Adding this triangle factor to $T$ we get a TCTF on
\[3(5+k\epsilon)t-3(\vass{A_1}+\vass{A_2})+3(\vass{A_1}+\vass{A_2})-12(1+2\epsilon)t=(3+(3k-24)\epsilon)t\]
vertices.
\smallskip
\underline{Case B}: $A_1$ is larger than $2(1+3\epsilon)t$ but all the other red components are smaller than $2(1+3\epsilon)t$.
Let $M_1$ be a blue matching in $A_1$ on $2\vass{A_1}-4(1+2\epsilon)t$ vertices. Let $U_1=A_1\setminus M_1$ and notice $\vass{U_1}\geq 4(1+2\epsilon)t-\vass{A_1}$. Because all the other red components are smaller than $2(1+3\epsilon)t$, we claim there exists $j$ such that $(1+3\epsilon)t< \vass{\bigcup_{i=2}^{j}A_i}\leq 2(1+3\epsilon )t$, and write $U_2=\bigcup_{i=2}^{j}A_i$. Indeed, if $|A_2|> (1+3\epsilon )t$ we can take $j=2$, while if not then we can increase $j$ sequentially until the lower bound is satisfied. Since in the latter situation we have $|A_j|\le|A_2|\le (1+3\epsilon)t$ the upper bound is not exceeded. Finally, let $W=V(G)\setminus (A_1\cup U_2)$ and note that $\vass{W}\geq (3+(k-6)\epsilon)t-\vass{A_1}$.
Because of the size of $U_2$, we can extend edges of the blue matching $M_1$ to form a triangle factor $T$ in $M_1\cup U_2$ over $3\vass{A_1}-6(1+2\epsilon)t$ vertices. We have that $\vass{U_2\setminus T}\geq 3(1+2\epsilon)t-\vass{A_1}$. Because $\vass{W},\vass{U_1}>\vass{U_2\setminus T}+4\epsilon$, we can find a blue triangle factor on $(U_1,U_2\setminus T, W)$ covering at least $3\vass{U_2\setminus T}$ vertices. Therefore combining this triangle factor with the one previously obtained over $M_1\cup T$ we have a TCTF over at least $3\vass{A_1}-6(1+2\epsilon)t +3(3(1+2\epsilon)t - \vass{A_1})= 3(1+2\epsilon)t$ vertices.
\smallskip
\underline{Case C}: Assume all connected components are smaller than $2(1+2\epsilon)t$. This means that we can partition $V(G)$ in three sets $U_1,U_2$ and $W$ such that $(1+3\epsilon)t< \vass{U_1},\vass{U_2}\leq 2(1+3\epsilon)t$ by choosing unions of components as in the previous case to get $U_1$ and $U_2$, and let $W$ be the union of the remaining components. Thus there are no red edges between any two of $U_1, U_2$ and $W$.
Because $\vass{W}=5(1+k\epsilon)t-\vass{U_1}+\vass{U_2}$ we have that all three sets $U_1, U_2$ and $W$ have size at least $(1+3\epsilon)t$ and that the largest of the three has at least $(1+6\epsilon)t$ vertices. We can find a blue matching between the smallest two of $U_1,U_2,W$ greedily of size $(1+2\epsilon)t$, and extend this to a blue TCTF of size $3(1+2\epsilon)t$ vertices greedily, using the largest component.
\end{proof}
\begin{lemma}\label{tripartitetctf}
For $n\in\bb{N}$ sufficiently large, let $G$ be a tripartite graph over $3n$ vertices with partition sets of the same size. Assume that every vertex has at least $\frac{3n}{4}$ neighbours in each of the two partition sets of which it is not part of. There exists a TCTF that covers every vertex of $G$.
Also, every pair of edges in $G$ is triangle connected.
\end{lemma}
\begin{proof}
Let $m=\frac{3n}{4}$ and $X, Y$ and $Z$ denote the sets which partition $G$.
We first use Hall's theorem to prove that there exists a perfect matching $M$ between $X$ and $Y$. Indeed, let $S$ be a subset of $X$. If $\vass{S}\leq m$, because every vertex in $S$ has at least $m$ neighbours in $Y$ we have that the neighbourhood of $S$ in $Y$ has size not smaller than the size of $S$ itself. If $\vass{S}>m$ observe that by inclusion-exclusion principle we have that every vertex in $Y$ has a neighbour in $S$.
We shall now define a bipartite support graph $H$ over the sets $M, Z$. We add an edge between $xy$ and $z$ if the vertices $xyz$ form a triangle in $G$. We can observe that the existence of a perfect matching in $H$ gives us a triangle factor that covers all vertices of $G$.
Let $xy$ be in $M$, we can notice that since both $x$ and $y$ have at least $m$ neighbours in $Z$ we have that at least $\frac{n}{2}$ of the vertices of $Z$ are neighbours of both $x$ and $y$. Therefore every edge of $M$ has minimum degree at least $\frac{n}{2}$ in $H$.
Also, every vertex in $Z$ has minimum degree at least $\frac{n}{2}$ in $H$, since in $G$ it has minimum degree at least $m$ in both $X$ and $Y$.
We can then repeat the above piece of proof and use Hall's theorem to prove that we can find a perfect matching in $H$ and therefore a perfect triangle factor in $G$.
Let us now show that every couple of edges in $G$ is triangle connected. Let us first observe that if $xy$ and $xy'$ are both edges with $x\in X$ and $y,y'\in Y$ then we have that $x,y,y'$ share a neighbour in $Z$ and therefore they are triangle connected. This implies that the set of edges between $X$ and $Y$ is in the same triangle-connected component. We can easily conclude noticing that every triangle has one edge in each of the components $(X,Y), (Y,Z)$ and $(Z,X)$ which are therefore all the same triangle-connected component.
\end{proof}
\begin{corollary}\label{corotripartitetctf}
For $n\in\bb{N}$ sufficiently large let $k,r\in\bb{N}$ such that $6r+4k<n$, let also $G$ be a tripartite graph over $3n$ vertices with partition sets $X, Y$ and $Z$ of the same size $n$. Moreover, assume every vertex in $G$ is adjacent to all but at most $k$ of the vertices in each of the two partition sets it is not a part of. Let us fix a 2-edge-colouring of $G$ such that $(X,Y)$, $(Y,Z)$ and $(X,Z)$ are $r$-red. We can find a red TCTF formed by at least $n-2r$ red triangles.
Also, all but at most $3r^2$ red edges of $G$ are in the same red triangle-connected component.
\end{corollary}
\begin{proof}
Let $X'\subseteq X, Y'\subseteq Y$ and $Z'\subseteq Z$ of size exactly $n'=n-2r$ such that every vertex in $X'\cup Y'\cup Z'$ has at most $r$ blue vertices in each of the other two components. We can apply Lemma \ref{tripartitetctf} to $G'=G^{Red}[X'\cup Y'\cup Z']$ considering that each vertex in $G'$ is adjacent to all but at most $r+k<\frac{3}{4}n'$ vertices in each of the two partitioning sets.
\end{proof}
\begin{lemma}\label{lemmasuppforlemma4}
There exists $\epsilon_0\in\mathbb{R}$ such that for all $ 0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a graph of minimum degree at least $\vass{G}-\epsilon t$ whose edges are 2-edge-coloured. If there exist in $G$ two disjoint sets $X$ and $Y$ of size respectively $(1+5\epsilon)t$ and $(5+200\epsilon)t$ such that $(X,Y)$ is $\epsilon t$-red, then $G$ contains a monochromatic TCTF on at least $3(1+\epsilon)t$ vertices.
\end{lemma}
\begin{proof}
Let $Y'$ be the set of vertices in $Y$ that have at least $|X|-\epsilon t$ red neighbours in $X$. We have that $\vass{Y'}\geq (5+100\epsilon)t$ and also $G[Y']$ has minimum degree at least $|Y'|-\epsilon t$. By Lemma \ref{lemmaconnmatchtctf} applied to $Y'$, we find either a blue TCTF of size $3(1+\epsilon)t$ or a red connected matching on $2(1+\epsilon)t$ vertices. In the first case we are done, so we can assume we have a red connected matching on $2(1+\epsilon)t$ vertices, let us denote it by $M$. By Lemma \ref{tripartitetctf} we can extend $M$ to a triangle factor $T$ of size at least $3(1+\epsilon)t$. We claim that this triangle factor is triangle connected. Indeed, every adjacent couple of red edges in $Y'$ is triangle connected since any three vertices in $Y'$ share a red neighbour in $X$. Since being triangle connected is a transitive property and because $M$ is red connected, we can conclude that $T$ is triangle connected.
\end{proof}
\section{General setting}\label{sec:genset}
To prove Lemma~\ref{MainLemma}, we will use a decomposition of $V(G)$ into red and blue cliques, and some associated notation. In this section, we describe the decomposition, define the notation, and prove that the decomposition exists under the assumptions of Lemma~\ref{MainLemma}.
\begin{setting}\label{mainsetforG}
Given $\epsilon,t>0$, let $m=\frac{1}{4}\vass{\log{\epsilon}}$ and let $k\in\mathbb{N}$ be arbitrary.
Given a graph $G$ with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$, suppose that $E(G)$ is 2-edge-coloured and that there is no monochromatic TCTF with at least $3(1+\epsilon)t$ vertices.
We fix a partition of $V(G)$ into a set $V_{\mathrm{bin}}$ of size at most $\epsilon^{1/2}t+\frac{40t}{\sqrt{m}}$ and a collection of at most $\frac{9t}{m}$ monochromatic cliques each of size between $2$ and $m$ such that the following holds.
For each vertex $u$ which is in a blue clique $C$ of the partition, we assume that at most $\tfrac{20t}{\sqrt{m}}$ blue edges go from $u$ to vertices in blue cliques of the partition which are not blue triangle connected to $C$. We assume a similar statement replacing red with blue. Moreover, the number of cliques of size less than $(1-\frac{1}{k})m$ is at most $\frac{400k}{|\log\epsilon|^{3/2}}t$.
We write $B_1$ for a blue triangle-connected component of blue cliques of the partition covering the largest number of vertices, $B_2$ for the next largest, and so on. We break ties arbitrarily, and define similarly $R_1$ for the largest red triangle-connected component of red cliques of the partition and so on. We write $B_{\ge3}:=B_3\cup B_4\cup\dots$, and $R_{\ge3}:=R_3\cup R_4\cup\dots$.
\end{setting}
It is important to note that while we will care about which vertices contain the triangles of a TCTF, we will not care which vertices are used for the triangle connections between these triangles: when we ask whether two (say red) edges are red triangle-connected, we will always mean red triangle-connected in the entire graph $G$. Thus `there is a red TCTF in $X$ of size $3s$' means that there is a set of $s$ vertex-disjoint red triangles contained in the set $X$, which are all in the same red triangle component of $G$. In particular, the set $B_1$ is a collection of blue cliques which are blue triangle-connected in $G$, the connections might well use vertices outside $B_1$.
In the following sections, we will often state lemmas referring to a `decomposition as in Setting~\ref{mainsetforG}'. When we do this, we intend to fix a specific decomposition which will remain unchanged in the proof, and statements we make refer only to this decomposition. Thus `there is no red TCTF of size $3s$ contained in the red cliques' should be understood as meaning that the union of the red cliques \emph{of the fixed partition} do not contain such a TCTF. It might be that there is a different partition which does contain such a TCTF.
\medskip
The idea of our proof of Lemma~\ref{MainLemma} is now roughly as follows. We suppose that $G$ contains no large monochromatic TCTF. Our initial aim is then to show that each of $B_1,B_2,R_1,R_2$ has roughly $2t$ vertices, while $B_3\cup R_3$ has roughly $t$ vertices, these give us the five large sets of the partition of Lemma~\ref{MainLemma}. We will see that once the size bounds are obtained, it is not too hard to show that the edge colours are as claimed. Our proof for the claimed size bounds will go over several steps of finding increasingly strong upper and lower bounds on these sizes.
\medskip
We obtain Setting~\ref{mainsetforG} by iterative application of Ramsey's theorem followed by removing a few vertices to $V_{\mathrm{bin}}$. The following Lemma \ref{lem:mainsetexists} states that this is always possible, provided $\epsilon$ is small enough and $t$ large enough.
\begin{claim}\label{lemma3path}
For $n$ sufficiently large, let $G$ be a graph over $2n$ vertices, and let $A$, $B$ be disjoint cliques of size $n$ in $G$. If there are more than $2(n-1)$ edges between $A$ and $B$, the graph is triangle connected.
\end{claim}
\begin{proof}
Equivalently, we can show that if $H$ is subgraph of $K_{n,n}$ without a path of length three, then $H$ has at most $2(n-1)$ edges. Assume $H$ is a subgraph of $K_{n,n}$ without paths of length three. In particular this means that every edge has one endpoint with degree exactly one. Therefore the number of edges in $H$ is at most equal to the number of vertices in $H$ with degree one. If we have less than $2n-2$ vertices of degree one we are done. If we have $2n$ vertices with degree exactly one we know that $H$ is a perfect matching. It cannot be the case that $2n-1$ vertices have degree exactly one. Therefore we covered all cases and we can conclude that the number of edges in $H$ is at most $2(n-1)$.
\end{proof}
\begin{lemma}\label{lem:mainsetexists}
There exists $\epsilon_0\in\mathbb{R}$ such that for all $ 0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ and $k\in\mathbb{N}$ the following holds. Given a graph $G$ with at least $(9-\epsilon)t$ vertices, with minimum degree at least $(9-2\epsilon)t$, whose edges are 2-edge-coloured, there exist sets $R_1,\dots$ and $B_1,\dots$ of monochromatic red and blue cliques respectively satisfying the properties of Setting \ref{mainsetforG}.
\end{lemma}
\begin{proof}
Let us start by proving that we can find disjoint monochromatic copies of $K_m$ covering all but at most $\epsilon^{\frac{1}{2}}t$ vertices of $G$.
First, notice that we do not want all cliques to be of the same colour, we just want monochromatic cliques (some might be red, some might be blue).
Let us start by selecting greedily as many monochromatic copies of $K_m$ as possible in $G$, this means that we start by selecting an arbitrary monochromatic $K_m$, then we remove its vertices and we repeat the process over the remaining vertices of $G$.
Let us assume by contradiction that when this process stops more than $\epsilon^{\frac{1}{2}}t$ vertices of $G$ remain. Let $W$ be a set of size $\epsilon^{\frac{1}{2}}t$ not containing any monochromatic clique. Because of the minimum degree condition over $G$, we have that each vertex of $G[W]$ has degree at least $(\epsilon^{\frac{1}{2}}-\epsilon)t$ and therefore $G[W]$ contains at least $\epsilon^{\frac{1}{2}}(\epsilon^{\frac{1}{2}}-\epsilon)t^2=\left(1-\frac{1}{\epsilon^{-\frac{1}{2}}}\right)(\epsilon^{\frac{1}{2}}t)^2$ edges.
By Turan's theorem, we have that $G[W]$ contains a (not necessarily monochromatic) clique $K$ of size $\epsilon^{-\frac{1}{2}}$. By a Ramsey's upper bound on diagonal Ramsey numbers we have that $R(m,m)\leq 4^m$, this value is smaller than $\epsilon^{-\frac{1}{2}}$ for $\epsilon$ small enough. Indeed, for $\epsilon<1$ we have $\epsilon=e^{-4m}$ and hence we can rewrite the inequality as $R(m,m)\leq 4^m\leq e^{2m}=\epsilon^{-\frac{1}{2}}$ which holds for $m$ large enough. Therefore we can find a monochromatic clique $K'$ in $W$.
This contradicts the stopping of our greedy algorithm.
We can now focus on the number of vertices in blue cliques that witness more than $\frac{20t}{\sqrt{m}}$ blue edges that have endpoints in distinct triangle-connected components of blue $K_m$.
\begin{itemize}
\item There are at most $\frac{9t}{m}$ disjoint copies of $K_m$ in $G$. This, combined with Claim \ref{lemma3path} gives us that at most $\frac{(10t)^2}{m}$ blue edges have endpoints in distinct triangle-connected components of blue $K_m$.
\item At most $\frac{(20t)^2}{m}$ vertices in blue cliques of $G$ witness a blue edge with its two extremities in two distinct triangle-connected components of blue $K_m$. Therefore at most $\frac{20t}{\sqrt{m}}$ vertices in blue cliques witness more than $\frac{20t}{\sqrt{m}}$ such edges.
\item We can do the same for red and obtain again at most $\frac{20t}{\sqrt{m}}$ vertices in red cliques that witness more than $\frac{20t}{\sqrt{m}}$ edges with their two extremities in two distinct triangle-connected components of red cliques.
\end{itemize}
Finally, for some given positive integer $k$, we want to count how many monochromatic cliques in $V(G)\setminus V_{\mathrm{bin}}$ can have less than $(1-\frac{1}{k})m$ vertices. Which is, we want to state at most how many cliques of $G$ can have more than $\frac{m}{k}$ vertices in $V_{\mathrm{bin}}$. It is not difficult to see that this number is less than $\frac{40t}{\sqrt{m}}\cdot \frac{k}{m}\leq\frac{400 k }{\vass{\log{\epsilon}}^{\frac{3}{2}}}t$.
Therefore at most $\frac{100 k }{\vass{\log{\epsilon}}^{\frac{1}{2}}}t$ vertices are in cliques of size at most $(1-\frac{1}{k})m$.
\end{proof}
\section{First upper bounds on the component size}
In this section, we prove that $|B_i|,|R_i|$ cannot be much bigger than $\tfrac73t$ (Lemma~\ref{upperboundsingle}) and that we cannot have both $B_1$ and $B_2$ (or $R_1$ and $R_2$) much bigger than $2t$ (Lemma~\ref{upperbounddouble}).
\begin{lemma}\label{upperboundsingle}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix any collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$. If $G$ has a set of blue triangle-connected cliques covering more than $(\frac{7}{3}+h)t$ vertices, then $G$ contains a monochromatic TCTF with $(1+\epsilon)t$ triangles. The same holds replacing blue with red.
\end{lemma}
\begin{proof}
Let $A$ be a triangle-connected set of blue cliques that covers more than $(\frac{7}{3}+h)t$ vertices. If $|A|\ge 3(1+\frac{50}{\vass{\log{\epsilon}}})t$ then we greedily construct a blue TCTF within $A$ that leaves out at most two vertices from each clique and obtain a blue TCTF covering at least $3(1+\epsilon)t$ vertices as desired, so we may now assume $|A|< 3(1+\frac{50}{\vass{\log{\epsilon}}})t$.
Because of this bound on the size of $A$, and the condition of Setting~\ref{mainsetforG} there are at most $\frac{40000}{\vass{\log{\epsilon}}^{\frac{3}{2}}} t$ cliques with less than $\frac{99}{100}m$ vertices, we have that there are at most
\[\frac{3(1+10\epsilon)t}{\frac{99}{100}m}+\frac{40000}{\vass{\log{\epsilon}}^{\frac{3}{2}}} t\leq \frac{16 t}{\vass{\log{\epsilon}}}\]
blue cliques in $A$ and at least $\vass{V(G)}-3(1+\frac{50}{\vass{\log{\epsilon}}})t$ vertices in $V(G)\setminus A$.
In succession for each blue clique in $A$, we greedily construct a blue triangle factor $T$ using one edge in the selected clique and one vertex outside of $A$. There are two possible cases.
\underline{Case A}: The greedy construction provides us with a set $T$ of $\frac{2}{3}(1+\epsilon)t$ triangles.
We can extend $T$ to a triangle factor $T'$ by adding triangles from within the cliques in $A$. When we stop, at most two vertices for each cliques are being unused and hence we obtained a blue TCTF covering at least
$$3\cdot \frac{2}{3}(1+\epsilon)t + \left(\big(\tfrac{7}{3}+h\big) - 2\cdot \frac{2}{3}(1+\epsilon) - 2\cdot \frac{16}{\vass{\log{\epsilon}}}\right)t$$
vertices. Note that this means that $T'$ covers at least $3(1+\epsilon) t$ vertices.
\underline{Case B}: The greedy construction stops before we get $\frac{2}{3}(1+\epsilon)t$ triangles.
Let $Y=V(G)\setminus (A\cup T)$. We have that
\begin{align*}
\vass{Y}&\geq (9-\epsilon)t-3(1+\frac{50}{\vass{\log{\epsilon}}})t-\frac{2}{3}(1+\epsilon)t\\
&\geq (5+h)t\ge\Big(5+\frac{20000}{\sqrt{\vass{\log\epsilon}}}\Big)t\,.
\end{align*}
Let us denote by $X$ the set of all the vertices in $A\setminus T$ which are in cliques that have at least three vertices in $A\setminus T$. At most $\frac{4}{3}(1+\epsilon)t+2\cdot \frac{16}{\vass{\log{\epsilon}}}t$ vertices are in $A$ but not in $X$. Therefore we have that
\[\vass{X}\geq \left(1+\frac{100}{\sqrt{\vass{\log\epsilon}}}\right)t\,.\]
Because we stopped the greedy procedure, we cannot extend $T$ using an edge in a clique of $X$ and a vertex in $Y$, therefore each vertex in $Y$ has at most one blue neighbour in each clique of $X$.
This means that there are at most $\frac{16 t}{\vass{\log{\epsilon}}}\cdot \vass{Y}< \frac{16 t}{\vass{\log{\epsilon}}}\cdot\left( 9-\frac{7}{3} \right)t\leq \frac{20^2t^2}{\vass{\log{\epsilon}}}$ blue edges between $X$ and $Y$. Hence we have that $(X,Y)$ is $\frac{20}{\sqrt{\vass{\log{\epsilon}}}}t$-red. We can now apply Lemma \ref{lemmasuppforlemma4} with input $\frac{20}{\sqrt{\vass{\log\epsilon}}}$. We conclude that $G$ contains a monochromatic TCTF on at least
\[3\Big(1+\frac{20}{\sqrt{\vass{\log\epsilon}}}\Big)t>3(1+\epsilon)t\]
vertices.
\end{proof}
\begin{lemma}\label{upperbounddouble}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix any collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$. If $G$ contains two disjoint sets of blue triangle-connected cliques, with each set of cliques covering more than $(2+h)t$ vertices, then $G$ contains a monochromatic TCTF with $(1+\epsilon)t$ triangles. The same holds replacing blue with red.
\end{lemma}
\begin{proof}
Let $A$ and $B$ be disjoint sets of triangle-connected blue cliques, each covering at least $(2+h)t$ vertices. We may suppose $h_0\le\tfrac1{30}$. Let $C$ denote the collection of all the remaining vertices in blue cliques, if any exist. By Lemma \ref{upperboundsingle}, either we have the desired monochromatic TCTF or both $A$ and $B$ are smaller than $\big(\tfrac73+h\big)t$. Therefore by Setting~\ref{mainsetforG} with $k=100$ they both contain at most the following number of blue cliques:
\[\frac{\frac{71}{30}t}{\frac{99}{100}\cdot\frac{1}{4}\vass{\log{\epsilon}}}+\frac{40000t}{\vass{\log{\epsilon}}^{\frac{3}{2}}}\leq \frac{10t}{\vass{\log{\epsilon}}}\,.\]
Moreover, by Claim \ref{lemma3path} there are less than $2m$ blue edges between any blue clique in $A$ and any clique in $B$. Therefore, between $A$ and $B$ there are less than $2m\cdot \frac{10t}{\vass{\log{\epsilon}}} \cdot \frac{10t}{\vass{\log{\epsilon}}}\leq \frac{50}{\vass{\log{\epsilon}}}t^2$ blue edges. Hence, $(A,B)$ is $\frac{8}{\sqrt{\vass{\log{\epsilon}}}}t$-red. Let us set $\lambda=\frac{8}{\sqrt{\vass{\log{\epsilon}}}}$.
Let us greedily build a blue triangle factor $T_A$ by extending blue edges in blue cliques of $A$ to blue triangles using vertices outside of $A$. Let $Y_A$ be the set of vertices in $V(G)\setminus A$ used in this way
and $A'$ the set of remaining vertices in $A$
. We can independently do the same construction with $B$ and obtain a triangle factor $T_B$ and some similar sets $Y_B$ and $B'$. Finally, let us denote $Z=V(G)\setminus(A\cup Y_A\cup B\cup Y_B)$.
Because we can extend $T_A$ to a blue TCTF that covers all but at most two vertices for each clique of $A$ {(and similarly for $B$)}, we have that $\vass{A\cup Y_A}, \vass{B\cup Y_B}\leq (3+3\epsilon+\frac{8}{m})t$. This implies $$\vass{Z}\geq \vass{V}-(\vass{A\cup Y_A}+\vass{B\cup Y_B})\geq (9-h)t-2(3+h)t=3(1-h)t\,.$$
We also have that $\vass{Y_A}, \vass{Y_B}\leq (1+\epsilon)t$, which implies that $\vass{A'}, \vass{B'} \geq (h-2\epsilon)t$.
Each vertex of $Z$ has at most one blue neighbour per clique in each of $A'$ and $B'$, since we cannot further extend $T_A$ or $T_B$. Since $\vass{Z}\leq 5t$ and there are at most $\frac{10 t}{\vass{\log{\epsilon}}}$ cliques in each of $A'$ and $B'$ we have that both $(A',Z)$ and $(B',Z)$ have at most $\frac{50}{\vass{\log{\epsilon}}}t^2$ blue edges and hence they are both $\lambda t$-red.
\begin{claim}
We claim that all red edges in $Z$ are triangle connected. Moreover, if $|C\cap Z|\ge \tfrac13t$ then we can find a TCTF in $(A, B, C\cap Z)$ on $\vass{C\cap Z}-ht$ triangles that is triangle connected to the red triangle-connected component of $Z$.
\end{claim}
\begin{claimproof}
Let $xy$ and $uv$ be two red edges in $Z$, let $N_A$ be the set of vertices in $A'$ red adjacent to all vertices $x,y,u$ and $v$ and let $N_B$ be defined similarly. To prove that $xy$ and $uv$ are triangle connected it suffices to show that there exists a red edge between $N_A$ and $N_B$. Because of the lower bound on the size of $A'$ and $B'$, because of the minimum degree condition and because every vertex in $Z$ is adjacent in red to all but at most $\frac{10t}{\vass{\log{\epsilon}}}$ of its neighbours in $A'$ and $B'$, we have that $\vass{N_A},\vass{N_B}\geq (h-2\epsilon)t-4\cdot \epsilon t-4\cdot \frac{10 t}{\vass{\log{\epsilon}}}\geq \frac{3h}{4}t$. Since $(A,B)$ is $\lambda t$-red, there is a red edge between $N_A$ and $N_B$. Therefore all the red edges in $Z$ are in the same triangle-connected component.
Let us now create a red TCTF $\Delta$ in $(A, B, C\cap Z)$ as follows. We first find a largest TCTF $\Delta'$ in $(A', B', C\cap Z)$. By Corollary \ref{corotripartitetctf}, we have that $\Delta'$ has at least $\frac{h}{2}t$ vertices, since we have a lower bound on both $\vass{A'}$ and $\vass{B'}$.
We can now use Corollary \ref{corotripartitetctf} to find a red TCTF in $(A\setminus \Delta', B\setminus \Delta', (C\cap Z)\setminus \Delta')$ that covers almost all $(C\cap Z)\setminus \Delta'$. Let us call $\Delta$ the union of the two triangle factors. By Lemma \ref{tripartitetctf} that $\Delta$ is triangle connected.
It now suffices to show that $\Delta'$ is triangle connected to the red triangle-connected component of $Z$. Let $xy$ be a red edge in $Z$, let $N_A$ be the set of vertices in $A'\cap \Delta'$ red adjacent to both $x$ and $y$, and let $N_B$ be defined similarly in $B'\cap\Delta'$. To prove that $xy$ and $\Delta'$ are triangle connected it suffices to show that there exists an edge of $\Delta'$ between $N_A$ and $N_B$. Because every vertex in $Z$ is adjacent in red to all but at most $\frac{10t}{\vass{\log{\epsilon}}}$ of its neighbours in $A'$ and $B'$, we have that $\vass{N_A},\vass{N_B}\geq \frac{99h}{100}t$. Since $\Delta'$ is a matching in $(A', B')$ of large size, some of its edges are between $N_A$ and $N_B$.
\end{claimproof}
$Z\setminus C$ can be extended to a set of triangle-connected red cliques of $G$, possibly adding vertices from $Y_A$ and $Y_B$. Therefore, we have $\vass{Z\setminus C}\leq \left(\frac{7}{3}+h\right)t$ and this in particular implies that $\vass{C\cap Z}\geq (\frac{2}{3}-4h)t$. We form a red TCTF as follows. We start by using our last claim to construct a TCTF $T_C$ over at least $\vass{C\cap Z}-h t\geq (\frac{2}{3}-5h)t$ triangles between $A, B$ and $C\cap Z$ that is also triangle connected to the red triangle-connected component of $Z$. We then extend this TCTF by taking triangles in cliques of $Z\setminus C$. This is enough to conclude.
\end{proof}
\section{Colours and connection, and the sharp upper bound}
In this section we begin by proving two lemmas which show that certain patterns of edges between triangle components imply triangle connections, which we need in both this section and the next. We then establish several inequalities about sizes of the components (Lemma~\ref{biguniquelemma}), most of which imply that various components cannot be too small. In particular, we establish the useful inequality $|B_2|\ge|B_{\ge3}|$, and similarly for red. Building on this, we finally prove the sharp upper bound we want: none of the components can contain much more than $2t$ vertices (Lemma~\ref{lemmalastupperbound}). These are the two statements we need to complete the proof of Lemma~\ref{MainLemma} in the next section.
\subsection{Colours and connection}
\begin{claim}\label{coroappendix1}
For any $h>0$ there exists $\epsilon>0$ such that if we use that $\epsilon$ for Setting \ref{mainsetforG} we have the following. Let $A, B$ be two disjoint sets of vertices in blue cliques such that there are no triangle-connected components with some vertices in $A$ and some vertices in $B$. Then the pair $(A,B)$ is $ht$-red. The same works for red.
\end{claim}
\begin{proof}
By Remark \ref{lem:mainsetexists}, in $G$ there are at most $\frac{9t}{\frac{99}{100}m}+\frac{\const{4}t}{\vass{\log{\epsilon}}^{\frac{3}{2}}}\leq \frac{40t}{\vass{\log{\epsilon}}}$ cliques. Therefore, by Claim \ref{lemma3path} we can have at most $2m\cdot\frac{20t}{\vass{\log{\epsilon}}}\cdot \frac{20t}{\vass{\log{\epsilon}}}\leq \frac{200 t^2}{\vass{\log{\epsilon}}}$ blue edges between $A$ and $B$. In particular this means that the pair $(A,B)$ is $\sqrt{\frac{200}{\vass{\log{\epsilon}}}}t$-red. For $\epsilon$ small enough we have the result we wanted.
\end{proof}
\begin{lemma}\label{lemma17}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix any collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$. Let $Y_1, Y_2, Y_3$ be subsets of size at least $10ht$ of vertices in distinct red triangle-connected clique components, and let $X$ be a set of size at least $ht$ of vertices in blue cliques which all have more than $2ht$ blue neighbours in two of the $Y_i$s. Then at least one of the blue edges in a clique of $X$ is triangle connected to the large blue TCTF in $(Y_1, Y_2, Y_3)$. Everything still works if we invert red and blue.
\end{lemma}
\begin{proof}
First, note that for $\epsilon$ small enough and by Claim \ref{coroappendix1} we have that each pair in $Y_1, Y_2, Y_3$ is $\frac{h^3}{2}t$-blue. Let $R_i$ be the set of vertices in $Y_i$ with more than $h^3t$-red edges in one of the other $Y_j$.
Without loss of generality let us assume that the set $S$ of vertices in $X$ with more than $2ht$ blue neighbours in both $Y_1$ and $Y_2$ has size at least $\frac{ht}{3}$. Then each vertex in $S$ has at least $(2h-h^3)t$ blue neighbours in both $Y_1\setminus R_1$ and $Y_2\setminus R_2$. Then we have a vertex $y_1$ in $Y_1\setminus R_1$ which is incident in blue to at least $(2h-h^3)t\cdot\frac{ht}{3}\cdot\frac{1}{9t}\geq\frac{1}{15}h^2t$ vertices in $S$. So for $t$ large enough $y_1$ is incident in blue to at least two vertices of $S$ that lie in the same clique, let us call two such vertices $x_1$ and $x_2$. Since $y_1$ has at least $\vass{Y_2}-(\epsilon+h+h^3)t\geq \vass{Y_2}-(2h+h^3)t$ blue neighbours in $Y_2\setminus R_2$, we have that $y_1$ and $x_1$ have a common blue neighbour $y_2$. This implies that $x_1x_2$ is blue-triangle connected to $y_1y_2$ and this by minimum degree condition means that $x_1x_2$ is triangle connected to the large blue TCTF over $(Y_1, Y_2, Y_3)$ given by Lemma \ref{tripartitetctf}.
\end{proof}
\begin{lemma}\label{lemma19}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix any collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$, and let $Y_1, Y_2$ be subsets of size at least $10ht$ of vertices in distinct red triangle-connected components, and let $X_1, X_2$ be subsets of size at least $10ht$ of vertices in distinct blue triangle-connected clique components.
Finally, assume that $X_1$ is $ht$-red to each of $Y_1$ and $Y_2$. Then at most $2ht$ vertices in $X_2$ have more than $2ht$ red neighbours in both $Y_1$ and $Y_2$. Everything still works if we invert red and blue.
\end{lemma}
\begin{proof}
First, note that for $\epsilon$ small enough and by Claim \ref{coroappendix1} we have that $(X_1,X_2)$ is $ht$-red.
Let $S$ be the set of vertices in $X_2$ which have more than $2ht$ red neighbours in both $Y_1$ and $Y_2$. Assume by contradiction $\vass{S}\geq 2ht$. Note that there is a vertex $x_1$ in $X_1$ which has at most $ht$ blue neighbours in each of $X_2$, $Y_1$ and $Y_2$, so $x_1$ is red-adjacent to some vertex $x_2\in S$. Now $x_1$ and $x_2$ have at least $\frac{h}{4}t$ common red neighbours in each $Y_i$ and therefore they have at least two common red neighbours from the same clique in each of the $Y_i$. But this is absurd because it would mean that a clique in $Y_1$ is triangle connected to a clique in $Y_2$.
\end{proof}
\subsection{Some lower bounds}
\begin{claim}\label{nuovoclaimsuireali}
Let $k$ be a positive integer and let $b_1\geq \dots{}\geq b_k>0$ be positive reals such that $\sum_{i> 1} b_i > b_1$. Then we can partition $\left\lbrace 1, \dots{}, k\right\rbrace$ into two sets $A$, $B$ such that if $\alpha:=\sum_{i\in A} b_i$ and $\beta:=\sum_{i\in B} b_i$ we have $2\alpha\geq \beta\geq \alpha$.
\end{claim}
\begin{proof}
We can construct such a partition greedily in two steps.
If $b_1+b_3\leq 2(b_2+b_4)$ we set $1, 3\in B$ and $2, 4\in A$. Otherwise we set $b_1\in B$ and $2, 3,\dots,\ell\in A$ with an $\ell$ such that $b_1>\sum_{i=2}^\ell b_i> \frac{b_1}{2}$ (such an $\ell$ exists because of the hypotheses and because $b_1>b_2+b_3$).
We now proceed by induction. Assume we already partitioned $1, \dots{}, i-1$ such that the requests of the lemma are satisfied and let $\alpha$ and $\beta$ be as in the statement of the lemma. If $2\alpha\geq \beta+b_i$ we can add $i\in B$. Otherwise, we have $\beta> 2\alpha-b_i \geq \alpha + b_i$, where the last inequality is given by the fact that the $b_i$ are ordered in decreasing order and $\vass{A}\geq 2$. In this second case we can add $i$ to the set $A$.
\end{proof}
\begin{lemma}\label{biguniquelemma
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix any collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$, and define $B_1,B_2,\dots$ and $R_1,R_2,\dots$ as in Setting~\ref{mainsetforG}.
\begin{enumerate}[label=(\roman*)]
\item\label{sublemma1} If $\vass{B_1}\leq \frac{7}{6}t$ then $\vass{\bigcup_{i}B_i}\leq \left(\frac{7}{2}+h\right)t$.
\item\label{sublemma2} If $\vass{B_1}\geq \frac{7}{6}t$ and $\vass{B_2}\leq \frac{7}{6}t$ then $\big|\bigcup_{i\neq 1}B_i\big|\leq \left(\frac{7}{3}+h\right)t$.
\item\label{sublemma3} If $\vass{B_1}, \vass{B_2}\geq \frac{7}{6}t$ then $\vass{\cup_{i}B_i}\leq (\frac{16}{3}+h)t$.
\item\label{sublemma4} We have $\frac{43}{12}t\leq \vass{\cup_i B_i}, \vass{\cup_i R_i}\leq (\frac{16}{3}+h)t$. We also have $\vass{B_1}> \frac{7}{6}t$.
\item\label{sublemma5} If $\vass{B_2}<\vass{\cup_{i\geq 3} B_i}$ then we can find a red TCTF in $\cup_i B_i$ of size at least $\frac{3}{2}\vass{\cup_{i\geq 2}B_i}-ht$.
\item\label{sublemma6} If $\vass{B_2}\leq \frac{8}{7}t$ then $\vass{\cup_{i}B_i}<(\frac{9}{2}-h)t$. Hence at most one of $B_2$ or $R_2$ can be smaller than $\frac{8}{7}t$.
\item\label{sublemma7} We have $\vass{B_2}\geq \vass{\cup_{i\geq3} B_i}$ and $\vass{R_2}\geq \vass{\cup_{i\geq 3}R_i}$.
\end{enumerate}
The corresponding results also hold for red and $R_1, R_2, R_3, \dots{}$.
\end{lemma}
\begin{proof}
We are going to prove these results in order, and we are sometimes going to use previous points already proved.
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma1}}:
Suppose for a contradiction that $\vass{B_1}\leq\tfrac76t$ and $\vass{\bigcup_{i}B_i}> \left(\frac{7}{2}+h\right)t$. Observe that by Setting~\ref{mainsetforG} with $k=100$, all but at most $\frac{40000}{\vass{\log{\epsilon}}^{\frac{1}{2}}}t$ vertices of $G$ are in cliques fixed in Setting~\ref{mainsetforG} with at least $\frac{99}{100}m$ vertices. We let for each $i$ the set $B'_i$ consist of all vertices in blue cliques of $B_i$ with at least $\frac{99}{100}m$ vertices.
We want to study how many edges have endpoints in two distinct $B'_i$. For any fixed $i$, the maximum number of blue edges that have one endpoint in $B'_i$ and the other in some $B'_j$ with $j\neq i$, is less than
\[2m\cdot\frac{\vass{B'_i}}{\frac{99}{100}m}\cdot\frac{\vass{\bigcup_{j\neq i}B'_j}}{\frac{99}{100}m}\leq 3\frac{\vass{B'_i}\cdot \vass{\bigcup_{j\neq i}B'_j}}{m}\leq \frac{27}{m}t\vass{B'_i}\,.\]
Let us now observe that the number of vertices in $B'_i$ that have more than $\frac{h}{100}t$ blue neighbours outside of $B'_i$ is at most $\frac{27}{m}t\vass{B'_i}\cdot\frac{100}{ht}\leq \frac{10^4}{mh}\vass{B'_i}$.
Let us remove from each $B'_i$ all the vertices with more than $\frac{h}{100}t$ blue neighbours in $\bigcup_{j\neq i}B'_j$, let us call the result $B_i''$. By the last observation, we have that
\begin{align*}
\Big|\bigcup_i B_i''\Big|&\geq \big(1-\tfrac{10^4}{mh}\big)\Big|\bigcup_i B'_i\Big|\\
&\geq \big(1-\tfrac{10^4}{mh}\big)\Big(\Big|\bigcup_i B_i\Big|-\frac{40000}{\vass{\log{\epsilon}}^{\frac{1}{2}}}t\Big)\\
&\geq \big(1-\tfrac{10^4}{mh}\big)\cdot \left(\tfrac{7}{2}+\tfrac{3h}{4}\right)t\\
&\geq \big(\tfrac{7}{2}+\tfrac{3h}{4}-\tfrac{4\cdot 10^4}{mh}-\tfrac{10^4}{m}\big)t\\
&\geq \big(\tfrac{7}{2}+\tfrac{h}{2}\big)t\,.
\end{align*}
In $G^{Red}\big[\bigcup_i B''_i\big]$ every vertex has red degree at least $\vass{\bigcup_i B'_i}-(\frac{7}{6}+\epsilon+\frac{h}{100})t$ which is more than $\frac{2}{3}\vass{\bigcup_i B''_i}$. So by Lemma \ref{lemmatwothirdsappe}, $G^{Red}[\cup_i B_i'']$ contains a red TCTF of size $\frac{7}{2}t$.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma2}}:
Let $B^\ast_1$ be a set of the fixed blue cliques in $B_1$ covering between $\tfrac76t-m$ and $\tfrac76t$ vertices. We may assume $|B_2|\le |B^\ast_1|$, by swapping these two sets of cliques if necessary.
Repeating what we did in Lemma~\ref{biguniquelemma}\ref{sublemma1} to the sets $B^\ast_1,B_2,B_3,\dots$, we obtain $\big|B^\ast_1\cup\bigcup_{i\ge 2}B_i\big|\le\big(\tfrac72+h\big)t$. Since $|B^\ast_1|\le\tfrac76t$, we have $\big|\bigcup_{i\ge 2}B_i\big|\le\big(\tfrac73+h\big)t$ as desired.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma3}}:
By Corollary \ref{corotripartitetctf} we have that $\vass{\cup_{i\geq 3} B_i}\leq (1+\frac{h}{3})t$ because otherwise we can find a red TCTF over more than $3(1+\epsilon)t$ vertices. By Lemmas \ref{upperboundsingle} and \ref{upperbounddouble} we have that $\vass{B_1}\leq (\frac{7+h}{3})t$ and $\vass{B_2}\leq (\frac{6+h}{3})t$.
Summing these bounds completes the proof.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma4}}:
By Lemmas ~\ref{biguniquelemma}\ref{sublemma1}, \ref{sublemma2}, ~\ref{sublemma3} we have that for any possible size of $B_1$ and $R_1$ we always have $\vass{\cup_i B_i}, \vass{\cup_i R_i}\leq (\frac{16}{3}+h)t$. Because $\vass{\cup_i B_i}+\vass{\cup_i R_i}\geq (9-h)t$ we therefore must have $\frac{43}{12}t\leq \vass{\cup_i B_i}, \vass{\cup_i R_i}$. By Lemma ~\ref{biguniquelemma}\ref{sublemma1} this implies that $\vass{B_1},\vass{R_1}> \frac{7}{6}t$.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma5}}:
Let us take a set of vertices $B_1'\subseteq B_1$ such that $\vass{B_1'}=\frac{1}{2}\vass{\cup_{i\geq 2} B_i}-\frac{1}{100}ht$ (we know that $B_1$ is large enough, indeed we know $\vass{B_1}\geq \frac{7}{6}t$ and it cannot be the case that $\vass{B_2}\geq \frac{7}{6}t$ because otherwise we would find a large red TCTF over $(B_1, B_2, \cup_{i\geq 3}B_i)$). By Claim \ref{coroappendix1} all but at most $\frac{1}{100}ht$ vertices of $B_1'$ have red degree in $G[B_1'\cup\bigcup_{i\geq 2}B_i]$ at least $\vass{\cup_{i\geq 2}B_i}-\frac{1}{100}ht$. Let $B_1''$ be a subset of size $\frac{1}{2}\vass{\cup_{i\geq 2} B_i}-\frac{2}{100}ht$ such that every vertex in $B_1''$ has red degree in $G[B_1''\cup\bigcup_{i\geq 2}B_i]$ at least $\vass{\cup_{i\geq 2}B_i}-\frac{1}{100}ht\geq \frac{2}{3}\vass{B_1''\cup\bigcup_{i\geq 2}B_i}$. Because every vertex in $\cup_{i\geq 2}B_i$ is in a triangle-connected component of size significantly smaller than $\frac{2}{3}\vass{B_1''\cup\bigcup_{i\geq 2}B_i}$ we can conclude by Lemma \ref{lemmatwothirdsappe} that we can find a red TCTF over all but at most two vertices of $B_1''\cup\bigcup_{i\geq 2}B_i$. Which is, we can find a red TCTF over at least $\frac{3}{2}\vass{\cup_{i\geq 2}B_i}-ht$ vertices.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma6}}:
Fix some $h>0$ arbitrarily small, depending on which we can choose our $\epsilon$. By Lemma ~\ref{biguniquelemma}\ref{sublemma1} we can assume $\vass{B_1}\geq \frac{7}{6}t$. Remember also that we have by Lemma \ref{upperboundsingle}, $(\frac{7}{3}+h)t\geq \vass{B_1}$.
Assume by contradiction $\vass{B_2}\leq \frac{8}{7}t$ and $\vass{\cup_{i}B_i}\geq (\frac{9}{2}-h)t$. Then we would have $\vass{\cup_{i\geq 3}B_i}\geq (\frac{9}{2}-h)t-(\frac{7}{3}+h)t-\frac{8}{7}t=(\frac{43}{42}-2h)t$. By Corollary \ref{corotripartitetctf} and Claim \ref{coroappendix1} it cannot be the case that $\vass{B_2}\geq (\frac{43}{42}-2h)t$ because otherwise we would find a large red TCTF over $(B_1, B_2, \cup_{i\geq 3}B_i)$. Therefore we must have $\vass{B_2}<\vass{B_3}$, and therefore by Lemma ~\ref{biguniquelemma}\ref{sublemma5} we must have that $\frac{3}{2}\vass{\cup_{i\geq 2}B_i}-ht< (3+h)t$ which is to say that $\vass{\cup_{i\geq 2}B_i}<\frac{25}{12}t$. We can conclude that $\vass{\cup_{i}B_i}<(\frac{7}{3}+h)t+\frac{25}{12}t<(\frac{9}{2}-h)t$.
\vspace{0.2cm}
\underline{Proof or ~\ref{biguniquelemma}\ref{sublemma7}}:
First, let us note that we cannot have both $\vass{B_2}< \vass{\cup_{i\geq3} B_i}$ and $\vass{R_2}< \vass{\cup_{i\geq3} R_i}$. Indeed, by ~\ref{biguniquelemma}\ref{sublemma6} at least one between $B_2$ and $R_2$ has cardinality at least $\tfrac{8}{7}t$. Let us say without loss of generality that $\vass{R_2}\geq \tfrac{8}{7}t$, then it cannot be $\vass{\cup_{i\geq 3}R_i}>\vass{R_2}$ because of Corollary \ref{corotripartitetctf}.
Let us now assume by contradiction that $\vass{\cup_{i\geq 3}B_i}>\vass{B_2}$. By Lemmas \ref{upperboundsingle} and \ref{upperbounddouble} we have that $\vass{R_1}\leq (\tfrac{7}{3}+h)t$ and $\vass{R_2}\leq (2+h)t$. Moreover, by Corollary \ref{corotripartitetctf} we have $\vass{R_3}\leq (1+h)t$. Therefore we have $\vass{B_2\bigcup\cup_{i\geq 3}B_i}\geq (\tfrac{4}{3}-5h)t$.
By Claim \ref{nuovoclaimsuireali}, since both $B_3$ and $B_4$ are non-trivial (by our contradiction hypothesis), we can partition the sets $B_i$ into collections $B_1'$, $B_2'$ and $B_3'$ such that $B_1'=B_1$ and $\vass{B_2'}\geq \vass{B_3'}$ and also $\vass{B_2'}\leq \tfrac{2}{3}\vass{\cup_{i\geq 2}B_i}$. In particular this means $2\vass{B_3'}\geq \vass{B_2'}\geq \vass{B_3'}$ and $\vass{B_2'}\geq (\tfrac23-5h)t$ and $\vass{B_3'}\geq (\tfrac{4}{9}-5h)t$.
Notice that by Lemma~\ref{biguniquelemma}\ref{sublemma5} we have $\vass{B_2\bigcup\cup_{i\geq 3}B_i}\leq (2-2h)t$.
We claim that no blue clique in $B_1'$ is triangle connected to the blue TCTF in $(R_1, R_2, R_3)$. Indeed we have that this would create a blue TCTF of size at least $3\vass{R_3}+\vass{B_1}$ and we have $\vass{R_3}\geq 9t-\vass{B_1}-\vass{B_2\bigcup\cup_{i\geq 3}B_i}-\vass{R_1}-\vass{R_2}\geq(\tfrac13-5h)t$ and $\vass{R_3}+\vass{B_1}\geq(\tfrac{8}{3}-4h)t$. Which implies that $3\vass{R_3}+\vass{B_1}>(3+h)t$.
In particular, by Lemma \ref{lemma17} this implies that all but at most $ht$ vertices in $B'_1$ have less than $2ht$ blue neighbours in two of the $R_1, R_2$ or $R_{\geq 3}$. This means that there is a set $T\subset B_1'$ of size at least $\frac13\big(\vass{B_1'}-ht\big)$ such that $(T, R_i), (T, R_j)$ are $ht$-red and $i,j\in \left\lbrace 1,2,\geq 3\right\rbrace$. Let us assume that $(T, R_2)$ is $ht$-red (if not, then we have $(T, R_1)$ is $ht$-red and this is strictly better in the following computations). We claim that $(R_2, B_2')$ and $(R_2, B_3')$ are $ht$-blue. Indeed by Lemma~\ref{biguniquelemma}\ref{sublemma5} and by the lower bound $\vass{B_2\bigcup\cup_{i\geq 3}B_i}\geq (\tfrac{4}{3}-5h)t$ we got earlier, we have a red TCTF in $B'_1\cup B'_2\cup B'_3$ of size at least $(2-10h)t$, since $\vass{R_2}\geq \tfrac87t$ we must have that each clique in $R_2$ is not triangle connected to the large TCTF between the $B_i$ components. By Lemma \ref{lemma17} and since $(T, R_2)$ is red, we get that $(R_2, B_2')$ and $(R_2, B_3')$ are $ht$-blue.
We now claim that there is a $B_i$ in $B_2'$ such that $(B_i, R_{\geq3})$ is $ht$-red, in particular, this means that each red clique in $R_{\geq 3}$ is in the same triangle connected component of $(B'_1, B'_2, B'_3)$. There exists such a $B_i$ because $B_2'$ is formed by at least two distinct blue triangle-connected components, which cannot therefore be triangle connected among themselves. But we also know that $(B_2', R_2)$ is $ht$-blue, so if we had that more than one blue component in $B_2'$ has blue neighbours in $R_3$ we would get that these blue components are triangle connected.
Now we claim that we must have $\vass{R_{\geq 3}}\geq (1-20h)t$. As observed above, there is a red TCTF in $B'_1\cup B'_2\cup B'_3$ of size at least $\tfrac32\vass{B_2'\cup B_3'}$, and its triangles are triangle-connected in red to $R_{\ge3}$, we have a red TCTF of size $\tfrac32\vass{B_2'\cup B_3'}+\vass{R_3}-ht$, moreover, we have $\vass{R_3\cup B_3'\cup B_2'}\geq 9-\vass{B_1'\cup R_1\cup R_2}\geq (\tfrac{7}{3}-10h)t$ which gives us a red TCTF over more than $(3+h)t$ vertices, unless $\vass{R_3}\geq (1-20h)t$.
In particular, we can say that we can find a blue TCTF in $(R_1, R_2, R_{\geq 3})$ of size at least $3(1-20h)t$. Since we have already that $(R_2, B'_2)$ and $(R_2, B_3')$ are $ht$-blue, and since we cannot extend the blue TCTF in $(R_1, R_2, R_{\geq 3})$ at all, this means that $(R_1, B_2')$ and $(R_1, B_3')$ must be $ht$-red, but this is absurd since it would create a red TCTF in $(B_1', B_2', B_3')\cup R_1$ of size at least $\tfrac32\vass{B_3'\cup B_2'}+\vass{R_1}-ht>3(1+h)t$.
\end{proof}
\subsection{The sharp upper bound}
\begin{lemma}\label{lemmalastupperbound}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix a collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$, and define $B_1,B_2,\dots$ as in Setting~\ref{mainsetforG}. We have that $\vass{B_1}, \vass{R_1}\leq (2+h)t$.
\end{lemma}
\begin{proof}
Let us denote with $B_{\geq 3}$ the set $\cup_{i\geq 3}B_i$ and similarly for red. By Lemmas \ref{upperboundsingle} and \ref{upperbounddouble} we can assume $\vass{B_1},\vass{R_1}\leq (\frac{7}{3}+h)t$ and $\vass{B_2},\vass{R_2}\leq (2+h)t$. Let us assume by contradiction that $\vass{B_1}\geq (2+h)t$. We construct greedily a blue TCTF $T_B$ as follows. Select an edge in a blue clique of $B_1$, and extend it (if possible) to a blue triangle using a vertex outside of $B_1$ not used yet in the process. We can repeat greedily until there are no blue edges in cliques of $B_1$ that can be extended outside of $T_B$. Let us denote with $Y_B$ the set of vertices $T_B\setminus B_1$ used to extend the edges in $B_1$, and let us denote with $B_1'$ the set $B_1\setminus T_B$ of remaining vertices.
Because $T_B$ is triangle connected, we have that the size of $T_B$ is smaller than $3(1+\epsilon)t$ and therefore in particular $\vass{B_1'}= \vass{B_1}-\vass{B_1\cap T_B}>\frac{h}{2}t$.
Let $h':=\min\left\lbrace\frac{\vass{B'_1}}{200t}, h\right\rbrace\geq h^{\tfrac32}$.
Because we stopped the greedy construction of $T_B$ only when we could not extend $T_B$ anymore, we have that every vertex in $V\setminus (B_1\cup Y_B)$ has at most as many blue neighbours in $B_1'$ as the number of cliques with at least two vertices that are in $B_1'$. This means that the number of blue edges in $(B_1', V\setminus (B_1\cup Y_B))$ is at most $7t\cdot (\frac{7}{3}\frac{1}{\frac{99}{100}m}+\frac{k}{\vass{\log{\epsilon}}^{\frac{3}{2}}})t\leq (\frac{5}{\sqrt{m}}t)^2$. Therefore we have that the pair $(B_1', V\setminus (B_1\cup Y_B))$ is $\lambda t$-red for $\lambda=\frac{5}{\sqrt{m}}$.
We now separate four cases.
\underline{Case A}: We already have $\vass{B_1}\geq (2+h)t$, and $\vass{B_1},\vass{R_1}\leq (\frac{7}{3}+h)t$, and $\vass{B_2},\vass{R_2}\leq (2+h)t$. Assume now $\vass{R_{\geq 3}}\leq ht$.
It follows that $\vass{\cup_{i} R_i}\leq (\frac{13}{3}+3h)t$, and from this it follows that $\vass{\cup B_i}\geq (\frac{14}{3}-4h)t$ and therefore $\vass{B_2\cup B_{\geq 3}}\geq (\frac{7}{3}-5h)t$. Since $\vass{B_2}\geq\vass{B_{\geq 3}}$ by ~\ref{biguniquelemma}\ref{sublemma7}, we have that $\vass{B_2}>(1+h)t$ and therefore by \ref{tripartitetctf}
we have $\vass{B_{\geq 3}}<(1+h)t$. By what stated above, we also get:
we get the following:
\begin{align*}\vass{B_{\geq 3}}\geq
\begin{cases}
(\frac{8}{3}-5h)t-\vass{R_1}\\
(\frac{7}{3}-5h)t-\vass{R_2}\geq (\frac{1}{3}-6h)t\\
(\frac{14}{3}-4h)t-\vass{B_1}-\vass{R_2}
\end{cases}
\end{align*}
Since $\vass{B_{\geq 3}}<(1+h)t$, we must have $\vass{R_2}>(\frac{4}{3}-6h)t$.
Let us call $C_B$ the red triangle-connected component in $(B_1, B_2, B_{\geq 3})$ that by Corollary \ref{tripartitetctf} contains almost all red edges of $(B_1, B_{\geq 3})$ and $(B_2, B_{\geq 3})$.
\begin{claim}
No red edge in a clique of $R_1$ or $R_2$ is red triangle connected to $C_B$.
\end{claim}
\begin{claimproof}
If $R_1$ were red triangle connected to $C_B$ we could extend a large red TCTF of size $3\vass{B_{\geq 3}}-ht$ (which is given us by Corollary \ref{tripartitetctf}) using vertices of $R_1$ and obtain a TCTF over
\begin{align*}
3\vass{B_{\geq 3}}+\vass{R_1}-2ht &\geq 3((\frac{8}{3}-5h)t-\vass{R_1})+\vass{R_1}-2ht\\
&=(8-17h)t-2\vass{R_1}>3(1+\epsilon)t
\end{align*}
vertices. It is also absurd that $R_2$ is red triangle connected to $C_B$. Indeed we would have:
\underline{\textit{Case 1}}: If $\vass{R_2}\leq \frac{17}{9}t$ then we have a red TCTF over $3\vass{B_{\geq 3}}+\vass{R_2}-ht\geq (7-16h)t-2\vass{R_2}>3(1+\epsilon)t$ vertices.
\underline{\textit{Case 2}}: If $\vass{R_2}\geq \frac{17}{9}t$, we can greedily construct a TCTF $T$ as follows. We select edges in red cliques of $R_2\setminus Y_B$ and we extend them to disjoint triangles using vertices of $B_1'$. Because $(R_2\setminus Y_B,B_1')$ is $\lambda t$-red we have that we can continue this process until we almost finish red edges in red cliques of $R_2\setminus Y_B$ (we can have at most $ht$ vertices remaining in $R_2\setminus Y_B$) or vertices in $B_1'$ with enough red neighbours in $R_2\setminus Y_B$ (if we stopped because of this we have that at most $h't$ vertices in $B_1'$ are not used, because otherwise we would have vertices with high red degree in $R_2\setminus Y_B$ and because $\vass{R_2\setminus Y_B}$ would be at least $h't$ we could find an edge in a clique between two neighbours of the same vertex of $B_1'$). At this point we can extend $T$ with triangles from cliques of $R_2$ and obtain a TCTF over at least $\min\left\lbrace \vass{R_2}+\vass{B_1'}, \frac{3}{2}\vass{R_2\setminus Y_B}+\vass{R_2\cap Y_B}\right\rbrace-3ht$ vertices. $T$ intersects $B_1$ in at most $t$ vertices, therefore we can again extend $T$ using the tripartition $(B_1\setminus T, B_2, B_{\geq 3})$, in this way we are adding at least $3\vass{B_{\geq 3}}-ht$ vertices since $\vass{B_3}\leq (1+h)t$, $\vass{B_2}> \frac{7}{6}$ and $\vass{B_1}\geq (2+h)t$. Therefore we end up with a red TCTF over $\min\left\lbrace \vass{R_2}+\vass{B_1'}, \frac{1}{2}\vass{R_2\setminus Y_B}+\vass{R_2}\right\rbrace+3\vass{B_{\geq 3}}-4ht=3\vass{B_{\geq 3}}+\vass{R_2}+\min\left\lbrace \vass{B_1'}, \frac{1}{2}\vass{R_2\setminus Y_B} \right\rbrace-4ht$ vertices. We can notice at this point that
\begin{align*}
\vass{B_1'}+\vass{B_{\geq 3}}&\geq \vass{B_{1}}+\vass{B_{\geq 3}}-(2+h)t\\
&\geq \frac{14}{3}t-4ht-\vass{B_2}-(2+h)t\\
&\geq (\frac{2}{3}-6h)t\,.
\end{align*}
Since $\vass{B_{\geq 3}}\geq (\frac{1}{3}6h)t$ and $\vass{R_2\setminus Y_B}\geq \frac{5}{6}t$, we are done. Indeed we have $3\vass{B_{\geq 3}}+\vass{R_2}+\min\left\lbrace \vass{B_1'}, \frac{1}{2}\vass{R_2\setminus Y_B} \right\rbrace-4ht\geq 3(\frac{1}{3}6h)t+\frac{17}{9}t+\frac{1}{3}t>3(1+h)t$.
\end{claimproof}
Now we know that neither $R_1$ nor $R_2$ are triangle connected to the large triangle-connected component of the tripartition $(B_1, B_2, B_{\geq 3})$.
In order to use Lemma \ref{lemma17} efficiently, we first need to remember that $B_1', R_1\setminus Y_B$ and $R_2\setminus Y_B$ are all non-trivial and that $(B_1',R_1\setminus Y_B)$ and $(B_1',R_2\setminus Y_B)$ are both $\lambda t$-red. Now we can use Lemma \ref{lemma17} to conclude that at most $2\lambda t$ vertices in $(R_1\cup R_2)\setminus Y_B$ can have more than $2\lambda t$ red neighbours in each of $B_2$ and $B_{\geq 3}$.
But this is absurd because of Lemma \ref{lemma19}.
\underline{Case B}: We already have $\vass{B_1}\geq (2+h)t$, and $\vass{B_1},\vass{R_1}\leq (\frac{7}{3}+h)t$, and $\vass{B_2},\vass{R_2}\leq (2+h)t$. Assume now that $\vass{B_{\geq 3}}\leq ht$.
We can also assume that $\vass{R_1}\leq (2+h)t$ because otherwise we would be in the same situation as case A under switching colours. By Corollary ~\ref{biguniquelemma}\ref{sublemma4} we have $\vass{\cup_i B_i}\geq \frac{43}{12}t$ and this implies $\vass{B_2}\geq \frac{6}{5}t$. We can consider that $\vass{R_2\cup R_{\geq 3}}\geq (9-h)t-\vass{R_1}-\vass{\cup_i B_i}\geq (\frac{8}{3}-5h)t$, which gives us $\vass{R_{\geq 3}}\geq (\frac{2}{3}-6h)t$. By Lemma ~\ref{biguniquelemma}\ref{sublemma7} we have $\vass{R_2}\geq (\frac{4}{3}-3h)t$. By Corollary \ref{corotripartitetctf} this also implies that there is a red TCTF on $(R_1, R_2, R_{\geq 3})$ covering at least $3\vass{R_{\geq 3}}-ht\geq (2-19h)t$ vertices. This gives us the upper bound $\vass{R_{\ge3}}\le \frac{1+h}t$. This also implies that $\vass{R_2}\geq (\tfrac53-6h)t$.
Since both $B_1$ and $B_2$ are bigger than $\frac{8}{7}t$ we have that neither $B_1$ nor $B_2$ can be blue triangle connected to the large TCTF over $(R_1, R_2, R_{\geq 3})$.
By Lemma \ref{lemma17} this means that at most $ht$ vertices from each of $B_1$ and $B_2$ can be blue adjacent to more than $2ht$ vertices in any two of $R_1, R_2$ or $R_{\geq 3}$. But we know also that $B'_1$, $R_1\setminus Y_B$ and $R_2\setminus Y_B$ are non trivial, and therefore $(B_1', R_1\setminus Y_B)$ and $(B_1', R_2\setminus Y_B)$ are $\lambda t$-red. Hence, by Lemma \ref{lemma19} it can not not be the case that there are more than $2ht$ vertices of $B_2$ with more than $2ht$ red neighbours in both $R_1\setminus Y_B$ and $R_2\setminus Y_B$. Therefore by Lemma \ref{lemma17} there are at most $3ht$ vertices in $B_2$ which have more than $2ht$ blue neighbours in $R_{\geq 3}$.
This means that we can find a set $S_1$ of at least $\frac{1}{2}\vass{B_2}-10ht$ vertices in $B_2$ such that every vertex in $S_1$ has at most $2ht$ blue neighbours both in $R_{\geq 3}$
and one of $R_1\setminus Y_B$ or $R_2\setminus Y_B$ (say $R_2$, it is the same if it was $R_1$). Therefore by applying Lemma \ref{lemma19} with $S_1$ and $B_1'$ on one side and $R_2$ and $R_{\geq 3}$ on the other side, we get that there are at most $6h't$ vertices in $B_1'$ which have more than $3h't$ red neighbours in $R_{\geq 3}$, and this means that $(B_1', R_{\geq 3})$ is $6h't$-blue.
By Lemma \ref{lemma17} we know that $(B_1', R_1)$ and $(B_1',R_2)$ are $9h't$-red, and in the same way we know that almost all the vertices of $B_2$ are $2h't$-red to one of $R_1$ or $R_2$. As an example, we are going to assume that we have a subset $S_2$ of $B_2$ of size at least $\frac{\vass{B_2}-20h't}{2}$ such that every vertex in $S_2$ has at most $2h't$ blue neighbours to $R_2$.
Therefore $(S_2, R_{\geq 3})$ and $(S_2, R_2)$ are $2h't$-red. Because $(B_1', R_1)$ and $(B_1', R_2)$ are both $9h't$-red, by lemma \ref{lemma19} we have that $(B_1, R_1)$ is $9h't$-red.
By Lemma \ref{lemma17} as above, at most $6h't$ vertices in $B_1$ can have more than $2h't$ blue neighbours in any two of $R_1$, $R_2$ and $R_{\geq3}$. We can find $S'\subseteq B_1$ of size at least $\frac{\vass{B_1}-20h't}{2}$ that is either $10h't$-red to $R_{\geq 3}$ or to $R_{2}$.
In the first case, we find a large red TCTF using triangles in $(S', S_2, R_{\geq 3})$ and then triangles in $B_1$. In the latter case, we can find a red TCTF on $(S_2, S', R_2)$ over at least $$2\cdot \min\left\lbrace\vass{S_2}, \vass{S'}, \vass{R_2}\right\rbrace+\vass{R_2}-20h't$$
vertices. We claim that this is enough, indeed we have $\vass{R_{\geq 3}}\leq (1+h')t$, and therefore we get the lower bound $(\frac{5}{3}-10h')t$ for $\vass{R_2}$ and $t-10h't$ for $\vass{S'}$.
\underline{Case C}: We already have $\vass{B_1}\geq (2+h)t$, and $\vass{B_1},\vass{R_1}\leq (\frac{7}{3}+h)t$, and $\vass{B_2},\vass{R_2}\leq (2+h)t$. Assume now $\vass{R_1}\leq (2+h)t$ and $\vass{B_{\geq 3}},\vass{R_{\geq 3}}\geq ht$.
We have two cases.
\underline{\textit{Case C.1}}: Let us assume $\vass{R_2}\leq \frac{8}{7}t$.
\begin{claim}
Neither $B_1$ nor $B_2$ is blue connected to the TCTF over $(R_1, R_2, R_{\geq 3})$. Also, $R_1$ is not triangle connected to $(B_1, B_2, B_{\geq 3})$.
\end{claim}
\begin{claimproof}
By Corollary ~\ref{biguniquelemma}\ref{sublemma4} we have that $\vass{R_1\cup R_2\cup R_{\geq 3}}\geq \frac{43}{12}t$ and hence $\vass{R_2\cup R_{\geq 3}}\geq (\frac{19}{12}-h)t$.
By Lemma~\ref{biguniquelemma}\ref{sublemma7} we have $\vass{R_2}\geq\vass{R_{\geq 3}}$ and by Lemma~\ref{biguniquelemma}\ref{sublemma6} we have $\vass{B_2}\geq \frac{8}{7}t$ and since $R_1,\dots,B_{\ge3}$ form a partition of $G$, we have $\vass{R_{\geq 3}}\geq (9-\frac{7}{3}-1-2-\frac{8}{7}-3h)t-\vass{B_2}>(\frac{5}{2}+h)t-\vass{B_2}$. By Corollary \ref{tripartitetctf} we can find a blue TCTF over $(R_1, R_2, R_{\geq 3})$ of size at least $3\vass{R_{\geq 3}}\geq\frac{15}{2}t-3\vass{B_2}$. In particular this implies that both $B_1$ and $B_2$ are not triangle connected to the blue TCTF over $(R_1, R_2, R_{\geq 3})$.
We now prove that $R_1$ is not triangle connected to $(B_1, B_2, B_{\geq 3})$.
If $\vass{R_2\cup R_{\geq 3}}> (\frac{8}{7}+1+h)t$,
then by Lemma~\ref{biguniquelemma}\ref{sublemma7} we have $\vass{R_2}>\vass{R_{\ge3}}$, since $\vass{R_2}\le\tfrac87t$ we have $\vass{R_{\ge3}}\ge(1+h)t$ and by Corollary~\ref{corotripartitetctf} we again obtain a blue TCTF of size $(3+h)t$.
If on the other hand we have $\vass{R_2\cup R_{\geq 3}}\leq (\frac{8}{7}+1+h)t$, it follows that $\vass{B_{\geq 3}}\geq (9-\frac{7}{3}-2-\frac{8}{7}-1-4h)t-\vass{R_1}$ which means $3\vass{B_{\geq 3}}+\vass{R_1}\geq \frac{24}{7}t$. Therefore it cannot be that $R_1$ is red triangle connected to the large TCTF over $(B_1, B_2, B_{\geq 3})$.
\end{claimproof}
Since $R_1$ is not connected to $(B_1, B_2, B_{\geq 3})$ we have by Lemma \ref{lemma17} that at most $h^5t$ vertices in $R_1$ have more than $2h^5t$ red neighbours in two of the $B_i$. Since $R_1\setminus Y_B$ is nontrivial we have that $(B_1', R_1\setminus Y_B)$ is $\lambda t$-red. Therefore we must have that $(R_1\setminus Y_B, B_2), (R_1\setminus Y_B,B_{\geq 3})$ are $h^2t$-blue. We can now apply Lemma \ref{lemma17} again knowing that $B_2$ is not blue triangle connected to the blue triangle component over $(R_1, R_2, R_{\geq 3})$ and therefore at most $h^5t$ vertices of $B_2$ have more than $2h^5t$ blue neighbours in two of the $R_i$. Hence, $(B_2, R_2)$ and $(B_2,R_{\geq 3})$ are $h^2t$-red.
Since they are not red triangle connected among themselves, we have that either $R_2$ or $R_{\geq 3}$ is not red triangle connected to the red triangle component over $(B_1, B_2, B_{\geq 3})$. Let $R_{2}$ the one not red triangle connected, and $R_{\geq 3}$ the other one (if the situation is reversed we get better bounds). Then by Lemma \ref{lemma17} we have that $R_2$ is $h^2t$-blue to $B_1$ and $B_{\geq 3}$, and therefore by the same Lemma we have that $(B_1, R_{\geq 3})$ is $h^2t$-red. Then $(B_1, B_2, B_{\geq 3}\cup R_{\geq 3})$ is a dense red tripartition with $\vass{B_1}, \vass{B_2}\geq\frac{8}{7}t$. We have $\vass{B_{\geq 3}\cup R_{\geq 3}}\geq (9-\frac{7}{3}-2-2-\frac{8}{7}-3h)t\geq \frac{3}{2}t$ which is enough to conclude by Corollary \ref{tripartitetctf}.
\underline{\textit{Case C.2}}: Let us now assume $\vass{R_2}\geq \frac{8}{7}t$.
Then both $R_1\setminus Y_B$ and $R_2\setminus Y_B$ are non trivial and $\lambda t$-red to $B_1'$. We cannot have that both $R_1$ and $R_2$ are red triangle connected to $(B_1, B_2, B_{\geq 3})$ (because otherwise they would be red triangle connected among themselves). By Lemma \ref{lemma17} this means that one between $R_1\setminus Y_B$ and $R_2\setminus Y_B$ must be $h^2t$-blue to both $B_2$ and $B_{\geq 3}$, we are going to work with the example in which $R_1\setminus Y_B$ is $h^2t$-blue to both $B_2$ and $B_{\geq 3}$ (it would be the same if we had $R_2$).
We cannot have both $B_2$ and $B_{\geq 3}$ to be blue triangle connected to $(R_1, R_2, R_{\geq 3})$ (otherwise they would be in the same connected component) and therefore we split our case depending on whether or not $B_2$ is blue triangle connected to $(R_1, R_2, R_{\geq 3})$.
Let us assume that it is so. Then $B_{\geq 3}$ is not blue triangle connected to $(R_1, R_2, R_{\geq 3})$ and so $(B_{\geq 3},R_2)$ and $(B_{\geq 3}, R_{\geq 3})$ are $h^2t$-red. By Lemma \ref{lemma17} this implies that $R_2$ is red triangle connected to $(B_1, B_2, B_{\geq 3})$ and therefore $R_{\geq 3}$ is not. Therefore $(R_{\geq 3}, B_1)$ and $(R_{\geq 3}, B_2)$ are $h^2t$-blue. Therefore $(B_1, R_1)$ and $(B_1,R_2)$ must be $h^2t$-red. Therefore we can find a blue TCTF over $3\vass{R_{\geq 3}}+\vass{B_2}$ vertices by taking triangles from $(R_1, R_2, R_{\geq 3})$ and $B_2$. We can also find a red TCTF over $3\vass{B_{\geq 3}}+\frac{3}{2}\vass{R_2}$ vertices by taking triangles from $(B_1, B_2, B_{\geq 3})$ and by taking edges from $R_2$ and extending them with vertices from $B_1$. We conclude by taking the average of the size of these two TCTFs.
Let us now assume that $B_2$ is not blue triangle connected to $(R_1, R_2, R_{\geq 3})$. Then $(B_2, R_2)$ and $(B_2, R_{\geq 3})$ are $h^2t$-red, since $(R_1\setminus Y_B, B_2)$ is $ht$ -blue, and this implies that $R_2$ is red triangle connected to $(B_1, B_2, B_{\geq 3})$. Therefore $R_{\geq 3}$ is $h^2t$-blue to $B_1$ and $B_{\geq 3}$ and so $B_{\geq 3}$ is blue triangle connected to $(R_1, R_2, R_{\geq 3})$. This also means that $B_1$ must be $h^2t$-red to both $R_1$ and $R_2$ in order not to be blue triangle connected to $(R_1, R_2, R_{\geq 3})$ but this leaves us with a dense red $(B_1, B_2, B_{\geq 3}\cup R_2)$.
\underline{Case D}: Let us assume that $\vass{R_1}\geq (2+h)t$ and both $B_{\geq 3}$ and $R_{\geq 3}$ contain more than $ht$ vertices (otherwise we are in the situation of case B). We can also assume without loss of generality that $\vass{B_{2}}\geq \vass{R_2}$ and therefore by Lemma \ref{biguniquelemma}\ref{sublemma6} we also have $\vass{B_2}\geq \tfrac{8}{7}t$.
We can greedily extend blue edges in cliques of $B_1$ to a blue TCTF $T_B$ by using vertices outside of $B_1$. Since $\vass{B_1}\geq (2+h)t$ we can either create a TCTF over more than $3(1+\epsilon)t$ vertices or we have to stop at some point. Since $\vass{B_1}\geq (2+h)t$, this means that $B_1\setminus T_B$ is non trivial. We can do the same with a red TCTF $T_R$ extending red edges in $R_1$ (since we are assuming $\vass{R_1}\geq (2+h)t$). Let us call $B_1':=B_1\setminus T_B$ and $R_1':=R_1\setminus T_R$. Since the TCTFs $T_R$ and $T_B$ are maximal, we have that $(B_1', V(G)\setminus T_B)$ is $ht$-red, while $(R_1, V(G)\setminus T_R)$ has to be $ht$-blue. In particular, there are non-trivial subsets $S_{B_1}\subseteq B_1$ of size at least $(1+\tfrac{h}{2})t$, $S_{B_2}\subseteq B_2$ of size at least $(\tfrac{1}{7}+\tfrac{h}{2})t$ and $S_{R_1}\subseteq R_1$ of size at least $(1+\tfrac{h}{2})t$ such that $(S_{B_1}, R_1')$ and $(S_{B_2}, R_1')$ are $ht$-blue and $(S_{R_1}, B_1')$ is $ht$-red.
There are two cases:
\underline{Case D.1}: $B_1$ is blue triangle connected to the large TCTF in $(R_1, R_2, R_{\geq 3})$. Then we know that $B_2$ and $B_{\geq 3}$ are not triangle connected to the same TCTF. In particular, since $(S_{B_2}, R_1')$ is $ht$-blue, we must have that both $(S_{B_2}, R_2)$ and $(S_{B_2}, R_{\geq 3})$ are $ht$-red. Now, either $R_1$ is red triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$ or not.
In the first case, we have that both $R_2$ and $R_{\geq 3}$ are not triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$. Because $(S_{B_2}, R_2)$ and $(S_{B_2}, R_{\geq 3})$ are $ht$-red, this means that $(R_{\geq 3}, B_{\geq 3}), (R_{\geq 3}, B_1)$ and $(R_{2}, B_{\geq 3}), (R_{2}, B_{1})$ are $ht$-blue, which is absurd because it would mean that $B_1$ and $B_{\geq 3}$ are in the same blue-connected component.
If $R_1$ is not red triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$, then $(S_{R_1}, B_{\geq 3})$ and $(S_{R_1}, B_2)$ have to be $ht$-blue. But now we get a contradiction since $(B_{\geq 3}, R_{\geq 3})$ and $(B_{\geq 3}, R_{2})$ need to be $ht$-red
or otherwise $B_{\geq 3}$ is going to be triangle connected to the blue TCTF in $(R_1, R_2, R_{\geq 3})$, and also $(B_{2}, R_{\geq 3})$ and $(B_{2}, R_{2})$ need to be $ht$-red
or otherwise $B_{2}$ is going to be triangle connected to the blue TCTF in $(R_1, R_2, R_{\geq 3})$. This is enough to say that $R_{2}$ and $R_{\geq 3}$ are in the same red-connected component.
\underline{Case D.2}: $B_1$ is not blue triangle connected to the large TCTF in $(R_1, R_2, R_{\geq 3})$ but $R_1$ is red triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$. Since $B_1$ is not blue triangle connected to the large TCTF in $(R_1, R_2, R_{\geq 3})$ and because $(S_{B_1}, R_1')$ is $ht$-blue, we have that $(S_{B_1}, R_2)$ and $(S_{B_1}, R_{\geq 3})$ are $ht$-red. Now, since $R_1$ is red triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$ we have that $R_2$ and $R_{\geq 3}$ are not, because $(S_{B_1}, R_2)$ and $(S_{B_1}, R_{\geq 3})$ are $ht$-red this implies that $(B_2, R_2), (B_{\geq 3}, R_2)$ and $(B_2, R_{\geq 3}), (B_{\geq 3}, R_{\geq 3})$ are $ht$-blue, which is absurd because it implies that both $B_2$ and $B_{\geq 3}$ are connected to the large TCTF in $(R_1, R_2, R_{\geq 3})$.
\underline{Case D.3}: $B_1$ is not blue triangle connected to the large TCTF in $(R_1, R_2, R_{\geq 3})$ and $R_1$ is not blue triangle connected to the large TCTF in $(B_1, B_2, B_{\geq 3})$. In which case we notice that the blue cliques in $B_1$ are not triangle connected to the large blue TCTF in $(R_1, R_2, R_{\geq 3})$ and similarly the red cliques in $R_1$ are not triangle connected to the large red TCTF in $(B_1, B_2, B_{\geq 3})$. In particular, this implies that $(S_{B_1}, R_{\geq 3})$ and $(S_{B_1}, R_2)$ are $ht$-red, because we have that $(S_{B_1}, R_1')$ is $ht$-blue and $(R_1', \cup_{i\geq 2}R_i)$ is $ht$-blue. Likewise, we have that $(S_{R_1}, B_{\geq 3})$ and $(S_{R_1}, B_2)$ are $ht$-blue. But this leaves us in a contradiction, indeed, neither $B_2$ nor $B_{\geq 3}$ can be triangle connected to $(R_1, R_2, R_3)$. Since $(S_{R_1}, B_{\geq 3})$ is $ht$-blue this means that $(R_2, B_{\geq 3})$ and $(R_{\geq 3}, B_{\geq 3})$ are $ht$-red. This is enough to get a contradiction, since we have $(R_2, B_{\geq 3})$ and $(R_{\geq 3}, B_{\geq 3})$ are $ht$-red but also $(S_{B_1}, R_{\geq 3})$ and $(S_{B_1}, R_2)$ are $ht$-red.
\end{proof}
\section{The colours of edges}\label{sec:finproof}
In this section we complete the proof of Lemma~\ref{MainLemma}. We first deduce an approximate version, proving that $B_{\ge3}\cup R_{\ge3}$ cannot have much more than $t$ vertices (which implies all components have roughly the correct size) and that most edges in various pairs have the `correct' colour. We then prove Lemma~\ref{MainLemma} by arguing that any edges with the `wrong' colour lead to triangle components which are much larger than they should be. The following is our approximate version.
\begin{lemma}\label{approxmainlemma}
There exists $h_0>0$ such that for every $0<h<h_0$ there exists $\epsilon_0>0$ such that for all $0<\epsilon<\epsilon_0$ there exists $t_0$ such that for every $t>t_0$ we have the following. Let $G$ be a 2-edge coloured graph with $(9-\epsilon)t$ vertices and minimum degree at least $(9-2\epsilon)t$. Fix a collection of red and blue cliques as in Setting \ref{mainsetforG} with parameters $\epsilon$ and $t$, and define $B_1,B_2,\dots$ as in Setting~\ref{mainsetforG}. Then it holds:
\begin{itemize}
\item $(2-h)t\leq \vass{B_1}, \vass{B_2}, \vass{R_1}, \vass{R_2}\leq (2+h)t$,
\item $(1-h)t\leq \vass{B_{\geq 3}\cup R_{\geq 3}}\leq (1+h)t$,
\item $\vass{G\setminus \cup_i (B_i\cup R_i)}\leq ht$,
\item The following pairs are $h^2t$-blue: $(B_1,R_1)$, $(B_2,R_2)$, $(R_1,R_2)$, $(R_1, B_{\geq 3}\cup R_{\geq 3})$ and $(R_2, B_{\geq 3} \cup R_{\geq 3})$,
\item The following pairs are $h^2t$-red: $(B_1,B_2)$, $(B_1, B_{\geq 3} \cup R_{\geq 3})$, $(B_2, B_{\geq 3} \cup R_{\geq 3})$, $(B_1,R_2)$ and $(B_2,R_1)$.
\end{itemize}
\end{lemma}
\begin{proof}
By Lemma \ref{lemmalastupperbound} we know that for $\epsilon>0$ small enough we have $\vass{B_1}, \vass{B_2},\vass{R_1},\vass{R_2}\leq (2+h^{\frac{3}{2}})t$ and therefore we have $\vass{B_{\geq 3} \cup R_{\geq 3}}\geq (1-5h^{\frac{3}{2}})t$. Without loss of generality, let us assume $\vass{B_{\geq 3}}\geq \vass{R_{\geq 3}}$.
\begin{claim}
We have that $R_1$ and $R_2$ are not red triangle connected to $(B_1, B_2, B_{\geq 3})$. Moreover, without loss of generality, we have $(R_1, B_1), (R_1, B_{\geq 3})$ and $(R_2, B_2), (R_2, B_{\geq 3})$ are $h^2t$-blue.
\end{claim}
\begin{claimproof}
Notice that $\vass{B_{1}}\geq \frac{(1-5h^{\frac{3}{2}})t}{2}$. Let us consider first that $R_1$ and $R_2$ are not red triangle connected to $(B_1, B_2, B_{\geq 3})$. Indeed, assume this is not the case and we have $3\vass{B_{\geq 3}}+\vass{R_i}<(3+h)t$ for some $i\in \left\lbrace 1,2\right\rbrace$. Then we have $\vass{R_1}+\vass{R_2}+\vass{B_{\geq 3}}<(3+h+2+h^{\frac{3}{2}})t-2\vass{B_{\geq 3}}<(4-3h)t$ which is clearly absurd because it implies $\vass{B_1}+\vass{B_2}+\vass{R_{\geq 3}}\geq (5+2h)t$.
We now claim that there is an ordering $i,j,k$ of $\left\lbrace 1, 2, \geq 3\right\rbrace$ such that $(R_1, B_i), (R_1, B_j)$ and $(R_2, B_k), (R_2, B_j)$ are $h^2t$-blue.
Indeed, by Lemma \ref{lemma17} we know that up to removing at most $h^5t$ vertices from each of $R_1$ and $R_2$, every vertex in $R_1\cup R_2$ has many blue edges in at least two among $\left\lbrace B_1, B_2, B_{\geq 3}\right\rbrace$. This means that we can partition (not in a unique way) almost all the vertices of $R_1$ among the sets $S^{R_1}_{B_h}$, where the vertices in $S^{R_1}_{B_h}$ have their red neighbour in $\cup_\ell B_\ell$ contained in $B_h$. We define similarly $S^{R_2}_{B_h}$. We claim that just one of the $S^{R_1}_{B_h}$ is not trivial.
Assume by contradiction that $S^{R_1}_{B_i}$ and $S^{R_1}_{B_j}$ have size at least $ht$. We cannot have that $S^{R_2}_{B_i}$ or $S^{R_2}_{B_j}$ have size at least $ht$, because otherwise we would have that $B_j$ and $B_k$ or $B_i$ and $B_k$ are connected respectively. Therefore we must have that $S^{R_2}_{B_k}$ contains almost all the vertices of $R_2$ and in particular is not trivial.
Therefore we have that $S^{R_1}_{B_i}$, $S^{R_1}_{B_j}$ and $S^{R_2}_{B_k}$ are not trivial, which gives us that both $B_i$ and $B_j$ are in the same triangle-connected component. This implies that just one of the $S_{B_h}^{R_1}$ is nontrivial, and by symmetry the same is true for $R_2$. Moreover, we have that $S_{B_{\geq 3}}^{R_i}$ is trivial, because otherwise we would find a large blue TCTF in $(B_1, B_2, R_i\cup B_{\geq3})$.
Finally, since by Lemma \ref{lemma19} we cannot have that $R_1$ and $R_2$ are $h^2t$-blue to the same pair, we know that each of $R_1$ and $R_2$ is $h^2t$-blue to $B_{\geq 3}$ and one between $B_1$ and $B_2$. We are going to assume without loss of generality that $(R_1, B_1)$ and $(R_1, B_{\geq 3})$ are $h^2t$-blue, and that $(R_2, B_2)$ and $(R_2, B_{\geq 3})$ are $h^2t$-blue, as we wanted.
\end{claimproof}
By the claim, we have that $(R_1, B_1)$, $(R_1, B_{\geq 3})$ and $(R_2, B_2)$, $(R_2, B_{\geq 3})$ are $h^2t$-blue. In particular, this means that we can find a blue TCTF in $(R_1, R_2, B_{\geq 3}\cup R_{\geq 3})$. This gives us immediately that $\vass{B_{\geq 3}\cup R_{\geq 3}}\leq (1+h^{\frac{3}{2}})t$ and in particular $\vass{B_1},\vass{B_2},\vass{R_1},\vass{R_2}\geq (2-h)t$.
Also, we get that $(B_1, R_2)$ and $(B_2, R_1)$ are $h^2t$-red. This holds because otherwise we would have both $B_{\geq 3}$ and $B_{2}$ are in the same connected component, indeed, $(B_{\geq 3}, R_1)$, $(B_{\geq 3}, R_2)$ and $(B_2, R_2)$ are $h^2t$ blue.
Assume now that $\vass{R_{\geq 3}}\geq h^{\tfrac{3}{2}t}$. We have that $(R_1, B_1), (R_1, B_{\geq 3})$ and $(R_2, B_2), (R_2, B_{\geq 3})$ are $h^2t$-blue, this gives us that $B_{\geq 3}$ is blue triangle connected to $(R_1, R_2, R_{\geq 3})$ (which is a non-trivial TCTF) which in turn gives us that $B_1$ and $B_2$ are not. From this last fact we can conclude that $(B_1, R_{\geq 3})$ and $(B_2, R_{\geq 3})$ are all $h^2t$-red.
So we have the construction that we wanted up to change the indices between $B_1$, $B_2$ and $R_1$, $R_2$ respectively.
\end{proof}
Let us now prove Lemma \ref{MainLemma} that we restate for convenience.
\MainLem*
\begin{proof}[Proof of Lemma~\ref{MainLemma}]
We are going to refine Lemma \ref{approxmainlemma} to obtain an exact result.
By Lemma \ref{approxmainlemma} we have that there exists $\delta_0>0$ such that for $\delta_0>h,\lambda>0$ there exists $\epsilon_0>0$ and $t_0\in\mathbb{N}$ such that for any $t>t_0$ and $\epsilon_0>\epsilon>0$ if $G$ is a 2-edge-coloured graph over $(9-\epsilon)t$ vertices with minimum degree at least $(9-2\epsilon)$ and without a monochromatic TCTF on at least $3(1+\epsilon)t$ vertices, then we can partition $V(G)$ in the sets $B_1, B_2, R_1, R_2, Z, T$ (where the $B_i$ and $R_i$ are as in Setting \ref{mainsetforG} and where where $Z=B_{\geq 3}\cup R_{\geq 3}$ and $T$ is the set of vertices which are not already counted) such that the following holds:
\begin{itemize}
\item $(2-h)t\leq \vass{B_1},\vass{B_2},\vass{R_1},\vass{R_2}\leq (2+h)t$,
\item $(1-h)t\leq \vass{Z}\leq (1+h)t$,
\item $\vass{T}\leq ht$,
\item The following pairs are $\lambda t$-blue: $(B_1,R_1)$, $(B_2,R_2)$, $(R_1,R_2)$, $(R_1, Z)$ and $(R_2, Z)$,
\item The following pairs are $\lambda t$-red: $(B_1,B_2)$, $(B_1, Z)$, $(B_2, Z)$, $(B_1,R_2)$ and $(B_2,R_1)$.
\end{itemize}
We first need to slightly prune our sets. We start by removing from $B_1$ (and putting in $T$) the vertices with more than $\frac{1}{8}\lambda$ red neighbours to $R_1$ and the vertices with more than $\lambda$ blue neighbours to either $B_2, R_2$ or $Z$. We do the same to $B_2$, $R_1$ and $R_2$ accordingly to the colour of the pairs we are considering.
Up to reducing $\epsilon_0$, we are still respecting all the bounds on the sizes that we need for Lemma \ref{MainLemma}, but we have a slightly better result on the state of the ``problematic'' edges. Indeed, we know that there are no vertices outside $T$ that witness more than $\lambda$ ``problematic'' edges.
We now just need to prove that $G[B_1]$, $G[B_2]$, $G[R_1]$, $G[R_2]$ and $(B_1,R_1)$, $(B_2,R_2)$, $(R_1, Z)$, $(R_2, Z)$, $(B_1, Z)$, $(B_2, Z)$, $(B_1,R_2)$, $(B_2,R_1)$ are entirely monochromatic.
The proof to show that $G[B_1]$, $G[B_2]$, $G[R_1]$, $G[R_2]$ are monochromatic have the same structure. Therefore without loss of generality we show that $G[B_1]$ is entirely blue. Assume by contradiction that we can find $u,v$ in $B_1$ such that $uv$ is red. By our earlier pruning we know that both $u$ and $v$ have at most $\frac{1}{8}\lambda$ blue neighbours in $R_2$. Therefore $uv$ is triangle connected to one of the red cliques of $R_2$ (and therefore to all red cliques of $R_2$). Let us now prove that $uv$ is also triangle connected to the large red TCTF in $(B_1, B_2, Z)$ (which is enough to conclude since we would then be able to find a large triangle-connected triangle component). Almost all the red edges in $(\left\lbrace u\right\rbrace, B_2)$ are triangle connected to $uv$, indeed, all but at most $\frac{1}{8}\lambda$ of them are in a red triangle with $uv$, the same holds for the red edges in $(\left\lbrace u\right\rbrace, Z)$. This means that there are at most $\lambda$ vertices in either $B_2$ or $Z$ that witness a red edge in $(B_2, Z)$ which is not triangle connected to $uv$. But this is absurd, as we mentioned before, since it implies that a large red TCTF in $(B_1, B_2, Z)$ is triangle connected to $uv$.
Let us now prove that $(B_1,R_1)$, $(B_2,R_2)$, $(R_1, Z)$, $(R_2, Z)$, $(B_1, Z)$, $(B_2, Z)$, $(B_1,R_2)$, $(B_2,R_1)$ are entirely monochromatic. The structure of these proofs is always the same, so without loss of generality we prove that $(B_1, R_1)$ is monochromatic. Assume it is not, and let $uv$ be a red edge between $B_1$ and $R_1$ (with $u\in B_1$). We prove that $uv$ is triangle connected both to one clique of $R_1$ (and therefore all cliques of $R_1$) and to the large red triangle-connected component in $(B_1, B_2, Z)$, which is absurd since this would give a large red TCTF.
We first show that $uv$ is triangle connected to $R_2$, let $w_1\in R_2$ such that $vw_1$ is an edge (which has to be red by our previous proof that $G[R_1]$ is entirely red). Then by our pruning we know that $u, v$ and $w_1$ share a red neighbour in $B_2$.
We now observe that if $w_2w_3$ is a red edge between $B_2$ and $Z$ (with $w_2\in B_2$) such that $w_2$ is a red neighbour of both $u$ and $v$ and $w_3$ is a red neighbour of $w_2$ and $u$, then $w_2w_3$ is triangle connected to $uv$. By the pruning we did earlier, we can say that most of the red edges between $B_2$ and $Z$ are triangle connected to $uv$, which is what we wanted.
Up to changing the roles of the clusters, the other proofs have the same structure.
\end{proof}
\section{Regularity Method: proofs of Lemma~\ref{regularityresult} and Theorem~\ref{thm:boundBW}}\label{sec:reg}
In this section we state the Regularity Lemma and Blow-up Lemma, and use them to deduce Lemma~\ref{regularityresult} and Theorem~\ref{thm:boundBW} from Lemma~\ref{MainLemma}.
\begin{definition}[density, $\varepsilon$-regular]
Let $G$ be a graph and let $X, Y$ be disjoint subsets in $V(G)$. We define the \emph{density} $d(X,Y)$ between $X$ and $Y$ to be:
$$d(X,Y):=\frac{e(X,Y)}{\vass{X}\vass{Y}}\,.$$
Given $\epsilon>0$, we say that $(X,Y)$ is $\epsilon$-regular if for every $X'\subseteq X,\ Y'\subseteq Y$ such that $\vass{X'}>\epsilon \vass{X}$ and $\vass{Y'}>\epsilon\vass{Y}$ we have
$\vass{d(X',Y')-d(X,Y)}<\epsilon$.
\end{definition}
We use the following version of the Regularity Lemma. We will apply this to the graph of red edges within $K_n$, and observe that if $(X,Y)$ is $\varepsilon$-regular in red edges then, since the blue edges are the complement of the red edges, it is also $\varepsilon$-regular in blue.
\begin{lemma}[Regularity Lemma]\label{RegularityLemma}
For every $\epsilon \in (0,1)$ there are $M, N_0\in\mathbb{N}$ such that the following holds. Let $G$ be a graph on $n \geq N_0$ vertices, then there is a partition $\left\lbrace V_0,\dots{}, V_m\right\rbrace$ of $V(G)$ such that the following conditions hold. We have $|V_0|\le\varepsilon^{-1}$ and $\varepsilon^{-1}\le m\le M$. We have $|V_1|=\dots=|V_m|$. Finally, for any given $i\in[m]$, for all but at most $\varepsilon m$ choices of $j\in[m]$ the pair $(V_i,V_j)$ is $\varepsilon$-regular in $G$.
\end{lemma}
This version follows from the original version of Szemer\'edi~\cite{SzeReg} (which is similar but bounds the total number of irregular pairs by $\varepsilon m^2$ rather than the number of irregular pairs meeting a part) applied with parameter $\tfrac18\varepsilon^2$, followed by removing parts incident to more than $\tfrac12\varepsilon m$ irregular pairs (of which there are at most $\tfrac12\varepsilon m$) to $V_0$; we leave the details to the reader.
Given $\varepsilon,d>0$, a $2$-edge-coloured complete graph $G$ and a partition obtained by applying Lemma~\ref{RegularityLemma} with parameter $\varepsilon$ to the subgraph of red edges, we define the \emph{$(\varepsilon,d)$-reduced graph} of $G$ (with respect to the partition) to be the graph $H$ on vertex set $[m]$, in which an edge $ij$ is present if it is $\epsilon$-regular, and assigned the colour red if its density in red is at least $1-d$, blue if its density in blue is at least $1-d$, and otherwise purple.
We will see that for the purposes of embedding a graph into $G$, we can treat purple edges as being either red or blue as we desire, so that a large TCTF in (red $\cup$ purple) edges, or in (blue $\cup$ purple) edges in the reduced graph implies the existence of the square paths and cycles in $G$ we need. In order to apply Lemma~\ref{MainLemma} in this setting, we deduce the following consequence, which roughly says that either we are done or we get essentially the same partition as in Lemma~\ref{MainLemma}, in particular there are very few purple edges.
\begin{lemma}\label{MainLemmaWithPurple}
For any $\delta>0$ there exists $\varepsilon>0$ such that for any $t\geq \tfrac{1}{\varepsilon}$, if $G$ is a $\left\lbrace\text{red, blue, purple}\right\rbrace$-edge-coloured graph on $(9-\varepsilon)t$ vertices with minimum degree at least $(9-2\varepsilon)t$, then either there is a choice of a colour between blue and red such that if we colour all the purple edges of that colour we can find a monochromatic TCTF on at least $3(1+\varepsilon)t$ vertices in $G$ or $V(G)$ can be partitioned in sets $\left\lbrace B_1, B_2, R_1, R_2, Z, T\right\rbrace$ such that the following hold.
\begin{enumerate}[label=(\alph*)]
\item\label{MLP:i} $(2-\delta)t\leq \vass{B_1}, \vass{B_2}, \vass{R_1}, \vass{R_2}\leq (2+\delta)t$,
\item\label{MLP:ii} $(1-\delta)t\leq \vass{Z}\leq (1+\delta)t$,
\item\label{MLP:iii} all the edges in $G[B_1]$ and $G[B_2]$ are blue, and all the edges in $G[R_1]$ and $G[R_2]$ are red,
\item\label{MLP:iv} the pairs $(B_1,R_1)$, $(B_2, R_2)$, $(R_1, Z)$ and $(R_2,Z)$ are entirely blue. Moreover, the pairs $(B_1,R_2)$, $(B_2, R_1)$, $(B_1, Z)$ and $(B_2,Z)$ are entirely red,
\item\label{MLP:v} the pair $(B_1, B_2)$ is $\delta t$-red, while the pair $(R_1, R_2)$ is $\delta t$-blue, and
\item\label{MLP:vi} $\vass{T}\leq \delta t$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\epsilon$ be given by Lemma~\ref{MainLemma} for input $h=\lambda=\tfrac1{1000}\delta$; without loss of generality we may suppose $\delta$ is sufficiently small for this application.
Let $G$ be a coloured graph satisfying the conditions of the lemma, and suppose there is neither a red-purple TCTF over $3(1+\varepsilon)t$ vertices nor a blue-purple TCTF over $3(1+\varepsilon)t$ vertices.
Let $G^r$ be the graph obtained from $G$ by recolouring the purple edges red, and similarly $G^b$ by recolouring them blue. Let $R_1^r,R_2^r,B_1^r,B_2^r,X^r,T^r$ be the partition obtained by applying Lemma~\ref{MainLemma} to $G^r$, and define similarly the partition for $G^b$ replacing $r$ with $b$. Observe that if we swap $R^1_r$ and $R^2_r$, and also $B^1_r$ and $B_2^r$, we still have a partition satisfying the conclusion of Lemma~\ref{MainLemma}. If $\big|R_2^r\cap R_1^b\big|>\big|R_1^r\cap R^1_b\big|$, we perform this swap (and in an abuse of notation continue to use the same letters for the swapped classes).
We define $R_i:=R_i^r\cap R_i^b$ and $B_i:=B_i^r\cap B_i^b$ for each $i=1,2$, and $Z:=Z_r\cap Z_b$ and finally $T:=V(G)\setminus(B_1\cup B_2\cup R_1\cup R_2\cup Z)$. We will now prove this partition satisfies the conclusions of the lemma. Observe that the statements in~\ref{MLP:iii},~\ref{MLP:iv} and~\ref{MLP:v} about sets or pairs being entirely red, or $\delta t$-red, follow directly from the same statements for the partition of $G^b$, and the corresponding ones about being blue from the partition of $G^r$; what remains is to prove these sets have the correct sizes.
To begin with, observe that all edges in $R_1^b$ are red in $G^b$ and so also in $G$. It follows that $R_1^b$ intersects $B_i^r$ in at most one vertex for each $i=1,2$, since otherwise $B_i^r$ would contain a red edge. Thus $R_1^b$ has at least $(2-\frac{1}{100}\delta)t-2$ vertices which are not in $B_1^r\cap B_2^r$. These vertices cannot all be in $T^r\cup Z^r$, which is too small, so $R_1^b$ has a vertex in at least one of $R_1^r$ and $R_2^r$. Now $R^1_b$ cannot have vertices in $Z^r$, since all edges from $Z^r$ to $R_1^r\cup R_2^r$ are not red. It follows that all but at most $\tfrac1{1000}\delta t+2$ vertices of $R^1_b$ are in $R_1^r\cup R_2^r$, and by the observation above there are at least as many vertices in $R_1^r$ as $R_2^r$. Since $(R_1^r,R_2^r)$ is $\tfrac1{1000}\delta t$-blue, and all edges in $R^1_b$ are red, we see $R^1_b$ has at most $\tfrac1{1000}\delta t$ vertices in $R_2^r$. Finally, we conclude $|R_1|\ge(2-\tfrac1{100}\delta)t$. We also have $|R_1|\le |R_1^r|\le(2+\tfrac1{1000}\delta)t$. By a similar argument (noting that $R_1^b$ and $R_2^b$ are disjoint) we obtain
\[(2-\tfrac{1}{100}\delta)t\le |R_i|\le (2+\tfrac1{1000}\delta)t\]
for each $i=1,2$.
We make a similar argument for $B_1^r$. As above, we can conclude that all but at most $\tfrac1{1000}\delta)t+2$ vertices of $B_1^r$ are in $B_1^b\cup B_2^b$. However we can now observe that all edges from $B_1^r$ to $R_1\subset R_1^r$ are blue, while the edges from $B_2^b$ to $R_1\subset R_1^b$ are red. It follows that $B_1^r$ is disjoint from $B_2^b$, and we obtain
\[(2-\tfrac{1}{100}\delta)t\le |B_i|\le (2+\tfrac1{1000}\delta)t\]
for $i=1$, and, by the similar argument, for $i=2$.
Now $Z^r$ and $Z^b$ are two sets of size at least $(1-\tfrac1{1000}\delta)t$ in $V(G)\setminus(R_1\cup R_2\cup B_1\cup B_2)$, which has size at most $(9-\varepsilon)t-4(2-\tfrac1{100}\delta)t\le t+\tfrac2{50}\delta t$. It follows their intersection $Z$ has size at least $(1-\tfrac{1}{10}\delta)t$, and at most $|Z^r|\le(1+\tfrac1{1000}\delta)t$. Finally, putting these size bounds together we have~\ref{MLP:i},~\ref{MLP:ii} and an upper bound on $|T|$ giving~\ref{MLP:vi}.
\end{proof}
To go with the above lemma, we state the following two embedding lemmas. The first is a corollary of~\cite[Lemma~7.1]{ABHKP}, though one could use the original Blow-up Lemma of Koml\'os, S\'ark\"ozy and Szemer\'edi~\cite{KSS} with some extra technical work in the proof of Theorem~\ref{thm:boundBW}. To deduce the following statement from~\cite[Lemma~7.1]{ABHKP}, we take $R'$ to be the graph with zero edges and $\Delta_{R'}=1$, we take $\kappa=2$, and we add to $H$ for each $i\in R$ a set of $|V_i|-|\phi^{-1}(i)|$ isolated vertices which (extending $\phi$) we map to $i$ and let be the buffer vertices $\tilde{X}_i$.
\begin{theorem}\label{thm:blowup}
Given $d,\gamma>0$ and $\Delta\in\mathbb{N}$, there exists $\varepsilon>0$ such that for any given $T$, the following holds for all $m\ge m_0$. Let $R$ be any graph on $[t]$, where $t\le T$. Let $V_1,\dots, V_t$ be disjoint vertex sets with $m\le |V_i|\le 2|V_j|$ for each $i,j\in[t]$, and suppose that $G$ is a graph on $V_1\cup\dots\cup V_t$ such that $(V_i,V_j)$ is an $\varepsilon$-regular pair of density at least $d$ for each $ij\in E(R)$. Suppose that $H$ is any graph with $\Delta(H)\le\Delta$ such that there exists a graph homomorphism $\phi:H\to R$ satisfying $\big|\phi^{-1}(i)\big|\le(1-\gamma)|V_i|$. Then $H$ is a subgraph of $G$.
\end{theorem}
The next is a consequence of the (original) Blow-up Lemma derived in~\cite{AllenBoettcherHladky2011}.
\begin{lemma}[Embedding Lemma, Allen, B\"{o}ttcher, Hladk\'{y}~\cite{AllenBoettcherHladky2011}]\label{EmbeddingLemma}
For all $d>0$ there exists $\epsilon_{EL}>0$ with the following property. Given $0<\epsilon<\epsilon_{EL}$, for every $m_{EL}\in\mathbb{N}$ there exists $n_{EL}\in \mathbb{N}$ such that the following holds for any graph $G$ on $n>n_{EL}$ vertices with $(\epsilon, d)$-reduced graph $R$ on $m\leq m_{EL}$ vertices.
Let $\xi(R)$ be the size of the largest TCTF in $R$, then for every $\ell\in \mathbb{N}$ with $3\ell\leq (1-d)\xi(R)\tfrac{n}{m}$ we have $C_{3\ell}^2\subset G$.
\end{lemma}
We are now in a position to prove Lemma~\ref{regularityresult}, which we restate for convenience.
\RegResult*
\begin{proof}
Given $\alpha>0$, let $d$ be such that $10000\alpha^{-2}d$ is returned by Lemma~\ref{MainLemmaWithPurple} when we use as input $\tfrac{\alpha^2}{10000}$. Let $\epsilon_{EL}$ be returned by Lemma~\ref{EmbeddingLemma} for input $d$, and let $\varepsilon=\min(\tfrac1{10}d,\varepsilon_{EL},\tfrac1{10000}\alpha^2)$.
Let now $N_0$ and $M$ be returned by Lemma \ref{RegularityLemma} with input $\varepsilon$. We let $n_{EL}$ be returned by Lemma \ref{EmbeddingLemma} for input $d$, $\varepsilon$ and $m_{EL}=M$. Finally, let $\delta=d$ and $n_0=\max(100\varepsilon^{-1}, N_0,N_1)$ be the constants returned by the lemma..
Let us now fix some $n>n_0$ and, for $N>(9-\delta)n$, a $2$-edge-colouring $G$ of $K_N$.
We apply Lemma \ref{RegularityLemma} with parameter as above to the red subgraph of $G$ to get a partition $V_0,\dots{}, V_m$ of $V(G)$, with $\varepsilon^{-1}\le m\le M$, as in Lemma~\ref{RegularityLemma}.
Let $H$ be the $(\varepsilon,d)$-reduced graph of $G$. Since any cluster $V_i$ is in at most $\varepsilon m$ irregular pairs, we have $\delta(H)\geq (1-\varepsilon)m-1$. Let $t=\tfrac{m}{9-10\alpha^{-1}d}$, so that $H$ has $(9-10\alpha^{-1}d)t$ vertices and, by choice of $\varepsilon$, minimum degree at least $(9-20\alpha^{-1}d)t$. By Lemma~\ref{MainLemmaWithPurple} with constants as above, one of the following occurs.
It could be that $H$ contains a red-purple TCTF over $3(1+10d)t=\tfrac{3(1+10d)}{9-10\alpha^{-1}d}m\ge\tfrac13(1+10d)m$ vertices. Applying Lemma~\ref{EmbeddingLemma} with constants as above, we conclude that $G$ contains a red $C^2_{3s}$ for each $s\le (1-d)\cdot \tfrac13(1+10d)\cdot(9-d)n\ge 3(1+d)n$. But then in particular $G$ contains a red copy of $C_{3n}^2$ and $P_{3n+2}^2$ and we are done. Similarly, if $H$ contains a blue-purple TCTF over $3(1+10d)t$ vertices then $G$ contains a blue copy of $C_{3n}^2$ and $P_{3n+2}^2$ and we are done.
Alternatively, by Lemma \ref{MainLemmaWithPurple} we get a partition of $V(H)$ in sets $B_1$, $B_2$, $R_1$, $R_2$, $Z''$ and $T$. We obtain from this a partition of $V(G)$, setting $X'_j=\bigcup_{i\in B_j}V_i$ and $Y'_j=\bigcup_{i\in R_j}V_i$ for $j=1,2$, setting $Z':=\bigcup_{i\in Z''}V_i$, and letting $R'$ be the remaining vertices. Since we applied Lemma~\ref{MainLemmaWithPurple} with input $\tfrac{\alpha^2}{10000}$ and by choice of $\varepsilon$, we have properties~\ref{conditionA} and~\ref{conditionB} of Lemma~\ref{approxmainlemma} with $\tfrac{1}{1000}\alpha^2$ instead of $\alpha$.
Since $(B_1,B_2)$ is $\tfrac1{10000}\alpha^2 t$-red, the number of blue edges in $G$ between $X'_1$ and $X'_2$ is at most
\[d|X'_1||X'_2|+\tfrac1{1000}\alpha^2 n|X'_2|+\tfrac{1}{1000}\alpha^2 n|X'_1|\le\tfrac{1}{200}\alpha^2 n^2 \,,\]
where the inequality uses $d\le\tfrac{1}{10000}\alpha^2$. In particular there are less than $\tfrac1{200}\alpha n$ vertices in $X'_1$ which have more than $\alpha n$ blue neighbours in $X'_2$, and similarly swapping $X'_1$ and $X'_2$. By the identical calculation, an analogous statement holds for $Y'_1$ and $Y'_2$ in red.
We now claim that at most $\varepsilon|X'_1|$ vertices in $X'_1$ send $\alpha n$ or more red edges to $Y'_1$. Suppose for a contradiction this statement is false. By averaging, there is a cluster $V_i$ with $i\in B_1$ such that a set $S$ of $\varepsilon|V_i|$ vertices of $V_i$ all send $\alpha n$ or more red edges to $Y'_1$. Since $V_i$ is in irregular pairs with at most $\varepsilon m$ other clusters, at most $2\varepsilon n$ red edges from each $s\in S$ go to clusters of $R_1$ that make irregular pairs with $V_i$. The remaining at least $\tfrac12\alpha n|S|$ edges from $S$ therefore go to the remaining less than $3m$ clusters $V_j$ with $j\in R_1$, which all form $\varepsilon$-regular pairs with $V_i$ that have density at most $d$ in red. Again by averaging, there is a cluster $V_j$ with $j\in R_1$ such that $(V_i,V_j)$ is $\varepsilon$-regular and has red density at most $d$, but also receives at least $\tfrac{\alpha n|S|}{6m}>2d|V_j||S|$ red edges from $S$. But this, since $\varepsilon<d$ and $|S|\ge\varepsilon|V_i|$, is a contradiction to regularity of $(V_i,V_j)$.
By a similar argument, at most $\varepsilon|X'_i|$ vertices in $X'_i$ send edges of the `wrong' colour to each $Y'_j$ or to $Z'$ or vice versa. We can modify the argument slightly to show that at most $\varepsilon|X'_1|$ vertices of $X'_1$ have more than $\alpha n$ red neighbours in $X'_1$: again we can find a set $S$ in a cluster $V_i$ with $i\in B'_1$ whose vertices all have more than $\alpha n$ red neighbours in $X'_1$, but we need to discard both red edges in irregular pairs at $V_i$ and also edges within $V_i$. Since $|V_i|\le\tfrac{m}{n}\le\varepsilon n$, there are in total at most $2\varepsilon n$ such edges, which is the same bound we used above and from this point the proof above works as written.
We now let $X_1$ be obtained from $X'_1$ by discarding all vertices which have more than $\alpha n$ edges of the `wrong' colour to any of $X'_i$ or $Y'_i$ or $Z'$. By the above calculations, in total we discard at most $4\varepsilon |X'_1|+\tfrac1{200}\alpha n\le\tfrac1{100}\alpha n$ vertices of $X'_1$. We define similarly $X_2,Y_1,Y_2,Z$, and similarly remove at most $\tfrac1{100}\alpha n$ vertices in each case. Finally, we let $R$ denote the set of all vertices not in $X_1\cup X_2\cup Y_1\cup Y_2\cup Z$. By construction, each $X_i$, $Y_i$ and $Z$ has the claimed size; and $|R|\le\alpha n$ follows since each vertex of $R$ was either in $V_0$, or $V_i$ for some $i\in T$, or removed from $X'_i$ or $Y'_i$ or $Z'$. There are at most $\varepsilon n+\tfrac{\alpha^2}{10000}n+5\cdot\tfrac1{100}\alpha n$ such.
Finally, by definition each vertex of $X_1$ has at most $\alpha n$ edges of the `wrong' colour to any of $X'_i,Y'_i$ or $Z'$, which are supersets of $X_i,Y_i,Z$ respectively, giving the required bounds on `wrong' coloured edges at $X_1$. The same holds for $X_2,Y_1,Y_2,Z$ by the similar argument.
\end{proof}
Finally, we prove Theorem~\ref{thm:boundBW}. First, we deduce from Lemma~\ref{MainLemma} that if $G$ satisfies the conditions of Theorem~\ref{thm:boundBW}, then the reduced graph $R$ of $G$ is an $m$-vertex graph which contains a monochromatic TCTF on nearly $\tfrac13m$ vertices. Suppose this is red. We then show how to construct a homomorphism from any given $H$ satisfying the conditions of Theorem~\ref{thm:boundBW} to the red subgraph of $R$ which does not overload any vertex $i$ of $R$, i.e.\ map too many vertices to $i$, and finally apply Theorem~\ref{thm:blowup} to find the desired monochromatic copy of $H$ in $G$.
The only tricky step of this sketch is to construct the required homomorphism. We split up $V(H)$ into \emph{chunks} and \emph{fragments}, which are intervals in the bandwidth ordering, alternating between chunks and fragments. Each fragment is of equal length and their total size is tiny compared to the size of a cluster, and the chunks are of equal length and much bigger than the fragments (but still much smaller than the size of a cluster). Given our TCTF in $R$, we put an order $T_1,\dots,T_k$ (arbitrarily) on the triangles of the TCTF, and fix for each $1\le i\le k-1$ a tight walk from $T_i$ to $T_{i+1}$. We assign each chunk of $H$ to some $T_i$ where $i$ is chosen uniformly and independently from $[k]$. We claim that it is possible to now construct a homomorphism where each chunk will be mapped entirely to its assigned triangle, using the fragments to connect up along the fixed tight walks, and that this homomorphism will with positive probability not overload any vertex of $R$: the point here will be to analyse the assignment of chunks, since the total size of all fragments is tiny.
\begin{proof}[Proof of Theorem~\ref{thm:boundBW}]
Given $\gamma>0$ and $\Delta$, we fix $h\le\tfrac{\gamma}{1000}$ and $\lambda>0$ (which will play no further r\^ole in this proof) sufficiently small for Lemma~\ref{MainLemma}, and let $2\varepsilon'$ and $t_0$ be the returned constants. We let $\varepsilon>0$ be returned by Theorem~\ref{thm:blowup} for input $d=\tfrac12$, $\tfrac{\gamma}{100}$ and $\Delta$. Without loss of generality, we may presume $\varepsilon<\tfrac1{10}\min(t_0^{-1},\varepsilon',\gamma)$. We input $\varepsilon$ and $d=\tfrac12$ to Theorem~\ref{RegularityLemma} and let $M,N_0$ be the returned constants. We input $T=M$ to Theorem~\ref{thm:blowup}, with the other parameters as above, and choose $N_1$ such that the returned constant $m_0\le N_1/M$.
We set $\rho=\tfrac{1}{60000}M^{-3}\gamma^2$ and $\beta=\tfrac{1}{100}M^{-4}\gamma$.
Suppose now $n\ge\max(N_0,N_1)$.
Let $N=(9+\gamma)n$. Given a $2$-edge-coloured $K_N$, we apply Lemma~\ref{RegularityLemma} with constants as above, to the graph of red edges in $K_N$, to obtain a partition $V(K_N)=V_0\cup V_1\cup\dots\cup V_m$, where $\varepsilon^{-1}\le m\le M$. By construction, the number of vertices in each part $V_i$ with $1\le i\le m$ is at least $\tfrac{(9+\gamma/2)n}{m}$.
Let $R$ be the corresponding coloured reduced graph on $[m]$, in which we colour a pair $ij$ red if $(V_i,V_j)$ is $\varepsilon$-regular and has density in red at least $\tfrac12$, blue if it is $\varepsilon$-regular and has density in blue strictly bigger than $\tfrac12$, and otherwise (i.e.\ if the pair is irregular) we do not put an edge $ij$. By construction, we have $\delta(R)\ge (1-\varepsilon)m$.
Let $t=m/(9-\varepsilon')$, so that $R$ has $(9-\varepsilon')t$ vertices and minimum degree at least $(9-2\varepsilon')t$. By Lemma~\ref{MainLemma}, either $R$ contains a monochromatic $3(1+\varepsilon')t$-vertex TCTF, or we obtain a partition of $V(R)$ as described in that lemma. In particular, there is a set $B_1$ of at least $(2-h)t$ vertices and a disjoint set $R_1$ of at least $(2-h)t$ vertices, such that any triangle with two vertices in $B_1$ and one in $R_1$ is monochromatic blue (and so all such triangles are in a blue triangle component). It follows that choosing $(1-h)t$ disjoint such triangles greedily we obtain a monochromatic TCTF with $3(1-h)t$ vertices. We see that in all cases $R$ contains a monochromatic TCTF on at least $3(1-h)t\ge\tfrac13(1-h)m=:3k$ vertices.
Fix such a TCTF, let its triangles be $T_1,\dots,T_k$ and suppose without loss of generality that it is red. By definition of red triangle-connectedness, for each $1\le i\le k-1$ there is a red triangle walk in $R$ from $T_i$ to $T_{i+1}$, and we fix for each $i$ one such $W_i$ chosen to be of minimum length. Thus $W_i$ is a sequence of triangles, starting with $T_i$ and ending with $T_{i+1}$, in which each pair of consecutive triangles shares two vertices. Finally, we assign labels $1,2,3$ to the vertices of all these triangles as follows: we label the vertices of $T_1$ in an arbitrary order, then assign labels to the successive triangles of $W_1,W_2,\dots,W_{k-1}$ in order as follows: when we assign labels to the next triangle, we keep the labels of the two vertices it shares with the previous triangle, and give the missing label to the third vertex. Note that a given vertex, or a given edge, might receive different labellings for different triangles, and indeed if a triangle appears in two different walks it might receive different labellings in the different walks.
Let $H$ be a graph with maximum degree at most $\Delta$, bandwidth at most $\beta n$, and a fixed $3$-vertex colouring in which no colour class has more than $n$ vertices. We split $V(H)$ into consecutive intervals $C_1,F_1,C_2,F_2,\dots,F_{s-1},C_s$ as follows: we let each $C_i$ (except perhaps the last two, which can be of any size) consist of $\rho n$ vertices, and each $F_i$ be of size $M^2\beta n$. For each $1\le i\le s$ we pick $\pi(i)\in[k]$ uniformly and independently at random. We now define a homomorphism $\psi:H\to R$ as follows. If $x$ is a vertex of the chunk $C_i$ for some $i$, and its colour in the fixed $3$-colouring of $H$ is $j\in[3]$, then we set $\psi(x)$ equal to the vertex of $T_{\pi(i)}$ with label $j$. We now describe how to construct $\psi$ on the fragment $F_1$; the same procedure is used for each subsequent fragment with the obvious updates. We separate $F_1$ into intervals of length $\beta n$. If $x$ is in the $i$th interval, and has colour $j$ in the $3$-colouring, then we set $\psi(x)$ equal to the vertex of the $i$th triangle after $T_1$ in $W_1$ with label $j$. If there is no such triangle (i.e.\ the walk has already reached $T_2$) then we set $\psi(x)$ equal to the vertex labelled $j$ in $T_2$. We claim that this last event occurs for the final interval. Indeed, if two triangles of $W_1$ both contain a given edge $e$ of $R$, then by minimality they are consecutive triangles in the walk, so $W_1$ has less than $M^2$ triangles.
We claim that this construction gives a homomorphism from $H$ to the red subgraph of $R$. Indeed, suppose $xy$ is an edge of $H$. Then $x$ and $y$ have different colours in the $3$-colouring, and they are separated by at most $\beta n$ in the bandwidth ordering. By construction, $x$ is assigned to a vertex of some triangle $T$ according to its colour. The vertex $y$ is assigned to a triangle $T'$ according to its colour; and either $T=T'$ or $T$ and $T'$ are consecutive triangles on one of the fixed walks, in particular they share two vertices and their labels are consistent on those two vertices. Either way, $x$ and $y$ are mapped to a red edge of $R$ (the only non-edge is if $T\neq T'$ and it goes between the two vertices of the symmetric difference of $T$ and $T'$, which both have the same label: but $x$ and $y$ have different colours).
We still need to justify that with high probability $\psi$ does not overload any vertex of $R$. To begin with, observe that the total number of vertices in the fragments is at most $M\cdot M^2\beta n=M^3\beta n\le\tfrac{\gamma n}{100m}$, which is much smaller than the size of any cluster $V_i$. In particular, if $i$ is not in any triangle of the TCTF, then $|\psi^{-1}(i)|<\tfrac12|V_i|$ as desired. Now consider the vertex $u$ of $T_i$ with label $j$. Apart from the at most $M^3\beta n$ vertices of the fragments, the vertices of $\psi^{-1}(u)$ are vertices of chunks with colour $j$. There are at most $n$ vertices in chunks of colour $j$ in total, and each such chunk has probability $1/k$ of being assigned to $T_i$. We see that the expected number of chunk vertices in $\psi^{-1}(u)$ is at most $n/k$. The probability that the actual number of such vertices exceeds $n/k$ by $s$ is by Hoeffding's inequality at most
\[\exp\big(-\tfrac{s^2}{2\cdot 3\rho^{-1}\cdot (\rho n)^2}\big)=\exp\big(-\tfrac{s^2}{6\rho n^2}\big)\,,\]
where we used that there are at most $3\rho^{-1}$ chunks, and the maximum contribution of a given chunk to $|\psi^{-1}(u)|$ is at most $\rho n$. Choosing $s=\tfrac{1}{100}\gamma M^{-1}n$, by choice of $\rho$ the probability that $|\psi^{-1}(u)|\ge n/k+s+M^3\beta n$ (the last term accounts for vertices in fragments) is at most $\exp(-M)$. In particular, with positive probability we have
\[|\psi^{-1}(u)|\le\tfrac{n}{k}+\tfrac{1}{100}\gamma\tfrac{n}{M}+M^3\beta n\]
for every $u\in V(R)$. Suppose this event occurs. Substituting our values for $\beta$, $k$ and finally $h$, we get
\[|\psi^{-1}(u)|\le\tfrac{9n}{(1-h)m}+\tfrac{1}{100}\gamma\tfrac{n}{m}+\tfrac{1}{100}\gamma\tfrac{n}{m}\le\tfrac{9n}{m}(1+2h)+\tfrac{1}{10}\gamma\tfrac{n}{m}\le\frac{(9+\tfrac{\gamma}{5})n}{m}\,.\]
Since $|V_u|\ge\tfrac{(9+\gamma/2)n}{m}$, as observed at the start of this proof, we have $|\psi^{-1}(u)|\le(1-\tfrac{\gamma}{100})|V_u|$ for every $u\in V(R)$. This is the required condition to apply Theorem~\ref{thm:blowup}.
Finally, by Theorem~\ref{thm:blowup} we conclude that there is a red copy of $H$ in the $2$-coloured $K_N$.
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}\label{sec:exact}
We are now ready to prove the main result of this paper, which we restate for convenience. Recall that we established the lower bound in Section~\ref{sec:note}, and what remains is to prove the corresponding upper bound. We give the full details for the square of a path, the square of a cycle case is similar.
\thmmain*
\begin{proof}[Proof of Theorem~\ref{thm:main}, upper bound for $P_{3n+1}^2$]
Let $n_0$ and $\delta$ be given by Lemma \ref{regularityresult} when we set $\alpha=\frac{1}{1000}$ (we are not trying to optimise this value) and then let us fix $n>\max(n_0, \frac{3}{\delta})$ and $N\geq 9n-3$. Let now $G$ be any 2-edge-colouring of $K_{N}$. We suppose for a contradiction that $G$ does not contain a monochromatic $P^2_{3n+1}$.
By Lemma \ref{regularityresult}, since $G$ does not contain a monochromatic $P_{3n+1}^2$, we have a partition $X_1, X_2, Y_1, Y_2, Z, R$ of $V(G)$ with the conditions \ref{conditionA}-\ref{conditionG}, which we fix.
We now want to refine these conditions by adapting repeatedly a greedy procedure. Since we are going to apply multiple times the same method, we will explain the greedy procedure and the arguments for existence only in the first instance.
\begin{claim}\label{claim91}
$X_1$ and $X_2$ are entirely blue, while $Y_1$ and $Y_2$ are entirely red.
\end{claim}
\begin{claimproof}
Assume by contradiction that there is a red edge $x_1x_1'$ in $X_1$. Since $x_1$ and $x_1'$ have at most $\alpha n$ blue neighbours in $Z$ and $(1-\alpha)n\leq \vass{X}$, we have that $x_1$ and $x_1'$ have a common red neighbour $z\in Z$. Similarly, by considering the common red neighbour of $x_1$ and $x_1'$ in $Y_2$, we can find $y_2, y_2' \in Y_2$ such that $y_2y_2', x_1y_2, x_1'y_2', x_1'y_2, x_1y_2'$ are all red.
We are now ready to extend the red path $P_0=y_2, y_2', x_1, x_1', z$ (whose square is also monochromatic red) to a path $P$ of length bigger than $3n+2$ such that $P^2$ is also monochromatic red. The idea is to greedily add to $P_0$ at least $\frac{3}{2}n$ vertices from $Y_2$ (using the fact that almost all the edges in $Y_2$ are red) and $2n$ vertices from $(X_1, X_2, Z)$.
In order to do that, it suffices to show that we can find a path $P_{Y_2}$ of length at least $\frac{3}{2}n$ in $Y_2$ that starts with $y_2y_2'$ and such that $P_{Y_2}^2$ is monochromatic red. Assume we have built already a path $P_{Y_2}=y_2, y_2',\dots{}, p_{\ell}$ with the aforementioned conditions, provided $\ell<\frac{3}{2}n$, we can extend $P_{Y_2}$ simply by appending a vertex $p_{\ell+1}$ that is in the common red neighbour of $p_{\ell}$ and $p_{\ell-1}$ in $Y_2\setminus P_{Y_2}$. But this is possible, indeed all but at most $\frac{2}{1000}n$ vertices in $Y_2$ have red edges to both $p_{\ell}$ and $p_{\ell-1}$.
We do a very similar procedure by greedily extending $P_{(X_1, X_2, Z)}$. Given a path $P_{(X_1, X_2, Z)}=x_1', z, \dots{}, p_{\ell}$ of length smaller than $2n$, we can extend it by taking a vertex in the common red neighbour of $p_{\ell}$ and $p_{\ell-1}$ and in the right component.
Since $P_0^2$ is monochromatic red, and since we showed how to extend the endpoints to form a long path whose square is also monochromatic, we are done.
The arguments for $X_2, Y_1$ and $Y_2$ are symmetric.
\end{claimproof}
\begin{claim}\label{claim92}
The pairs $(X_1, Z)$ and $(X_2, Z)$ are entirely red, while the pairs $(Y_1, Z)$ and $(Y_2, Z)$ are entirely blue.
\end{claim}
\begin{claimproof}
Assume by contradiction that there is a blue edge $x_1z$ between $X_1$ and $Z$. Let $y_1\in Y_1$ be in the common blue neighbourhood of $x_1$ and $z$ (which exists by arguments similar to the ones above).
Take $x_1'\in X_1\setminus\left\lbrace x_1\right\rbrace$ in the common blue neighbourhood of $y_1$ and $x_1$ and let $P_0=z,y_1,x_1,x_1'$. We have that $P_0^2$ is blue monochromatic. Also, we can greedily extend $P_0$ to a path $P$ such that $P^2$ is also blue monochromatic and $\vass{P}>3n$ by extending $x_1, x_1'$ to a path of length at least $\frac{3}{2}n$ in $X_1$ and extending the $zy_1$ end in $(Y_1, Y_2, Z)$ by at least $2n$ vertices.
The argument for the other pairs is symmetric.
\end{claimproof}
\begin{claim}\label{claim93}
The pairs $(X_1, Y_1)$ and $(X_2, Y_2)$ are entirely blue, while the pairs $(X_1, Y_2)$ and $(X_2, Y_1)$ are entirely red.
\end{claim}
\begin{claimproof}
Assume by contradiction that there is a red edge $x_1y_1$ in $(X_1, Y_1)$. The vertices $x_1$ and $y_1$ share a red neighbour $x_2$ in $X_2$. We can also find in $Y_1\setminus \left\lbrace y_1\right\rbrace$ a common red neighbour $y_1'$ of $y_1$ and $x_2$.
We can start with the path $P_0=x_1, x_2, y_1, y_1'$, and then extend it using vertices in $Y_1$ on one side and vertices of $(X_1, X_2, Z)$ on the other, until we get a path $P$ such that $P^2$ is monochromatic red and covers at least $3n+2$ vertices.
The argument for the other pairs is symmetric.
\end{claimproof}
\begin{claim}\label{claim94}
The pair $(X_1, X_2)$ has no blue $P_4$, while the pair $(Y_1, Y_2)$ has no red $P_4$.
\end{claim}
\begin{claimproof}
Assume $x_1x_2x_1'x_2'$ formed a blue $P_4$ in $(X_1, X_2)$. Since $X_1$ and $X_2$ are entirely blue, the edges $x_1x_1'$ and $x_2x_2'$ are blue. Each of these edges is the beginning of a square of a path covering the respective part. These join together to form a square of a path that is longer than allowed.
The argument for the other pair is symmetric.
\end{claimproof}
From the claims above we can see that in the situation depicted by Lemma \ref{regularityresult} we have $\vass{X_1}, \vass{X_2}, \vass{Y_1}, \vass{Y_2}\leq 2n-1$. We can now partition the vertices of the remainder set $R$ depending on their neighbourhoods as follows.
\begin{itemize}
\item[1)] Let us denote with $R_Z$ the set of vertices in $R$ with more than $\frac{n}{4}$ red neighbours both in $X_1$ and $X_2$,
\item[2)] for $i=1,2$ let $R_i$ be the vertices in $R$ with more than $\frac{n}{4}$ blue neighbours in both $X_i$ and $Y_i$,
\item[3)] let $R_{12}$ denote the vertices in $R$ with more than $\frac{n}{4}$ red neighbours in both $X_1$ and $Y_2$,
\item[4)] let $R_{21}$ denote the vertices in $R$ with more than $\frac{n}{4}$ red neighbours in both $X_2$ and $Y_1$,
\item[5)] let $R^*$ denote any vertices in $R$ that are not in any of the above sets.
\end{itemize}
\begin{claim}\label{claim95}
Vertices in $R^*$ have at least $\frac{3}{2}n$ blue neighbours in each $X_i$ and at least $\frac{3}{2}n$ red neighbours in each $Y_i$. Moreover, $\vass{R^*}\leq 1$.
\end{claim}
\begin{claimproof}
The first part of the claim is true by construction. Let us now assume that there are two vertices $u$ and $v$ in $R^*$. Then $u$ and $v$ have more than $\frac{n}{2}$ common blue neighbours in each $X_i$ and at least $\frac{n}{2}$ common red neighbours in each $Y_i$. Therefore if $uv$ is blue it will create a blue square of a path with vertices from $X_1$ and $X_2$, while if it is red it will join long red squares of paths in $Y_1$ and $Y_2$.
\end{claimproof}
\begin{claim}\label{cl:96}
We have the following bounds: $\vass{X_1\cup R_1}, \vass{X_2\cup R_2}, \vass{Y_1\cup R_{21}}, \vass{Y_2\cup R_{12}}\leq 2n-1$.
\end{claim}
\begin{claimproof}
Assume by contradiction that $\vass{X_1\cup R_1}\geq 2n$. Recall that in previous claims we proved that all the edges in $X_1$ and $(X_1, Y_1)$ are blue. Let us label the vertices in $R_1$ by $r_1,\dots{}, r_{\ell}$. Recall that $\ell=\vass{R_1}\leq \vass{R}\leq \alpha n$.
Since every vertex in $R_1$ has at least $\frac{n}{4}$ blue neighbours in both $X_1$ and $Y_1$ we can find disjoint blue triangles $T_1,\dots{}, T_{\ell}$ where triangle $T_i$ contains the vertices $r_i,x_i,y_i$ with $x_i\in X_1$ and $y_i\in Y_1$. We next find for each $i\in[\ell]$ vertices $a_i,b_i,c_i,a'_i,b'_i,c'_i$ as follows. We let $c_i$ be a blue neighbour of $r_i$ in $Y_1$, and $a_i,b_i\in X_1$, we let $a'_1$ be a neighbour in $X_1$ of $r_1$, $b'_1$ be in $X_1$, and $c'_1$ be in $Y_1$. Observe that since $\ell\le\alpha n$, we can ensure that all these vertices are different.
By construction, the vertex ordering $P_0=(a_1,b_1,c_1,x_1,r_1,y_1,a'_1,b'_1,c'_1,\dots)$, where we repeat the same letter ordering for $i=2$ and so on afterwards, is a blue square path. We extend $P_0$ further by choosing distinct vertices from $X_1$, $X_1$ and then $Y_1$ in this order, until no unused vertices remain in $X_1$. As $\vass{X_1\cup R_1}\geq 2n$, what we obtain is a blue square path with at least $3n$ vertices, if $\vass{X_1\cup R_1}\geq 2n+1$ we obtain at least $3n+1$ vertices. We can extend $P_0$ by one more vertex by adding a so far unused vertex of $Y_1$ at the start of the ordering. This gives the required $3n+1$-vertex square path (and $3n+2$ vertices if $\vass{X_1\cup R_1}\geq 2n+1$). The arguments for the other pairs of sets are the same.
\end{claimproof}
\begin{claim}\label{claim97}
We have that $\vass{Z\cup R_Z}\leq n-1$.
\end{claim}
\begin{claimproof}
Let as assume that $\vass{Z\cup R_Z}\geq n$ and let us label the vertices in $R_Z$ by $r_1,\dots{}, r_{\ell}$. Since $(X_1, X_2)$ has no blue path on $4$ vertices, there are at most $40$ vertices in $X_1\cup X_2$ with more than $\frac{n}{20}$ blue neighbours in the opposite part. Call the set of these vertices $X_{\text{bad}}$. Since each vertex in $R_Z$ has more than $\frac{n}{4}$ red neighbours in each $X_i$, we can find disjoint red triangles $T_1, \dots{}, T_{\ell}$ such that each $T_i$ uses $r_i$, a vertex $x_i^1\in X_{1}\setminus X_{\text{bad}}$ and a vertex $x_i^2\in X_2\setminus X_{\text{bad}}$.
The idea is now to find for each $i\in [\ell]$ vertices $a_i, a_i'\in X_1$, $b_i, b_i'\in Z$, $c_i, c_i'\in X_2$ such that for every $i\in [\ell-1]$ we have that $(x_i^1, r_i, x_i^2, a_i, b_i, c_i, a_i', b_i', c_i', x_{i+1}^1, r_{i+1}, x_{i+1}^2)$ is a red square of a path. But this can be done greedily since $\ell\leq \alpha n$. We now build the path $P_0=(x_1^1, r_1, x_1^2, a_1, b_1, c_1, a_1', b_1', c_1', x_{2}^1, \dots{}, x_{\ell}^2)$ which by construction has the property that $P_0^2$ is red.
We can extend $P_0$ by choosing distinct vertices from $X_1$, $Z$ and then $X_2$ in this order, until no unused vertices remain in $Z$. As $\vass{Z\cup R_Z}\geq n$, what we obtain is a red square path with at least $3n$ vertices.
\end{claimproof}
Putting the bounds from the last three claims together, we see $|G|\le 1+4(2n-1)+n-1=9n-4$, which contradiction completes the proof.
\end{proof}
The proof for $P_{3n+2}^2$ is almost verbatim as above (we actually worked with $P_{3n+2}^2$ in most of the claims), with the exception that in Claim~\ref{cl:96} we obtain the upper bound $|X_1\cup R_1|\le 2n$, as explained in the proof of that claim, and consequently a final upper bound $|G|\le 1+4(2n)+n-1=9n$ for a contradiction.
\begin{proof}[Sketch proof of cycle case of Theorem~\ref{thm:main}]
In order to prove that for $n$ large enough we have $R(C_{3n}^2,C_{3n}^2)=9n-3$, is suffices to modify our previous proof. We start by constructing the same partition we built at the beginning of the proof of Theorem \ref{thm:main} to get the sets $X_1, X_2, Y_1, Y_2, Z, R$. Now, by using the same technique introduced in Claim \ref{claim91} we can prove some weakened for of Claims \ref{claim91}, \ref{claim92}, \ref{claim93}, \ref{claim94}. Which is, we can prove that in $X_1$ we cannot find two disjoint red edges (the same holds for $X_2$), in $Y_1$ we cannot find two disjoint blue edges (the same holds for $Y_2$). Similarly, we cannot find two disjoint edges of the wrong colours in none of the following pairs: $(X_1, Z)$, $(X_2, Z)$, $(Y_1, Z)$, $(Y_2, Z)$, $(X_1, Y_1)$, $(X_2, Y_2), (X_1, Y_2), (X_2, Y_1)$. Moreover, we cannot find two vertex-disjoint $P_4$ of the wrong colour in $(X_1, X_2)$ nor in $(Y_1, Y_2)$.\\
To
From these results and the previously proved Lemma \ref{regularityresult}, we can see that also in this case we have $\vass{X_1}, \vass{X_2}, \vass{Y_1}, \vass{Y_2}\leq 2n-1$. Now we can define the same partition of $R$ in sets $R_Z, R_1, R_2, R_{12}, R_{21}$ and $R^*$.
Let us point out that from this modified version of Claim \ref{claim91} we have that there are two vertices $a, b\in X_1$ such that all edges in $G[X_1\setminus \left\lbrace a,b\right\rbrace]$ are blue. In particular, from Claims \ref{claim91}, \ref{claim92}, \ref{claim93}, \ref{claim94} we get that up to moving at most $10$ vertices from $X_1$ to $R_1$ (and similarly from $X_2$ to $R_2$, from $Y_1$ to $R_{21}$, from $Y_2$ to $R_{12}$ and from $Z$ to $R_Z$) all the vertices in $X_1$ (and similarly in $X_2, Y_1, Y_2, Z$) are incident only to edges of the right colour in $G[X_1\cup X_2\cup Y_1\cup Y_2\cup Z]$, with the possible exception of edges in $(X_1, X_2)$ and $(Y_1, Y_2)$.\\
We now aim to explain how to modify Claim \ref{claim95} to hold for cycles and how to modify the proof of Claims \ref{cl:96}, \ref{claim97}. The first part of Claim \ref{claim95} holds by construction without any modifications. The second part of Claim \ref{claim95} needs to be modified to state that we cannot find two parallel edges of the same colour in $R^*$. Indeed, otherwise we could find a long monochromatic blue cycle $C$ such that $C^2$ is also blue by using vertices from $X_1$, $X_2$ and the two blue edges in $R^*$. In particular, this implies that $\vass{R^*}\leq 4$.
As a guide to show how to modify the proofs of Claim \ref{cl:96} and \ref{claim97}, we give a sketch of the modifications needed for Claim \ref{cl:96}. If we assume by contradiction that $\vass{X_1\cup R_1}\geq 2n$ we can almost verbatim repeat the same proof, having care of extending our path $P_0$ in both directions and making sure that the two endpoints of $P_0$ and their neighbours are adjacent in blue to each other. This is possible because $G[X_1]$ is entirely blue as claimed above.
\begin{claim}
If $R^*$ contains a blue edge, then $\vass{X_1\cup R_1}, \vass{X_2\cup R_2}\leq 2n-2$ (same holds for red, $Y_1\cup R_{21}$ and $Y_2\cup R_{12}$).
\end{claim}
\begin{claimproof}
Assume $R^*$ contains a blue edge $uv$, then $\vass{X_1\cup R_1}\leq 2n-2$ (the arguments for the other cases are the same). In order to prove this, it suffices to show that there exists a maximal matching $T$ in $X_1$ such that we can build a blue cycle $C$ that covers all the edges of $X_1$, the two vertices $u, v\in R^*$ and, for each edge in $T$, an extra vertex in $Y_1$. This can be done because by Claim \ref{claim91} and Lemma \ref{regularityresult} there is a vertex $w\in X_1$ such that the red neighbourhood of $w$ in $X_1$ has size at most $\alpha n$, but $G[X_1\setminus w]$ has at most one red edge and because $u$ and $v$ have both at least $\frac{3}{2}n$ blue neighbours in $X_1$. Therefore, it is possible to build a cycle by replicating the construction in Claim \ref{cl:96} and by carefully adding the edge $uv$ to the cycle.
\end{claimproof}
This suffices to conclude. Indeed, if $\vass{R^*}\leq 3$ then we still have $\vass{X_1\cup R_1\cup R^*\cup X_2\cup R_2}\leq 4n-1$, while if $\vass{R^*}=4$ then we have both a red and a blue edge in $R^*$ (since we cannot have two vertex-disjoint edges of the same colour). In this case we have the following inequalities: $\vass{X_1\cup R_1}, \vass{X_2\cup R_2}, \vass{Y_1\cup R_{21}}, \vass{Y_2\cup R_{12}} \leq 2n-2$, which are enough to obtain the wanted bound.
\end{proof}
\printbibliography
\end{document}
|
2,869,038,154,899 | arxiv | \section{Introduction}
The so-called isochrone spherical models were introduced by Michel H\'enon in the fifties \cite{MH1,MH2,MH3} in the study of the dynamics of globular clusters, see \cite{JBY} for a brief review on the subject. {Globular clusters
are dense, roughly spherically symmetric, distributions of stars whose
dynamics is usually described through an averaged gravitational potential, leading naturally to the study of general central potentials. Henon’s isochrone models are intimately related to
the classical Bertrand's theorem.}
Strictly speaking, an isochrone
model in Henon's sense is a Newtonian spherically symmetric gravitational field for which the radial periods of bounded orbits do not depend on their angular momenta.
{The Newtonian and the harmonic oscillator potentials}, which according to
Bertrand's theorem are the only central potentials for which all bounded trajectories are periodic (closed), are also isochrone in
Henon's sense. {In fact,
Bertrand’s theorem can be considered as a refinement of the concept of isochrone potential,
the Newtonian and the harmonic oscillator potentials are the only isochrone potential with
closed orbits,
see Section 4.4 of \cite{PPD}.
}
In principle, non-relativistic gravitational configurations
as the isochrone spherical models
might be effectively attained in globular clusters
as a result of a dynamical mechanism called resonant relaxation, as suggested by H\'enon in his original works \cite{MH1,MH2,MH3}, {see also \cite{Relax,BTGD} for more modern approaches to
this subject.}
For further recent developments on the dynamics of isochrone potentials, see \cite{PPD,PPP,PPC,RP}.
The family of Henon's spherical isochrone potentials includes, besides the two cases of Bertrand's theorem,
three other central potentials, namely
the so-called H\'enon potential
\begin{equation}
\label{he}
V_{\rm He}(r) = -\frac{k}{b+\sqrt{b^2 + r^2}},
\end{equation}
and, respectively, the bounded and hollowed potentials
\begin{eqnarray}
\label{bo}
V_{\rm bo}(r) &=& \frac{k}{b + \sqrt{b^2 - r^2}}, \\
\label{ho}
V_{\rm ho}(r) &=& -\frac{k}{r^2}\sqrt{ r^2- b^2 },
\end{eqnarray}
where $b$ and $k$ are positive constants. It is clear that the
potentials (\ref{bo}) and (\ref{ho}) are not defined for all $r$ and that the
Newtonian potential arises from the
limit $b\to 0$ of $V_{\rm He}$ or $V_{\rm ho}$. The isochrone potentials (\ref{he}) and (\ref{ho}) are asymptotically
Newtonian for large $r$, while (\ref{he}) and (\ref{bo}) are effectively
harmonic near the center
$r=0$.
Some years ago, Perlick \cite{Perlick}
introduced the notion of a Bertrand spacetime, namely a spherically symmetric and
static spacetime in which any bounded trajectory of test bodies is periodic, clearly
extending the classical Bertrand's theorem to the realm of General Relativity.
From Bertrand's theorem, we have also that
the azimuthal angle for the Newtonian and harmonic potential cases is given by $\Theta = \frac{\pi}{\beta}$ with, respectively, $\beta = 1$ and $\beta=2$. We remind that the azimuthal angle
corresponds to the angular variation between the closest and farthest points to the center
of a bounded trajectory.
Rather surprisingly, Perlick showed that it is always possible to construct a static spherically symmetrical spacetime in which all test body bounded trajectories
are periodic for any rational value of the parameter $\beta$.
We know that
any spherically symmetric and
static spacetime can be cast in a spherical Schwarzschild coordinate system where its metric takes the form
\begin{equation}
\label{metric}
g_{ab}dx^adx^b = -f(r)dt^2 + \frac{ dr^2}{h(r)} + r^2d\Omega^2,
\end{equation}
where $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ denotes the usual metric on {the unit sphere} $S^2$.
There are two types of Bertrand spacetimes.
For the first type, we have
\begin{eqnarray}
\label{bertr1f}
{\frac{1}{f(r)}} &=& G+ \sqrt{\kappa+ r^{-2}}, \\
{h(r)} &=& \beta^2\left(1+\kappa r^2\right),
\label{bertr1h}
\end{eqnarray}
whereas the second case corresponds to
\begin{eqnarray}
\label{bertr2f}
{\frac{1}{f(r)}} &=& G\mp \frac{r^2}{1-Dr^2 \pm \sqrt{(1-Dr^2)^2 - Kr^4}}, \\
{h(r)} &=& \frac{\beta^2\left((1-Dr^2)^2 - Kr^4\right)}{2\left(1-Dr^2 \pm \sqrt{(1-Dr^2)^2 - Kr^4}\right)},
\label{bertr2h}
\end{eqnarray}
with arbitrary constants $G$, $\kappa$, $D$, and $K$. For an enlightening geometrical interpretation of the Bertrand spacetimes and their relation with the standard harmonic and Newtonian potentials on curved 3-dimensional manifolds, see \cite{esp}. For the Bertrand spacetimes of the first type, the Newtonian limit is
obtained by setting
$\kappa = 0$ (locally flat condition) and $\beta = 1$ (absence of conical singularity at the origin),
and clearly corresponds to the $r^{-1}$ interaction. On the other hand, for the second type of spacetimes,
the locally flat condition and the absence of conical singularity requires, respectively, $D=K=0$
and $\beta = 2$, and we have eventually the harmonic potential case.
In the present paper, we introduce the relativistic version of the
Henon's isochrone models, which we denominate ``isochrone spacetimes''. {They
correspond to the static spherically symmetric spacetimes
(\ref{metric}) in which all bounded timelike trajectories are isochrone in Henon's sense, {\em i.e.}, their radial periods do not depend
on their angular momenta. As in the non-relativistic Newtonian case \cite{RP}, the isochrony condition
for a static spherically symmetric spacetime
is
equivalent to demand that the relativistic radial action be additively separable in the two pertinent
constants of motion, namely the energy and the angular momentum. However, as we will see, in clear contrast with the
Newtonian case, such condition is not a very stringent restriction in the relativistic domain.} Besides the rather trivial extension of the Bertrand spacetimes for
the case of real $\beta$, there are many other families of isochrone spacetimes.
We propose a procedure to generate such new spacetimes by means of an algebraic equation and
present explicitly some families of solutions. In particular,
we present a family whose Newtonian limit corresponds to Henon’s isochrone potentials (\ref{he}),
(\ref{bo}) and (\ref{ho}).
Some of these spacetimes are asymptotically flat and others have regular centers in an analogous way
of their Newtonian counterparts and, hence, they could be useful as a relativistic extension of the original Henon's proposal for the study of globular clusters.
However, isochrone spacetimes generically violate the weak energy condition and may exhibit
naked singularities and, consequently, their physical interpretation in General Relativity is rather challenging.
In the next section, we will introduce the isochrone spacetimes and present a procedure to generate them by exploring an algebraic equation. We will present explicitly three large families of these spacetimes and discuss
briefly some of their main properties.
The third and last section is devoted to some concluding remarks, including the issue of the violation of the weak
energy conditions and some brief comments about the causal structure of isochrone spacetimes.
\section{The isochrone spacetimes}
The pertinent Lagrangian for the motion of test bodies in a
static spherically symmetrical spacetime with metric (\ref{metric})
reads
\begin{equation}
\label{lagr}
\mathcal{L} = -f \dot{t}^2 + \frac{\dot{r}^2}{h} +
r^2\dot\phi^2,
\end{equation}
{with the dot denoting the usual derivation with respect to the proper time
of the timelike geodesic,}
where we have, without loss of generality due the spherical symmetry, restricted the motion to the equatorial plane. {The geodesic equation derived form (\ref{lagr}) admits 3 constants of
motion, which are}
\begin{equation}
{f\dot t} = 1, \quad \ell = r^2 \dot\phi, \quad
E = -\frac{1}{f} + \frac{\dot r^2}{h} + \frac{\ell^2}{r^2},
\end{equation}
with $\ell$ and $E$ interpreted, respectively, as the trajectory angular momentum and
energy. This is a simple one-dimensional motion problem and the orbit radial period can be determined from the conserved quantities
by exploring elementary methods. We have
\begin{equation}
\label{RF}
T(E,\ell) = 2 \int_{r_{\rm min}}^{r_{\rm max}} \frac{dr}{\sqrt{h(r)}\sqrt{E-U_{ {\ell}}(r)}},
\end{equation}
where $r_{\rm min}$ and $r_{\rm max}$ are the usual return points of the effective potential
{\begin{equation}
\label{unov}
U_\ell(r)= - \frac{1}{f(r)}+ \frac{\ell^2}{r^{2}},
\end{equation}
which} is expected to have a local minimum at $r_0$ corresponding to the circular orbits. We also
assume $U_{ {\ell}}''(r_0)>0$, { {\em i.e.}, the dynamical stability of the
circular orbit.} Moreover, all functions here are assumed to be sufficiently smooth.
The other important quantity for our purposes in this work is the azimuthal angle, which is given
by
\begin{equation}
\label{apsidal}
\Theta(E,\ell) = {\ell} \int_{r_{\rm min}}^{r_{\rm max}} \frac{dr}{r^2\sqrt{h(r)}\sqrt{E-U_{ {\ell}}(r)}}
\end{equation}
{
and
corresponds to the angular variation between the closest and farthest points to the center
of a bounded trajectory.
Notice that the apsidal angle, another common quantity used in this kind of analysis, is defined as the angular variation during
one radial period and, hence, is twice the azimuthal angle.
Although the effective potential (\ref{unov}) is indeed the relativistic counterpart
of the sum of the Newtonian
gravitational potential and the centrifugal barrier term, the presence of the factor
$h(r)$ in (\ref{RF}) and (\ref{apsidal}) spoils any other useful analogy here.
From (\ref{metric}), we see that $h(r)=1$ basically reduces the problem to the Newtonian one,
since in this case we can interpret $V(r) = -1/f(r)$ as a Newtonian potential in a spatially flat
spacetime. Hence, the genuine relativistic problem demands a non-constant $h(r)$, which
implies in a spatially curved
spacetime, preventing the direct use of any
Newtonian result in the present case. In particular, there is no place here for the introduction of the
so-called Henon variables and, consequently, the identification of the parabolic properties of the
isochrone potentials, see \cite{PPD}.
For further details on the interpretation of the effective potential (\ref{unov}) on spatially
curved manifolds, see \cite{esp}.
}
The isochrony condition
$\frac{\partial T}{\partial \ell} = 0$ is equivalent to demand $\frac{\partial \Theta}{\partial E} = 0$
since we have
$
T = 2\frac{\partial \mathcal{A}_{r}}{\partial E}$ and $
\Theta = -\frac{1}{2}\frac{\partial \mathcal{A}_{r}}{\partial \ell}$,
with $\mathcal{A}_{r}$ standing for the so-called radial action of the problem
\begin{equation}
\mathcal{A}_{r}(E,\ell) = 2\int_{r_{\rm min}}^{r_{\rm max}} \frac{\sqrt{E-
U_{ {\ell}} (r)} dr}{\sqrt{h(r)}}.
\end{equation}
{Notice that, as in the Newtonian case \cite{RP}, the isochrony condition for
static spherically
symmetrical spacetimes
is fully
equivalent to demand that the relativistic radial action be additively separable, {\em i.e.},
$\mathcal{A}_{r}(E,\ell) = \mathcal{B}(E) + \mathcal{C}(\ell)$. In other words, we have
$\frac{\partial T}{\partial \ell} = 0$, and consequently $\frac{\partial \Theta}{\partial E} = 0$, if and only if the radial action is additively separable
in the constants $E$ and $\ell$. Unfortunately, however, due to the generic presence of spatial curvature (non constant $h(r)$),
the relativistic isochrony condition is not enough to single out relativistic isochrone potentials as it happens in the Newtonian case. }
{
For the sake of notation simplicity, we will drop hereafter the $\ell$ index
denoting dependence in $U(r)$.
It is convenient now to follow \cite{Perlick} and
introduce} a different integration variable $R$ such that
\begin{equation}
\label{change}
\frac{dR}{R^2} = \frac{dr}{r^2\sqrt{h}},
\end{equation}
in terms of which the azimuthal angle (\ref{apsidal}) reads
\begin{equation}
\label{apsidalR}
\Theta(E,\ell) = {\ell} \int_{R_{\rm min}}^{R_{\rm max}} \frac{dR}{R^2\sqrt{E-U(R)}} ,
\end{equation}
where the effective potential is now given by
\begin{equation}
\label{Ueff}
U(R) = \ell^2 v(R) - w(R),
\end{equation}
with
{
\begin{equation}
\label{vR}
v(R) = \frac{1}{r^2(R)}, \quad
w(R) = \frac{1}{f(r(R))},
\end{equation} }
\noindent and
\begin{equation}
\label{h(r)}
h(r) = \left(\frac{rv'R^2}{2}\right)^2 = \left(\frac{R^2}{r^2R'} \right)^2,
\end{equation}
{where the prime $'$ denotes the derivative of the function for its
considered variable.
Despite equation (\ref{apsidalR}) in the new radial variable $R$ is formally identical to the equivalent Newtonian
one, the effective potential (\ref{Ueff}) is quite different. In particular, the term corresponding
to the centrifugal barrier now is an arbitrary function and not a fixed $R^{-2}$ term as we would have in the
Newtonian case. It is clear that the problem in General Relativity is much less restricted and that,
in principle, many other solutions are indeed possible, exactly in the same way of the
Bertrand spacetime problem.}
We stress that the spacetime functions $v(R)$ and $w(R)$ are independent of the trajectory initial conditions and, hence, cannot depend on $\ell$ nor $E$. Also, the condition $R'(r)>0$ tacitly assumed in (\ref{change}) and its direct consequence $v'(R)<0$ from (\ref{vR})
will be important to select the correct solutions in the subsequent analysis.
In terms of the new radial variable $R$, the metric (\ref{metric}) is given by
\begin{equation}
\label{metric1}
g_{ab}dx^adx^b = - \frac{dt^2}{w(R)} + \frac{ dR^2}{\left(R^2v(R) \right)^2} + \frac{d\Omega^2}{v(R)}.
\end{equation}
Let us now decompose the motion range $[R_{\rm min},R_{\rm max}]$ into the branches $R_-\le R_0 = R(r_0)$ and $R_+\ge R_0$, where $U(R)$ in inverted in each of these branches, see Fig. \ref{fig1} and
\cite{Perlick}.
\begin{figure}[t]
\begin{center}
\input{plot.tex}
\end{center}
\caption{ {Aspect of a generic effective potential $U(R)$ near its local minimum at $R=R_0$. The potential
is inverted in each of the two branches
$R_-(U)\le R_0 $ and $R_+(U)\ge R_0$. It is clear that $R_+ - R_-$ must be a function of $U$.
However, in the vicinity of $R_0$, $U$ is well described by a parabola since we assume $U''_0 = U''(R_0)>0$ and,
consequently, $ R_+ - R_- = \frac{2\sqrt{2}}{\sqrt{U''_0}}\sqrt{U - U_0} $ near $R_0$, which is the main
motivation for the proposed expression (\ref{RR1}).
}}
\label{fig1}
\end{figure}
We can write (\ref{apsidalR}) as
\begin{equation}
\label{apsidal1}
\Theta = {\ell} \int_{U_0}^{E} \frac{1}{ \sqrt{E-U }} \frac{d }{dU}
\left(\frac{1}{R_-} - \frac{1}{R_+}
\right)dU,
\end{equation}
where $U_0=U(R_0)$.
The isochrony condition corresponds to require
an azimuthal angle independent of $E$, {\em i.e.}, $ \Theta(E,\ell) = \frac{\pi}{\beta_\ell}$. Eq. (\ref{apsidal1}) can be inverted by using the Abel equation \cite{Perlick}, leading in this
case to
\begin{equation}
\label{RR}
\frac{1}{R_-} - \frac{1}{R_+} = \frac{ {2}}{\ell\beta_\ell}\sqrt{U-U_0}.
\end{equation}
We follow now the same approach \cite{AJP} we have used recently for deriving the classical
isochrone potentials (\ref{he}), (\ref{bo}), and (\ref{ho}). By inspecting the analytical properties of the potential
$U(R)$ near its minimal at $R=R_0$, see Fig. \ref{fig1}, we can write
\begin{equation}
\label{RR1}
R_+ - R_- = \frac{\sqrt{U-U_0}}{F(U)},
\end{equation}
where $F(U)$ is an arbitrary function such that $F(U_0) = \frac{\sqrt{U''_0}}{2\sqrt{2}}$,
with $U''_0=U''(R_0)$.
Notice that since $F(U)$ is assumed to be arbitrary, there is indeed no loss of generality
in choosing the form (\ref{RR1}) for the function corresponding to
$R_+(U) - R_-(U)$. Nevertheless, this choice is a very convenient one since, from (\ref{RR}) and
(\ref{RR1}), we will have
\begin{equation}
\label{expr}
\sqrt{U-U_0} = F(U)R_+ - \frac{A}{R_+} = -\left( F(U)R_- - \frac{A}{R_-} \right),
\end{equation}
implying finally
\begin{equation}
\label{exprf}
U-U_0 = \left( F(U)R - \frac{A}{R} \right)^2,
\end{equation}
valid for both branches, with
\begin{equation}
A=\frac{\ell \beta_\ell}{2}.
\end{equation}
From (\ref{RR}) and
(\ref{RR1}) we have also the useful relation
\begin{equation}
\ell^2\beta^2_\ell = \frac{R_0^4U''_0}{2}.
\end{equation}
In summary, an isochrone effective potential $U(R)$ must obey the algebraic relation (\ref{exprf})
for a function $F(U)$ such that (\ref{RR1}) holds. Hence, we indeed have a procedure to generate
isochrone effective potentials: once a function $F(U)$ is given, we can formally
solve (\ref{exprf}) for $U$ and obtain the corresponding isocrone effective potential. However, of course,
we cannot always get explicit solutions with the required properties for this equation. Fortunately, we can do it for some algebraic choices for $F(U)$. Let us consider now these explicit relevant examples.
\subsection{First case: constant $F(U)$ }
\label{secconst}
The simplest choice is a constant $F(U)=\alpha$. From (\ref{exprf}), we have in this case
simply
\begin{equation}
\label{linear}
U(R) = BR^2 + \frac{C}{R^2} +D,
\end{equation}
with $B=\alpha^2$, $C=A^2 = \ell^2\beta^2_\ell/4$, and $U_0= \alpha \ell\beta_\ell - D$. The parameter
$\alpha$ is completely arbitrary and could depend, in principle, also on $\ell$ and, hence, the decomposition of of $U(R)$ in
``centrifugal'' and ``central potential'' parts as (\ref{Ueff}) can involve
the first two terms of (\ref{linear}) rather arbitrarily. The most general
solution corresponds to set
\begin{equation}
\label{vr}
v(R) = B_1R^2 + \frac{C_1}{R^2} + D_1,
\end{equation}
and
\begin{equation}
\label{wr}
w(R) = B_2R^2 + \frac{C_2}{R^2}+ D_2.
\end{equation}
Of course, we are exploring an underlying linear structure of the
problem, we are considering $v(R)$ and $w(R)$ as generic linear combinations of the three terms
of (\ref{linear}) such that $B=B_1\ell^2+B_2$, $C= C_1\ell^2+C_2$, and $D=D_1\ell^2 +D_2$.
This will be valid for all cases considered here.
Once $v(R)$ is given,
we can recast the original spherical coordinates (\ref{metric}) by using
(\ref{vR}) and (\ref{vr}). Assuming $K=4B_1C_1\ne 0$, we will have
\begin{equation}
R^2 = \frac{1-D_1r^2 \mp \sqrt{(1-D_1r^2)^2 - Kr^4}}{2B_1r^2},
\end{equation}
leading to
\begin{equation}
h = \frac{2C_1\left((1-D_1r^2)^2 - Kr^4\right)}{1-D_1r^2 \pm \sqrt{(1-D_1r^2)^2 - Kr^4}}.
\label{bertr2hh}
\end{equation}
Notice that (\ref{bertr2h}) and (\ref{bertr2hh}) are identical, up to the definition
of the constant $C_1$. For the determination of $f(r)$, we will explore the linear structure of the problem and consider
$w(R) = w(R) -\xi v(R) + \xi/r^2$, where (\ref{vR}) was invoked, with $\xi = C_2/C_1$.
Hence, without loss of generality, one can consider instead the function
(\ref{wr}) its equivalent form
\begin{equation}
\label{wr1}
w(R) = {B'_2}{R^2}+ D'_2 + \frac{\xi}{r^2},
\end{equation}
with $B_2' = B_2 - \xi B_1 \ne 0 $ and $D'_2 = D_2 - \xi D_1 $,
leading finally to
\begin{equation}
\label{bertr2ff}
{\frac{1}{f}} = G\mp \frac{\rho r^2}{1-D_1r^2 \pm \sqrt{(1-D_1r^2)^2 - Kr^4}} + \frac{\xi}{r^2},
\end{equation}
where $G=D_2'$ and $\rho = 2B'_2C_1$. The extra $r^{-2}$ term in (\ref{wr1}) and (\ref{bertr2ff}) can be also understood
as a manifestation of the
well-known gauge invariance of the problem: if $U(R)$ is an isochrone potential, then
$\tilde U(R) = U(R) + \Delta + \Lambda v(R)$, for any constants $\Delta$ and $\Lambda$, will be also isochrone, but with the
parameters $E$ and
$\ell$ redefined accordingly.
The spacetime functions (\ref{bertr2hh}) and
(\ref{bertr2ff}) define our first class of isochrone spacetimes, for which we have
\begin{equation}
\beta_\ell^2 = 4C_1\left( 1 + \frac{\xi}{\ell^2} \right).
\end{equation}
The Newtonian limit for this family is obtained by setting $D_1 = K = 0$ and $C_1=1$, and a simple inspection of (\ref{bertr2ff}) reveals that it corresponds to the harmonic potential plus an extra $r^{-2}$ interaction term. The Bertrand spacetimes of the second type arise by considering the case $\xi = 0$
and a rational $\sqrt{C_1}$. The extra $r^{-2}$ term is associated with a spacetime singularity at $r=0$ as one can see by noticing that $f \sim r^{2}$ for $r\to 0$ and $\xi \ne 0$, and
that the scalar curvature of the metric (\ref{metric}) in this case reads
\begin{equation}
\mathcal{R} \sim \frac{2-3\left(rh' + 2h \right)}{r^2}
\end{equation}
for $r \to 0$, diverging for the two possibilities of (\ref{bertr2hh}). The presence of this singularity
for $\xi\ne 0$ is generic for all families of isochrone spacetimes discussed here.
Let us now inspect the particular cases with $K=0$. First, of course,
one cannot have both $B_1 = 0$ and $C_1 = 0$ simultaneously, since it would correspond to no centrifugal barrier at all, which is incompatible with (\ref{vR}).
For the case $B_1 = 0$ and $C_1\ne 0$, the expressions (\ref{bertr2hh}) and (\ref{bertr2ff}) are
still valid in the limit $B_1\to 0$, choosing the pertinent signs.
The case $B_1\ne 0$ and $C_1=0$ is also possible but
somehow distinct, since it does not have a Newtonian limit. It corresponds to a second family of isochrone spacetimes for which
\begin{equation}
\label{hh3}
h = \frac{(1-D_1r^2)^3}{B_1r^4}
\end{equation}
and
\begin{equation}
\label{ff3}
{\frac{1}{f}} = G + \frac{\rho r^2}{1-D_1r^2} + \frac{\xi}{r^2}.
\end{equation}
For this family, we have
\begin{equation}
\beta_\ell^2 = \frac{4\rho}{B_1\ell^2}.
\end{equation}
Since $\rho\ne 0$, otherwise $U(R)$ would not have a local minimum, there is no limit
of constant $\beta_\ell$ for these spacetimes and, consequently, they do not have a Bertrand limit.
The clear possibility of having $h(r_*)=0$, with $r_*>0$, for $D_1>0$ in (\ref{hh3}) (and also in (\ref{bertr2hh})) and its implication for the causal
structure of the underlying spacetime deserve some comments. We will address these points in the
last section.
Finally, notice the condition $B'_2\ne 0$ assumed
in (\ref{wr}) is necessary by a rather subtle reason. If $B'_2 = 0$, the effective potential has only
the centrifugal barrier term, and the requirement of a local minimum $U(R)$ will be equivalent
of a local minimum of $v(R)$, but due to (\ref{h(r)}), we will have $h(r_0)=0$. We will also return to this point
in the last section.
\subsection{Second case: linear $F(U)$ }
\label{seclinear}
The second simplest choice for our problem is, of course, the linear $F(U)$ case.
Since we can always add a constant to $U$, one can consider without loss of generality $F(U) = \alpha U$. We will proceed
along the same lines of the preceding case
and solve (\ref{exprf}). We get
\begin{equation}
\label{Ulin}
U(R) = \frac{B\sqrt{p + \epsilon R^2}}{R^2} + \frac{C}{R^2} +D,
\end{equation}
where $U_0 = -\epsilon \alpha^2B^2 - D$, with $\epsilon$ assuming two possible values $\epsilon = \pm 1$. The other parameters in the potential (\ref{Ulin}) are such that
\begin{eqnarray}
\label{Ulinp}
p&=& \frac{4A\alpha +1}{4\alpha^4B^2},\\
\label{UlinC}
C &=& \frac{2A\alpha +1}{2\alpha^2},
\end{eqnarray}
and it is clear that one requires $p>0$ for $\epsilon = -1$.
The signs of $B$ and $C$ must also be conveniently chosen to guarantee a local minimum for the
effective potential $U(R)$.
One can advance from (\ref{Ulin}) that
the flat space limit ($R=r$) of $U(R)$ will comprehend all Henon's isochrone potentials (\ref{he}),
(\ref{bo}) and
(\ref{ho}), but occasionally with some extra $r^{-2}$ terms. As in the preceding case, let us first consider
the linear combinations
\begin{equation}
\label{vr2}
v(R) = B_1 \frac{\sqrt{p + \epsilon R^2}}{R^2} + \frac{C_1}{R^2} + D_1,
\end{equation}
and
\begin{equation}
w(R) = B'_2 \frac{\sqrt{p + \epsilon R^2}}{R^2} + D'_2 + \frac{\xi}{r^2},
\end{equation}
with $C_1\ne 0$ and $B'_2\ne 0$.
Reintroducing the original spherical coordinates, we have
\begin{equation}
R^2 = \epsilon C_1\left( \left(\frac{ r +\mu \Phi(r) }{ \Xi(r) -\epsilon \nu r} \right)^2 -\mu\right),
\end{equation}
where $\mu = p/C_1$, $\nu = B_1/2\sqrt{C_1}$,
\begin{equation}
\Phi(r) = \epsilon\left( \frac{1}{r} - D_1 r\right),
\end{equation}
and
\begin{equation}
\Xi^2(r) = \epsilon +\kappa r^2 + \mu \Phi^2(r) ,
\end{equation}
with $
\kappa = \nu^2- \epsilon D_1,
$
leading to
\begin{equation}
\label{hlin}
h = \frac{ \epsilon C_1 r \left(r+ \mu \Phi(r) + 2\epsilon \nu \mu \left( \Xi(r) - \epsilon\nu r\right) \right) \Xi^2(r)}{ \left( r + \mu\Phi(r) \right)^2}
\end{equation}
and
\begin{equation}
\label{flin}
{\frac{1}{f}} = G + \frac{\epsilon\rho\left( r +\mu \Phi(r) \right)\left( \Xi(r) - \epsilon \nu r\right) }{\left( r +\mu \Phi(r) \right)^2 - \mu \left( \Xi(r) - \epsilon \nu r\right)^2} + \frac{\xi}{r^2},
\end{equation}
The general expression for the azimuthal angle of the metric with the functions (\ref{hlin}) and (\ref{flin}) is rather
complicated.
For $\mu\ne 0$, we have from (\ref{Ulinp})
and (\ref{UlinC})
\begin{equation}
\alpha^2 = \frac{1}{2C_1\zeta_\ell}\left(1-\sqrt{1-\zeta_\ell} \right),
\end{equation}
with
\begin{equation}
\zeta_\ell = \mu \left(2\nu + \frac{ \rho }{\ell^2 + \xi} \right)^2,
\end{equation}
leading to
\begin{equation}
\label{bb3}
\beta_\ell^2 = 2C_1\left( 1 + \frac{\xi}{\ell^2} \right) \left(\frac{1}{\zeta_\ell} - 1 \right)\left(
1-\sqrt{1-\zeta_\ell}
\right).
\end{equation}
The case $\mu=0$ follows straightforwardly from the $\mu\to 0$ limit of (\ref{bb3}).
The Newtonian limit of this family of isochrone spacetimes corresponds to
set $\kappa = \nu = 0$ (locally flat condition) and $C_1=0$ (no conic singularity at the origin), and
it corresponds to the classic Henon's potentials (\ref{he}), (\ref{bo}), and (\ref{ho}), with
an extra $r^{-2}$ interaction. Furthermore, the parameter $\xi$ can be properly chosen to reproduce (\ref{he}), (\ref{bo}), and (\ref{ho}) exactly, eliminating the singularity at the origin. In this sense,
this family of isochrone spacetimes can be considered the relativistic version of the Henon's original
potentials. The Bertrand limit corresponds to the case $\mu = \xi = 0$ and a rational $\sqrt{C_1}$, and
coincide with the Perlick first family defined by (\ref{bertr1f}) and (\ref{bertr1h}). The family defined by
the functions
(\ref{hlin}) and (\ref{flin}) has some asymptotically flat spacetimes, they correspond to the cases
with $\Xi$ constant for large $r$, which demands $\epsilon=1$ and $ \mu D_1^2 + \kappa = \mu D_1^2 - D_1 + \nu^2 = 0$, and in contrast with the Bertrand family defined by (\ref{bertr1f}) and (\ref{bertr1h}),
these asymptotically flat isochrone spacetimes can admit curved spatial sections.
As in the preceding case, we are left with the $C_1=0$ case, which does not have a Newtonian limit. We have
for this case
\begin{equation}
h = \frac{B_1r\left(\sqrt{B_1^2r^2+4p \Phi^2} -\epsilon B_1r \right)\left(B_1^2r^2+4p \Phi^2 \right)}{8p\Phi^2},
\end{equation}
\begin{equation}
{\frac{1}{f}} = G + \frac{\rho \left( \sqrt{B_1^2r^2+4p \Phi^2} -\epsilon B_1r\right)}{2p r} + \frac{\xi}{r^2},
\end{equation}
and
\begin{equation}
\beta_\ell^2 = \frac{C_2}{\ell^2} \left(\frac{1}{\bar \zeta_\ell} - 1 \right)\left(
1-\sqrt{1-\bar\zeta_\ell}
\right),
\end{equation}
with
$
\bar\zeta_\ell = p(\ell^2 +\xi)^2/\rho^2.
$
This family does not admit asymptotically flat spacetimes or Bertrand limit.
\subsection{Third case: quadratic $F(U)$}
Our last explicit case is also the next natural one: the quadratic $F(U)$. Again, thanks to the gauge invariance of the problem, we can
consider without loss of generality $F(U) = \alpha( U^2 + \gamma)$. Equation (\ref{exprf}) in this case
will be an irreducible fourth-order polynomial, but it admits the simple solution
\begin{equation}
U(R) = B\sqrt{\frac{1}{R} + p} + \frac{C}{R} + D,
\end{equation}
where $\alpha^2 = \frac{1}{4B^2C}$, $\gamma = -B^2p$, $U_0 = -\frac{B^2}{4C} -Cp -D$, and
\begin{equation}
A = \frac{C^\frac{3}{2}}{2B}.
\end{equation}
The corresponding spacetimes are not asymptotically flat and have no Newtonian limit. We have for
$C_1\ne 0$:
\begin{eqnarray}
h &=& \frac{C_1^2r^2 \Omega^2 }{4\left(B_1 + \Omega \right)^2} , \\
{\frac{1}{f}}& =& G + \rho \sqrt{ \Omega^2 + B_1\left(B_1 + 2\Omega \right) } + \frac{\xi}{r^2},
\end{eqnarray}
with
\begin{equation}
\Omega^2 = B_1^2 + 4C_1\left( C_1 p -D + \frac{1}{r^2}\right),
\end{equation}
and
\begin{equation}
\beta_\ell^2 = \frac{\sqrt{C_1}\left( \ell^2 + \xi \right)^{\frac{3}{2}}}{ \ell^2\left(2\rho+ \frac{B_1}{C_1}\left(\ell^2 + \xi\right) \right) }.
\end{equation}
For case with $C_1=0$, we have
\begin{eqnarray}
h &=& \left(\frac{B_1^2r^3 }{4\left(1-D_1r^2 \right)} \right)^2, \\
{\frac{1}{f}}& =& G + \frac{\rho\left( \left(1-D_1r^2 \right)^2 -B_1^2p \right)}{B_1^2r^4} + \frac{\xi}{r^2},
\end{eqnarray}
and
\begin{equation}
\beta_\ell^2 = \frac{\rho^\frac{3}{2}}{2B_1^2(\ell^2+\xi)}.
\end{equation}
These families do not admit asymptotically flat spacetimes and have no
Bertrand limit.
\section{Final Remarks}
We have introduced the notion of isochrone spacetimes and presented a procedure to generate
them by means of the algebraic equation (\ref{exprf}). We have obtained explicit
expressions for the families of spacetimes corresponding to the cases with constant, linear, and quadratic $F(U)$, but many others solutions are indeed possible.
We remind that we are looking for solutions of (\ref{exprf}) allowing for, at least, a three-dimensional
vector space that will
give origin to
the two functions $v(R)$ and $w(R)$ of the effective potential (\ref{Ueff}). With the help of Maple,
we could find explicit solutions for (\ref{exprf}) with the required properties for $F(U) = \alpha\sqrt{U}+\gamma$, $F(U) = \alpha U^{-1}+ \gamma$, and $F(U) = \frac{\alpha U}{U+\gamma}$, among others, but the resulting expressions are too cumbersome to be useful in our context. For cases with
$F(U)$ involving
any rational expressions with non-linear polynomials,
the problem of solving (\ref{exprf}) reduces to find the roots of higher order irreducible polynomials in $U$, and in some cases the solutions are indeed compatible with our requirements, despite
their typical intricate expressions. We do not expect any compatible solution for non-algebraic functions $F(U)$, but we could not prove that they indeed do not exist.
{Nevertheless, the constant and linear $F(U)$ cases are particularly relevant here because they have the Bertrand
spacetimes \cite{Perlick} as special limits, see Fig. \ref{fig2}.}
\begin{figure}
\tikzstyle{naveqs} = [ draw , text centered, text width=3.5cm, rounded corners]
\resizebox{.48\textwidth}{!}{
\begin{tikzpicture}
\matrix[row sep=4cm,column sep=1cm, ampersand replacement=\&] {%
\node (p1) [naveqs] {Isochrone spacetime $F(U)=\alpha$}; \& \& \node (p2) [naveqs] {Bertrand spacetime \\ of the second type }; \\
\& \node [naveqs](p3) {Harmonic potential}; \& \\
};
\draw (p1) edge [->,>=stealth,shorten <=2pt, shorten >=2pt,thick] node[above]{Closed orbits} (p2) ;
\draw (p1) edge [->,>=stealth,shorten <=2pt, shorten >=2pt,thick] node[sloped, above]{Newtonian limit} (p3) ;
\draw (p2) edge [->,>=stealth,shorten <=2pt, shorten >=2pt, thick] node[sloped, above]{Newtonian limit} (p3);
\end{tikzpicture}
}
\resizebox{.48\textwidth}{!}{
\begin{tikzpicture}
\matrix[row sep=4cm,column sep=2.8cm , ampersand replacement=\&] {%
\node (p1) [naveqs] {Isochrone spacetime $F(U)=\alpha U$}; \& \& \node (p2) [naveqs] {Bertrand spacetime \\ of the first type }; \\
\node [naveqs](p3) {Hen\'on potentials}; \& \& \node [naveqs](p4) {Newtonian potential}; \\
};
\draw (p1) edge [->,>=stealth,shorten <=2pt, shorten >=2pt,thick] node[above]{Closed orbits} (p2) ;
\draw (p1) edge [->,>=stealth,shorten <=2pt, shorten >=2pt,thick] node[sloped, above]{Newtonian limit} (p3) ;
\draw (p2) edge [->,>=stealth,shorten <=2pt, shorten >=2pt, thick] node[sloped, above]{Newtonian limit} (p4);
\draw (p3) edge [->,>=stealth,shorten <=2pt, shorten >=2pt,thick] node[above]{Closed orbits} (p4) ;
\end{tikzpicture}}
\caption{\label{fig2} {
Schematic representation of the physically most relevant isochrone spacetimes. Left: the constant $F(U)$
family of section \ref{secconst}. It has the Bertrand spacetimes of the second type
(\ref{bertr2f}) and (\ref{bertr2h}) as its limit with closed orbits, and both have the harmonic
potential as their Newtonian limit. Right: The linear $F(U)$
family of section
\ref{seclinear}. Its particular cases with closed orbits are the
Bertrand spacetimes of the first type
(\ref{bertr1f}) and (\ref{bertr1h}), and its respective Newtonian limit corresponds to the Hen\'on isochrone potentials (\ref{he}),
(\ref{bo}) and (\ref{ho}). The Newtonian potential arises as the Newtonian limit of the
Bertrand spacetimes of the first type and as the closed orbits case of the isochrone potentials.
}}
\end{figure}
{
Moreover, the linear case has as
Newtonian limit the Henon’s isochrone potentials (\ref{he}),
(\ref{bo}) and (\ref{ho}), and, hence, such family of isochrone spacetimes could be useful as a relativistic extension of the original Henon's proposal for the study of globular clusters. They
might also generalize some recent models for galactic dark matter \cite{astroB} based in the Bertrand spacetimes of the
first type. It is worth stressing that only the constant and linear $F(U)$ cases have isochrone
Newtonian limits and, hence, they are probably the most relevant isochrone spacetimes.}
The causal structure of isochrone spacetimes certainly deserves a deeper investigation, but some
issues are already clear. One can see from all explicit families of spacetime of the previous section that
one can have, for some choice of parameters, $h(r_*)=0$ with $r_*>0$. In such a case, from the metric (\ref{metric}), we see that $r=r_*$ is a null sphere and, hence, a candidate to be an event horizon provided no spacetime
singularity is present at
$r=r_*$. Incidentally, this is the reason why one cannot have $B_2'=0$ in the discussions of the last section, since in
such a case we would have $r_0 = r_*$, a circular null orbit, and our analysis is focused from the beginning only on
timelike geodesics.
From (\ref{h(r)}), we see that $r_*$ corresponds to a critical
point of $v(R)$, and from (\ref{metric1}) we see that no evident spacetime singularity is expected at the
critical points of $v(R)$, provided $w(R)$ be regular there. Hence, despite that $R$ defined in (\ref{change}) is a tortoise-like
coordinate, there should be a possibility of extending the isochrone spacetime across $r_*$, giving
origin to a black hole spacetime in the case of absence of singularities at $r =r_*$. The possibility of
having an isochrone black hole is certainly instigating from a theoretical point of view. As an illustration, let us consider the case with $C_1=0$ for a constant $F(U)$, which corresponds to the
functions (\ref{hh3}) and (\ref{ff3}), with the parameters $G=\xi=0$ and $\rho^\frac{1}{2} = B_1^\frac{1}{4} = D_1^\frac{1}{2} = r_*^{-1}$ for the sake of simplicity,
\begin{equation}
\label{new}
g_{ab}dx^adx^b = -\frac{r_*^2-r^2}{r^2}dt^2 + \frac{r_*^2r^4 }{(r_*^2- r^2)^3} dr^2 + r^2d\Omega^2.
\end{equation}
Both the scalar curvature and the Kretschmann invariant suggest that the null suface $r=r_*$ is
not singular, and hence the metric (\ref{new}) could be extended naturally for $r>r_*$. Of course, this metric
is not static in this exterior region, and hence our discussion on isochrone orbits does not apply there.
On the other hand, we can also see form the curvature invariants that there is a naked spacetime singularity in the interior region at $r=0$.
We finishing noticing that the isochrone spacetimes generically violates the weak energy condition, as one can see
from the Einstein tensor evaluated for the metric (\ref{metric}). We have for the temporal component
\begin{equation}
G_{00} = \frac{f(r)\left( 1 -\frac{d}{dr} rh(r) \right)}{r^2},
\end{equation}
and it is clear that one can have $G_{00}<0$ for sufficiently large $r$ for all families we have
considered in this paper. This raises a pertinent question about the physical interpretation of the isochrone spacetimes, since in the context of General Relativity they would require exotic matter. Nevertheless, in the context
of modified gravity, the energy conditions can be considerably different from those ones of General Relativity \cite{EC}, and perhaps some of these
theories could have isochrone spacetimes as physically viable solutions.
\section*{Acknowledgment}
AS acknowledges the financial support of
CNPq and FAPESP (Brazil) through the grants 302674/2018-7 and 21/09293-7, respectively,
and thanks Vitor Cardoso and Jos\'e S. Lemos for
the warm hospitality at the Center for Astrophysics and Gravitation of the
University of Lisbon, where this work was finished.
\section*{Data availability}
This paper has no associated data.
\section*{References}
|
2,869,038,154,900 | arxiv | \section{Introduction}
In their seminal paper \cite{DiPerna-Lions}, R. J. DiPerna and P.-L. Lions proved the existence and uniqueness of solutions to transport equations on $\mathbb{R}^d$.
We recall here a slightly simplified version of their statement.
\begin{theorem}[DiPerna-Lions]
Let $d \geq 1$ be an integer.
Let $1 \leq p \leq \infty$ and $p'$ its H\"older conjugate.
Let $a_0$ be in $L^p(\mathbb{R}^d)$.
Let $v$ be a fixed divergence free vector field in $L^1_{loc}(\mathbb{R}_+, \dot{W}^{1,p'}(\mathbb{R}^d))$.
Then there exists a unique distributional solution $a$ in $L^{\infty}(\mathbb{R}_+,L^p(\mathbb{R}^d))$ of the Cauchy problem
\begin{equation}
\left \{
\begin{array}{c c}
\partial_t a + \nabla \cdot (a v) = 0 \\
a (0) = a_0, \\
\end{array}
\right.
\end{equation}
with the initial condition understood in the sense of $\mathcal{C}^0(\mathbb{R}_+, \mathcal{D}'(\mathbb{R}^d))$.
We recall that $a$ is a distributional solution of the aforementioned Cauchy problem if and only if,
for any $\varphi$ belonging to $\mathcal{D}(\mathbb{R}_+ \times \mathbb{R}^d)$ and any $T > 0$, there holds
\begin{equation}
\int_0^T \int_{\mathbb{R}^d} a(t,x) \left(\partial_t \varphi(t,x) + v(t,x) \cdot \nabla \varphi(t,x) \right) dx dt
= \int_{\mathbb{R}^d} a(T,x) \varphi(T,x) dx - \int_{\mathbb{R}^d} a_0(x) \varphi(0,x) dx.
\end{equation}
\end{theorem}
Beyond this theorem, many authors have since proved similar existence and (non-)uniqueness theorems, see for instance
\cite{AmbrosioBV}, \cite{AmbrosioCrippa}, \cite{BouchutJames1}, \cite{BouchutJames2}, \cite{BouchutJames3}, \cite{Depauw}, \cite{LeBrisLions}, \cite{LeFlochXin},
\cite{Lerner} and references therein.
In particular, the papers \cite{BouchutJames1}, \cite{BouchutJames2} and \cite{BouchutJames3} use a duality method which is close in spirit to our results.
Our key result, which relies on the maximum principle for the \emph{adjoint} equation, is both more general and more restrictive than the DiPerna-Lions theorem.
The generality comes from the wider range of exponents allowed, along with the affordability of additional scaling-invariant and/or dissipative terms in the equation.
We thus extend the result from \cite{NoteAuCRAS}, where the setting was restricted to the $L^2_{t,x}$ case and no right-hand side was considered.
On the other hand, we do not fully extend the original theorem, since we are unable to prove the existence of solutions in the uniqueness classes.
Here is the statement.
\begin{theorem}
Let $d \geq 1$ be an integer.
Let $\nu \geq 0$ be a positive parameter.
Let $1 \leq p,q \leq \infty$ be real numbers with Hölder conjugates $p'$ and $q'$.
Let $v = v(t,x)$ be a fixed, divergence free vector field in $L^{p'}(\mathbb{R}_+, \dot{W}^{1,q'}(\mathbb{R}^d))$.
Given a time $T^* > 0$, let $a$ be in $L^p([0,T^*], L^q(\mathbb{R}^d))$.
Assume that $a$ is a distributional solution of the Cauchy problem
\begin{equation}
(C) \left \{
\begin{array}{c c}
\partial_t a + \nabla \cdot (a v) - \nu \Delta a = 0 \\
a (0) = 0, \\
\end{array}
\right.
\end{equation}
with the initial condition understood in the sense of $\mathcal{C}^0([0,T^*], \mathcal{D}'(\mathbb{R}^d))$.
That is, we assume that, for any function $\varphi$ in $\mathcal{D}(\mathbb{R}_+ \times \mathbb{R}^d)$ and any $T > 0$, there holds
\begin{equation}
\int_{\mathbb{R}_+ \times \mathbb{R}^d} a(t,x) \left(\partial_t \varphi(t,x) + v(t,x) \cdot \nabla \varphi(t,x) + \nu \Delta \varphi(t,x) \right) dx dt
= \int_{\mathbb{R}^d} u(T,x) \varphi(T,x) dx.
\end{equation}
Then $a$ is identically zero on $[0,T^*] \times \mathbb{R}^d$.
\label{Uniqueness}
\end{theorem}
Though one may fear that the lack of existence might render the theorem unapplicable in practice, it does not.
For instance, when working with the Navier-Stokes equations, the vorticity of a Leray solution only belongs, a priori, to
$$L^{\infty}(\mathbb{R}_+, \dot{H}^{-1}(\mathbb{R}^d)) \cap L^2(\mathbb{R}_+ \times \mathbb{R}^d).$$
In particular, the only Lebesgue-type space to which this vorticity belongs is $L^2(\mathbb{R}_+ \times \mathbb{R}^d)$.
Our theorem is well suited for solutions possessing \emph{a priori} no integrable derivative whatsoever.
As such, our theorem appears a regularization tool.
The philosophy is that, if an equation has smooth solutions, then any sufficiently integrable \emph{weak} solution is automatically smooth.
We illustrate our theorem with an application to the regularity result of J. Serrin \cite{Serrin} and subsequent authors
\cite{BeiraoDaVeiga}, \cite{CaffKohnNiren}, \cite{CheminZhang}, \cite{FabesJonesRiviere}, \cite{FabreLebeau}, \cite{Giga}, \cite{IskauSereginSverak}, \cite{Struwe}, \cite{vonWahl}.
We warn the reader that we did \emph{not} prove that the Leray solutions are unique in their class and will not claim so.
Indeed, the uniqueness stated in Theorem \ref{UniquenessSerrin} is purely linear.
In particular, it does not use the link between the vorticity and the exterior fields.
It does not rely either on the divergence freeness of the vorticity.
The key point in our proof is the maximum principle of the \emph{adjoint equation}.
The validity of the maximum principle partially depends on the vorticity equation having only differential operators rather than pseudodifferential ones.
Another standpoint on this theorem, which we owe to a private communication from N. Masmoudi, is that we now have two ways to recover the vorticity field $\Omega$ from the velocity.
We may either we use the defining identity
$$\Omega := \nabla \wedge u$$
or that $\Omega$ is the unique solution of the linear problem
$$
(NSV) \left \{
\begin{array}{c c}
\partial_t \Omega + \nabla \cdot (\Omega \otimes u) - \Delta \Omega = \nabla \cdot ( u \otimes \Omega)\\
\Omega (0) = \nabla \wedge u_0. \\
\end{array}
\right.
$$
The second choice makes a strong use of the peculiar algebra of the Navier-Stokes equations, while the first one is general and requires no other assumption on $u$ than the divergence-free condition.
Thus, we may hope to garner more information from the vorticity uniqueness, even though it may seem circuitous.
Embodied by Theorem \ref{ExistenceSerrin} is our new approach to the Serrin-type regularity results, relying on finer algebraic properties of the equation than
its belonging to the semilinear heat equations family.
\section{Results}
Let us comment a bit on the strategy we shall use.
First, because $a$ lies in a low-regularity class of distributions, energy-type estimates seem out of reach.
Thus, a duality argument is much more adapted to our situation.
Given the assumptions on $a$, which for instance imply that $\Delta a$ is in $L^p(\mathbb{R}_+, \dot{W}^{-2, q}(\mathbb{R}^d))$, we need to prove the following existence result.
\begin{theorem}
Let $\nu \geq 0$ be a positive real number.
Let $v = v(t,x)$ be a fixed, divergence free vector field in $L^{p'}(\mathbb{R}_+, \dot{W}^{1,q'}(\mathbb{R}^d))$.
Let $\varphi_0$ be a smooth, compactly supported function in $\mathbb{R}^d$.
There exists a function $\varphi$ in $L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d)$ solving
\begin{equation}
(C') \left \{
\begin{array}{c c}
\partial_t \varphi - \nabla \cdot ( \varphi v) - \nu \Delta \varphi = 0 \\
\varphi (0) = \varphi_0 \\
\end{array}
\right.
\end{equation}
in the sense of distributions and satisfying the estimate
$$\|\varphi(t)\|_{L^{\infty}(\mathbb{R}^d)} \leq \|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)}. $$
\label{Existence}
\end{theorem}
Picking some positive time $T > 0$ and considering $\varphi(T - \cdot)$ instead of $\varphi$,
Theorem \ref{Existence} amounts to build, for $T > 0$, a solution on $[0,T] \times \mathbb{R}^d$ of the Cauchy problem
\begin{equation}
(-C') \left \{
\begin{array}{c c}
- \partial_t \varphi - \nabla\cdot( \varphi v) - \nu\Delta \varphi = 0 \\
\varphi (T) = \varphi_0 . \\
\end{array}
\right.
\end{equation}
This theorem is a slight generalization of the analogue theorem in the Note \cite{NoteAuCRAS}.
The proof we provide here follows the same lines but retains only the key estimate, which is the boundedness of the solution.
The additional estimate in the Note was inessential and had the inconvenient to degenerate when the viscosity coefficient is small.
In contrast, the boundedness is unaffected by such changes.
The techniques used in the proof of Theorem \ref{Existence} are robust.
This robustness is encouraging for future work, as many generalizations are possible depending on the needs.
We will not try to list them all ; instead, we give some examples of possible adaptations to other contexts.
The most direct one is its analogue for diagonal systems, for uniqueness in this case reduces to applying the scalar case to each component of the solution.
Alternatively, one may add various linear, scaling invariant terms on the right hand side, or any dissipative term (such as a fractional laplacian) on the left hand side.
Also, in view of application to compressible fluid mechanics, the main theorems remain true without the divergence freeness of the transport field
provided that the negative part of its divergence belongs to $L^1(\mathbb{R}_+, L^{\infty}(\mathbb{R}^d))$.
This extension was already present in the original paper \cite{DiPerna-Lions} from R.J. DiPerna and P.-L. Lions.
Among these numerous variants, a particular one stands out.
It applies to a restricted family of equations, which are essentially the Navier-Stokes equations with frozen coefficients.
These equations are obtained from $(C)$ by adding a linear, non diagonal term on the right-hand side, of a peculiar form.
The purpose of this variant is to provide a different proof of the renowned Serrin theorem.
We now state it.
\begin{theorem}
Let $d \geq 3$ be an integer.
Let $\nu > 0$ be a positive real number.
Let $2 \leq p < \infty$ and $d < q \leq \infty$ be real numbers satisfying $\frac 2p + \frac dq = 1$.
Let $v$ be a fixed divergence free vector field in $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d))$.
Let $w$ be a fixed vector field in $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d)) \cap L^p(\mathbb{R}_+, L^q(\mathbb{R}^d))$.
Let $a$ be in $L^2(\mathbb{R}_+ \times \mathbb{R}^d)$.
Assume that $a$ is a distributional solution of the Cauchy problem
\begin{equation}
(C_{NS}) \left \{
\begin{array}{c c}
\partial_t a + \nabla \cdot (a \otimes v) - \nu \Delta a = \nabla \cdot (w \otimes a) \\
a (0) = 0, \\
\end{array}
\right.
\end{equation}
with the initial condition understood in the sense of $\mathcal{C}^0([0,T], \mathcal{D}'(\mathbb{R}^d))$.
Then $a$ is identically zero on $\mathbb{R}_+ \times \mathbb{R}^d$.
\label{UniquenessSerrin}
\end{theorem}
This time, the addition of a non diagonal -- though scaling invariant -- term induces some notable changes, because of two algebraic facts which we wish to emphasize.
The first one relates to the divergence freeness of the solution when dealing with the Navier-Stokes equations.
Indeed, we have in this case the equality
$$\nabla \cdot (w \otimes \Omega) = \Omega \cdot \nabla w $$
and both sides make sense as distributions.
However, since we forget the divergence freeness of $a$ when we compute the adjoint equation,
it is of utmost importance to write the equation with the right-hand side written in its divergence form $\nabla \cdot (w \otimes a)$.
This divergence form is the only one with which we are able to get a essential bound (or generalized maximum principle) for the adjoint equation, an absolutely crucial feature of our proof.
The second one stems from the vectorial nature of the solution $a$, which complexifies the integration by parts of the term $|a|^{r-2}a \Delta a$.
As it is well-known, adding a laplacian term in a partial differential equation has a smoothing effect on solutions.
However, when $r$ grows, the smoothing effect concentrates mostly on $|a|^2$ and not on the full solution $a$.
While this may look like a trivial observation to the accustomed reader, it is precisely what prevents us from removing the scale-invariant assumption that $w$ belongs to
$L^p(\mathbb{R}_+, L^q(\mathbb{R}^d))$.
If we were able to lift it -- which we believe we cannot, owing to the numerical results of J. Guillod and V. V. \v{S}ver\'ak in \cite{GuillodSverak} --,
then a linear uniqueness statement for Leray solutions would hold.
\begin{remark}
Theorem \ref{UniquenessSerrin} also holds in the limit case $(p,q) = (d,\infty)$, provided that $w$ satisfies the smallness condition
$$\|w\|_{L^{\infty}(\mathbb{R}_+, L^d(\mathbb{R}^d))} < \frac{2\nu}{C},$$
where $C$ is the Sobolev constant associated to the embedding $\dot{H}^1(\mathbb{R}^d) \hookrightarrow L^{\frac{2d}{d-2}}(\mathbb{R}^d)$.
\end{remark}
To prove Theorem \ref{UniquenessSerrin}, we will need, as for Theorem \ref{Uniqueness}, a dual existence result, which we state.
\begin{theorem}
Let $d \geq 3$ be an integer.
Let $\nu > 0$ be a positive real number.
Let $2 \leq p < \infty$ and $d < q \leq \infty$ be real numbers satisfying $\frac 2p + \frac dq = 1$.
Let $v$ be a fixed divergence free vector field in $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d))$.
Let $w$ be a fixed vector field in $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d)) \cap L^p(\mathbb{R}_+, L^q(\mathbb{R}^d))$.
There exists a solution $\varphi$ to the following Cauchy problem
\begin{equation}
(C'_{NS}) \left \{
\begin{array}{c c}
\partial_t \varphi - \nabla \cdot (\varphi \otimes v) - \nu \Delta \varphi = - ^t \nabla \varphi \cdot a \\
\varphi (0) = \varphi_0 \in \mathcal{D}(\mathbb{R}^d) \\
\end{array}
\right.
\end{equation}
satisfying in addition, for almost every $t > 0$,
\begin{equation}
\|\varphi(t)\|_{L^{\infty}(\mathbb{R}^d)} \leq \|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)} \exp\left[\frac{C^p}{p\nu^{p-2}} \int_0^t \|w(s)\|_{L^q(\mathbb{R}^d)}^p ds \right].
\label{GrowingMaximumPriciple}
\end{equation}
Above, $C$ denotes a constant depending only on the dimension $d$.
\label{ExistenceSerrin}
\end{theorem}
In the right hand side of the main equation, the quantity $ - ^t \nabla \varphi \cdot a$ is a shorthand for
$$ - \nabla (\varphi \cdot a) + ^t \nabla a \cdot \varphi $$
and this last expression makes sense in $L^2(\mathbb{R}_+, \dot{H}^{-1}_{loc}(\mathbb{R}^d)) + L^2(\mathbb{R}_+ \times \mathbb{R}^d)$ provided that $\varphi$ is bounded in space-time.
Using coordinates, the different terms expand respectively as
$$(^t \nabla \varphi \cdot a)_i = \sum_{j=1}^d \partial_i \varphi_j a_j ; $$
$$(\nabla (\varphi \cdot a))_i = \sum_{j=1}^d \partial_i(\varphi_j a_j) ; $$
$$(^t \nabla a \cdot \varphi)_i = \sum_{j=1}^d \partial_i a_j \varphi_j.$$
Although the left-hand sides of $(C_{NS})$ and its adjoint equation $(C'_{NS})$ are almost identical, their right-hand sides are different.
This discrepancy has striking consequences on their global behaviour, in that $(C'_{NS})$ does possess a generalized maximum principle, while $(C_{NS})$ does not.
That fact is the core of our paper, without which no conclusion on the Navier-Stokes and Euler equations could have been drawn.
Conversely, we are able to prove a uniqueness result for $(C_{NS})$ while we do not expect any analogous result for $(C'_{NS})$, at least at the present time.
As a consequence of Theorem \ref{UniquenessSerrin}, we give an alternative proof of the Serrin theorem in most cases.
This new proof has the advantage of making a stronger use of the algebra of the Navier-Stokes equations than the previous one.
To avoid technical details which would only obscure the proof, we choose to present it in the case of the three dimensional torus.
An analogue exists when the regularity assumption is written on the whole space $\mathbb{R}^3$, or a subdomain thereof, with a similar proof and some minor adjustments.
We recall the theorem of J. Serrin in its improved form by Y. Giga in \cite{Giga}, written with integrability assumptions on the Leray solution.
\begin{theorem}[J. Serrin]
Let $u = u(t,x)$ be a Leray solution of the Navier-Stokes equations
$$
(NS) \left \{
\begin{array}{c c}
\partial_t u + \nabla \cdot (u \otimes u) - \Delta u = - \nabla p \\
\text{div }u = 0 \\
u (0) = u_0 \in L^2(\mathbb{T}^3) \\
\end{array}
\right.
$$
on $\mathbb{R}_+ \times \mathbb{T}^3$.
Assume the existence of times $T_2 > T_1 > 0$ and exponents $2 \leq p < \infty, 3 < q \leq \infty$ such that $u$ belongs to $L^p(]T_1,T_2[, L^q(\mathbb{T}^3))$.
Then $u$ belongs to $\mathcal{C}^{\infty}(]T_1,T_2[ \times \mathbb{T}^3)$.
\label{NouvellePreuveSerrin}
\end{theorem}
Besides reproving in a novel way the results of J. Serrin and his continuators, an immediate corollary of Theorem \ref{ExistenceSerrin} is the following.
\begin{theorem}
Let $d \geq 3$ be an integer.
Let $\nu$ and $T$ be strictly positive real numbers.
Let $u$ be a strong solution of the Navier-Stokes equations
$$
\left \{
\begin{array}{c c}
\partial_t u + \nabla \cdot (u \otimes u) - \nu \Delta u = - \nabla p \\
\text{div }u = 0 \\
\end{array}
\right.
$$
on $]0,T[ \times \mathbb{R}^d$.
Then, there exists a constant $C$ depending only on $d$ such that for any $0 < t < T$ and any $2 \leq p < \infty$, $d < q \leq \infty$ satisfying $\frac 2p + \frac dq = 1$, there holds
\begin{equation}
\|\Omega(t)\|_{L^1(\mathbb{R}^d)} \leq \|\Omega(0)\|_{L^1(\mathbb{R}^d)} \exp\left[\frac{C^p}{p\nu^{p-2}} \int_0^t \|u(s)\|_{L^q(\mathbb{R}^d)}^p ds \right].
\end{equation}
\label{BorneL1VorticiteForte}
\end{theorem}
Finally, applying Theorem \ref{Uniqueness} to the 2D Euler equations on the torus, one gets the following statement.
\begin{theorem}
Let $p \geq 2$ be a real number.
Let $u$ be a weak solution of the Euler equations starting from zero initial data
$$
\left \{
\begin{array}{c c}
\partial_t u + \nabla \cdot (u \otimes u) = - \nabla p \\
\text{div }u = 0 \\
u(0) = 0
\end{array}
\right.
$$
and assume that $\omega := \text{curl }u$ belongs to $L^{\infty}(\mathbb{R}_+, L^p(\mathbb{T}^2))$.
Then $u$ is identically zero on $\mathbb{R}_+ \times \mathbb{T}^2$.
\label{UniquenessZeroEuler}
\end{theorem}
\section{Proofs}
We state here a commutator lemma, similar to Lemma II.1 in \cite{DiPerna-Lions}, which we will use in the proof of Theorem \ref{Uniqueness}.
\begin{lemma}
Let $T > 0$.
Let $v$ be a fixed, divergence free vector field in $L^{p'}(\mathbb{R}_+, \dot{W}^{1,q'}(\mathbb{R}^d))$.
Let $a$ be a fixed function in $L^p(\mathbb{R}_+, L^q (\mathbb{R}^d))$.
Let $\rho = \rho(x)$ be some smooth, positive and compactly supported function on $\mathbb{R}^d$.
Normalize $\rho$ to have unit norm in $L^1(\mathbb{R}^d)$ and define $\rho_{\varepsilon} := \varepsilon^{-d} \rho\left(\frac{\cdot}{\varepsilon}\right)$.
Define the commutator $C^{\varepsilon}$ by
$$
C^{\varepsilon}(t,x) := v(t,x) \cdot (\nabla \rho_{\varepsilon} \ast a(t))(x) - (\nabla \rho_{\varepsilon} \ast (v(t) a(t)))(x) .
$$
Then, as $\varepsilon \to 0$,
$$\|C^{\varepsilon}\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \to 0. $$
\label{LemmeCommutateur}
\end{lemma}
\begin{proof}
For almost all $(t,x)$ in $\mathbb{R}_+ \times \mathbb{R}^d$, we have
$$C^{\varepsilon}(t,x) = \int_{\mathbb{R}^d} \frac{1}{\varepsilon^d}a(t,y) \frac{v(t,x)-v(t,y)}{\varepsilon} \cdot \nabla \rho\left(\frac{x-y}{\varepsilon}\right) dy.$$
Performing the change of variable $y = x + \varepsilon z$ yields
$$C^{\varepsilon}(t,x) = \int_{\mathbb{R}^d} a(t,x+\varepsilon z) \frac{v(t,x)-v(t,x+\varepsilon z)}{\varepsilon} \cdot \nabla \rho(z) dz.$$
Using the Taylor formula
$$v(\cdot,x+\varepsilon z) - v(\cdot,x) = \int_0^1 \nabla v(\cdot,x+r\varepsilon z) \cdot (\varepsilon z) dr, $$
which is true for smooth functions and extends to $\dot{W}^{1,q'}(\mathbb{R}^d)$ thanks to the continuity of both sides on this space
and owing to Fubini's theorem to exchange integrals, we get the nicer formula
$$C^{\varepsilon}(t,x) = - \int_0^1 \int_{\mathbb{R}^d} a(t,x+\varepsilon z) \nabla v(t,x+r\varepsilon z) : ( \nabla \rho(z) \otimes z) dz dr,$$
where $:$ denotes the contraction of rank two tensors.
Because $q$ and $q'$ are dual H\"older exponents, at least one of them is finite.
We assume for instance that $q < \infty$, the case $q' < \infty$ being completely similar.
Let
$$\widetilde{C}^{\varepsilon}(t,x) := - \int_0^1 \int_{\mathbb{R}^d} a(t,x+r\varepsilon z) \nabla v(t,x+r\varepsilon z) : ( \nabla \rho(z) \otimes z) dz dr. $$
We claim that, as $\varepsilon \to 0$,
$$\|C^{\varepsilon}-\widetilde{C}^{\varepsilon}\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \to 0.$$
Integrating both in space and time and owing to H\"older's inequality, we have
\begin{multline*}
\|C^{\varepsilon}-\widetilde{C}^{\varepsilon}\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \leq \\
\int_0^1 \int_{\mathbb{R}^d} \int_0^{\infty}
\|a(t,\cdot+\varepsilon z)-a(t,\cdot+r\varepsilon z)\|_{L^q(\mathbb{R}^d)} \|\nabla v(t)\|_{L^{q'}(\mathbb{R}^d)} |\nabla \rho(z) \otimes z| dt dz dr.
\end{multline*}
Since $a \in L^p(\mathbb{R}_+, L^q(\mathbb{R}^d))$ and $q < \infty$, for almost any $t \in \mathbb{R}_+$, for all $z \in \mathbb{R}^d$ and $r \in [0,1]$,
$$\|a(t,\cdot+\varepsilon z)-a(t,\cdot+r\varepsilon z)\|_{L^q(\mathbb{R}^d)} \to 0 $$
as $\varepsilon \to 0$.
Thanks to the uniform bound
\begin{multline*}
\|a(t,\cdot+\varepsilon z)-a(t,\cdot+r\varepsilon z)\|_{L^q(\mathbb{R}^d)}\|\nabla v(t)\|_{L^{q'}(\mathbb{R}^d)} |\nabla \rho(z) \otimes z| \leq \\
2 \|a(t)\|_{L^q(\mathbb{R}^d)}\|\nabla v(t)\|_{L^{q'}(\mathbb{R}^d)} |\nabla \rho(z) \otimes z|,
\end{multline*}
we may invoke the dominated convergence theorem to get the desired claim.
From this point on, we denote by $U(t,x)$ the quantity $a(t,x) \nabla v(t,x)$.
We notice that $U$ is a fixed function in $L^1(\mathbb{R}_+ \times \mathbb{R}^d)$ and that, by definition,
$$\widetilde{C}^{\varepsilon}(t,x) = - \int_0^1 \int_{\mathbb{R}^d} U(t,x+r\varepsilon z) : ( \nabla \rho(z) \otimes z) dz dr.$$
The normalization on $\rho$ yields the identity
$$- \int_{\mathbb{R}^d} \nabla \rho(z)\otimes z dz = \left(\int_{\mathbb{R}^d} \rho(z) dz\right) I_d = I_d, $$
where $I_d$ is the $d-$dimensional identity matrix.
This identity in turn entails that
$$\widetilde{C}^0(t,x) = a(t,x) \nabla v(t,x) : I_d = a(t,x) \text{ div }v(t,x) = 0. $$
A second application of the dominated convergence theorem to the function $U$ gives
$$\|\widetilde{C}^{\varepsilon} - \widetilde{C}^0\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \to 0 $$
as $\varepsilon \to 0$, from which the lemma follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Existence}]
Let us choose some mollifying kernel $\rho = \rho(x)$ and denote $v_{\delta} := \rho_{\delta} \ast v$, where $\rho_{\delta}(x) := \delta^{-d} \rho(\frac{x}{\delta})$.
Let $(C'_{\delta})$ be the Cauchy problem $(C')$ where we replaced $v$ by $v_{\delta}$.
The existence of a (smooth) solution $\varphi^{\delta}$ to $(C'_{\delta})$ is then easily obtained thanks to, for instance, a Friedrichs method combined with heat kernel estimates.
We now turn to the $L^{\infty}$ bound uniform in $\delta$.
Let $r \geq 2$ be a real number.
Multiplying the equation on $\varphi^{\delta}$ by $\varphi^{\delta} |\varphi^{\delta}|^{r-2}$ and integrating in space and time, we get
$$
\frac 1r \|\varphi^{\delta}(t)\|_{L^r(\mathbb{R}^d)}^r + (r-1) \int_0^t \| \nabla \varphi^{\delta}(s) |\varphi^{\delta}(s)|^{\frac{r-2}{2}} \|_{L^2(\mathbb{R}^d)}^2 ds
= \frac 1r \|\varphi_0\|_{L^r(\mathbb{R}^d)}^r.
$$
Discarding the gradient term, taking $r$-th root in both sides and letting $r$ go to infinity gives
\begin{equation}
\|\varphi^{\delta}(t)\|_{L^{\infty}(\mathbb{R}^d)} \leq \|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)}.
\label{BorneInfinie}
\end{equation}
Thus, the family $(\varphi^{\delta})_{\delta}$ is bounded in $L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d)$.
Up to an extraction, $(\varphi^{\delta})_{\delta}$ converges weak$-\ast$ in $L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d)$ to some function $\varphi$.
As a consequence, because $v_{\delta} \to v$ strongly in $L^1_{loc}(\mathbb{R}_+ \times \mathbb{R}^d)$ as $\delta \to 0$, the following convergences hold :
$$\Delta \varphi^{\delta} \rightharpoonup^{\ast} \Delta \varphi \text{ in } L^{\infty}(\mathbb{R}_+, \dot{W}^{-2,\infty} (\mathbb{R}^d)) ;$$
$$ \varphi^{\delta}v^{\delta} \rightharpoonup \varphi v \text{ in } L^1_{loc}(\mathbb{R}_+\times \mathbb{R}^d). $$
In particular, such a $\varphi$ is a distributional solution of $(C')$ with the desired regularity.
\end{proof}
We are now in position to prove the main theorem of this paper.
\begin{proof}[Proof of Theorem \ref{Uniqueness}]
Let $\rho = \rho(x)$ be a radial mollifying kernel and define $\rho_{\varepsilon}(x) := \varepsilon^{-d} \rho(\frac{x}{\varepsilon})$.
Convolving the equation on $a$ by $\rho_{\varepsilon}$ gives, denoting $a_{\varepsilon} := \rho_{\varepsilon} \ast a$,
$$
(C_{\varepsilon}) \ \ \partial_t a_{\varepsilon} + \nabla \cdot (a_{\varepsilon} v) - \nu\Delta a_{\varepsilon}
= C^{\varepsilon},
$$
where the commutator $C^{\varepsilon}$ has been defined in Lemma \ref{LemmeCommutateur}.
Notice that even without any smoothing in time, $a_{\varepsilon}$, $\partial_t a_{\varepsilon}$ lie respectively in $L^{\infty}(\mathbb{R}_+, \mathcal{C}^{\infty}(\mathbb{R}^d))$ and
$L^1(\mathbb{R}_+, \mathcal{C}^{\infty}(\mathbb{R}^d))$, which is enough to make the upcoming computations rigorous.
In what follows, we let $\varphi^{\delta}$ be a solution of the Cauchy problem $(-C'_{\delta})$, where $(-C'_{\delta})$ is $(-C')$ (defined in Theorem \ref{ExistenceSerrin}) with $v$ replaced by $v_{\delta}$.
Let us now multiply, for $\delta, \varepsilon > 0$ the equation $(C_{\varepsilon})$ by $\varphi^{\delta}$ and integrate in space and time.
After integrating by parts (which is justified by the high regularity of the terms we have written), we get
$$
\int_0^T \int_{\mathbb{R}^d} \partial_t a_{\varepsilon}(s,x) \varphi^{\delta}(s,x) dx ds =
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} -
\int_0^T \int_{\mathbb{R}^d} a_{\varepsilon}(s,x) \partial_t \varphi^{\delta}(s,x) dx ds.
$$
From this identity, it follows that
\begin{multline*}
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)}
= \int_0^T \int_{\mathbb{R}^d} \varphi^{\delta}(s,x) C^{\varepsilon}(s,x) dx ds \\ -
\int_0^T \int_{\mathbb{R}^d} a_{\varepsilon}(s,x) \left(- \partial_t \varphi^{\delta}(s,x) - \nabla \cdot (v(s,x)\varphi^{\delta}(s,x)) - \nu\Delta \varphi^{\delta}(s,x) \right) dx ds.
\end{multline*}
From Lemma \ref{LemmeCommutateur}, we know in particular that $C^{\varepsilon}$ belongs to $L^1(\mathbb{R}_+ \times \mathbb{R}^d)$ for each fixed $\varepsilon > 0$.
Thus, in the limit $\delta \to 0$, we have, for each $\varepsilon > 0$,
$$\int_0^T \int_{\mathbb{R}^d} \varphi^{\delta}(s,x) C^{\varepsilon}(s,x) dx ds \to \int_0^T \int_{\mathbb{R}^d} \varphi(s,x) C^{\varepsilon}(s,x) dx ds. $$
On the other hand, the definition of $\varphi^{\delta}$ gives
$$- \partial_t \varphi^{\delta} - \nabla \cdot (v \varphi^{\delta}) - \nu\Delta \varphi^{\delta} = \nabla \cdot ((v_{\delta} - v) \varphi^{\delta}). $$
Thus, the last integral in the above equation may be rewritten, integrating by parts,
$$- \int_0^T \int_{\mathbb{R}^d} \varphi^{\delta} (v_{\delta} - v)\cdot \nabla a_{\varepsilon}(s,x) dx ds. $$
For each fixed $\varepsilon$, the assumption on $a$ entails that $\nabla a_{\varepsilon}$ belongs to $L^p(\mathbb{R}_+,L^q(\mathbb{R}^d))$.
Furthermore, it is an easy exercise to show that
$$\|v_{\delta} - v\|_{L^{p'}(\mathbb{R}_+,L^{q'}(\mathbb{R}^d))} \leq
\delta \|\nabla v\|_{L^{p'}(\mathbb{R}_+,L^{q'}(\mathbb{R}^d))} \||\cdot|\rho\|_{L^1(\mathbb{R}^d)}. $$
Now, taking the limit $\delta \to 0$ while keeping $\varepsilon > 0$ fixed, we have
$$
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} = \int_0^T \int_{\mathbb{R}^d} \varphi(s,x) C^{\varepsilon}(s,x) dx ds .
$$
Taking the limit $\varepsilon \to 0$ and using Lemma \ref{LemmeCommutateur}, we finally obtain
$$
\langle a(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} = 0.
$$
This being true for any test function $\varphi_0$, $a(T)$ is the zero distribution and finally $a \equiv 0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{ExistenceSerrin}]
The proof of this Theorem is very similar to that of Theorem 3.1 in \cite{Struwe}.
We nevertheless reproduce it in our cse for the sake of completeness.
For simplicity, we reduce to the case $\nu = 1$.
Let $\rho = \rho(x)$ be a radial mollifying kernel and let us denote $\rho_{\delta}(x) = \delta^{-d}\rho(\frac{x}{\delta})$.
Let $w_{\delta} = \rho_{\delta} \ast w$ and $v_{\delta} = \rho_{\delta} \ast v$.
Let $(C'_{\delta})$ be the Cauchy problem $(C')$ with $w,v$ replaced by $w_{\delta}, v_{\delta}$ respectively.
The existence of a smooth solution $\varphi^{\delta}$ to the Cauchy problem $(C'_{\delta})$ is easy and thus omitted.
We focus on the the relevant estimates.
Let $r \geq 2$ be a real number.
We first take the scalr product of the equation on $\varphi^{\delta}$ by $\varphi^{\delta}$ and carefully rearrange the laplacian term to get the following equation on $|\varphi^{\delta}|^2$
\begin{equation}
\frac 12 \partial|\varphi^{\delta}|^2 + \frac 12 v_{\delta} \cdot \nabla |\varphi^{\delta}|^2 - \frac{\nu}{2} \Delta |\varphi^{\delta}|^2 + \nu|\nabla \varphi^{\delta}|^2
= \varphi^{\delta} \cdot (^t \nabla \varphi^{\delta} \cdot w_{\delta}).
\end{equation}
For notational convenience, we let $\psi^{(\delta)} := |\varphi^{\delta}|^2$ in the sequel.
Now, multiplying this new equation by $|\varphi^{\delta}|^{r-2}$ and integrating in space and time, we get
\begin{multline*}
\frac 1r \|\varphi^{\delta}(t)\|_{L^r(\mathbb{R}^d)}^r + \frac{r-2}{4}\nu\int_0^t \| (\psi^{(\delta)})^{\frac{r-4}{4}} \nabla \psi^{(\delta)}\|_{L^2(\mathbb{R}^d)}^2 ds + \nu \int_0^t \| \nabla \varphi^{\delta}(s) |\varphi^{\delta}(s)|^{\frac{r-2}{2}} \|_{L^2(\mathbb{R}^d)}^2 ds \\
= \frac 1r \|\varphi_0\|_{L^r(\mathbb{R}^d)}^r - \int_0^t \int_{\mathbb{R}^d}|\varphi^{\delta}|^{r-2} \varphi^{\delta} \cdot (^t \nabla \varphi^{\delta} \cdot w_{\delta}) dx ds .
\end{multline*}
Denote by $I(t)$ the integral on the right hand side.
Rewriting
$$I(t) = \int_0^t \int_{\mathbb{R}^d}(\psi^{(\delta)})^{\frac r4} |\varphi^{\delta}|^{\frac{r-4}{2}}\varphi^{\delta} \cdot (^t \nabla \varphi^{\delta} \cdot w_{\delta}) dx ds, $$
the H\"older inequality yields
$$
|I(t)| \leq \int_0^t \|\nabla \varphi^{\delta}(s) |\varphi^{\delta}(s)|^{\frac{r-2}{2}}\|_{L^2(\mathbb{R}^d)} \|(\psi^{(\delta)})^{\frac r4}(s)\|_{L^{\widetilde{q}}(\mathbb{R}^d)}
\|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)} ds,
$$
where $\widetilde{q}$ is defined by $\frac 12 + \frac 1q + \frac{1}{\widetilde{q}} = 1$.
By the Sobolev embedding $\dot{H}^{1-\frac 2p}(\mathbb{R}^d) \hookrightarrow L^{\widetilde{q}}(\mathbb{R}^d)$, there exists a constant $C = C(p,d)$ such that
$$\|(\psi^{(\delta)})^{\frac r4}(s)\|_{L^{\widetilde{q}}(\mathbb{R}^d)} \leq C \|(\psi^{(\delta)})^{\frac r4}(s)\|_{\dot{H}^{1-\frac 2p}(\mathbb{R}^d)}. $$
Since $d \geq 3$ and $0 \leq 1- \frac 2p < 1$, we may choose $C$ uniformly in $p$ for fixed $d$.
Interpolating $\dot{H}^{1-\frac 2p}$ between $L^2$ and $\dot{H}^1$ gives
\begin{multline*}
\|(\psi^{(\delta)})^{\frac r4}(s)\|_{\dot{H}^{1-\frac 2p}(\mathbb{R}^d)} \leq
\|(\psi^{(\delta)})^{\frac r4}(s)\|_{L^2(\mathbb{R}^d)}^{\frac 2p} \|\nabla (\psi^{(\delta)})^{\frac r4}(s)\|_{L^2(\mathbb{R}^d)}^{1-\frac 2p} \\
= \|\varphi^{\delta}(s)\|_{L^r(\mathbb{R}^d)}^{\frac rp} \|\nabla (\psi^{(\delta)})^{\frac r4}(s)\|_{L^2(\mathbb{R}^d)}^{1-\frac 2p}.
\end{multline*}
As $\nabla (\psi^{(\delta)})^{\frac r4} = \frac r4 (\psi^{(\delta)})^{\frac{r-4}{4}}\nabla \psi^{(\delta)},$ we may now bound $|I(t)|$ from above by
$$
C \int_0^t \|\nabla \varphi^{\delta}(s) |\varphi^{\delta}(s)|^{\frac{r-2}{2}}\|_{L^2(\mathbb{R}^d)}
\left(\frac r4 \|(\psi^{(\delta)})^{\frac{r-4}{4}}(s)\nabla \psi^{(\delta)}(s)\|_{L^2(\mathbb{R}^d)}\right)^{1-\frac 2p}
\|\varphi^{\delta}(s)\|_{L^r(\mathbb{R}^d)}^{\frac rp}
\|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)} ds.
$$
The Young inequality for real numbers yields, with $\widetilde{p}$ defined in the same way as $\widetilde{q}$,
\begin{multline*}
|I(t)| \leq \frac{\nu}{2} \int_0^t \|\nabla \varphi^{\delta}(s) |\varphi^{\delta}(s)|^{\frac{r-2}{2}}\|_{L^2(\mathbb{R}^d)}^2 ds
+ \frac{r\nu }{4\widetilde{p}} \int_0^t
\|(\psi^{(\delta)})^{\frac{r-4}{4}}(s)\nabla \psi^{(\delta)}(s)\|_{L^2(\mathbb{R}^d)}^2 ds \\
+ \frac{C^p}{p\nu^{p-2}}\int_0^t \|\varphi^{\delta}(s)\|_{L^r(\mathbb{R}^d)}^r \|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)}^p ds.
\end{multline*}
Absorbing the first two terms in the left-hand side of the inequality gives
$$\frac 1r \|\varphi^{\delta}(t)\|_{L^r(\mathbb{R}^d)}^r
\leq \frac 1r \|\varphi_0\|_{L^r(\mathbb{R}^d)}^r + \frac{C^p}{p\nu^{p-2}}\int_0^t \|\varphi^{\delta}(s)\|_{L^r(\mathbb{R}^d)}^r \|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)}^p ds. $$
By Gr\"onwall's inequality,
$$\|\varphi^{\delta}(t)\|_{L^r(\mathbb{R}^d)}
\leq \|\varphi_0\|_{L^r(\mathbb{R}^d)} \exp\left(\frac{C^p}{p\nu^{p-2}}\int_0^t \|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)}^p ds \right). $$
Letting $r$ go to infinity and using the trivial bound $\|w_{\delta}(s)\|_{L^q(\mathbb{R}^d)} \leq \|w(s)\|_{L^q(\mathbb{R}^d)}$ yields
\begin{equation}
\|\varphi^{\delta}(t)\|_{L^{\infty}(\mathbb{R}^d)}
\leq \|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)} \exp\left(\frac{C^p}{p\nu^{p-2}}\int_0^t \|w(s)\|_{L^q(\mathbb{R}^d)}^p ds \right).
\end{equation}
It only remains to take the limit $\delta\to 0$.
As the family $(\varphi^{\delta})_{\delta}$ is a bounded subset in $L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d)$, up to an extraction, there exists $\varphi$ in
$L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d)$ such that
$$\varphi^{\delta} \rightharpoonup^{\ast} \varphi \qquad \text{in } L^{\infty}(\mathbb{R}_+ \times \mathbb{R}^d) \text{ as } \delta \to 0. $$
By Fatou's lemma, the bound
\begin{equation}
\|\varphi(t)\|_{L^{\infty}(\mathbb{R}^d)}
\leq \|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)} \exp\left(\frac{C^p}{p\nu^{p-2}}\int_0^t \|w(s)\|_{L^q(\mathbb{R}^d)}^p ds \right)
\end{equation}
follows.
On the other hand, since $v$ and $w$ belong to $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d))$, it is clear that
$$v_{\delta}, w_{\delta} \longrightarrow v, w \qquad \text{strongly in } L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{R}^d)) \text{ as } \delta \to 0. $$
Hence, taking the limit $\delta \to 0$ in the equation on $\varphi^{\delta}$, we see that $\varphi$ indeed satisfies the adjoint equation and the proof is over.
\end{proof}
We now turn to the proof of the uniqueness theorem.
\begin{proof}[Proof of Theorem \ref{UniquenessSerrin}]
Let $\rho = \rho(x)$ be a radial mollifying kernel and define $\rho_{\varepsilon}(x) := \varepsilon^{-d} \rho(\frac{x}{\varepsilon})$.
Convolving the equation on $a$ by $\rho_{\varepsilon}$ gives, denoting $a_{\varepsilon} := \rho_{\varepsilon} \ast a$,
$$
(C_{\varepsilon}) \ \ \partial_t a_{\varepsilon} + \nabla \cdot (a_{\varepsilon} \otimes v) - \nu\Delta a_{\varepsilon}
= \nabla \cdot (w \otimes a_{\varepsilon}) + C^{\varepsilon} + D^{\varepsilon},
$$
where the commutator $C^{\varepsilon}$ has been defined in Lemma \ref{LemmeCommutateur}.
The second commutator is defined by
$$D^{\varepsilon} := \rho_{\varepsilon} \ast \nabla \cdot (a \otimes w) - \nabla \cdot (w \otimes a_{\varepsilon}).$$
Similarly to what we proved for $C^{\varepsilon}$, we have
$$\|D^{\varepsilon}\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \to 0 \text{ as $\varepsilon \to 0$.} $$
Notice that even without any smoothing in time, $a_{\varepsilon}$, $\partial_t a_{\varepsilon}$ lie respectively in $L^{\infty}(\mathbb{R}_+, \mathcal{C}^{\infty}(\mathbb{R}^d))$ and
$L^1(\mathbb{R}_+, \mathcal{C}^{\infty}(\mathbb{R}^d))$, which is enough to make the upcoming computations rigorous.
In what follows, we let $\varphi^{\delta}$ be a solution of the Cauchy problem $(-C'_{\delta})$, with $(-C'_{\delta})$ being $(-C')$ where $v$ and $a$ are replaced by $v_{\delta}$ and $a_{\delta}$.
Let us now multiply, for $\delta, \varepsilon > 0$ the equation $(C_{\varepsilon})$ by $\varphi^{\delta}$ and integrate in space and time.
After integrating by parts (which is justified by the high regularity of the terms we have written), we get
$$
\int_0^T \int_{\mathbb{R}^d} \partial_t a_{\varepsilon}(s,x) \varphi^{\delta}(s,x) dx ds =
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} -
\int_0^T \int_{\mathbb{R}^d} a_{\varepsilon}(s,x) \partial_t \varphi^{\delta}(s,x) dx ds.
$$
From this identity, it follows that
\begin{multline*}
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)}
= \int_0^T \int_{\mathbb{R}^d} \varphi^{\delta}(s,x) ( C^{\varepsilon} + D^{\varepsilon})(s,x) dx ds \: +\\
\int_0^T \int_{\mathbb{R}^d} a_{\varepsilon}(s,x)
\left(\partial_t \varphi^{\delta}(s,x)+ \nabla \cdot (v(s,x)\varphi^{\delta}(s,x))+ \nu\Delta\varphi^{\delta}(s,x) - ^t \nabla \varphi^{\delta}(s,x) \cdot w(s,x) \right) dx ds.
\end{multline*}
From Lemma \ref{LemmeCommutateur}, we know in particular that $C^{\varepsilon}$ belongs to $L^1(\mathbb{R}_+ \times \mathbb{R}^d)$ for each fixed $\varepsilon > 0$ and the same goes for $D^{\varepsilon}$.
Thus, in the limit $\delta \to 0$, we have, for each $\varepsilon > 0$,
$$\int_0^T \int_{\mathbb{R}^d} \varphi^{\delta}(s,x) (C^{\varepsilon} + D^{\varepsilon})(s,x) dx ds \to \int_0^T \int_{\mathbb{R}^d} \varphi(s,x) (C^{\varepsilon} + D^{\varepsilon})(s,x) dx ds. $$
On the other hand, the definition of $\varphi^{\delta}$ gives
$$- \partial_t \varphi^{\delta} - \nabla \cdot (v \varphi^{\delta}) - \nu\Delta \varphi^{\delta} + ^t \nabla \varphi^{\delta} \cdot w
= \nabla \cdot ((v_{\delta} - v) \varphi^{\delta}) + ^t \nabla \varphi^{\delta} \cdot (w - w_{\delta}). $$
Thus, the last integral in the above equation may be rewritten, integrating by parts,
$$- \int_0^T \int_{\mathbb{R}^d} \varphi^{\delta} (v_{\delta} - v)\cdot \nabla a_{\varepsilon}(s,x) dx ds
+ \int_0^T \int_{\mathbb{R}^d} \varphi^{\delta}(s,x) \nabla \cdot ( (w-w_{\delta})(s,x) \otimes a_{\varepsilon}(s,x)) dx ds. $$
For each fixed $\varepsilon$, the assumption on $a$ entails that $\nabla a_{\varepsilon}$ belongs to $L^2(\mathbb{R}_+ \times \mathbb{R}^d)$.
Furthermore, it is an easy exercise to show that
$$\|v_{\delta} - v\|_{L^2(\mathbb{R}_+ \times \mathbb{R}^d)} \leq
\delta \|\nabla v\|_{L^2(\mathbb{R}_+ \times \mathbb{R}^d)} \||\cdot|\rho\|_{L^1(\mathbb{R}^d)} $$
and
$$\|\nabla \cdot ((w - w_{\delta}) \otimes a_{\varepsilon})\|_{L^1(\mathbb{R}_+ \times \mathbb{R}^d)} \to 0 \text{ as } \delta \to 0, \text{ for any fixed } \varepsilon. $$
Now, taking the limit $\delta \to 0$ while keeping $\varepsilon > 0$ fixed, we have
$$
\langle a_{\varepsilon}(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} = \int_0^T \int_{\mathbb{R}^d} \varphi(s,x) (C^{\varepsilon} + D^{\varepsilon})(s,x) dx ds .
$$
Taking the limit $\varepsilon \to 0$ and using Lemma \ref{LemmeCommutateur}, we finally obtain
$$
\langle a(T), \varphi_0 \rangle_{\mathcal{D}'(\mathbb{R}^d), \mathcal{D}(\mathbb{R}^d)} = 0.
$$
This being true for any test function $\varphi_0$, $a(T)$ is the zero distribution and finally $a \equiv 0$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{NouvellePreuveSerrin}]
Let $\Omega := \nabla \wedge u$ and $\Omega_0 := \nabla \wedge u_0$.
The equation on $\Omega$ writes
$$
(NSV) \left \{
\begin{array}{c c}
\partial_t \Omega + \nabla \cdot (\Omega \otimes u) - \Delta \Omega = \nabla \cdot ( u \otimes \Omega)\\
\Omega (0) = \Omega_0. \\
\end{array}
\right.
$$
Let $\chi = \chi(t)$ be a smooth cutoff in time supported inside $]T_1,T_2[$.
Let $\varphi = \varphi(t)$ be another smooth cutoff such that
$$\text{supp } \chi \subset \{\varphi \equiv 1 \}. $$
Denoting $\Omega' = \chi \Omega$ and $u' = \varphi u$, we have
$$
(NSV') \left \{
\begin{array}{c c}
\partial_t \Omega' + \nabla \cdot (\Omega' \otimes u) - \Delta \Omega' = \nabla \cdot ( u' \otimes \Omega') + \Omega \partial_t \chi\\
\Omega' (0) = 0. \\
\end{array}
\right.
$$
Following the same lines as for Theorem \ref{ExistenceSerrin}, we sketch a way to build a solution $\Omega''$ of
$$
\left \{
\begin{array}{c c}
\partial_t \Omega'' + \nabla \cdot (\Omega'' \otimes u) - \Delta \Omega'' = \nabla \cdot ( u' \otimes \Omega'') + \Omega \partial_t \chi\\
\Omega'' (0) = 0. \\
\end{array}
\right.
$$
belonging to
$$L^{\infty}(\mathbb{R}_+, L^2(\mathbb{T}^3)) \cap L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3)). $$
For $\delta > 0$, let $u_{\delta}, u'_{\delta}$ and $\Omega_{\delta}$ be smooth space mollifications of $u,u'$ and $\Omega$ respectively.
By the Friedrichs method and heat kernel estimates, there exists a smooth solution $\Omega''_{\delta}$ of
$$
\left \{
\begin{array}{c c}
\partial_t \Omega''_{\delta} + \nabla \cdot (\Omega''_{\delta} \otimes u_{\delta}) - \Delta \Omega''_{\delta}
= \nabla \cdot ( u'_{\delta} \otimes \Omega''_{\delta}) + \Omega_{\delta} \partial_t \chi\\
\Omega''_{\delta} (0) = 0. \\
\end{array}
\right.
$$
Performing an energy estimate in $L^2(\mathbb{T}^3)$ gives
\begin{multline*}
\frac 12 \|\Omega''_{\delta}(t)\|_{L^2(\mathbb{T}^3)}^2 + \int_0^t \|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds \\
= \int_0^t \int_{\mathbb{T}^3} \Omega''_{\delta}(x,s) \cdot
\left( \nabla \cdot \left(u'_{\delta}(x,s) \otimes \Omega''_{\delta}(x,s) \right) + \Omega_{\delta}(x,s) \partial_t \chi(s) \right) dx ds.
\end{multline*}
The right-hand side decomposes in two terms, which we estimate separately.
For the first one, we integrate by parts and use H\"older inequality to get
\begin{multline*}
\int_0^t \int_{\mathbb{T}^3} \Omega''_{\delta}(x,s) \cdot
\left( \nabla \cdot \left(u'_{\delta}(x,s) \otimes \Omega''_{\delta}(x,s) \right) \right) dx ds \\
= - \int_0^t \int_{\mathbb{T}^3} \nabla \Omega''_{\delta}(x,s) :
\left(u'_{\delta}(x,s) \otimes \Omega''_{\delta}(x,s) \right) dx ds \\
\leq \int_0^t \|\nabla \Omega''_{\delta}(s) \|_{L^2(\mathbb{T}^3)} \| u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)}
\|\Omega''_{\delta}(s)\|_{L^{\tilde{q}}(\mathbb{T}^3)} ds,
\end{multline*}
where $\tilde{q}$ is defined by
$$\frac{1}{\tilde{q}} = \frac 12 - \frac 1q. $$
The Sobolev embedding $\dot{H}^{\frac 3q}(\mathbb{T}^3) \hookrightarrow L^{\tilde{q}}(\mathbb{T}^3)$ gives
$$
\|\Omega''_{\delta}(s)\|_{L^{\tilde{q}}(\mathbb{T}^3)}
\lesssim \|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{\frac 3q} \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{\frac 2p}.
$$
Hence,
\begin{multline*}
\int_0^t \int_{\mathbb{T}^3} \Omega''_{\delta}(x,s) \cdot
\left( \nabla \cdot \left(u'_{\delta}(x,s) \otimes \Omega''_{\delta}(x,s) \right) \right) dx ds \\
\lesssim \int_0^t \|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{1+\frac 3q}
\|u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)} \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{\frac 2p} ds.
\end{multline*}
Young inequality entails the existence of a constant $C$ depending only on $q$ such that
\begin{multline*}
\|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{1+\frac 3q}
\| u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)} \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^{\frac 2p} \\
\leq \frac 12 \|\nabla \Omega''_{\delta}(s) \|_{L^2(\mathbb{T}^3)}^2
+ \frac C2 \| u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)}^p \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2
\end{multline*}
The second term is easier to bound.
Indeed, thanks to the Cauchy-Schwarz inequality,
\begin{multline*}
\int_0^t \int_{\mathbb{T}^3} \Omega''_{\delta}(x,s) \cdot \Omega_{\delta}(x,s) \partial_t \chi(s) ds
\leq \int_0^t \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)} \|\Omega_{\delta}(s)\|_{L^2(\mathbb{T}^3)} |\partial_t \chi(s)| ds \\
\leq \frac 12 \int_0^t \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds
+ \frac 12 \|\partial_t \chi\|_{L^{\infty}(\mathbb{R}_+)}^2\int_0^t \|\Omega_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds.
\end{multline*}
Gathering these estimates, we have shown that, for some constant $C$ depending only on $q$,
\begin{multline*}
\|\Omega''_{\delta}(t)\|_{L^2(\mathbb{T}^3)}^2 + \int_0^t \|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds \\
\leq \int_0^t \left(1+ C \| u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)}^p\right) \|\Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds
+ \|\partial_t \chi\|_{L^{\infty}(\mathbb{R}_+)}^2 \|\Omega_{\delta}\|_{L^2(\mathbb{R}_+ \times \mathbb{T}^3)}^2.
\end{multline*}
Since $u'_{\delta}$ and $\Omega_{\delta}$ are mollifications of $u'$ and $\Omega$ respectively, for any $s \in \mathbb{R}_+$ and any $\delta > 0$, there holds
$$\|u'_{\delta}(s)\|_{L^q(\mathbb{T}^3)} \leq \|u'(s)\|_{L^q(\mathbb{T}^3)}$$
and
$$\|\Omega_{\delta}(s)\|_{L^2(\mathbb{T}^3)} \leq \|\Omega(s)\|_{L^2(\mathbb{T}^3)}.$$
Combining these facts to Gr\"onwall's inequality entails the bound
\begin{multline*}
\|\Omega''_{\delta}(t)\|_{L^2(\mathbb{T}^3)}^2 + \int_0^t \|\nabla \Omega''_{\delta}(s)\|_{L^2(\mathbb{T}^3)}^2 ds \\
\leq \|\partial_t \chi\|_{L^{\infty}(\mathbb{R}_+)}^2 \|\Omega\|_{L^2(\mathbb{R}_+ \times \mathbb{T}^3)}^2
\exp\left(t+ C\int_0^t \| u'(s)\|_{L^q(\mathbb{T}^3)}^p ds \right).
\end{multline*}
It only remains to pass to the limit.
From the uniform $L^{\infty}(\mathbb{R}_+, L^2(\mathbb{T}^3)) \cap L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3))$ bound on the $(\Omega''_{\delta})_{\delta}$, there exists some
$\Omega'' \in L^{\infty}(\mathbb{R}_+, L^2(\mathbb{T}^3)) \cap L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3))$ such that, up to an extraction,
$$\Omega''_{\delta} \rightharpoonup \Omega''\text{ weakly in } L^{\infty}(\mathbb{R}_+, L^2(\mathbb{T}^3)) \cap L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3)) \text{ as } \delta \to 0 $$
This weak convergence allows us to pass to the limit in the equation on $\Omega''_{\delta}$, thanks to the strong convergences
$$u_{\delta}, u'_{\delta} \to u, u' \text{ strongly in } L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3)) \text{ as } \delta \to 0. $$
Such an $\Omega''$ thus belongs to
$$L^{\infty}(\mathbb{R}_+, L^2(\mathbb{T}^3)) \cap L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3))$$
and solves, as required,
$$
\left \{
\begin{array}{c c}
\partial_t \Omega'' + \nabla \cdot (\Omega'' \otimes u) - \Delta \Omega'' = \nabla \cdot ( u' \otimes \Omega'') + \Omega \partial_t \chi\\
\Omega'' (0) = 0. \\
\end{array}
\right.
$$
Now, letting $\widetilde{\Omega} := \Omega' - \Omega''$, we see that $\widetilde{\Omega}$ solves
$$
(NSV^0) \left \{
\begin{array}{c c}
\partial_t \widetilde{\Omega} + \nabla \cdot (\widetilde{\Omega} \otimes u) - \Delta \widetilde{\Omega} = \nabla \cdot ( u' \otimes \widetilde{\Omega})\\
\widetilde{\Omega} (0) = 0. \\
\end{array}
\right.
$$
We recall that $u$ and $u'$ belong to $L^2(\mathbb{R}_+, \dot{H}^1(\mathbb{T}^3))$ and by assumption, $u'$ further belongs to $L^p(\mathbb{R}_+, L^q(\mathbb{T}^d))$.
Moreover, the high regularity of $\Omega''$ and the fact that $u$ is a Leray solution of the Navier-Stokes equations together entail that $\widetilde{\Omega}$ belongs to
$L^2(\mathbb{R}_+ \times \mathbb{T}^3)$.
These regularity assumptions allow us to invoke Theorem \ref{UniquenessSerrin}, from which we deduce that $\widetilde{\Omega} \equiv 0$.
It follows that
$$\Omega \in L^{\infty}_{loc}(]T_1,T_2[, L^2(\mathbb{T}^3)) \cap L^2_{loc}(]T_1,T_2[, \dot{H}^1(\mathbb{T}^3)). $$
Since $u$ is the inverse curl of $\Omega$, the above regularity on $\Omega$ is equivalent to
$$u\in L^{\infty}_{loc}(]T_1,T_2[, \dot{H}^1(\mathbb{T}^3)) \cap L^2_{loc}(]T_1,T_2[, \dot{H}^2(\mathbb{T}^3)). $$
From here, improving again the regularity on $\Omega$ and $u$ relies on an induction procedure, which is tedious to write thoroughly but not difficult.
We need to prove that, for all $s \in \mathbb{N}$, we have
$$\Omega \in L^{\infty}_{loc}(]T_1,T_2[, \dot{H}^s(\mathbb{T}^3)) \cap L^2_{loc}(]T_1,T_2[, \dot{H}^{s+1}(\mathbb{T}^3)), $$
which is equivalent to requiring
$$u\in L^{\infty}_{loc}(]T_1,T_2[, \dot{H}^{s+1}(\mathbb{T}^3)) \cap L^2_{loc}(]T_1,T_2[, \dot{H}^{s+2}(\mathbb{T}^3)). $$
The case $s = 0$ is exactly what we just proved.
To go from the step $s$ to the step $s+1$, we simply compute all the space derivatives of order $s+1$ of the equation on $\Omega'$.
More precisely, denoting by $\partial^{s+1}$ a generic space derivative of order $s+1$, we have
$$\partial_t \partial^{s+1} \Omega' + \nabla \cdot (u \otimes \partial^{s+1} \Omega') - \Delta \partial^{s+1} \Omega'
= \nabla (\partial^{s+1}\Omega' \otimes u) + (\text{l.o.t in } \Omega').$$
Performing an energy estimate in $L^2(\mathbb{T}^3)$ as above and using Theorem \ref{UniquenessSerrin}, we get
$$\partial^{s+1}\Omega' \in L^{\infty}_{loc}(]T_1,T_2[, L^2(\mathbb{T}^3)) \cap L^2_{loc}(]T_1,T_2[, \dot{H}^1(\mathbb{T}^3)),$$
which is what we wanted.
Time derivatives may now be handled by a similar induction argument, which we will not write.
This closes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{BorneL1VorticiteForte}]
Given the assumptions we made, we compute the vorticity equation by taking the curl on each side of the Navoer-Stokes equations.
Let $s < t$ be two real numbers in $]0,T[$.
Let $\varphi : [0,t-s] \times \mathbb{R}^d \to \mathbb{R}^d$ be a solution of the adjoint equation satisfying the bound \eqref{GrowingMaximumPriciple}.
Imitating the proof of Theorem \ref{UniquenessSerrin} for the time interval $[s,t]$, we arrive at
\begin{equation}
\langle \Omega(t), \varphi_0 \rangle_{L^1(\mathbb{R}^d), L^{\infty}(\mathbb{R}^d)} = \langle \Omega(s), \varphi(t-s)\rangle_{L^1(\mathbb{R}^d), L^{\infty}(\mathbb{R}^d)}.
\end{equation}
Thanks to the bound \eqref{GrowingMaximumPriciple}, for any $\varphi_0$ in $\mathcal{D}(\mathbb{R}^d)$, we have
\begin{equation}
|\langle \Omega(t), \varphi_0 \rangle_{L^1(\mathbb{R}^d), L^{\infty}(\mathbb{R}^d)}| \leq
\|\Omega(s)\|_{L^1(\mathbb{R}^d)}\|\varphi_0\|_{L^{\infty}(\mathbb{R}^d)} \exp\left[\frac{C^p}{p\nu^{p-2}} \int_s^t \|u(s)\|_{L^q(\mathbb{R}^d)}^p ds \right].
\end{equation}
Taking the supremum over all possible $\varphi_0$ and letting $s \to 0$ yields the result.
\end{proof}
\subsection*{Acknowledgements}
The author is grateful to L. Sz\'ekelyihidi both for his kind invitation to Universit\"at Leipzig and the fruitful discussions which led to notable improvements of the paper.
\textbf{TODO : virer l'approximation et ne montrer qu'une estimation a priori pour l'existence}
|
2,869,038,154,901 | arxiv | \section{Introduction}
Solving quadratic eigenvalue problems commonly leads to numerical instabilities, and is computationally inefficient. In the literature, there are certain numerical solution strategies to linearize the polynomial eigenvalue problems to increase the computational efficiency, without sacrificing accuracy \citep{datta2010numerical}. However, in certain applications, the derivatives of the eigenvalues are also required. In order to simultaneously obtain the eigenvalues and their derivatives, a novel solution strategy is proposed: first the eigenvalue problem is linearized using the Generalized Reduction method, after which dual numbers are used to perform Automatic Differentiation. We refer to this method as GRAD.
As our guiding example, we consider simulations of guided waves using the semi analytical finite element (SAFE) approach.
Guided waves are generally employed for SHM and ultrasonic inspection applications, owing to the fact that these types of waves can travel over long distances \citep{segers2020probing}. As such, large areas can be inspected with a limited number of sensors \citep{wang2018sparse}. One of the most common methods to simulate guided wave propagation in solid plates, bars or tubes, is the SAFE approach, because of its accuracy and robustness to compute the Lamb wave mode spectra \citep{bartoli2006modeling,mazzotti2014computation}.
However, for inverse problem applications, such as the (visco)elastic parameter characterization of composite laminates, or for quasi-real time structural health monitoring applications,
the total computational time becomes dominated by the evaluation speed of the forward model. Therefore fast and accurate evaluation of the forward model is important.
In the present study, GRAD is implemented to calculate phase, group and energy velocities of Lamb waves in solid plates. It is shown that the computational speed and efficiency is greatly enhanced by utilizing GRAD.
\section{Model}
As a guiding example for the implementation of GRAD, the quadratic eigenvalue problem of the SAFE method is considered; although the method can be extended to higher-order polynomial eigenvalue problems, or other quadratic eigenvalue problems such as Legendre polynomials \citep{bou2013legendre}, or higher order shear deformation theories \citep{orta2021modelling} as well.
The SAFE method, and its accompanying parameters, are well defined in the literature \citep{bartoli2006modeling,treyssede2008elastic}. The quadratic eigenvalue problem of the SAFE method can generally be expressed as \citep{treyssede2008elastic}
\begin{equation}
\bqty{ K_3 k^2 + (K_2-K_2^T) k i + (K_1 - M \omega^2) }u(x,t) = 0,
\label{eq:poly_eig}
\end{equation}
where $u(x, t) = u_0 \exp[ik x-i\omega t]$ is a plane wave with initial displacement vector $u_0$, the $K_j$ are stiffness matrices, $M$ is mass matrix, $k$ is the wavenumber ($k=2 \pi / \lambda$, with $\lambda$ the wavelength), and $\omega$ is the angular frequency ($\omega= 2 \pi f$, with $f$ the frequency). In this equation, we are not exclusively interested in the solutions for $k$, which would give us the allowable phase velocities at a certain frequency and thus the dispersion curves, but also in ${\partial \omega}/{\partial k}$, the group velocity of the modes. While there are several existing techniques to solve for $k$ [6], in this paper we focus on demonstrating how these techniques can be easily extended to include the computation of ${\partial \omega}/{\partial k}$, up to the machine precision, by using dual numbers.
\subsection{Generalized reduction method}
The most straightforward solution strategy is to first linearize Eq.~\ref{eq:poly_eig} using the generalized reduction method \citep{datta2010numerical}. If the matrix $K_3$ is non-singular, we can exploit this to rewrite the quadratic eigenvalue equation as a linear system, by defining
\begin{equation}
\underbrace{\begin{bmatrix}
0 & \mathbb{1}\\
K_3^{-1} (K_1-M\omega^2) & K_3^{-1} (K_2-K_2^T)
\end{bmatrix}}_{A(\omega)}
\underbrace{\begin{bmatrix}
u(x,t) \\
i k u(x,t) \\
\end{bmatrix}}_{v(\omega)}
= i k \underbrace{\begin{bmatrix}
u(x,t) \\
i k u(x,t) \\
\end{bmatrix}}_{v(\omega)}
\label{eq:linearized_eigen}
\end{equation}
As Eq. \ref{eq:poly_eig} and Eq. \ref{eq:linearized_eigen} are equivalent, the eigenvalues $k$ are identical for both. Therefore, the eigenvalues of Eq. \ref{eq:poly_eig} can be obtained by finding the eigenvalues of the matrix $A(\omega)$, which can be found using conventional methods.
However, we are not only interested in the eigenvalues $k$, but also wish to simultaneously obtain $\partial k / \partial \omega$.
In order to calculate the latter, dual numbers are used.
\subsection{Automatic differentiation of eigenvalues}
Dual numbers rose to prominence in the realm of automatic differentiation \citep{revels2016forwardmode}. A dual number is an expression of the form $a + b \epsilon$, where $a,b \in \mathbb{C}$, and $\epsilon$ is defined to satisfy $\epsilon^2=0$.
Assuming the matrix function $A(\omega)$ is analytical in some neighborhood of $\omega$, it permits the Taylor expansion
\begin{equation}
A(\omega+\epsilon) = A(\omega) + \epsilon A'(\omega).
\label{eq:taylor_series}
\end{equation}
It follows that $\Dual [ A(\omega + \epsilon) ] = A'(\omega)$, where $\Dual$ selects the dual part.
Assuming the matrix $A = A(\omega)$ is diagonalizable as $A = v \Lambda v^{-1}$, where
$v = [ v_1, v_2 \ldots, v_n ]$ is a square matrix whose columns are the linearly independent eigenvectors $v_i$, and $\Lambda_{ij}(\omega) = k_i(\omega) \delta_{ij}$ is the diagonal matrix containing the eigenvalues,
we define the vector of eigenvalues as $\vec{k}(\omega) = \text{diag}(\Lambda(\omega))$. Then the derivatives of the eigenvalues are given by \citep{lancaster1964eigenvalues}:
\begin{equation}
\vec{k}' =\dv{\vec{k}}{\omega} = \text{diag}\pqty{v^{-1}(\omega) A'(\omega) v(\omega)}.
\label{eq:der_eigenvalues}
\end{equation}
Since $A'(\omega) = \Dual[A(\omega+\epsilon)]$, these derivatives can be computed with machine precision, provided $A(\omega+\epsilon)$ can be evaluated. In general this can be done using an automatic differentiation library \citep{revels2016forwardmode}, but
in the particular case of $A(\omega)$ as defined in Eq. \ref{eq:linearized_eigen}, $A(\omega+\epsilon)$ can be evaluated analytically:
\begin{equation}
A'(\omega) = \begin{bmatrix}
0 & 0\\
2K_3^{-1} M\omega & 0 \\
\end{bmatrix}.
\end{equation}
This enables us to find $k_j' = {\partial k_j}/{\partial \omega}$ using Eq. \ref{eq:der_eigenvalues}, after which
the corresponding group velocity $\partial{\omega}/\partial{k_j}$ is $1 / k_j'$.
\newpage
\subsection{Calculation of energy velocity}
As previously mentioned in the literature \citep{bartoli2006modeling,treyssede2008elastic}, the group velocity definition is no longer valid in waveguides in attenuative media. For damped waves, the wavenumbers become complex, where the imaginary part carries the attenuation information. Using the group velocity definition, the derivative of the real part of the complex wavenumber yields nonphysical solutions such as infinite velocities at some frequencies. At this point, the energy velocity $V_e$ is considered the appropriate property for damped media. The general expression of the energy velocity reads \citep{treyssede2008elastic,mazzotti2012guided}:
\begin{equation}
V_e(\omega) =\frac{\Im(u^H 2 \omega (K_2^T+K_3 i k)u)}{\Re(u^H(K_3 k^2 + (K_2 - K_2^T)i k + K_1 + M \omega^2)u)} \label{eq:simplified_energy}
\end{equation}
However, the eigenvectors $A(\omega)$, as defined in Eq. \ref{eq:linearized_eigen}, are $v_j~=~\smqty[ u_j & i k_j u_j ]^T$. This allows Eq. \ref{eq:simplified_energy} to be further simplified to
\begin{align}
V_e(\omega) = \frac{2\omega\Im\bqty{ \text{diag}(v^H A_1 v)}}{\Re \bqty{ \text{diag}(v^H A_2 v)}}, \label{eq:energy_vel}
\end{align}
where $A_1 = \sbmqty{K_2^T & 0\\ -K_3 & 0 }$ and $A_2 = \sbmqty{K_1+M\omega^2 & 0\\ -(K_2-K_2^T) & K_3}$.
Note that the energy velocity exactly reverts to the group velocity in the case of undamped wave propagation.
\section{Results}
To validate the proposed group and energy velocity computation model, a numerical study is conducted for which the GRAD results are compared with the conventional SAFE software GUIGUW \citep{bocchini2011graphical}. The propagation characteristics of Lamb waves are examined for a homogenized purely elastic orthotropic carbon/epoxy (C/E) composite plate and a visco-elastic version of the same plate \citep{bartoli2006modeling}, with material stiffness (and viscosity) tensor components as listed in Table \ref{table:materials}. The material density of the C/E material is $\rho=1571$ kg/m$^3$ and the plate thickness is 1 mm.
\begin{figure}[htb]
\subfigure[ ]{\includegraphics[trim={2.5cm 1cm 2.4cm 2cm},clip=true, scale=0.25, angle=0]{figure1a.pdf}}
\subfigure[ ]{\includegraphics[trim={2.5cm 1cm 2.4cm 2cm},clip=true, scale=0.25, angle=0]{figure1b.pdf}}
\subfigure[ ]{\includegraphics[trim={2.5cm 1cm 2.4cm 2cm},clip=true, scale=0.25, angle=0]{figure1c.pdf}}
\subfigure[ ]{\includegraphics[trim={2.5cm 1cm 2.4cm 2cm},clip=true, scale=0.25, angle=0]{figure1d.pdf}}
\caption{Comparison between GRAD and GUIGUW for C/E where $\phi=0$ (in-plane angle) (a) Undamped phase velocities, (b) Damped phase velocities, (c) Group velocities, and (d) Energy Velocities.}
\label{fig:composite}
\end{figure}
\begin{table}[htb]
\caption{Elastic and viscous properties of C/E composite lamina (in GPa).}
\begin{center}
\begin{tabular}{cccccccccc}
\hline
$Material$ & $C_{11}$ & $C_{12}$ & $C_{13}$ & $C_{22}$ & $C_{23}$ & $C_{33}$ & $C_{44}$ & $C_{55}$ & $C_{66}$ \\ \cline{1-10}
C/E Lamina \citep{bartoli2006modeling} & 132 & 6.9 & 12.3 & 5.9 & 5.5 & 12.1 & 3.32 & 6.21 & 6.15 \\ \cline{1-10}
$Material$ & $\eta_{11}$ & $\eta_{12}$ & $\eta_{13}$ & $\eta_{22}$ & $\eta_{23}$ & $\eta_{33}$ & $\eta_{44}$ & $\eta_{55}$ & $\eta_{66}$ \\ \cline{1-10}
C/E Lamina \citep{bartoli2006modeling} & 0.4 &0.001 & 0.016 & 0.037 & 0.021 & 0.043 & 0.009 & 0.015 & 0.02 \\ \cline{1-10}
\hline
\end{tabular}
\label{table:materials}
\end{center}
\end{table}
The results show excellent agreement between GUIGUW and GRAD, both for the undamped and the damped case in C/E (see Fig. \ref{fig:composite}). Note that the visco-elasticity has a negligible effect on the phase velocities, whereas the difference between the group and energy velocity values is substantial, as expected.
As the mass and stiffness matrices do not change with frequency, the use of frequency domain solutions, as well as the new compact formulaes (Eq. \ref{eq:linearized_eigen}, Eq. \ref{eq:der_eigenvalues} and Eq. \ref{eq:energy_vel}), are essential for the computational speed. In addition, the original quadratic eigenvalue problem has been reduced into a standard eigenvalue problem which requires less computational power compared to other methods. The average solution times using the proposed algorithms (average over 50 simulations) are listed in Table \ref{table:time} for computation on a workstation with Intel\textregistered Core\texttrademark i7-8700 CPU $@$ 3.20 GHz and 32 GB ram. The calculation times may change based on the computer hardware, the number of elements used in the discretization through thickness, and the number of solution points. For the most time consuming case (40 elements, 500 solution points), the solution time for GRAD only measured 112 seconds, whereas GUIGUW required 228 seconds, which demonstrates the computational efficiency of the suggested algorithms.
\begin{table}[H]
\caption{Averaged solution times in seconds (50 simulations).}
\centering
\begin{tabular}{ccccc}
\hline
& & \multicolumn{3}{c}{\footnotesize{$\#$ of solution points} } \\
& & \multicolumn{1}{c}{100} & \multicolumn{1}{c}{250} & \multicolumn{1}{c}{500} \\
\hline
\parbox[t]{2mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\footnotesize{$\#$ of elements}}}} &5 & 0.3373 & 0.7903 & 1.5716 \\
&10 & 1.2028 & 3.0140 & 3.0140 \\
&20 & 5.1893 & 13.0273 & 26.1161 \\
&40 & 22.7285 & 55.7703 & 112.0324\\
\hline
\end{tabular}
\label{table:time}
\end{table}
\section{Conclusion}
A novel solution strategy for the computation of eigenvalues and their derivatives in quadratic eigenvalue problems was presented. The presented strategy first linearizes the eigenvalue problem with the generalized reduction method, after which the derivatives of the eigenvalues are obtained by using dual numbers.
This method was then used to calculate Lamb wave phase, group and energy velocities in the SAFE method.
The proposed GRAD method can also be applied to different methods, such as Legendre polynomials, higher order shear deformation theory, etc.
The introduced concepts roughly doubled the computational speed and efficiency in comparison with the semi analytical finite element method, and can be used for similar polynomial eigenvalue problems in the domain of complex wave propagation.
\section*{Acknowledgements}
The authors gratefully acknowledge the financial support from the Fund for Scientific Research-Flanders (FWO Vlaanderen, grants G066618N, G0B9515N, 1S45216N and 12T5418N), KU Leuven IF project C14/16/067 and the NVIDIA corporation.
\bibliographystyle{elsarticle-num}
|
2,869,038,154,902 | arxiv | \section{Introduction}
Convolutional neural network (CNN) based super resolution (SR) \cite{dong2015image} is getting popular in recent years because of better reconstructed high resolution images over traditional interpolation methods. These SR models are getting deeper and wider and use complicated structure to get better performance\cite{zhang2018image, ahn2018fast, dai2019second, zhao2020efficient} as shown in Fig.~\ref{performance versus PSNR}. However, SR models suffer from high computational complexity and memory bandwidth because their feature sizes are not getting smaller over layers and they need to process large input size. Thus, hardware acceleration is demanded for real-time applications.
Various hardware accelerators have been proposed\cite{9223656, Yen2020RealtimeSR, 9159619}. However, current designs only use plain networks like the widely used FSRCNN\cite{dong2016accelerating} instead of recent large size and complicated models due to the high model complexity, which limits their reconstructed image quality. In addition, most of the designs use layer-by-layer processing that will need to store intermediate data to DRAM and load it back for each layer or large buffer size, which is extremely bad for SR applications due to their large feature size.
To address the above problems, we propose a full model block convolution based super resolution accelerator (BSRA) with hardware efficient pixel attention (HPAN) model. The HPAN model uses pixel attention for better image quality than FSRCNN but keeps the structure simple for small model size and low complexity. The accelerator uses block convolution for the whole model instead of certain layers as in \cite{li2021block} to enable the whole model fusion to reduce the external memory bandwidth to model input and output only and needs a small on-chip buffer size. The required convolution and pixel attention are well supported by the proposed distributed weight PE array. The final implementation can achieve real-time full HD image throughput with TSMC 40nm CMOS process.
\section{Proposed SR Model with Hardware Efficient Pixel Attention}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{performance_versus_PSNR.png}
\caption{Image quality vs model size for different SR models}
\label{performance versus PSNR}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[scale=0.4]{model_structure.png}
\caption{The proposed HPAN model}
\label{The block diagram of proposed neural network}
\end{figure*}
\subsection{Related Work}
Since the first CNN based SR model, SRCNN~\cite{dong2015image}, was proposed, many deep learning skills are added to the super resolution model to get better image quality. CARN~\cite{ahn2018fast} cascades the residual in residual blocks to get multilevel representations. RCAN~\cite{zhang2018image}, which is over 400 layers, uses residual blocks and channel attention to learn more channel-wise information. SAN~\cite{dai2019second} proposes non-locally enhanced residual group to gain long distance and structure message. These models are getting deeper and wider for better images.
Fig.~\ref{performance versus PSNR} the scatter chart of the SR model size and its image quality. This figure shows that the quality of the reconstructed image is increasing with model size, but the increment of parameters is not efficient at million scale. Take FSRCNN~\cite{dong2016accelerating} and RCAN~\cite{zhang2018image} for example, the difference in performance on Set5 between them is only 1.3 dB, but the model size of the latter one is about 1300 larger than the former one. The model size increment is very inefficient so that most of hardware designs \cite{9223656, Yen2020RealtimeSR, 9159619} use plain and shallow networks just like SRCNN~\cite{dong2015image} and FSRCNN~\cite{dong2016accelerating}. For example, in ~\cite{9159619}, they uses a classification network to judge whether image tiles are with more high frequency signal and send them to appropriate plain networks. This mechanism can reduce the computation overhead of the SR part, but the classification part is much bigger than the SR part. Therefore, this paper tries to find a better architecture which has a trade-off between model size and reconstructed image quality for single image super resolution.
\subsection{Proposed HPAN Model}
Fig.~\ref{The block diagram of proposed neural network} shows the proposed HPAN model based on convolution pixel attention mechanism and the number of parameters is just about 26K. The model design is based on the following observations. First, for the overall model structure, skip or dense connection is popular in most of the models to learn the residual between high resolution (HR) images and SR images, but this will need extra memory access for hardware implementation. Thus, instead of the residual architecture, we take the attention mechanism into consideration. In Fig.~\ref{performance versus PSNR}, RCAN~\cite{zhang2018image} and SAN~\cite{dai2019second} are the best two models in performance with different attention methods. However, the trend line shows that pixel attention (PAN)~\cite{zhao2020efficient} is better. Thus, we use pixel attention to construct our model. However, ~\cite{zhao2020efficient} has a local skip connection in the basic block and a global interpolated image addition. Besides, in the up-sampling stage, the final image is not generated immediately because the interpolated features still need to be tuned by some convolution layers and thus needs large computation. These are not hardware friendly.
Based on the above observations, we propose the hardware-friendly pixel attention model with 26K parameters as shown in Fig.~\ref{The block diagram of proposed neural network}. The model consists of three stages: feature extraction, mapping, and reconstruction. The feature extraction stage uses one convolution layer. The mapping stage is based on the Clamping Pixel Attention Block (CPAB) in (\ref{eq:cpab}) to extract features. After cascading some CPABs, we use transpose convolution to reconstruct the image directly to prevent the problem mentioned before.
The representation of CPAB in Fig.~\ref{The block diagram of proposed neural network} is shown as follows.
\begin{equation}
\label{eq:clamping}
Clamp(f_{in}) =
\left\{
\begin{array}{lr}
+255, & f_{in} \geq +255 \\
-255, & f_{in} \leq -255 \\
f_{in}, & \text{otherwise}
\end{array}
\right.
\end{equation}
\begin{equation}
\label{eq:sigmoid}
D(f_{in}) = sigmoid(f_{in}/256)
\end{equation}
\begin{equation}
\label{eq:pa}
PA(f_{in}) = M_{ew}(f_{in},Clamp( D(CL_{1}(f_{in}))))
\end{equation}
\begin{equation}
\label{eq:cpab}
CPAB(f_{in}) = CL_{3}( PA( CL_{1}( f_{in} ) ) )
\end{equation}
where $f_{in}$ is input feature, and $Clamp()$, $D()$, $M_{ew}(,)$, $PA()$, and $CL_{i}()$ denote clamping function, divisor, element-wise multiplication, pixel attention and convolution with kernel size $i$, respectively. In this model, with (\ref{eq:sigmoid}), the weight and activation distribution will be more reasonable after training because the range of pixel intensity is from 0 to 1 or 0 to 255. This is applied to every convolution to ensure the proper range. Besides, since the value of the attention map is smaller than 1, the absolute value of multiplication results will not exceed 255. The divisor in (\ref{eq:sigmoid}) chooses 256 for hardware-friendly design as a simple shifting operation.
\section{Proposed Architecture}
\subsection{Overview}
Fig.~\ref{System architecture} shows the proposed system architecture. This design gets weights and input images by accessing external memory and stores them into weight and feature SRAM buffers. Both have 32 banks to store kernels of different layers and intermediate results, respectively, for the computing core. This core consists of 32 processing element (PE) arrays to process 32 channels of input. Then the outputs of different channels are summed in a local accumulator and stored in the partial sum buffer. These partial sums will be further accumulated in the selective adder for convolution. The results will be further processed by sigmoid and clamping function according to the proposed model. The results will be sent to the multiplexer for attention mechanism or feature memory for the next layer.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{System_architecture.png}
\caption{System architecture}
\label{System architecture}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{PE_array.png}
\caption{PE arrays}
\label{PE array}
\end{figure}
\subsection{PE Array with Distributed Weight}
Fig.~\ref{PE array} shows the proposed one PE array, which consists of six PE lines (PEL0 to PEL5) with six PEs in each PELx. Each PE is a multiplier with registers. In the figure, each PE array is for one channel of convolution in a layer. In this way, the input feature maps and weights share the same spatial location in the same PE array. Therefore, the multipliers in each PE get different features according to their location in the spatial space. In other words, 36 input pixels are allocated to 36 PEs. As for the multiplicands in one PE array, there are two choices, one is convolutional kernel weights and the other is pixel-wise attention mask. If the multiplicands are kernel weights, the weights will broadcast vertically. If the multiplicands are an attention mask, the mask values will be distributed in PE array. With this distributed weight scheme, this design can support convolution as well as pixel attention. After completing the multiplications, the 36 results are transferred to the next stage to be accumulated.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{accumulator.png}
\caption{The three-stage accumulator.}
\label{Accumulator_stage}
\end{figure}
\subsection{Three-Stage Accumulator}
Fig.~\ref{Accumulator_stage} shows the accumulator to accumulate the partial sum, which is partitioned into three-stage pipelines for minimizing the critical path and attention mechanism. In which, for our 32 channel computation, eight partial sums at the same spatial location but different channels are summed up at the first stage, and their results are further accumulated to complete a 32 channel summation at the second stage. More channel summations are done at the third stage for the final convolutional results. The selective adder at the third stage will select the proper input based on convolution or attention operation for the correct output.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{Convolution_computation.png}
\caption {An example of the proposed data flow. In which, for data allocation, the same colored geometric shapes share the same weights. If kernel size is $k$, after $k$ clock cycle, the cached feature should be updated.}
\label{Data flow chart}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{Image_result.png}
\caption{Comparison of reconstructed images from the Set14 and Urban100 datasets. The images from left to right are original ground truth, enlarged ground truth, bicubic interpolation, our floating-point result and quantization result with block convolution, respectively.}
\label{image results}
\end{figure*}
\subsection{Data Flow of Convolution}
Fig.~\ref{Data flow chart} shows an example data flow with one PE array with 4 PEs in each PEL for simplicity. In the proposed data flow, the kernel weights are broadcast vertically along the PE array row and the feature inputs are transferred to PEs depending on their spatial locations. With this allocation, the results of PEs, $P^{sum}_{i,j}$, are belong to the output pixel $O_{i,j}$ at the first cycle. At the next cycle, the weights in the PE array are circularly shifted right. The results $P^{sum}_{i,j}$ are belong to $O_{i-1,j}$. At the third cycle, the weights are shifted again and the results are belonging to $O_{i-2,j}$. When the first row of weights are fully used, the second row of weights are loaded into the PE array. By repeating this processing flow, all partial sums can be generated. These partial sums will be accumulated in the accumulator and stored in the partial sum buffer. After nine clock cycles, the feature input will be shifted down by four pixels and will be loaded for another round of computation until the bottom of this tile. After that, the feature input will be shifted right by six pixels for computation. With this, the feature map can be reused to reduce the memory bandwidth due to the large input in the SR model.
\subsection{Block Convolution for Whole Model Fusion Execution}
Because the feature size of intermediate layers is as large as the input of the SR model, the required on-chip buffer size will grow up quickly as the input size is increased. This problem cannot be solved well by commonly used tile-based processing \cite{9159619} due to large boundary data storage or computation between tiles. Thus, this paper adopts the block convolution ~\cite{li2021block} by splitting the input into nonoverlapped tiles and processing them with suitable padding without boundary information. However, the previous approach in \cite{li2021block} limits this to only certain layers and needs a large tile size due to information loss at the boundary. In contrast, the proposed network with pixel level attention could reduce this impact to minimum. We apply this to all layers with small tile size ($40,48$) , and enable the whole model fused execution by processing one tile from input to output directly. All intermediate data can be stored in small on-chip buffers without external DRAM access as other SR accelerators. With this, the external DRAM access can be reduced to the model input and output only.
\section{Experimental Results}
\subsection{Model Simulation Result}
Fig.\ref{image results} shows the simulation results and comparisons. The results with bicubic interpolation will lose a lot of detail and become blur, especially at the stripe boundary. In contrast, the proposed network can reconstruct images correctly and the generated results are more clear than the traditional ones. The results are almost kept the same even when we quantize the weights and activations and include the full model block convolution without handling boundary information.
Table~\ref{comparison with others} shows performance comparisons with other works. In which, VDSR and PAN have better performance than ours, but they also need 10 and 25 times of model parameters than ours. The performance difference between VDSR and ours is marginal for several datasets. In contrast, our model size is close to the lightweight models but has much better performance. Our model has achieved a good balance between model size and performance, which has also been illustrated in Fig.~\ref{performance versus PSNR}.
\begin{table}[t]
\caption{Comparison of several software approaches on commonly used datasets (Urb.100 is Urban100 and M.109 is Manga109).}
\centering
\begin{tabular}{lllllll}
\hline
Algorithm & Param. & Set5 & Set14 & B100 & Urb.100 & M.109 \\
\hline
Bicubic & - & 33.66 & 30.25 & 29.57 & 26.89 & 30.86 \\
SRCNN-Ex & 57,184 & 36.66 & 32.45 & 31.36 & 29.50 & 35.60 \\
FSRCNN & 12,464 & 37.00 & 32.63 & 31.50 & 29.88 & 36.67 \\
This work & \textbf{25,920} & \textbf{37.38} & \textbf{32.91} & \textbf{31.69} & \textbf{30.29} & \textbf{37.00}
\\
\hline
VDSR & 665,000 & 37.53 & 33.05 & 31.90 & 30.77 & 37.22 \\
PAN & 261,000 & 38.00 & 33.59 & 32.18 & 32.01 & 38.70 \\
\hline
\end{tabular}
\label{comparison with others}
\end{table}
\begin{table}
\centering
\caption{Comparison with other hardware works}
\begin{tabular}{|c|c|c|c|}
\hline
Work & \cite{Yen2020RealtimeSR} & \cite{9159619} & This work \\
\hline
Method & Improved IDN & S/L FSRCNN & Pixel Attention \\
\hline
Resolution & FHD & FHD & FHD \\
\hline
Process & 32 nm & 65 nm & 40nm \\
\hline
Frequency & 200 MHz & 200 MHz & 471 MHz \\
\hline
Parameters & 12,726 & 16,660\footnotemark[1] & 25,920 \\
\hline
PSNR (dB) & 36.96 & 37.06 & 37.18 \\
\hline
Precision(W/A) & - / 12 & 8 / 16 & 11 / 18 \\
\hline
Memory Size & - & 572 KB & 232 KB \\
\hline
Gate Count & 2614K & - & 2837K \\
\hline
External BW & I/O + & I/O + partial & I/O only \\
&intermediate & intermediate & \\
\hline
FPS & 60 & 31.8 & 30 \\
\hline
\end{tabular}
\label{Comparison with other hardware works}
\footnotemark[1]{SR model only without classification model}\\
\end{table}
\subsection{Hardware Implementation}
Table~\ref{Comparison with other hardware works} shows hardware implementation results of TSMC 40nm CMOS process and comparison. This design can achieve real-time full HD image reconstruction with 30 frames per second when running at 471 MHz. Compared to other designs, this design can save significant intermediate feature map I/O to DRAM with smaller on-chip buffer size and better reconstruction image quality.
\section{Conclusion}
This paper proposes an SR accelerator with hardware efficient pixel attention model to reconstruct images with 37.38 dB PSNR on Set5 using 25.9 K parameters, which is the first hardware work without just plain network. Besides, we also reduce the external memory access to model I/O only with full model block convolution and layer fusion. Such convolution and pixel attention computation have been well supported by PE arrays with distributed weights. The final implementation can achieve real-time full HD image reconstruction with smaller buffer size and bandwidth but better image quality than other SR accelerators.
\section*{Acknowledgment}
This work was supported by the Ministry of Science and Technology, Taiwan, under Grant 109-2634-F-009-022, 109-2639-E-009-001, 110-2221-E-A49-148-MY3 and 110-2622-8-009-018-SB, and TSMC.
\bibliographystyle{IEEEtran}
|
2,869,038,154,903 | arxiv | \section{Introduction}
\textbf{\textit{Introduction.}} Semiconductor quantum dots (QD) are regarded as a key building block in quantum information science and technology. One of their notable functionalities is the generation of quantum entangled photon pairs \cite{Benson_PRL00}, which will provide long-distance fully secured quantum key distribution \cite{Gisin_RMP02} and ultrahigh-resolution imaging \cite{Okano_SciRep15}. A fundamental prerequisite for entangled-pair generation is the elimination of structural asymmetry in self-assembled dots \cite{Santori_PRB02}. The use of a $C_\text{3v}$ symmetric (111) surface as a growth substrate is an efficient and scalable way of creating highly symmetric dots, as proposed theoretically \cite{Sing_PRL09,Schliwa_PRB09}, and demonstrated experimentally \cite{Mano_APEX10,Juska_NatPhot13}. Although standard QD growth based on the Stranski-Krastanov (S-K) mode is not applicable to QD formation on (111) surfaces, droplet epitaxy makes it possible to grow QDs on Ga-rich (111)A oriented surfaces \cite{Mano_APEX10}. A great reduction in anisotropy-induced fine structure splitting (FSS) was observed in GaAs/AlGaAs QDs on GaAs(111)A, which led to the generation of highly entangled photon pairs where the fidelity to the maximally entangled state was 86\% \cite{kuroda_PRB13}.
With the aim of extending the emission wavelength to optical fiber telecommunication wavelengths, we have recently demonstrated the droplet epitaxial growth of InAs QDs embedded in InAlAs using InP(111)A substrates \cite{Ha_APL14}. Their emission spectra covered the O ($\lambda \sim 1.3$~$\mu$m) and C ($\lambda \sim 1.55$~$\mu$m) telecommunication bands. The QDs revealed the probability of finding ideal dots with zero FSS as high as 2\% \cite{Liu_PRB14}, which suggests the possibility of actually using a QD as a quantum light device. However, the peak emission wavelength of these dots was shorter than 1.5 $\mu$m, and there are only small numbers of dots in the C telecommunication band, which meets the highest technological demand.
In this paper we report on the further wavelength extension of InAs dots on InP(111)A that we realized by using the ternary alloy In(Al,Ga)As which is lattice matched to InP, as an energy tunable barrier. Thanks to the reduced barrier height of In(Al,Ga)As compared with that of InAlAs, the emission wavelength of these QDs becomes sufficiently long without significant changes in morphology (see Fig. 1(a) for the concept image). As a result, we are able to systematically control the emission wavelength of symmetric QDs over the O, C, and L telecommunication bands.
\begin{figure}
\includegraphics[width=8cm]{Fig_1_Re}%
\caption{\label{fig_concept} (Color online) (a) Schematic of wavelength tuning of InAs quantum dots using a ternary alloy InAlGaAs barrier. (b) Layer sequence of grown samples. }
\end{figure}
\textbf{\textit{Experimental methods.}} We prepared a series of InAs QD samples embedded in different barriers, namely In$_{0.52}$Al$_{0.48}$As, In$_{0.52}$Al$_{0.24}$Ga$_{0.24}$As, and In$_{0.52}$Al$_{0.12}$Ga$_{0.36}$As, all of which are lattice matched to InP. The three samples are denoted respectively as samples a, b, and c. They were grown on a semi-insulating Fe-doped InP(111)A substrate. Figure 1(b) shows the sample structure.
We carried out the growth sequence described below using a solid-source molecular beam epitaxy machine. First, we grew a 100-nm-thick InAlGaAs bottom layer at 470~$^{\circ}$C. Then, we deposited 0.4 monolayers (ML) of indium with a flux of 0.2~ML/s at 320~$^{\circ}$C. This stage led to the formation of indium droplets. Next, we supplied an As$_4$ flux of $3 \times 10^{-5}$~Torr at 270~$^{\circ}$C to crystallize the indium droplets into InAs QDs. While As$_4$ was being supplied we observed the reflection high-energy electron diffraction image, which changed from a halo pattern to a spotty pattern. Following QD growth, we annealed the sample at 370~$^{\circ}$C for 5 min under a weak As$_4$ flux. We then capped the InAs QDs with a 75-nm-thick InAlGaAs layer at 370~$^{\circ}$C. The alloy composition of the capping layer was the same as that of the bottom barrier layer. Finally, we annealed the samples at 470~$^{\circ}$C for 5 min to improve crystal quality. We also prepared samples with InAs QDs on the top InAlGaAs surface without capping for morphology analysis.
The morphology of InAs QDs was studied using atomic force microscopy (AFM). For optical characterization, photoluminescence (PL) spectra were measured using the 532 nm line of a continuous wave diode-pumped laser as an excitation source. The spectra were analyzed using an InGaAs diode array detector with a sensitivity between 0.9 to 1.7 $\mu$m, or a PbS photoconductive detector with a sensitivity of up to 2.5 $\mu$m, depending on the target wavelength. The experiments were performed using a temperature variable closed cycle cryostat, whose base temperature was 9 K.
\textbf{\textit{Results and discussions.}} Figure 2(a), (b), and (c) show AFM top views of uncapped QDs. They reveal the formation of well-isolated QDs with densities of (a) $3 \times 10^9$, (b) $6 \times 10^9$, and (c) $5 \times 10^9$~cm$^{-2}$. Figure 2(d) shows a cross-sectional profile of a typical QD in sample c. Identical cross-sections along orthogonal in-plane directions [01-1] and [-211] support the view that QDs have a laterally symmetric shape without any elongation. This observation is in stark contrast to widely studied S-K grown InAs QDs on InP(100), which exhibit strongly elongated shapes that appear like wires or dashes \cite{Brault_ASS00,Yang_JAP02}. The formation of symmetric QDs is a direct consequence of the use of an InP(111) substrate, which has $C_\text{3v}$ point group symmetry.
\begin{figure}
\includegraphics[width=6cm]{Fig_2_Re}%
\caption{\label{fig_afm}%
(Color online) (a, b, and c) AFM top views of samples a, b, and c, respectively. (b) Cross-sectional profile of a typical QD in sample c along the [01-1] and [-211] in-plane directions.}
\end{figure}
Figure 3(a), (b), and (c) summarize the QD diameter ($D$) and height ($H$) statistics in each sample. The diameter of sample a is distributed around an average value of 38~nm with a standard deviation of 10~nm, expressed as $D = 38\,(\pm 10)$~nm. The height is distributed with $H = 2.9\,(\pm 1.0)$~nm. Thus, the QDs have a flat disk-like shape with heights of 6-10 ML. Note that one monolayer along the [111] direction has a thickness of 0.35~nm in InP. The values for sample b are $D = 30\,(\pm 8)$~nm and $H = 2.6\,(\pm 0.6)$~nm, and those for sample c are $D = 32\,(\pm 8)$~nm and $H = 2.2\,(\pm 0.5)$~nm. Thus, the QD size and aspect ratio are essentially independent of the bottom surface, as InAlAs (sample a) and In(Al,Ga)As (samples b and c) have the same lattice constant and similar reconstructed surfaces. The solid line is a linear fit to the height-diameter distribution of sample a, i.e.,
\begin{equation}
H + 0.38 = 0.88 D \quad \text{(nm)}
\end{equation}
The above dependence is also plotted on the distributions of samples b and c, and will be used as a model structure for numerical simulations.
\begin{figure}
\includegraphics[width=8.7cm]{Fig_3_v2}%
\caption{\label{fig_distribution}%
(a, b, and c) Statistics of the diameters and heights of QDs in samples a, b, and c, respectively. The solid line is a linear fit to the statistics of sample a, shown in Eq.~(1). }
\end{figure}
Figure 4 shows the low temperature PL spectra of samples a, b, and c. They were observed at 9 K. Sample a exhibits a spectrum that covers wavelengths between 1.3 and 1.5 $\mu$m (Fig.~3(a)). The spectrum consists of split peaks, which are attributed to different families of QDs with heights varying in ML steps. The presence of split peaks suggests that the disk-like QDs have an abrupt and atomically flat top interface, as has been confirmed with transmission electron microscopy in similar dot samples \cite{Ha_APL14}.
The vertical lines in Fig. 4 are numerically simulated exciton energies of strained InAs QDs with different ML heights. For simplicity we assume that QDs have a truncated pyramidal shape with analytic height and base variations in Eq.~(1), which we determined by AFM statistical analysis. The calculation was based on the k.p perturbation method with three-dimensional strain modeling (see supplementary information for details) \cite{Ramirez_PRB10,SJC_PRB15}. The theoretical level series well reproduces the experimental spectral peaks. The highest PL peaks are attributed to QDs with heights of 7 and 8 ML, which is consistent with the AFM statistics result.
Figure 4(b) shows the PL spectrum of sample b. It exhibits a spectral shift to longer wavelengths compared with sample a. The spectral red shift occurs due to the use of a QD barrier with a narrower band gap. The main PL peaks are attributed to QDs with heights between 6 and 8 ML, as observed for sample a. It should be emphasized that the spectrum of sample b successfully covers a wavelength of 1.5~$\mu$m, which has the benefit of a low transmission loss for silica telecommunication fibers. Figure 4(c) shows the spectrum of sample c, which exhibits a further red shift. The PL wavelength extends beyond 1.8~$\mu$m, which covers the telecommunication L band (and even the U band). These results demonstrate the practical usefulness of our wavelength tuning technique for QD telecommunication applications.
\begin{figure}
\includegraphics[width=7cm]{Fig_4_v2}%
\caption{\label{fig_spctr}%
PL spectra of InAs QDs in samples a, b, and c at 9 K. The spectra of samples a and b were analyzed using an InGaAs detector, and that of sample c was analyzed using a PbS detector. The vertical lines are the simulated exciton energies of InAs truncated pyramidal dots with heights ranging from 5 to 9 ML.}
\end{figure}
Figure 5(a) shows the PL spectra of sample a at different temperatures. The intensity decreases with increasing temperature, and the multiple peaks shift in unison to a longer wavelength. Note that the signals remained even at 300~K. Figure~5(b) shows the spectral series of sample b, which exhibits a larger intensity reduction with temperature than sample a. The signals almost disappear at temperatures higher than 200~K. Sample c shows a further large intensity reduction, as shown in Fig.~5(c). The observed temperature quenching is associated with the charge carrier escaping from the QDs. In sample a, the large band offset yields strong carrier confinement and high emission stability against thermalization. On the other hand, in samples b and c, the narrow band-gap barriers lead to shallow carrier confinement and a lower emission yield at high temperatures.
\begin{figure}
\includegraphics[width=8.5cm]{Fig_5_v2}%
\caption{\label{fig_quench}%
(Color online) (a, b, and c) PL spectra of samples a, b, and c, respectively, at different temperatures. (d) Dependence of spectrally integrated PL intensities on inverse temperature. The intensities are normalized to those at the low temperature limit. The solid lines are the results of fitting with the Arrhenius-type relaxation model in Eq.~(2).}
\end{figure}
We discuss the impact of carrier thermalization on PL quantitatively using the Arrhenius-type relaxation model. For simplicity we deal with spectrally integrated intensities. Figure 5(d) shows PL intensities as a function of inverse temperature. We analyze the PL intensity data using the following function,
\begin{equation}
I = \frac{I_0}{1+A_1\exp{(-E_1/kT)}+A_2\exp{(-E_2/kT)}}, \quad \text{where} \, E_1 > E_2.
\end{equation}
The model includes two relaxation channels, which have activation energies of $E_1$ and $E_2$ \cite{Bimberg_PRB71,Jahan_JAP13}. As $E_1 > E_2$, the $E_1$ energy specifies PL behaviors at relatively high temperatures, and the $E_2$ energy specifies those at relatively low temperatures. Through fitting, the $E_1$ and $E_2$ energies are extracted for each sample, and summarized in Table 1.
\begin{table}[b]
\caption{Activation energies $E_1$ and $E_2$ obtained from fitting to spectrally integrated PL intensities, conduction and valence band offset energies \cite{Hybertsen_APL91} used for calculation, and electron and hole quantization energies calculated for truncated pyramidal QDs with different heights. }
\begin{ruledtabular}
\begin{tabular}{lcccc}
& & sample a & sample b & sample c \\
& & (meV) & (meV) & (meV) \\ \hline
Activation energy & $E_1$ & 210 & 148 & 18 \\
& $E_2$ & 30 & 30 & -- \\ \hline
Conduction band offset & & 786.0 & 523.5 & 392.3 \\ \hline
Electron quantization energy & 6 ML & 530.9 & 423 & 352.1 \\
& 7 ML & 483.8 & 395.4 & 335.5 \\
& 8 ML & 445.2 & 372 & 320.9 \\ \hline
Valence band offset & & 311.0 & 223.5& 179.8 \\ \hline
Hole quantization energy & 6 ML & 68.2 & 55 & 46.5 \\
& 7 ML & 44.1 & 34.4 & 28.2 \\
& 8 ML & 26.3 &18.8 & 14.2 \\
\end{tabular}
\end{ruledtabular}
\end{table}
The thermal PL quenching in sample~a is described with $E_1 = 210$~meV and $E_2 = 30$~meV. Note that PL quenching is predominantly governed by the $E_1$ term in Eq.~2, and additional minor quenching in a limited temperature range below 150~K is associated with the $E_2$ term. Note that the observed $E_1$ value is consistent with the theoretical expectation. Carrier escape from QDs is characterized by the energy difference between the band-offset energy and the single-carrier quantization energy (see Table 1 for the calculated values). In sample~a, 7 ML high QDs have carrier escape energies of 302 meV for electrons, and 267 meV for holes, and these values agree fairly well with the observed $E_1$ value. The much smaller $E_2$ value comes with another nonradiative channel, possibly related to defects or impurity centers in the barrier.
The PL quenching in sample~b exhibits values of $E_1 = 148$~meV and $E_2 = 30$~meV. Again the $E_1$ value agrees with the theoretical carrier escape energies of 129~meV for electrons and 189~meV for holes in 7~ML high QDs. Steep PL quenching in sample~c is described sufficiently well with a single activation energy $E_1 = 18$~meV, whereas the theoretical energy for carrier escape is 56 meV for electrons and 151 meV for holes. Such shallow QDs possibly suffer from several quenching mechanisms present in the barrier and at the interface, and exhibit lower activation energies than expected simply with carrier confinement.
\textbf{\textit{Conclusions.}} The fabrication of telecom-compatible 1.55~$\mu$m quantum dots has remained a challenge. InP is regarded as an ideal substrate on which to grow InAs QDs that emit at 1.55 $\mu$m, although QDs on InP(100) generally exhibit highly elongated shapes that appear like wires or dashes. The use of high-index InP(311)B yields more symmetric QDs \cite{Akahane_APL08}. However, the dots tend to be very dense, and strong inter-dot coupling makes their application to single dot devices difficult. Here we have successfully demonstrated the simultaneous realization of a symmetric shape and true 1.55 $\mu$m emission from InAs QDs using a $C_\text{3v}$ symmetric InP(111)A substrate. The emission wavelength was systematically tuned by changing the ternary alloy composition of an InAlGaAs barrier without any change in morphology. Thermal quenching is dominantly associated with single carrier escape from QDs. The incorporation of QDs in a double heterostructure possibly keeps the charge carriers in the vicinity of the dots, and might improve high temperature PL efficiency.
This work was partly supported by Grant-in-Aid from the Japan Society of Promotion of Science.
|
2,869,038,154,904 | arxiv | \section{Introduction}
The gas in the solar atmosphere goes from mostly neutral in the
photosphere to highly ionized in the corona.
In the dynamic interface of these two regimes, the
chromosphere and transition region, atoms will ionize
and recombine on various timescales. The ionization will be out of
equilibrium if the ionization/recombination timescale is longer than
the typical hydrodynamic timescale. The ionization balance of
sufficiently abundant atomic species affect the energetics of the
atmosphere, as an ionized atom stores energy that would otherwise
increase the temperature of the gas.
Stellar atmosphere simulations that use the simplifying assumption of
statistical equilibrium (SE) might therefore miss potential
effects of non-equilibrium ionization.
In a numerical model of the solar atmosphere, treating ionization-recombination
processes in detail requires solving the complete radiative transfer
problem - a non-linear, non-local problem which in 3D is too
computationally demanding for present day computers to handle. In simpler geometry,
however, the
situation is different: \cite{carlsson_stein1992, carlsson_stein1995,
carlsson_stein1997} carried out 1D simulations
of a dynamic solar atmosphere where the non-equilibrium ionization and
recombination of abundant elements was included, and they found that the
effects of this are indeed important for the thermodynamic structures of
the atmosphere. In a follow-up study, it was shown that the
relaxation timescales of hydrogen ionization and recombination are long compared to the dynamic
timescales, especially in the cool post-shock phase \citep[hereafter referred to as
CS2002]{carlsson_stein2002}.
This leads to a lower ionization degree in
chromospheric shocks and a higher ionization degree between the shocks,
compared to the statistical equilibrium solution, since the hydrogen
populations do not have time to adjust to the rapidly changing conditions.
Based on a simplified method for treating the radiative transition rates of hydrogen
\citep{sollum1999}, an experiment with hydrogen ionization
was carried out by \cite{leenaarts2006} in 3D. This study confirmed that
hydrogen is out of equilibrium in the chromosphere.
The experiment was repeated by
\cite{leenaarts2007} in 2D, but this time the ionization was included also in the
equation-of-state (EOS), resulting in larger temperature variations in and between
the shocks propagating in the chromosphere than what was found with an
EOS assuming local thermodynamic equilibrium (LTE). Non-equilibrium formation of $\mathrm{H}_2$ was later
included in this method \citep{leenaarts2011} and is currently a part
of the Bifrost stellar atmosphere code
\citep{gudiksen2011}.
\cite{leenaarts2011} pointed out that their
model had a temperature plateau ($\sim 10$~kK ) in the upper
chromosphere and that it most likely was associated with the LTE
treatment of helium in their EOS. Our goal is to realistically treat
the non-equilibrium ionization of both hydrogen and helium in
Bifrost. With such a model we plan to perform studies of the
formation of the spectral lines \ion{He}{1}\ 10830 (formed in the chromosphere) and \ion{He}{2}\
304 (formed in the transition region), both of which are often used diagnostics,
e.g. SDO/AIA \citep{lemen2012}, STEREO's SECCHI/EUVI
\citep{howard2008}, VTT/TIP II: \citep{collados2007}, NST/NIRIS
\citep{cao2012}.
The 10830 line is an absorption line that forms when continuum photons from the
photosphere are scattered or absorbed in the chromosphere by neutral
helium atoms occupying the metastable $2s \,^{3}\!S$ state. This state is
mostly populated by recombination cascades that follow from the photoionization of
neutral helium atoms by the coronal EUV incident
radiation \citep{avrett1994, mauas2005, centeno2008}.
For this reason the line
maps out the boundaries of coronal
holes, which are regions where the coronal EUV emission is weaker
\citep{sheeley1980, harvey2002}. \ion{He}{1}\ 10830 is a valuable line also
for studying active regions where the EUV emission is strong. An
example is \cite{ji2012} who used high resolution imaging data from
NST to study heating events in small scale magnetic loops.
Another application is the study of magnetic fields:
\cite{xu2012} used inversions of the the full Stokes vectors of the
photospheric \ion{Si}{1} 10827 and the chromospheric \ion{He}{1}\ 10830 to
study the magnetic field associated with an active region
filament. Owing to its diagnostic potential, the 10830 line is one of
the candidate lines for the planned Solar-C mission.
\ion{He}{2}\ 304 is an optically thick line that forms in the transition
region. It is an important source of the impinging EUV radiation
absorbed by the chromosphere, hence it might be important for the
spectrum of \ion{He}{1}\ \citep{andretta2003} as well as for the energy balance
of the chromosphere and transition region.
Its formation is still debated. As noted by \cite{jordan1975},
the helium line intensities are larger than what is predicted by
models constructed from observations of other EUV lines. This author introduced the idea of
high energy electrons mixing with cold ions, typically associated
with a steep temperature gradient, to enhance the predicted
intensity. \cite{laming1992} elaborated on a similar idea involving
high energy electrons
in a burst model to match observed intensities. An alternative
view of the 304 formation is the photoioniziation-recombination picture,
where coronal EUV incident radiation photoionizes \ion{He}{2}\ leading to
a recombination cascade ultimately ending up in a 304 photon
\citep{zirin1988}. Both of these processes may be of importance.
\cite{andretta2003} showed that at least for the
quiet sun, there are not enough coronal EUV photons produced
to account for all of the 304 emission.
The line is often used for
the study of filaments and prominences \citep{liewer2009, bi2012,
labrosse2012}. Another recent example of its use is for the study of
spicules \citep{murawski2011} and heating events associated with them
\citep{depontieu2011}.
In this paper we describe a series of 1D radiation-hydrodynamics
simulations similar to those of CS2002. Based on the results we derive
a simplified helium model atom suitable for treating non-equilibrium
helium ionization in 3D numerical models. Additionally, we perform an
initial investigation of non-equilibrium ioniziation effects on the 10830 and
304 spectral lines. In Section \ref{section:method} we describe the
code, simulation setup and the assumptions for the construction of the
simplified model atom, in Section \ref{sec:results} we describe our
results. Finally, in Section \ref{section:conclusions} we draw
conclusions.
\section{Method}\label{section:method}
We use the 1D radiation-hydrodynamics code RADYN
\citep[][CS2002]{carlsson_stein1992, carlsson_stein1995, carlsson_stein1997},
which solves the equations of mass, momentum, charge
and energy conservation, as well as the rate equations, on an adaptive grid.
The simulations include a detailed treatment of
non-equilibrium excitation, ionization and radiative transfer from the
atomic species H, He and Ca. Other elements are also included, but
their contribution to the ionization energy and background opacity is
based on the LTE assumption and read from
a table produced by the Uppsala opacity program
\citep{gustafssen1973}. We devote special attention to the internal
energy balance equation:
\begin{eqnarray}
e=\frac{3kT}{2} \left( n_\mathrm{e}+\sum_{i,j} n_{i,j} \right)
+\sum_{i,j} n_{i,j} \chi_{i,j}
\label{eq:energy}
\end{eqnarray}
where $e$, $n_\mathrm{e}$, $n_{i,j}$ and $\chi_{i,j}$ are the internal energy, electron
density, population density and excitation/ionization energy
corresponding to the $i$-th state of the $j$-th ion. The two terms on the right side
represent the contributions from thermal energy and
ionization/excitation energy.
Hydrogen and singly ionized calcium are
modeled with six level atoms and helium is modeled with a 33 level atom.
Each line is described with 31-101 frequency points, whereas 6-90
frequency points are used for the various continua.
The simulations carried out in this study are very similar to the one
carried out in CS2002 with the major difference being the very detailed helium model atom included here. For more technical information about the
code, we refer to this paper and the references therein.
\subsection{Model atoms}
The hydrogen and calcium model atoms are identical to the ones used in
CS2002.
The helium model atom is a reduced version of an atomic model
extracted from HAOS-DIPER\footnote{The HAO Spectral Diagnostic
Package for Emitted Radiation:
http://www.hao.ucar.edu/modeling/haos-diper}.
This original model has 75
energy levels: the ground state of neutral helium plus 48 excited states (14
singlet and 34 triplet) up to $n=5$, the ground state of
\ion{He}{2}\ and 24 excited states up to $n=5$, and \ion{He}{3}. The energies of
the \ion{He}{1}\ and \ion{He}{2}\ states
are from the National Institute of Standards and Technology
(NIST)\footnote{www.nist.gov} database and \cite{sugar1979}, respectively.
The model atom has 311
transitions: 255 lines and 56 continua. Line oscillator strengths for
neutral helium are from NIST, and for the transitions in
\ion{He}{2}\ from \cite{parpia1982}. All
photoionization cross sections are from the OPACITY
project\footnote{http://cdsweb.u-strasbg.fr/topbase/TheOP.html}. The collisional
rates of bound-bound neutral helium transitions are from
\cite{sawey1993} and the neutral bound-free rates from excited states are
modeled with Seaton's
semi empirical formula for neutrals \citep[Section 18]{allen1973}.
For \ion{He}{2}\ the bound-bound collisional rates are taken from
CHIANTI \citep{dere2009} and the bound-free collisional rates from excited
states are from \cite{burgess1983}. The collisional ionization rate from the
ground states of both \ion{He}{1}\ and \ion{He}{2}\ is
from \cite{ar1985}.
We reduce the number of levels in the atom from 75 to 33 by merging
the neutral helium singlet $n=4$ states into one representative
state. Similarly, the neutral helium singlet $n=5$ states, triplet $n=4$ states and
triplet $n=5$ states are merged into three representative states. Also the
\ion{He}{2}\ $n=4$ and $n=5$ states are merged into two
representative states.
The merging of levels is done by the method described in
\cite{bard2008}, i. e. energies of merged levels are weighted
averages and transition probabilities are computed under the
assumption that the original levels have similar energies
and identical departure coefficients (ratio between population density in non-LTE and LTE).
\subsection{Boundary conditions}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_incrad.eps}
\caption{EUV incident radiation field at the upper boundary
of the computational domain. The data has been binned at the same
resolution as the employed frequency grid, while preserving the
frequency-integrated energy flux.}
\label{fig:fig_incrad}
\end{figure}
The boundary conditions are the same as those in CS2002. Both the
upper and lower boundaries are transmitting. The lower boundary is
located at a fixed geometrical depth, corresponding to 480~km below
$\tau_{500}=1$ in the initial atmosphere. The incoming
characteristics at the lower boundary are prescribed, and they
result in waves that propagate up through the atmosphere. The
upper boundary is located at 10 Mm and has a fixed temperature of
$10^6$~K representing a corona.
The upper boundary is irradiated by a fixed EUV incident
radiation field (see Figure \ref{fig:fig_incrad}) that represents
illumination by the corona. This radiation field is equal to that
derived in \cite{wahlstrom1994} based on data from
\cite{tobiska1991}.
\subsection{Simulations}
\label{sec:simulations}
We carried out three simulations, each running for 3600~s of solar time. The
simulations differ only in the treatment of helium. We used the following
setups: the 33-level model atom with
non-equilibrium population densities (referred to as the NE-run),
the 33-level atom with statistical equilibrium population densities (referred to as
the SE-run) and a 3-level atom (derived in Section \ref{section:simplified_atom})
with non-equilibrium population densities (referred to as the NE3-run).
Several smaller simulations are carried out in addition to the three
main runs in order to determine the relaxation timescales of the helium
ionization/recombination processes. The initial atmospheres of these
runs are snapshots very similar to those found in the NE-run (as they are
from a simulation where a 9 level He model atom was used instead of
the 33 level model atom). These initial atmospheres contain the
statistical equilibrium (SE) solution
for the thermodynamic state of the atmosphere. The
temperature is then increased by 1\% and the populations are allowed to
adjust, during which all other quantities are forced to remain
constant with zero velocity.
In addition, we computed the SE solution of the perturbed atmospheres.
\begin{figure}
\includegraphics[width=\columnwidth]{fig2_ionfracs.eps}
\caption{Panel {\it a}--{\it c}: Helium SE (black) and LTE (red) ion fractions of the
initial atmosphere. The $^{10}$log of the
column mass [g cm$^{-2}$] is indicated in the upper panel.
Panel {\it d}: comparison of the
temperatures assuming
SE (black) and LTE (red) helium ionization in the energy equation
(Eq. \ref{eq:energy}). Panel {\it e}: The terms of
Eq. \ref{eq:energy} assuming SE (black) and LTE (red). The temperature is
sensitive to the levels of helium ionization.}
\label{fig:fig2_ionfracs}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_rateflow1.eps}
\caption{Net transition rates in the \ion{He}{1}\ - \ion{He}{2}\
system of the initial atmosphere. The red and black arrows represent
collisional and radiative transitions, respectively. Their
thickness is proportional to their absolute value; the
maximum net rate is denoted in the figure as Rate. The driver of
the system is the photoionization from the ground state of neutral
helium, which is balanced mainly by radiative
recombination cascades. This picture is qualitatively valid from the temperature
minimum at a height of 0.9~Mm to the transition region at 1.5~Mm.}
\label{fig:hei_transitions}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_rateflow2.eps}
\caption{Net transition rates in the \ion{He}{2}\ - \ion{He}{3}\ system at the
base of the transition region in the same
format as Figure~\ref{fig:hei_transitions}. Similar to
the \ion{He}{1}\ - \ion{He}{2}\ system,
photoionization from the ground state, balanced by radiative
recombination cascades dominate in setting the ionization degree. At
higher temperatures, collisional ionization becomes more
important.}
\label{fig:heii_transitions}
\end{figure}
\subsection{Initial atmosphere}\label{sec:initial_atmosphere}
The initial atmosphere for all runs is at rest and all population densities are in
SE. Figure \ref{fig:fig2_ionfracs} gives a comparison of the ion
fractions, $f=n_{\mathrm{ion}}/n_{\mathrm{total}}$, and the
corresponding LTE values.
At low heights helium is mostly neutral, and only from the
upper chromosphere and upwards ($z>1.2$~Mm) do we find \ion{He}{2}\
fractions above $10^{-3}$. The \ion{He}{2}\ fraction peaks somewhere in the transition region
above which effectively all helium is in the form of \ion{He}{3}. LTE is a
decent approximation in the photosphere, but it fails to reproduce
realistic ion fractions from the chromosphere and up since LTE does
not take into account radiative transitions, whose most important
contribution is the photoionization caused by coronal EUV radiation
\citep{zirin1975}.
Since we suspect that the ionization state of helium has an effect on the
energy balance of the atmospheric gas, we carry out a rough
numeric test of the temperature's sensitivity to the ionization state
of helium. This is done in the following way: we fix the internal energy of the initial
atmosphere, set the helium population densities to their LTE value and
then re-solve the internal energy equation (Eq. \ref{eq:energy}) together with the
charge conservation equation (as ionized helium is a significant source of
electrons).
We compare the energy balance between thermal energy
and ionization energy in the SE and LTE cases in
the two lower panels of Figure \ref{fig:fig2_ionfracs} (note the different height
scale). Moving upwards from the photosphere, the temperature in the SE
and LTE cases starts to deviate in the upper chromosphere.
The density of \ion{He}{2}\ ions is several orders of magnitude higher in SE
than compared to what is predicted by LTE.
This leads to a higher ionization energy, which, since the total
energy is fixed, must result in a decrease of the thermal energy.
Continuing into the transition region the
LTE temperature drops abruptly before it rises to the coronal
value. The sudden drop happens because helium in LTE goes from
neutral to fully ionized in a very short temperature interval centered
at about 20~kK. This leads to an increase in the ionization energy and a
subsequent fall in the thermal energy that is not present with helium
in SE.
This illustrates that the temperature in the solar chromosphere and
transition region is sensitive to the helium
ionization balance.
The ionization state in the upper chromosphere of the initial model atmosphere
is mostly set by
photoionization from the ground states of \ion{He}{1}\ and \ion{He}{2}\ followed by
a radiative recombination cascade through the
excited states back to the ground states. This is illustrated in Figure
\ref{fig:hei_transitions} and \ref{fig:heii_transitions}.
\subsection{Simplified model atom}\label{section:simplified_atom}
Figures~\ref{fig:hei_transitions} and \ref{fig:heii_transitions}
do not show how important the photoionization from the excited states
is, only that it is less frequent than radiative recombination.
We make the simplifying assumption that the excited states
act solely as intermediate steps in recombination cascades, and that
they do not serve as states from which photoionization is taking
place. Based on this assumption, we set up a model atom with only
three levels: ground-state \ion{He}{1}\, ground-state \ion{He}{2}\ and \ion{He}{3}.
In addition to the ordinary collisional and radiative rates between
these levels, we add an extra recombination rate for each of the two
ions, \ion{He}{2}\ and \ion{He}{3}. This extra rate is meant to model the net
recombination to excited states.
The number of photoionizations ($n_iR_{ic}$), spontaneous
recombinations ($n_cR_{ci}^{\mathrm{sp}}$) and induced
recombinations ($n_cR_{ci}^{\mathrm{in}}$) between
a level $i$ and its overlying continuum level $c$
can be expressed as
\begin{eqnarray}
n_iR_{ic} &=& n_i f_{ic}(J_{\nu}) \label{eq:photoionization} \\
n_cR_{ci}^{\mathrm{in}} &=& n_c n_\mathrm{e}
f_{ci}^{\mathrm{in}}(T,J_{\nu}) \label{eq:indrec} \\
n_cR_{ci}^{\mathrm{sp}} &=& n_c n_\mathrm{e}
f_{ci}^{\mathrm{sp}}(T), \label{eq:spontrec}
\end{eqnarray}
where $n_i$ and $n_c$ are the population densities of the
$i$-th and $c$-th level and $J_{\nu}$ is the
(frequency-dependent) mean intensity. The functions $f_{ic}(J_{\nu})$,
$f_{ci}^{\mathrm{in}}(T,J_{\nu})$ and $f_{ci}^{\mathrm{sp}}(T)$ depend
on temperature and mean intensity
\citep{mihalas1978}. The net recombination from the
$c$-th ionization stage to excited states of the $(c-1)$-th ionization stage,
$P_c$, can now be expressed as
\begin{eqnarray}
P_c=n_c n_\mathrm{e}\sum_i(f_{ci}^{\mathrm{sp}} + f_{ci}^{\mathrm{in}})
- \sum_in_i f_{ic},
\end{eqnarray}
where $i$ in the sum represents the excited states.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{fig_recrates.eps}
\end{center}
\caption{Effective recombination rate coefficient, $r_c$, as a function
of temperature for neutral (upper panel) and singly ionized helium
(lower panel). The grey lines correspond to different snapshots
from the NE-run simulation and the black line represents the value chosen for
the model atom. Negative values
correspond to net photoionization from excited states.}
\label{fig:recrate_falc}
\end{figure}
We want to model this with an effective recombination rate
$P_c=n_c n_\mathrm{e}r_c$, where $r_c$
is dependent on
temperature only. This leads to the following expression for $r_c$:
\begin{eqnarray}
r_c = \sum_{i}(f_{ci}^{\mathrm{sp}}(T) + f_{ci}^{\mathrm{in}}(T, J_{\nu})) - \frac{\sum_in_if_{ic}(J_{\nu})}{n_c n_\mathrm{e}}.
\label{eq:effrec}
\end{eqnarray}
If photoionization from excited states and induced recombination are
negligible, $r_c$ will be dominated by spontaneous recombination that is dependent on temperature only.
In Figure \ref{fig:recrate_falc} we show $r_c$ for all snapshots of the NE-run.
The upper panel shows recombination from \ion{He}{2}\ to \ion{He}{1}. At the low temperature end ($T=5$~kK) there is a steep downturn and a horizontal extension toward higher temperatures at $r_2=0$. These are points in the photosphere. Between $T=20$~kK and $T=60$~kK there is net photoionization (negative $r_2$). We choose to model $r_2$ with its time average, except at the low temperature end, where we used an increase in recombination with decreasing temperature. We neglect the net photoionization between $T=20$~kK and $T=60$~kK, as there is very little neutral helium at those temperatures; the error we make is thus small. Similarly we ignore the downturn at the low temperature end. These points correspond to the photosphere where there is little \ion{He}{2}\ and the high density there causes the rates to be dominated by collisional processes.
The lower panel of Figure~\ref{fig:recrate_falc} shows $r_3$. There is very little spread, except in the low temperature end, which confirms the radiative recombination cascade scenario
illustrated in Figure \ref{fig:heii_transitions}. We chose the time-averaged values of $r_3$ for the 3-level model atom, with an extrapolation at the low temperature end.
\section{Results}\label{sec:results}
\subsection{Time-dependent ionization with the full model atom}
\begin{figure*}
\includegraphics[width=\textwidth]{fig_processes.eps}
\caption{Ion fractions and net rates for two representative NE-run
snapshots. Ion fractions from NE3-run are shown for comparison.
The lines with plus signs represent collisional processes.
Each row corresponds to a snapshot. The left column shows
the conditions in the chromosphere and rates into the ground state of
\ion{He}{2}. The right column shows the conditions in the transition
region and the rates into \ion{He}{3}. To be able to better compare the
structure of the transition region in the two runs, we have set
$z=0$~Mm at $T=$100~kK.}
\label{fig:processes}
\end{figure*}
As waves propagate upwards in the atmosphere, the thermodynamic conditions change
too fast for the ion densities to adjust, resulting in
non-equilibrium ion fractions.
Details of the ion fractions and net transition rates from two
snapshots of the NE-run are shown in Figure \ref{fig:processes}. The simulations
reveal, as far as the helium transitions are concerned, two types of
situations: pre-shock and post-shock, which are represented
in the two chosen snapshots.
In the chromospheric pre-shock phase ($t=2930$~s,
$z=[-0.08,-0.02]$~$\mathrm{Mm}$, where $z=0$~Mm is defined to be where the temperature is 100 kK) there is a net
ionization. This is driven by the EUV incident radiation which is
photoionizing neutral helium. Counteracting radiative recombination
to excited states is taking place, but the rates are not large
enough to balance the photoionization.
When the shock passes and compresses the gas, the electron
density increases by both hydrogen ionizing and the compression
itself (CS2002).
The radiative recombination rate coefficient
depends linearly on electron density, whereas the photoionization
rate coefficient is independent of electron density
(Eqs. \ref{eq:photoionization}--\ref{eq:spontrec}). This
results in an increase of the net recombination in the post-shock
phase ($z<-0.08$~Mm). Collisions play only a minor role in the
chromosphere.
In the snapshot $t=2950$~s the post-shock phase is slowly adjusting
itself back to the pre-shock phase where photoionization dominates,
concurrent with the depletion of electrons
which is due to the recombination of hydrogen.
The effects of \ion{He}{3}\ are most important in the transition region where
$T \sim 10-100$~kK, as the relative amount of neutral hydrogen and
helium here is small. Where $T<50$~kK, \ion{He}{2}\ is behaving similarly to
\ion{He}{1}: photoionizations from the ground state of \ion{He}{2},
driven by the EUV incident radiation, is roughly balanced by
radiative recombinations to the excited states. At $T>50$~kK, however,
collisional ionization from the ground state of \ion{He}{2}\ replaces
photoionization as the dominating rate, peaking at $T=90$~kK. Also
note the non-negligible collisional ionization from the
meta-stable \ion{He}{2}\ $2s$ state.
In the transition region pre-shock phase
($t=2930$~s) the radiative recombination dominates, causing a net
recombination. When the shock passes, the
temperature rises due to the compression, and the higher temperature results
in net ionization as the net collisional ionization rates grow large ($t=2950$~s).
When the plasma cools down, the net rates adjust themselves back towards
the pre-shock phase.
\subsubsection{Time-dependent ionization with the simplified model atom}
Ionization fractions obtained with the simplified model atom (from the
NE3-run) are shown together with the NE-run ion fractions in Figure
\ref{fig:processes}. We have replaced 30 atomic states in the
simplified model atom with two extra recombination
rates. Processes involving the removed states are not taken into
account, and this leads to a small shift in the position of the
transition region compared to the NE-run. We have set $z=0$~Mm where
$T=100$~kK in both simulations to be able to compare the
structure of the transition region directly from the figure.
The simplified model atom reproduces the ion fraction from the NE-run
very well, despite the neglected effects. Processes not taken into
account include 3-body recombination to- and
ionization from the excited states of both \ion{He}{1}\ and \ion{He}{2}. Figure
\ref{fig:processes} shows that these effects are subordinate.
\subsection{Relaxation timescales}\label{section:timescales}
\begin{figure}
\includegraphics[width=\columnwidth]{timescale.eps}
\caption{The temperature profiles of three snapshots (upper panel)
and their corresponding relaxation timescales (middle
and lower panel). The middle panel shows the whole computational domain, the lower panel is a blowup of the chromosphere and transition region.
Each snapshot is shown in its own color (red, green and blue) in all panels.
The maximum and minimum timescales obtained
are shown in thick black. The timescale obtained with the simplified
model atom is show for one atmosphere only (thin blue curve with plus signs).}
\label{fig:timescale}
\end{figure}
We define a relaxation timescale in the same way as in CS2002. As
described in Section \ref{sec:simulations}, we have initial ($n_0$),
time-dependent ($n(t)$) and final ($n_{\infty}$) population densities
from representative snapshots where the temperature is perturbed. The
relaxation timescales, $\tau_R$, are obtained numerically by fitting
the time-dependent population density of the ground state of \ion{He}{2}\ to
the general solution of a rate equation, assuming two levels
and constant transition rate coefficients:
\begin{eqnarray}
n(t)=n_{\infty} - (n_{\infty} - n_{0})e^{-t/\tau_R}.
\label{eq:timescales}
\end{eqnarray}
Figure \ref{fig:timescale} shows the the relaxation timescales of the
three representative snapshots featuring both the pre-shock phase and
the post-shock phase. In the corona and photosphere, large rates result
in short relaxation timescales. The relaxation timescale is largest around the
photosphere/chromosphere interface at a column mass of
$^{10}\log{m_c}\approx -3$~g~cm$^{-2}$ with typically a few hundreds of
seconds. In the chromosphere it ranges from a couple of tens and up to
about a hundred seconds.
A wave front is building up at
$^{10}\log{m_c}=-2.7$~g~cm$^{-2}$ in the red atmosphere. At this
stage, a shock has just passed through the transition region and the
chromospheric relaxation timescale is adjusting itself back to the
pre-shock values of about 60-100~s. The shock propagates through the
chromosphere (blue line) and reduces the timescale to its post-shock
value of a few tens of seconds (green line).
In the transition region at $^{10}\log{m_c}=-5.6$~g~cm$^{-2}$ there is
a sharp spike in $\tau_R$, marking the position of where the
ionization shifts from a balance between \ion{He}{1}\ and \ion{He}{2}\ to a balance
between \ion{He}{2}\ and \ion{He}{3}. The definition of $\tau_R$ is based on a two
level system, and will not be very meaningful when three levels are
important at the same time. However, the spike itself is
within the region where the balancing is between \ion{He}{2}\
and \ion{He}{3}, i.e. Eq. \ref{eq:timescales} is a decent approximation and
provides a meaningful estimate.
The simplified model atom reproduces the relaxation timescales we
get with the full 33-level atom very well. The model atom includes the
most important transition processes - clearly these are also the
dominant in setting the timescales of the system.
\subsubsection{Timescales of various processes}
We assume constant transition rates and use the analysis tools
developed in \cite{judge2005}. The set of rate equations in a
static atmosphere constitutes a set of first order differential
equations that can be expressed as a matrix equation:
\begin{eqnarray}
\dot{{\bf n}} = P{\bf n},
\label{eq:rateeq}
\end{eqnarray}
with the solution
\begin{eqnarray}
{\bf n} = \sum_i c_i {\bf v}_i e^{\lambda_i t},
\label{eq:rateeq_solution}
\end{eqnarray}
where $P$ is the rate matrix, ${\bf n}$ is a vector containing the
population densities, ${\bf v}_i$ and $\lambda_i$ are the
corresponding eigenvectors and eigenvalues of the rate matrix and the
coefficients, $c_i$,
are constants depending on the initial conditions. The eigenvectors
represent different relaxation processes. One of the eigenvalues is
zero and the corresponding eigenvector is proportional to the SE
solution. Each of the processes has a corresponding relaxation
timescale, given by $\tau_i=-1/\lambda_i$. The
smallest non-zero eigenvalue (in absolute value since they are all
negative) corresponds to the slowest relaxation process and thereby
determines the relaxation timescale of the system as a whole.
By inspecting the eigenvectors in the 33-level model atom we find that
the balancing between the ground states of \ion{He}{1}\ and \ion{He}{2}\
is the slowest process in the chromosphere. The
second slowest, by a factor of 5-6, is the balancing between the
ground state of \ion{He}{2}\ and \ion{He}{3}. This is
valid in both the pre and post-shock phases. In the transition region
where $T=60$~kK (corresponding to the peak in the
relaxation timescale, see Figure \ref{fig:timescale}) balancing between
the ground state of
\ion{He}{2}\ and \ion{He}{3}\ is the slowest
process. All other processes happen on timescales less
than a second.
\subsection{Non-equilibrium effects}
The relaxation timescales of helium in the chromosphere and transition
region ranges from a few tens of seconds to about one hundred
seconds. When shock waves propagate through these regions the
hydrodynamic conditions may change too fast for the ionization balance
to adjust. Statistical equilibrium is an often-used assumption when
producing synthetic spectra, but this may not be a good assumption for
the spectral lines of helium if there are rapid changes
in the solar atmosphere. To obtain a feel for the types of
errors such an assumption would cause, we perform an initial
investigation of the non-equilibrium effects of \ion{He}{1}\ 10830 and \ion{He}{2}\ 304.
\subsubsection{\ion{He}{2}\ 304}
\begin{figure*}
\includegraphics[width=\textwidth]{fig_int304.eps}
\caption{Formation of the \ion{He}{2}\ 304
line. From left to right: Emergent intensity in the NE-run, emergent intensity in the
SE-run, line core intensities, temperature at line core optical depth unity and fraction of
\ion{He}{2}\ at line core optical depth unity ($\tau=1$). In the three rightmost
panels the NE-run values are black and SE-run values are red. The
line core is defined as the wavelength where the emergent intensity is at
its maximum and indicated by the solid lines in the two leftmost
panels. Also indicated in these panels is the plasma velocity at
$\tau=1$ (dotted lines), which
overlaps almost completely with the doppler shift of
the line core.}
\label{fig:304_int}
\end{figure*}
The resonance line \ion{He}{2}\ 304 has two components,
$1s\,^2\!S^\mathrm{e}_{1/2}-2p\,^{2}\!P^{\mathrm{o}}_{1/2,3/2}$
(excluding the forbidden
$1s\,^2\!S^\mathrm{e}_{1/2}-2s\,^{2}\!S^{\mathrm{e}}_{1/2}$), treated
separately in our simulations.
We investigated the
component with the largest oscillator strength ($1s
\,^2\!S^\mathrm{e}_{1/2}-2p\,^{2}\!P^{\mathrm{o}}_{3/2}$).
Figure \ref{fig:304_int} compares the formation of the 304 line in the
NE-run and the SE-run. A shockwave-induced sawtooth pattern is present in both runs, and
the intensity of the line generally increases as the shock front passes
the transition region where the line forms.
We first discuss the NE line formation.
During the shock front passage the temperature increases on a
timescale shorter than the ionization-recombination timescale (see Section
\ref{section:timescales}), and the \ion{He}{2}\ is therefore not ionized away. The compression work done by the shock is converted mainly into thermal energy and not to ionization energy, and the temperature increases strongly.
Investigation of the rates show that the line
photons are mainly produced by collisional excitation from the ground
state. The large temperature rise leads to increased collisional excitation and thus strong emission in the 304 line as the shock passes the transition region. This is why the line core intensity and temperature at the
formation height are so strongly correlated.
In contrast, in the SE-run \ion{He}{2}\ is instantaneously ionized as the shock front passes.
This has two
consequences, both of which reduce the number of 304 photons produced. First, the
temperature increase is smaller since the increase of energy goes into ionizing helium. Second, the instantaneous ionization to \ion{He}{3}\ lowers the 304 opacity, and shifts the
$\tau=1$ height down, where the temperature is lower. The line intensity in the SE-run is thus lower than in the NE-run.
A good example of this can be seen at $t=2810$~s.
The plasma velocity at the formation height nicely coincides with the
doppler shift of the line. This suggests that the line is formed in
a thin atmospheric layer.
\subsubsection{\ion{He}{1}\ 10830}
\begin{figure}
\includegraphics[width=\columnwidth]{fig_10830.eps}
\caption{Formation of the \ion{He}{1}\ 10830 line profile at
$t=1650$~s. Upper panel: Emergent intensity for the NE-run (solid)
and SE-run (dashed). The red and blue diamonds indicate the
wavelengths used in the lower panel. Lower panel: velocity in the
atmosphere (solid black, scale to the left) and opacity at the two
selected wavelengths (blue and red solid and dashed, scale to the
right). The blue curves correspond to the frequency of the blue
diamond in the upper panel (solid for NE, dashed for SE), the red
curves for the frequency of the red diamond. The red and blue
frequencies are also indicated on the velocity scale by the horizontal
dotted lines in the lower panel. In addition we show the temperature in grey.
By coincidence the line profiles resemble the 10830
blend, so we stress that the profiles shown here are only
the strongest component of the line.
The red
dip in the line is formed in the pre-shock phase where matter is
falling downward, and the blue dip is formed in the shock wave.}
\label{fig:10830}
\end{figure}
The \ion{He}{1}\ 10830 line forms when continuum photons from the
photosphere are scattered and absorbed in the chromosphere by neutral helium atoms
residing in the $2s\,^{3}\!S^{\mathrm{e}}_1$ state in the triplet system
of \ion{He}{1}. This is the lower level of the three 10830 transitions, $2s\,^{3}\!S^{\mathrm{e}}_1-
2p\,^{3}\!P^{\mathrm{o}}_{0,1,2}$.
The triplet states are populated through recombination of \ion{He}{2}\
\citep{avrett1994, mauas2005}. This
recombination happens on very short timescales compared to the
timescales of the ionization (see Section
\ref{section:timescales}). This means that the $2s\,^{3}\!S^\mathrm{e}_1$
population density, and hence the opacity, is always adjusted to the
amount of \ion{He}{2}.
Our simulations treat each of the three components of
the 10830 line separately, and we choose to investigate the component
with the largest oscillator strength ($2s\,^{3}\!S^{\mathrm{e}}_1-
2p\,^{3}\!P^{\mathrm{o}}_{2}$).
The upper panel of Figure \ref{fig:10830} shows the line profiles for both NE and SE runs at $t=1650$~s. The line profile has two depression cores at positive and negative doppler shift relative to the rest wavelength. The NE profile has a much deeper blue-shifted depression then the SE profile.
The lower panel shows the opacity at the wavelengths of the maximum depression together with the structure of the atmosphere. The red-shifted absorption is formed in the downflowing pre-shock material, the blue shifted component in the upflowing material behind the shock front. The doppler shift of the minima of the absorption components is nearly equal to the gas velocity in the line-forming region.
The opacity of the blue-shifted component in the SE-run is much lower
than in the NE-run, explaining the depth difference of the depression cores.
In this region
ionization equilibration timescales are around 50~s (see Figure
\ref{fig:timescale}). The helium population densities of the SE-run
equilibrates instantaneously to the increased electron density
associated with the shock, resulting in a smaller amount of
\ion{He}{2}. Since \ion{He}{2}\ is acting as a reservoir
for the triplet states, the 10830 opacity decreases
accordingly.
\section{Conclusions}\label{section:conclusions}
We have carried out several 1D radiation-hydrodynamic simulations in
order to study the processes determining helium ionization and the
timescales at which they work, and we have performed an initial investigation
of the formation of the \ion{He}{1}\ 10830 and \ion{He}{2}\ 304 lines.
Helium ionization is far from LTE in the upper chromosphere and
transition region. Photoionization and
collisional ionization from the ground states are the determining
processes in setting the level of helium ionization. These processes
work on timescales of up to 100~s. Thermodynamic conditions are
sensitive to the ionization fraction of helium. Using an equation of
state with helium ionization assumed to be in LTE in numerical models will therefore give
erroneous results. To rectify this we have constructed a simplified
3-level helium model atom, based on the driving mechanisms, that
reproduces the non-equilibrium ionization fractions quite well. The rate equations
of the simplified model atom are simple enough for inclusion in
multi dimensional stellar atmosphere codes. We plan to include
non-equilibrium ionization of helium in the 3D radiation-MHD code
Bifrost.
We have shown that the
formation of \ion{He}{1}\ 10830 and \ion{He}{2}\ 304 lines is sensitive to non-equilibrium effects.
Both lines show behavior that is not reproduced when SE is assumed.
We therefore recommend exercising caution
when observations are interpreted on the basis of SE computations.
\acknowledgements
This research was supported by the Research Council of Norway through the grant ``Solar Atmospheric
Modelling''
and by the European Research Council under the European Union's
Seventh Framework Programme (FP7/2007-2013) / ERC Grant
agreement nr. 291058. We would like to thank the referee for helpful comments
in the preparation of this manuscript.
\bibliographystyle{apj}
|
2,869,038,154,905 | arxiv | \section{Introduction}
Regular languages are precisely the behaviours of finite automata. A machine-\penalty0\hskip0pt\relax independent
characterization of regularity is the starting point of algebraic
automata theory (see e.g.~\cite{pin13}): one defines recognition via
preimages of monoid morphisms $f: \Sigma^* \to M$, where $M$ is a finite monoid, and every regular language is recognized in this way by
its syntactic monoid. This suggests to investigate how operations on regular languages relate to operations on monoids. Recall that a \emph{pseudovariety of monoids} is a class of finite monoids closed under finite products, submonoids and quotients (homomorphic images), and a \emph{variety of regular languages} is a class of regular languages closed under the boolean operations (union, intersection and complement), left and right derivatives\footnote[1]{The left and right derivatives of a language $L\subseteq \Sigma^*$ are $w^{-1}L = \{u\in\Sigma^*: wu\in L\}$ and $Lw^{-1}=\{u\in \Sigma^*: uw\in L\}$ for $w\in \Sigma^*$, respectively.} and preimages of monoid morphisms $\Sigma^*\rightarrow\Gamma^*$.
Eilenberg's variety theorem \cite{Eilenberg76}, a cornerstone of automata theory, establishes a lattice isomorphism
\[ \text{varieties of regular languages} ~\cong~ \text{pseudovarieties of monoids}.\] Numerous variations of this correspondence are known, e.g. weakening the closure properties in the definition of a variety \cite{pin95,polak01}, or replacing regular languages by formal power series \cite{reut80}. Recently Gehrke, Grigorieff and
Pin~\cite{ggp08,ggp10} proved a ``local''
version of Eilenberg's theorem: for every fixed alphabet
$\Sigma$, there is a lattice isomorphism between \emph{local varieties of regular languages} (sets of regular languages over $\Sigma$ closed under boolean operations and derivatives) and \emph{local pseudovarieties of monoids} (sets of $\Sigma$-generated finite monoids closed under quotients and subdirect products). At the heart of this result lies the use of Stone duality to relate the boolean algebra of regular languages over $\Sigma$, equipped with left and right derivatives, to the free $\Sigma$-generated profinite monoid.
In this paper we generalize the local Eilenberg theorem to the level of an abstract duality. Our approach starts with the observation that all concepts involved in this theorem are inherently categorical:
\begin{enumerate}[(1)]
\item The boolean algebra $\mathsf{Reg}_\Sigma$ of all regular languages over $\Sigma$ naturally carries the structure of a deterministic automaton whose transitions $L\xrightarrow{a} a^{-1}L$ for $a\in \Sigma$ are given by left derivation and whose final states are the languages containing the empty word. In other words, $\mathsf{Reg}_\Sigma$ is a coalgebra for the functor $T_\Sigma Q = \mathbb{2} \times Q^\Sigma$ on the category of boolean algebras, where $\mathbb{2}=\{0,1\}$ is the two-chain. The coalgebra $\mathsf{Reg}_\Sigma$ can be captured abstractly as the \emph{rational fixpoint} $\rho T_\Sigma$ of $T_\Sigma$, i.e., the terminal locally finite
$T_\Sigma$-coalgebra \cite{m10}.
\item Monoids are precisely the monoid objects in the category of sets, viewed as a monoidal category w.r.t. the cartesian product.
\item The categories of boolean algebras and sets occurring in (1) and (2) are locally finite varieties of algebras (that is, their finitely generated algebras are finite), and the full subcategories of \emph{finite} boolean algebras and \emph{finite} sets are dually equivalent via Stone duality.
\end{enumerate}
Inspired by (3), we call two locally finite varieties $\mathcal{C}$ und $\mathcal{D}$ of (possibly ordered) algebras \emph{predual} if the respective full subcategories $\mathcal{C}_f$ and $\mathcal{D}_f$ of finite algebras are dually equivalent. Our aim is to prove a local Eilenberg theorem for an abstract pair of predual varieties $\mathcal{C}$ and $\mathcal{D}$, the classical case being covered by taking $\mathcal{C}$ = boolean algebras and $\mathcal{D}$ = sets. In this setting deterministic automata are modeled both as coalgebras for the functor
\[T_\Sigma:\mathcal{C}\rightarrow\mathcal{C},\quad T_\Sigma Q = \mathbb{2} \times Q^\Sigma,\]
and as algebras for the functor
\[L_\Sigma: \mathcal{D}\rightarrow\mathcal{D},\quad L_\Sigma A = \mathbb{1} + \coprod_{\Sigma} A,\]
where $\mathbb{2}$ is a two-element algebra in $\mathcal{C}$ and $\mathbb{1}$ is its dual finite algebra in $\mathcal{D}$. These functors are \emph{predual} in the sense that their restrictions $T_\Sigma: \mathcal{C}_f\rightarrow \mathcal{C}_f$ and $L_\Sigma: \mathcal{D}_f\rightarrow \mathcal{D}_f$ to finite algebras are dual, and therefore the categories of finite $T_\Sigma$-coalgebras and finite $L_\Sigma$-algebras are dually equivalent. As a first approximation to the local Eilenberg theorem, we consider the rational fixpoint $\rho T_\Sigma$ for $T_\Sigma$ -- which is always carried by the automaton $\mathsf{Reg}_\Sigma$ of regular languages -- and the initial algebra $\mu L_\Sigma$ for $L_\Sigma$ and establish a lattice isomorphism
\[\text{subcoalgebras of $\rho T_\Sigma$ ~$\cong$~ ideal completion of the poset of finite quotient algebras of $\mu L_\Sigma$}.\]
This is ``almost'' the
desired general local Eilenberg theorem. For the classical case ($\mathcal{C}=$ boolean algebras and $\mathcal{D}$ = sets) one has $\rho T_\Sigma = \mathsf{Reg}_\Sigma$ and $\mu L_\Sigma = \Sigma^*$, and the above isomorphism states that the boolean subalgebras of $\mathsf{Reg}_\Sigma$ closed under \emph{left}
derivatives correspond to sets of finite quotient
algebras of $\Sigma^*$ closed under quotients and
subdirect products. What is missing is the closure under \emph{right} derivatives on the coalgebra side, and quotient algebras of $\Sigma^*$ which are
\emph{monoids} on the algebra side.
The final step is to prove that the above isomorphism restricts to one between \emph{local varieties of regular languages in $\mathcal{C}$} (i.e., subcoalgebras of $\rho T_\Sigma$ closed under right derivatives) and
\emph{local pseudovarieties of $\mathcal{D}$-monoids}. Here a \emph{$\mathcal{D}$-monoid} is a monoid object in the monoidal category $(\mathcal{D},\otimes,\Psi 1)$ where $\otimes$ is the tensor product of algebras and $\Psi 1$ is the free algebra on one generator. In more elementary terms, a $\mathcal{D}$-monoid is an algebra $A$ in $\mathcal{D}$ equipped with a ``bilinear'' monoid
multiplication $A\times A \xrightarrow{\circ} A$, which means that the maps $a\circ -$
and $- \circ a$ are $\mathcal{D}$-morphisms for all $a\in A$. For example, $\mathcal{D}$-monoids in $\mathcal{D}=$ sets, posets, join-semilattices and vector spaces over $\mathds{Z}_2$ are monoids, ordered monoids, idempotent semirings and $\mathds{Z}_2$-algebras (in the sense of algebras over a field), respectively. In all these examples the monoidal category $(\mathcal{D}, \otimes, \Psi 1)$ is \emph{closed}: the set $\mathcal{D}(A,B)$ of homomorphisms between two algebras $A$ and $B$ is an algebra in $\mathcal{D}$ with the pointwise algebraic structure. Our main result is the \\
\noindent\fcolorbox{lightgray}{lightgray}{\parbox{0.98\textwidth}{%
\textbf{General Local Eilenberg Theorem.} Let $\mathcal{C}$ and $\mathcal{D}$ be predual locally finite varieties of algebras, where the algebras in $\mathcal{D}$ are possibly ordered. Suppose further that $\mathcal{D}$ is monoidal closed w.r.t. tensor product, epimorphisms in $\mathcal{D}$ are surjective, and the free algebra in $\mathcal{D}$ on one generator is dual to a two-element algebra in $\mathcal{C}$. Then there is a lattice isomorphism
\[\text{\textbf{local varieties of regular languages in $\boldsymbol{\mathcal{C}}$ ~$\boldsymbol{\cong}$~ local pseudovarieties of $\boldsymbol{\mathcal{D}}$-monoids.}}\]
}}
{~}\\~\\
By applying this to Stone duality ($\mathcal{C}= $ boolean algebras and $\mathcal{D}=$ sets) we recover the ``classical'' local Eilenberg theorem. Birkhoff duality ($\mathcal{C}$ = distributive lattices and $\mathcal{D}= $ posets) gives another result of Gehrke et. al, namely a lattice isomorphism between \emph{local lattice varieties of regular languages} (subsets of $\mathsf{Reg}_\Sigma$ closed under union, intersection and derivatives) and local pseudovarieties of ordered monoids. Finally, by taking $\mathcal{C}=\mathcal{D}= $ join-semilattices and $\mathcal{C}=\mathcal{D}=$ vector spaces over $\mathds{Z}_2$, respectively, we obtain two new local Eilenberg theorems. The first one establishes a lattice isomorphism between \emph{local semilattice varieties of regular languages} (subsets of $\mathsf{Reg}_\Sigma$ closed under union and derivatives) and local pseudovarieties of idempotent semirings, and the second one gives an isomorphism between \emph{local linear varieties of regular languages} (subsets of $\mathsf{Reg}_\Sigma$ closed under symmetric difference and derivatives) and local pseudovarieties of $\mathds{Z}_2$-algebras.
As a consequence of the General Local Eilenberg Theorem we also gain a generalized view of profinite monoids. The dual equivalence between $\mathcal{C}_f$ and $\mathcal{D}_f$ lifts to a dual equivalence between $\mathcal{C}$ and a category $\hat \mathcal{D}$ arising as a profinite completion of $\mathcal{D}_f$. In the classical case we have $\mathcal{C}$ = boolean algebras, $\mathcal{D}$ = sets and $\hat \mathcal{D}$ = Stone spaces, and the dual equivalence between $\mathcal{C}$ and $\hat \mathcal{D}$ is given by Stone duality. Then the $\hat \mathcal{D}$-object dual to the rational fixpoint $\rho T_\Sigma \in \mathcal{C}$ can be equipped with a monoid structure that makes it the free profinite $\mathcal{D}$-monoid on $\Sigma$.
\noindent\fcolorbox{lightgray}{lightgray}{\parbox{0.98\textwidth}{%
\textbf{Theorem.} Under the assumptions of the General Local Eilenberg Theorem, the free profinite $\mathcal{D}$-monoid on $\Sigma$ is dual to the rational fixpoint $\rho T_\Sigma$.
}}
{~}\\~\\
\noindent This extends the corresponding results of Gehrke, Grigorieff and Pin \cite{ggp08} for $\mathcal{D}$ = sets and $\mathcal{D}$ = posets.\\
\noindent The present paper is a revised and extended version of the conference paper \cite{ammu14}, providing full proofs and more detailed examples. In comparison to \emph{loc. cit.} we work with a slightly modified categorical framework in order to simplify the presentation, see Section \ref{sec:oldasm}.
\paragraph{Related work.} Our paper is inspired by the work of Gehrke, Grigorieff and
Pin~\cite{ggp08,ggp10} who showed that the algebraic operation of the free
profinite monoid on $\Sigma$ dualizes to the derivative operations on
the boolean algebra of regular languages (and similarly for the free
ordered profinite monoid on $\Sigma$). Previously, the duality between
the boolean algebra of regular languages and the Stone space of
profinite words appeared (implicitly) in work by
Almeida~\cite{Almeida94} and
was formulated by Pippenger~\cite{Pippenger97} in terms of Stone duality.
A categorical approach to the duality theory of regular languages has been suggested by Rhodes and Steinberg \cite{rs2008}. They introduce the notion of a boolean bialgebra, and prove the equivalence of bialgebras and profinite semigroups. The precise connection to their work is yet to be investigated.
Another related work is Pol\'ak~\cite{polak01} and Reutenauer \cite{reut80}. They
consider what we treat as the example of join-semilattices and vector spaces, respectively, and
obtain a (non-local) Eilenberg theorem for these cases. To the
best of our knowledge the local version we prove does not follow from
the global version, and so we believe that our result is new.
The origin of all the above work is, of course, Eilenberg's
theorem~\cite{Eilenberg76}. Later Rei\-terman~\cite{Reiterman82} proved another characterization of
pseudovarieties of monoids in the spirit of Birkhoff's classical variety
theorem. Reiterman's theorem states
that any pseudovariety of monoids can be characterized by
profinite equations (i.e., pairs of elements of a free profinite monoid). We do not treat profinite
equations in the present paper.
\section{The Rational Fixpoint}
The aim of this section is to recall the rational fixpoint of a functor, which provides a an abstract coalgebraic view of the set of regular languages. As a prerequisite, we need a categorical notion of ``finite automaton'', and so we will work with categories where enough ``finite'' objects exist -- viz. \emph{locally finitely presentable} categories \cite{ar94}.
\begin{defn}
\label{def:lfp}
\begin{enumerate}[(a)]
\item An object $X$ of a category $\mathcal{C}$ is \emph{finitely presentable} if the hom-functor $\mathcal{C}(X,-):\mathcal{C}\rightarrow\mathsf{Set}$ is finitary (i.e., preserves filtered colimits). Let $\mathcal{C}_{f}$ denote the full subcategory of all finitely presentable objects of $\mathcal{C}$.
\item $\mathcal{C}$ is \emph{locally finitely presentable} if it is cocomplete, $\mathcal{C}_{f}$ is small up to isomorphism and every object of $\mathcal{C}$ is a filtered colimit of finitely presentable objects.
\end{enumerate}
\end{defn}
\begin{expl}\label{ex:lfp} Let $\Gamma$ be a finitary signature, that is, a set of operation symbols with finite arity.
\begin{enumerate}[(1)]
\item Denote by $\mathsf{Alg}\,{\Gamma}$ the category of $\Gamma$-algebras and $\Gamma$-homomorphisms. A \emph{variety of algebras} is a full subcategory of $\mathsf{Alg}\,{\Gamma}$ closed under products, subalgebras (represented by injective homomorphisms) and homomorphic images (represented by surjective homomorphisms). Equivalently, by Birkhoff's theorem \cite{Birkhoff35}, varieties of algebras are precisely the classes of algebras definable by equations of the form $s=t$, where $s$ and $t$ are $\Gamma$-terms. Every variety of algebras is locally finitely presentable \cite[Corollary 3.7]{ar94}.
\item Similarly, let $\mathsf{Alg}\,_{\leq} \Gamma$ be the category of ordered $\Gamma$-algebras. Its objects are $\Gamma$-algebras carrying a poset structure for which every $\Gamma$-operation is monotone, and its morphisms are monotone $\Gamma$-homomorphisms. A \emph{variety of ordered algebras} is a full subcategory of $\mathsf{Alg}\,_\leq \Gamma$ closed under products, subalgebras and homomorphic images. Here subalgebras are represented by embeddings (injective $\Gamma$-homomorphisms that are both monotone and order-reflecting), and homomorphic images are represented by surjective $\Sigma$-homomorphisms that are monotone but not necessarily order-reflecting.
Bloom \cite{Bloom1976} proved an ordered analogue of Birkhoff's theorem: varieties of ordered algebras are precisely the classes of ordered algebras definable by inequalities $s\leq t$ between $\Gamma$-terms. From this it is easy to see that every variety of ordered algebras is finitary monadic over the locally finitely presentable\xspace category of posets, and hence locally finitely presentable\xspace \cite[Theorem and Remark 2.78]{ar94}.
\end{enumerate}
In our applications we will work with the varieties in the table below. Observe that all these varieties are \emph{locally finite}, that is, their finitely presentable objects are precisely the finite algebras.
\begin{center}
\begin{tabular}{|lll|}
\hline\rule[11pt]{0pt}{0pt}
$\mathcal{C}$ & objects & morphisms \\
\hline
$\mathsf{Set}$ & sets & functions\\
$\mathsf{BA}$ & boolean algebras & boolean morphisms\\
$\mathsf{DL}_{01}$ & distributive lattices with $0$ and $1\quad$ & lattice morphisms preserving $0$ and $1$\\
${\mathsf{JSL}_0}$ & join-semilattices with $0$ & semilattice morphisms preserving $0$\\
$\Vect{\mathds{Z}_2}\quad$ & vector spaces over the field $\mathds{Z}_2$ & linear maps\\
$\mathsf{Pos}$ & partially ordered sets & monotone functions\\
\hline
\end{tabular}
\end{center}
\end{expl}
{~}
\begin{rem}
For the rest of this paper the term ``variety'' refers to both varieties of algebras and varieties of ordered algebras.
\end{rem}
\begin{notation}
Fix a locally finitely presentable category $\mathcal{C}$ and a finitary endofunctor $T: \mathcal{C}\rightarrow \mathcal{C}$.
\end{notation}
\begin{defn}
\label{nota:fpcoalg}
A \emph{$T$-coalgebra} is a pair $(Q,\gamma)$ of a $\mathcal{C}$-object $Q$ and a $\mathcal{C}$-morphism $\gamma: Q\rightarrow TQ$. A \emph{homomorphism}
\[ h: (Q,\gamma) \rightarrow (Q',\gamma') \]
of $T$-coalgebras is a $\mathcal{C}$-morphism $h: Q\rightarrow Q'$ with $\gamma' \cdot h = Th\cdot \gamma$. We denote by
$\Coalg{T}$ the category of all $T$-coalgebras and their homomorphisms, and by $\FCoalg{T}$ the full subcategory of $T$-coalgebras $(Q,\gamma)$ with finitely presentable carrier $Q$ (in the case where $\mathcal{C}$ is a locally finite variety, these are precisely the finite coalgebras).
\end{defn}
\begin{expl}\label{exp:tcoalg}
Given a finite alphabet $\Sigma$ and an object $\mathbb{2}$ in $\mathcal{C}$, the endofunctor
\[ T_\Sigma = \mathbb{2}\times \mathsf{Id}^\Sigma = \mathbb{2} \times \mathsf{Id}\times \mathsf{Id}\times \ldots \times \mathsf{Id}\] of $\mathcal{C}$ is finitary since in any locally finitely presentable\xspace category filtered colimits commute with finite products. If $\mathcal{C}$ is a locally finite variety and $\mathbb{2}$ is a two-element algebra in $\mathcal{C}$, then $T_\Sigma$-coalgebras are deterministic automata, see e.g. \cite{Rutten2000}. Indeed, by the universal property of the product, to give a coalgebra $Q\xrightarrow{\gamma} T_\Sigma Q=\mathbb{2} \times Q^\Sigma$ means precisely to give an algebra $Q$ (of states), morphisms $\gamma_a:Q\rightarrow Q$ for every $a\in\Sigma$ (representing $a$-transitions) and a morphism $f:Q\rightarrow \mathbb{2}$ (representing final states). Here are two special cases:
\begin{enumerate}[(a)]
\item The usual concept of a deterministic automaton (without initial states) is captured as a coalgebra for $T_\Sigma$ where $\mathcal{C}=\mathsf{Set}$ and $\mathbb{2}=\{0,1\}$.
An important example of a $T_\Sigma$-coalgebra is the automaton $\mathsf{Reg}_\Sigma$ of regular languages. Its states are the regular languages over $\Sigma$, its transitions are
\[ \gamma_a(L) = a^{-1}L \qquad \text{for all $L\in\mathsf{Reg}_\Sigma$ and $a\in\Sigma$,} \]
and the final states are precisely the languages containing the empty word $\varepsilon$.
\item Analogously, consider $T_\Sigma$ as an endofunctor of
$\mathcal{C}=\mathsf{BA}$ with $\mathbb{2}=\{0,1\}$ (the two-element boolean algebra).
A coalgebra for $T_\Sigma$ is a deterministic automaton with a boolean algebra structure on the state set $Q$. Moreover, the transition maps $\gamma_a:Q\rightarrow Q$ are boolean homomorphisms, and the final states (given by the inverse image of $1$ under $f:Q\rightarrow \mathbb{2}$) form an ultrafilter. The above automaton $\mathsf{Reg}_\Sigma$ is also a $T_\Sigma$-coalgebra in $\mathsf{BA}$: the set of regular languages is a boolean algebra w.r.t. the usual set-theoretic operations, left derivatives preserve these operations, and the languages containing $\varepsilon$ form a principal ultrafilter.
\end{enumerate}
\end{expl}
\begin{defn}\label{rem:rat_fix}
\begin{enumerate}[(a)] \item A coalgebra is called \emph{locally finitely presentable} if it is a filtered colimit of coalgebras with finitely presentable carrier. The full subcategory of $\Coalg{T}$ of all locally finitely presentable coalgebras is denoted $\Coalglfp{T}$.
\item The \emph{rational fixpoint} of $T$ is the filtered colimit \[r: \rho T\to T(\rho T)\] of \emph{all} coalgebras with finitely presentable carrier, i.e., the colimit of the diagram $\FCoalg{T}\rightarrowtail \Coalg{T}$.
\end{enumerate}
\end{defn}
The term ``rational fixpoint'' is justified by item (a) in the theorem below.
\begin{thm}[see \cite{m10}]\label{thm:lfp_indcomp}
\begin{enumerate}[(a)]
\vspace{-0.2cm}\item $r$ is an isomorphism.
\item $\rho T$ is the terminal locally finitely presentable $T$-coalgebra, i.e., every locally finitely presentable $T$-coalgebra has a unique coalgebra homomorphism into $\rho T$.
\end{enumerate}
\end{thm}
\begin{expl}
\label{exp:ratfix}
The rational fixpoint of $T_\Sigma:\mathsf{Set}\rightarrow\mathsf{Set}$ is the automaton
$\rho T_\Sigma = \mathsf{Reg}_\Sigma$
of Example \ref{exp:tcoalg}(a), see \cite{amv_atwork}. For any locally finitely presentable $T_\Sigma$-coalgebra $(Q,\gamma)$, the unique homomorphism $(Q,\gamma)\rightarrow \rho T_\Sigma$ maps each state $q\in Q$ to its accepted language
\[ L_q = \{a_1\ldots a_n \in \Sigma^* ~:~ q \xrightarrow{a_1} q_1 \xrightarrow{a_2} q_2 \rightarrow \cdots \xrightarrow{a_n} q_n \text{ for some final state } q_n\}.\]
\end{expl}
This example can be generalized:
\begin{thm}\label{thm:rhotlift}
Suppose that $\mathcal{C}$ is a locally finite variety and $T$ lifts a finitary functor $T_0$ on $\mathsf{Set}$, that is, the following diagram (where $U$ denotes the forgetful functor) commutes:
\[
\xymatrix{
\mathcal{C} \ar[r]^T \ar[d]_{U}& \mathcal{C} \ar[d]^{U}\\
\mathsf{Set} \ar[r]_{T_0} & \mathsf{Set}
}
\]
Then the functor $\mathbb{U}: \Coalg{T}\rightarrow \Coalg{T_0}$ given by \[ Q \xrightarrow{\gamma} TQ \quad\mapsto\quad UQ \xrightarrow{U\gamma} UTQ = T_0 UQ\]
preserves the rational fixpoint, i.e.,
\[\mathbb{U}(\rho T) \cong \rho T_0.\]
\end{thm}
\begin{proof}
The functor $\mathbb{U}$ is finitary (since filtered colimits of $T$-coalgebras are formed on the level of $\mathcal{C}$ and hence on the level of $\mathsf{Set}$) and restricts to finite coalgebras, so we have a commutative square
\[
\xymatrix{
\Coalg{T} \ar[r]^{\mathbb{U}} & \Coalg{T_0}\\
\FCoalg{T} \ar@{>->}[u]^I \ar[r]_{\mathbb{V}} & \FCoalg{T_0} \ar@{>->}[u]_{I_0}
}
\]
where $I$ and $I_0$ are the inclusion functors. We will prove below that $\mathbb{V}$ is cofinal, from which the claim follows:
\begin{align*}
\mathbb{U}(\rho T) &= \mathbb{U}(\mathop{\mathrm{colim}} I) & \text{def. } \rho T\\
&\cong \mathop{\mathrm{colim}}(\mathbb{U}I) & \text{$\mathbb{U}$ finitary}\\
&\cong \mathop{\mathrm{colim}}(I_0 \mathbb{V}) & \text{$\mathbb{U}I = I_0 \mathbb{V}$}\\
&\cong \mathop{\mathrm{colim}}(I_0) & \text{$\mathbb{V}$ cofinal}\\
&= \rho T_0 & \text{def. $\rho T_0$}
\end{align*}
The cofinality of $\mathbb{V}$ amounts to proving that
\begin{enumerate}[(1)]
\item for every finite $T_0$-coalgebra $(Q,\gamma)$ there exists a $T_0$-coalgebra homomorphism $(Q,\gamma)\rightarrow \mathbb{V}(Q',\gamma')$ for some finite $T$-coalgebra $(Q',\gamma')$, and
\item any two such coalgebra homomorphisms are connected by a zig-zag.
\end{enumerate}
Proof of (1). Let $\Phi:\mathsf{Set}\rightarrow\mathcal{C}$ be the left adjoint of the forgetful functor $U: \mathcal{C}\rightarrow \mathsf{Set}$, and denote the unit and counit of the adjunction by $\eta$ and $\varepsilon$, respectively. Given a finite $T_0$-coalgebra $Q\xrightarrow{\gamma} T_0 Q$ form the ``free'' $T$-coalgebra
\[ \Phi Q \xrightarrow{\Phi \gamma} \Phi T_0 Q \xrightarrow{\Phi T_0 \eta_Q} \Phi T_0 U\Phi Q = \Phi UT \Phi Q \xrightarrow{\varepsilon_{T\Phi Q}} T\Phi Q.\]
Note that $\Phi Q$ is finite because $\mathcal{C}$ is locally finite. Then \[\eta_Q: (Q,\gamma) \rightarrow \mathbb{V}(\Phi Q, \varepsilon_{T\Phi Q}\cdot \Phi T_0\eta_Q \cdot \Phi \gamma)\] is a coalgebra homomorphism. Indeed, the diagram below commutes by the naturality of $\eta$ and the triangle identity $U\varepsilon \cdot \eta U = \mathsf{id}$:
\[
\xymatrix{
Q \ar[rrrr]^\gamma \ar[dd]_{\eta_Q} &&&& T_0 Q \ar[dd]^{T_0\eta_Q} \ar@/_3ex/[ddlll]_{\eta_{T_0 Q}} \ar[dl]_{T_0\eta_Q}\\
&&& T_0U\Phi Q \ar@{=}[dr] \ar[dl]_{\eta_{T_0U\Phi Q}} &\\
U\Phi Q \ar[r]_{U\Phi \gamma} & U\Phi T_0 Q \ar[r]_<<<<<{U\Phi T_0\eta_Q} & U\Phi T_0U\Phi Q = U\Phi UT\Phi Q \ar[rr]_{U\varepsilon_{T\Phi Q}}&& UT\Phi Q = T_0 U\Phi Q
}
\]
Proof of (2). Given any coalgebra homomorphism $h: (Q,\gamma)\rightarrow \mathbb{V}(Q',\gamma')$ there exists a unique $\mathcal{D}$-morphism $\overline h: \Phi Q \rightarrow Q'$ with $U\overline{h}\cdot \eta_Q = h$ by the universal property of $\eta$. We claim that $\overline{h}$ is a coalgebra homomorphism
\[\overline{h}: (\Phi Q, \varepsilon_{T\Phi Q}\cdot \Phi T_0\eta_Q \cdot \Phi \gamma) \rightarrow (Q',\gamma').\]
Indeed, the lower square in the diagram below commutes when precomposed with $\eta_Q$, from which the equation $\gamma' \cdot \overline{h} = T\overline{h} \circ \varepsilon_{T\Phi Q}\cdot \Phi T_0\eta_Q \cdot \Phi \gamma$ follows.
\[
\xymatrix{
Q \ar[rrr]^\gamma \ar[d]_{\eta_Q} \ar@/_8ex/[dd]_h &&& T_0 Q \ar[d]^{T_0\eta_Q} \ar@/^8ex/[dd]^{T_0 h}\\
U\Phi Q \ar[d]_{U\overline{h}} \ar[rrr]_{U(\varepsilon_{T\Phi Q}\cdot \Phi T_0\eta_Q \cdot \Phi \gamma)} &&& T_0 U\Phi Q \ar[d]^{T_0U\overline{h}}\\
UQ' \ar[rrr]_{U\gamma'} &&& T_0 UQ'
}
\]
Now given two coalgebra homomorphisms $h: (Q,\gamma)\rightarrow \mathbb{V}(Q',\gamma')$ and $k: (Q, \gamma) \to \mathbb{V}(Q'', \gamma'')$, the desired zig-zag in $\Coalgfp{T}$ is
$\xymatrix@[email protected]{
Q' & \Phi Q \ar[l]_{\overline h} \ar[r]^-{\overline k} & Q''.
}$\qed
\end{proof}
\begin{cor}\label{cor:ratlift}
Let $\mathcal{C}$ be a locally finite variety with a two-element algebra $\mathbb{2}$. Then the rational fixpoint of $T_\Sigma = \mathbb{2} \times \mathsf{Id}^\Sigma: \mathcal{C}\rightarrow \mathcal{C}$ is carried by the automaton $\mathsf{Reg}_\Sigma$ of Example \ref{exp:tcoalg}(a). For any locally finitely presentable $T_\Sigma$-coalgebra $(Q,\gamma)$, the unique homomorphism $(Q,\gamma)\rightarrow \rho T_\Sigma$ maps each state $q\in Q$ to its accepted language.
\end{cor}
\begin{proof}
Apply Theorem \ref{thm:rhotlift} to $T=T_\Sigma$ and $T_0 = \{0,1\}\times \mathsf{Id}^\Sigma$. Since $\rho T_0 = \mathsf{Reg}_\Sigma$ by Example \ref{exp:ratfix}, the claim follows. \qed
\end{proof}
Next we will show that the locally finitely presentable $T$-coalgebras arise as a ``free completion'' of the coalgebras with finitely presentable carrier (Theorem \ref{thm:lfpindcomp} below).
\begin{rem}\label{rem:indcomp}
\begin{enumerate}[(a)]\item Recall that the \emph{free completion under filtered colimits} of a small category $\mathcal{A}$ is a full embedding $\mathcal{A}\hookrightarrow \mathsf{Ind}\,{\mathcal{A}}$ such that $\mathsf{Ind}\,{\mathcal{A}}$ has filtered colimits and every functor $F: \mathcal{A} \rightarrow \mathcal{B}$ into a category $\mathcal{B}$ with filtered colimits has a finitary extension $\overline{F}: \mathsf{Ind}\,{\mathcal{A}}\rightarrow \mathcal{B}$, unique up to natural isomorphim:
\[
\xymatrix{
\mathcal{A} \ar@{>->}[r] \ar[dr]_F & \mathsf{Ind}\,{\mathcal{A}} \ar@{-->}[d]^{\overline{F}} \\
& \mathcal{B}
}
\]
This determines $\mathsf{Ind}\,{\mathcal{A}}$ up to equivalence. If $\mathcal{A}$ has finite colimits then $\mathsf{Ind}\,{\mathcal{A}}$ is locally finitely presentable\xspace and $(\mathsf{Ind}\,{\mathcal{A}})_{f} \cong \mathcal{A}$. Conversely, every locally finitely presentable\xspace category $\mathcal{C}$ arises in this way: $\mathcal{C}\cong\IndC{\mathcal{C}_{f}}$.
\item If $\mathcal{A}$ is a join-semilattice then $\mathsf{Ind}\,{\mathcal{A}}$ is its ideal completion, see Remark \ref{rem:idcomp}.
\end{enumerate}
\end{rem}
\begin{lem}
\label{lem:coref}
Let $\mathcal{B}$ be a cocomplete category and $J : \mathcal{A} \rightarrowtail \mathcal{B}$ be a small full subcategory of finitely presentable objects closed under finite colimits. Then the unique finitary extension $J^* : \IndC{\mathcal{A}} \to \mathcal{B}$ forms a full coreflective subcategory.
\end{lem}
\begin{proof}
By the theorem in Section~VI.1.8 of Johnstone \cite{j82}, we
know that $J^*$ is a full embedding so that $\IndC\mathcal{A}$ can be
identified with the full subcategory of $\mathcal{B}$ given by all
filtered colimits of objects from $\mathcal{A}$. We will show that this
full subcategory is coreflective. Let $B$ be an object of $\mathcal{B}$
and define $\overline B$ to be the colimit of the filtered diagram
\[
\xymatrix@1{
\mathcal{A} /B \ar[r] & \mathcal{A} \ar@{ (->}[r] & \mathcal{B},
}
\]
where the first arrow is the canonical projection functor and the
second one the inclusion functor. We denote the corresponding colimit injections
by
\[
\mathsf{in}_f: A \to \overline B \qquad \text{for every $f: A\to B$ in $\mathcal{A} /B$.}
\]
Clearly, the objects in $\mathcal{A}/B$ form a cocone on the above diagram
and so we have a unique morphism $b: \overline B \to B$ such that
\[
b \cdot \mathsf{in}_f = f \qquad \text{for every $f: A \to B$ in $\mathcal{A}/B$.}
\]
We will now prove this morphism $b$ to be couniversal. To this end, let
$A$ be an object of $\IndC\mathcal{A}$, i.\,e.,
\[
A = \mathop{\mathrm{colim}}\limits_{i \in I} A_i
\]
is a filtered colimit in $\mathcal{B}$ of objects from $\mathcal{A}$ with
colimit injections $a_i: A_i \to A$, $i \in I$. Given a morphism $f:
A \to B$ in $\mathcal{B}$, the morphism $f \cdot a_i: A_i \to B$ is an
object of $\mathcal{A}/B$, and for each connecting morphism $a_{i,j}: A_i
\to A_j$ we have
\[
(f \cdot a_j) \cdot a_{i,j} = f \cdot a_i.
\]
Thus, $a_{i,j}$ is a morphism in $\mathcal{A} /B$ and so
\[
\mathsf{in}_{f \cdot a_j} \cdot a_{i,j} = \mathsf{in}_{f\cdot a_i},
\]
i.\,e., the morphisms $\mathsf{in}_{f \cdot a_i}: A_i \to \overline B$ form a
cocone. So we get a unique $\overline f: A \to \overline B$ such that
\[
\overline f \cdot a_i = \mathsf{in}_{f \cdot a_i} \qquad \text{for every $i \in I$.}
\]
Now the following diagram commutes:
\[
\xymatrix{
&
\overline B
\ar[r]^-b & B
\\
A_i
\ar[ru]^-{\mathsf{in}_{f \cdot a_i}}
\ar[r]_-{a_i}
&
A
\ar[u]_{\overline f}
\ar[ru]_f
}
\]
Indeed, the outside and left-hand triangle commute, and so the
right-hand one commutes when precomposed with every $a_i$, $i \in
I$, whence that triangle commutes since the colimit injections $a_i$
form a jointly epimorphic family.
We still need to show that $\overline f$ is unique such that $b \cdot \overline f =
f$. So assume that $\overline f: A \to \overline B$ is any such morphism. Fix $i
\in I$. Then, since $\overline B$ is a filtered colimit and $A_i$ is
finitely presentable, it follows that there exists some $g: A_i' \to
B$ in $\mathcal{A}/B$ and some morphism $\overline f': A_i \to A_i'$ such that
the square below commutes:
\[
\xymatrix{
A_i'
\ar[r]^-{\mathsf{in}_g}
&
\overline B
\\
A_i
\ar[u]^{\overline f '}
\ar[r]_-{a_i}
&
A
\ar[u]_{\overline f}
}
\]
It follows that $\overline f'$ is a connecting morphism in $\mathcal{A}/B$ from
$f \cdot a_i$ to $g$:
\[
g \cdot \overline f' = b \cdot \mathsf{in}_g \cdot \overline f' = b \cdot \overline f \cdot a_i = f \cdot a_i.
\]
Therefore we get $\mathsf{in}_g \cdot \overline f' = \mathsf{in}_{f \cdot a_i}$ so that
\[
\overline f \cdot a_i = \mathsf{in}_g \cdot \overline f' = \mathsf{in}_{f \cdot a_i},
\]
which determines $\overline f$ uniquely. This completes the proof. \qed
\end{proof}
\begin{thm}\label{thm:lfpindcomp}
$\Coalglfp{T}$ is the $\mathsf{Ind}\,$-completion of $\FCoalg{T}$ and forms a coreflective subcategory of $\Coalg{T}$.
\end{thm}
\begin{proof}
We apply the previous lemma to $\mathcal{A} = \Coalgfp{T}$ and $\mathcal{B} = \Coalg{T}$. Then $\mathcal{B}$ is clearly cocomplete, and $\mathcal{A}$ is closed under finite colimits (since colimits of coalgebras are constructed in the base category and finitely presentable objects are closed under finite colimits). Moreover, as shown in ~\cite{ap04}, every $T$-coalgebra with finitely presentable carrier is a finitely presentable object of $\Coalg{T}$.
Hence Lemma~\ref{lem:coref} yields that $J^*: \IndC{\Coalgfp{T}}\hookrightarrow \Coalg{T}$ is a full coreflective subcategory. The definition of $J^*$ is that it takes formal filtered diagrams of objects in $\Coalgfp{T}$ and constructs their colimits. Therefore its image is precisely $\Coalglfp{T}$, which gives the desired equivalence $\Coalglfp{T} \cong \IndC{\FCoalg{T}}$ and that $\Coalglfp{T}$ is coreflective.\qed
\end{proof}
\section{Algebraic and Coalgebraic Recognition}\label{sec:algcoalgrec}
We are ready to present our first take on the local Eilenberg theorem. At the heart of our approach lies the investigation of a duality for our categories of interest (e.g. Stone duality between finite boolean algebras and finite sets) and the induced algebra-coalgebra duality.
\begin{defn}
Two categories $\mathcal{C}$ and $\mathcal{D}$ are called \emph{predual} if their full subcategories $\mathcal{C}_f$ and $\mathcal{D}_f$ of finitely presentable objects are dually equivalent, that is, $\mathcal{C}_f \cong \mathcal{D}_f^{op}$.
\end{defn}
\begin{expl}\label{expl:dualpairs}
The pairs of locally finite varieties listed in the table below are predual.
\begin{center}
\begin{tabular}{|ll|}
\hline\rule[11pt]{0pt}{0pt}
$\mathcal{C}\quad\quad\quad$ & $\mathcal{D}\quad\quad\quad$ \\%& $P\quad\quad\quad$ & $P^{-1}\quad\quad\quad$\\
\hline
$\mathsf{BA}$ & $\mathsf{Set}$\\
$\mathsf{DL}_{01}$ & $\mathsf{Pos}$\\
${\mathsf{JSL}_0}$ & ${\mathsf{JSL}_0}$\\
$\Vect{\mathds{Z}_2}\quad$ & $\Vect{\mathds{Z}_2}$\\
\hline
\end{tabular}
\end{center}
In more detail:
\begin{enumerate}[(a)]
\item The categories $\mathsf{BA}$ and $\mathsf{Set}$ are predual via Stone duality. The equivalence $\mathsf{BA}_f \xrightarrow{\cong} \mathsf{Set}_f^{op}$ assigns to each finite boolean algebra the set of all atoms, and its associated equivalence $\mathsf{Set}_f^{op} \xrightarrow{\cong} \mathsf{BA}_f$ sends each finite set to the boolean algebra of all subsets.
\item The categories $\mathsf{DL}_{01}$ and $\mathsf{Pos}$ are predual via Birkhoff duality. The equivalence $\mathsf{DL}_{01,f} \xrightarrow{\cong} \mathsf{Pos}_f^{op}$ assigns to each finite distributive lattice the subposet of all join-irreducible elements, and its associated equivalence $\mathsf{Pos}_f^{op} \xrightarrow{\cong} \mathsf{DL}_{01,f}$ sends each finite poset to the lattice of all down-closed subsets.
\item The category ${\mathsf{JSL}_0}$ is self-predual. The equivalence $\mathsf{JSL}_{0,f} \xrightarrow{\cong} \mathsf{JSL}_{0,f}^{op}$ sends each finite join-semilattice $X$ to its dual poset $X^{op}$.
\item The category $\Vect{\mathds{Z}_2}$ is self-predual. The equivalence $\Vect_f{\mathds{Z}_2} \xrightarrow{\cong} (\Vect_f{\mathds{Z}_2})^{op}$ sends each finite $\mathds{Z}_2$-vector space $X$ to the dual space $X^* = \hom(X,\mathds{Z}_2)$ of all linear maps from $X$ to $\mathds{Z}_2$.
\end{enumerate}
\end{expl}
\begin{defn}
Let $\mathcal{C}$ and $\mathcal{D}$ be predual categories. Two functors $T: \mathcal{C} \rightarrow \mathcal{C}$ and $L: \mathcal{D} \rightarrow \mathcal{D}$ are called \emph{predual} if they restrict to functors $T_f: \mathcal{C}_f\rightarrow\mathcal{C}_f$ and $L_f: \mathcal{D}_f\rightarrow\mathcal{D}_f$ and these restrictions are dual, i.e., the following diagram commutes up to natural isomorphism:
\[
\xymatrix{
\mathcal{D}_f^{op} \ar[r]^{L_f^{op}} & \mathcal{D}_f^{op} \\
\mathcal{C}_f \ar[u]^{\cong} \ar[r]_{T_f} & \mathcal{C}_f \ar[u]_{\cong}
}
\]
\end{defn}
\begin{assumptions}\label{asm:sec3}
For the remainder of this paper we fix the following data:
\begin{enumerate}[(a)]
\item $\mathcal{C}$ is a locally finite variety of algebras and $\mathcal{D}$ is a locally finite variety of algebras or ordered algebras.
\item $\mathcal{C}$ and $\mathcal{D}$ are predual with equivalence functors
\[ \widehat{(\mathord{-})} : \mathcal{C}_f \xrightarrow{\cong} \mathcal{D}_f^{op} \quad\text{and}\quad \overline{(\mathord{-})}: \mathcal{D}_f^{op} \xrightarrow{\cong} \mathcal{C}_f.\]
\item $T: \mathcal{C}\rightarrow \mathcal{C}$ and $L:\mathcal{D}\rightarrow\mathcal{D}$ are predual finitary functors. Moreover, $T$ preserves monomorphisms and intersections and $L$ preserves epimorphisms.
\end{enumerate}
\end{assumptions}
\begin{expl}\label{exp:TL}
The endofunctor $T_\Sigma = \mathbb{2}\times \mathsf{Id}^\Sigma$ of $\mathcal{C}$ (see Example \ref{exp:tcoalg}) has the predual endofunctor \[L_\Sigma = \mathbb{1} + \coprod_\Sigma \mathsf{Id}\] of $\mathcal{D}$, where $\mathbb{1} = \widehat{\mathbb{2}}$. Clearly both functors are finitary, $T_\Sigma$ preserves monomorphisms and intersections and $L_\Sigma$ preserves epimorphisms.
\end{expl}
\begin{defn}
\label{not:lalg}
An \emph{$L$-algebra} $(A,\alpha)$ consists of a $\mathcal{D}$-object $A$ and a $\mathcal{D}$-morphism $\alpha:LA\rightarrow A$. A \emph{homomorphism}
\[ h: (A,\alpha) \rightarrow (A',\alpha') \]
of $L$-algebras is a $\mathcal{D}$-morphism $h: A\rightarrow A'$ with $h\cdot \alpha = \alpha'\cdot Lh$.
We denote by $\mathsf{Alg}\,{L}$ the category of $L$-algebras, and by $\FAlg{L}$ the full subcategory of finite $L$-algebras, that is, algebras $(A,\alpha)$ with finite carrier $A$.
\end{defn}
\begin{expl} Algebras for the functor $L_\Sigma$ of Example \ref{exp:TL} correspond to deterministic automata in $\mathcal{D}$ with an initial state but without final states. Indeed, by the universal property of the coproduct an $L_\Sigma$-algebra $L_\Sigma A = \mathbb{1} + \coprod_\Sigma A \xrightarrow{\alpha} A$ is determined by a morphism $i: \mathbb{1} \rightarrow A$ (specifying an initial state) and morphisms $\alpha_a: A\rightarrow A$ for each $a\in \Sigma$ (specifying the $a$-transitions). Here are two special cases:
\begin{enumerate}[(a)]
\item If $\mathcal{C}=\mathsf{BA}$ and $\mathcal{D} = \mathsf{Set}$ then $\mathbb{1}=\widehat{\mathbb{2}}$ is the one-element set since the two-element boolean algebra $\mathbb{2}$ has one atom, so $L_\Sigma$-algebras are precisely the (classical) deterministic automata with an initial state but without final states.
\item Similarly, if $\mathcal{C}=\mathsf{DL}_{01}$ and $\mathcal{D} = \mathsf{Pos}$ then $\mathbb{1}$ is the one-element poset, and $L_\Sigma$-algebras correspond to \emph{ordered} deterministic automata with an initial state but without final states.
\end{enumerate}
\end{expl}
\begin{rem}\label{rem:algcoalgdual}
The categories $\FCoalg{T}$ and $\Algfp{L}$ are dually equivalent. Indeed, the equivalence functor $\widehat{(\mathord{-})}: \mathcal{C}_f \xrightarrow{\cong} \mathcal{D}_f^{op}$ lifts to an equivalence $\FCoalg{T}\xrightarrow{\cong} (\Algfp{L})^{op}$ given by
\[ (Q\xrightarrow{\gamma} TQ) \quad\mapsto\quad (L\widehat{Q} = \widehat{TQ} \xrightarrow{\widehat \gamma} \widehat Q). \]
\end{rem}
\begin{notation}\label{not:order}
\begin{enumerate}
\item By a \emph{subcoalgebra} of a $T$-coalgebra $(Q,\gamma)$ is meant one represented by a homomorphism $m:(Q',\gamma')\rightarrowtail(Q,\gamma)$ with $m$ monic in $\mathcal{C}$. Subcoalgebras are ordered as usual: $m\leq m'$ iff $m$ factorizes through $m'$ in $\Coalg{T}$. We denote by $\Sub{\rho T}$
the poset of all subcoalgebras of $\rho T$, and by
$\FPSub{\rho T}$
the subposet of all \emph{finite subcoalgebras} of $\rho T$.
\item
Likewise, a \emph{quotient algebra} of an $L$-algebra is one represented by an epimorphism in $\mathcal{D}$. Quotient algebras are ordered by $e\leq e'$ iff $e$ factorizes through $e'$ in $\mathsf{Alg}\,{L}$. For the initial $L$-algebra $\mu L$, which exists because $L$ is finitary, we have the posets $\Quofp{\mu L}\subseteq \mathsf{Quo}(\mu L)$ of all (finite) quotient algebras of $\mu L$.
\end{enumerate}
\end{notation}
\begin{rem}\label{rem:algfact}
\begin{enumerate}[(a)]
\item Since $L$ preserves epimorphisms, $\mathsf{Alg}\,{ L}$ has a factorization system consisting of homomorphisms carried by epimorphisms and strong monomorphisms in $\mathcal{D}$, respectively. Indeed,
given any homomorphism $h: (A, \alpha) \to (B, \beta)$ of
$ L$-algebras take its (epi, strong mono)-factorization $h = m \cdot e$ in $\mathcal{D}$
and then use that $ L$ preserves epis and diagonalization to obtain an
algebra structure on the domain of $m$ such that
$m$ and $e$ are $ L$-algebra homomorphisms:
\[
\xymatrix{
L A
\ar@{->>}[d]_{ L e}
\ar[r]^-\alpha
&
A
\ar@{->>}[d]^e
\\
L C
\ar[d]_{ L m}
\ar@{-->}[r]
&
C
\ar@{ >->}[d]^m
\\
L B
\ar[r]_-{\beta}
&
B
}
\]
Moreover, given a commutative square of $L$-algebra homomorphisms, where $e$ is an epimorphism and $m$ is a strong monomorphism in $\mathcal{D}$, the unique diagonal $d$ is easily seen to be a homomorphism of $L$-algebras.
\[
\xymatrix{
(A,\alpha) \ar@{->>}[r]^e \ar[d]_f & (B,\beta)\ar[d]^g \ar[dl]^d \\
(C,\gamma) \ar@{>->}[r]_m & (D,\gamma)
}
\]
\item Dually, since $T$ preserves monomorphisms, $\Coalg{T}$ has a factorization system of homomorphisms carried by strong epimorphisms and monomorphisms in $\mathcal{C}$.
\end{enumerate}
\end{rem}
Using factorizations, locally finitely presentable coalgebras can be described in terms of subcoalgebras.
\begin{prop}\label{prop:locfin}
\begin{enumerate}[(a)]
\item A $T$-coalgebra is locally finitely presentable iff it is locally finite, i.e., every state is contained in a some finite subcoalgebra.
\item Every subcoalgebra of a locally finite coalgebra is locally finite.
\end{enumerate}
\end{prop}
\begin{proof}
(a) Let $(Q,\gamma)$ be a locally finitely presentable coalgebra, i.e., it arises as a filtered $(Q_i,\gamma_i)\xrightarrow{c_i} (Q,\gamma)$ ($i\in I$) of finite coalgebras $(Q_i,\gamma_i)$. Since filtered colimits of $T$-coalgebras are formed on the level of $\mathcal{C}$ and hence on the level of $\mathsf{Set}$, the maps $c_i$ are jointly surjective. It follows that every state $q\in Q$ is contained in $c_i[Q_i]$ for some $i$, and hence in the subcoalgebra of $(Q,\gamma)$ obtained by factorizing $c_i$ as in Remark \ref{rem:algfact}(b).
Conversely, suppose that every state is contained in some finite subcoalgebra of $(Q,\gamma)$. Then the filtered cocone $(Q_i,\gamma_i)\rightarrowtail (Q,\gamma)$ ($i\in I$) of all finite subcoalgebras of $(Q,\gamma)$ is jointly surjective. This implies that $Q_i \rightarrowtail Q$ ($i\in I$) is a filtered colimit in $\mathsf{Set}$ and hence also in $\mathcal{C}$ and $\Coalg{T}$.
(b) Let $Q$ be a subcoalgebra of a locally finite coalgebra $Q'$. Then every state $q\in Q$ is contained in some finite subcoalgebra $Q''$ of $Q'$. Since $T$ preserves intersections, $Q''\cap Q$ is a finite subcoalgebra of $Q$ containing $q$, so $Q$ is locally finite.\qed
\end{proof}
\begin{expl}
A coalgebra for the functor $T_\Sigma Q= \mathbb{2} \times Q^\Sigma$ on $\mathcal{C}$ is locally finitely presentable iff from every state only finitely many states are reachable by transitions.
\end{expl}
\begin{prop}\label{prop:subquolat}
$\Sub{\rho T}$ and $\mathsf{Quo}(\mu L)$ are complete lattices, and $\FPSub{\rho T}$
and $\Quofp{\mu L}$ are join-subsemilattices.
\end{prop}
\begin{proof}
Given a family of subcoalgebras $m_i: (Q_i,\gamma_i)\rightarrowtail \rho T$ ($i\in I$) it is easy to see that their join in $\Sub{\rho T}$ is the subcoalgebra $m: (Q,\gamma) \rightarrowtail \rho T$ obtained by factorizing the homomorphism $[m_i]: \coprod_i (Q_i,\gamma_i) \rightarrow \rho T$ as in Remark \ref{rem:algfact}(b).
\[
\xymatrix{
\coprod_i (Q_i,\gamma_i) \ar[rr]^{[m_i]} \ar@{->>}[dr] && \rho T\\
& (Q,\gamma) \ar@{>->}[ur]_m &
}
\]
If $I$ and all $(Q_i,\gamma_i)$ are finite, then $(Q,\gamma)$ is clearly also finite. This proves that $\Sub{\rho T}$ is a complete lattice and $\FPSub{\rho T}$ is a join-subsemilattice. The corresponding statements about $\mathsf{Quo}(\mu L)$ and $\Quofp{\mu L}$ are shown by a dual argument.\qed
\end{proof}
\begin{prop}
\label{prop:subquo}
The semilattices $\FPSub{\rho T}$ and $\Quofp{\mu L}$ are isomorphic. The isomorphism is given by
\[ (m: (Q,\gamma)\rightarrowtail \rho T) \quad\mapsto\quad (e: \mu L \twoheadrightarrow (\widehat Q, \widehat \gamma))\]
where $e$ is the unique $L$-algebra homomorphism defined by the initiality of $\mu L$.
\end{prop}
\begin{proof}
Consider any finite $T$-coalgebra $(Q,\gamma)$ and its dual finite $L$-algebra $(\widehat Q, \widehat \gamma)$, see Remark \ref{rem:algcoalgdual}. Since $\rho T$ is the terminal locally finite $T$-coalgebra and $\mu L$ is the initial $L$-algebra, there are unique homomorphisms
\[(Q,\gamma)\xrightarrow{m} \rho T\quad\text{and}\quad \mu L \xrightarrow{e} (\widehat Q, \widehat \gamma).\]
We will prove that
\[
\text{$m$ is monic (in $\mathcal{C}$)}
\quad \text{iff} \quad
\text{$e$ is epic (in $\mathcal{D}$).}
\]
from which the claim immediately follows. Assume first that $m$ is monic in $\mathcal{C}$, and let $e=e_2\cdot e_1$ be the factorization of $e$ as in Remark \ref{rem:algfact}(a):
\[
\xymatrix{
\mu L \ar[rr]^e \ar@{->>}[dr]_{e_1} && (\widehat Q,\widehat \gamma)\\
& (A,\alpha) \ar@{>->}[ur]_{e_2} &
}
\]
Since $e_2$ is injective, the algebra $(A,\alpha)$ is finite. Moreover, the strong $\mathcal{D}$-monomorphism $e_2$ is also strongly monic in $\mathcal{D}_{f}$ because the full embedding $\mathcal{D}_{f}\hookrightarrow \mathcal{D}$ preserves epis (since $\mathcal{D}_{f}$ is closed under finite colimits). Hence the dual morphism $\overline{e_2}$ is strongly epic in $\mathcal{C}_{f}$. Since $\rho T$ is the terminal locally finite $T$-coalgebra and $(\overline{A},\overline{\alpha})$ is finite, there exists a unique coalgebra homomorphism $f$ making the triangle below commute:
\[
\xymatrix{
(\overline{\widehat{Q}},\overline{\widehat{\gamma}}) \cong (Q,\gamma) \ar@{>->}[r]^>>>>>m \ar@{->>}[d]_{\overline{e_2}} & \rho T\\
(\overline A,\overline \alpha) \ar[ur]_f&
}
\]
By assumption $m$ is monic, so $\overline{e_2}$ is monic in $\mathcal{C}$ and hence in $\mathcal{C}_{f}$. But $\overline{e_2}$ is also strongly epic in $\mathcal{C}_{f}$, and hence an isomorphism (both in $\mathcal{C}_f$ and $\mathcal{C}$). It follows that $e_2$ is an isomorphism (both in $\mathcal{D}_f$ and $\mathcal{D}$), so $e=e_2\cdot e_1$ is epic in $\mathcal{D}$.
\noindent The converse direction is proved by a symmetric argument.\qed
\end{proof}
\begin{rem}\label{rem:idcomp}
Recall that the \emph{ideal completion} $\Ideal{A}$ of a join-semilattice $A$ is the complete lattice of all ideals (= join-closed downsets) of $A$ ordered by inclusion. Up to isomorphism $\Ideal{A}$ is characterized as a complete lattice containing $A$ such that:
\begin{enumerate}[(1)]
\item every element of $\Ideal{A}$ is a directed join of elements of $A$, and
\item the elements of $A$ are compact in $\Ideal{A}$: if $x\in A$ lies under a directed join of elements $y_i\in \Ideal{A}$, then $x\leq y_i$ for some $i$.
\end{enumerate}
\end{rem}
\begin{thm}\label{thm:latiso}
$\Sub{\rho T}$ is the ideal completion of $\Quofp{\mu L}$.
\end{thm}
\begin{proof}
Since $\FPSub{\rho T} \cong \Quofp{\mu L}$ by Proposition \ref{prop:subquo}, it suffices to prove that $\Sub{\rho T}$ (which forms a complete lattice by Proposition \ref{prop:subquolat}) is the ideal completion of $\FPSub{\mu L}$. To this end we verify the properties (1) and (2) of Remark \ref{rem:idcomp}.
(1) We need to prove that every subcoalgebra $m: (Q,\gamma)\rightarrowtail \rho T$ is a directed join of subcoalgebras in $\FPSub{\rho T}$. The coalgebra $(Q,\gamma)$ is locally finite, being a subcoalgebra of the locally finite coalgebra $\rho T$ (see Proposition \ref{prop:locfin}), and hence a filtered colimit $c_i: (Q_i,\gamma_i)\rightarrow(Q,\gamma)$ ($i\in I$) of coalgebras in $\Coalgfp{T}$. Factorize each $c_i$ as in Remark \ref{rem:algfact}(b):
\[
\xymatrix{
(Q_i,\gamma_i) \ar@{->>}[dr]_{e_i} \ar[rr]^{c_i} && (Q,\gamma) \ar@{>->}[r]^m & \rho T\\
& (Q_i', \gamma_i') \ar@{>->}[ur]_{m_i} &
}
\]
Then $n_i=m\cdot m_i$ ($i\in I$) is a directed set in $\FPSub{\rho T}$. Since $(c_i)$ is colimit cocone, the morphism $[c_i]: \coprod_{i\in I} Q_i\rightarrow Q$ is a strong epimorphism in $\mathcal{C}$. This implies that $\bigcup m_i = id_Q$, hence $\bigcup n_i = m$.
(2) We show that every finite subcoalgebra $m: (Q,\gamma)\rightarrowtail \rho T$ is compact in $\Sub{\rho T}$. So let $n=\bigcup_{i\in I} n_i$ be a directed union of subcoalgebras $n_i: (Q_i,\gamma_i)\hookrightarrow \rho T$ in $\Sub{\rho T}$. Then in $\mathcal{C}$ we also have a directed union $n=\bigcup n_i$ because coproducts in $\Coalg{T}$ are formed on the level of $\mathcal{C}$. Indeed, $\bigcup n_i$ is formed from $[n_i]$ by an image factorization, see the proof of Proposition \ref{prop:subquolat}, and recall from Remark \ref{rem:algfact} that image factorizations in $\Coalg{T}$ are the liftings of (strong epi, mono)-factorizations in $\mathcal{C}$.
Since $Q$ is finite, from $m\subseteq n$ is follows that $m\subseteq n_i$ for some $i\in I$. That is, there exists a morphism $f: Q\rightarrow Q_i$ in $\mathcal{C}$ with $m=n_i\cdot f$. It remains to verify that $f$ is a $T$-coalgebra homomorphism:
\[
\xymatrix{
Q \ar[r]^\gamma \ar@{>->}[d]_f & TQ \ar@{>->}[d]^{Tf} \\
Q_i \ar@{>->}[d]_{n_i} \ar[r]^{\gamma_i} & TQ_i \ar@{>->}[d]^{Tn_i} \\
\rho T \ar[r]_{r} & T(\rho T)
}
\]
Since the lower square and the outer square commute, it follows that the upper square commutes when composed with $Tn_i$. By assumption $T$ preserves monomorphisms, so we can conclude that upper square commutes.\qed
\end{proof}
\begin{expl}
Let $T_\Sigma: \mathsf{BA} \rightarrow \mathsf{BA}$ and $L_\Sigma: \mathsf{Set}\rightarrow\mathsf{Set}$ as in Example \ref{exp:TL}. The rational fixpoint of $T_\Sigma$ is the boolean algebra $\mathsf{Reg}_\Sigma$ with transitions $L \xrightarrow{a} a^{-1}L$, see Corollary \ref{cor:ratlift}, and the initial algebra of $L_\Sigma$ is the automaton $\Sigma^*$ with initial state $\varepsilon$ and transitions $w \xrightarrow{a} wa$ for $a\in \Sigma$. Hence the previous theorem gives a one-to-one correspondence between
\begin{enumerate}[(i)]
\item sets of regular languages over $\Sigma$ closed under boolean operations and left derivatives, and
\item ideals of $\Quofp{\Sigma^*}$, i.e., sets of quotient automata of $\Sigma^*$ closed under quotients and joins.
\end{enumerate}
This correspondence is refined in the following section.
\end{expl}
\section{The Local Eilenberg Theorem}\label{sec:locvar}
In this section we establish our main result, the generalized local Eilenberg theorem. We continue to work under the Assumptions \ref{asm:sec3} -- that is, a locally finite variety of algebras $\mathcal{C}$ and a predual locally finite variety of (possibly ordered) algebras $\mathcal{D}$ are given -- and restrict our attention to deterministic automata, modeled as coalgebras and algebras for the predual functors
\[ T_\Sigma=\mathbb{2}\times \mathsf{Id}^\Sigma:\mathcal{C}\rightarrow\mathcal{C} \quad\text{and}\quad L_\Sigma = \mathbb{1} + \coprod_\Sigma \mathsf{Id}:\mathcal{D}\rightarrow\mathcal{D},\]
respectively. Here $\mathbb{2}=\{0,1\}$ is a fixed two-element algebra in $\mathcal{C}$ and $\mathbb{1} = \widehat{\mathbb{2}}$ is its dual algebra in $\mathcal{D}$.
The crucial step towards Eilenberg's theorem is to prove that the isomorphism
\[ \FPSub{\rho T_\Sigma} \cong \Quofp{\mu L_\Sigma} \]
of Proposition \ref{prop:subquo} restricts to one between the finite subcoalgebras of $\rho T_\Sigma$ closed under right derivatives and the finite quotient algebras of $\mu L_\Sigma$ whose transitions are induced by a monoid structure. To this end we will characterize right derivatives and monoids from a categorical perspective and show that they are dual concepts (Sections \ref{sec:rightder} and \ref{sec:monoids}). The general local Eilenberg theorem is proved in Section \ref{sec:proofloceil}.
\subsection{Right derivatives}\label{sec:rightder}
The closure of the regular languages under right derivatives is usually proved via the following automata construction: suppose a deterministic $\Sigma$-automaton in $\mathsf{Set}$ (with states $Q$ and final states $F\subseteq Q$) accepts a language $L\subseteq \Sigma^*$. Then given $w\in\Sigma^*$ replace the set of final states by
\[ F' = \{q\in Q: q \xrightarrow{w} q' \text{ for some } q'\in F \}.\]
The resulting automaton accepts the right derivative $Lw^{-1} = \{u\in \Sigma^*: uw\in L\}$ of $L$. This construction generalizes to arbitrary $T_\Sigma$-coalgebras:
\begin{notation}\label{not:rqc} $T_\Sigma$-coalgebras $Q\xrightarrow{\gamma} \mathbb{2}\times Q^\Sigma$ are represented as triples
\[(Q,\gamma_a: Q\rightarrow Q, f:Q\rightarrow\mathbb{2}),\]
see Example \ref{exp:tcoalg}. For each $T_\Sigma$-coalgebra $Q=(Q,\gamma_a,f)$ and $w\in\Sigma^*$ we put
\[ Q_w := (Q,\gamma_a, f\cdot \gamma_w)\]
where, as usual, $\gamma_w = \gamma_{a_n}\cdot \cdots \cdot \gamma_{a_1}$ for $w=a_1\cdots a_n$.
\end{notation}
\begin{rem}\label{rem:rqcmor}
A $\mathcal{C}$-morphism $h: Q\rightarrow Q'$ is a $T_\Sigma$-coalgebra homomorphism $h: (Q,\gamma_a,f)\rightarrow (Q',\gamma_a',f')$ iff the following diagram commutes for all $a\in\Sigma$:
\[
\xymatrix{
Q \ar[r]^{\gamma_a} \ar[d]_h & Q \ar[d]^h \ar[dr]^f & \\
Q' \ar[r]_{\gamma_a'} & Q' \ar[r]_{f'} & \mathbb{2}
}
\]
In this case also the square below commutes, which implies that $h$ is a $T_\Sigma$-coalgebra homomorphism $h: Q_w\rightarrow Q'_w$ for all $w\in\Sigma^*$.
\[
\xymatrix{
Q \ar[r]^{\gamma_w} \ar[d]_h & Q \ar[d]^h \\
Q' \ar[r]_{\gamma_w'} & Q'
}
\]
\end{rem}
\begin{proposition}\label{prop:rqc2}
A subcoalgebra $Q$ of $\rho T_\Sigma$ is closed under right derivatives (i.e., $L\in Q$ implies $Lw^{-1}\in Q$ for each $w\in\Sigma^*$) iff there exists a coalgebra homomorphism from $Q_w$ to $Q$ for each $w\in\Sigma^*$.
\end{proposition}
\begin{proof}
A subcoalgebra $Q$ of $\rho T_\Sigma$ is up to isomorphism a set of regular languages over $\Sigma$ carrying a $\mathcal{C}$-algebraic structure and closed under left derivatives, see Corollary \ref{cor:ratlift}.
The ordering of subcoalgebras is induced by inclusion of sets, i.e.,\ $Q \leq Q'$ in $\Sub{\rho T_\Sigma}$ iff $Q \subseteq Q'$.
Suppose that a coalgebra homomorphism $\alpha_w : Q_w \to Q$ exists for each $w \in \Sigma^*$. The languages accepted by $Q_w$ are precisely $\{ L w^{-1} : L \in Q \}$ because we have moved the final states backwards along all $w$-paths. The morphism $\alpha_w$ assigns to each state of $Q_w$ its accepted language, so $\{ L w^{-1} : L \in Q \} \subseteq Q$. Hence $Q$ is closed under right derivatives.
Conversely suppose that $Q \rightarrowtail \rho T_\Sigma$ is closed under right derivatives. Clearly $Q_w$ is locally finite for each $w\in \Sigma^*$, so there are unique homomorphisms $Q_w\rightarrow \rho T_\Sigma$ and $\coprod_w Q_w \rightarrow \rho T_\Sigma$. Factorize them as in Remark \ref{rem:algfact}(b):
\[
Q_w \twoheadrightarrow {\widetilde{Q_w}} \rightarrowtail \rho T_\Sigma
\qquad\qquad
\coprod_{w \in \Sigma^*} Q_w \twoheadrightarrow {\widetilde{Q}} \rightarrowtail \rho T_\Sigma
\]
It is now easy to see (using the factorization system) that
\[
{\widetilde{Q}} = \bigvee \{ {\widetilde{Q_w}} : w \in \Sigma^* \}
\]
in $\Sub{\rho T_\Sigma}$. Since $Q$ is closed under right derivatives we have $\widetilde{Q_w}=\{ Lw^{-1} : L \in Q \} \subseteq Q$ and hence $\widetilde{Q} \subseteq Q$. Since $Q = Q_\varepsilon$ the reverse inclusion holds, so $Q = \widetilde{Q}$. Therefore we obtain a coalgebra homomorphism $Q_w \xrightarrow{in_w} \coprod_{w \in \Sigma^*} Q_w \twoheadrightarrow \widetilde{Q} = Q$.\qed
\end{proof}
\begin{lemma}
\label{lem:finsub_in_finrqc}
Every finite subcoalgebra of $\rho T_\Sigma$ is contained in a finite subcoalgebra of $\rho T_\Sigma$ closed under right derivatives.
\end{lemma}
\begin{proof}
Let $Q \rightarrowtail \rho T_\Sigma$ be an finite subcoalgebra of $\rho T_\Sigma$. Since $\mathcal{C}(Q,Q)$ is finite, there exists a finite set $W\subseteq \Sigma^*$ of words such that for every $u \in \Sigma^*$ there exists $w \in W$ with $Q_u = Q_w$. Then the coalgebra $\coprod_{w \in W} Q_w$ is finite because the coproduct is constructed on the level of $\mathcal{C}$, and $\mathcal{C}_f$ is closed under finite colimits as $\mathcal{C}$ is locally finite. Factorize the unique homomorphism $\coprod_{w\in W} Q_w\rightarrow \rho T_\Sigma$ as is Remark \ref{rem:algfact}(b):
\[
\xymatrix@=0pt{
\coprod_{w \in W} Q_w \ar@{>>}[rrr]^-e &&& \; {Q'} \; \ar@{>->}[rrr]^-m &&& \rho T_\Sigma\\
&&&&&&&\\
Q=Q_\varepsilon \ar[uu]^{in_w} \ar@/_2ex/@{>-->}[uurrr] \ar@{>->}@/_3ex/[uurrrrrr] &&&&&&&
}
\]
The coalgebra ${Q'}$ is clearly also finite. Moreover, if $w$ is chosen such that $Q=Q_\varepsilon=Q_w$, the above diagram yields $Q\rightarrowtail Q'$.
It remains to show that ${Q'}$ is closed under right derivatives. First observe that, for each $w \in \Sigma^*$, one has
\[
(\coprod_{w' \in W} Q_{w'})_w
= \coprod_{w' \in W} (Q_{w'})_w
= \coprod_{w' \in W} Q_{ww'}
\]
because $(-)_w$ commutes with coproducts and $f \circ \gamma_{w'} \circ \gamma_w = f \circ \gamma_{ww'}$. Consider the diagram of coalgebra homomorphisms
\[
\xymatrix{
\coprod_{w' \in W} Q_{ww'} = (\coprod_{w' \in W} Q_{w'})_w
\ar@{->>}[r]^-{e}
\ar[d]_-h
&
Q_w'
\ar@{->}[dd]^-\beta
\ar@{-->}[ddl]^{\alpha_w}
\\
\coprod_{w' \in W} Q_{w'}
\ar@{->>}[d]_e
\\
Q'
\ar@{>->}[r]_m
&
\rho T_\Sigma
}
\]
The strong epic $e$ and mono $m$ were defined above, $\beta$ is the final morphism, the topmost $e$ exists by Remark \ref{rem:rqcmor}, and $h$ exists because each $Q_{ww'}$ equals some $Q_v$ with $v \in W$. Hence the square commutes by finality, and diagonal fill-in gives a coalgebra homomorphism $\alpha_w: Q'_w\rightarrow Q'$. Since $w$ was an arbitrary word we deduce from Proposition \ref{prop:rqc2} that $Q'$ is closed under right derivatives.
\qed
\end{proof}
\begin{cor}\label{cor:finsub_in_finrqc}
Let $Q\rightarrowtail Q' \rightarrowtail \rho T_\Sigma$ be subcoalgebras where $Q$ is finite and $Q'$ is closed under right derivatives. Then $Q$ is contained in a finite subcolgebra of $Q'$ closed under right derivatives.
\end{cor}
\begin{proof}
By the previous lemma there exists a finite subcoalgebra $Q''\rightarrowtail \rho T_\Sigma$ that contains $Q$ and is closed under right derivatives. Since $T_\Sigma$ preserves intersections, $Q''\cap Q'$ is a subcoalgebra of $Q'$ with the desired properties.
\end{proof}
\begin{notation}
$\Subrqc{\rho T_\Sigma}$ and $\FPSubrqc{\rho T_\Sigma}$ are the posets of (finite) subcoalgebras of $\rho T_\Sigma$ closed under right derivatives.
\end{notation}
\begin{proposition}\label{prop:rqcidcomp}
$\Subrqc{\rho T_\Sigma}$ is the ideal completion of $\FPSubrqc{\rho T_\Sigma}$.
\end{proposition}
\begin{proof}
Since $T_\Sigma$ preserves intersections, $\Subrqc{\rho T_\Sigma}$ forms a complete lattice whose meet is set-theoretic intersection. It remains to check the conditions (1) and (2) of Remark \ref{rem:idcomp}.
For (2), just note that directed joins are directed unions of subcoalgebras, so that every finite subcoalgebra closed under right derivatives is compact. For (1) we use that every subcoalgebra $Q \rightarrowtail \rho T_\Sigma$ is locally finite, see Proposition \ref{prop:locfin}, and hence arises as the directed union of its finite subcoalgebras. But by Corollary \ref{cor:finsub_in_finrqc} the poset of all finite subcoalgebras of $Q$ contains the ones closed unter right derivatives as a final subposet, so $Q$ is also the directed union of its finite subcoalgebras closed under right derivatives.
\qed
\end{proof}
\subsection{$\boldsymbol{\mathcal{D}}$-Monoids} \label{sec:monoids}
By Proposition \ref{prop:rqc2} closure under right derivatives of a subcoalgebra $Q\rightarrowtail \rho T_\Sigma$ is characterized by the existence of $T_\Sigma$-coalgebra homomorphisms $Q_w \rightarrow Q$. In this section we investigate the dual $L_\Sigma$-algebra homomorphisms $\widehat{Q}\rightarrow \widehat{Q_w}$ and show that they define a monoid structure on $\widehat{Q}$. This requires the following additional assumptions on the variety $\mathcal{D}$:
\begin{assumptions}\label{asm:geneilenberg} For the rest of Section \ref{sec:locvar} we assume that
\begin{enumerate}[(a)]
\item epimorphisms in $\mathcal{D}$ are surjective;
\item for any two algebras $A$ and $B$ in $\mathcal{D}$, the set $[A,B]$ of homomorphisms from $A$ to $B$ is an algebra in $\mathcal{D}$ with the pointwise algebraic structure, i.e., a subalgebra of the power $B^{\und{A}} = \prod_{x\in A} B$;
\item $\mathbb{1} = \widehat \mathbb{2}$ is a free $\mathcal{D}$-algebra on the one-element set $1$, that is,
\[ \mathbb{1} \cong \Psi 1 \]
for the left adjoint $\Psi: \mathsf{Set}\rightarrow \mathcal{D}$ to the forgetful functor $\mathcal{D}\rightarrow\mathsf{Set}$.
\end{enumerate}
\end{assumptions}
\begin{rem}\label{rem:dmonoidal}
Here is a more categorical view of Assumption \ref{asm:geneilenberg}(b). Given algebras $A$, $B$ and $C$ in $\mathcal{D}$, a \emph{bimorphism} is a function $f: A\times B \rightarrow C$ such that $f(a,\mathord{-}): B\rightarrow C$ and $f(\mathord{-},b): A\rightarrow C$ are $\mathcal{D}$-morphisms for all $a\in A$ and $b\in B$. A \emph{tensor product} of $A$ and $B$ is a universal bimorphism $t: A\times B \rightarrow A\otimes B$, i.e., for every bimorphism $f: A\times B \rightarrow C$ there is a unique $\mathcal{D}$-morphism $f'$ making the diagram below commute.
\[
\xymatrix{
A\times B \ar[dr]_f \ar[r]^t & A\otimes B \ar@{-->}[d]^{f'}\\
& C
}
\]
The tensor product exists in every variety $\mathcal{D}$ and turns it into a symmetric monoidal category $(\mathcal{D},\otimes,\Psi 1)$, see \cite{BN1976}. Assumption \ref{asm:geneilenberg} (b) then states precisely that $\mathcal{D}$ is monoidal closed.
\end{rem}
\begin{expl}
The varieties $\mathcal{D}=\mathsf{Set},\mathsf{Pos},{\mathsf{JSL}_0},\Vect{\mathds{Z}_2}$ of Example \ref{expl:dualpairs} meet the Assumptions \ref{asm:geneilenberg}.
\end{expl}
\begin{rem}\label{rem:ev}
\begin{enumerate}[(1)]
\item For all algebras $A$ and $B$ in $\mathcal{D}$ and $x\in A$ let $\mathsf{ev}_x$ be the composite
\[ [A,B]\rightarrowtail B^{\und{A}} \xrightarrow{\pi_x} B.\]
The morphism $\mathsf{ev}_x$ is evaluation at $x$, i.e.,
\[\mathsf{ev}_x(f) = f(x)\quad \text{ for all } f\in [A,B].\]
\item For any two $\mathcal{D}$-morphisms $f:B\rightarrow B'$ and $g: A \rightarrow A'$, the maps
\[c_f: [A,B]\rightarrow [A,B']\quad\text{and}\quad c_g': [A,B]\rightarrow [A',B]\]
given by composition with $f$ and $g$, respectively, are $\mathcal{D}$-morphisms.
\end{enumerate}
\end{rem}
The assumption that $\mathcal{D}$ is monoidal closed gives rise to inductive definition and proof principles that we shall use extensively.
\begin{definition}[Inductive Extension Principle]\label{def:ext}
Let $(g_i: A\rightarrow B)_{i\in I}$ be a set-indexed family of morphisms between two fixed $\mathcal{D}$-objects $A$ and $B$. Its \emph{inductive extension} is the family $(g_x: A\rightarrow B)_{x\in \Psi I}$ defined as follows:
\begin{enumerate}[(1)]
\item Extend the function $g: I \rightarrow [A,B]$, $i\mapsto g_i$, to a $\mathcal{D}$-morphism $\overline g: \Psi I \rightarrow[A,B]$:
\[
\xymatrix{
I\ar[d]_{\eta} \ar[dr]^-{g} &\\
\Psi I \ar@{-->}[r]_<<<<<{\overline g} & [A,B]
}
\]
\item Put $g_x := \overline g(x)$ for all $x\in\Psi I$.
\end{enumerate}
\end{definition}
\begin{lemma}\label{lem:ext}
\begin{enumerate}[(a)]
\item In Definition \ref{def:ext} we have $g_{\eta i}=g_i$ for all $i\in I$.
\item For all sets $I$ and $\mathcal{D}$-objects $A$, the family $(\mathsf{ev}_x : [\Psi I, A]\rightarrow A)_{x\in \und{\Psi I}}$ is the inductive extension of $(\mathsf{ev}_{\eta i} : [\Psi I,A]\rightarrow A)_{i\in I}$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(a)]
\item For all $x\in A$ we have:
\begin{align*}
\und{g_{\eta i}}(x) &= \und{\mathsf{ev}_{x}}(g_{\eta i}) & \text{def. } & \mathsf{ev}\\
&= \und{\mathsf{ev}_{x}}(\und{\overline g}(\eta i)) & \text{def. } & g_{\eta i}\\
&= \und{\mathsf{ev}_{x}}(g_i) & \text{def. } & \overline g\\
&= \und{g_i}(x) & \text{def. } & \mathsf{ev}
\end{align*}
\item Extend the function $g: I\rightarrow \und{[[\Psi I, A],A]}$, $i\mapsto \mathsf{ev}_{\eta i}$, to a $\mathcal{D}$-morphism $\bar g$:
\[
\xymatrix{
I\ar[d]_{\eta} \ar[dr]^-{g} &\\
\und{\Psi I} \ar[r]_-{\und{\overline g}} & \und{[[\Psi I,A],A]}
}
\]
We need to show that the extended family consists of $g_x = \mathsf{ev}_x$ for all $x\in\und{\Psi I}$. To this end, we first prove the equation
\[ \mathsf{ev}_f\cdot \overline g = f \quad\text{for all } f: \Psi I\rightarrow A.\]
It suffices to prove $\und{\mathsf{ev}_f\cdot \overline g}\cdot\eta = \und{f}\cdot\eta$, and indeed we have for all $i\in I$:
\begin{align*}
\und{\mathsf{ev}_f}(\und{\overline g}(\eta i)) &= \und{\mathsf{ev}_f}(gi) & \text{def. } & \overline g\\
&= \und{\mathsf{ev}_f}(\mathsf{ev}_{\eta i}) & \text{def. } & g\\
&= \und{\mathsf{ev}_{\eta i}}(f) & \text{def. } & \mathsf{ev}\\
&= \und{f}(\eta i) & \text{def. } & \mathsf{ev}
\end{align*}
Therefore, for all $x\in\und{\Psi I}$ and $f: \Psi I\rightarrow A$,
\begin{align*}
\und{g_x}(f) &= \und{\mathsf{ev}_f}(g_x) & \text{def. } \mathsf{ev}\\
&= \und{\mathsf{ev}_f}(\und{\overline g}(x)) & \text{def. } g_x\\
&= \und{f}(x) & \text{eqn. above}\\
&= \und{\mathsf{ev}_x}(f) & \text{def. } \mathsf{ev}
\end{align*}
so $g_x=\mathsf{ev}_x$ as claimed. \qed
\end{enumerate}
\end{proof}
\begin{lemma}[Inductive Proof Principle]\label{lem:indproof}
Let $f,f'$, $h,h'$ and $g_i, g_i'$ ($i\in I$) be $\mathcal{D}$-morphisms as in the diagram $(\ast)$ below.
\[
\xymatrix@=10pt{
& B \ar@{}[ddr]|{(\ast)} \ar[r]^{g_i} & C \ar[dr]^h & && & B \ar@{}[ddr]|{(\ast\ast)} \ar[r]^{g_x} & C \ar[dr]^h & \\
A \ar[ur]^f \ar[dr]_{f'} & & & D && A \ar[ur]^f \ar[dr]_{f'} & & & D \\
& B' \ar[r]_{g_i'} & C' \ar[ur]_{h'} & && & B' \ar[r]_{g_x'} & C' \ar[ur]_{h'} &
}
\]
If $(\ast)$ commutes for all $i\in I$, then $(\ast\ast)$ commutes for all $x\in\Psi I$.
\end{lemma}
\begin{proof}
Suppose that $h\cdot g_i\cdot f = h'\cdot g_i' \cdot f'$ for all $i\in I$. We first prove the equation
\[ c_f'\cdot c_h \cdot \overline g = c_{f'}'\cdot c_{h'} \cdot \overline{g'} \]
where $c_h$, $c_{h'}$, $c_f'$ and $c_{f'}'$ are the $\mathcal{D}$-morphisms of Remark \ref{rem:ev}(b) and $\overline g, \overline {g'}: \Psi I \rightarrow [B,C]$ are as in Definition \ref{def:ext}. Indeed, we have for all $i\in I$:
\begin{align*}
\und{c_f'}(\und{c_h}(\und{\overline g}(\eta i))) &= \und{c_f'}(\und{c_h}(g_i)) & \text{def. } \overline g\\
&= h\cdot g_i \cdot f & \text{def. } c_h, c_f'\\
&= h'\cdot g_i' \cdot f' & \text{assumption}\\
&= \und{c_{f'}'}(\und{c_{h'}}(\und{\overline{g'}}(\eta i))) & \text{compute backwards}
\end{align*}
We conclude that, for all $x\in\Psi I$,
\begin{align*}
h\cdot g_x \cdot f &= \und{c_f'}(\und{c_h}(g_x)) & \text{def. } c_h, c_f'\\
&= \und{c_f'}(\und{c_h}(\und{\overline g}(x))) & \text{def. } g_x\\
&= \und{c_{f'}'}(\und{c_{h'}}(\und{\overline {g'}}(x))) & \text{eqn. above}\\
&= h'\cdot g_x' \cdot f' & \text{compute backwards} \tag*{\qed}
\end{align*}
\end{proof}
\begin{definition}\label{def:bimonoid}
A \emph{$\mathcal{D}-$monoid} $(A,\circ,i)$ consists of an algebra $A$ in $\mathcal{D}$, a bimorphism $\circ: A\times A\rightarrow A$ and an element $i: 1\rightarrow A$ subject to the usual monoid axioms, i.e., the multiplication $\circ$ is associative and $i$ is its unit. A \emph{morphism} of $\mathcal{D}$-monoids
\[ h: (A,\circ,i) \rightarrow (A',\circ',i') \]
is a $\mathcal{D}$-morphism $h:A\rightarrow A'$ that is also a monoid morphism, i.e, it preserves the multiplication and the unit. We denote by $\mathsf{Mon}(\DCat)$ the category of $\mathcal{D}$-monoids, and by $\mathsf{Mon}_{f}(\DCat)$ the full subcategory of all finite $\mathcal{D}$-monoids.
\end{definition}
\begin{rem}
$\mathcal{D}$-monoids are precisely the monoid objects in the monoidal category $(\mathcal{D},\otimes,\Psi 1)$, see Remark \ref{rem:dmonoidal}.
\end{rem}
\begin{expl}
$\mathcal{D}$-monoids in $\mathcal{D}=\mathsf{Set},\mathsf{Pos},{\mathsf{JSL}_0},\Vect{\mathds{Z}_2}$ correspond to monoids, ordered monoids, idempotent semirings and $\mathds{Z}_2$-algebras, respectively.
\end{expl}
\begin{lemma}\label{lem:bimprops}
\begin{enumerate}[(a)]
\item $\mathsf{Mon}(\DCat)$ is complete and $\mathsf{Mon}_{f}(\DCat)$ is finitely complete, with limits formed on the level of $\mathcal{D}$.
\item The (epi, strong mono)-factorization system of $\mathcal{D}$ lifts to $\mathsf{Mon}(\DCat)$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(a)]
\item Let $V:\mathsf{Mon}(\DCat)\rightarrow\mathcal{D}$ denote the forgetful functor. Given a diagram $D: \mathcal{S}\rightarrow \mathsf{Mon}(\DCat)$ where $D_s=(A_s,\circ_s, i_s)$ for $s\in\mathcal{S}$, form the limit cone $(L\xrightarrow{p_s} VD_s)_{s\in\mathsf{S}}$ in $\mathcal{D}$ (which is also a limit cone in $\mathsf{Set}$ as limits in $\mathcal{D}$ are formed one the level of underlying sets). Since \[(\und{L}\times\und{L} \xrightarrow{\und{p_s}\times\und{p_s}} \und{VD_s}\times \und{VD_s} \xrightarrow{\circ_s} \und{VD_s})_{s\in\mathcal{S}}\quad\text{and}\quad (1\xrightarrow{i_s} \und{VD_s})_{s\in\mathcal{S}}\]
are also cones (in $\mathsf{Set}$) over $\und{VD}$, there are unique mediating maps $\circ: \und{L}\times\und{L}\rightarrow \und{L}$ and $i: 1\rightarrow\und{L}$. It is now easy to verify that $(L,\circ,i)$ is $\mathcal{D}$-monoid and that $((L,\circ, i)\rightarrow D_s)_{s\in\mathcal{S}}$ forms a limit cone over $D$ in $\mathsf{Mon}(\DCat)$. This proves the completeness of $\mathsf{Mon}(\DCat)$. The proof that $\mathsf{Mon}_{f}(\DCat)$ is finitely complete is identical -- just start with a finite diagram $D$ and use that $\mathcal{D}_{f}$ is closed under finite limits.
\item Let $f : (M,\bullet,i) \to (M',\bullet',i')$ be a morphism of $\mathcal{D}$-monoids and $M \xrightarrow{e} M_0 \xrightarrow{m} M'$ be its (epi, strong mono)-factorization in $\mathcal{D}$. Then $\und{m}$ is injective and $\und{e}$ is surjective by Assumption \ref{asm:geneilenberg}(a). Consequently, there exists a unique monoid structure $\star$ and $i_0$ on $\und{M_0}$ for which $\und{e}$ and $\und{m}$ are monoid morphisms. All that needs proving is that for every element $x \in \und{M_0}$ the function $x \star -$ is an endomorphism of $M_0$ in $\mathcal{D}$ (and similarly for $- \star x$). Let $d' : M' \to M'$ be the $\mathcal{D}$-morphism
\[
\und{d'} = x' \bullet' -
\qquad
\text{for $x' = \und{m}(x)$.}
\]
Since $\und{e}$ is surjective, we have $y \in \und{M}$ with $\und{e}(y) = x$ and we denote by $d : M \to M$ the $\mathcal{D}$-morphism $\und{d} = y \bullet -$. Then $f \cdot d = d' \cdot f$ because, for all $z\in M$,
\[
\begin{array}{rcl@{\qquad\qquad}p{5cm}}
\und{f \cdot d}(z) & = & \und{f}(y \bullet z) & def.~$d$\\
& = & \und{f}(y) \bullet' \und f (z) & $\und f$ monoid morphism \\
& = & x' \bullet' \und f (z) & since $\und f(y) = x'$ \\
& = & \und{d'}(\und f (z)) & def.~$d'$ \\
& = & \und{d' \cdot f}(z).
\end{array}
\]
The unique diagonal fill in
\[
\xymatrix@=10pt{
M \ar[d]_d \ar[rr]^e && M_0 \ar[d]^m \ar@{-->}[ddll]_{d_0}
\\
M \ar[d]_e && M' \ar[d]^{d'}
\\
M_0 \ar[rr]_m && M'
}
\]
yields a $\mathcal{D}$-morphism $d_0$ with $\und{d_0} = x \star -$. Indeed, given $p \in \und{M_0}$, choose $q \in \und{M}$ such that $p = \und{e}(q)$. Then:
\[
\und{d_0}(p) = \und{d_0 \cdot e}(q) = \und{e} \cdot \und{d}(q) = \und{e}(y \bullet q) = \und{e}(y) \star \und{e}(q) = x \star p. \tag*{\qed}
\]
\end{enumerate}
\end{proof}
Our next goal is to show that the free monoid $(\Sigma^*,\cdot,\varepsilon)$ on $\Sigma$ in $\mathsf{Set}$ extends to a free $\mathcal{D}$-monoid $(\Psi \Sigma^*, \bullet, \eta\varepsilon)$ on $\Sigma$.
\begin{definition}
For every word $w\in\Sigma^*$ the endomaps $w\cdot -$ and $-\cdot w$ of $\Sigma^*$ yield unique $\mathcal{D}$-endomorphisms $l_w$ and $r_w$ of $\Psi\Sigma^*$ making the squares below commute:
\[
\xymatrix{
\Sigma^* \ar[r]^{w \cdot -} \ar[d]_{\eta} & \Sigma^* \ar[d]^{\eta} & \Sigma^* \ar[r]^{-\cdot w} \ar[d]_{\eta} & \Sigma^* \ar[d]^{\eta}\\
\und{\Psi\Sigma^*} \ar[r]_{\und{l_w}} & \und{\Psi\Sigma^*} & \und{\Psi\Sigma^*} \ar[r]_{\und{r_w}} & \und{\Psi\Sigma^*}
}
\]
Let $l_x, r_x: \Psi\Sigma^*\rightarrow \Psi\Sigma^*$ ($x\in \und{\Psi\Sigma^*}$) be the arrows obtained by inductively extending the families $(l_w)_{w\in\Sigma^*}$ and $(r_w)_{w\in\Sigma^*}$, respectively.
\end{definition}
\begin{lemma}\label{lem:lrprops} For all $x,y\in \und{\Psi\Sigma^*}$ the following equations hold:
\begin{enumerate}[(a)]
\item $r_x\cdot l_y = l_y \cdot r_x$
\item $\und{r_y}(x)=\und{l_x}(y)$
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}[(a)]
\item Consider the diagrams below:
\[
\xymatrix{
\Psi\Sigma^* \ar [r]^{r_w} \ar[d]_{l_v} & \Psi\Sigma^* \ar[d]^{l_v} & \Psi\Sigma^* \ar [r]^{r_x} \ar[d]_{l_v} & \Psi\Sigma^* \ar[d]^{l_v} & \Psi\Sigma^* \ar [r]^{r_x} \ar[d]_{l_y} & \Psi\Sigma^* \ar[d]^{l_y} & \\
\Psi\Sigma^* \ar[r]_{r_w} & \Psi\Sigma^* & \Psi\Sigma^* \ar[r]_{r_x} & \Psi\Sigma^* & \Psi\Sigma^* \ar[r]_{r_x} & \Psi\Sigma^*
}
\]
The left square commutes for all $v,w\in\Sigma^*$: indeed, for all $u\in\Sigma^*$ we have
\[ \und{l_v}(\und{r_w}(\eta u)) = \eta(vuw) = \und{r_w}(\und{l_v}(\eta u)) \]
by the definition of $l_v$ and $r_w$. By induction it follows that the middle square commutes for all $v\in\Sigma^*$ and $x\in\und{\Psi\Sigma^*}$. Using induction again we conclude that the right square commutes for all $x,y\in\und{\Psi\Sigma^*}$.
\item We first prove that the following diagram commutes for all $y\in \und{\Psi\Sigma^*}$ (where $\overline l$ is defined as shown in Definition~\ref{def:ext}(1)):
\[
\xymatrix{
\Psi\Sigma^* \ar[r]^-{\overline l} \ar@{=}[d] & [\Psi\Sigma^*,\Psi\Sigma^*] \ar[d]^{\mathsf{ev}_y}\\
\Psi\Sigma^* \ar[r]_-{r_y} & \Psi\Sigma^*
}
\]
By Lemma \ref{lem:ext}(b) and the induction principle it suffices to prove that it commutes for $y=\eta w$ where $w\in\Sigma^*$. In fact, we have for all $v\in\Sigma^*$:
\begin{align*}
\und{\mathsf{ev}_{\eta w}}(\und{\overline l}(\eta v)) &= \und{\mathsf{ev}_{\eta w}}(l_v) & \text{def. } \overline l\\
&= \und{l_v}(\eta w) & \text{def. } \mathsf{ev}\\
&= \eta(vw) & \text{def. } l_v\\
&= \und{r_w}(\eta v) & \text{def. } r_w\\
&= \und{r_{\eta w}}(\eta v) & \text{Lemma \ref{lem:ext}(a)}
\end{align*}
and therefore $\mathsf{ev}_{\eta w}\cdot \overline l = r_{\eta w}$. It follows that, for all $x,y\in\und{\Psi\Sigma^*}$:
\begin{align*}
\und{r_y}(x) &= \und{\mathsf{ev}_y}(\und{\overline l}(x)) & \text{diagram above}\\
&= \und{\mathsf{ev}_y}(l_x) & \text{def. } l_x\\
&= \und{l_x}(y) &\text{def. } \mathsf{ev} \tag*{\qed}
\end{align*}
\end{enumerate}
\end{proof}
\begin{definition}
We define a multiplication on $\und{\Psi\Sigma^*}$ as follows:
\[ x\bullet y := \und{r_y}(x) = \und{l_x}(y) \quad\text{ for all } x,y\in \und{\Psi\Sigma^*}.\]
\end{definition}
\begin{proposition}\label{prop:freebim}
$(\Psi\Sigma^*,\bullet,\eta\varepsilon)$ is the free $\mathcal{D}$-monoid on $\Sigma$: for any $\mathcal{D}$-monoid $(A,\circ,i)$ and any function $f: \Sigma\rightarrow \und{A}$, there is a unique extension to a $\mathcal{D}$-monoid morphism $\overline f: \Psi\Sigma^*\rightarrow A$.
\[
\xymatrix{
\Sigma^* \ar[r]^-\eta & \und{\Psi\Sigma^*} \ar@{-->}[d]^{\und{\overline f}}\\
\Sigma \ar@{ >->}[u] \ar[r]_f & \und{A}
}
\]
\end{proposition}
\begin{proof}
We first show that $(\Psi\Sigma^*,\bullet,\eta\varepsilon)$ is a $\mathcal{D}$-monoid. Indeed:
\begin{enumerate}[(1)]
\item $\bullet$ is associative: for all $x,y,z\in\und{\Psi\Sigma^*}$ we have
\[x\bullet (y\bullet z) = \und{l_x}(\und{r_z}(y)) = \und{r_z}(\und{l_x}(y)) = (x\bullet y)\bullet z\]
using the definition of $\bullet$ and Lemma \ref{lem:lrprops}(a).
\item $\eta \varepsilon$ is the neutral element: for all $x\in\und{\Psi\Sigma^*}$ we have
\[x\bullet \eta \varepsilon = \und{r_{\eta\varepsilon}}(x) =\und{r_\varepsilon}(x) = \und{\mathsf{id}}(x) = x\]
and symmetrically $\eta\varepsilon \bullet x = x$.
\item $\bullet$ is a $\mathcal{D}$-bimorphism since, for all $x\in\und{\Psi\Sigma^*}$, the functions $l_x = x\bullet -$ and $r_x = -\bullet x$ are $\mathcal{D}$-morphisms.
\end{enumerate}
It remains to verify the universal property. Given $f$ as in the diagram above, one can first extend $f$ to a monoid morphism $f': \Sigma^*\rightarrow \und{A}$ (using that $\Sigma^*$ is the free monoid on $\Sigma$) and then extend $f'$ to a $\mathcal{D}$-morphism $\overline f:\Psi\Sigma^*\rightarrow A$ with $\und{\overline f}\cdot \eta = f'$ (by the universal property of $\eta$). We only need to verify that $\und{\overline f}$ is a monoid morphism. Firstly, $\und{\overline f}$ preserves the unit:
\begin{align*}
\und{\overline f}(\eta \varepsilon) &= f'(\varepsilon) & \text{def. } \overline f\\
&= i & f' \text{ monoid morphism}
\end{align*}
To prove that $\und{\overline f}$ also preserves the multiplication, consider the squares below where $l_x', r_w': A\rightarrow A$ are the $\mathcal{D}$-morphisms $\und{l_x'}=\und{\overline f}(x)\circ -$ and $\und{r_w'}=-\circ \und{\overline f}(\eta w)$.
\[
\xymatrix{
\Psi\Sigma^* \ar[r]^{r_w} \ar[d]_{\overline f} & \Psi\Sigma^* \ar[d]^{\overline f} & \Psi\Sigma^* \ar[r]^{l_x} \ar[d]_{\overline f} & \Psi\Sigma^* \ar[d]^{\overline f} \\
A \ar[r]_{r_w'} & A & A \ar[r]_{l_x'} & A
}
\]
The left square commutes for all $w\in\Sigma^*$ because, for all $v\in\Sigma^*$,
\begin{align*}
\und{\overline f}(\und{r_w}(\eta v)) &= \und{\overline f}(\eta(vw)) & \text{def. } r_w\\
&= f'(vw) & \text{def. } \overline f\\
&= f'(v)\circ f'(w) & f' \text{ monoid morphism}\\
&= \und{\overline f}(\eta v)\circ \und{\overline f}(\eta w) & \text{def. } \overline f\\
&= \und{r_{w}'}(\und{\overline f}(\eta v)) & \text{def. } r_w'
\end{align*}
Then also the right square commutes for all $x\in\und{\Psi\Sigma^*}$ because, for all $w\in\Sigma^*$,
\begin{align*}
\und{\overline f}(\und{l_x}(\eta w)) &= \und{\overline f}(\und{r_w}(x)) & \text{Lemma \ref{lem:lrprops}(b)}\\
&= \und{r_w'}(\und{\overline f}(x)) & \text{left square}\\
&= \und{\overline f}(x)\circ \und{\overline f}(\eta w) & \text{def. } r_w'\\
&= \und{l_x'}(\und{\overline f}(\eta w)) & \text{def. } l_x'
\end{align*}
We conclude that, for all $x,y\in\und{\Psi\Sigma^*}$,
\begin{align*}
\und{\overline f}(x\bullet y) &= \und{\overline f}(\und{l_x}(y)) & \text{def. } \bullet\\
&= \und{l_x'}(\und{\overline f}(y)) & \text{right square}\\
&= \und{\overline f}(x)\circ \und{\overline f}(y) & \text{def. } l_x'\tag*{\qed}
\end{align*}
\end{proof}
\begin{expl}
\begin{enumerate}[(a)]
\item For $\mathcal{D}=\mathsf{Set}$ or $\mathsf{Pos}$ we have $\Psi\Sigma^*=\Sigma^*$ (discretely ordered in the case $\mathcal{D}=\mathsf{Pos}$). The monoid multiplication is concatenation of words, and the unit is $\varepsilon$.
\item For $\mathcal{D}={\mathsf{JSL}_0}$ we have $\Psi\Sigma^*=\Pow_\omega \Sigma^*$, the semilattice of all finite languages over $\Sigma$ w.r.t. union. The monoid multiplication is concatenation of languages, and the unit is the language $\{\varepsilon\}$.
\item For $\mathcal{D}=\Vect{\mathds{Z}_2}$ we have $\Psi\Sigma^*=\mathcal{P} \Sigma^*$, the vector space of all finite languages over $\Sigma$ where vector addition is symmetric difference. The monoid multiplication is the $\mathds{Z}_2$-weighted concatenation $L\otimes L'$ of languages, (i.e., $L \otimes L'$ consists of all words $w$ having an odd number of factorizations $w=uu'$ with $u\in L$ and $u'\in L'$), and the unit is again $\{\varepsilon\}$.
\end{enumerate}
\end{expl}
\begin{definition}\begin{enumerate}[(a)]
\item A \emph{$\Sigma$-generated $\mathcal{D}$-monoid} is a quotient of $\Psi\Sigma^*$, i.e., a $\mathcal{D}$-monoid morphism $e: \Psi\Sigma^*\twoheadrightarrow A$ with $e$ epic in $\mathcal{D}$. A \emph{morphism} between two $\Sigma$-generated $\mathcal{D}$-monoids $e: \Psi\Sigma^*\twoheadrightarrow A$ and $e': \Psi\Sigma^*\rightarrow A'$ is a generator-preserving $\mathcal{D}$-monoid morphism $f: A\rightarrow A'$, i.e., $f\cdot e=e'$.
\item We denote by $\Sigma\text{-}\mathsf{Mon}(\DCat)$ the poset of all $\Sigma$-generated $\mathcal{D}$-monoids under the usual quotient ordering (see Notation \ref{not:order}), and by $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$ the subposet of all $\Sigma$-generated finite $\mathcal{D}$-monoids.
\end{enumerate}
\end{definition}
\begin{rem}\label{rem:subdirprod}
The poset $\Sigma\text{-}\mathsf{Mon}(\DCat)$ is a complete lattice -- the join of a family $e_i: \Psi\Sigma^*\twoheadrightarrow A_i$ ($i\in I$) of $\Sigma$-generated $\mathcal{D}$-monoids is their \emph{subdirect product} $S$, obtained by forming their product in $\mathsf{Mon}(\DCat)$ and the (strong epi, mono)-factorization of the morphism $\langle e_i\rangle$:
\[
\xymatrix{
\Psi\Sigma^* \ar@{->>}[d] \ar[drr]^{\langle e_i\rangle} \ar@{->>}[rr]^{e_i} && A_i\\
S \ar@{>->}[rr] && \prod A_i \ar[u]_{\pi_i}
}
\]
Indeed, this follows from the fact that $\mathsf{Mon}(\DCat)$ is complete and inherits the factorization system of $\mathcal{D}$, see Lemma \ref{lem:bimprops}. Analogously, since $\mathsf{Mon}_{f}(\DCat)$ is finitely complete, $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$ is a join-semilattice, in fact a join-subsemilattice of $\Sigma\text{-}\mathsf{Mon}(\DCat)$.
\end{rem}
$\Sigma$-generated monoids are closely related to algebras for the functor $L_\Sigma$.
\begin{notation}\label{not:mon}
In analogy to Notation \ref{not:rqc} we represent $L_\Sigma$-algebras $\mathbb{1} + \coprod_\Sigma A \xrightarrow{\alpha} A$ as triples
\[ A = (A, \alpha_a: A\rightarrow A, i: \mathbb{1}\to A).\]
For any $L_\Sigma$-algebra $A=(A,\alpha_a,i)$ and $w\in\Sigma^*$ we put
\[ A_w := (A,\alpha_a, \alpha_w\cdot i)\]
where $ \alpha_w = \alpha_{a_n}\cdot \cdots \cdot \alpha_{a_1}$ for $w=a_1\cdots a_n\in\Sigma^*$.
\end{notation}
\begin{rem}\label{rem:algmor}
Dually to Remark \ref{rem:rqcmor}, a $\mathcal{D}$-morphism $h: A\rightarrow A'$ is an $L_\Sigma$-algebra homomorphism $h: (A,\alpha_a,i)\rightarrow (A',\alpha_a',i')$ iff the following diagram commutes for all $a\in\Sigma$:
\[
\xymatrix{
\mathbb{1} \ar[r]^i \ar[dr]_{i'} & A \ar[r]^{\alpha_a} \ar[d]_h & A \ar[d]^h \\
& A' \ar[r]_{\alpha_a'} & A'
}
\]
In this case also the square below commutes, which implies that $h$ is $L_\Sigma$-coalgebra homomorphism $h: A_w\rightarrow A'_w$ for all $w\in\Sigma^*$.
\[
\xymatrix{
A \ar[r]^{\alpha_w} \ar[d]_h & A \ar[d]^h \\
A' \ar[r]_{\alpha_w'} & A'
}
\]
\end{rem}
\begin{rem}\label{rem:bimtoalg}
Every $\Sigma$-generated $\mathcal{D}$-monoid $e:\Psi\Sigma^*\twoheadrightarrow (A,\circ,i)$ induces an $ L_\Sigma$-algebra $\widetilde A=(A,\alpha_a,\overline i)$ where $\alpha_a = -\circ \und{e}(\eta a): A\rightarrow A$ and $\overline i: \Psi 1 = \mathbb{1}\rightarrow A$ is the free extension of $i:1\rightarrow \und{A}$.
In particular, for $e=\mathsf{id}$ we obtain
\[\widetilde{\Psi\Sigma^*}=(\Psi\Sigma^*,r_a,\overline{\eta \varepsilon}).\]
Clearly every generator-preserving morphism $f:A\rightarrow B$ of $\Sigma$-generated $\mathcal{D}$-monoids is also an $L_\Sigma$-algebra homomorphism $f: \widetilde A\rightarrow \widetilde B$.
\end{rem}
\begin{proposition}\label{prop:initalg}
$\widetilde{\Psi\Sigma^*}=\mu L_\Sigma$.
\end{proposition}
\begin{proof}
Consider the functor $ L_\Sigma^0=1+\mathsf{Id}^\Sigma$ on $\mathsf{Set}$. Since $ L_\Sigma^0$ and $ L_\Sigma$ are finitary, their initial algebras arise as colimits of the respective initial chains:
\[
\xymatrix@=10pt{
\emptyset \ar[r] & 1 \ar[r] & 1 + \coprod_\Sigma 1 \ar[r] & \dots
& \qquad &
0 \ar[r] & \mathbb{1} \ar[r] & \mathbb{1} + \coprod_\Sigma \mathbb{1} \ar[r] & \dots
}
\]
Now $\Psi$ preserves colimits (being a left adjoint) and $\mathbb{1} = \Psi 1$, so $\Psi$ maps the initial chain of $ L_\Sigma^0$ to the initial chain of $ L_\Sigma$ and preserves the colimit of the left chain, which implies that $\Psi(\mu L_\Sigma^0)=\mu L_\Sigma$. Since $\mu L_\Sigma^0=(\Sigma^*,-\cdot a,\varepsilon)$ we have $\mu L_\Sigma=(\Psi\Sigma^*, \Psi(-\cdot a), \overline{\eta\varepsilon})$. Moreover $\Psi(-\cdot a)=r_a$ by the definition of $r_a$, so
\[\mu L_\Sigma=(\Psi\Sigma^*, r_a, \overline{\eta\varepsilon}) = \widetilde{\Psi\Sigma^*}. \tag*{\qed}\]
\end{proof}
\begin{prop}\label{prop:monalgiso}
$\Sigma\text{-}\mathsf{Mon}(\DCat)$ is a sublattice of $\mathsf{Quo}(\mu L_\Sigma)$, and $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$ is a subsemilattice of $\Quofp{\mu L_\Sigma}$.
\end{prop}
\begin{proof}
Remark \ref{rem:bimtoalg} and Proposition \ref{prop:initalg} show that every (finite) $\Sigma$-generated $\mathcal{D}$-monoid $e: \Psi\Sigma^* \twoheadrightarrow A$ induces a (finite) quotient algebra $e: \mu L_\Sigma \twoheadrightarrow \widetilde A$ of $\mu L_\Sigma$ carried by the same morphism $e$. We show that the map $\widetilde{(-)}: \Sigma\text{-}\mathsf{Mon}(\DCat) \rightarrow \mathsf{Quo}(\mu L_\Sigma)$ and its restriction $\widetilde{(-)}: \Sigma\text{-}\mathsf{Mon}_{f}(\DCat) \rightarrow \Quofp{\mu L_\Sigma}$ are order-embeddings. Clearly $\widetilde{(-)}$ is injective and monotone.
To show that $\widetilde{(-)}$ it is order-reflecting, consider a commutative diagram as below, where $e$ and $e'$ are $\mathcal{D}$-monoid morphisms and $f: \widetilde{A} \rightarrow \widetilde{A'}$ is an $L_\Sigma$-algebra homomorphism.
\[
\xymatrix{
&\mu L_\Sigma=\Psi\Sigma^* \ar@{->>}[dl]_e \ar@{->>}[dr]^{e'} &\\
A \ar[rr]_f && A'
}
\]
We need to show that $\und{f}$ is a monoid morphism. Let $x',y'\in \und{A}$ and choose $x,y\in\und{\Psi\Sigma^*}$ with $e(x)=x'$ and $e(y)=y'$, using that $e$ is surjective by Assumption \ref{asm:geneilenberg}(a). Then
\[ \und{f}(x'\circ y') = \und{f}(\und{e}(x\bullet y)) = \und{e'}(x\bullet y) = \und{e'}x\circ' \und{e'}y = \und{fe}(x)\circ' \und{fe}(y) = \und{f}x'\circ' \und{f}y'\]
and moreover $\und{f}$ preserves the unit because $f$ is an $ L_\Sigma$-algebra homomorphism. \qed
\end{proof}
Hence $\Sigma\text{-}\mathsf{Mon}(\DCat)$ is isomorphic to a sublattice of $\mathsf{Quo}(\mu L_\Sigma)$. The following proposition characterizes its elements in terms of $L_\Sigma$-algebra homomorphisms.
\begin{proposition}\label{prop:bimtoalg}
A quotient algebra $e: \mu L_\Sigma \twoheadrightarrow (A,\alpha_a,i)$ of $\mu L_\Sigma$ is induced by a $\Sigma$-generated $\mathcal{D}$-monoid if and only if there exists an $L_\Sigma$-algebra homomorphism from $A$ to $A_w$ for each $w\in\Sigma^*$.
\end{proposition}
\begin{proof}
($\Rightarrow$) Suppose that $e: \mu L_\Sigma \twoheadrightarrow (A,\alpha_a, i)$ is induced by some $\Sigma$-generated $\mathcal{D}$-monoid $e: \Psi\Sigma^* \twoheadrightarrow (A,\circ,i')$, that is, $\alpha_a=-\circ \und{e}(\eta a)$ and $i=\overline{i'}$. For each $w\in\Sigma^*$ consider the $\mathcal{D}$-morphism
\[\beta_w = \und{e}(\eta w)\circ - : A\rightarrow A.\] We claim that $\beta_w$ is an $ L_\Sigma$-algebra homomorphism
\[ \beta_w: (A,\alpha_a, i) \rightarrow (A,\alpha_a,\alpha_w \cdot i),\]
that is, a homomorphism $\beta_w: A\rightarrow A_w$. In fact, we have for each $x\in \und{A}$ and $a\in\Sigma$:
\begin{align*}
\und{\beta_w}(\und{\alpha_a}(x)) &= \und{e}(\eta w)\circ(x\circ \und{e}(\eta a)) &\text{def. } \beta_w,~\alpha_a\\
&= (\und{e}(\eta w)\circ x)\circ \und{e}(\eta a) & \text{associativity}\\
&= \und{\alpha_a}(\und{\beta_w}(x)) & \text{def. } \beta_w,~\alpha_a
\end{align*}
so $\beta_w\cdot \alpha_a = \alpha_a\cdot\beta_w$. To prove preservation of initial states (i.e., $\beta_w\cdot i=\alpha_w\cdot i$) it suffices to prove $\und{\beta_w\cdot i}\cdot \eta_1=\und{\alpha_w\cdot i}\cdot\eta_1$ where $\eta_1: 1\rightarrow \und{\Psi 1} = \und{\mathbb{1}}$ is the unit. We compute:
\begin{align*}
\und{\beta_w\cdot i}\cdot \eta_1(*)&= \und{\beta_w}(i') & i = \overline{i'}\\
&= \und{e}(\eta w)\circ i' & \text{def. } \beta_w\\
&= \und{e}(\eta w) & i' \text{ neutral element of } A\\
&= \und{e}(\eta \varepsilon \bullet \eta w) & \eta \varepsilon \text{ neutral element of } \Psi\Sigma^*\\
&= \und{e}(\und{r_w}(\eta \varepsilon)) & \text{def. } \bullet\\
&= \und{\alpha_w}(\und{e}(\eta \varepsilon))& e \text{ } L_\Sigma\text{-algebra homomorphism}\\
&= \und{\alpha_w}(i') & e \text{ monoid morphism}\\
&= \und{\alpha_w}(\und{ i}(\eta_1(*))) & i = \overline{i'}\\
&= \und{\alpha_w\cdot i}\cdot\eta_1(*)
\end{align*}
Hence $\beta_w: A\rightarrow A_w$ is an $L_\Sigma$-algebra homomorphism as claimed.
($\Leftarrow$) Suppose that an $L_\Sigma$-algebra homomorphism $\beta_w: A\rightarrow A_w$ is given for each $w\in\Sigma^*$, and let $(\beta_x: A\rightarrow A)_{x\in\und{\Psi\Sigma^*}}$ and $(\alpha_x: A\rightarrow A)_{x\in\und{\Psi\Sigma^*}}$ be the inductive extensions of the families $(\beta_w: A\rightarrow A)_{w\in\Sigma^*}$ and $(\alpha_w: A\rightarrow A)_{w\in\Sigma}$ of $\mathcal{D}$-morphisms.
\begin{enumerate}[(1)]
\item For all $x\in\und{\Psi\Sigma^*}$, let $A_x:=(A,\alpha_a, \alpha_x\cdot i)$. We claim that $\beta_x: A\rightarrow A_x$ is an $ L_\Sigma$-algebra homomorphism, which means that the following squares commute:
\[
\xymatrix{
A \ar[r]^{\beta_x} \ar[d]_{\alpha_a} & A \ar[d]^{\alpha_a} & \mathbb{1} \ar[r]^i \ar[d]_{i} & A \ar[d]^{\beta_x}\\
A \ar[r]_{\beta_x} & A & A \ar[r]_{\alpha_x} & A
}
\]
Indeed, they clearly commute if $x=\eta(w)$ for some $w\in\Sigma^*$ because $\beta_{\eta w}=\beta_w: A\rightarrow A_w$ is an $ L_\Sigma$-algebra homomorphism, and therefore they commute for all $x$ by the induction principle (Lemma \ref{lem:indproof}).
\item We prove the equation
\[
\und{e}(x\bullet y) = \und{\alpha_y}(\und{e}(x)) \quad \text{for all } x,y\in\und{\Psi\Sigma^*},
\]
where $\bullet$ is the multiplication of the free $\mathcal{D}$-monoid $\Psi\Sigma^* = \mu L_\Sigma$. Observe first that the following diagram commutes for all $y\in\Psi\Sigma^*$:
\[
\xymatrix{
\Psi\Sigma^* \ar[r]^{r_y} \ar[d]_{e} & \Psi\Sigma^* \ar[d]^{e} \\
A \ar[r]_{\alpha_y} & A
}
\]
In fact, it commutes if $y=\eta w$ for some $w\in\Sigma^*$ because $e$ is an $ L_\Sigma$-algebra homomorphism, so it commutes for all $y$ by induction. Therefore
\[ \und{e}(x\bullet y) = \und{e}(\und{r_y}(x)) = \und{\alpha_y}(\und{e}(x)).\]
\item We prove the equation
\[ \und{e}(x\bullet y) = \und{\beta_x}(\und{e}(y)) \quad \text{for all }x,y\in\und{\Psi\Sigma^*}.\]
First note that $l_x: \mu L_\Sigma \rightarrow (\mu L_\Sigma)_x$ is an $ L_\Sigma$-algebra homomorphism: we have $l_x \cdot r_a = r_a \cdot l_x$ by Lemma~\ref{lem:lrprops}(a) and $l_x \cdot \overline{\eta\varepsilon} = r_x \cdot \overline{\eta\varepsilon}$ because
\[
\und{l_x \cdot \overline{\eta\epsilon}} \cdot \eta_1(*) = \und {l_x}(\eta\epsilon) = x \bullet \eta \epsilon = \eta \epsilon \bullet x =
\und{r_x}(\eta\epsilon) = \und{rx \cdot \overline{\eta\epsilon}} \cdot \eta_1(*).
\]
Since also $\beta_x: A\rightarrow A_x$ is an $ L_\Sigma$-algebra homomorphism by (1), the following diagram commutes by initiality of $\mu{\hat{L}}_\Sigma$:
\[
\xymatrix{
\mu L_\Sigma \ar[r]^-{l_x} \ar[d]_e & (\mu L_\Sigma)_x \ar[d]^e \\
A \ar[r]_{\beta_x} & A_x
}
\]
Therefore
\[ \und{e}(x\bullet y) = \und{e}(\und{l_x}(y)) = \und{\beta_x}(\und{e}(y)) .\]
\item We define the desired monoid structure on $\und{A}$. The unit is $i\cdot \eta_1(\ast)\in A$, and the multiplication is given as follows: for all $x',y'\in \und{A}$, choose $x,y\in \und{\Psi\Sigma^*}$ with $x'=\und{e}(x)$ and $y'=\und{e}(y)$ (using that $\und{e}$ is surjective by Assumption \ref{asm:geneilenberg}(a)) and put
\[ x'\circ y' := \und{e}(x\bullet y).\]
We need to prove that $x'\circ y'$ is well-defined, i.e., independent of the choice of $x$ and $y$. In fact, by (2) above, $x'\circ y'$ is independent of the choice of $x$ and moreover $- \circ y' = \alpha_y$. Analogously (3) states that $x'\circ y'$ is independent of the choice of $y$ and that $x' \circ - = \beta_x$. It follows that $\circ: A\times A \rightarrow A$ is a well-defined $\mathcal{D}$-bimorphism, and by definition we have $e(x\bullet y) = e(x)\circ e(y)$ for all $x,y\in \Psi\Sigma^*$. The associative and unit laws for $\circ$ hold in $A$ because they hold in in $\Psi\Sigma^*$ and $\und{e}$ is surjective, concluding the proof that $(A,\circ, i\cdot \eta_1(\ast))$ is a $\mathcal{D}$-monoid. Moreover, $\und{e}$ is clearly a monoid morphism \[e: \Psi\Sigma^*\twoheadrightarrow (A,\circ,i\cdot \eta_1(\ast)),\] and the quotient algebra of $\mu L_\Sigma$ it induces is precisely $(A,\alpha_a,i)$. For the latter we need to show $- \circ e(\eta a) = \alpha_a$ for all $a\in \Sigma$. Given $x'\in A$, we choose $x\in \Psi\Sigma^*$ with $e(x)=x'$ and compute
\[ x' \circ e(\eta a) = e(x\bullet \eta a) = e(r_a(x)) = \alpha_a(e(x)) = \alpha_a(x'), \]
using the definitions of $\circ$ and $\bullet$ and the fact the $e$ is an $L_\Sigma$-algebra homomorphism.\qed
\end{enumerate}
\end{proof}
\subsection{The Local Eilenberg Theorem} \label{sec:proofloceil}
We can now put our (co-)algebraic characterizations of right derivatives and monoids together to prove the general local Eilenberg theorem. The key result is
\begin{prop}\label{prop:loceilfin}
The semilattices $\FPSubrqc{\rho T_\Sigma}$ and $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$ are isomorphic.
\end{prop}
\begin{proof}
We show that the isomorphism $\FPSub{\rho T_\Sigma}\cong \Quofp{\rho T_\Sigma}$ of Proposition \ref{prop:subquo} restricts to an isomorphism $\FPSubrqc{\rho T_\Sigma} \cong \Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$. Indeed, a finite subcoalgebra $(Q,\gamma_a,f)\rightarrowtail \rho T_\Sigma$ is closed under right derivatives iff a $T_\Sigma$ coalgebra homomorphism $Q_w \rightarrow Q$ exists for each $w\in \Sigma^*$ (see Proposition \ref{prop:rqc2}). By Remark \ref{rem:algcoalgdual} the $L_\Sigma$-algebra dual to $Q=(Q,\gamma_a,f)$ is $\widehat{Q}=(\widehat{Q},\widehat{\gamma_a},\widehat{f})$, and the one dual to $Q_w = (Q,\gamma_a,f\cdot \gamma_w)$ is \[\widehat{Q_w}= (\widehat{Q}, \widehat{\gamma_a}, \widehat{\gamma_{w}}\cdot \widehat{f}) = \widehat{Q}_{w^r}\]where $w^r$ is the reversed word of $w$. Indeed, for $w=a_1\ldots a_n$ we have $\widehat{\gamma_w} = \widehat{\gamma_{a_1}} \cdot \cdots \cdot \widehat{\gamma_{a_n}}$. Hence by duality an $L_\Sigma$-algebra homomorphism $\widehat{Q} \rightarrow \widehat{Q}_{w^r}$ exists for each $w\in\Sigma^*$, which by Proposition \ref{prop:monalgiso} and \ref{prop:bimtoalg} means precisely that $\widehat{Q} \in \Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$.\qed
\end{proof}
\begin{definition}
A \emph{local variety of regular languages in $\mathcal{C}$} is a subcoalgebra of $\rho T_\Sigma$ closed under right derivatives.
\end{definition}
\begin{expl}
\begin{enumerate}[(a)]
\item A \emph{local variety of regular languages} is a set of regular languages over $\Sigma$ closed under the boolean operations and derivatives and containing $\emptyset$. This is the case $\mathcal{C}=\mathsf{BA}$.
\item A \emph{local lattice variety of regular languages} is a set of regular languages over $\Sigma$ closed under union, intersection and derivatives and containing $\emptyset$ and $\Sigma^*$. This is the case $\mathcal{C}=\mathsf{DL}_{01}$.
\item A \emph{local semilattice variety of regular languages} is a set of regular languages over $\Sigma$ closed under union and derivatives and containing $\emptyset$. This is the case $\mathcal{C}={\mathsf{JSL}_0}$.
\item A \emph{local linear variety of regular languages} is a set of regular languages over $\Sigma$ closed under symmetric difference and derivatives and containing $\emptyset$. This is the case $\mathcal{C}=\Vect{\mathds{Z}_2}$.
\end{enumerate}
\end{expl}
\begin{definition}
A \emph{local pseudovariety of $\mathcal{D}$-monoids} is a set of finite $\Sigma$-generated $\mathcal{D}$-monoids closed under subdirect products and quotients, i.e., an ideal in the join-semilattice $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$.
\end{definition}
This leads to the main result of this paper. For convenience, we recall all assumptions used so far in the statement of the theorem.
\begin{theorem}[General Local Eilenberg Theorem]\label{thm:loceil}
Let $\mathcal{C}$ and $\mathcal{D}$ be predual locally finite varieties of algebras, where the algebras in $\mathcal{D}$ are possibly ordered. Suppose further that $\mathcal{D}$ is monoidal closed w.r.t. tensor product, epimorphisms in $\mathcal{D}$ are surjective, and the free algebra in $\mathcal{D}$ on one generator is dual to a two-element algebra in $\mathcal{C}$. Then there is a lattice isomorphism
\[\text{local varieties of regular languages in $\mathcal{C}$} ~\cong \text{
local pseudovarieties of $\mathcal{D}$-monoids.}\]
\end{theorem}
\begin{proof}
By Proposition \ref{prop:loceilfin} we have a semilattice isomorphism
\[\FPSubrqc{\rho T_\Sigma} \cong \Sigma\text{-}\mathsf{Mon}_{f}(\DCat)\]
Taking ideal completions on both sides yields a complete lattice isomorphism
\[\Ideal{\FPSubrqc{\rho T_\Sigma}} \cong \Ideal{\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)}\]
The ideals of the join-semilattice $\Sigma\text{-}\mathsf{Mon}_{f}(\DCat)$ are by definition precisely the pseudovarieties of $\mathcal{D}$-monoids. Moreover, Proposition \ref{prop:rqcidcomp} yields
\[\Ideal{\FPSubrqc{\rho T_\Sigma}} \cong \Subrqc{\rho T_\Sigma}, \]
and the elements of $\Subrqc{\rho T_\Sigma}$ are by definition precisely the local varieties of regular languages in $\mathcal{C}$.\qed
\end{proof}
\begin{cor} By instantiating Theorem \ref{thm:loceil} to the categories of Example \ref{expl:dualpairs} we obtain the following lattice isomorphisms:
\end{cor}
\begin{center}
\begin{tabular}{|lllll|}
\hline\rule[11pt]{0pt}{0pt}
$\mathcal{C}$ & $\mathcal{D}$ & local varieties of regular languages &$\cong$& local pseudovarieties of \dots \\
\hline
$\mathsf{BA}$&$\mathsf{Set}$ &local varieties &$\cong$& monoids\\
$\mathsf{DL}_{01}$&$\mathsf{Pos}$ &local lattice varieties &$\cong$& ordered monoids\\
${\mathsf{JSL}_0}$&${\mathsf{JSL}_0}$ &local semilattice varieties &$\cong$& idempotent semirings\\
$\Vect{\mathds{Z}_2}$&$\Vect{\mathds{Z}_2}$ & local linear varieties &$\cong$& $\mathds{Z}_2$-algebras\\
\hline
\end{tabular}
\end{center}
The first two isomorphisms were proved in \cite{ggp08, ggp10}, the last two are new local Eilenberg correspondences.
\section{Profinite Monoids}
\label{sec:dual}
In \cite{ggp08,ggp10} Gehrke, Grigorieff and Pin demonstrated that the boolean algebra $\mathsf{Reg}_\Sigma$, equipped with left and right derivatives, is dual to the free $\Sigma$-generated profinite monoid. In this section we generalize this result to our categorical setting (Assumptions \ref{asm:sec3}). For this purpose we will introduce below a category $\hat \mathcal{D}$ that is dually equivalent (rather than just predual) to $\mathcal{C}$, and arises as a ``profinite'' completion of $\mathcal{D}_f$.
\begin{defn} \begin{enumerate}[(a)]
\item Dually to Definition \ref{def:lfp}, an object $X$ of a category $\mathcal{B}$ is called \emph{cofinitely presentable} if the hom-functor $\mathcal{B}(-,X):\mathcal{B}\rightarrow\mathsf{Set}^{op}$ is cofinitary, i.e., preserves cofiltered limits. The full subcategory of all cofinitely presentable objects is denoted by $\mathcal{B}_{cfp}$. The category $\mathcal{B}$ is \emph{locally cofinitely presentable} if $\mathcal{B}_{cfp}$ is essentially small, $\mathcal{B}$ is complete and every object arises as a cofiltered limit of cofinitely presentable objects.
\item The dual of $\mathsf{Ind}$ is denoted by $\mathsf{Pro}$: the \emph{free completion under cofiltered limits} of a small category $\mathcal{A}$ is a full embedding $\mathcal{A}\hookrightarrow \mathsf{Pro}\,{\mathcal{A}}$ such that $\mathsf{Pro}\,{\mathcal{A}}$ has cofiltered limits and every functor $F: \mathcal{A} \rightarrow \mathcal{B}$ into a category $\mathcal{B}$ with cofiltered colimits has an essentially unique cofinitary extension $\overline{F}: \mathsf{Pro}\,{\mathcal{A}}\rightarrow \mathcal{B}$:
\[
\xymatrix{
\mathcal{A} \ar@{>->}[r] \ar[dr]_F & \mathsf{Pro}\,{\mathcal{A}} \ar@{-->}[d]^{\overline{F}} \\
& \mathcal{B}
}
\]
Note that $(\mathsf{Pro}\, \mathcal{A})^{op} \cong \IndC{\mathcal{A}^{op}}$. If $\mathcal{A}$ has finite limits then $\mathsf{Pro}\,{\mathcal{A}}$ is locally cofinitely presentable and $(\mathsf{Pro}\,{\mathcal{A}})_{cfp} \cong \mathcal{A}$. Conversely, for every locally cofinitely presentable category $\mathcal{B}$ one has $\mathcal{B}\cong\ProC{\mathcal{B}_{cfp}}$.
\end{enumerate}
\end{defn}
\begin{notation}
Let $\hat \mathcal{D}$ denote the free completion of $\mathcal{D}_f$ under cofiltered limits, i.e.,
\[ \hat \mathcal{D} = \ProC{\mathcal{D}_f}.\]
\end{notation}
\begin{rem}
$\hat \mathcal{D}$ is dually equivalent to $\mathcal{C}$ since
\[ \mathcal{C} = \IndC{\mathcal{C}_f} \cong \IndC{\mathcal{D}_f^{op}} \cong \ProC{\mathcal{D}_f}^{op} =\hat \mathcal{D}^{op}.\]
Moreover, the equivalence functors
\[ \widehat{(\mathord{-})} : \mathcal{C}_f \xrightarrow{\cong} \mathcal{D}_f^{op} \quad\text{and}\quad \overline{(\mathord{-})}: \mathcal{D}_f^{op} \xrightarrow{\cong} \mathcal{C}_f\]
extend essentially uniquely to equivalences -- also called $\widehat{(\mathord{-})}$ and $\overline{(\mathord{-})}$ -- between $\mathcal{C}$ and $\hat \mathcal{D}^{op}$:
\[
\xymatrix{
\mathcal{C} = \IndC{\mathcal{C}_{f}} \ar[r]^<<<<<{\widehat{(\mathord{-})}} & \hat \mathcal{D}^{op} && \mathcal{C} & \ProC{\mathcal{D}_f}^{op} = \hat \mathcal{D}^{op} \ar[l]_>>>>>>{\overline{(\mathord{-})}}\\
\mathcal{C}_f \ar@{>->}[u] \ar[r]_{\widehat{(\mathord{-})}} & \mathcal{D}_f^{op} \ar@{>->}[u] && \mathcal{C}_f \ar@{>->}[u] & \mathcal{D}_f^{op} \ar@{>->}[u] \ar[l]^{\overline{(\mathord{-})}}
}
\]
\end{rem}
\begin{expl}\label{expl:dualpairs2}
For the predual varieties $\mathcal{C}$ and $\mathcal{D}$ of Example \ref{expl:dualpairs} we have the following categories $\hat \mathcal{D}$:
\begin{center}
\begin{tabular}{|lll|}
\hline\rule[11pt]{0pt}{0pt}
$\mathcal{C}\quad\quad\quad$ & $\mathcal{D}\quad\quad\quad$ & $\hat \mathcal{D}\quad\quad\quad$ \\%& $P\quad\quad\quad$ & $P^{-1}\quad\quad\quad$\\
\hline
$\mathsf{BA}$ & $\mathsf{Set}$ & $\mathsf{Stone}$\\
$\mathsf{DL}_{01}$ & $\mathsf{Pos}$ & $\mathsf{Priest}$ \\
${\mathsf{JSL}_0}$ & ${\mathsf{JSL}_0}$ & ${\mathsf{JSL}_0}$ in $\mathsf{Stone}$\\
$\Vect{\mathds{Z}_2}$ & $\Vect{\mathds{Z}_2}$ & $\Vect{\mathds{Z}_2}$ in $\mathsf{Stone}$\\
\hline
\end{tabular}
\end{center}
In more detail:
\begin{enumerate}[(a)]
\item For $\mathcal{C}=\mathsf{BA}$ and $\mathcal{D}=\mathsf{Set}$, we have the classical Stone duality: $\hat \mathcal{D}$ is the category $\mathsf{Stone}$ of Stone spaces (i.e., compact Hausdorff spaces with a base of clopen sets) and continuous maps.
The equivalence functor $\overline{(\mathord{-})}: \mathsf{Stone}^{op} \rightarrow\mathsf{BA}$ assigns to each Stone space the boolean algebra of clopen sets, and its associated equivalence $\widehat{(\mathord{-})}: \mathsf{BA} \rightarrow \mathsf{Stone}^{op}$ assigns to each boolean algebra the Stone space of all ultrafilters.
\item For the category $\mathcal{C}=\mathsf{DL}_{01}$ and $\mathcal{D}=\mathsf{Pos}$ we have the classical Priestley duality: $\hat \mathcal{D}$ is the category $\mathsf{Priest}$ of Priestley spaces (i.e., ordered Stone spaces such that given $x\not\leq y$ there is a clopen set containing $x$ but not $y$) and continuous monotone maps. The equivalence functor $\overline{(\mathord{-})}: \mathsf{Priest}^{op}\rightarrow\mathsf{DL}_{01}$ assigns to each Priestley space the lattice of all clopen upsets, and its associated equivalence $\widehat{(\mathord{-})}: \mathsf{DL}_{01} \rightarrow \mathsf{Priest}^{op}$ assigns to each distributive lattice the Priestley space of all prime filters.
\item For $\mathcal{C}=\mathcal{D}={\mathsf{JSL}_0}$ the dual category $\hat \mathcal{D}$ is the category of join-semilattices in $\mathsf{Stone}$.
Similarly, for $\mathcal{C}=\mathcal{D}=\Vect{\mathds{Z}_2}$ the dual category $\hat \mathcal{D}$ is the category of $\mathds{Z}_2$-vector spaces in $\mathsf{Stone}$, see \cite{j82}.
\end{enumerate}
\end{expl}
\begin{defn}
We denote by ${\hat{L}}:\hat \mathcal{D}\rightarrow\hat \mathcal{D}$ the dual of the functor $T:\mathcal{C}\rightarrow\mathcal{C}$, i.e., the essentially unique functor for which the following diagram commutes up to natural isomorphism:
\[
\xymatrix{
\hat \mathcal{D}^{op} \ar[r]^{{\hat{L}}^{op}} \ar@{}[dr]|{\cong} & \hat \mathcal{D}^{op} \\
\mathcal{C} \ar[u]^{\widehat{(\mathord{-})}} \ar[r]_{T} & \mathcal{C} \ar[u]_{\widehat{(\mathord{-})}}
}
\]
\end{defn}
\begin{rem}\label{rem:algcoalgdual2}
In analogy to Remark \ref{rem:algcoalgdual}, the categories $\Coalg{T}$ and $\mathsf{Alg}\,{{\hat{L}}}$ are dually equivalent: the equivalence $\widehat{(\mathord{-})}: \mathcal{C} \xrightarrow{\cong} \mathcal{D}^{op}$ lifts to an equivalence $\Coalg{T} \xrightarrow{\cong} (\mathsf{Alg}\,{{\hat{L}}})^{op}$ given by
\[ (Q\xrightarrow{\gamma} TQ) \quad\mapsto\quad ({\hat{L}}\widehat{Q} = \widehat{TQ} \xrightarrow{\widehat \gamma} \widehat Q). \]
\end{rem}
\begin{expl}\label{exp:dualfunc}
The dual of the endofunctor $T_\Sigma Q = \mathbb{2} \times Q^\Sigma$ of $\mathcal{C}$, see Example \ref{exp:tcoalg}, is the endofunctor of $\mathcal{D}$
\[ {\hat{L}}_\Sigma Z = \mathbb{1} + \coprod_\Sigma Z\]
where $\mathbb{1}=\widehat{\mathbb{2}}$. In $\hat \mathcal{D}=\mathsf{Stone}$ the object $\mathbb{1}$ is the one-element space. Hence, by the universal property of the coproduct, an ${\hat{L}}_\Sigma$-algebra ${\hat{L}}_\Sigma Z = \mathbb{1} + \coprod_\Sigma Z \rightarrow Z$ is a deterministic $\Sigma$-automaton (without final states) in $\mathsf{Stone}$, given by a Stone space $Z$ of states, continuous transition functions $\alpha_a: Z\rightarrow Z$ for $a\in\Sigma$, and an initial state $\mathbb{1}\rightarrow Z$. Analogously for the other dualities of Example \ref{expl:dualpairs2}.
\end{expl}
\begin{notation}
The category of all ${\hat{L}}$-algebras with a cofinitely presentable carrier (shortly \emph{cfp-algebras}) is denoted by $\Algcfp{{\hat{L}}}$.
\end{notation}
\begin{rem}
Note that $\Algcfp{{\hat{L}}} \cong \FAlg{L}$ because the restrictions of ${\hat{L}}$ and $L$ to $\hat \mathcal{D}_{cfp} \cong \mathcal{D}_f$ are naturally isomorphic.
\end{rem}
\begin{defn}
Dually to Definition \ref{rem:rat_fix} an ${\hat{L}}$-algebra is called \emph{locally cofinitely presentable} if it is a cofiltered limit of cfp-algebras.
\end{defn}
\begin{rem}\label{rem:lcpalg}
The category of all locally cofinitely presentable\xspace algebras is equivalent to
$\ProC{\Algcfp{{\hat{L}}}}$.
This is the dual of Theorem \ref{thm:lfpindcomp}. The initial object $\tau {\hat{L}}$ is what one can call the \emph{dual of the rational fixpoint}. By the dual of Definition \ref{rem:rat_fix}, one can construct $\tau {\hat{L}}$ as the limit of all algebras in $\mathsf{Alg}\,{{\hat{L}}}$ with cofinitely presentable carrier.
\end{rem}
\begin{expl}
\begin{enumerate}[(a)]
\item For $\mathcal{C}=\mathsf{BA}$ and $\hat \mathcal{D}=\mathsf{Stone}$, we have
\[\tau {\hat{L}}_\Sigma = \text{ultrafilters of regular languages.}\]
\item Analogously, for $\mathcal{C}=\mathsf{DL}_{01}$ and $\hat \mathcal{D}=\mathsf{Priest}$, we have \[\tau {\hat{L}}_\Sigma = \text{prime filters of regular languages.}\]
\end{enumerate}
\end{expl}
\begin{defn}
\label{not:F}
We denote by $F:\mathcal{D}\rightarrow\hat \mathcal{D}$ the essentially unique finitary functor for which
\[
\xymatrix@=5pt{
& \ar@{_{(}->}[dl]\mathcal{D}_f \ar@{^{(}->}[dr]& \\
\mathcal{D}=\IndC{\mathcal{D}_{f}}\ar[rr]_F & & \ProC{\mathcal{D}_{f}}=\hat \mathcal{D}
}
\]
commutes, and by $U:\hat \mathcal{D}\rightarrow\mathcal{D}$ the essentially unique cofinitary functor for which
\[
\xymatrix@=5pt{
& \ar@{_{(}->}[dl]\mathcal{D}_{f} \ar@{^{(}->}[dr]& \\
\hat \mathcal{D}=\ProC{\mathcal{D}_f}\ar[rr]_U & & \IndC{\mathcal{D}_f}=\mathcal{D}
}
\]
commutes.
\end{defn}
\begin{lem}\label{lem:ufadj}
The functors $F$ and $U$ are well-defined and $F$ is a left adjoint to $U$.
\end{lem}
\begin{proof}
$F$ is well-defined because $\hat \mathcal{D}$ has filtered colimits: being locally cofinitely presentable\xspace it is cocomplete. Analogously for $U$. Furthermore, $F$ and $U$ form an adjunction: every object $A\in\mathcal{D}$ is a filtered colimit
\[ A=\mathop{\mathrm{colim}}_{i\in I} A_i \text{ with } A_i\in\mathcal{D}_{f}.\]
This implies, since $FA_i=A_i$, that
\[ FA = \mathop{\mathrm{colim}}_{i\in I} A_i \text{ in } \hat \mathcal{D}. \]
Analogously, every object $B\in \hat \mathcal{D}$ is a cofiltered limit
\[ B=\lim_{j\in J} B_j \text{ with } B_j\in \mathcal{D}_{f}. \]
This implies, since $UB_j=B_j$, that,
\[ UB = \lim_{j\in J} B_j \text{ in } \mathcal{D}.\]
Consequently we have the desired natural isomorphism
\[ \hat \mathcal{D}(FA,B) \cong \lim_{i\in I}\lim_{j\in J} \mathcal{D}_{fp}(A_i, B_j) \cong \mathcal{D}(A, UB). \tag*{\qed}\]
\end{proof}
\begin{expl}
\begin{enumerate}
\item
For $\mathcal{C} = \mathsf{BA}$, $\mathcal{D} = \mathsf{Set}$ and $\hat \mathcal{D} = \mathsf{Stone}$, the functor $F : \mathsf{Set} \to \mathsf{Stone}$ is the Stone-\v{C}ech compactification and $U : \mathsf{Stone} \to \mathsf{Set}$ is the forgetful functor.
\item
For $\mathcal{C} = \mathsf{DL}_{01}$, $\mathcal{D} = \mathsf{Pos}$ and $\hat \mathcal{D} = \mathsf{Priest}$, the functor $F : \mathsf{Pos} \to \mathsf{Priest}$ constructs the free Priestley space on a poset and $U : \mathsf{Priest} \to \mathsf{Pos}$ is the forgetful functor.
\end{enumerate}
\end{expl}
\begin{defn}
Let $\hat{U}: \ProC{\Algcfp{{\hat{L}}}}\rightarrow\mathsf{Alg}\,{L}$ denote the essentially unique cofinitary functor that makes the triangle below commute:
\[
\xymatrix@=5pt{
& \ar@{ )->}[dl]\Algcfp{{\hat{L}}}\cong\Algfp{L} \ar@{ (->}[dr]& \\
\ProC{\Algcfp{{\hat{L}}}}\ar[rr]_(.55){\hat{U}} & & \mathsf{Alg}\,{L}
}
\]
\end{defn}
\begin{expl}
For $T_\Sigma Q = \mathbb{2} \times Q^\Sigma : \mathsf{BA} \to \mathsf{BA}$ we have ${\hat{L}}_\Sigma Z
= \mathbb{1} + \coprod_\Sigma Z: \mathsf{Stone} \to \mathsf{Stone}$ and $L_\Sigma A = \mathbb{1}
+ \coprod_\Sigma A : \mathsf{Set} \to \mathsf{Set}$. The objects of
$\ProC{\Algcfp{{\hat{L}}_\Sigma}}$ are the locally cofinitely presentable\xspace ${\hat{L}}_\Sigma$-algebras, and the
functor $\hat{U} : \ProC{\Algcfp{{\hat{L}}_\Sigma}} \to \mathsf{Alg}\,{L_\Sigma}$
simply forgets the topology on the carrier of an ${\hat{L}}_\Sigma$-algebra.
\end{expl}
\begin{prop}\label{prop:hu_adj}
$\hat{U}$ is a right adjoint.
\end{prop}
\begin{proof}
We have a commutative square
\[
\xymatrix@=5pt{
\ProC{\Algcfp{{\hat{L}}}} \ar[dd] \ar[rr]^(.65){\hat{U}} && \mathsf{Alg}\,{ L} \ar[dd]\\
&&\\
\mathsf{Pro}\, \hat \mathcal{D}_{cfp} = \hat \mathcal{D} \ar[rr]_(.7)U && \mathcal{D}
}
\]
where the vertical functors are the obvious forgetful functors. Recall that limits in $\mathsf{Alg}\,{ L}$ are formed on the level of $\mathcal{D}$, analogously for limits in $\ProC{\Algcfp{{\hat{L}}}}$. Since $U$ preserves limits by Lemma \ref{lem:ufadj}, we conclude that so does $\hat{U}$. By the Special Adjoint Functor Theorem, $\hat{U}$ is a right adjoint: the category $\ProC{\Algcfp{{\hat{L}}}}$ is complete, $\Algcfp{{\hat{L}}}$ is its (essentially small) cogenerator, and $\ProC{\Algcfp{{\hat{L}}}}$ is wellpowered. Indeed, it is dual to a locally finitely presentable\xspace category which is co-wellpowered by \cite[Theorem 1.58]{ar94}.\qed
\end{proof}
\begin{rem}
It follows that the left adjoint $\hat{F}$ of $\hat{U}$ maps the initial $L$-algebra to the initial locally cofinitely presentable\xspace ${\hat{L}}$-algebra: $\hat{F}(\mu L) = \tau {\hat{L}}$.
One can prove that $\hat{F}$ assigns to
every $L$-algebra $\alpha: LA \rightarrow A$ the limit of the diagram of all
its quotients in $\Algfp{L}=\Algcfp{{\hat{L}}}$. Thus, we see that $\tau {\hat{L}}$
can be constructed as the limit (taken in $\mathsf{Alg}\,{{\hat{L}}}$) of all finite quotient $L$-algebras of $\mu L$. This construction generalizes a similar one given by Gehrke \cite{gehrke13}.
\end{rem}
\begin{rem} Under the Assumptions \ref{asm:geneilenberg} of the previous section we obtain a generalization of the result of Gehrke, Grigorieff and Pin \cite{ggp08,ggp10} that $\mathsf{Reg}_\Sigma$ endowed with boolean operations and derivatives is dual to the free profinite monoid on $\Sigma$.
By Proposition \ref{prop:loceilfin} and Lemma \ref{lem:finsub_in_finrqc} the finite $\Sigma$-generated $\mathcal{D}$-monoids form a cofinal subposet of $\Quofp{\mu L_\Sigma}$. Thus, the corresponding diagrams have the same limit in $\AlgCat{{\hat{L}}_\Sigma}$. Hence by the previous remark $\tau {\hat{L}}_\Sigma$ is also the limit of the directed diagram of all finite $\Sigma$-generated $\mathcal{D}$-monoids. Since $\hat{U}$ preserves this limit and the forgetful functor $\mathsf{Mon}{\mathcal{D}}$ creates limits (see Lemma \ref{lem:bimprops}) it follows that $\hat{U}(\tau {\hat{L}}_\Sigma)$ carries the structure of a $\mathcal{D}$-monoid and it is then easy to see that it is the free profinite $\mathcal{D}$-monoid on $\Sigma$: for every $\mathcal{D}$-monoid morphism $e: \Psi\Sigma^*\rightarrow M$ into a finite $\mathcal{D}$-monoid $M$, there exists a unique ${\hat{L}}_\Sigma$-algebra homomorphism $\overline{e}: \tau {\hat{L}}_\Sigma \rightarrow M$ such that $\hat{U}\overline{e}$ is a $\mathcal{D}$-monoid morphism and the diagram below commutes:
\[
\xymatrix{
\Psi\Sigma^* \ar[r]^<<<<{\hat \eta} \ar@{->}[dr]_e & \hat U \hat F (\mu L_\Sigma) = \hat U (\tau {\hat{L}}_\Sigma) \ar[d]^{\hat U\overline{e}}\\
& M = \hat U \hat F M
}
\]
Here $\hat\eta$ is the unit of the adjunction $\hat F\dashv \hat U$. In summary:
\end{rem}
\begin{thm}
Under the assumptions of the General Local Eilenberg Theorem, $\tau {\hat{L}}_\Sigma$ is the free profinite $\mathcal{D}$-monoid on $\Sigma$.
\end{thm}
\section{A Categorical Framework}\label{sec:oldasm}
Although we have assumed $\mathcal{C}$ and $\mathcal{D}$ to be locally finite varieties throughout this paper, our methodology was purely categorical (rather than algebraic) in spirit. In fact, all our results and their proofs can be adapted to the following categorical setting:
\begin{enumerate}
\item $\mathcal{C}$ and $\mathcal{D}$ are predual categories, i.e., the categories $\mathcal{C}_f$ and $\mathcal{D}_f$ of finitely presentable objects are dually equivalent.
\item $\mathcal{C}$ has the following additional properties:
\begin{enumerate}
\item $\mathcal{C}$ is locally finitely presentable\xspace.
\item $\mathcal{C}_f$ is closed under strong epimorphisms.
\item $\mathcal{C}$ is concrete, i.e., a faithful functor $\under{\mathord{-}}_{\mathcal{C}}: \mathcal{C}\rightarrow\mathsf{Set}$ is given.
\item $\under{\mathord{-}}_{\mathcal{C}}$ is a finitary right adjoint and maps finitely presentable objects to finite sets.
\item $\under{\mathord{-}}_{\mathcal{C}}$ is amnestic, i.e., every subset $B\subseteq \under{A}_{\mathcal{C}}$, where $A$ is an object of $\mathcal{C}$, is carried by at most one subobject of $A$ in $\mathcal{C}$.
\end{enumerate}
\item $\mathcal{D}$ has the following additional properties:
\begin{enumerate}
\item $\mathcal{D}$ is locally finitely presentable\xspace.
\item $\mathcal{D}_f$ is closed under strong monomorphisms and finite products.
\item $\mathcal{D}$ is concrete, i.e., a faithful functor $\under{\mathord{-}}_{\mathcal{D}}: \mathcal{D}\rightarrow\mathsf{Set}$ is given.
\item $\under{\mathord{-}}_{\mathcal{D}}$ is a finitary right adjoint and maps finitely presentable objects to finite sets.
\item $\under{\mathord{-}}_{\mathcal{D}}$ preserves epimorphisms.
\item For any two objects $A$ and $B$ of $\mathcal{D}$, there exists an embedding $[A,B]\xto{e_{A,B}} B^{\under{A}}$ given by hom-sets, i.e.,
\begin{enumerate}
\item $[A,B]$ has the underlying set $\mathcal{D}(A,B)$,
\item $e_{A,B}$ is a monomorphism,
\item $\under{e_{A,B}}$ takes $f: A\rightarrow B$ to its underlying function $|f|\in \under{B}^{\under{A}}$, and
\item whenever a $\mathcal{D}$-morphism $h: X\rightarrow [A,B]$ factorizes through $e_{A,B}$ in $\mathsf{Set}$, it factorizes through $e_{A,B}$ in $\mathcal{D}$.
\end{enumerate}
\end{enumerate}
\item The $\mathcal{D}$-object $\mathbb{1} = \Psi 1$, where $\Psi: \mathsf{Set}\rightarrow\mathcal{D}$ is the left adjoint of $\under{\mathord{-}}_{\mathcal{D}}$, is dual to an object $\mathbb{2}$ of $\mathcal{C}$ with two-element underlying set.
\item $T: \mathcal{C} \rightarrow \mathcal{C}$ and $L: \mathcal{D}\rightarrow \mathcal{D}$ are finitary predual functors, i.e., they restrict to functors $T_f: \mathcal{C}_f\rightarrow\mathcal{C}_f$ and $L_f: \mathcal{D}_f\rightarrow\mathcal{D}_f$ and these restrictions are dual. Moreover, $T$ preserves monomorphisms and preimages, and $L$ preserves epimorphisms. In Section \ref{sec:locvar} one works with the functors $T= T_\Sigma = \mathbb{2}\times\mathsf{Id}^\Sigma$ on $\mathcal{C}$ and $L = L_\Sigma = \mathbb{1} + \coprod_\Sigma \mathsf{Id}$ on $\mathcal{D}$.
\end{enumerate}
\section{Conclusions and Future Work}
Inspired by recent work of Gehrke, Grigorieff and Pin \cite{ggp08, ggp10} we have proved a
generalized local Eilenberg theorem, parametric in a pair of dual categories $\mathcal{C}$ and $\hat \mathcal{D}$ and a type of coalgebras $T:\mathcal{C}\rightarrow\mathcal{C}$. By instantiating our framework to deterministic automata, i.e., the functor $T_\Sigma=\mathbb{2}\times \mathsf{Id}^\Sigma$ on $\mathcal{C}=\mathsf{BA}$, $\mathsf{DL}_{01}$, ${\mathsf{JSL}_0}$ and $\Vect{\mathds{Z}_2}$, we derived the local Eilenberg theorems for (ordered) monoids as in ~\cite{ggp08}, as well as two new local Eilenberg theorems for idempotent semirings and $\mathds{Z}_2$-algebras.
There remain a number of open points for further work. Firstly, our
general approach should be extended to the
ordinary (non-local) version of Eilenberg's theorem. Secondly, for different functors $T$ on the categories we have
considered our approach should provide the means to relate varieties of rational behaviours of $T$ with
varieties of appropriate algebras. In this way, we hope to obtain
Eilenberg theorems for systems such as Mealy and Moore automata, but also weighted or probabilistic automata -- ideally, such results would be proved
uniformly for a certain class of functors.
Another very interesting aspect we have not treated in this paper are profinite
equations and syntactic presentations of varieties (of $\mathcal{D}$-monoids or
regular languages, resp.) as in the work of Gehrke, Grigorieff and
Pin~\cite{ggp08}. An important role in studying profinite equations will be played by the ${\hat{L}}$-algebra $\tau {\hat{L}}$, the dual of the rational fixpoint, that we identified as the free profinite
$\mathcal{D}$-monoid. A profinite equation is then a pair of elements of $\tau {\hat{L}}$. We
intend to investigate this in future work.
\bibliographystyle{splncs03}
|
2,869,038,154,906 | arxiv | \section{Introduction}
The discovery of gravitational waves (GW) has given us the new probe for observing the universe~\cite{PhysRevLett.116.061102}.
The typical strain sensitivity, $h$, of second generation interferometric detectors, such as Advanced LIGO~\cite{0264-9381-32-7-074001}, Advanced Virgo~\cite{0264-9381-32-2-024001}, and KAGRA~\cite{0264-9381-29-12-124007, PhysRevD.88.043007}, is around $10^{-23}/\sqrt{\mathrm{Hz}}$ at 100 Hz.
In GW150914 event data analysis, it has been shown that the calibration errors give significant impact on the sky localization accuracy.
The 90~\% sky confidence region gets larger from $150~\mathrm{deg}^2$ to $610~\mathrm{deg}^2$ by introducing the calibration uncertainties of 10~\% in amplitude and 10 degrees phase~\cite{PhysRevD.93.122004}, and eventually got smaller to $230~\mathrm{deg}^2$ with the improved calibration uncertainties~\cite{PhysRevX.6.041015, PhysRevD.96.102001}.
Using GW signals from compact binary coalescences events, researchers can derive several parameters of the source objects such as masses, spins, luminosity distance, orbital inclination and the sky location.
The precision of these derived parameters is potentially limited by the calibration accuracy. As the number of detected sources increases and events with higher signal-to-noise ratio (SNR) are detected, calibration uncertainty
will become the dominant source of errors when extracting physical informations from the signals.
Testing general relativity has been demonstrated with the GW events from binary black hole mergers~\cite{PhysRevLett.116.221101}. In most of the analysis, the effect of calibration uncertainties on the detection and parameter estimation of GW events have focused on placing constrains on the calibration accuracy by modeling the calibration errors as smooth and random frequency-dependent fluctuations. By a semi-analytic approach to explicitly relate systematic errors in calibration parameters to the GW signal parameters, it has been shown that for events with SNR$\sim20$, calibration accuracy of a few percent is required for certain parameters such as optical gain and actuation strength in order to achieve noise-limited systematics~\cite{Hall:2017off}.
Upper limits and observations of continuous GW waves such as rapidly rotating neutron stars and stochastic background of unresolvable sources depend on calibration uncertainties. The associated uncertainties on the upper limits of continuous waves amount to $\sim20~\%$ by including 10~\% amplitude calibration uncertainty~\cite{0004-637X-839-1-12,PhysRevD.96.122004}.
The suppression of the calibration error also improve burst GW reconstruction.
Especially, the precise correction of the frequency dependence will remove the
biases on arrival time and polarization components. These parameters affects on
estimations for a source direction and rotational axis of Super Novae core, respectively.
In the idea to estimate a mass of an isolated neutron star using gravitational waves~\cite{PhysRevD.91.084032,1742-6596-716-1-012026}, it must determine the phase difference precisely between once and twice spin frequency modes. For many known pulsar cases, these frequencies are around the unity gain frequency where the transfer function phase changes steeply. Also, since the continuous wave measurement would use a long-duration data sets as a order of years, the robustness and stableness of calibrations is essentially important.
In particular, the uncertainty in the absolute amplitude of the GW signal
propagates directly into the estimation of the distance to the sources.
The rate that compact binary system coalescences in the universe if drawn from detected events. The SNR by the searches are quadratically sensitive to the calibration errors since they are maximized over arrival time, waveform phase, and the template banks. The amplitude calibration uncertainty of 10~\% and the derived uncertainty of the luminosity distances of the sources corresponds to an approximately 30~\% uncertainty in volume and will dominate over the statistical uncertainty~\cite{2041-8205-833-1-L1}.
The detection of a GW signal from the GW170817 Binary Neutron Star (BNS) system, along with a concurrent electromagnetic (EM) signal, began a new era of multi-messenger astronomy~\cite{PhysRevLett.119.161101}. These observations
allow us to use GW170817 as a standard
siren~\cite{Abbott:2017xzu,Schutz_1986,Holz_2005,Nissanke_2010} with witch we can
determine the absolute luminosity distance to the source directly from the
GW signals. Assuming an event rate of 3000 Gpc$^{-3}$yr$^{-1}$
which is consistent with the 90~\%
confidence interval for GW170817~\cite{PhysRevLett.119.161101},
we expect that GW signals will be detected from about
50 BNS standard sirens dureing the next few observing runs.
These observations can constrain the Hubble constant ($H_0$) to 2~\% error or less~\cite{Feeney:2018mkj}, and eventually resolve
the 3-$\sigma$ tension in $H_0$ measurements between Cephied-SN distance
ladder~\cite{Riess_2016} and CMB data when assuming the $\Lambda$CDM
model.~\cite{2016-planck} Systematic errors in the calibration of
the absolute GW signal amplitude must be suppressed less than 1~\% to
achieve higher-precision $H_0$ measurements using GW standard sirens.
Laser interferometers measure change in distance along the two interferometer arms. Fluctuations in the degree of freedom of the differential arm length (DARM) are suppressed by a DARM control loop. The reconstruction of the DARM fluctuation at the observation frequency is affected by the GWs. The gravitational waveform can be reconstructed from the calibrated error and control signals of this DARM loop. To calibrate these signals, accurate physical models of the actuator and sensing function are essential. These models require measurements of the transfer function and monitoring of the time dependency of the transfer function using continuous sine waves (calibration lines). The residual of the time-dependent model corresponds to the uncertainty of the observaton.
To reduce the systematic uncertainty in the calibration, we need to inject well-parameterized calibration lines for the photon calibrator (Pcal) or other calibration sources for monitoring the time variation of the interferometer responce. The Pcal was developed by the Glasgow and GEO600 reserch groups~\cite{CLUBLEY200185,MOSSAVI20061}, followed by Advanced LIGO which particularly improved Pcals for calibrating the time-dependent response of interferometers~\cite{0264-9381-32-2-024001, doi:10.1063/1.4967303,0264-9381-27-8-084024,0264-9381-26-24-245011,0264-9381-32-2-024001}. However, Pcal still face a challenges in finding of the absolute amplitude calibration because of the uncertainty in the laser power standards published by different national metrology institutes~\cite{EUROMET}. The absolute power between these institutes vary by a few \% \footnote{Figure. 9 at page 46. from \cite{EUROMET} shows the absolute power measurement between the standard institute from nine countries. The systematic discrepancies between nine countries are as large as 3.5 \%.}.
Gravity field calibrator (Gcal) is one of the most promising candidates to be able to solve the uncertainty problem of the absolute laser amplitude calibrations. The technology has been developed and tested by Forward and Miller~\cite{doi:10.1063/1.1709366}, Weber~\cite{PhysRevLett.18.795,PhysRev.167.1145}, University of Tokyo~\cite{Hirakawa,1347-4065-19-3-L123,1347-4065-20-7-L498,PhysRevD.26.729,PhysRevD.32.342} and Rome university group~\cite{Astone1991, Astone1998}. Related techniques using Gcal are discussed in Matone {\it et. al.} and Raffai {\it et. al.}~\cite{0264-9381-24-9-005, PhysRevD.84.082002}. The device can modulate a test mass using a gravity gradient generated by a rotor that depending on the masses, distance, frequency, radius, and the gravity constant.
This paper proposes new method for achieving sub-percent uncertainty in the absolute amplitude calibration of the GW detectors. The method combines Pcal and Gcal.
Section~\ref{sec:Pcal} explains the methods used for Pcal. In section~\ref{sec:Gcal}, we discuss the principle of a multipole moment of gravity and how it is modulated to derive a calibration signal.
We demonstrate how to calibrate absolute displacement using two calibrators in concert in section \ref{sec:PGCAL}, and in section~\ref{sec:EST}, we discuss the contributions of the systematic error and estimate the current technological limits on the gravitational wave observation from typical physical assumptions.
\section{Photon calibrator} \label{sec:Pcal}
Pcal exploit how the photon radiation pressure from power-modulated laser beams reflects from a test mass. The periodic photon recoil applies a periodic force to test mass~\cite{doi:10.1063/1.4967303}.
Advanced LIGO, Advanced Virgo and KAGRA employ Pcals for the calibration of the interferometer response~\cite{0264-9381-34-1-015002, KAGRA_Pcal,0264-9381-32-2-024001}. All of them use laser of the same wave length, 1047~nm, to actuate the test mass. The test mass displacement is described as
\begin{equation}
x = \frac{2P \cos{\theta}}{c} s(\omega)\left(1+\frac{M}{I}\vec{a} \cdot \vec{b} \right) , \label{eq:pcal}
\end{equation}
where $P$ is the absolute laser power, $\theta$ is the incident angle of the Pcal laser, $M$ is the mass of test mass, $\omega$ is the angular frequency of the laser power modulation, and $\vec{a}$ and $\vec{b}$ are the position vectors of the Pcal laser beams. A schematic view of device is shown in Fig.~\ref{fig:Pcal}. $I=Mh^2/12+Mr^2/4$ is the moment of inertia, where $h$ and $r$ are the thickness and radius of the test mass, respectively. $s(\omega)$ is the transfer function between force and displacement. We can regard the value of $s(\omega)$ as $1/(M \omega^2)$ at the frequency above 20 Hz, as the test mass behaves as a free mass in this regime.
The amplitude of the laser power noise is stabilized to be less than the design sensitivity. As shown in Fig.~\ref{fig:Pcal}, the power stabilized laser is mounted on the transmitter module. The power of the photo detector responses at the transmitter module, $V_{\mathrm{TxPD}}$, and receiver module, $V_{\mathrm{RxPD}}$ are monitored for differences.
The largest relative
uncertainty of photon calibrator is that of laser power.
Advanced LIGO and KAGRA use a working standard to cross-calibrate the relative interferometer responces. The relative uncertainty of each calibrator is 0.51 \%~\cite{doi:10.1063/1.4967303}.
The second largest relative uncertainty is the optical efficiency of the optical path in the calibrator. We calibrate the injected power from the exterior of the vacuum chamber. Therefore, we need to consider the difference in optical efficiency due to the transmittance of the vacuum window and the reflectance of the mirrors. The measured uncertainty of the optical efficiency in the Advanced LIGO is 0.37 \%.
For absolute calibration, the photo detector, following the so called gold standard, is calibrated using the laser power standard maintained by the National Institute of Standards and Technology (NIST) in Boulder, CO~\cite{taylor:1994:GEEU} in the U.S.
The working standard responces for Hanford, Livingston and KAGRA GW detectors are calibrated to this gold standard.
However, a comparison of the accuracies of the absolute laser power standards maintained by each national standard institute shows a few~\%
uncertainty~\cite{EUROMET}.
This uncertainty leads to the serious systematic error in the distance calibrations propagated from the uncertainty of the absolute calibration.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{Pcal.eps}
\caption{Schematic view of photon calibrator. The stabilized laser is placed on the transmitter module. The signal injected to the test mass is monitored through the difference in photo detector responce power between the transmitter module and receiver module,$V_{TxPD}$ and $V_{RxPD}$. The geometrical factor is characterized in term of the position vectors of the photon calibrator beams, $\vec{a}=\vec{a_1}+\vec{a_2}$, and the main beam, $\vec{b}$.}
\label{fig:Pcal}
\end{center}
\end{figure}
\begin{table}
\begin{center}
\caption{Specification summary of Advanced LIGO, Advanced Virgo and KAGRA photon calibrator. \label{pcal}}
\footnotesize
\begin{tabular}{cccc}
\hline
& KAGRA& Advanced LIGO& Advanced Virgo \\
\hline
Mirror material & Sapphire & Silica & Silica \\
Mirror mass & 23 kg & 40 kg & 40 kg \\
Mirror diameter & 220 mm & 340 mm & 350 mm \\
Mirror thickness & 150 mm & 200 mm & 200 mm \\
Distance from Pcal & 36 m & 8 m & 1.5 m \\
to test mass &&& \\
Pcal laser power & 20 W & 2~W & 3 W \\
Pcal laser frequency & 1047 nm & 1047 nm &1047 nm\\
Incident angle& 0.72 deg & 8.75 deg &30 deg \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Gravity field calibrator} \label{sec:Gcal}
To address this problem of uncertainty in the the absolute calibration, we propose a new calibration method that combines Pcal and Gcal.
The Gcal generates a dynamic gravity field by rotating the multipole masses with a rotor placed in a vacuum chamber that isolates acoustic noise. To monitor the frequency of this rotation, an encoder with
a 16-bit analog to digital converter is included.
Next, we calculate the displacement of the test mass in the dynamic gravity field generated by a multipole moment with N masses.
The calculation assumes a free-mass of a test mass and a set of masses mounted on a disk as shown in Fig ~\ref{fig:dim}.
The rotating the masses $m$ are arranged around the rotor at radius, $r$.
The distance between the center of this rotor and the test mass mirror is assumed $d$.
We rotate the disk rotates at the angular frequency $\omega_{\mathrm{rot}}=2\pi f_{\mathrm{rot}}$.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{dim.eps}
\caption{Schematic of Gcal. The rotor is placed at the same height as the test mass and at a distance of $d$. Multipole masses spinning around the rotor generate a varying gravitational potential at the position of test mass.}
\label{fig:dim}
\end{center}
\end{figure}
We estimate the equation of motion of the test mass as it is moved by the dynamic gravity field.
First, we calculate the distance between the test mass and the N pieces of masses arranged around the rotor.
The distance between i-th mass and the center of test mass is written as
\begin{equation}
L_i=d \sqrt{1+\left( \frac{r}{d} \right)^2 -2\left( \frac{r}{d} \right) \cos{\phi_i} },
\end{equation}
where the angle of the i-th mass is assumed to be $\phi_i=\omega_{\mathrm{rot}} t + 2\pi i/N$.
The gravitational potential at the center of test mass can be described as
\begin{eqnarray}
V &=& \Sigma^N_{i=0} V_i, \\
&=& -GMm \Sigma^N_{i=0}L_i^{-1},\\ \label{eq:vpot}
&=&-\frac{G\!M\!m\!}{d} \Sigma^N_{i=0} \Sigma^{\infty}_{n=0}\! \left(\! \frac{r}{d}\! \right)^n
\!P_n\! \left(\! \cos{\!\left( \! \omega_{\mathrm{rot}} t \!+\!\frac{2 \pi i}{N}\right)\!}\right),
\end{eqnarray}
where $P_n$ is the Legendre polynomial, and $V_i$ is the potential of a mass. The equation of motion of the test mass is
\begin{eqnarray}
Ma&=&\left| \frac{\partial V}{\partial{d}} \right| =\frac{GMm}{d^2}\Sigma^N_{i=0} \Sigma^{\infty}_{n=0}(n+1) \left( \frac{r}{d} \right)^n \nonumber P_n\left(\cos{\left(\omega_{\mathrm{rot}} t +\frac{2 \pi i}{N}\right)}\right),
\end{eqnarray}
where $a$ is the acceleration of the test mass.
We arrange the masses around the rotor in a superposition of quadrupole and hexapole arrangements, as shown in Fig.~\ref{fig:hex}. A hole is placed between each mass. These holes is effectively double the magnitude of the gravity gradient. Therefore, the equation of motion of the test mass is
\begin{eqnarray}
Ma&=&\left| \frac{\partial V}{\partial{d}} \right| =\frac{2GMm}{d^2}\Sigma^N_{i=0} \Sigma^{\infty}_{n=0}(n+1) \left( \frac{r}{d} \right)^n \nonumber P_n\left(\cos{\left(\omega_{\mathrm{rot}} t +\frac{2 \pi i}{N}\right)}\right). \label{eq:EOM}
\end{eqnarray}
Next, we will calculate the displacements of the quadrupole and hexapole rotor masses in sections ~\ref{Quad} and ~\ref{Hexa}.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{Hexapole.eps}
\caption{Configuration of the rotor with quadrupole and hexapole mass distributions. $m_{\mathrm{q}}$ and $m_{\mathrm{h}}$ are the masses of quadrupole and hexapole masses. $r_{\mathrm{q}}$ and $r_{\mathrm{h}}$ are the radii of the quadrupole and arrangements hexapole.}
\label{fig:hex}
\end{center}
\end{figure}
\subsection{Displacement of test mass driven by quadrupole mass distribution} \label{Quad}
We calculate the displacement of the quadrupole mass distribution with two pieces and two holes so $N=2$.
The masses and radii of the quadrupole arrangement are $m_{\mathrm{q}}$ and $r_{\mathrm{q}}$.
The equation of motion for the test mass is
\begin{eqnarray}
Ma&=&\frac{2GMm_{\mathrm{q}}}{d^2}\Sigma^{\infty}_{n=0}(n+1) \left( \frac{r_{\mathrm{q}}}{d} \right)^n \nonumber \Sigma^1_{i=0} P_n\left(\cos{\left(\omega_{\mathrm{rot}} t +\pi i \right)}\right).
\end{eqnarray}
If we assume $r \ll d$, the displacement of the time-dependent lower harmonics can be written as
\begin{equation}
x=\Sigma_{k=1}^{\infty}x_{k\mathrm{f}}\cos(k\omega_{\mathrm{rot}} t)\sim x_{\mathrm{2f}}\cos(2\omega_{\mathrm{rot}} t)=x_{\mathrm{2f}}\cos{\omega t},
\end{equation}
where $k$ is the number of the harmonics.
The amplitude of the 2-f rotation is then
\begin{equation}
x_{2\mathrm{f}}=9\frac{GMm_{\mathrm{q}}r_{\mathrm{q}}^2}{d^4}s(\omega). \label{2f}
\end{equation}
\subsection{Displacement of test mass driven by hexapole mass distribution} \label{Hexa}
We also calculate the displacement of the hexapole mass distribution with three holes as $N=3$.
The masses and radii of the hexapole distribution are $m_{\mathrm{h}}$ and $r_{\mathrm{h}}$.
The equation of motion of test mass driven by this arrangement alone is
\begin{eqnarray}
Ma &=& \frac{2GMm_{\mathrm{h}}}{d^2}\Sigma^{\infty}_{n=0}(n+1) \left( \frac{r_{\mathrm{h}}}{d} \right)^n \nonumber \Sigma^2_{i=0} P_n \left(\cos{\left(\omega_{\mathrm{rot}} t+\frac{2\pi i}{3} \right)} \right).
\end{eqnarray}
If we assume $r \ll d$, the displacement of the time-dependent lower harmonics can be written as
\begin{equation}
x=\Sigma_{k=1}^{\infty}x_{k\mathrm{f}}\cos(k\omega_{\mathrm{rot}} t)\sim x_{3\mathrm{f}}\cos(3\omega_{\mathrm{rot}} t)=x_{\mathrm{3f}}\cos{\omega t},
\end{equation}
where amplitude of 3-f is described as
\begin{equation}
x_{3\mathrm{f}}=15\frac{GMm_{\mathrm{h}}r_{\mathrm{h}}^3}{d^5}s(\omega). \label{3f}
\end{equation}
\section{Absolute power calibration with both photon and Gravity field calibrator} \label{sec:PGCAL}
This section discusses how to combine the calibration signals from Pcal and Gcal to allow absolute laser power calibration using an interferometer.
Figure~\ref{fig:IFO} diagrams the combined calibration system.
First, the test mass is driven by the Gcal. The $x_{\mathrm{2f}}$ and $x_{\mathrm{3f}}$ signals are measured from the response of the interferometer. Second, this interferometer signal is sent to the excitation port of the Pcal. This signal acts as a reference signal for feedback control, as shown in Fig.~\ref{fig:IFO}. The Pcal then cancels out the displacement modulated by the Gcal.
Third, the voltage responses of the transmitter and the receiver module photodetectors are measured. The output signal of the transmitter module, $V_{\mathrm{TxPD}}$ and receiver module, $V_{\mathrm{RxPD}}$ should correspond to the displacement caused by the Gcal. By using Eq~(\ref{eq:pcal}),(\ref{2f}), and (\ref{3f}), the modulated signal powers are
\begin{eqnarray}
P_{\mathrm{2f}}=\frac{9}{2} \frac{Gcm_{\mathrm{q}}Mr_{\mathrm{q}}^2}{d^4cos\theta}\frac{1}{1+\frac{M}{I}\vec{a}\cdot \vec{b}}, \label{P2f} \\
P_{\mathrm{3f}}= \frac{15}{2}\frac{Gcm_{\mathrm{h}}Mr_{\mathrm{h}}^3}{d^5cos\theta}\frac{1}{1+\frac{M}{I}\vec{a}\cdot \vec{b}}. \label{P3f}
\end{eqnarray}
Fourth, we demodulate the signal of the transmitter and receiver modules using the measured encoder signal from the Gcal.
The demodulated signals are
\begin{eqnarray}
V_{\mathrm{2f}}^{\mathrm{T}}=\rho_{\mathrm{T}}P_{\mathrm{2f}}, \\
V_{\mathrm{2f}}^{\mathrm{R}}=\rho_{\mathrm{R}}P_{\mathrm{2f}}, \\
V_{\mathrm{3f}}^{\mathrm{T}}=\rho_{\mathrm{T}}P_{\mathrm{3f}}, \\
V_{\mathrm{3f}}^{\mathrm{R}}=\rho_{\mathrm{R}}P_{\mathrm{3f}},
\end{eqnarray}
where $\rho_{\mathrm{T}}$ and $\rho_{\mathrm{R}}$ are the transfer functions from power to the photo detector output voltages at the transmitter and receiver modules.
Therefore, we can measure the distance from the ratio of responses of the 2-f and 3-f components:
\begin{equation}
d=\frac{5}{3} \frac{V_{\mathrm{2f}}^{\mathrm{T}}}{V_{\mathrm{3f}}^{\mathrm{T}}}\frac{m_{\mathrm{h}}}{m_{\mathrm{q}}}\frac{r_{\mathrm{h}}^{3}}{r_{\mathrm{q}}^{2}}=\frac{5}{3} \frac{V_{\mathrm{2f}}^{\mathrm{R}}}{V_{\mathrm{3f}}^{\mathrm{R}}} \frac{m_{\mathrm{h}}}{m_{\mathrm{q}}}\frac{r_{\mathrm{h}}^{3}}{r_{\mathrm{q}}^{2}}. \label{d}
\end{equation}
Finally, we calculate the displacement formula for the Pcal calibrated by Gcal. We substitute the Eq. (\ref{2f}) to Eq. (\ref{eq:pcal}) to obtain the following equation for displacement:
\begin{eqnarray}
x&=&\frac{2P \cos{\theta}}{c} s(\omega)\left(1+\frac{M}{I}\vec{a} \cdot \vec{b} \right), \\
&=&9\frac{Gm_{\mathrm{q}} M r_{\mathrm{q}}^2}{d^4}\frac{P}{P_{\mathrm{2f}}}s(\omega) , \\
&=&\frac{729}{625} \frac{GM m^5_{\mathrm{q}} r_{\mathrm{q}}^{10}}{m^4_{\mathrm{h}} r_{\mathrm{h}}^{12} } \frac{{V_{\mathrm{3f}}^{R}}^4}{{V_{\mathrm{2f}}^{R}}^5}V_{\mathrm{in}} s(\omega) , \label{pcal_new}
\end{eqnarray}
where we assumed that $P(\omega)=\rho_{\mathrm{R}} V_{\mathrm{in}}$, and $V_{\mathrm{in}}$ is the amplitude of the input voltage.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{IFO.eps}
\caption{Test apparatus for the absolute calibration. The Gcal is placed behind the test mass. The frequency of the Gcal is monitored with the encoder output. The error signal for the differential arm length of the interferometer is sent to the reference port of the photon calibrator for canceling the modulation of the dynamic gravity field with feedback with transfer function $G$. Output signals from the photon calibrator are synchronized with the forces driven by the Gcal. The output signals are demodulated with 2-f and 3-f signals monitored by the encoder.}
\label{fig:IFO}
\end{center}
\end{figure}
The factor $(GMm_{\mathrm{q}}^5 r_{\mathrm{q}}^{10})/(m_{\mathrm{h}}^4 r_{\mathrm{h}}^{12})$ can be measured before the calibration. $V^R_{\mathrm{3f}}/V^R_{\mathrm{2f}}$ is measured during the calibration of the Gcal and Pcal. The interval of the calibration signals between the Pcal and Gcal depend on the stability of the photon calibrator laser power. The Advanced LIGO experiment calibrates the absolute laser power using the working standard monthly. Therefore, The Gcal should be run monthly or more frequently. The present method reconstructs the Pcal signal from Gcal signals. Therefore, the Gcal does not need to be operated during observation runs.
During operation, the Gcal would contaminate the noise floor by adding acoustic and/or vibration noise. However, we can minimize this noise effect by controlling the rotation frequency. The above analysis has not considered to the noise added by the Gcal during observation runs, as we only propose that the Gcal be used to calibrate the absolute displacement before the observations.
The demodulation technique allows us to reduce the systematic error introduced by rotation . When the modulation of Gcal is canceled using Pcal, the transfer functions of the Gcal and Pcal are also canceled. Therefore, the estimated displacement of the test mass does not depend on the frequency of the rotation.
\section{Estimation of uncertainty} \label{sec:EST}
In this section, we evaluate the accuracy of the estimated displacement, and discuss the effects on systematic error by changing the operating frequency and distance. After that, we discuss the uncertainty in the displacement of the mirror.
The following discussion assumes the basic parameters of the KAGRA experiment listed in Table~\ref{pcal}, and the parameters of the Gcal as listed in Table~\ref{sus}.
\begin{table}
\begin{center}
\caption{\label{sus}Assumed parameters. $G$ is gravity constant~\cite{RevModPhys.88.035009}. $\theta$ is incident angle of the Pcal beams. $M$ is mass of test mass. $1+\frac{I}{M}\vec{a}\cdot \vec{b}$ is geometrical factor.}
\footnotesize
\begin{tabular}{ccc}
\hline
&Value&Relative uncertainty \\
\hline
$G$&$6.67408 \times 10^{-11}~\mathrm{m^3kg^{-1}sec^{-2}}$&0.0047 \%\\
$\cos{\theta}$ &1.000& 0.07~\%\\
$M$ &22.89~kg & 0.02~\%\\
$m_{\mathrm{q}}$&4.485~kg & 0.004~\%\\
$m_{\mathrm{h}}$& 4.485~kg &0.004~\%\\
$r_{\mathrm{q}}$&0.200~m & 0.010~\%\\
$r_{\mathrm{h}}$& 0.125~m & 0.016~\%\\
$1+\frac{I}{M}\vec{a}\cdot \vec{b}$& 1&0.3~\% \\
\hline
\end{tabular}\\
\end{center}
\end{table}
\subsection{Systematic error of higher order terms}
To achieve the precision less than 1 \%, we need to consider the effect of higer-order Legendre polynomials at the position of the test mass. This is because higher-order polynomials also affect the 2-f and 3-f components. The $n$-th order Legendre polynomial is calculated with Eq.(\ref{eq:EOM}). The effect of higher-order factors is mitigated by the factor $(r/d)^n$. Tables~\ref{tab:N2} and \ref{tab:N3} show the calculated displacements of the higher order terms.
To investigate the higher order effects, we compare the estimated test mass displacement between the Legendre polynomial approximation and numerical calculations of $\frac{\partial V}{\partial{d}}$ and Eq.(\ref{eq:vpot}). The ratio of two calculations of the test mass displacement is shown in Figs.~\ref{fig:FEM-2f} and ~\ref{fig:FEM-3f} for the quadrupole ($N=2$) and hexapole ($N=3$) components, respectively, as a function of the distance, $d$.
The results show that the effect of higher-order of polynomials is less than that of systematic error. The mirror therefore needs to be placed at least 2~m away from the rotating mass. Then the sum of the first and second order equations can be used to suppress the systematic error well below 1 \% as shown in Figs.~\ref{fig:FEM-2f} and \ref{fig:FEM-3f}.
If we place the Gcal near the KAGRA end test mass, the distance of 2~m is reasonable. The rotor could be mounted outside of the vacuum chamber. In the following calculations, we assume $d= $~2~m for the simplification of the discussion.
The analytical calculation of the displacement of the test mass in Eq.(\ref{eq:EOM}) assumes that the rotor masses and the test mass can be approximated as point masses.
We compared the results of this analytical calculation with the numerical integral of the displacements generated by the actual dimensions of the rotor
with the parameters shown in Table~\ref{sus}, and confirmed that the analytical formula is sufficiently at $d= $~2~m.
\begin{table}
\begin{center}
\caption{Calculated quadrupole ($N=2$) displacement. $n$ is the order of the Legendre polynomial, where $\omega=n\omega_{\mathrm{rot}}$. \label{tab:N2}}
\footnotesize
\begin{tabular}{cccccccc}
\hline
& n=1 & n=2& n=3 &n=4&n=5&n=6&n=7 \\
\hline
1-f&0&0&0&0&0&0&0 \\
2-f&0&$9 \frac{Gmr^2}{d^4\omega^2}$&0&$\frac{25}{4} \frac{Gmr^4}{d^6\omega^2}$&0&$\frac{735}{128} \frac{Gmr^6}{d^8\omega^2}$&0 \\
3-f&0&0&0&0&0&0&0\\
4-f&0&0&0&$\frac{175}{16} \frac{Gmr^4}{d^6\omega^2}$&0& $\frac{273}{32} \frac{Gmr^6}{d^8\omega^2}$&0 \\
5-f&0&0&0&0&0&0&0 \\
6-f&0&0&0&0&0&$\frac{1617}{128} \frac{Gmr^6}{d^8\omega^2}$&0 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Calculated hexapole ($N=3$) displacement. $n$ is the order of the Legendre polynomial, where $\omega=n\omega_{\mathrm{rot}}$. \label{tab:N3}}
\footnotesize
\begin{tabular}{cccccccc}
\hline
& n=1 & n=2& n=3 &n=4&n=5&n=6&n=7 \\
\hline
1-f&0&0&0&0&0&0&0 \\
2-f&0&0&0&0&0&0&0 \\
3-f&0&0&$15\frac{Gmr^3}{d^5\omega^2}$&0&$\frac{315}{32}\frac{Gmr^5}{d^7\omega^2}$&0& $\frac{567}{64} \frac{Gmr^7}{d^9 \omega^2}$\\
4-f&0&0&0&0&0&0&0 \\
5-f&0&0&0&0&0&0&0 \\
6-f&0&0&0&0&0&$\frac{4851}{256} \frac{Gmr^6}{d^8\omega^2}$&0 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{2f.eps}
\caption{Ratio of Legendre polynomial approximations with the numerical calculations of $\frac{\partial V}{\partial{d}}$ and Eq.(~\ref{eq:vpot})
on the test mass displacement for the quadrupole ($N=2$) component as a function of the distance. Dotted, dashed and solid lines correspond to first-order only, second-orders, and third-order approximations, respectively.
The analytical results are listed in Table~\ref{tab:N2}. To achieve precision less than 1~\%, the higher-order terms need to be included.}
\label{fig:FEM-2f}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{3f.eps}
\caption{Ratio of the Legendre polynomial approximation to the numerical calculations of $\frac{\partial V}{\partial{d}}$ and Eq.(~\ref{eq:vpot})
on the test mass displacement for the hexapole distribution ($N=3$) component as a function of distance. The dotted, dashed and solid lines correspond to the first-order only, second-orders, and third-orders, respectively.
The analytical result is listed in Table~\ref{tab:N3}. To achieve the precision less than 1~\%, the higher-order terms need to be included..}
\label{fig:FEM-3f}
\end{center}
\end{figure}
\subsection{Systematic error of the transfer function}
The Gcal modulates the test mass mirror with gravitational potential gradient. However, this gradient also actuates the masses of suspension system as shown in Fig.~\ref{fig:cryo}. We simulated the transfer function with the assumption of the cryogenic suspension system installed in KAGRA~\cite{0264-9381-34-22-225001}.The transfer function was calculated using the rigid-body suspension simulation code, called SUMCON~\cite{SUMCON}. We estimated the total displacement by superimposing the displacements driven by both mass distributions. Figure \ref{fig:ratio} shows the displacement ratio between the motion signal and the free-mass motion as a function of frequency. The simulation result is in good agreement with the free-mass motion at this frequencies larger than 20~Hz. The low frequency structures correspond to the resonant peak of the suspension system. Therefore, we can neglect this intermediate-mass effect and regard as motion at frequency over 20~Hz as free-mass motion.
Therefore, we need to operate the rotor at speeds larger than 20 Hz to achieve error less than 0.1~\%.
We assumed the rotation frequency to be 16~Hz, which corresponds to 32~Hz and 48~Hz for the effective frequency of the 2-f and 3-f components.
This assumption applies to the discussion in the next section.
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{Cryo.eps}
\caption{Schematic of the suspension system. The parameters of the heights and masses are marked with their assumed values. }
\label{fig:cryo}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=12cm]{dx_Gcal_ratio.eps}
\caption{Displacement ratio of the transfer function for mult-pendulums by changing modulation frequency. The relations of the modulation frequency, $f$, modulation angular frequency, $\omega$, and rotation angular frequency $\omega_{\mathrm{rot}}$ are described as $n\omega_{\mathrm{rot}}=\omega=2\pi f$. }
\label{fig:ratio}
\end{center}
\end{figure}
\subsection{Uncertainty of displacement and laser power}
In this section, we estimate the typical displacement based on the result in Table.~\ref{sus}. We neglect Legendre polynomials of degree higher than twwo in the following discussion to simplify the discussion, though they are relevant in the calculations.
The estimated 2-f and 3-f displacements are described as
\footnotesize
\begin{eqnarray}
x^{\mathrm{rms}}_{\mathrm{2f}}&=&1.18 \times 10^{-16}\mathrm{[m]} \times \left( \frac{G}{6.67408 \times 10^{-11} \mathrm{[m^3kg^{-1}sec^{-2}]}} \right) \nonumber \\
&\times&\! \left( \! \frac{m_{\mathrm{q}}}{4.485 \mathrm{[kg]}} \!\right) \! \times \!\left( \!\frac{r_{\mathrm{q}}}{0.200 \mathrm{[m]}} \! \right)^2 \! \times \! \left( \! \frac{2\mathrm{[m]}}{d} \! \right)^4 \! \times \! \left( \! \frac{2\pi \! \times \! 32\mathrm{[Hz]}}{\omega} \! \right)^2,\\
x^{\mathrm{rms}}_{\mathrm{3f}}&=&2.13 \times 10^{-18}\mathrm{[m]} \times \left( \frac{G}{6.6742 \times 10^{-11} \mathrm{[m^3kg^{-1}sec^{-2}]}} \right) \nonumber \\
&\times& \! \left( \! \frac{m_{\mathrm{h}}}{4.485 \mathrm{[kg]}}\! \right) \! \times \! \left( \!\frac{r_{\mathrm{h}}}{0.125 \mathrm{[m]}} \! \right)^3
\! \times \! \left(\! \frac{2\mathrm{[m]}}{d} \! \right)^5 \! \times \! \left( \! \frac{2\pi\!\times \! 48\mathrm{[Hz]}}{\omega} \! \right)^2.
\end{eqnarray}
\normalsize
We define the SNR in the term of the ratio of RMS displacement of the design noise spectrum density for the interferometer of KAGRA at 32~Hz for 2-f and 48~Hz for 3-f.
Using this result, we estimate the SNR of the peaks.
\footnotesize
\begin{eqnarray}
\!S\!N\!R_{\mathrm{2f}}&=&392 \times \left(\frac{3.0 \times 10^{-19} [\mathrm{m/\sqrt{Hz}}]}{n_{\mathrm{32Hz}}} \right) \times \left(\frac{T}{1 [\mathrm{sec}]} \right)^{\frac{1}{2}} \left(\frac{x_{\mathrm{2f}}^{\mathrm{rms}}}{1.178 \times 10^{-16}\mathrm{[m]} } \right), \\
\!S\!N\!R_{\mathrm{3f}}&=&73 \times \left(\frac{2.9 \times 10^{-20} [\mathrm{m/\sqrt{Hz}}]}{n_{\mathrm{48Hz}}} \right) \times \left(\frac{T}{1 [\mathrm{sec}]} \right)^{\frac{1}{2}} \left(\frac{x_{\mathrm{2f}}^{\mathrm{rms}}}{2.130 \times 10^{-18}\mathrm{[m] }} \right),
\end{eqnarray}
\normalsize
where $T$ is the integration time. If we integrate a signal larger than 3 min, we can measure $V^{\mathrm{R}}_{2f}$ and $V^{\mathrm{R}}_{3f}$ with sufficiently high SNR that systematic error can be reduced to less than 0.1~\%.
This method used to measure the absolute laser power as well. The estimated powers are
\footnotesize
\begin{eqnarray}
P_{\mathrm{2f}}&=&0.023 ~\mathrm{[W]}\times \left( \frac{G}{6.6742 \times 10^{-11} \mathrm{[m^3kg^{-1}sec^{-2}]}} \right) \times \left( \frac{m_{\mathrm{q}}}{4.485 \mathrm{[kg]}} \right) \times \left( \frac{r_{\mathrm{q}}}{0.200 \mathrm{[m]}} \right)^2 \times \left( \frac{2\mathrm{[m]}}{d} \right)^4 \nonumber \\ &&\times \left( \frac{1}{\cos{\theta}} \right) \times \left( \frac{1}{1+\frac{M}{I}\vec{a}\cdot \vec{b}} \right)^2,\\
P_{\mathrm{3f}}&=&0.00095~\mathrm{[W]} \times \left( \frac{G}{6.6742 \times 10^{-11} \mathrm{[m^3kg^{-1}sec^{-2}]}} \right) \times \left( \frac{m_{\mathrm{h}}}{4.485 \mathrm{[kg]}} \right) \times \left( \frac{r_{\mathrm{h}}}{0.125 \mathrm{[m]}} \right)^3 \times \left( \frac{2\mathrm{[m]}}{d} \right)^5 \nonumber \\ &&\times \left( \frac{1}{\cos{\theta}} \right) \times \left( \frac{1}{1+\frac{M}{I}\vec{a}\cdot \vec{b}} \right)^2.
\end{eqnarray}
\normalsize
We estimate the laser power with following equations:
\footnotesize
\begin{eqnarray}
\left( \frac{\delta P_{\mathrm{2f}}}{P_{\mathrm{2f}}} \right)^2 &\sim& \!16 \! \left( \frac{\delta V^{\mathrm{R}}_{{\mathrm{2f}}}}{V^{\mathrm{R}}_{{\mathrm{2f}}}} \!\right)^2+16\left( \frac{\delta V^{\mathrm{R}}_{{\mathrm{3f}}}}{V^{\mathrm{R}}_{{\mathrm{3f}}}} \right)^2 +\left( \frac{\delta P_{\mathrm{sys}}}{P_{\mathrm{sys}}} \right)^2, \label{dP2f} \\
\left( \frac{\delta P_{\mathrm{3f}}}{P_{\mathrm{3f}}} \right)^2 &\sim& \!16 \! \left( \frac{\delta V^{\mathrm{R}}_{{\mathrm{2f}}}}{V^{\mathrm{R}}_{{\mathrm{2f}}}} \right)^2+16\left( \frac{\delta V^{\mathrm{R}}_{{\mathrm{3f}}}}{V^{\mathrm{R}}_{{\mathrm{3f}}}} \right)^2+\left( \frac{\delta P_{\mathrm{sys}}}{P_{\mathrm{sys}}} \right)^2, \label{dP3f}
\end{eqnarray}
\normalsize
where $\delta P_{\mathrm{sys}}/P_{\mathrm{sys}}$ is the relative systematic error of the power due to the machining tolerance of the rotor masses and radiuses, which are calculated by
\begin{eqnarray}
\frac{\delta P_{\mathrm{sys}}}{P_{\mathrm{sys}}}&\sim& \frac{\delta G}{G} + \frac{\delta M}{M} +\frac{\delta \cos{\theta}}{\cos{\theta}}+ \frac{\delta\left( 1+\frac{M}{I}\vec{a}\cdot \vec{b} \right)}{\! \left( \! 1+\frac{M}{I}\vec{a}\cdot \vec{b} \! \right)}
+\frac{12}{\sqrt{6}} \frac{\delta r_{\mathrm{h}}}{r_{\mathrm{h}}} +\frac{10}{2} \frac{\delta r_{\mathrm{q}}}{r_{\mathrm{q}}} +\frac{5}{2} \frac{\delta m_{\mathrm{q}}}{m_{\mathrm{q}}} +\!\frac{4}{\sqrt{6}} \! \frac{\delta m_{\mathrm{h}}}{m_{\mathrm{h}}}.
\end{eqnarray}
We next consider the mitigating effect of the systematic error of the masses and radiuses due to the tolerance and uncertainty of the measurement instruments. The values of the masses and radiuses vary slightly with the tolerance of the fabrication process. The errors in $m_{\mathrm{q}}$, $r_{\mathrm{q}}$, $m_{\mathrm{h}}$, and $r_{\mathrm{h}}$ are mitigated by the factor of $1/\sqrt{6}$ and $1/\sqrt{4}$.
The uncertainty in the quadrupole and hexapole masses are limited by the accuracy of the electronic balance. In this case, we modeled masses made of Tungsten. The density of Tungsten is $19.25~\mathrm{g/cm^3}$. The diameter and thickness of the mass are 0.06~m and 0.08~m, respectively. Therefore, the mass of the rotor mass is 4.485~kg. We assumed that the CG-6000 electronic balance is used to weigh these means, with tolerance of 0.2~g~\cite{CG6000}. Therefore, the relative uncertainty in the mass of the rotor mass is 0.004~\%.
The rotor disk can be machined by Numerical Control milling. Dimensional accuracy
of less than 0.02 mm can typically be achieved with this process. For measuring the shape, we assume that a three-dimension coordinate measuring machine (CMM) will be employed~\cite{Inoue:16}. The precision of CMM is $2~\mathrm{\mu m}$. This indicates that we can measure the shape of the rotor and masses with sufficiently low uncertainty using the CMM.
The estimated relative uncertainties of the laser powers are 0.52~\%. One of the largest uncertainties is the geometrical factor of the Pcal laser. The geometrical factor uncertainty is assumed to be 0.3 \%, which is the same number as the instrument used in Advanced LIGO.
Finally, assuming that the statistical fluctuations of $V_{in}$, $s(\omega)$, $V^{\mathrm{R}}_{{\mathrm{2f}}}$, and $V^{\mathrm{R}}_{{\mathrm{3f}}}$ are independent for each measurement and therefore can be added in quadrature, the estimated relative uncertainty in the displacement measurements is written as
\footnotesize
\begin{equation}
\left( \frac{\delta x}{x} \right)^2 \!\sim \! \left( \!\frac{\delta V_{in}}{V_{in}}\! \right)^2+\left(\! \frac{\delta s(\omega)}{s(\omega)} \! \right)^2\!+\!25\!\left(\!\frac{\delta V^{\mathrm{R}}_{{\mathrm{2f}}}}{V^{\mathrm{R}}_{{\mathrm{2f}}}}\! \right)^2+16\!\left(\! \frac{\delta V^{\mathrm{R}}_{{\mathrm{3f}}}}{V^{\mathrm{R}}_{{\mathrm{3f}}}}\! \right)^2\!+ \left(\! \frac{\delta x_{\mathrm{sys}}}{x_{\mathrm{sys}}} \! \right)^2 \label{deltax},
\end{equation}
\normalsize
where $\delta x_{\mathrm{sys}}/x_{\mathrm{sys}}$ is the relative systematic error of the displacement which cannot be added in quadrature. This factor is written as
\footnotesize
\begin{equation}
\frac{\delta x_{\mathrm{sys}}}{x_{\mathrm{sys}}}=\frac{\delta G}{G} + \frac{\delta M}{M} +\frac{12}{\sqrt{6}} \frac{\delta r_{\mathrm{h}}}{r_{\mathrm{h}}} +\frac{10}{2} \frac{\delta r_{\mathrm{q}}}{r_{\mathrm{q}}} +\frac{5}{2} \frac{\delta m_{\mathrm{q}}}{m_{\mathrm{q}}} +\!\frac{4}{\sqrt{6}} \! \frac{\delta m_{\mathrm{h}}}{m_{\mathrm{h}}}.
\end{equation}
\normalsize
We assumed the mitigation factors of radiuses and masses discussed above in this calculation.
To reduce the noise of the displacement measurement, we need to reduce the uncertainty in the shape of the rotor and masses.
The uncertainties in $V^{\mathrm{R}}_{\mathrm{2f}}$,$V^{\mathrm{R}}_{\mathrm{3f}}$, $V^{\mathrm{R}}_{0}$ are much less than that of other contributions. We can reduce the uncertainty of these values using long integration times with statistical measures. Each of the uncertainty is listed in Table~\ref{sus}. The estimated total uncertainty of the displacement measurement is 0.17~\%.
\section{Conclusion}
Pcal are used in Advanced LIGO, Advanced Virgo and KAGRA. These devices are used to calibrate the interferometer response, and the uncertainty in the calibration affects the estimation of parameters of the GW source. In particular, the distance to the source strongly depends on the absolute laser power of the photon calibrator. In previous studies, the gold standard, in which the interferometer response is calibrated to the NIST laser power standard, was used for the absolute laser power calibration of the photon calibrator. However, the current standard for absolute laser power vary by a few~\% between different countries's metrology institutes. This uncertainty propagates directly to the calculation of the GW detector's absolute displacement.
To address this problem, we proposed a combined calibration method that uses both a Pcal and a Gcal. The Gcal modulated the test mass using a dynamic gravity field. When canceling the displacement of the test mass using the Pcal, the Gcal was used to calibrate the interferometer responce.
This method had the advantage of offering a direct comparison between the amplitudes of the injected power and gravity field modulation at the test mass. Without the proposed gravity-field calibrator, the uncertainties of the optical efficiency through the window and mirrors and the geometrical factor of the laser position need to considered, because the working standard calibration is measured outside of the chamber. However, the method of gravity field can compare the displacement directly. Using this method, the uncertainty of the optical efficiency is avoided when calibrating the absolute laser power. The estimated laser power uncertainty with this method is 0.52~\%. This result suggests that a new power calibration standard can be proposed that gains threefold improvement over the current standards.
Finally, we estimated the uncertainty of absolute calibration with the proposed method. The estimated absolute uncertainty in the displacement measurement is 0.17~\%, which is a tenfold improvement on previous studies. This uncertainty affects the estimation of the distance to the gravitational wave source. This estimated uncertainty brings the precision of the Hubble constant to less than 1~\%. This may address the tension between the Cephied-SN distance ladder~\cite{Riess_2016} and CMB data assuming a $\Lambda$CDM
model~\cite{2016-planck}.
\acknowledgments
We thank Richard Savage and Darkhan Tuyrnbayev for discussion of the photon calibrator. We would like to express our gratitude to Prof. Takaaki Kajita and Prof. Henry Wong. We would like to thank the KEK Cryogenics Science Center for the support. YI and SH are supported by Academia Sinica and Ministry of Science and Technology (MOST) under Grants No. CDA-106-M06, MOST106-2628-M-007-005 and MOST106-2112-M-001-016 in Taiwan. This work was supported by JSPS KAKENHI Grant Numbers 17H106133 and 17H01135. KAGRA project is supported by MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, MEXT Grant-in-Aid for Scientific Research on Innovative Areas 24103005, JSPS Core-to-Core Program, A. Advanced Research Networks, and the joint research program of the Institute for Cosmic Ray Research, University of Tokyo.
|
2,869,038,154,907 | arxiv | \section{Introduction}
Albert algebras are exceptional Jordan algebras. Over fields, any simple Jordan algebra is either \emph{special}, i.e.\ a subalgebra of the anticommutator algebra $B^+$
of an associative algebra $B$, or an Albert algebra \cite{MZ}. Albert algebras over rings have attracted recent interest, as is seen in the extensive survey \cite{P}, which contains a historic account to which we refer the interested reader.
Over an algebraically closed field, there is a unique Albert algebra up to isomorphism; this both
testifies to the remarkable character of these algebras, and indicates that the framework of torsors, descent and cohomology is suitable for studying them over a general base.
Furthermore, the assignment $A\mapsto \mathbf{Aut}(A)$ defines a correspondence between Albert algebras and groups of type $\mathrm F_4$. Thus understanding
Albert algebras provides a precise understanding of such groups, and vice versa. This is analogous to the relationship between octonion algebras and groups of type $\mathrm G_2$.
In the present work, we
approach Albert algebras using appropriate torsors (principal homogeneous spaces), building on this interplay between groups and algebras as well as the relation that Albert
algebras have to octonion algebras, triality for groups of type $\mathrm D_4$, and cubic norms. Thanks to this approach, we are able to shed new light on classical constructions, generalise them, and, expressing the objects as twists by appropriate torsors, we are able to obtain results that the classical methods
have fallen short of providing.
We begin by giving a sketch of the key ideas of the present paper,
with emphasis on the torsors
involved. In this introduction, we abuse notation by referring to groups simply by their type; this is done for the sake of overview and does not imply uniqueness of groups of a certain type.
Precise notation will be used in the respective sections.
An indication that Albert algebras over general unital, commutative rings can be expected to behave much less orderly than they do over fields is that this is true for octonion algebras.
Indeed, Gille showed in \cite{G1}, by means of $\mathrm G_2$-torsors, that octonion
algebras over rings are not determined by their norm forms. In \cite{AG} the author, together with Gille, showed, using triality, how twisting an octonion algebra $C$ by
such a torsor precisely gives the classical construction
of isotopes of $C$. Abusing notation, these papers dealt with the torsor\footnote{In this notation for torsors, the arrow denotes the structure map from the total space to the base, and
the group labelling the arrow is the structure group.}
\[\xymatrix@R-10pt{\mathrm D_4\ar[d]^{\mathrm G_2}\\ \mathbf{S}_C^2}\]
where $\mathrm D_4$ and $\mathrm G_2$ are (simply connected) groups of the corresponding types, and $\mathbf{S}_C$ is the octonionic unit sphere in $C$, which is an affine scheme.
Inspired by this, we set out, in the current work, to examine Albert algebras from this point of view.
We start in Section \ref{S2} with the question of isotopy, where we show that the isotopes of an Albert algebra $A$ are precisely the twists of $A$ by the torsor defined by the projection
of the isometry group of the cubic norm $N$ of $A$, which is simply connected of type $\mathrm E_6$, to the unit sphere of $N$. This is a torsor under the automorphism group of $A$; thus it is of the type
\[\xymatrix@R-10pt{\mathrm E_6\ar[d]^{\mathrm F_4}\\ \mathbf{S}_N}\]
with $\mathbf S_N$ an affine scheme. The non-triviality of this torsor for certain Albert algebras is therefore known, since it is
known that over e.g.\ $\mathbb R$ the Albert algebra $H_3(\mathbb O)$ of hermitian $3\times 3$-matrices over the real division octonion algebra $\mathbb O$
has non-isomorphic isotopes. However, over any field, isotopes of the split Albert algebra are isomorphic. We prove that this is not the case over rings in general.
In Section \ref{S3} we turn to the inclusion of groups of type $\mathrm D_4$ into the automorphism group of a \emph{reduced} Albert algebra, i.e.\ one of the form $H_3(C,\Gamma)$ for
an octonion algebra $C$ and a triple $\Gamma$ of invertible scalars. (This is a slight deformation of the algebra $H_3(C,\mathbf{1})=H_3(C)$ of hermitian matrices.) In \cite{ALM} it was shown that over fields,
this inclusion defines a $\mathrm D_4$-torsor where the base is the scheme of frames of idempotents of the Albert algebra, and this result easily generalises to rings. We show that over rings this torsor is non-trivial in general, even under the condition
that the Albert algebra be split. This torsor is of the type
\[\xymatrix@R-10pt{\mathrm F_4\ar[d]^{\mathrm D_4}\\ \mathbf{F}_A}\]
with $\mathbf F_A$ an affine scheme. In a sense, therefore, this torsor bridges the gap between the two torsors above. The natural question that arises then is what objects it parametrizes.
To answer it, we observe that the definition of a reduced Albert algebra becomes more clean if the symmetric algebras of para-octonions are used instead of octonion algebras. This is in
line with a recent philosophy followed in \cite{KMRT} and \cite{AG}. From there, we are able to generalise the construction of reduced Albert algebras by noting that one may replace the para-octonion algebra by
any \emph{composition of quadratic forms} of rank 8. Such objects generalise (para)-octonion algebras, since they consist of a triple $(C_1,C_2,C_3)$ of quadratic spaces, each of constant
rank 8 as a locally free module, with a map
(the composition) $C_3\times C_2\to C_1$ that is compatible with the quadratic forms. If $C_1=C_2=C_3=C$, where $C$ is an octonion algebra, then the multiplication of $C$ is a composition
of quadratic forms, and every composition of rank 8 is locally isomorphic to a compositions arising from an octonion algebra in this way. We show that the $\mathrm D_4$-torsor in question classifies
those compositions of quadratic spaces that give rise to isomorphic Albert algebras via this generalised construction. Non-triviality of the torsor means that two non-isomorphic compositions may give
rise to isomorphic Albert algebras. More precisely we show that even the algebra $H_3(C)$ can arise from compositions not isomorphic to the one obtained from $C$, even in the split case.
This gives rise to the yet more precise question of the extent to which a reduced Albert algebra $A=H_3(C)$ determines $C$, which we treat in Section \ref{S4}. Over a field $k$, the isomorphism class of $A$ (in fact even the
isotopy class) determines $C$ up to isomorphism; this goes back to the 1957 paper \cite{AJ} of Albert and Jacobson. In cohomological terms, the map
\[H^1(k,\mathbf{Aut}(C))\to H^1(k,\mathbf{Aut}(A)),\]
induced by the natural inclusion $\mathbf{Aut}(C)\to\mathbf{Aut}(A)$, has trivial kernel. Since this inclusion factors through the simply connected group $\mathbf{Spin}(q_C)$ of the quadratic form $q_C$ of $C$,
we first show how the aforementioned result of \cite{G1}, augmented by results from \cite{AG}, readily implies that in general there exists an octonion algebra $C'\not\simeq C$ with $H_3(C')\simeq H_3(C)$,
namely by taking $C'$ having the same quadratic form as $C$. This corresponds to the torsor
\[\xymatrix@R-10pt{\mathrm F_4\ar[d]^{\mathrm G_2}\\ \mathrm F_4/\mathrm G_2}\]
being non-trivial. One may therefore ask if this is the only obstruction, i.e.\ if $H_3(C)$ at least determines the quadratic form of $C$ up to isometry. The main result of Section \ref{S4} is that this question
has a negative answer, which we show by proving that the torsor
\[\xymatrix@R-10pt{\mathrm F_4\overset{\mathrm G_2}{\wedge}\mathrm D_4\ar[d]^{\mathrm D_4}\\ \mathrm F_4/\mathrm G_2}\]
is non-trivial, where $\wedge$ denotes the contracted product.
A conclusion of our work is thus that the behaviour of Albert algebras over rings is significantly more intricate than it is over fields, even when the algebras are reduced or split, and that
these intricacies can be better understood using appropriate torsors. The combination of these two conclusions hopefully serves as an indication for future investigations.
\subsection{Preliminaries on Albert Algebras}
Throughout, $R$ is a unital, commutative ring. By an $R$-ring we mean a unital, commutative and associative $R$-algebra, and by an $R$-field we mean an $R$-ring that is a field. If $M$
is an $R$-module and $S$ is an $R$-ring, we write $M_S$ for the $S$-module $M\otimes_R S$ obtained by base change. All sheaves and cohomology sets involved will be with respect to the fppf topology, and we therefore omit the subscript fppf universally.
A \emph{Jordan algebra} over $R$ is an $R$-module $A$ endowed with a distinguished element $1$ and a quadratic map $U:A\to \mathrm{End}_R(A), x\mapsto U_x$, satisfying the identities
\[\begin{array}{llll}
U_1=\mathrm{Id}_A, & U_{U_xy}=U_xU_yU_x, & \text{and} & U_x\{yxz\}=\{xy(U_xz)\}
\end{array}
\]
for all $x,y,z\in A_S$ whenever $S$ is an $R$-ring, where $\{abc\}=(U_{a+c}-U_a-U_c)b$ is the \emph{triple product}. The map $U$ is called the $U$-operator of $A$ and replaces, in a sense,
the multiplication in linear Jordan algebras, and the element $1$ is the \emph{unity} of $A$.
\begin{Ex} The algebras we will consider will be \emph{cubic Jordan algebras}. Recall from \cite{P} that a \emph{cubic norm structure} over $R$ is an $R$-module $A$ endowed
with a base point $1_A$, a cubic form $N=N_A:A\to R$, known as the \emph{norm},
and a quadratic map $A\to A$ denoted as $x\mapsto x^\sharp$ and known as the \emph{adjoint}. These are required to satisfy certain regularity and compatibility properties (see \cite[5.2]{P}) and give rise to the bilinear trace
$T=T_A:A\times A\to R$, and the bilinear cross product $A\times A\to A$ given by $(x,y)\mapsto x\times y:=(x+y)^\sharp-x^\sharp-y^\sharp$. The cubic Jordan algebra associated to this
cubic norm structure is the module $A$ with unity $1=1_A$ and $U$-operator given by
\begin{equation}\label{U}U_xy=T(x,y)x-x^\sharp\times y.\end{equation}
\end{Ex}
An element $p\in A$ is invertible precisely when $N(p)\in R^*$, in which case $p^{-1}=N(p)^{-1}p^\sharp$.
We refer to \cite{P} for a thorough discussion of cubic Jordan algebras. An \emph{Albert algebra} over $R$ is a cubic Jordan $R$-algebra $A$ the underlying module of which is projective of constant rank 27, and such that the Jordan
algebra $A\otimes_R k$ is simple for every $R$-field $k$. The notion of an Albert algebra is stable under base change.
\subsection{Preliminaries on Torsors}
We recall and slightly extend some known facts about torsors that will be useful later. Let $E$ be a $G$-torsor over $X$, where $E$ and $X$ are $R$-schemes, $X$ is affine, and
$G$ is an $R$-group scheme. Recall that the topology in
question is always the fppf topology. In particular, we have a map $\Pi:E\to X$ and a compatible action of $G$ on $E$ (to fix a convention, we assume $G$ acts on the \emph{right}.)
Recall that $E$ is trivial if there is a $G$-equivariant morphism $E\to X\times G$ over $X$, which is equivalent to the projection $\Pi$ admitting a section \cite[III.4.1.5]{DG}.
\begin{Rk}\label{Rtrivial} If $E$ is trivial, then clearly $\Pi_S:E(S)\to X(S)$ is surjective for every $R$-ring $S$. We will need the converse, which holds as well: if $E$ is
non-trivial, then there is an $R$-ring $A$ such that $\Pi_A$ is not surjective. Indeed, if $X=\mathrm{Spec}(A)$ for an $R$-ring $A$, then $X(A)=\operatorname{Hom}(A,A)$ contains the generic element $\mathrm{Id}_A$.
If $\mathrm{Id}_A=\Pi(f)$ for some $f\in E(A)$, then the morphism $X=\mathrm{Spec}(A)\to E$ corresponding to $f$ is a section.
\end{Rk}
A certain class of torsors will be of particular interest to us.
\begin{Rk}\label{Rquotient} If $G$ is an $R$-group scheme, and $H$ is a subgroup scheme of $G$,
then the inclusion $i:H\to G$ induces a map $i^*:H^1(R,H)\to H^1(R,G)$; explicitly, by \cite[III.4.4.1]{DG}, $i^*$ maps the class of an $H$-torsor $E$ to the class of the $G$-torsor
$E\wedge^H G$, where $\wedge^H$ denotes the contracted product over $H$. This is the quotient sheaf of $E\times G$ by the relation $\sim$ defined, on each $E(S)\times G(S)$,
by $(e,g)\sim(e',g')$ whenever $(eh,g)\sim(e',hg')$ for some $h\in H(S)$, and the structural projection of $E\wedge^H G$ is the projection on the first component. By \cite[III.4.4.5]{DG} (or \cite[2.4.3]{G2}), the kernel of $i^*$ is in bijection with the set of orbits
of the left action of $G(R)$ on $(G/H)(R)$, where $G/H$ is the fppf quotient. This bijection is given by assigning to the orbit of $x\in (G/H)(R)$ the class of the $H$-torsor
$\Pi^{-1}(x)$ where $\Pi$ is the quotient projection.
\end{Rk}
In order to produce examples of non-trivial torsors, we will use the following straight-forward generalisation of an argument from \cite{G1}.
\begin{Lma}\label{Lhomotopy} Let $K\in\{\mathbb R,\mathbb C\}$, let $X$ be an affine $K$-scheme, $G$ an affine algebraic $K$-group, $E$ a $G$-torsor over $X$, and assume that $E$ has a $K$-point $e$. If for some $n\in\mathbb N$ the
homotopy group $\pi_n(G(K),1_G)$ is not a direct factor of the homotopy group $\pi_n(E(K),e)$, then the torsor $E$ is non-trivial.
\end{Lma}
The condition of the existence of a $K$-point does not imply that $E$ is trivial. Indeed this condition is fulfilled whenever $E$ is an affine algebraic $K$-group, $G$ a closed subgroup, and
$X=E/G$, with $e$ being the neutral element of $E$.
\begin{proof} We mimic the proof of \cite{G1}. Assume that $E$ is trivial. Then there is a $G$-equivariant morphism $E\to X\times G$ over $X$. This implies the existence of a map
$\rho: E\to G$ that is a retraction of the inclusion $G\to E$ defined by $g\mapsto e_Sg$ for every $K$-ring $S$ and $g\in G(S)$. This in particular implies that the corresponding
inclusion $G(K)\to E(K)$ admits a continuous retraction. It follows that $\pi_n(G(K),1_G)$ is a direct factor of the homotopy group $\pi_n(E(K),e)$. The proof is complete.
\end{proof}
\subsection{Acknowledgements} I am indebted to Philippe Gille for many fruitful discussions and valuable remarks. I am also grateful to Erhard Neher and
Holger Petersson for their interest and encouragement and for enriching conversations and comments.
\section{Isometries, Isotopes and $\mathrm{F}_4$-torsors}\label{S2}
\subsection{Isometry and Automorphism Groups}
Let $A$ be an Albert algebra over $R$ with norm $N$. We begin by determining the groups $\mathbf{Aut}(A)$ and $\mathbf{Isom}(N)$. While this is done to varying degree in the existing literature, we
include the proofs of the below results as there is no single reference to which we can refer. Recall that the subgroup $\mathbf{Isom}(N)$ of $\mathbf{GL}(A)$ is defined, for each $R$-ring $S$, by
\[\mathbf{Isom}(N)(S)=\{\phi\in \mathbf{GL}(A)(S)|N_S\circ\phi=N_S\},\]
where $N_S$ is the norm of $A_S=A\otimes S$. Let further $\mathbf{S}_N$ be the cubic sphere of $A$, i.e.\ the $R$-group functor defined, for each $R$-ring $S$, by
\[\mathbf{S}_N(S)=\{x\in A_S|N_S(x)=1\}.\]
Clearly, $\mathbf{Isom}(N)$ acts on $\mathbf{S}_N$ by $\phi\cdot x=\phi(x)$ for each $R$-ring $S$, $x\in\mathbf{S}_N(S)$ and $\phi\in\mathbf{Isom}(N)(S)$. We shall refer to this simply as \emph{the action of $\mathbf{Isom}(N)$ on $\mathbf{S}_N$}.
\begin{Prp} The group $\mathbf{Aut}(A)$ is a semisimple algebraic group of type $\mathrm F_4$, and $\mathbf{Isom}(N)$ is a semisimple
simply connected algebraic group of type $\mathrm E_6$.
\end{Prp}
In the case where $R$ is a field of characteristic different from 2 and 3, the result is due to \cite[Theorems 7.2.1 and 7.3.2]{SV}
It is known (see \cite[Theorem 17]{P}) that there is a faithfully flat $R$-ring $S$ such that
$A_S$ is split, hence isomorphic to $A^s\otimes_\mathbb{Z} S$, where $A^s$ is the split $\mathbb{Z}$-Albert algebra $H_3(C^s)$ with norm $N^s$, $C^s$ being the split octonion algebra over $\mathbb{Z}$.
To unburden notation, we set $\mathbf{G}=\mathbf{Isom}(N)$ and $\mathbf{H}=\mathbf{Aut}(A)$, which are group schemes over $R$, and $\mathbf{G}^s=\mathbf{Isom}(N^s)$ and $\mathbf{H}^s=\mathbf{Aut}(A^s)$, which are group schemes over $\mathbb{Z}$.
\begin{proof} Since $\mathbf{G}$ and $\mathbf{H}$ are forms of $\mathbf{G}^s_R$ and $\mathbf{H}^s_R$, respectively, for the fppf topology, it suffices to establish the claims for $\mathbf{G}^s$ and $\mathbf{H}^s$, i.e.\ to
prove that they are smooth affine $\mathbb{Z}$-group schemes with
connected, semisimple, simply connected geometric fibres of the appropriate types. Now $\mathbf{G}^s$ and $\mathbf{H}^s$ are affine since they are closed subschemes of the affine scheme $\mathbf{GL}(A^s)$. To prove smoothness, it suffices, by \cite[Lemma B.1]{AG}, to show that $\mathbf{G}^s$ and $\mathbf{H}^s$
are finitely presented and fibre-wise (i.e.\ over any field) smooth, connected and equidimensional. Finite presentation is clear.
Since a group over a field is smooth if and only if its scalar extension to
an algebraic closure is, smoothness of the fibres follows from the lemma below, which also implies that the fibres are connected and equidimensional.
Thus the groups are smooth, and the lemma also implies that they have semisimple, simply connected geometric fibres of the appropriate types. The proof is then complete.
\end{proof}
\begin{Lma}\label{L1} Assume that $R=k$ is an algebraically closed field. Then $\mathbf{Aut}(A)$ is a semisimple algebraic group of type $\mathrm F_4$, and $\mathbf{Isom}(N)$ is a semisimple
simply connected algebraic group of type $\mathrm E_6$.
\end{Lma}
\begin{proof} First we address smoothness. The group $\mathbf{Aut}(A)$ is smooth (see e.g.\ \cite{ALM}). For $\mathbf{Isom}(N)$, consider the exact sequence of
$k$-group schemes
\[\xymatrix{1\ar[r]&\mathbf{Isom}(N)\ar[r]^f&\mathbf{Str}(A)\ar[r]^\mu&\mathbb{G}_{\mathrm{m}}\ar[r]&1},\]
where the structure group $\mathbf{Str}(A)$ is the group of norm similarities with respect to $N$ (the field being infinite) and $\mu$ is the multiplier map $\phi\mapsto\mu_\phi=N(\phi(1))$.
By \cite[Corollary 6.6]{L}, $\mathbf{Str}(A)$ is smooth. For $\mathbf{Isom}(N)$ to be smooth it is necessary and sufficient that the differential $d\mu$ be a surjective map of Lie algebras.
But $\mathrm{Id}+\varepsilon\mathrm{Id}\in\mathbf{Str}(A)(k[\varepsilon])$ (where $\varepsilon^2=0$) and $\mu_{\mathrm{Id}+\varepsilon\mathrm{Id}}=1+\varepsilon$. Thus $\mathrm{Id}\in\mathrm{Lie}(\mathbf{Str}(A))$, and its image under $d\mu$ is $1\in k=\mathrm{Lie}(\mathbb{G}_{\mathrm{m}})$. Surjectivity
follows by linearity.
Let $j$ be the inversion on $A$, i.e.\ the birational map on $A$ defined by $x\mapsto x^{-1}$ for all $x\in A^*$. Then $\Sigma=(A,j,1_A)$ is an H-structure in the sense of
\cite{M}, as well as a J-structure in the sense of \cite{S}. By \cite{S}, the structure group $\mathbf{S}(j)$ of $j$ is a smooth closed subgroup of $\mathbf{GL}(A)$, and so is the
automorphism group of $\Sigma$ as a J-structure. By definition a linear automorphism of $A$ is an automorphism of $\Sigma$ as a J-structure if and only if it is an automorphism of $\Sigma$ as an
H-structure. By \cite{M} we thus have $\mathbf{Aut}(\Sigma)(k)\simeq\mathbf{Aut}(A)(k)$ and $\mathbf{S}(j)(k)\simeq\mathbf{Str}(A)(k)$. Thus since $k$ is algebraically closed, and all groups involved are smooth,
$\mathbf{Aut}(A)\simeq\mathbf{Aut}(\Sigma)$ and $\mathbf{Str}(A)\simeq\mathbf{S}(j)$.
The statement about $\mathbf{Aut}(A)$ then follows from \cite[14.20]{S}, as does the statement that $\mathbf{Str}(A)$ is the product of its one-dimensional centre and a semisimple, simply
connected algebraic group $\mathbf{G}'$ of type $\mathrm E_6$. From this and the above exact sequence, it follows that $f$ induces an isomorphism $\mathbf{Isom}(N)^\circ\to \mathbf{G}'$, and it remains to be
shown that $\mathbf{Isom}(N)$ is connected. By Proposition 3.7 and Lemma 3.9 of \cite{ALM}, the group $\mathbf{Str}(A)(k)$ acts transitively on $A^*$ ($k$ being algebraically closed). If $a\in A^*$ has norm 1
and $\phi\in\mathbf{Str}(A)(k)$ satisfies $\phi(1)=a$, then necessarily $\phi\in\mathbf{Isom}(N)(k)$. Thus $\mathbf{Isom}(N)$ acts transitively on $\mathbf{S}_N$, which is an irreducible $k$-variety since $N$ is irreducible.
(The irreducibility of $N$ follows from the fact that its restriction on $M_3(k)$ is the usual determinant, which is irreducible, as shown in e.g.\ \cite[Theorem 7.2]{J2}.) By \cite{ALM}, the stabiliser of $1$ is $\mathbf{Aut}(A)$, which
is connected. Thus $\mathbf{Isom}(N)$ is connected. Hence it is a semisimple, simply connected group of type $\mathrm E_6$, and the proof is complete.
\end{proof}
\subsection{Isotopes}
If $p\in A^*$, then the \emph{$p$-isotope} $A^{(p)}$ of $A$ is the Jordan algebra with unity $p^{-1}$ and $U$-operator $x\mapsto U^{(p)}_x:=U_xU_p$, where $U$ is the
$U$-operator of $A$. It is known that $A^{(p)}$ is an Albert algebra, and one can check that the map $\lambda U_p:A^{(\lambda U_pp)}\to A^{(p)}$ is an isomorphism
for any $\lambda\in R^*$, in particular for $\lambda=N(p)^{-1}$, in which case $N(\lambda U_pp)=1$. Thus up to isomorphism one may assume that $N(p)=1$. In that case the norm of $A^{(p)}$ coincides
with that of $A$, and the adjoint is given by $x\mapsto U_{p^{-1}}x^\sharp$. In this section we will show that isotopes of Albert algebras are twists by a certain torsor. To begin with, the following
proposition provides the torsor.
\begin{Prp}\label{Psphere} The stabiliser of $1_A$ with respect to the action of $\mathbf{Isom}(N)$ on $\mathbf{S}_N$ is $\mathbf{Aut}(A)$, and the fppf quotient sheaf $\mathbf{Isom}(N)/\mathbf{Aut}(A)$ is representable by a smooth scheme. Furthermore, the map $\Pi:\mathbf{Isom}(N)\to\mathbf{S}_N$, defined by $\phi\mapsto \phi(1)^{-1}$ for any $R$-ring $S$ and $\phi\in\mathbf{Isom}(N)(S)$, induces an isomorphism between the quotient
and $\mathbf{S}_N$.
\end{Prp}
\begin{proof} By the results of \cite{ALM} quoted in the proof of Lemma \ref{L1}, the action is transitive (in the sense of being transitive on geometric fibres)
and the stabiliser of $1$ is $\mathbf{Aut}(A)$. Then since $\mathbf{Aut}(A)$ is flat (in fact smooth) and $\mathbf{Isom}(N)$ is smooth, $\mathbf{Isom}(N)/\mathbf{Aut}(A)$ is representable by a smooth scheme by
\cite[XVI.2.2 and VIB.9.2]{SGA3}. The induced map
is then an isomorphism by \cite[III.3.2.1]{DG}.
\end{proof}
This defines an $\mathbf{Aut}(A)$-torsor over $\mathbf{S}_N$ for the fppf topology,
and we denote by $\mathbf{E}^p$ the fibre of $p\in \mathbf{S}_N(R)$, which is an $\mathbf{Aut}(A)$-torsor over $\mathrm{Spec}(R)$. The next theorem shows that the isotopes of $A$ are precisely the twists of $A$
by this torsor. Before formulating it, we shall
define precisely what we mean by a twist of a (quadratic) Jordan algebra.
Let $A$ be a Jordan algebra and $\mathbf{E}$ a (right) $\mathbf{Aut}(A)$-torsor over $\mathrm{Spec}(R)$, and denote by $\mathbf{W}(A)$ the vector group scheme defined by
$\mathbf{W}(A)(S)=A_S$ for
every $R$-ring $S$. The \emph{twist $\mathbf{E}\wedge A$ of $A$ by $\mathbf{E}$} is the sheaf of Jordan algebras associated to the presheaf
\[S\mapsto (\mathbf{E}(S)\times A_S)/\sim\]
defined as follows: for each $R$-ring $S$, the equivalence relation $\sim$ is defined by $(\phi,a)\sim(\psi,b)$ if and only if $(\phi \rho^{-1},\rho(a))=(\psi,b)$ for some
$\rho\in\mathbf{Aut}(A)(S)$. Denoting the equivalence classes by square brackets, the additive structure is given by
\[[\phi,x]+[\psi,y]=[\phi,x+\phi^{-1}\psi(y)],\]
the $S$-module structure by
\[\lambda[\phi,x]=[\phi,\lambda x],\]
the unity by $[\phi,1_A]$, and the $U$-operator by
\[U_{[\phi,x]}[\psi,y]=[\phi,U_x\phi^{-1}\psi(y)],\]
for any $\phi,\psi\in \mathbf{E}(S)$, $x,y\in A_S$ and $\lambda\in S$. It is straight-forward to check that these expressions are well-defined on equivalence classes, and satisfy the axioms
of a Jordan algebra.
\begin{Rk} If $\phi,\psi\in \mathbf{E}(S)$, then $\rho=\phi^{-1}\psi\in\mathbf{Aut}(A)(S)$. Thus
\[[\psi,y]=[\psi\rho^{-1},\rho(y)]=[\phi,\phi^{-1}\psi(y)].\]
Thus the above definition is equivalent to the definition
\[U_{[\phi,x]}[\phi,y]=[\phi,U_xy].\]
Note further that in an algebra with bilinear multiplication, the twist of the multiplication is defined by
\[(\phi,x)(\psi,y)=(\phi,x\phi^{-1}\psi(y)).\]
If $2\in R^*$, then $A$ is endowed with a bilinear multiplication, and the $U$-operator is defied by
\[U_xy=2x(xy)-x^2y,\]
and one easily checks that the twist of the $U$-operator defined above is precisely the one obtained in this way from the twist of the bilinear multiplication.
\end{Rk}
\begin{Thm} Let $p\in \mathbf{S}_N(R)$. The map
\[\Theta_S:[\phi,x]\to \phi(x)\]
for each $R$-ring $S$, $\phi\in\mathbf{E}^p(S)$, and $x\in A_S$, defines a natural isomorphism of algebras $\mathbf{E}^p\wedge A\to A^{(p)}$.
\end{Thm}
The reader may wish to compare this result to \cite[Theorem 4.6]{AG}.
\begin{proof} The proof that this is a well-defined, injective linear map from $(\mathbf{E}^p(S)\times A_S)/\sim$ to $A_S^{(p)}$ is straight-forward and analogous to that of \cite[Theorem 4.6]{AG}.
The same goes for surjectivity whenever $\mathbf{E}^p(S)\neq\emptyset$, and for naturality. This proves that the map in question is a an isomorphism of sheaves of modules. It is a morphism of algebras
since $\Theta_S([\phi,1])=\phi(1)=p^{-1}=1_{A^{(p)}}$, and
\[\Theta_S(U_{[\phi,x]}[\psi,y])=\Theta_S([\phi,U_x\phi^{-1}\psi(y)])=\phi(U_x\phi^{-1}\psi(y)).\]
Now $\phi\in\mathbf{E}^p(S)$ implies that $\phi\in\mathbf{Isom}(N)(S)$ with $\phi(1_A)=p^{-1}=1_{A^{(p)}}$. Thus $\phi:A\overset{\sim}{\rightarrow} A^{(p)}$, and
\[U_{\phi(x)}^{(p)}\phi(\phi^{-1}\psi(y))=U_{\phi(x)}^{(p)}\psi(y)=U_{\Theta_S[\phi,x]}^{(p)}\Theta_S[\psi,y],\]
which completes the proof.
\end{proof}
\subsection{Non-Trivial Isotopes}
It is well-known that the above torsor is non-trivial in general; indeed, over $\mathbb R$, the Albert algebras form two isotopy classes, one of which consists of two isomorphism
classes (see \cite[13.7]{P}). However, by e.g.\ \cite[Proposition 20]{P}, any isotope of the split Albert algebra over a field $K$ is isomorphic it. As the next result shows, this is
not the case over rings.
\begin{Thm}\label{TF4} There exists a smooth $\mathbb C$-ring $S$ such that the split Albert $S$-algebra admits a non-isomorphic isotope.
\end{Thm}
\begin{proof} Let $A$ be the (split) complex Albert algebra, and denote its cubic norm by $N$.
By Proposition \ref{Psphere}, the affine scheme $\mathbf{S}_N$ is smooth. In view of this it suffices, by Remark \ref{Rtrivial}, to show that the $\mathbf{Aut}(A)$-torsor $\mathbf{Isom}(N)\to\mathbf{S}_N$ of Proposition \ref{Psphere} is non-trivial.
To this end it suffices, by Lemma \ref{Lhomotopy}, to show that for some $n$,
$\pi_n(\mathbf{Aut}(A)(\mathbb C))$ is not a direct factor of $\pi_n(\mathbf{Isom}(N)(\mathbb C))$. (The choice of base point is immaterial by connectedness.)
Let $C$ be the real division octonion algebra, and set $A_0=H_3(C,\mathbf{1})$, with norm $N_0$, noting that $A=A_0\otimes_{\mathbb R}\mathbb C$.
Then $\mathbf{Aut}(A_0)$ (resp.\ $\mathbf{Isom}(N_0)$) is an anisotropic real group of type $\mathrm F_4$ (resp.\ $\mathrm E_6$).
The Lie group $\mathbf{Aut}(A)(\mathbb C)=\mathbf{Aut}(A_0)(\mathbb C)$ (resp.\ $\mathbf{Isom}(N)(\mathbb C)=\mathbf{Isom}(N_0)(\mathbb C)$) contains $\mathbf{Aut}(A_0)(\mathbb R)$ (resp.\ $\mathbf{Isom}(N_0)(\mathbb R)$) as a maximal compact subgroup.
By Cartan decomposition, $\mathbf{Aut}(A)(\mathbb C)$ is thus homeomorphic to $\mathbf{Aut}(A_0)(\mathbb R)\times\mathbb R^k$ and $\mathbf{Isom}(N)(\mathbb C)$ is homeomorphic to $\mathbf{Isom}(N_0)(\mathbb R)
\times\mathbb R^l$ for suitable $k$ and $l$. By \cite{Mi}, $\pi_8(\mathbf{Aut}(A_0)(\mathbb R))$, and hence also $\pi_8(\mathbf{Aut}(A)(\mathbb C))$,
is cyclic or order 2, while by \cite{BS}, $\pi_8(\mathbf{Isom}(N_0)(\mathbb R))$, and thus also $\pi_8(\mathbf{Isom}(N^s)(\mathbb C))$, is trivial. This completes the proof.
\end{proof}
\section{Compositions of Quadratic Forms and $\mathrm{D}_4$-torsors}\label{S3}
In this section, we will define and consider Albert algebras arising from compositions of quadratic forms, and describe these in terms of certain $\mathrm{D}_4$-torsors.
\subsection{Composition Algebras and Compositions of Quadratic Forms}
An Albert algebra over $R$ is called \emph{reduced} if it is isomorphic to the algebra $H_3(C,\Gamma)$ of twisted hermitian matrices over an octonion $R$-algebra $C$, with
$\Gamma=(\gamma_1,\gamma_2,\gamma_3)\in (R^*)^3$. Before giving the definition of $H_3(C,\Gamma)$, we note that the formulae can be simplified slightly if they are expressed in terms of the \emph{para-octonion algebra obtained from $C$}. This is the algebra with underlying
quadratic module $C$, and multiplication given by
\[x\cdot y=\bar x\ \bar y,\]
where $a\mapsto \bar a$ denotes the usual involution on $C$ and juxtaposition the multiplication of $C$. This algebra is not unital, but it is
a symmetric composition algebra in the sense of the following definition.
\begin{Def} A \emph{composition algebra} $(C,q)$ over $R$ is a faithfully projective $R$-module $C$ endowed with a bilinear map $C\times C\to C, (x,y)\mapsto x\cdot y$ (the multiplication) and a
nonsingular quadratic form $q=q_C$ satisfying $q(x\cdot y)=q(x)q(y)$ for all $x,y\in C$. The algebra is called \emph{symmetric} if the symmetric bilinear form
\[(x,y)\mapsto \langle x,y\rangle=q(x+y)-q(x)-q(y)\]
satisfies
\[\langle x\cdot y, z\rangle=\langle x,y\cdot z\rangle\]
for all $x,y,z\in C$.
\end{Def}
We shall only be interested in symmetric composition algebras of \emph{constant rank 8}. Examples of these include para-octonion algebras and (over fields) Okubo algebras. In this case, the module being faithfully projective amounts to it being projective (of constant rank 8), and
non-singularity of $q$ is equivalent to non-degeneracy of $\langle,\rangle$. Abusing notation, we shall refer to a composition algebra $(C,q)$ simply by $C$.
We shall begin by writing the definition of reduced Albert algebras in terms of symmetric composition algebras. Let thus $C$ be a symmetric composition algebra over $R$ of
constant rank 8. We will often consider the $R$-module $R^3\times C^3$. Following the notation of \cite{P}, we write an
element $(\alpha_1,\alpha_2,\alpha_3,u_1,u_2,u_3)\in R^3\times C^3$ as $\sum \alpha_i e_i +\sum u_i[jl]$ where the second sum runs over all cyclic permutations $(i,j,l)$ of
$(1,2,3)$. We write $\Delta$ for the trilinear form on $C$ defined by
\[\Delta(u_1,u_2,u_3)=\langle u_3\cdot u_2,u_1\rangle,\]
noting that by symmetry of $C$, it is invariant under cyclic permutations.
\begin{Prp}\label{PComp} Let $C$ be a symmetric composition algebra over $R$ of constant rank 8 and let $\Gamma=(\gamma_1,\gamma_2,\gamma_3)\in R^3$ with $\gamma_1\gamma_2\gamma_3\in R^*$.
Then $R^3\times C^3$ is a cubic norm structure with base point $e=\sum e_i$ and norm $N$ and adjoint map $\sharp$ defined by
\[N(x)=\alpha_1\alpha_2\alpha_3+\gamma_1\gamma_2\gamma_3\Delta(u_1,u_2,u_3)-\sum\gamma
a_j\gamma_l\alpha_i q(u_i)\]
and
\[x^\sharp=\sum(\alpha_j\alpha_l-\gamma_j\gamma_lq(u_i))e_i + \sum (\gamma_iu_l\cdot u_j-\alpha_iu_i)[jl],\]
where $x=\sum \alpha_i e_i +\sum u_i[jl]$.
Moreover, the bilinear trace $T$ of this cubic norm structure satisfies
\[T(x,y)=\sum \alpha_i\beta_i+\sum \gamma_j\gamma_l\langle u_i,v_i\rangle,\]
where $y=\sum \beta_i e_i +\sum v_i[jl]$.
\end{Prp}
\begin{proof} It is straight-forward to verify that $N$, $\sharp$ and $1$ satisfy the axioms of \cite{P}, and that the bilinear trace defined in \cite{P} is equal to $T$.
\end{proof}
We denote the Jordan algebra of the above cubic form by $H(C,\Gamma)$.
\begin{Rk} If $C$ is a para-octonion algebra obtained from an octonion algebra $O$, then $H(C,\Gamma)=H_3(O,\Gamma)$, and the above definition coincides with the definition
of $H_3(O,\Gamma)$ in \cite{P}.
\end{Rk}
For a para-octonion algebra $C$, we write $\mathbf{RT}(C)$ for the affine group scheme defined by
\[\mathbf{RT}(C)(S)=\{(t_1,t_2,t_3)\in\mathbf{SO}(q_C)(S)^3|t_1(x\cdot y)=t_3(x)\cdot t_2(y)\}.\]
Up to renaming the indices, by \cite{AG}, this is precisely the semi-simple simply connected group of type $\mathrm D_4$ denoted in \cite{AG} by $\mathbf{RT}(O)$, where $O$ is the
octonion algebra from which $C$ is obtained.
For later use, we recall the definition of a composition of quadratic forms, generalising that of a composition algebra.
\begin{Def} A \emph{composition of quadratic forms over $R$} is a heptuple
\[\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m),\]
where $C_1$, $C_2$ and $C_3$ are projective
$R$-modules of the same rank, and each $q_i$ is a non-singular quadratic form on $C_i$, and where
$m:C_3\times C_2\to C_1$ is a bilinear map satisfying $q_1(m(x,y))=q_3(x)q_2(y)$ for any $x\in C_3$ and $y\in C_2$. We call the rank of $C_i$ the \emph{rank} of $\mathcal{M}$. If
$\mathcal{M}'=(C'_1,C'_2,C'_3,q'_1,q'_2,q'_3,m')$ is another composition, then a \emph{morphism of compositions of quadratic forms} is a triple $(t_1,t_2,t_3)$ of isometries $t_i:C_i\to C'_i$
such that $m'(t_3(x),t_2(y))=t_1m(x,y)$ for each $x\in C_3$ and $y\in C_2$.
\end{Def}
It is clear that this notion is stable under base change. The notions of isomorphisms and automorphisms are clear, and we thus obtain an affine group scheme $\mathbf{Aut}(\mathcal{M})$.
\begin{Ex} Let $C$ be a composition algebra over $R$, with quadratic form $q_C$.
The multiplication map $m_C:C\times C\to C$ makes $\mathcal{M}_C:=(C,C,C,q_C,q_C,q_C,m_C)$ into a composition of quadratic forms. If $C$ is a para-octonion algebra, then by definition, $\mathbf{Aut}(\mathcal{M}_C)=\mathbf{RT}(C)$.
\end{Ex}
Given a composition of quadratic forms $\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m)$ of constant rank 8, the regularity of $q_2$ and $q_3$ implies that the map $m$ determines maps $m_2:C_1\times C_3\to C_2$ and $m_3:C_2\times C_1\to C_3$ by
\begin{equation}\label{companions}
\begin{array}{l}
\langle m_2(x_1,x_3),x_2\rangle_2=\langle x_1, m(x_3,x_2)\rangle_1,\\ \langle x_3, m_3(x_2,x_1)\rangle_3=\langle m(x_3,x_2),x_1\rangle_1
\end{array}
\end{equation}
where $\langle,\rangle_i$ is the bilinear form corresponding to the quadratic form $q_i$. The significance of these is due to the following proposition.
\begin{Prp}\label{PExt} Let $\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m)$ be a composition of quadratic forms over $R$ of constant rank 8.
Then $\mathcal{M}\otimes_R S\simeq_S\mathcal{M}_C\otimes_R S$ for some para-octonion $R$-algebra $C$ and some faithfully flat $R$-ring $S$. Under any isomorphism $\mathcal{M}\otimes_R S\to\mathcal{M}_C\otimes_R S$,
the multiplication on $C_S$ restricts to the
maps $m_i:C_l\times C_j\to C_i$ for every
cyclic permutation $(i,j,l)$ of $(1,2,3)$, where $m_1=m$ and $m_2$ and $m_3$ are as in \eqref{companions}.
\end{Prp}
To simplify notation we shall write $\mathcal{M}\otimes_R S$ as $(C_{1S},C_{2S},C_{3S},q_{1S},q_{2S},q_{3S},m_S)$.
\begin{Rk} It suffices to prove that $\mathcal{M}\otimes_R S\simeq \mathcal{M}_C$ for some para-octonion algebra $C$ \emph{over $S$}. Indeed, assume this is the case. If $O$
is the octonion
$S$-algebra from which $C$ is obtained, then $O\otimes_S T\simeq \mathrm{Zor}(R)\otimes_R T$ for some faithfully flat $S$-ring $T$, where $\mathrm{Zor}(R)$ is the split (Zorn) octonion $R$-algebra. Hence
$C\otimes_S T\simeq Z\otimes_R T$, where $Z$ is the para-octonion $R$-algebra obtained from $\mathrm{Zor}(R)$, and \emph{a fortiori}
$\mathcal{M}_C\otimes_S T\simeq \mathcal{M}_{Z\otimes S}\otimes_S T\simeq \mathcal{M}_Z\otimes_R T $. Altogether,
\[\mathcal{M}\otimes_R T\simeq (\mathcal{M}\otimes_R S)\otimes_S T\simeq\mathcal{M}_C\otimes_S T\simeq\mathcal{M}_Z\otimes_R T,\]
and the claim follows since $T$ is faithfully flat over $R$.
\end{Rk}
\begin{proof}[Proof of Proposition \ref{PExt}] Choose $S$ such that there are $a\in C_{3S}$ and $b\in C_{2S}$ with $q_{3S}(a)=q_{2S}(b)=1$. (This is possible since a regular quadratic form is split by a faithfully
flat extension.) Define
\[\begin{array}{ll}
f:C_{2S}\to C_{1S},& x\mapsto m_S(a,x),\\
g:C_{3S}\to C_{1S},& x\mapsto m_S(x,b).
\end{array}
\]
Then $q_{1S}f(x)=q_{2S}(x)$ and $q_{1S}g(x)=q_{3S}(x)$, whence $f$ and $g$ are isometries, hence invertible. Thus
\[(x,y)\mapsto x\cdot y:=m_S(g^{-1}(x),f^{-1}(y))\]
defines a bilinear multiplication on $C_{1S}$. Since $f^{-1}$ and $g^{-1}$ are isometries, we have
\[q_{1S}(x\cdot y)=q_{3S}g^{-1}(x)q_{2S}f^{-1}(y)=q_{1S}(x)q_{1S}(y).\]
Thus $q_{1S}$ is multiplicative with respect to the multiplication $\cdot$. Setting
$e=m_S(a,b)$ we have
$e=f(b)=g(a)$. Thus $e\cdot x=ff^{-1}(x)=x$, and likewise $x\cdot e=x$, for all $x\in C_{1S}$. Thus the multiplication is unital, whence it makes
$C_{1S}$ into an octonion algebra over $S$ with quadratic form $q_{1S}$, and
\[(\mathrm{Id},f,g):\mathcal{M}\otimes S\to \mathcal{M}_{C_{1S}}\]
is an isomorphism of compositions of quadratic forms since $g(x)\cdot f(y)=m_S(x,y)$ by construction. Denoting by $\kappa$ the involution on the octonion algebra
$C_{1S}$ and by $C$ the corresponding
para-octonion algebra, we have an isomorphism
\[(\mathrm{Id},\kappa f,\kappa g): \mathcal{M}\otimes S\to\mathcal{M}_C,\]
and the first statement is proved.
To prove the second statement for $i=1$, let $x\in C_3$ and $y\in C_2$. Since any isomorphism $\mathcal{M}\otimes_R S\to\mathcal{M}_C\otimes_R S$ takes $m_S$ to the multiplication of $C_S$, it
suffices to show that $z:=m_S(x\otimes 1,y\otimes 1)\in C_{1S}$ descends to $C_1$, which is equivalent to
$z\otimes 1$ being fixed by the standard descend datum
\[\begin{array}{ll}\theta:C_1\otimes S\otimes S\to C_1\otimes S\otimes S,& c\otimes \mu\otimes\nu\mapsto c\otimes \nu\otimes\mu\end{array}.\]
This in turn follows from the fact that
\[m_S(x\otimes 1,y\otimes 1)=m(x,y)\otimes 1\otimes 1.\]
To prove the statement for $i=3$, we let instead $x\in C_2$ and $y\in C_1$. We must show that for any symmetric composition algebra $C$ with multiplication $\bullet$ over $S$ and any
isomorphism of compositions of quadratic forms
\[(\phi_1,\phi_2,\phi_3): \mathcal{M}\otimes S\to\mathcal{M}_C\]
we have
\[\phi_3^{-1}(\phi_2(x\otimes 1)\bullet\phi_1(y\otimes 1))=m_3(x,y)\otimes 1.\]
Since $q_{3S}$ is non-degenerate, it suffices for this to show that for any $z\in C_3$,
\[\langle\phi_3^{-1}(\phi_2(x\otimes 1)\bullet\phi_1(y\otimes 1)),z\otimes 1\rangle_{q_{3S}}=\langle m_3(x,y)\otimes 1,z\otimes 1\rangle_{q_{3S}}.\]
Since $\phi_3:C_{3S}\to C$ is an isometry, the left hand side equals
\[\langle\phi_2(x\otimes 1)\bullet\phi_1(y\otimes 1),\phi_3(z\otimes 1)\rangle_{q_C}\]
which, by symmetry, is equal to
\[\langle\phi_3(z\otimes 1)\bullet\phi_2(x\otimes 1),\phi_1(y\otimes 1)\rangle_{q_C}=\langle \phi_1m_S(z\otimes 1,x\otimes 1),\phi_1(y\otimes 1)\rangle_{q_C}.\]
Since $\phi_1$ is an isometry, this is equal to
\[\langle m_S(z\otimes 1,x\otimes 1),y\otimes 1\rangle_{q_{1S}}=\langle m(z,x),y\rangle_1\otimes 1,\]
which by definition of $m_3$ is equal to
\[\langle m_3(x,y),z\rangle_3\otimes 1=\langle m_3(x,y)\otimes 1,z\otimes 1\rangle_{q_{3S}}\]
as desired. The statement for $i=2$ is deduced analogously, and the proof is complete.
\end{proof}
\subsection{Albert Algebras Constructed from Compositions}\label{Storsor}
Given a composition of quadratic forms
\[\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m)\]
of constant rank 8, consider the $R$-module $R^3\times C_1\times C_2\times C_3$, whose elements we will write, in analogy to the above, as $x=\sum \alpha_i e_i +\sum u_i[jl]$, where $\alpha_i\in R$
and $u_i\in C_i$, and where the second sum runs over all cyclic permutations $(i,j,l)$ of $(1,2,3)$.
\begin{Prp}\label{PCOQF} The module $R^3\times C_1\times C_2\times C_3$ is a cubic norm structure with base point $e=\sum e_i$ and norm $N$ and adjoint map $\sharp$ defined by
\[N(x)=\alpha_1\alpha_2\alpha_3+\gamma_1\gamma_2\gamma_3\langle m(u_3,u_2),u_1\rangle_1-\sum\gamma a_j\gamma_l\alpha_i q_i(u_i)\]
and
\[x^\sharp=\sum(\alpha_j\alpha_l-\gamma_j\gamma_lq_i(u_i))e_i + \sum (\gamma_i m_i(u_l,u_j)-\alpha_iu_i)[jl].\]
Moreover, the bilinear trace $T$ of this cubic norm structure satisfies
\[T(x,y)=\sum \alpha_i\beta_i+\sum \gamma_j\gamma_l\langle u_i,v_i\rangle_i,\]
where $y=\sum \beta_i e_i +\sum v_i[jl]$. The cubic Jordan algebra of this cubic norm structure is an Albert algebra.
\end{Prp}
We denote this cubic Jordan algebra by $H(\mathcal{M},\Gamma)$. If $\mathcal{M}=\mathcal{M}_C$ for some symmetric composition algebra $C$, then $H(\mathcal{M},\Gamma)=H(C,\Gamma)$.
\begin{proof} We need to verify the identities (1) of \cite[5.2]{P} in all scalar extensions of $R$. It suffices to do this upon replacing $R$ by a faithfully flat $R$-ring $S$.
By Proposition \ref{PExt}, we can choose $S$ so that $\mathcal{M}\otimes S\simeq \mathcal{M}_C$ for some symmetric composition $S$-algebra $C$. Extending any isomorphism
$\mathcal{M}\otimes S\simeq \mathcal{M}_C$ by the
identity on $S^3$, we obtain a linear bijection $H(\mathcal{M},\Gamma)\otimes S=H(\mathcal{M}\otimes S,\Gamma)\to H(C,\Gamma)$ that transforms the expressions
for the norm, adjoint and trace into those of Proposition \ref{PComp}, from which the first two statements therefore follow. We conclude with \cite[Theorem 17]{P} and Proposition \ref{PComp}.
\end{proof}
\begin{Rk} A related, but different, construction over fields is the Springer construction, which provides an Albert algebra given a \emph{twisted composition}, see \cite[38.6]{KMRT}. Twisted compositions
over a field $k$ involve a quadratic space over a cubic \'etale algebra over $k$; eight-dimensional (symmetric) composition algebras over $k$ give rise to twisted compositions using the split
\'etale algebra $k\times k\times k$. Note however that the automorphism group of such a twisted composition is the semi-direct product of a simply connected group of type $\mathrm D_4$
with the symmetric group $S_3$ \cite[36.5]{KMRT}, and therefore is not the required notion for our purposes. Generalising the notion of twisted compositions to the ring setting
is a possible topic for a future investigation.
\end{Rk}
The following lemma is known for classical reduced Albert algebras over fields, and generalises, mutatis mutandis, to the case at hand.
\begin{Lma}\label{Lincl} Let $\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m)$ be a composition of quadratic forms of constant rank 8 over $R$, and let $\Gamma\in (R^*)^3$. The map $\iota:\mathbf{Aut}(\mathcal{M})\to\mathbf{Aut}(H(\mathcal{M},\Gamma))$ defined, for each $R$-ring $S$ and each $(t_1,t_2,t_3)\in\mathbf{Aut}(\mathcal{M})(S)$, by
\[\iota(t_1,t_2,t_3)\left(\sum \alpha_i e_{ii} + \sum u_i[jl]\right)=\sum \alpha_i e_{ii} + \sum t_i(u_i)[jl]\]
is a monomorphism of groups.
\end{Lma}
\begin{proof} Set $A=H(\mathcal{M},\Gamma)$. We first show that $\iota_S(t_1,t_2,t_3)$ is an automorphism of $A_S$ for each $(t_1,t_2,t_3)\in \mathbf{Aut}(\mathcal M)(S)$, $S$ an $R$-ring.
As it is linear, by \cite[Corollary 18]{P}, it suffices to show that it fixes the unity and preserves
the cubic norm. The first condition is satisfied by construction; as for the second, recall that
\[N(x)=\alpha_1\alpha_2\alpha_3+\gamma_1\gamma_2\gamma_3\langle m(u_3,u_2),u_1\rangle_1-\sum\gamma a_j\gamma_l\alpha_i q_i(u_i);\]
thus since $t_i$ is an isometry with respect to $q_i$ for each $i$, the norm is preserved by $\iota_S(t_1,t_2,t_3)$ if and only if
\[\langle m(t_3(u_3),t_2(u_2)),t_1(u_1)\rangle_1=\langle m(u_3,u_2),u_1\rangle_1.\]
By definition of $\mathbf{Aut}(\mathcal{M})$, the left hand side is equal to $\langle t_1m(u_3,u_2),t_1(u_1)\rangle_1$, which is equal to the right hand side since $t_1$ is an isometry.
The map $\iota$ is further clearly natural, respects composition and is injective, whence the claim follows.
\end{proof}
To understand the quotient $\mathbf{Aut}(A)/\iota(\mathbf{Aut}(\mathcal{M}))$, we will extend the approach of \cite{ALM}.
\begin{Def} Let $A$ an Albert algebra over $R$. A \emph{frame for $A$} is a triple $(c_1,c_2,c_3)\in A^3$ such that
\begin{enumerate}
\item $U_{c_i}1=c_i$, $U_{c_i}c_j=\delta_{ij}$ and $\sum_i c_i=1$, and
\item the module $U_{c_i}A$ has rank 1.
\end{enumerate}
\end{Def}
The first condition states that $(c_1,c_2,c_3)$ is a complete set of idempotents, and the space $U_{c}A$, for an idempotent $c\in A$, is the \emph{Peirce 2-space} of $c$, which is known to be
a direct summand of $A$ by Peirce decomposition, and hence is projective. The arguments of \cite[4.5]{ALM} (using \cite[6.12]{CTS}) show that the functor $\mathbf{F}=\mathbf{F}_A$ from the category of $R$-rings to that of sets, defined by
$\mathbf{F}(S)$ being the set of all frames for $A_S$, is an affine $R$-scheme. It should be mentioned that there is an extensive theory of frames in Jordan algebras,
in which we shall not delve, the notion introduced above being the particular case we need for our purposes. The interested reader may consult \cite{PR}.
\begin{Ex}\label{Edist} If $A=H(\mathcal{M},\Gamma)$, then the triple $\mathbf{e}=(e_1,e_2,e_3)$ is a frame for $A$, which we will call the \emph{distinguished frame} of $A$.
\end{Ex}
\begin{Prp}\label{stabidem} Let $A=H(\mathcal{M},\Gamma)$. Under the natural action of $\mathbf{Aut}(A)$ on $\mathbf{F}_A$ the stabiliser of $(e_1,e_2,e_3)$ is $\iota(\mathbf{Aut}(\mathcal{M}))$.
\end{Prp}
\begin{proof} Let $\mathbf{H}\subseteq\mathbf{Aut} A$ be the stabiliser of $(e_1,e_3,e_3)$. This is a finitely presented group scheme and, by construction, it is clear that
$\iota(\mathbf{Aut}(\mathcal{M}))\subseteq \mathbf{H}$. To check that this inclusion is an equality, it suffices to do so locally with respect to the fppf topology.
We may thus assume that $\mathcal{M}=\mathcal{M}_C$ for a para-octonion algebra $C$. Recall that then, $\mathbf{Aut}(\mathcal{M})=\mathbf{RT}(C)$.
By \cite[4.6]{ALM}, the inclusion $\mathbf{RT}(C)\to\mathbf{H}$ is an isomorphism on each geometric fibre in the case $\Gamma=\mathbf{1}$, and in the general case
since over an algebraically closed field, $\mathbf{Aut}(H(C,\Gamma))\simeq\mathbf{Aut}(H(C,\mathbf{1}))$ via an isomorphism that maps $(e_1,e_2,e_3)$ to itself and commutes with the inclusion of
$\mathbf{RT}(C)$. Since $\mathbf{RT}(C)$ is smooth, hence flat, and of finite presentation,
we conclude with the fibre-wise isomorphism criterion \cite[$_4$.17.9.5]{EGAIV} and the fact that the property of being an isomorphism over fields
is preserved under descent from an algebraic closure.
\end{proof}
\begin{Prp}\label{Ptorsor} Let $A=H(\mathcal M,\Gamma)$. The map $\Sigma:\mathbf{Aut}(A)\to\mathbf{F}_A$, defined by $\rho\mapsto (\phi(e_i))_i$ for any $R$-ring $S$ and $\rho\in\mathbf{Aut}(A)(S)$, induces an isomorphism between the fppf quotient
$\mathbf{Aut}(A)/\iota(\mathbf{Aut}(\mathcal{M}))$ and $\mathbf{F}_A$.
\end{Prp}
\begin{proof} With respect to the fppf topology, $\mathbf{Aut}(\mathcal{M})$ is locally isomorphic to the smooth group $\mathbf{RT}(C)$ for a para-octonion $R$-algebra $C$. Therefore $\mathbf{Aut}(\mathcal{M})$
is smooth, and the quotient is representable by a scheme, which is smooth since so is $\mathbf{Aut}(A)$. By the fibre-wise isomorphism criterion \cite[$_4$17.9.5]{EGAIV} and the fact that the property of being an isomorphism over fields
is preserved under descent from an algebraic closure, it suffices to check that the induced map is
an isomorphism on geometric fibres. We may thus assume that $R$ is an algebraically closed field. But then $\mathcal{M}\simeq\mathcal{M}_C$ for some para-octonion algebra $C$, and
$\mathbf{Aut}(H(\mathcal{M},\Gamma))\simeq \mathbf{Aut}(H(C,\Gamma))\simeq\mathbf{Aut}(H(C,\mathbf{1}))$. It thus suffices to consider the case where the quotient
in question is $\mathbf{Aut}(H(C,\mathbf{1}))/\mathbf{RT}(C)$, in which we conclude with \cite[4.6]{ALM}.
\end{proof}
Similarly to the previous section, this defines an $\mathbf{RT}(C)$-torsor over $\mathbf{F}_A$ for the fppf topology,
and we denote by $\mathbf{E}^\mathbf{c}$ the fibre of $\mathbf{c}=(c_1,c_2,c_3)\in \mathbf{F}_A(R)$. The main result of this section is that this torsor is in general non-trivial, even in the split case.
\begin{Thm}\label{TD4} If $C$ is the para-octonion algebra of the (split) complex octonion algebra $O$, then
there exists a smooth $\mathbb C$-ring $S$ and
a composition $\mathcal{M}$ of quadratic forms
over $S$ with $\mathcal{M}\not\simeq\mathcal{M}_{C_S}$ and such that $H(\mathcal{M},\mathbf{1})$ is isomorphic to $H_3(O,\mathbf{1})=H(\mathcal{M}_C,\mathbf{1})$.
\end{Thm}
\begin{proof} We have $O=O_0\otimes_{\mathbb R}\mathbb C$, where $O_0$ is the real division octonion algebra. The quadratic form of $O_0$
is the Euclidean form $x\mapsto x_1^2+\cdots + x_8^2$. Let $C_0$ be the corresponding real para-octonion algebra, so that $C=C_0\otimes_{\mathbb R}\mathbb C$. It is known (see e.g.\ \cite{AG}) that
$\mathbf{RT}(C_0)\simeq \mathbf{Spin}_{8,\mathbb R}$, which is anisotropic, and from \cite{J1} we know that $\mathbf{Aut}(H_3(O_0,\mathbf{1}))$ is anisotropic of type $\mathrm F_4$.
From \cite{Mi} we know that
$\pi_n(\mathbf{Spin}_8(\mathbb R))$ contains $\pi_n(\mathbf{Spin}_7(\mathbb R))$ as a direct factor, that $\pi_7(\mathbf{Spin}_7(\mathbb R))\simeq \mathbb Z$ and that the seventh homotopy group of the
compact group of type $\mathrm F_4$ is trivial, and thus does not contain a direct factor isomorphic to $\pi_7(\mathbf{Spin}_{8,\mathbb R}(\mathbb R))$. Now the complex Lie group
$\mathbf{RT}(C)(\mathbb C)\simeq\mathbf{Spin}_{8,\mathbb R}(\mathbb C)$ contains $\mathbf{Spin}_{8,\mathbb R}(\mathbb R)$ as a maximal compact subgroup, and $\mathbf{Aut}(H_3(O,\mathbf{1}))(\mathbb C)$ contains
$\mathbf{Aut}(H_3(O_0,\mathbf{1}))(\mathbb R)$
as a maximal compact subgroup. As in the proof of Theorem \ref{TF4}, it follows by Cartan decomposition that the homotopy groups of these complex Lie groups coincide with those of their
maximal compact subgroups. Since $\mathbf{F}_A$ is affine, we conclude with Lemma \ref{Lhomotopy} and Remark
\ref{Rtrivial}, since the quotient of a smooth group by a
smooth group is smooth.
\end{proof}
\subsection{Compositions of Quadratic Forms and Circle Products}
The above theorem states that $H(\mathcal{M},\mathbf{1})\simeq H(\mathcal{M}',\mathbf{1})$ may occur when $\mathcal{M}\not\simeq \mathcal{M}'$, and Proposition \ref{Ptorsor} implies that those $\mathcal{M}'$ with
$H(\mathcal{M}',\mathbf{1})\simeq H(\mathcal{M},\mathbf{1})$ are obtained as the twists $\mathbf{E}^\mathbf{c}\wedge \mathcal{M}$ as $\mathbf{c}$ runs through the frames of $H(\mathcal{M},\mathbf{1})$. In this section, we set out to
describe these twists explicitly. To this end, we shall use the circle product of Jordan algebras, defined by $x\circ y=\{x1y\}$ in terms of the triple product.
\begin{Rk} If $2\in R^*$, then $J$ carries the structure of a linear Jordan algebra with bilinear product $(x,y)\mapsto xy$ defined by $2xy=x\circ y$.
\end{Rk}
Let $\mathbf{c}=(c_1,c_2,c_3)\in \mathbf{F}_A(R)$, and for any permutation $(i,j,l)$ of $(1,2,3)$, set
\[C_{l}^\mathbf{c}=A_1(c_i)\cap A_1(c_j),\]
where the Peirce 1-space of an idempotent $c$ is defined as
\[A_1(c)=\{x\in A| c\circ x=x\}.\]
\begin{Lma} The restriction of the quadratic trace $S:x\mapsto T(x^\sharp,1)$ to $C_i^\mathbf{c}$ is a non-degenerate quadratic form of constant rank 8.
\end{Lma}
\begin{proof} The quadratic trace being a quadratic form, so is its restriction to $C_i^\mathbf{c}$. Since non-degeneracy and rank are invariant under faithfully flat descent,
it suffices to prove the statement
after a faithfully flat extension $S$ of $R$. By Proposition \ref{Ptorsor}, one can choose $S$ such that $c_i=\phi(e_i)$ for some $\phi\in\mathbf{Aut}(A)(S)$.
Thus $C_i^\mathbf{c}\otimes S$ is the image
of ${C_{i}}_S$, which is of rank 8, and the quadratic form in question is the transfer of the restriction of the quadratic trace to ${C_{i}}_S$, which by Proposition \ref{PCOQF} is an invertible scalar multiple of
the non-degenerate form $q_i$. This proves the statement.
\end{proof}
We write $q_i^\mathbf{c}$ for the \emph{negative of} the restriction of the quadratic trace to $C_i^\mathbf{c}$. The choice of sign is made for the next proposition to hold, which describes the
essential ingredient in our twist.
\begin{Rk} For convenience we introduce some notation. Let $\Gamma=(\gamma_1,\gamma_2,\gamma_3)\in (R^*)^3$. If $\mathcal{M}=(C_1,C_2,C_3,q_1,q_2,q_3,m)$ is a composition of quadratic forms, then so is
$(C_1,C_2,C_3,\gamma_2\gamma_3 q_1,\gamma_1\gamma_3 q_2,\gamma_1\gamma_2 q_3,\gamma_1 m)$. We shall denote this composition by $\Gamma\mathcal{M}$.
\end{Rk}
\begin{Prp}\label{Pdeform} The circle product of $A=H(\mathcal{M},\Gamma)$ defines a composition of quadratic forms
$\mathcal{M}^\mathbf{c}=(C_1^\mathbf{c},C_2^\mathbf{c},C_3^\mathbf{c},q_1^\mathbf{c},q_2^\mathbf{c},q_3^\mathbf{c},m_\circ^\mathbf{c})$ of rank 8, where $m_\circ^\mathbf{c}$ is the
restriction of $\circ$ to $C_3^\mathbf{c}\times C_2^\mathbf{c}$. If $\mathbf{c}=(e_1,e_2,e_3)$, then $\mathcal{M}^\mathbf{c}$ is isomorphic to $\Gamma\mathcal{M}$.
In that case the automorphism group of $\mathcal{M}^\mathbf{c}$ is $\iota(\mathbf{Aut}(\mathcal{M}))$.
\end{Prp}
From this it follows that in particular, if $\Gamma=\mathbf{1}$ and $\mathbf{c}=(e_1,e_2,e_3)$, then $\mathcal{M}^\mathbf{c}$ is isomorphic to $\mathcal{M}$; if in addition $\mathcal{M}=\mathcal{M}_C$
for some symmetric composition algebra $C$, then $\mathcal{M}^\mathbf{c}$ has automorphism group $\iota(\mathbf{RT}(C))$.
\begin{proof} If $x\in C_3^\mathbf{c}$ and $y\in C_2^\mathbf{c}$, then from \cite[Theorem 31.12(c)]{PR} it follows that $x\circ y\in C_1^\mathbf{c}$. To show the compatibility of
quadratic forms, we must show that $S(x\circ y)=-S(x)S(y)$. We first assume that $\mathbf{c}=(e_1,e_2,e_3)$. Writing $x=u_3[12]$ and $y=u_2[31]$ we find, from
\eqref{U} and the fact that $x$ and $y$ are orthogonal to 1 with respect to the bilinear trace, that
\[x\circ y=-(x\times y)\times 1.\]
From the formula for the adjoint in Proposition \ref{PCOQF} we have
$x\times y=\gamma_1m(u_3,u_2)[23]$,
and using the same formula again we find
\[x\circ y=-(x\times y)\times 1=\gamma_1m(u_3,u_2)[23].\]
By an easy computation, one finds that the quadratic trace satisfies $S(u_i[jl])=-\gamma_j\gamma_lq_i(u_i)$, from which it follows that
\[S(x\circ y)=-\gamma_2\gamma_3q_1(\gamma_1 m(u_3,u_2))=-\gamma_1^2\gamma_2\gamma_3q_3(u_3)q_2(u_2)=-S(x)S(y)\]
as desired. This shows the first statement for this particular choice of $\mathbf{c}$. For the general case, let $\lambda=S(x\circ y)+S(x)S(y)\in R$. We need to show that $\lambda=0$, for which it suffices
to show that $\lambda\otimes 1=0$ in $R\otimes_R R'=R'$, where $R'$ is a faithfully flat $R$-ring. Now for any $z\in A$, writing $S'$, $T'$ and $\sharp'$ respectively for the quadratic trace, linear
trace, and adjoint of $A_{R'}$, we have $(z\otimes 1)^{\sharp'}=z^{\sharp}\otimes 1$ by \cite[33.1]{PR} and thence by linearity of $T$,
\[S'(z\otimes1)=T'((z\otimes 1)^{\sharp'})=T'(z^{\sharp}\otimes 1)=T(z^{\sharp})\otimes 1=S(z)\otimes 1.\]
This, together with the bilinearity of the circle product, implies that
\[\lambda\otimes 1=S'((x\otimes 1)\circ(y\otimes 1))+S'(x\otimes1)S'(y\otimes1).\]
By Proposition \ref{Ptorsor} we may choose $R'$ such that
$\mathbf{Aut}(A)(R')$ acts transitively on the set of frames of $A_{R'}$. Thus there is $\phi\in\mathbf{Aut}(A)(R')$ with $\phi(c_i\otimes 1)=e_i\otimes 1$, and since $\phi$ preserves $S'$ and respects the circle
product, we have
\[\lambda\otimes 1=S'(\phi(x\otimes 1)\circ\phi(y\otimes 1))+S'(\phi(x\otimes1))S'(\phi(y\otimes1)).\]
Since $\phi(C_i^\mathbf{c}\otimes R')=C_i\otimes R'$, we are back to the special case treated above, from which we conclude that $\lambda\otimes 1=0$ as desired. This proves the first statement.
The second statement follows since by the above, $\mathcal{M}^\mathbf{c}\simeq\Gamma\mathcal{M}$ for this choice of $\mathbf{c}$, and the third statement follows since the quadratic forms and the composition of $\mathcal{M}^\mathbf{c}$
are scalar multiples of those of $\mathcal{M}$.
\end{proof}
We can now give an explicit construction of the twist of a composition of quadratic forms by the above torsor.
\begin{Def} Let $\mathcal{M}$ be a composition of quadratic forms of rank 8, set $A=H(\mathcal{M},\Gamma)$ and let $\mathbf{c}=(c_1,c_2,c_3)$ be a frame in $A$. The \emph{$(\mathbf{c},\Gamma)$-deformation}
of $\mathcal{M}$
is the composition $\mathcal{M}^\mathbf{c}$ of quadratic forms, defined in Proposition \ref{Pdeform}.
\end{Def}
Note that $\mathcal{M}^\mathbf{c}$ is determined uniquely by the datum $(\mathcal{M},\mathbf{c},\Gamma)$. We will compare this to the twist $\mathbf{E}^\mathbf{c}\wedge(\Gamma\mathcal{M})$ of $\Gamma\mathcal{M}$ by the the torsor
$\mathbf{E}^\mathbf{c}$ defined in the end of Section \ref{Storsor}.
\begin{Rk} If $\mathbf{E}$ is any $\mathbf{Aut}(\mathcal{M})$-torsor over $\mathrm{Spec}(R)$, then the twist $\mathbf{E}\wedge\mathcal{M}$ is defined in analogy to twists of modules and algebras. Concretely, it
is the fppf sheaf associated to the presheaf whose $S$-points, for each $R$-ring $S$, is the composition of quadratic forms
constructed as follows. For each $k\in\{1,2,3\}$, define the equivalence relation $\sim_k$ on $\mathbf{E}(S)\times {C_k}_S$ by $(\phi,x)\sim_k (\phi',x')$
precisely when there exists $(t_1,t_2,t_3)\in\mathbf{Aut}(\mathcal{M})$ such that
\[(\phi,t_k(x))=(\phi'\cdot(t_1,t_2,t_3),x')),\]
where the right action $\cdot$ of $\mathbf{Aut}(\mathcal{M})$ on $\mathbf{E}$ is the one defining the torsor. Denote the equivalence
class of $(\phi,x)$ under $\sim_k$ by $[\phi,x]$. For each $k$, the set $(\mathbf{E}(S)\times {C_k}_S)/\sim_k$ is a quadratic $S$-module under the scalar multiplication
\[\lambda[\phi,x]=[\phi, \lambda x]\]
and addition
\[[\phi,x]+[\psi,y]=[\phi, x+t_k(y)],\]
where $(t_1,t_2,t_3)$ is uniquely defined by $\psi=\phi\cdot(t_1,t_2,t_3)$, and quadratic form $[\phi,x]\mapsto q_i(x)=S(x)$. Moreover we have a map
\[(\mathbf{E}(S)\times {C_3}_S)/\sim_3\times (\mathbf{E}(S)\times {C_2}_S)/\sim_2\ \longrightarrow\ (\mathbf{E}(S)\times {C_1}_S)/\sim_1\]
defined by
\[([\phi,x],[\psi,y])\mapsto[\phi,m(x,t_2(y))],\]
where $t_2$ is given by $\psi=\phi\cdot(t_1,t_2,t_3)$. These data define the desired composition of quadratic forms.
Noting that $\mathbf{Aut}(\mathcal{M})=\mathbf{Aut}(\Gamma\mathcal{M})$, we can do the same after replacing $\mathcal{M}$ by $\Gamma\mathcal{M}$.
\end{Rk}
\begin{Thm} Let $\mathcal{M}$ be a composition of quadratic forms of rank 8 and let $\mathbf{c}=(c_1,c_2,c_3)$ be a frame in $S=H(\mathcal{M},\Gamma)$.
The twist $\mathbf{E}^\mathbf{c}\wedge(\Gamma\mathcal{M})$ is canonically isomorphic to $\mathcal{M}^\mathbf{c}$.
\end{Thm}
\begin{proof} By Proposition \ref{Pdeform}, we may and will identify $\Gamma\mathcal{M}$ with $\mathcal{M}^\mathbf{e}$ for the distinguished frame $\mathbf{e}$ from Example \ref{Edist},
upon which $\mathbf{Aut}(\mathcal{M})$ acts via the inclusion $\iota$ of Lemma \ref{Lincl}.
Consider the map $\Theta:\mathbf{E}^\mathbf{c}\wedge(\Gamma\mathcal{M})\to\mathcal{M^\mathbf{c}}$ induced by the maps
\[\begin{array}{ll}
(\mathbf{E}(S)\times {C_k}_S)/\sim_k\ \to C_k^\mathbf{c}(S), & [\phi,x]\mapsto \phi(x),
\end{array}
\]
which is well-defined by definition of $\sim_k$ and the fact that if $\phi\in\mathbf{Aut}(A)(S)$ maps $\mathbf{e}$ to $\mathbf{c}$, then it maps $C_k^\mathbf{e}$ to $C_k^\mathbf{c}$. It is straight-forwardly checked that these maps are a well-defined linear
bijections of modules whenever $\mathbf{E}^\mathbf{c}(S)\neq\emptyset$. From Proposition \ref{Pdeform} it follows
that they respect the quadratic forms and are compatible with the composition. We conclude with the definition of the sheaf associated to a presheaf.
\end{proof}
\section{Non-Isomorphic and Non-Isometric Coordinate Algebras}\label{S4}
Over fields, a celebrated theorem that goes back, in its essential form, to Albert and Jacobson \cite{AJ} (see \cite[Theorem 21]{P} for a more general statement) implies that for two
octonion algebras $C$ and $C'$, we have
\[H_3(C,\mathbf{1})\simeq H_3(C',\mathbf{1})\Longleftrightarrow q_C\simeq q_{C'}\Longleftrightarrow C\simeq C'.\]
By \cite{G1}, the second equivalence does not hold over arbitrary rings. In this section we consider the first equivalence and show that this also
fails over rings in general. We begin by showing how a combination of known results implies that the first and third conditions are not equivalent over rings. (For visual clarity, we will, in
this section, often use the notation $\overline{C}$ to denote the para-octonion algebra obtained from the octonion algebra $C$.)
\begin{Prp}\label{Pnontriv} There exists a smooth $\mathbb C$-ring $S$ and two octonion $S$-algebras $C$ and $C'$ such that $C\not\simeq C'$ and $H_3(C,\mathbf{1})\simeq H_3(C',\mathbf{1})$.
\end{Prp}
\begin{proof} Set $A=H_3(C,\mathbf{1})$. We need to show that the map
\[H^1(S,\mathbf{Aut}(C)){\longrightarrow} H^1(S,\mathbf{Aut} (A)),\]
induced by the subgroup inclusion, has non-trivial kernel for a smooth $\mathbb C$-ring $S$ and octonion $S$-algebra $C$. This inclusion is the composition of the inclusions $i: \mathbf{RT}(\overline{C})\to \mathbf{Aut}(A)$ of Lemma \ref{Lincl} (noting that
$\mathbf{RT}(\overline{C})=\mathbf{Aut}(\mathcal{M}_{\overline{C}})$) and
$j:\mathbf{Aut}(C)\to\mathbf{RT}(\overline{C})$ defined by $t\mapsto (t,t,t)$ (see \cite[3.5]{AG}). Thus the map of cohomologies factors as
\[H^1(S,\mathbf{Aut}(C))\overset{j^*}{\longrightarrow}H^1(S,\mathbf{RT}(\overline{C}))\overset{i^*}{\longrightarrow} H^1(S,\mathbf{Aut} A)\]
and has non-trivial kernel since by \cite[4.3]{AG} (which is a variant of \cite[3.5]{G1}), so does $j^*$, for an appropriate (smooth) choice of $S$. This completes the proof.
\end{proof}
From \cite[6.6]{AG} we know that two octonion algebras $C$ and $C'$
have isometric norms if and only if the corresponding compositions $\mathcal{M}_{\overline{C}}$ and $\mathcal{M}_{\overline{C'}}$ of quadratic forms are isomorphic. In the previous section we showed that non-isomorphic compositions of quadratic forms
may give rise to isomorphic Albert algebras. To achieve our goal for this section, we need to refine this statement and show that non-isomorphic compositions of quadratic forms
of the form $\mathcal{M}_{\overline{C}}$ may give rise to isomorphic Albert algebras.
Let thus $C$ be an octonion algebra over $R$ and $\overline{C}$ the para-octonion algebra obtained from $C$. Recall that $H_3(C,\mathbf{1})=H(\mathcal{M}_{\overline{C}},\mathbf{1})$, and
consider the inclusions $i: \mathbf{RT}(\overline{C})=\mathbf{Aut}(\mathcal{M}_{\overline{C}})\to \mathbf{Aut}(A)$ and
$j:\mathbf{Aut}(C)\to\mathbf{RT}(\overline{C})$ from the proof of Proposition \ref{Pnontriv}, and, for every $R$-ring $S$, the induced maps
\[H^1(S,\mathbf{Aut}(C))\overset{j^*}{\longrightarrow}H^1(S,\mathbf{RT}(\overline{C}))\overset{i^*}{\longrightarrow} H^1(S,\mathbf{Aut} A).\]
To show that there exists an octonion $S$-algebra $C'$ with $\mathcal{M}_{\overline{C}}\not\simeq \mathcal{M}_{\overline{C'}}$ and
$A=H_3(C,\mathbf{1})\simeq H_3(C',\mathbf{1})$, we need to prove that for some $R$-ring $S$ there is a non-trivial element $\eta\in\mathrm{Ker}(i^*)\cap \mathrm{Im}(j^*)=j^*(\mathrm{Ker}(i^*j^*))$.
Let $Z=\mathbf{Aut}(A)/\mathbf{Aut}(C)$ and $Y=\mathbf{Aut}(A)/\mathbf{RT}(\overline{C})$ be the respective fppf quotients with quotient projections $\pi_Z:\mathbf{Aut}(A)\to Z$ and $\pi_Y:\mathbf{Aut}(A)\to Y$.
We will show that for some $z\in Z(R)$, the $\mathbf{RT}(\overline{C})$-torsor
\[\pi_Z^{-1}(z)\wedge^{\mathbf{Aut}(C)}\mathbf{RT}(\overline{C})\]
is non-trivial for an appropriate choice of $R$. By Remark \ref{Rquotient}, the class of this torsor is the image under $j^*$ of the class of the torsor $\pi_Z^{-1}(z)$. This implies, together with Remark
\ref{Rtrivial}, that
for some $R$-ring $S$, this torsor has no $S$-point, and hence there is an octonion $S$-algebra $C'$ with the desired properties. We will therefore prove that the $\mathbf{RT}(\overline{C})$-torsor
\begin{equation}\label{torsor}
\xymatrix{\mathbf{Aut}(A)\wedge^{\mathbf{Aut}(C)}\mathbf{RT}(\overline{C})\ar[d]^{\mathbf{RT}(\overline{C})}\\ Z}
\end{equation}
is non-trivial, i.e.\ does not admit a global section. The following proposition says that this indeed the case in general. We refer the reader to Remark \ref{Rquotient} for the definition of contracted products.
\begin{Thm}\label{TG2} The torsor \eqref{torsor} is non-trivial over $\mathbb R$ for the real division octonion algebra $C$.
\end{Thm}
The proof makes use of the following basic observation.
\begin{Lma} Let $G$ be a group scheme over $R$, $H$ a subgroup scheme of $G$, and $K$ a subgroup scheme of $H$. Then $G\wedge^K H$ and $G\times H/K$ are isomorphic as $H$-torsors over
$G/K$. Here the right action of $H$ on $G\times H/K$ is given by $(g,x)\cdot h=(gh,h^{-1}x)$ and the structural projection $G\times H/K\to G/K$ is induced by the product map $G\times H\to G, (g,h)\mapsto gh$.
\end{Lma}
\begin{proof} It is straight-forwardly verified that the $H$-action and structural projection of $G\times H/K$ are well-defined; they make $G\times H/K$ an $H$-torsor over $G/K$ since
if over the $R$-ring $S$, the $K$-torsor $G\to G/K$ admits a section $\sigma$, then the structural projection of $G\times H/K$ admits the section $x\mapsto (\sigma(x),e)$, where
$e$ is
the image of the neutral element of $H$ in $H/K$. To show that the two $H$-torsors in question are isomorphic over $G/K$, consider the map
\[\Theta:G\wedge^K H\to G\times H/K.\]
induced by the map $G\times H\to G\times H/K$ defined by $(g,h)\mapsto (gh,\overline{h^{-1}})$, where $\overline{h^{-1}}$ is the quotient image of $h^{-1}$. This map is invariant with respect
to the quotient defining the contracted product. The fact that the induced map is $H$-invariant and lifts the identity of $G/K$ can be straight-forwardly
verified on the level of presheaves. We conclude with the universal property the sheaf associated to a presheaf and the fact that any morphism of torsors is an isomorphism.
\end{proof}
\begin{proof}[Proof of Theorem \ref{TG2}] The lemma above implies that the two $\mathbf{RT}(\overline{C})$-torsors
\[\mathbf{Aut}(A)\wedge^{\mathbf{Aut}(C)}\mathbf{RT}(\overline{C})\]
and
\[\mathbf{Aut}(A)\times \mathbf{RT}(\overline{C})/\mathbf{Aut}(C)\]
over $\mathbf{Aut}(A)/\mathbf{Aut}(C)$, which is representable by an affine scheme by \cite[6.12]{CTS}, are isomorphic. By \cite[Theorem 4.1]{AG}, the quotient $\mathbf{RT}(\overline{C})/\mathbf{Aut}(C)$ is isomorphic to the product of two copies of the Euclidean
7-sphere $\mathbb{S}_7$. Using Lemma \ref{Lhomotopy} and arguing as in the proof of Theorem \ref{TD4}, it thus suffices to show that for some $n$,
$\pi_n(\mathbf{Spin}_8(\mathbb R))$ is not a direct
factor in
\[\Gamma=\pi_n(\mathbf{Aut}(A)(\mathbb R))\times\pi_n(\mathbb{S}_7(\mathbb R))^2.\]
Now $\mathbf{Aut}(A)(\mathbb R)$ is compact of type $\mathrm F_4$. It is known that $\pi_{14}(\mathbb{S}_7)\simeq \mathbb{Z}_{120}$, while from \cite{Mi} we get $\pi_{14}(\mathbf{Aut}(A)(\mathbb R))\simeq \mathbb{Z}_2$. Thus in particular
$\Gamma$ has no direct factor isomorphic to $\mathbb{Z}_7$, which, by \cite{Mi} is a direct factor of $\pi_{14}(\mathbf{Spin}_8(\mathbb R))$. This completes the proof.
\end{proof}
\begin{Cor} There exists a smooth $\mathbb{R}$-ring $S$ and octonion $S$-algebras $C$ and $C'$ such that
$H_3(C,\mathbf{1})\simeq H_3(C',\mathbf{1})$ and $q_C\not\simeq q_{C'}$.
\end{Cor}
\begin{proof} Let $C_0$ be the real division octonion algebra. Since $\mathbf{Aut}(H_3(C_0,\mathbf{1}))$ and $\mathbf{Aut}(C_0)$ are smooth, so is their quotient, and it follows from
Theorem \ref{TG2} and Remark \ref{Rtrivial} that there exists a smooth $\mathbb R$-ring $S$ and an octonion $S$-algebra $C'$ with $H_3(C)\simeq H_3(C')$ and $\mathcal{M}_{\overline{C}}\not
\simeq M_{\overline{C'}}$, where $C=C_0\otimes_{\mathbb R} S$. If $q_C\simeq q_{C'}$,
then the class of $C'$ is in the kernel of the map $H^1(S,\mathbf{Aut}(C))\to H^1(S,\mathbf{O}(q_C))$ induced by the inclusion $\mathbf{Aut}(C)\to\mathbf{O}(q_C)$. But by \cite[Theorem 6.6]{AG}, this kernel
coincides with the kernel of
\[j^*:H^1(R,\mathbf{Aut}(C))\to H^1(R,\mathbf{RT}(\overline{C})).\]
Thus the class of $C'$ is in the kernel of $j^*$, whence $\mathcal{M}_{\overline{C}}\simeq M_{\overline{C'}}$, a
contradiction. This completes the proof.
\end{proof}
|
2,869,038,154,908 | arxiv | \section{Introduction}
The basic problem alluded to the title is as follows:\\
\textit{Given a Hilbert module ${\mathscr M}$ and a submodule ${\mathscr M}_0$ over the algebra of holomorphic functions ${\mathcal A}(\Omega)$ on a bounded domain $\Omega$ in ${\mathbb C}^m$, satisfying the exact sequence $$0 \rightarrow {\mathscr M}_0 \overset{i}{\rightarrow} {\mathscr M} \overset{\pi}{\rightarrow} {\mathscr M}_q \rightarrow 0,$$ where $i$ is the inclusion map, $\pi$ is the quotient map and ${\mathscr M}_q$ is the quotient module ${\mathscr M}\ominus{\mathscr M}_0$, is it possible to determine ${\mathscr M}_q$ in terms of ${\mathscr M}$ and ${\mathscr M}_0$? One can make this general question more precise by asking if one can assign some computable invariants on ${\mathscr M}_q$ in terms of ${\mathscr M}$ and the submodule ${\mathscr M}_0$.}
\smallskip
For a quasi-free Hilbert module ${\mathscr M}$ \cite[Section 2]{QFRHM}, \cite[Page 3]{OQFHM} over ${\mathcal A}(\Omega)$, the quotient module ${\mathscr M}_q$ obtained from the submodule ${\mathscr M}_0$, where ${\mathscr M}_0$ is the maximal set of functions in ${\mathscr M}$ vanishing along a smooth hypersurface in $\Omega$, was first studied by R.G. Douglas and G. Misra in \cite{GIRHM}. In fact, they considered a quasi-free Hilbert module ${\mathscr M}$ of rank $1$ and described a geometric invariant of the quotient module ${\mathscr M}_q$, namely, the fundamental class of the variety $\mathcal{Z}$ \cite[Page 61]{PAG} and they described the fundamental class $[\mathcal{Z}]$ in terms of the curvatures of the line bundles (Remark $\ref{modvec}$) obtained from ${\mathscr M}$ and ${\mathscr M}_0$. Later in the paper \cite{GIRQMGRID}, this result was extended to quotient modules corresponding to the submodules consisting of complex valued functions in ${\mathcal A}(\Omega)$ vanishing along a complex algebraic variety of complete intersection of finitely many smooth hypersurfaces. It was the paper \cite{OQMAM} where quotient modules arising from submodules, ${\mathscr M}^k_0$, of functions in ${\mathcal A}(\Omega)$ vanishing on a smooth complex hypersurface of order $k \geq 2$. The module of jets corresponding to a Hilbert module was introduced by means of the jet construction \cite[Page 372]{OQMAM} and was showed that the quotient module considered there could be thought of as the module of jets restricted to the hypersurface. Thus, in the first half of \cite{OQMAM} a model for the quotient module obtained from submodules ${\mathscr M}^k_0$ was provied while the later half was devoted in finding geometric invariants of the quotient modules in terms of the fundamental class of the hypersurface generalizing the results of \cite{GIRHM}. In \cite{EQHMI} a complete set of unitary invariants for quotient modules with $k=2$ was determined and in a subsequent paper \cite{EQHMII} they described complete unitary invariants for quotient modules with an arbitrary $k$. It was shown in \cite{EQHMII} that two quotient modules $\mathcal{Q}:={\mathscr M}\ominus{\mathscr M}^k_0$ and $\tilde{\mathcal{Q}}:=\tilde{{\mathscr M}}\ominus\tilde{{\mathscr M}}^k_0$ are unitarily equivalent if and only if the line bundles $E_{{\mathscr M}}$ and $E_{\tilde{{\mathscr M}}}$ arising from ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively, are isomorphic in certain sense \cite[Definition 4.2]{EQHMII}. Moreover, for $k=2$, a complete set of unitary invariants for quotient modules were obtained which are the tangential and transverse components of the curvature of the line bundle $E_{{\mathscr M}}$ relative to hypersurface $\mathcal{Z}$ and the second fundamental form for the inclusion $E_{{\mathscr M}}\subset J^{(2)}_1E_{{\mathscr M}}$ where $J^{(2)}_1E_{{\mathscr M}}$ is the second order jet bundle of $E_{{\mathscr M}}$ relative to $\mathcal{Z}$ \cite[Section 3]{EQHMII}. More recently, results in the paper \cite{EQHMII} have been generalized to the quotient modules obtained from submodules of vector valued holomorphic functions on $\Omega$ vanishing along a smooth complex hypersurface in $ \Omega$ by L.~Chen and R.G.~Douglas in \cite{ALOTCD}. So to this extent it is natural to consider the case where quotient modules are obtained from submodules of vector valued holomorphic functions on $\Omega \subset {\mathbb C}^m$ vanishing of order $k\geq 2$ on a smooth complex analytic set of codimension at least $2$.
\smallskip
In this article, we intend to study such quotient modules to obtain a canonical model for them. We then make use of these canonical models to describe the complete set of unitary invariants of those quotient modules. In order to accomplish our goal we consider a quasi-free Hilbert module ${\mathscr M}$ (Section $\ref{prelim}$) over ${\mathcal A}(\Omega)$ consisting of vector valued holomorphic functions on $\Omega$ with the module action obtained by point wise multiplication and go on to describe submodules ${\mathscr M}_0$ of interest in Section $\ref{SM}$. We first define the order of vanishing of a vector valued holomorphic function along a connected complex submanifold $\mathcal{Z}$ of arbitrary codimension which is the key ingredient of the definition of ${\mathscr M}_0$. Since the definition of the order of vanishing is not canonical it becomes difficult to calculate the same for a given holomorphic function. However, we provide an equivalent condition to define the order of vanishing which is easy to compute. We then, following the technique introduced in the paper \cite{OQMAM}, describe the jet construction on ${\mathscr M}$ relative to the submanifold $\mathcal{Z}$ to identify the quotient module to the restriction of a Hilbert module $J({\mathscr M})$ (we refer the readers to \eqref{jcon} for definition) to $\mathcal{Z}$.
\smallskip
The identification, mentioned above, enables us to associate a natural geometric object to the quotient module, namely, the $k$-th order jet bundle relative to $\mathcal{Z}$ (Section $\ref{JB}$) of the vector bundle associated to the module ${\mathscr M}$. Thus, we pass to the geometric counter part of our study of quotient modules to be able to provide some geometric invariants for them. These jet bundles are canonically associated to the module of jets $J({\mathscr M})$ of the module ${\mathscr M}$. Then using the technique of normalised frame \cite[Lemma 2.3]{DACM} of a {\bf h} we successfully describe a complete set of unitary invariants for quotient modules. Since the jet bundles of our interest are holomorphically trivial (that is, they possess a global holomorphic frame) we always have a bundle isomorphism between any two of them which does not depend on the base manifold. But it is, a priori, not true that such an isomorphism also preserves the Hermitian metric of the jet bundle mentioned above. In our case, while the jet bundles are associated to equivalent quotient modules, we do have isometric bundle isomorphism between corresponding jet bundles which does not depend on the base manifold. More precisely, we show that two such quotient modules are unitarily equivalent if and only if there exists a constant isometric jet bundle isomorphism between the corresponding jet bundles restricted to the submanifold $\mathcal{Z}$ (Theorem $\ref{jean1}$) with the aid of normalized frame (we refer readers to Proposition $\ref{prop1}$ for definition). We then make use of this result to determine the unitary invariants of aforementioned quotient modules.
\smallskip
Finally, we describe the quotient module obtained from the submodule of functions in weighted Bergman module ${\mathcal H}^{(\alpha,\beta,\gamma)}$ over ${\mathcal A}({\mathbb D}^3)$ vanishing to order $2$ along the diagonal set of ${\mathbb D}^3$. Furthermore, with the help of our main theorem (Theorem $\ref{mthm}$) we determine unitary equivalence classes of weighted Bergman modules over ${\mathcal A}({\mathbb D}^m)$ in terms of quotient modules arising from the submodules of functions vanishing to order $2$ along the diagonal set of ${\mathbb D}^m$. Thus, the results presented in this article extend most of the results in the papers \cite{EQHMII}, \cite{OQMAM}, \cite{EQHMI} and \cite{ALOTCD} to the case of quotient modules arising from submodules of vector valued holomorphic functions on a bounded domain in ${\mathbb C}^m$ which vanishes along a smooth irreducible complex analytic set of order at least $2$.
\smallskip
The present article is organized in the following way. In Section $\ref{prelim}$, we recall some basic definitions and introduce few notations which will be used through out this note. Then a complete description of the submodule ${\mathscr M}_0$ of interest is presented in Section $\ref{SM}$. Section $\ref{QM}$ is devoted to study quotient modules obtained from submodules introduced in Section $\ref{SM}$. There we provide a canonical model for such quotient modules and in the subsequent section, Section $\ref{JB}$, describe the complete set of unitary invariants of those quotient modules. We then finish this article by presenting some examples and applications in Section $\ref{exap}$.
\section{Preliminaries}\label{prelim}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$ and ${\mathcal A}(\Omega)$ be the unital Banach algebra obtained as the norm closure with respect to the supremum norm on $\overline{\Omega}$ of all functions holomorphic on a neighbourhood of $\overline{\Omega}$. A complex Hilbert space ${\mathcal H}$ is said to be a Hilbert module over ${\mathcal A}(\Omega)$ with module map ${\mathcal A}(\Omega)\times{\mathcal H} \overset{\pi}\rightarrow {\mathcal H}$ by point wise multiplication such that the module action ${\mathcal A}(\Omega)\times {\mathcal H} \overset{\pi}\rightarrow {\mathcal H}$ is norm continuous. We say that a Hilbert module ${\mathcal H}$ over ${\mathcal A}(\Omega)$ is contractive if $\pi$ is a contraction.
Suppose that ${\mathcal H}_1$ and ${\mathcal H}_2$ are two Hilbert modules over ${\mathcal A}(\Omega)$ with module actions $(f,h_i)\mapsto M^{(i)}_{f}(h_i) $, $i=1,2$. Then a Hilbert space isomorphism $\Phi: {\mathcal H}_1 \rightarrow {\mathcal H}_2$ is said to be module isomorphism if $\Phi(M^{(1)}_{f}(h_1))=M^{(2)}_{f}(\Phi(h_1))$ and we denote ${\mathcal H}_1 \simeq_{{\mathcal A}(\Omega)} {\mathcal H}_2$.
\medskip
In this article, we study quotient modules obtained from certain submodules of quasi-free Hilbert modules. So we recall that a Hilbert space ${\mathcal H}$ is said to be a quasi-free Hilbert module over ${\mathcal A}(\Omega)$ of rank $r$, for $1\leq r \leq \infty$ and a bounded domain $\Omega\subset {\mathbb C}^m$, if ${\mathcal H}$ is a Hilbert space completion of the algebraic tensor product ${\mathcal A}(\Omega)\otimes_{alg}{\mathbb C}^r$ and following conditions happen to be true:
\begin{enumerate}
\item[(i)] the evaluation operators $e_{w}:{\mathcal H} \rightarrow {\mathbb C}^r$ defined by $h\mapsto h(w)$ are uniformly bounded on $\Omega$,
\item[(ii)] a sequence $\{h_k\}\subset{\mathcal A}(\Omega)\otimes_{alg}{\mathbb C}^r$ that is Cauchy in the norm in ${\mathcal H}$ converges to $0$ in the norm in ${\mathcal H}$ if and only if $e_{w}(h_k)$ converges to $0$ in ${\mathbb C}^r$ for $w\in\Omega$, and
\item[(iii)] multiplication by functions in ${\mathcal A}(\Omega)$ define a bounded operator on ${\mathcal H}$.
\end{enumerate}
\smallskip
The condition (i) and (ii) together make the completion ${\mathcal H}$ a functional Hilbert space over ${\mathcal A}(\Omega)$ \cite[Page 347]{TRK}. Moreover, condition (i) ensures that the Hilbert space ${\mathcal H}$ possesses an $r\times r$ matrix valued reproducing kernel thanks to Riesz representation theorem. Finally, condition (iii) along with (i) make ${\mathcal H}$ a Hilbert module over ${\mathcal A}(\Omega)$ in the sense of \cite[Definition 1.2]{HMFA}. Thus, a quasi-free Hilbert module over ${\mathcal A}(\Omega)$ of rank $r$ gives rise to a reproducing kernel Hilbert module of ${\mathbb C}^r$ valued holomorphic functions over ${\mathcal A}(\Omega)$.
\medskip
For $\Omega \subset {\mathbb C}$, we recall the definition of the Cowen-Douglas class $B_n(\Omega)$ consisting of operators $T$ on a Hilbert space ${\mathcal H}$ for which each $w \in \Omega$ is an eigenvalue of uniform multiplicity $n$ of $T$, the eigenvectors span the Hilbert space ${\mathcal H}$ and ran$(T-wI_{{\mathcal H}})$ is closed for $w \in \Omega$. Later the definition was adapted to the case of an $m$-tuple of commuting operators $\bf T$ acting on a Hilbert space ${\mathcal H}$, first in the paper \cite{OPOSE}, and then in the paper \cite{GBKCD} from slightly different point of view which emphasized the role of the reproducing kernel. Let us now define the class $B_n(\Omega)$ for $\Omega \subset {\mathbb C}^m$ a bounded domain.
\begin{defn}\label{CDC}
The $m$-tuple $\bf T=(T_1,\hdots,T_m)$ is in $B_n(\Omega)$ if
\begin{enumerate}
\item[(i)] ran$D_{{\bf T}-w}$ is closed for all $w \in \Omega $ where $D_{\bf T}: {\mathcal H} \rightarrow {\mathcal H} \otimes {{\mathbb C}}^n $ is defined by $D_{\bf T} h =( {T_1} h,...,{T_m} h), h \in {\mathcal H} $;
\item[(ii)] span$\{\ker D_{{\bf T}-w}:w \in \Omega\}$ is dense in ${\mathcal H}$ and
\item[(iii)] $\dim\ker D_{{\bf T}-w}=n$ for all $w \in \Omega$.
\end{enumerate}
\end{defn}
It was then shown that each of these $m$-tuples $\bf T$ determines a Hermitian holomorphic vector bundle $E$ of rank $n$ on $\Omega$ and that two $m$-tuples of operators in $B_n(\Omega)$ are unitarily equivalent if and only if the corresponding vector bundles are locally equivalent. In case of $n=1$, this is a question of equivalence of two Hermitian holomorphic line bundles and hence is a question of equality of the curvatures of those line bundles. However, no such simple characterization is known if the rank of the bundle is strictly greater than $1$.
\begin{rem}\label{modvec}
Let us consider a quasi-free Hilbert module ${\mathcal H}$ of rank $r$ over the algebra ${\mathcal A}(\Omega)$. Then, as mentioned earlier, ${\mathcal H}$ is a reproducing kernel Hilbert module with reproducing kernel $K$ on $\Omega$. Let $\bf M$ be the $m$-tuple of multiplication operators $(M_1,\hdots,M_m)$ acting on ${\mathcal H}$ as multiplication by coordinate functions. It then follows from the reproducing property of $K$ that
\begin{eqnarray}\label{mo}{M_i}^*K(.,w)\eta=\overline{w}_iK(.,w)\eta,~\text{for}~\eta \in {\mathbb C}^r,~ w \in \Omega,~ 1\leq i \leq m.\end{eqnarray}
As a Consequence, the dimension of the joint eigenspace of $\bf M^*$ at $\overline{w}$ is at least $r$. Moreover, from the definition of quasi-free Hilbert modules and holomorphic functional calculus we see that the joint eigenspace at $\overline{w}$ must have dimension exactly $r$ for $w \in \Omega$. Thus, $s(\overline{w}):=\{K(.,w)\sigma_1,\hdots,K(.,w)\sigma_r\}$ defines a global holomorphic frame for the vector bundle $E \rightarrow \Omega^*$ with fibre at $\overline{w} \in \Omega^*:=\{\overline{w}:w \in \Omega\}$, $E_{\overline{w}}:= \text{span}~s(\overline{w})=\ker D_{{\bf M}^*-\overline{w}}$ where $\{\sigma_j\}_{j=1}^r$ is the standard ordered basis for ${\mathbb C}^r$.
We also note that the third condition in the definition of quasi-free Hilbert modules implies that ran$D_{{\bf M}^*-\overline{w}}$ is dense in ${\mathcal H}$. Thus, $\bf M$ satisfies every condition of the definition of $B_r(\Omega^*)$ except the first one. This difference was discussed in the paper \cite{OQFHM} where the notion of a quasi-free Hilbert module was introduced. While most of our examples lie in the class $B_r(\Omega)$, our methods work even with the weaker hypothesis that the modules are quasi-free of rank $r$ over the algebra ${\mathcal A}(\Omega)$.
\end{rem}
In this article, we are interested to present some geometric invariant of quotient modules obtained from certain class of submodules of a quasi-free Hilbert module over ${\mathcal A}(\Omega)$. We, therefore, need some geometric tools from complex differential geometry. For the sake of completeness let us recall some basic notions from complex differential geometry following \cite{CGOT} and Chapter 3 of \cite{DACM}.
\smallskip
Let $E$ be a Hermitian holomorphic vector bundle of rank $n$ over a complex manifold $M$ of dimension $m$ with the Chern connection $D$. Then a simple calculation shows that with respect to a local frame $s=\{e_1,\hdots,e_n\}$ of $E$ the Chern connection $D$ takes the form
\begin{eqnarray} D(s)=\partial H(s)\cdot H(s)^{-1}\end{eqnarray} where $H(s)$ is the Grammian matrix of the frame $s$. From now on, by a connection on a {\bf h} we will mean the Chern connection.
\smallskip
The curvature of $E$ is defined as $\mathcal K:=D^2=D\circ D$ and $\mathcal K$ is an element in ${\mathcal E}^2(M)\otimes \text{Hom}(E,E)$ where ${\mathcal E}^2(M)$ is the collection of smooth $2$-forms on $M$. Consequently, in a local coordinate chart of $M$ one can write $\mathcal K$ as $$\mathcal K(\sigma)=\sum_{i,j=1}^m \mathcal K_{i\overline{j}}dz_i\wedge d\overline{z}_j, \,\,\,\sigma \in {\mathcal E}^0(M,E).$$ Since $\mathcal K \in {\mathcal E}^2(M)\otimes \text{Hom}(E,E)$ we have $\mathcal K_{i\overline{j}}$ are also bundle map for $i,j=1,\hdots,m$. As before we can also express the curvature tensor $\mathcal K$ with respect to a local frame as follows
\begin{eqnarray}\label{locK}\mathcal K(s)=\bar\partial(\partial H(s)\cdot H(s)^{-1})~ \text{and equivalently},~
\mathcal K_{i\overline{j}}(s)=\bar\partial_j(\partial_i H(s)\cdot H(s)^{-1})\end{eqnarray} where $\partial_i=\frac{\partial}{\partial z_i}$ and $\bar\partial_j=\frac{\partial}{\partial\overline{z}_j}$. It is well known that the curvature operator is self adjoint (\cite[(2.15.4)]{CGOT}) in the sense that $\mathcal K_{i\overline{j}}=\mathcal K^*_{j\overline{i}}$.
\smallskip
Now following \cite[Lemma 2.10]{CGOT}, for a given local frame $s$ of $E$ and a $C^{\infty}$ bundle map $\Phi: E \rightarrow \tilde{E}$, we have that
\begin{eqnarray}\label{covd1}\Phi_{z_i}(s)=\partial_{z_i}\Phi(s)-[\partial_{z_i}H(s)\cdot H(s)^{-1},\Phi(s)]~~\text{and}~~\Phi_{\overline{z}_i}(s)=\partial_{\overline{z}_i}\Phi(s)\end{eqnarray}
where $\Phi(s),\Phi_{z_i}(s),\Phi_{\overline{z}_j}(s)$ are matrix representations of $\Phi, \Phi_{z_i},\Phi_{\overline{z}_j}$, respectively, with respect to the local frame $s$ and $[A,B]$ denotes the commutator of matrices $A$ and $B$.
\smallskip
In the following lemma we calculate the covariant derivatives of curvature tensor which will be useful in Section $\ref{JB}$. The proof of the following lemma, for $d=1$, is well known \cite[Proposition 2.18]{CGOT}. Although the similar set of arguments used there with more than one variables yields the proof of the following lemma, we present a sketch of the proof for the sake of completeness.
\begin{lem}\label{cume}
Let $E$ be a Hermitian holomorphic vector bundle over $\Omega$ in ${\mathbb C}^m$ with a fixed holomorphic frame $S:=\{s_1,\hdots,s_r\}$ whose Grammian matrix is $H$. Then
\begin{enumerate}
\item[(i)] For $1 \leq d \leq m$, $\alpha,\beta \in (\mathbb{N} \cup \{(0)\})^d$, and $i,j=1,\hdots,d$, the $r\times r$ matrices \linebreak $(\mathcal K_{i\overline{j}}(S))_{{z_1}^{\alpha_1}\cdots{z_d}^{\alpha_d}{\overline{z}_1}^{\beta_1}\cdots{\overline{z}_d}^{\beta_d}}$ can be expressed in terms of $H^{-1}$ and ${\partial_1}^{p_1}\cdots{\partial_d}^{p_d}{\bar\partial_1}^{q_1}\cdots{\bar\partial_d}^{q_d}H$, $0 \leq p_l \leq \alpha_l+1$, $0 \leq q_l \leq \beta_l+1$, $l=1,\hdots,d$.\\
\item[(ii)] Given $1 \leq d \leq m$, $\alpha,\beta \in (\mathbb{N} \cup \{(0)\})^d$, ${\partial_1}^{\alpha_1}\cdots{\partial_d}^{\alpha_d}{\bar\partial_1}^{\beta_1}\cdots{\bar\partial_d}^{\beta_d}H$ can be written in terms of $H^{-1}$, ${\partial_1}^{p_1}\cdots{\partial_d}^{p_d}H$, ${\bar\partial_1}^{q_1}\cdots{\bar\partial_d}^{q_d}H$, for $0 \leq p_l \leq \alpha_l$, $0 \leq q_l \leq \beta_l$, \linebreak and $(\mathcal K_{i\overline{j}})_{{z_1}^{r_1}\cdots{z_d}^{r_d}{\overline{z}_1}^{s_1}\cdots{\overline{z}_d}^{s_d}}$, for $0 \leq r_l \leq \alpha_l-1$, $0 \leq s_l \leq \beta_l-1$, $l=1,\hdots,d$, $i,j=1,\hdots,d$.
\end{enumerate}
\end{lem}
\begin{proof}
Let $E$, $S$ and $H$ be as above. Then, for $j=1,\hdots,m$, we have
\begin{eqnarray}\label{dm} \partial_{z_j}H^{-1}=-H^{-1}\cdot\partial_{z_j}H\cdot H^{-1}~\text{ and }~
\partial_{\overline{z}_j}H^{-1}=-H^{-1}\cdot\partial_{\overline{z}_j}H\cdot H^{-1}.\end{eqnarray}
Now from the definition of curvature we obtain, for $i,j=1,\hdots,d$,
$$
\mathcal K_{i\overline{j}} = \partial_{\overline{z}_j}(H^{-1}\cdot\partial_{z_i}H) = -H^{-1}\cdot\partial_{\overline{z}_j}H\cdot H^{-1}\cdot\partial_{z_i}H+
H^{-1}\cdot\partial_{\overline{z}_j}\partial_{z_i}H
$$
which also implies that
\begin{eqnarray}\label{ddbm}\partial_{\overline{z}_j}\partial_{z_i}H=H\cdot\mathcal K_{i\overline{j}}+
\partial_{\overline{z}_j}H\cdot H^{-1}\cdot \partial_{z_i}H.\end{eqnarray}
Then repeated application of Leibnitz rule together with the equations in $\eqref{dm}$ provide the desired expression in (i). Further, (ii) can also be obtained as before by using Leibnitz rule and formulas $\eqref{dm}\text{ and }\eqref{ddbm} $ repeatedly.
\end{proof}
\medskip
\subsection{Notations and Conventions}
We finish this introduction with a list of notations and conventions those will be useful through out the paper.
\begin{enumerate}
\item In this article, we are intended to study quotient Hilbert modules obtained from submodules of quasi-free Hilbert modules over ${\mathcal A}(\Omega)$ for a bounded domain $\Omega\in {\mathbb C}^m$. So from now on we assume that our Hilbert modules are quasi-free unless and otherwise stated.
\item\label{Res} Let ${\mathcal H}$ be a Hilbert module over ${\mathcal A}(\Omega)$ consisting of holomorphic functions on $\Omega$ and ${\mathcal H}_0 \subset {\mathcal H} $ be a subspace which is also a Hilbert module over $\Omega$. Assume that ${\mathcal A}(\Omega)$ acts on ${\mathcal H}$ by point wise multiplication and ${\mathcal H}_q$ be the quotient module ${\mathcal H}\ominus{\mathcal H}_0$. Let $U \subset \Omega$ be an open connected subset. Then from the identity theorem for holomorphic functions of several complex variables we have ${\mathcal H} \simeq_{{\mathcal A}(\Omega)} {{\mathcal H}}|_{\text{res}U}$, ${\mathcal H}_0 \simeq_{{\mathcal A}(\Omega)}{{\mathcal H}_0}|_{\text{res}U}$, and hence ${\mathcal H}_q \simeq_{{\mathcal A}(\Omega)}{{\mathcal H}_q}|_{\text{res}U}$ where ${\mathcal H} \simeq_{{\mathcal A}(\Omega)}{{\mathcal H}}|_{\text{res}U}=\{h|_U: h \in {\mathcal H} \}$. Indeed, the restriction map $R:{\mathcal H}\rightarrow{\mathcal H}|_{\text{res}U}$ defined by $f\mapsto f|_U$ is an onto map whose kernel is trivial thanks to the identity theorem and hence the inner product $\langle R(f),R(g)\rangle:=\langle f,g\rangle$ on ${\mathcal H}|_{\text{res}U}$ turns $R$ to a unitary map. Then one can make ${\mathcal H}|_{\text{res}U}$ to a Hilbert module by restricting the module action of ${\mathcal A}(\Omega)$ to the open set $U$ and note that $R$ intertwines the module actions. Thus, ${\mathcal H}$ and ${\mathcal H}|_{\text{res}U}$ are unitarily equivalent as modules and we also have ${\mathcal H}_0 \simeq{{\mathcal H}_0}|_{\text{res}U}$, ${\mathcal H}_q \simeq{{\mathcal H}_q}|_{\text{res}U}$ as modules. We, therefore, may cut down the domain $\Omega$ to a suitable open subset $U$, if necessary, and pretend $U$ to be $\Omega$.
\item Let $1\leq d\leq m$, $k\in \mathbb{N}$, $N= {d+k-1 \choose k-1}-1$, $I_N:=\{0,1,\hdots,N\}$ and $A:=\{\alpha=(\alpha_1,\hdots,\alpha_d)\in (\mathbb{N} \cup \{0\})^d:0\leq |\alpha|\leq k-1\}$ where $|\alpha|=\alpha_1+\cdots+\alpha_d$. Then consider the bijection $\theta: A \rightarrow I_N$ defined by
\begin{eqnarray}\label{bij} \theta(\alpha):=\sum_{j=1}^{d-1}\frac{1}{j!}\left(|\alpha|-\sum_{i=1}^{d-j}\alpha_i\right)_j + \frac{1}{d!}(|\alpha|)_d \end{eqnarray}
where $(z)_t$ is the Pochhammer symbols defined as, for any complex number $z$ and a natural number $t$, $(z)_t=z(z+1)\cdots(z+t-1)$. Then we put an order on $A$ by pulling back the usual order on $I_N$ via the bijection $\theta$, that is, $$(\alpha_1,\hdots,\alpha_d)\leq(\alpha'_1,\hdots,\alpha'_d)\text{ if and only if }\theta(\alpha_1,\hdots,\alpha_d)\leq\theta(\alpha'_1,\hdots,\alpha'_d).$$
Note that the order induced by $\theta$ is nothing but the graded colexicographic ordering on $A$. Here we also point out that $A=\cup_{t=0}^{k-1}A_t$ where $A_t:=\{\alpha \in A:|\alpha|=t\}$. Therefore, one can have a natural bijection between $A_t$ and $I_{N_t}$ where $N_t$ is the cardinality of the set $A_t$, namely, \begin{eqnarray}\label{bijs}\theta_t:=\theta|{A_t}:A_t \rightarrow \theta(A_t).\end{eqnarray} These new set of bijections will be useful in the next section.
\item From now on, for $\alpha \in A$ and $\theta(\alpha_1,\hdots,\alpha_d)=l$, we use following notations
\begin{eqnarray}\label{not}
\partial^l~(\text{respectively, }\bar\partial^l) &:=& \partial^{\alpha}~(\text{respectively, }\bar\partial^{\alpha})\\
\nonumber &:=& \frac{\partial^{|\alpha|}}{\partial{z_1}^{\alpha_1}\cdots\partial{z_d}^{\alpha_d}}~\left(\text{respectively, }\frac{\bar\partial^{|\alpha|}}{\bar\partial{z_1}^{\alpha_1}\cdots\bar\partial{z_d}^{\alpha_d}}\right)
\end{eqnarray}
unless and otherwise stated, where $\partial_i=\frac{\partial}{\partial z_i}$, $i=1,\hdots,d$. In this context, note that since $\theta$ is a bijection there exists unique $(\alpha_1,\hdots,\alpha_d)\in A$ for every $l \in I_N$, and we are denoting $\partial^{\theta^{-1}(l)}$ as $\partial^l$.
\end{enumerate}
\section{The Submodule ${{\mathscr M}}_0$}\label{SM}
Let ${\mathscr M}$ be a quasi-free Hilbert module of rank $r$ over ${\mathcal A}(\Omega)$ and denote the elements of ${\mathscr M}$ as $h=(h_1,\hdots,h_r)$ where $h_j\in {\mathcal A}(\Omega)$, $1 \leq j \leq r$. In this section, we define the submodule ${\mathscr M}_0$ of ${\mathscr M}$. So we begin by recalling some elementary definitions regarding complex analytic varieties.
\begin{defn}
Let $\Omega \subset {\mathbb C}^m$ be a bounded domain. Then a subset $\mathcal{Z} \subset\Omega$ is called an analytic set if, for any point $p \in \Omega$, there is a connected open {neighbourhood} $U$ of $p$ in $\Omega$ and finitely many holomorphic functions $\phi_1,\hdots,\phi_d$ on $U$ such that $$U\cap\mathcal{Z}=\{q \in U: \phi_j(q)=0,\text{ }1\leq j\leq d\}.$$
\end{defn}
\begin{defn}\label{sav}
An analytic set $\mathcal{Z}\subset \Omega$ is said to be regular of codimension $d$ at $p \in \mathcal{Z}$ if there is an open {neighbourhood} $U_p \subset \Omega$ and holomorphic functions $\phi_1,\hdots,\phi_d$ on $U_p$ such that
\begin{enumerate}
\item [(a)] $\mathcal{Z} \cap U_p = \{q \in \Omega : \phi_1(q)=\cdots=\phi_d(q)=0\}$,
\item [(b)] the rank of the Jacobian matrix of the mapping $q \mapsto (\phi_1(q),\hdots,\phi_d(q))$ at $p$ is $d$.
\end{enumerate}
\end{defn}
An analytic set is said to be irreducible if it can not be decomposed as union of two analytic sets. It is known in literature that any smooth analytic set is irreducible if and only if it is connected with respect to the subspace topology \cite[page 20]{PAG}.
\medskip
Here we point out that such an analytic set $\mathcal{Z}$ is a regular complex submanifold of codimension $d$ in $\Omega$ thanks to following well known fact \cite[page 161]{FHFTCM}.
\begin{prop}\label{defun}
An analytic set $\mathcal{Z}$ is regular of codimension $d$ at $p\in M$ in a complex manifold $M$ of dimension $m$ if and only if there is a complex coordinate chart $(U,\phi)$ of $M$ such that $B:=\phi(U)$ is an open subset of ${\mathbb C}^m$ with $\phi(p)=0$ and $\phi(U\cap\mathcal{Z})=\{\lambda=(\lambda_1,...,\lambda_m) \in B:\lambda_1=\cdots=\lambda_d=0\}$.
\end{prop}
\begin{rem}\label{defc}
In this article, we are interested in smooth irreducible analytic sets $\mathcal{Z}$ of codimension $d$ in some bounded domain $\Omega$ in ${\mathbb C}^m$. So from the Definition $\ref{sav}$ and the Proposition $\ref{defun}$ we have, for each point $p \in \mathcal{Z}$, there is a coordinate chart $(U,\phi)$ at $p$ of $\Omega$ satisfying following properties:
\begin{enumerate}
\item [(a)] $\phi(p)=0$ with $\phi(U\cap\mathcal{Z})=\{\lambda=(\lambda_1,...,\lambda_m) \in B:\lambda_1=\cdots=\lambda_d=0\}$,
\item [(b)] the rank of the Jacobian matrix of the mapping $q \mapsto (\phi_1(q),\hdots,\phi_d(q))$ at $p$ is $d$.
\end{enumerate}
\end{rem}
We are now about to define the order of vanishing of a holomorphic function along a smooth analytic set. Our definition is essentially a direct generalization of the definition given in \cite{OQMAM} to define the order of vanishing of a holomorphic function along a smooth complex hypersurface.
\smallskip
\begin{defn}
Let $\Omega$ and $\mathcal{Z}$ be as above and $f:\Omega\rightarrow{\mathbb C}$ be a holomorphic function. Then $f$ is said to have zero of order $k$ at some point $p\in \mathcal{Z}$ if there exists a coordinate chart $(U,\phi)$ at $p$ of $\Omega$ satisfying the properties (a) and (b) in the Remark $\ref{defc}$ such that \begin{eqnarray}\label{ordv}[f]\in I_{\mathcal{Z}}^{k-1}\,\,\,\text{but}\,\,\,[f] \notin I_{\mathcal{Z}}^k\end{eqnarray} where $[f]$ is the germ of $f$ at $p$ and $I_{\mathcal{Z}}$ is the ideal in $\mathcal{O}_{m,p}$ generated by $[\phi_1],\hdots,[\phi_d]$.
\end{defn}
\begin{rem}\label{coind}
Note that the above defintion is independent of the choice of coordinate chart at $p$. Indeed, for two such charts $(U_1,\phi_1)$ and $(U_2,\phi_2)$ with the properties listed in Remark $\ref{defc}$, $\phi_1$ and $\phi_2\circ\phi_1^{-1}$, respectively, induce isomorphisms $\Phi_1:\mathcal{O}_{m,p}\rightarrow\mathcal{O}_{m,0}$ defined by $\Phi_1([g])=[g\circ\phi^{-1}]$ and $\Phi:\mathcal{O}_{m,0}\rightarrow\mathcal{O}_{m,0}$ with $\Phi([g\circ\phi_1^{-1}])=[g\circ\phi_2^{-1}]$. As a consequence, it turns out that $f$ satisfies $\eqref{ordv}$ if and only if $[f\circ\phi_1^{-1}]\in I_1^{k-1}$ but $[f\circ\phi_1^{-1}]\notin I_1^{k}$ which is again equivalent to the fact that $[f\circ\phi_2^{-1}]\in I_2^{k-1}$ but $[f\circ\phi_2^{-1}]\notin I_2^{k}$ where $I_j$ is the ideal generated by the germs $[\lambda^j_1],\hdots,[\lambda^j_d]$, $j=1,2$, with local coordinates $\lambda^j_1,\hdots,\lambda^j_m$ of ${\mathbb C}^m$ corresponding to $\phi_j$.
\end{rem}
\begin{defn}
Let ${\mathscr M}$ be a quasi-free Hilbert module of rank $r$ over ${\mathcal A}(\Omega)$. Then the submodule ${{\mathscr M}}_0$ is defined as
$${{\mathscr M}}_0:=\{h \in {\mathscr M}: h_j\,\text{ has zero of order }k\text{ at every }q \in \mathcal{Z},\,\,1\leq j\leq r\}.$$
\end{defn}
\begin{lem}\label{lem}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$, $\mathcal{Z}$ be a complex submanifold in $\Omega$ and $f: \Omega \rightarrow {\mathbb C}$ be a holomorphic function. Then, for each point $p \in \mathcal{Z}$, $f$ vanishes to order $k$ at $p$ along $\mathcal{Z}$ if and only if $k$\def\mathcal K{{\mathscr K}} is the largest integer such that
$$ \partial^{\alpha}_{\lambda}(f \circ {\phi}^{-1})|_{\phi(U \cap \mathcal{Z})} := \frac{\partial^{|\alpha|}}{\partial{\lambda_1}^{\alpha_1}\cdots\partial{\lambda_d}^{\alpha_d}}(f \circ {\phi}^{-1})|_{\phi(U \cap \mathcal{Z})}=0\,\,\,\text{for}\,\,\,0 \leq |\alpha| \leq k-1,$$
where $\alpha=(\alpha_1,\hdots,\alpha_d)$ and $|\alpha|=\alpha_1+\cdots+\alpha_d$, for some coordinate chart $(U,\phi)$ as in the Remark $\ref{defc}$.
\end{lem}
In general, there are no global defining functions $\phi_1,\hdots,\phi_d$ for a smooth irreducible analytic set $\mathcal{Z}$. But since it has been shown at the end of the Section $\ref{prelim}$ that the modules and the submodules of interest can be localized we can work with a small enough open set $U \subset \Omega$ intersecting $\mathcal{Z}$. So from now on we consider a fixed {neighbourhood} $U \subset \Omega$ of $p$ with $U \cap \mathcal{Z} \neq \varnothing$ and defining functions $\phi_1,\hdots,\phi_d$ satisfying conditions (a) and (b) in Remark $\ref{defc}$. Since the Jacobian matrix of the mapping $z \mapsto (\phi_1(z),\hdots,\phi_d(z))$ has rank $d$ at $p$, by rearranging the coordinates in ${\mathbb C}^m$, we can assume that $\mathcal{D}_1(p):=(({\partial}_j{\phi}_i|_{p}))^d_{i,j=1}$ is invertible. Then it is easily seen that $\mathcal{D}_1(z)$ is invertible on some {neighbourhood} of $p$ in $U$. Abusing the notation, let us denote this {neighbourhood} by the same letter $U$. Now we consider the mapping $\phi:U \rightarrow \phi(U)$ defined as $\phi(z)=(\phi_1(z),\hdots,\phi_d(z),z_{d+1},\hdots,z_m)$ and note that $\phi$ is a biholomorphism from $U$ onto $\phi(U)$ with $\phi(p)=0$ and $\phi(U\cap\mathcal{Z})=\{\lambda=(\lambda_1,...,\lambda_m) \in \phi(U):\lambda_1=\cdots=\lambda_d=0\}$. Thus, once we fix a chart as above and pretend that $U=\Omega$, the submodule ${\mathscr M}_0$ may be described as $${\mathscr M}_0 = \left\{h \in {\mathscr M}: \frac{\partial^{|\alpha|}}{\partial{\lambda_1}^{\alpha_1}\cdots\partial{\lambda_d}^{\alpha_d}}(h_j \circ {\phi}^{-1})(\lambda)|_{\phi(\mathcal{Z})}=0\,\,\,\text{for}\,\,\,0 \leq |\alpha| \leq k-1,\,\,1\leq j \leq r\right\}.$$
\smallskip
At this stage, we introduce a definition which separates out the coordinate chart described above and will be useful through out this article.
\begin{defn}\label{ad}
Let $\Omega$ be a domain in ${\mathbb C}^m$ and $\mathcal{Z} \subset \Omega$ be a complex submanifold of codimension $d$. Then, for any point $p \in \mathcal{Z}$, we call a coordinate chart $(U,\phi)$ of $\mathcal{Z}$ around $p$ an $\mathit{admissible}$ coordinate chart if the biholomorphism $\phi:U \rightarrow \phi(U)$ takes the form $\phi(z)=(\phi_1(z),\hdots,\phi_d(z),z_{d+1},\hdots,z_m)$ with $\phi(p)=0$ and $\phi(U\cap\mathcal{Z})=\{\lambda=(\lambda_1,...,\lambda_m) \in \phi(U):\lambda_1=\cdots=\lambda_d=0\}$ for some holomorphic functions $\phi_1,\hdots,\phi_d$ on $U$.
\end{defn}
Now we should note that even in this local description of the submodule there is a choice of normal directions to the submanifold $\mathcal{Z}$ involved. The following proposition ensures that in this local picture two different sets of normal directions to $\mathcal{Z}$ give rise to equivalent submodules. At this point, let us recall some elementary definitions and properties of the ring of polynomial functions on a finite dimensional complex vector space which will be useful in the course of the proof of following proposition.
\smallskip
For any complex vector space $V$ of dimension $d$, we denote by ${\mathbb C}[V]$ the ring of polynomial functions on $V$. Let us recall that $f:V\rightarrow {\mathbb C}$ is an element of ${\mathbb C}[V]$ means that, for any basis $\{e_1,\hdots,e_d\}$ of $V$, there exists some polynomial $\phi \in {\mathbb C}[x_1,\hdots,x_d]$ such that $f(\alpha_1 e_1+\cdots+\alpha_d e_d)=\phi(\alpha_1,\hdots,\alpha_d)$ for all $(\alpha_1,\hdots,\alpha_d)\in {\mathbb C}^d$. In other words, $f$ is a polynomial into the elements $x_1=e^*_1,\hdots,x_d=e^*_d$ of the dual basis. It is then clear that
\begin{eqnarray}\label{eqpf} {\mathbb C}[V] \simeq S(V^*) \simeq {\mathbb C}[x_1,\hdots,x_d]\end{eqnarray} where $S(V^*)$ is the graded vector space of all symmetric tensors on $V^*$. Note that ${\mathbb C}[V]$ is an algebra over ${\mathbb C}$.
\smallskip
A polynomial function $f$ on $V$ is said to be homogeneous of degree $t$ if $f(\alpha v)={\alpha}^t f(v)$ for all $\alpha \in {\mathbb C}$ and $v \in V$. We denote ${\mathbb C}[V]_t$ the subspace of ${\mathbb C}[V]$ of homogeneous polynomial functions of degree $t$. In particular, $C[V]_0={\mathbb C}$, ${\mathbb C}[V]_1=V^*$ and ${\mathbb C}[V]_t$ is canonically identified in the first isomorphism in $\eqref{eqpf}$ with the $t$-th symmetric power $S^t(V^*)$, and it can also be identified with the subspace of ${\mathbb C}[x_1,\hdots,x_d]$ generated by the monomials $x_1^{t_1}\cdots x_d^{t_d}$ with $t_1+\cdots+t_d=t$ via the second isomorphism of $\eqref{eqpf}$.
\begin{prop}\label{coch}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$, $\mathcal{Z}$ be a complex submanifold in $\Omega$ and $f: \Omega \rightarrow {\mathbb C}$ be a holomorphic function. Then, for each point $p \in \mathcal{Z}$ there exists an admissible coordinate chart $(U,\phi)$ of $\Omega$ at $p$ such that
$$\partial_\lambda^{\alpha}(f\circ\phi^{-1})(\lambda)|_{\phi(p)}=0,~0 \leq |\alpha| \leq k-1~\text{if and only if}~{\partial}^{\alpha}f(z)|_{p}=0,~0 \leq |\alpha| \leq k-1$$
where $\alpha=(\alpha_1,\hdots,\alpha_d) \in (\mathbb{N} \cup \{0\})^d$, $\lambda=(\lambda_1,\hdots,\lambda_m)$ denotes the standard coordinates on $\phi(U)\subset {\mathbb C}^m$, and $\partial^{\alpha}$ denotes the differential operator $\frac{\partial^{|\alpha|}}{\partial{z_1}^{\alpha_1}\cdots\partial{z_d}^{\alpha_d}}$.
\end{prop}
\begin{proof}
Let us consider an admissible coordinate system $(U,\phi)$ (Definition $\ref{ad}$) at $p \in \mathcal{Z} \subset \Omega$ of $\Omega$, that is, $\phi:U \rightarrow \phi(U)$ defined as $\phi(z)=(\phi_1(z),\hdots,\phi_d(z),z_{d+1},\hdots,z_m)$.
\smallskip
For $q \in U$, let $V_q$ and $V_{\phi(q)}$ be tangent spaces at $q$ and $\phi(q)$ to $({\mathbb C}^d\times\{0\})\cap U$ and $({\mathbb C}^d\times\{0\})\cap \phi(U)$, respectively. We denote the standard ordered basis of $V_q$ by ${\mathcal B}_1(q):=\{\frac{\partial}{\partial z_j}|_q \}_{j=1}^d$ and that of $V_{\phi(q)}$ by ${\mathcal B}_1(\phi(q)):=\{\frac{\partial}{\partial \lambda_j}|_{\phi(q)} \}_{j=1}^d$. Then it is easily seen that $\phi$ induces a linear transformation from $V^*_q$ onto $V^*_{\phi(q)}$, namely, $L_1(q):V^*_q \rightarrow V^*_{\phi(q)}$ defined by $$L_1(q)(dz_j)=\sum^d_{i=1}(\partial_j\phi_i(q))d\lambda_i$$ where $\{dz_j\}_{j=1}^d$ and $\{d\lambda_j\}_{j=1}^d$ are dual bases of ${\mathcal B}_1(q)$ and ${\mathcal B}_1(\phi(q))$, respectively.
\smallskip
Now we consider the ring of polynomial functions ${\mathbb C}[V_q]$ and ${\mathbb C}[V_{\phi(q)}]$ on $V_q$ and $V_{\phi(q)}$, respectively, and observe, in view of the first isomorphism in $\eqref{eqpf}$, that $L_1(q)$ canonically induces linear mappings $L_t(q):S^t(V^*_q)\rightarrow S^t(V^*_{\phi(q)}) $ defined by $$L_t(q)(dz_1^{\alpha_1}\otimes\cdots\otimes dz_d^{\alpha_d})=L_1(q)(dz_1)^{\alpha_1}\otimes\cdots\otimes L_1(q)(dz_d)^{\alpha_d}$$ where $\alpha=(\alpha_1,\hdots,\alpha_d)\in (\mathbb{N}\cup\{0\})^d$ with $|\alpha|=\alpha_1+\cdots+\alpha_d=t$ and by $dz_j^{\alpha_j}$ (respectively, by $L_1(q)(dz_j)^{\alpha_j}$) we mean that the $\alpha_j$-th symmetric power of $dz_j$ (respectively, $L_1(q)(dz_j)$).
\smallskip
Let ${\mathcal B}_t(q):=\{dz_1^{\alpha_1}\otimes\cdots\otimes dz_d^{\alpha_d}:|\alpha|=t\}$ and ${\mathcal B}_t(\phi(q)):=\{d\lambda_1^{\alpha_1}\otimes\cdots\otimes d\lambda_d^{\alpha_d}:|\alpha|=t\}$ be bases for vector spaces $S^t(V^*_q)$ and $S^t(V^*_{\phi(q)})$, respectively, and make them ordered bases with respect to the order induced by the bijection $\theta_t$ $\eqref{bijs}$. We denote the matrix of $L_t(q)$ represented with respect to the basis ${\mathcal B}_t(q)$ and ${\mathcal B}_t(\phi(q))$ as $\mathcal{D}_t(q)$, for $t\in \mathbb{N} \cup \{0\}$. Note that since $L_t(p)$ is a vector space isomorphism for each $t\in \mathbb{N} \cup \{0\}$ the matrices $\mathcal{D}_t(p)$'s are invertible.
\pagebreak
In this set up we claim, for $z \in U$ with $\phi(z)=\lambda\in\phi(U)$, that
\begin{eqnarray}\label{cvf}
A_{k,\phi}(z)\cdot
\left(
\begin{array}{c}
f \circ \phi^{-1}(\lambda)\\
\partial^1_{\lambda}f \circ \phi^{-1}(\lambda)\\
\vdots\\
\partial^N_{\lambda}f \circ \phi^{-1}(\lambda)\\
\end{array}
\right)
&=&
\left(
\begin{array}{c}
f(z)\\
\partial^1 f(z)\\
\vdots\\
{\partial^N}f(z)\\
\end{array}
\right)
\end{eqnarray}
where $\partial^t_{\lambda}$ stands for the differential operator $\frac{\partial^{|\alpha|}}{\partial\lambda_1^{\alpha_1}\cdots\partial\lambda_d^{\alpha_d}}$ with $(\alpha_1,\hdots,\alpha_d)=\theta^{-1}(t)$, $A_{k,\phi}(z)$ is the block lower triangular matrix with $1$, $\mathcal{D}_1(z)$, $\hdots$, $\mathcal{D}_{k-1}(z)$ as the diagonal blocks and $N={d+k-1 \choose k-1}-1$.
\smallskip
We prove this claim with the help of mathematical induction on $k$. Here we note that the base case is the direct consequence of change of variables formula. So let the equation $\eqref{cvf}$ hold true for $t=l$, $1 \leq l \leq k-1$ and we need to prove that $\eqref{cvf}$ holds for $t=l+1$. Now, for $\alpha=(\alpha_1,\hdots,\alpha_d)$ with $|\alpha|=l$ and $\theta_l(\alpha_1,\hdots,\alpha_d)=i$, the induction hypothesis yields that
\begin{eqnarray*}
\partial^{\alpha}f(z) &=& \sum_{|\beta|=l}(\mathcal{D}_l(z))_{i\theta_l(\beta)}\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\lambda)+
\text{ other terms}
\end{eqnarray*}
where $\mathcal{D}_l(z)$ is the matrix $(((\mathcal{D}_l(z))_{\theta_l(\alpha)\theta_l(\beta)}))_{|\alpha|=l,|\beta|=l}$. Therefore, differentiating both sides of the above equation with respect to the $z_j$-th coordinate and using the Leibnitz rule we have, for an arbitrary but fixed point $q \in U$,
\begin{eqnarray*}
\partial_j\partial^{\alpha}f(z)|_{z=q} &=& \sum_{|\beta|=l}\partial_j(\mathcal{D}_l(z))_{i\theta_l(\beta)}|_{z=q}\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\\ &+& \sum_{|\beta|=l}(\mathcal{D}_l(q))_{i\theta_l(\beta)}\left(\sum_{s=1}^d \partial_j\phi_s(q)\partial_{\lambda_s}\right)\partial_{\lambda}^{\beta}
f\circ\phi^{-1}(\lambda)|_{\lambda=\phi(q)} + \partial_j(\text{ other terms})\\
&=& \sum_{|\beta|=l}(\mathcal{D}_l(q))_{i\theta_l(\beta)}\partial_{\lambda}^{\beta}\left(\sum_{s=1}^d \partial_j\phi_s(q)\partial_{\lambda_s}f\circ\phi^{-1}(\lambda)\right)\big|_{\lambda=\phi(q)}\\ &+& (\text{ other terms involving }\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\text{ with }|\alpha|\leq l)
\end{eqnarray*}
Let us now note that the rings of polynomial functions $S(V^*_q)$ and $S(V^*_{\phi(q)})$ can be canonically identified with the algebras of linear partial differential operators with constant coefficients, namely, $\Gamma_q:S(V^*_q)\simeq_{{\mathbb C}}\{\sum_{\alpha}a_{\alpha}\partial_1^{\alpha_1}\cdots\partial_d^{\alpha_d}:a_{\alpha}\in {\mathbb C}\}$ under the correspondence $dz_1^{\alpha_1}\otimes\cdots\otimes dz_d^{\alpha_d}\overset{\Gamma_q}{\mapsto} \partial^{\alpha_1}_1\cdots\partial^{\alpha_d}_d$ and similarly, $\Gamma_{\phi(q)}:S(V^*_{\phi(q)})\simeq_{{\mathbb C}}\{\sum_{\alpha}a_{\alpha}\partial_{\lambda_1}^{\alpha_1}\cdots\partial_{\lambda_d}^{\alpha_d}:a_{\alpha}\in {\mathbb C}\}$ via the mapping $d\lambda_1^{\alpha_1}\otimes\cdots\otimes d\lambda_d^{\alpha_d}\overset{\Gamma_{\phi(q)}}{\longmapsto} \partial^{\alpha_1}_{\lambda_1}\cdots\partial^{\alpha_d}_{\lambda_d}$. Then with respect to the above identification we have
\begin{eqnarray*}
\partial^{\alpha+\varepsilon_j}f(q) &=& \sum_{|\beta|=l}(\mathcal{D}_l(q))_{i\theta_l(\beta)}\partial_{\lambda}^{\beta}\left(\sum_{s=1}^d \partial_j\phi_s(q)\partial_{\lambda_s}f\circ\phi^{-1}(\lambda)\right)\big|_{\lambda=\phi(q)}\\ &+& (\text{ other terms involving }\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\text{ with }|\alpha|\leq l)\\
&=&(\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_1))^{\alpha_1}\cdots
(\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_j))^{\alpha_j}\cdots
(\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_d))^{\alpha_d}\\ && (\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_j))f\circ\phi^{-1}(\lambda)\big|_{\lambda=\phi(q)}\\ &+& (\text{ other terms involving }\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\text{ with }|\alpha|\leq l)\\
&=& (\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_1))^{\alpha_1}\cdots
(\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_j))^{\alpha_j+1}\cdots
(\Gamma_{\phi(q)}L_1(q)\Gamma_q^{-1}(\partial_d))^{\alpha_d}\\ && f\circ\phi^{-1}(\lambda)\big|_{\lambda=\phi(q)}+(\text{ other terms involving }\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\text{ with }|\alpha|\leq l)\\
&=& \sum_{|\beta|=l+1}(\mathcal{D}_{l+1}(q))_{\theta(\alpha+\varepsilon_j)\theta_l(\beta)}\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\\ &+& (\text{ other terms involving }\partial_{\lambda}^{\beta}f\circ\phi^{-1}(\phi(q))\text{ with }|\alpha|\leq l).\\
\end{eqnarray*}
Since $q$ was chosen to be arbitrary in $U$ we are done with the claim. Thus, $A_{k,\phi}(z)$ is invertible if and only if $\mathcal{D}_1(z)$, $\hdots$, $\mathcal{D}_{k-1}(z)$ are simultaneously invertible which is the case for $z \in U$. Hence it completes the proof.
\end{proof}
Thus, from the above proposition and Remark $\ref{coind}$ we have another characterization of the submodule ${{\mathscr M}}_0$ as follows:
$${{\mathscr M}}_0=\{h \in {\mathscr M}:{\partial_1}^{\alpha_1}\cdots{\partial_d}^{\alpha_d}(h_j)|_{\mathcal{Z}}=0,\,0 \leq |\alpha| \leq k-1,\,\,1 \leq j \leq r\}.$$
\begin{rem}\label{remres}
Let $U$ be an open subset of $\Omega$ such that $U \cap \mathcal{Z}$ is non-empty. Then recall that the restriction map $R:{\mathscr M}\rightarrow{\mathscr M}|_{\text{res}U}$ defined by $f\mapsto f|_U$ is an unitary map with respect to the prescribed inner product on ${\mathscr M}|_{\text{res}U}$ ($\eqref{Res}$ in Section $\ref{prelim}$). Moreover, it is now clear from the definition of ${\mathscr M}_0$ that $R({\mathscr M}_0)$ and $R({\mathscr M})_0$ are unitarily equivalent where $R({\mathscr M})_0:=\{f \in R({\mathscr M}):f\text{ vanishes along }U\cap \mathcal{Z}\text{ to order }k\}$. Consequently, $R({\mathscr M}_q)$ and $R({\mathscr M})_q(:=R({\mathscr M})\ominus R({\mathscr M})_0)$ are also unitarily equivalent Hilbert modules. So restricting ourselves to an admissible coordinate chart $(U,\phi)$ around some point $p \in \mathcal{Z} \subset \Omega$, it is enough to study these modules with respect to the new coordinate system obtained by $\phi$. We now elaborate upon this fact.
\end{rem}
\smallskip
So let us consider the module, $\phi^*({\mathscr M}|_{\text{res}U})$ which is, by definition, $$\phi^*({\mathscr M}|_{\text{res}U}):=\{f|_{U}\circ\phi^{-1}:f \in {\mathscr M}\}$$ and note that it is a module over ${\mathcal A}(\Omega)$ with the module action $g\cdot(f|_{U}\circ\phi^{-1}):=(gf)|_{U}\circ\phi^{-1}$, for $g \in {\mathcal A}(\Omega)$. Then it is evident that the modules $\phi^*({\mathscr M}|_{\text{res}U})$ and ${\mathscr M}$ are isomorphic via the isomorphism $\Phi:{\mathscr M} \rightarrow \phi^*({\mathscr M}|_{\text{res}U})$ defined by $f \mapsto f|_{U}\circ\phi^{-1}$. So, defining an inner product as $$\<f|_{U}\circ\phi^{-1},g|_{U} \circ\phi^{-1}\>_{\phi^*({\mathscr M}|_{\text{res}U})}:=\<f,g\>_{{\mathscr M}}$$ we see that $\phi^*({\mathscr M}|_{\text{res}U})$ is unitarily equivalent to ${\mathscr M}$ as Hilbert modules. Since ${\mathscr M}$ is a reproducing kernel Hilbert module with a reproducing kernel, say, $K$ so is $\phi^*({\mathscr M}|_{\text{res}U})$ with the kernel function $K'$ defined by $K'(u,v)=K(\phi^{-1}(u),\phi^{-1}(v))$, for $u,v \in \phi(U)$. It is also easily seen that the multiplication operators $M_{z_1},\hdots,M_{z_m}$ on ${\mathscr M}$ are simultaneously unitarily equivalent to $M_{u_1},\hdots,M_{u_m}$ on $\phi^*({\mathscr M}|_{\text{res}U})$. Indeed, for $\phi^{-1}=(\psi_1,\hdots,\psi_m)$, we note that $\psi_i(\phi(z_1,\hdots,z_m))=z_i$, $i=1,\hdots,m$ and therefore, we have, for $i=1,\hdots,m$ and $f \in {\mathscr M}$,
\begin{eqnarray*}
\Phi^{-1}M_{u_i}\Phi(f) &=& \Phi^{-1}(M_{u_i}(f|_{U}\circ\phi^{-1}))\\
&=& \Phi^{-1}((u_i\circ\phi^{-1})\cdot(f|_{U}\circ\phi^{-1}))\\
&=& M_{z_i}f|_{U}.
\end{eqnarray*}
Furthermore, Proposition \ref{coch} together with the Remark \ref{remres} ensure that the submodules ${\mathscr M}_0$ and $\phi^*R({\mathscr M}_0)$ are also unitarily equivalent via the same map as mentioned earlier. As a consequence along with the help of the Remark \ref{remres} we have the following Proposition.
\begin{prop}\label{eococh}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$, $\mathcal{Z}$ be a complex connected submanifold in $\Omega$ and ${\mathscr M}^1$, ${\mathscr M}^2$ be two quasi-free Hilbert modules of rank $r$ over ${\mathcal A}(\Omega)$. Let ${\mathscr M}^1_0$ and ${\mathscr M}^2_0$ be submodules of ${\mathscr M}^1$ and ${\mathscr M}^2$, respectively, consisting of holomorphic functions vanishing of order $k-1$ along $\mathcal{Z}$. Assume that $(U,\phi)$ is an admissible coordinate system around some point $p \in \mathcal{Z}$. Then ${\mathscr M}^1_0$ is untarily equivalent to ${\mathscr M}^2_0$ as Hilbert modules if and only if $\phi^*({\mathscr M}^1_0|_{\text{res}U})$ is unitarily equivalent to $\phi^*({\mathscr M}^2_0|_{\text{res}U})$. In other words, the following diagram commutes.
$$\begin{CD}
{\mathscr M}^1_0 @>R>> {\mathscr M}^1_0|_{\text{res}U} @>\Phi>> \phi^*({\mathscr M}^1_0|_{\text{res}U})\\
@VVV @VVV @VVV\\
{\mathscr M}^2_0 @>R>> {\mathscr M}^2_0|_{\text{res}U} @>\Phi>> \phi^*({\mathscr M}^2_0|_{\text{res}U})
\end{CD}$$
\end{prop}
\smallskip
In this article, we are interested in studying equivalence classes of quotient modules obtained from the aforementioned submodules of a quasi-free Hilbert modules. In the next section, we define the quotient modules which are the central object of this paper.
\section{Quotient Module ${{\mathscr M}}_q$}\label{QM}
We start with a quasi-free Hilbert module ${\mathscr M}$ over ${\mathcal A}(\Omega)$ of rank $r \geq 1$ and the submodule ${{\mathscr M}}_0 \subset {\mathscr M}$ consisting of ${\mathbb C}^r$-valued holomorphic functions on $\Omega$ vanishing to order $k$ along an irreducible smooth complex analytic set $\mathcal{Z} \subset \Omega$ of codimension $d$, $d\geq 2$. In this setting we are interested in studying the quotient module $${{\mathscr M}}_q:={\mathscr M}/{{\mathscr M}}_0={\mathscr M} \ominus {{\mathscr M}}_0,$$ in other words, we have following exact sequence
\begin{eqnarray}\label{exact} 0 \rightarrow {\mathscr M}_0 \overset{i}{\rightarrow} {\mathscr M} \overset{P}{\rightarrow} {\mathscr M}_q \rightarrow 0\end{eqnarray} where $i$ is the inclusion map and $P$ is the quotient map. Now, for $f \in {\mathcal A}(\Omega)$ and $h \in {\mathscr M}$, we define the module action on the quotient module ${\mathscr M}_q$ as
\begin{eqnarray} fP(h)=P(fh),\end{eqnarray} here we mean $(fh_1,\hdots,fh_r)$ by $fh$.
\medskip
In order to study quotient modules we first describe the jet construction relative to the submanifold $\mathcal{Z}$ following \cite{OQMAM}. Suppose that ${\mathcal H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions on $\Omega$ taking values in ${\mathbb C}^r$ with a reproducing kernel $K$. Let $N = {d+k-1 \choose k-1}-1$, $\{equivalent_l\}^N_{l=0}$ be the standard ordered basis of ${\mathbb C}^{N+1}$, $\{\sigma_i\}_{i=1}^r$ be the standard ordered basis of ${\mathbb C}^r$ and recall that $\partial_1,...,\partial_d$ are the partial derivative operators with respect to $z_1,...,z_d$ variables, respectively. For $h \in {\mathcal H}$, recalling the notations introduced $\eqref{not}$ let us define
$$ \textbf{h}:=\sum_{i=1}^r\left(\sum_{l=0}^{N}\partial^l h_i \otimes equivalent_l\right)\otimes \sigma_i$$ and we consider the space
$J({\mathcal H}):=\{\textbf{h}:h \in {\mathcal H}\} \subset {\mathcal H}\otimes {\mathbb C}^{(N+1)r}$. Consequently, we have the mapping \begin{eqnarray}\label{jcon} J:{\mathcal H} \rightarrow J({\mathcal H}) \text{ defined by } h \mapsto \textbf{h}.\end{eqnarray} Since $J$ is injective we define an inner product on $J({\mathcal H})$ making $J$ to be an unitary transformation as follows $$\<J(h_1),J(h_2)\>_{J({\mathcal H})}:=\<h_1,h_2\>_{{\mathcal H}}.$$
\medskip
Since ${\mathcal H}$ is a reproducing kernel Hilbert space it is natural to expect that $J({\mathcal H})$ is also a reproducing kernel Hilbert space . So we calculate the reproducing kernel of $J({\mathcal H})$.
\begin{prop}
The reproducing kernel $JK:\Omega \times \Omega \rightarrow M_{(N+1)r}({\mathbb C})$ for the Hilbert space $J({\mathcal H})$ is given by the formula
\begin{eqnarray}\label{matJK}
(JK)^{kl}_{ij}(z,w)=\partial^k\bar\partial^l K_{ij}(z,w)\,\,\,\text{for}\,\,\,0\leq l,k\leq N,\,\,1\leq i,j \leq r.
\end{eqnarray}
\end{prop}
\begin{proof}
Let $\{e_n\}_{n\geq 1}$ be an orthonormal basis of ${\mathcal H}$ with $e_n=\sum_{i=1}^r e^i_n\otimes \sigma_i$ where for every $n$ $e^i_n$ is a holomorphic function on $\Omega$. Since $J$ is a unitary operator $\{Je_n\}_{n\geq 1}$ is an orthonormal basis for $J({\mathcal H})$. Thus, we have
$$JK(z,w)=\sum_{n=1}^{\infty}Je_n(z)(Je_n(w))^*$$ where $Je_n(w)^*:{\mathbb C}^{(N+1)r} \rightarrow {\mathbb C} $ is defined by $(Je_n(w))^*(\zeta):=\<\zeta,Je_n(w)\>_{{\mathbb C}^{(N+1)r}}$ and $Je_n(z):{\mathbb C} \rightarrow {\mathbb C}^{(N+1)r}$ is defined as $x \mapsto x\cdot Je_n(z)$. Then note that, for $w \in \Omega$, $Je_n(w)^*(equivalent_l\otimes\sigma_j)=\bar\partial^l\overline{e^j_n(w)}$ and hence we can write \begin{eqnarray}\label{JK}JK(z,w)equivalent_l\otimes\sigma_j=\sum_{n=1}^{\infty}Je_n(z)\bar\partial^l\overline{e^j_n(w)},~~ 0 \leq l \leq N, 1 \leq j \leq r.\end{eqnarray} Therefore, equation $\eqref{JK}$ together with the calculation below imply the identity in $\eqref{matJK}$. $$\<\sum_{n=1}^{\infty}Je_n(z)\bar\partial^l\overline{e^j_n(w)},equivalent_k\otimes\sigma_i\>_{{\mathbb C}^{(N+1)r}}=\sum_{n=1}^{\infty}\bar\partial^l\overline{e^j_n(w)}\partial^k e^i_n(z)=\partial^k\bar\partial^l K_{ij}(z,w).$$
Since $\textbf{h}$ can uniquely be expressed as $\textbf{h}=\sum_{n=1}^{\infty}a_n Je_n$ with $\{a_n\}_{n=1}^{\infty}\subset{\mathbb C}$, using $\eqref{JK}$, we have
$$ \<\textbf{h},JK(.,w)equivalent_l\otimes\sigma_j\>_{J({\mathcal H})} =
\sum_{n=1}^{\infty}a_n\partial^l e^j_n(w) = \partial^l(\sum_{n=1}^{\infty}a_n e^j_n(w))
= \partial^l h_j(w).$$
Here we note that the second equality in above two equations hold due to the fact that the series in first equality converges uniformly on compact subsets of $\Omega$. This completes the proof.
\end{proof}
Thus, we have shown that $J({\mathcal H})$ is a reproducing kernel Hilbert space on $\Omega$. Now we want to make $J({\mathcal H})$ to be a Hilbert module over ${\mathcal A}(\Omega)$. So let us define an action of ${\mathcal A}(\Omega)$ on $J({\mathcal H})$ so that $J$ becomes a module isomorphism. We define, for $f \in {\mathcal A}(\Omega)$ and $\textbf{h} \in J({\mathcal H})$, $J_f: J({\mathcal H}) \rightarrow J({\mathcal H})$ by $J_f(\textbf{h}):=\mathcal{J}(f).\textbf{h}$ where $\mathcal{J}(f)$ is an $(N+1) \times (N+1)$ complex matrix defined as follows
\begin{eqnarray}\label{modac}\mathcal{J}(f)_{lj}:= {\alpha \choose \beta}\partial^{\alpha-\beta}f := {\alpha_1 \choose \beta_1}\cdots{\alpha_d \choose \beta_d}\partial^{\alpha-\beta}f\end{eqnarray} with $\alpha=(\alpha_1,\hdots,\alpha_d)=\theta^{-1}(l)$ and $\beta=(\beta_1,\hdots,\beta_d)=\theta^{-1}(j)$ and $\textbf{h}$ can be thought of an $(N+1)\times r$ matrix with $\textbf{h}_i:= \sum_{l=0}^{N}\partial^l h_i \otimes equivalent_l$, $1 \leq i \leq r$, as column vectors. Note that this is a lower triangular matrix and it takes the following matrix form
\[ \mathcal J(f)=
\left(
\begin{array}{ccccc}
f \\
& \ddots & & \text{\huge0}\\
\vdots & \mathcal{J}(f)_{lj} & \ddots \\
& & & \\
\partial^N f & \hdots & \hdots & & f
\end{array}
\right)
\]
Thus, with the above definition, $J({\mathcal H})$ becomes a Hilbert module over ${\mathcal A}(\Omega)$ and $J$ is a module isomorphism between ${\mathcal H}$ and $J({\mathcal H})$ as it is clear from following simple calculation which is essentially an application of Leibniz rule. For $1 \leq i \leq r$, we have
\begin{eqnarray*}
J(f\cdot h_i) &=& \sum_{l=0}^N\partial^l(f \cdot h_i)\otimesequivalent_l\\
&=& \sum_{l=0}^N\sum_{\beta_1=0}^{\alpha_1}\cdots\sum_{\beta_d=0}^{\alpha_d}\left({\alpha_1 \choose \beta_1}\cdots{\alpha_d \choose \beta_d}\partial_1^{\alpha_1-\beta_1}\cdots\partial_d^{\alpha_d-\beta_d}f\cdot \partial_1^{\beta_1}\cdots\partial_d^{\beta_d}h_i\right)\otimes equivalent_l\\
&=& \mathcal J(f)\cdot \textbf{h}_i\\
\end{eqnarray*}
which shows that $J(f\cdot h)=\mathcal{J}(f)\cdot\textbf{h}$.
\smallskip
Applying the above construction to the Hilbert module ${\mathscr M}$ we have the module of jets $J({\mathscr M})$. As in the case of Hilbert submodule ${\mathscr M}_0$ of the Hilbert module ${\mathscr M}$ it is clear that the subspace $$J({\mathscr M})_0:=\{\textbf{h}\in J({\mathscr M}):\textbf{h}|_{\mathcal{Z}}=0\}$$ is a submodule of $J({\mathscr M})$. Let $J({\mathscr M})_q$ be the quotient module obtained by taking orthogonal complement of $J({\mathscr M})_0$ in $J({\mathscr M})$, that is, $J({\mathscr M})_q:=J({\mathscr M})\ominus J({\mathscr M})_0$. The following theorem provides the equivalence of two quotient modules ${\mathscr M}_q$ and $J({\mathscr M})_q$.
\smallskip
\begin{thm}\label{qejq}
${{\mathscr M}}_q$ and $J({\mathscr M})_q$ are isomorphic as modules over ${\mathcal A}(\Omega)$.
\end{thm}
\begin{proof}
Let us begin with a naturally arising spanning set of ${{\mathscr M}}_q$. Since, for $w \in \Omega$ and $\zeta \in {\mathbb C}^r$, $K(.,w)\zeta \in {\mathscr M} $ it is evident that $\bar\partial^{\alpha}K(.,w)\zeta \in {\mathscr M} $ for $\zeta \in {\mathbb C}^r$ and $\alpha \in A$. Thus, using the reproducing property of $K$, we note that
$\<h,\bar\partial^{\alpha}K(.,w)\zeta\> = \<\partial^{\alpha}h(w),\zeta\>$
where $h(\cdot)=\sum_{n=1}^{\infty}a_ne_n(\cdot)$ is any vector in ${\mathscr M}$ and $\{e_n(\cdot)\}_{n=1}^{\infty}$ is an orthonormal basis of ${\mathscr M}$.
\medskip
Then from the above calculation together with the identity obtained by differentiating both sides of the equation $\eqref{mo}$ we have, for $\zeta \in {\mathbb C}^r$, $w \in \mathcal{Z}$, $\alpha \in A$, and $h \in {{\mathscr M}}_0$, that
$$\<h,\bar\partial^{\alpha}K(.,w)\zeta\>=\<\partial^{\alpha}h(w),\zeta\>=0$$ which in turn implies that $D:=\{\bar\partial^{\alpha}K(.,w)\sigma_i:w \in \mathcal{Z},1\leq i \leq r,\alpha \in A\}$ is contained in ${{{\mathscr M}}_0}^{\perp}$, that is, $(\text{span}D)^{\perp} \subset {{\mathscr M}}_0$ and ${{\mathscr M}}_0 \subset {D}^{\perp} =(\text{span}D)^{\perp}$. Consequently, $D$ is a spanning set for ${{\mathscr M}}_q$.
\smallskip
Now we claim that $J(D)$ spans $J({{\mathscr M}})_q$. In order to establish the claim we follow the same strategy as before, in other words, we first show that $J(D)^{\perp}=J({\mathscr M})_0.$ So let us recall the definition of the operator $J$ and try to understand the set $J(D)$. Note that, by definition of the map $J$, we have
\begin{eqnarray*}
J(D) &=& \{J(\bar\partial^{\alpha}K(.,w)\sigma_i): w \in \mathcal{Z},1\leq i \leq r, \alpha \in A\}\\
&=& \{JK(.,w)\varepsilon_j\otimes\sigma_i: 0 \leq j \leq N, 1\leq i \leq r, w \in \mathcal{Z}\}.\end{eqnarray*}
As before, for $\textbf{h} \in J({\mathscr M})$, $1 \leq i \leq r$, and $w \in \Omega$, from the reproducing property of $JK$ we have
\begin{eqnarray}
\<\textbf{h},JK(.,w)\varepsilon_j\otimes\sigma_i\>_{J({\mathscr M})} = \<\textbf{h}(w),\varepsilon_j\otimes\sigma_i\>_{{\mathbb C}^{(N+1)r}} = \partial^j h_i(w),~ 0 \leq j \leq N,
\end{eqnarray}
which justifies our claim. Hence we conclude that $J(D)$ spans the quotient space $J({\mathscr M})_q$, that is, $$J({\mathscr M}_q) = J(\text{span}D) = \text{span}J(D) = J({\mathscr M})_q.$$
Now in course of completion of our proof it remains to check that $J$ is a module isomorphism from ${{\mathscr M}}_q$ onto $J({\mathscr M})_q$. In other words, we need to verify the following identity
$$J \circ P \circ M_f = (JP)\circ J_f\circ J,$$ for $f \in {\mathcal A}(\Omega)$, which is equivalent to show that $$J{M^*_f}P = {J^*_f}(JP)J$$ where $JP:J({\mathscr M})\rightarrow J({\mathscr M})_q$ is the orthogonal projection operator. Since it amounts to show that $J$ intertwines the module actions on $D$ and both $P$ and $JP$ are identity on $D$ and $J(D)$, respectively, it is enough to prove that $$J{M^*_f} = {J^*_f}J,\,\,\,\text{for}\,\,f \in {\mathcal A}(\Omega), \,\,\,\text{on}\,\,D.$$
\smallskip
Let $\alpha=(\alpha_1,\cdots,\alpha_d) \in A $, $1 \leq i \leq r$ and $\bar\partial^{\alpha}K(.,w)\sigma_i \in D$ (we refer the readers $\eqref{not}$ for the notation $\bar\partial^{\alpha}$). For $f \in {\mathcal A}(\Omega)$, $w \in \mathcal{Z}\subset\Omega$, we have \begin{eqnarray} {M^*_f}K(.,w)\sigma_i=\overline{f(w)}K(.,w)\sigma_i.\end{eqnarray} Then differentiating both sides of the above equation and using induction on degree of the differentiation and adopting the notation introduced in $\eqref{not}$ we obtain
\begin{eqnarray} {M_f}^*\bar\partial^{\alpha}K(.,w)\sigma_i &=& \sum_{\beta_1=0}^{\alpha_1}\cdots\sum_{\beta_d=0}^{\alpha_d}{\bar\partial_1}^{\alpha_1-\beta_1}\cdots{\bar\partial_d}^{\alpha_d-\beta_d}\overline{f(w)}{\bar\partial_1}^{\beta_1}\cdots{\bar\partial_d}^{\beta_d}K(.,w)\sigma_i.\end{eqnarray} Therefore,
\begin{eqnarray*}
J({M_f}^*\bar\partial^{\alpha}K(.,w)\sigma_i) &=& \sum_{l=0}^{N}\partial^l \left[\sum_{\beta_1=0}^{\alpha_1}\cdots\sum_{\beta_d=0}^{\alpha_d}{\bar\partial_1}^{\alpha_1-\beta_1}\cdots{\bar\partial_d}^{\alpha_d-\beta_d}\overline{f(w)}{\bar\partial_1}^{\beta_1}\cdots{\bar\partial_d}^{\beta_d}K(.,w)\right]\otimes \varepsilon_l\otimes \sigma_i\\
&=& \sum_{\beta_1=0}^{\alpha_1}\cdots\sum_{\beta_d=0}^{\alpha_d}{\bar\partial_1}^{\alpha_1-\beta_1}\cdots{\bar\partial_d}^{\alpha_d-\beta_d}\overline{f(w)}JK(.,w)
(\varepsilon_{\theta(\beta)}\otimes\sigma_i),\,\,\,\,\beta=(\beta_1,\hdots,\beta_d)\\
&=& JK(.,w)(\mathcal{J}(f)(w))^*(\varepsilon_{\theta(\alpha)}\otimes\sigma_i).\\
\end{eqnarray*}
Thus, for $\textbf{h} \in J({\mathscr M})$, $\zeta \in {\mathbb C}^{(N+1)r}$ and $w \in \Omega$, we have
$$
\<\textbf{h},{J^*_f}JK(.,w)\cdot \zeta\>_{J({\mathscr M})}
= \<\textbf{h}(w),\mathcal{J}(f)(w)^*\zeta\>_{{\mathbb C}^{(N+1)r}}
= \<\textbf{h},JK(.,w)\mathcal{J}(f)(w)^*\zeta\>_{J({\mathscr M})}.
$$
This completes the roof.
\end{proof}
\medskip
\begin{rem}
Note that as mentioned in \cite{OQMAM} the Theorem $\ref{qejq}$ is equivalent to the fact that the following diagram of exact sequences is commutative.
$$\begin{CD}
0 @>>> {\mathscr M}_0 @>i>> {\mathscr M} @>P>> {\mathscr M}_q @>>> 0\\
@. @VVV @VVV @VVV\\
0 @>>> J({\mathscr M})_0 @>i>> J({\mathscr M}) @>JP>> J({\mathscr M})_q @>>> 0
\end{CD}$$
\end{rem}
\smallskip
In \cite{TRK} it was shown that for a reproducing kernel Hilbert space ${\mathcal H}$ with scalar valued reproducing kernel $K$ on some set $W$, the restriction of $K$ on a subset $W_1$ of $W$ is also a reproducing kernel and restriction of $K$ to $W_1$ constitutes a reproducing kernel Hilbert space which is isomorphic to the quotient space ${\mathcal H} \ominus {\mathcal H}_0$ where ${\mathcal H}_0:=\{f \in {\mathcal H}:f|_{W_1}=0\}$. Here, adopting the proof from \cite{TRK} for our case with vector valued kernel, we have the following theorem. Since this result is well known (Theorem 3.3, \cite{OQMAM}) for the case while the codimension of the submanifold, $\mathcal{Z}$, is one and using the techniques used in that proof in a similar way one can the following theorem, we omit the proof.
\begin{thm}\label{qeres}
The normed linear space $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ is a Hilbert space and the Hilbert spaces $J({\mathscr M})_q$ and $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ are unitarily equivalent. Consequently, the reproducing kernel $K_1$ for $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ is the restriction of the kernel $JK$ to the submanifold $\mathcal{Z}$. Moreover, $J({\mathscr M})_q$ and $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ are isomorphic as modules over ${\mathcal A}(\Omega)$.
\end{thm}
\begin{thm}\label{qm}
The quotient module ${{\mathscr M}}_q$ is equivalent to the module $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ over ${\mathcal A}(\Omega)$.
\end{thm}
\begin{proof}
It is obvious from Theorem $\ref{qejq}$ and Theorem $\ref{qeres}$.
\end{proof}
We now provide a necessary condition for equivalence of two quotient modules in the following theorem.
\begin{thm}\label{eqm}
Let ${\mathscr M}$ and $\tilde{{\mathscr M}}$ be Hilbert modules in $B_1(\Omega)$ and ${\mathscr M}_0$ and $\tilde{{\mathscr M}}_0$ be the submodules of functions in ${\mathcal A}(\Omega)$ vanishing along $\mathcal{Z}$ to order $k$. If ${\mathscr M}$ and $\tilde{{\mathscr M}}$ are equivalent as Hilbert modules over ${\mathcal A}(\Omega)$ then the corresponding quotient modules ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are also equivalent as Hilbert modules over ${\mathcal A}(\Omega)$.
\end{thm}
\begin{proof}
Let us begin with a unitary module map $T: {\mathscr M} \rightarrow \tilde{{\mathscr M}}$. Then, following \cite{GBKCD}, we have that there is a non-vanishing holomorphic function $\psi:\Omega \rightarrow {\mathbb C}$ such that $T=T_{\psi}$ where $T_{\psi}: {\mathscr M} \rightarrow \tilde{{\mathscr M}}$ defined by $T_{\psi}f=\psi f$.
\smallskip
Now recalling the definition $\eqref{jcon}$ of the unitary operator $J$ we note that $T_{\psi}$ gives rise to the module map $J_{\psi}: J({\mathscr M})\rightarrow J(\tilde{{\mathscr M}})$ by the formula $J_{\psi}:=J\circ T_{\psi} \circ J^*$ and $$J_{\psi}(\textbf{h})=J\circ T_{\psi} \circ J^*(\textbf{h})=J(\psi h)=\mathcal J(\psi)\textbf{h}$$ which is actually a unitary module map. Since $\psi$ is non-vanishing the definition $\eqref{modac}$ of $\mathcal J(\psi)$ ensures that $J({\mathscr M})_0$ gets mapped onto $J(\tilde{{\mathscr M}})_0$ by $J_{\psi}$ and hence $J({\mathscr M})_q$ is equivalent to $J(\tilde{{\mathscr M}})_q$. Thus, we are done thanks to Theorem $\ref{qejq}$.
\end{proof}
\begin{rem}\label{rem}
Let us now clarify the module action of ${\mathcal A}(\Omega)$ on the quotient module ${{\mathscr M}}_q$ before proceeding further. To facilitate this action we, following \cite[page 384]{OQMAM}, consider the algebra of holomorphic functions on $\Omega$ taking values in ${\mathbb C}^{N+1}$ with $N={d+k-1\choose k-1}-1$,
$$J{\mathcal A}(\Omega):=\{\mathcal J f:f \in {\mathcal A}(\Omega)\} \subset {\mathcal A}(\Omega) \otimes {\mathbb C}^{(N+1) \times (N+1)}$$ with the multiplication defined by the usual matrix multiplication, namely, $(\mathcal J f\cdot\mathcal J g)(z):=\mathcal J f(z)\mathcal J g(z)$. Then from $\eqref{modac}$ it is clear that $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ is a module over the algebra $J{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$ obtained by restricting $J{\mathcal A}(\Omega)$ to the submanifold $\mathcal{Z}$. Note that $J$ defines an algebra isomorphism from ${\mathcal A}(\Omega)$ onto $J{\mathcal A}(\Omega)$ and intertwines the restriction operators $R_1:{\mathcal A}(\Omega)\rightarrow{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$ and $R_2:J{\mathcal A}(\Omega)\rightarrow J{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$. Consequently, $J:{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}\rightarrow J{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$ is also an algebra isomorphism. So $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ can be thought of as a Hilbert module over $J{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$.
\smallskip
On the other hand, considering the inclusion $i:\mathcal{Z}\rightarrow \Omega$ we see that $i$ induces a map $i^*:J{\mathcal A}(\Omega)\rightarrow J{\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$ defined by $i^*(\mathcal J f)(z)=\mathcal J f(i(z))$, for $z \in \mathcal{Z}$. Now one can make $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ to a module over the algebra $J{\mathcal A}(\Omega)$ by pushing it forward under the map $i^*$, that is,
$$\mathcal J f\cdot\textbf{h}|_{\mathcal{Z}}:=i^*(\mathcal J f)\textbf{h}|_{\mathcal{Z}}.$$ Thus, recalling the fact that $J$ defines an algebra isomorphism between ${\mathcal A}(\Omega)$ and $J{\mathcal A}(\Omega)$, we can think of $J({\mathscr M})_q$ as a module over ${\mathcal A}(\Omega)$.
\end{rem}
Since the similar construction can be done for the Hilbert modules ${\mathscr M} \in B_r(\Omega)$ with submodules ${\mathscr M}_0$ consisting of holomorphic functions ${\mathcal A}(\Omega)$ vanishing along $\mathcal{Z}$ to order $k$, one can also ask whether the quotient modules arising from such submodules are in $B_{(N+1)r}(\mathcal{Z})$. In the following theorem, we give an affirmative answer of this for a simple class of Hilbert modules in $B_1(\Omega)$.
\begin{thm}
Let $\Omega \subset {\mathbb C}^m$
be a bounded domain containing the origin and $\mathsf{Z}\subset \Omega$ be the coordinate plane defined by $\mathsf{Z}:=\{z=(z_1,\hdots,z_m) \in \Omega:z_1=\cdots=z_d=0\}$. We also assume that ${\mathscr M}$ is a reproducing kernel Hilbert space with the property that the reproducing kernel $K$ has diagonal power series expansion, that is, for $z,w \in \Omega$
$$K(z,w)=\sum_{\alpha \geq 0}a_{\alpha}(z-z_0)^{\alpha}\overline{(w-w_0)}^{\alpha},$$
for some $z_0,w_0 \in \mathsf{Z}$. Then the quotient module ${\mathscr M}_q$ restricted to a module over ${\mathcal A}(\mathsf{Z})$ lies in $B_{N+1}(\mathsf{Z})$ provided ${\mathscr M} \in B_1(\Omega)$.
\end{thm}
\begin{proof}
In view of Theorem $\ref{qm}$ and Remark $\ref{rem}$, it is enough to prove that the module of jets, $J({\mathscr M})$ restricted to $\mathsf{Z}$ is in $B_{(N+1)}(\mathsf{Z})$. Let $JK|_{\mathsf{Z}}$ be the reproducing kernel of $J({\mathscr M})|_{\text{res}\mathsf{Z}}$. Then $JK|_{\mathsf{Z}}$ has the following power series expansion at $(z_0,w_0)\in \mathsf{Z}$:
$$JK|_{\mathsf{Z}}(\tilde{z},\tilde{w})=\sum_{\lambda,\mu \geq 0}A_{\lambda\mu}(\tilde{z}-z_0)^{\lambda}\overline{(\tilde{w}-w_0)}^{\mu}$$ where $\tilde{z},\tilde{w} \in \mathsf{Z}$, $\lambda,\mu \in (\mathbb{N} \cup \{0\})^{m-d}$, and $A_{\lambda\mu} \in M_{N+1}({\mathbb C})$ are defined by the following formula
$$A_{\lambda\mu}=\partial_{d+1}^{\lambda_1}\cdots\partial_{m}^{\lambda_{m-d}}\bar\partial_{d+1}^{\mu_1}\cdots\bar\partial_{m}^{\mu_{m-d}}JK|_{\mathsf{Z}}(z_0,w_0).$$
Therefore, using the definition of $JK|_{\mathsf{Z}}$ we get, for $0 \leq l,k \leq N$, that
\begin{eqnarray}\label{eq} (A_{\lambda\mu})_{lk} &=& \partial_{d+1}^{\lambda_1}\cdots\partial_{m}^{\lambda_{m-d}}\bar\partial_{d+1}^{\mu_1}\cdots\bar\partial_{m}^{\mu_{m-d}}\partial_{1}^{\alpha_1}\cdots\partial_{d}^{\alpha_{d}}\bar\partial_{1}^{\beta_1}
\cdots\bar\partial_{d}^{\beta_{d}}K(z_0,w_0) \end{eqnarray}
where $(A_{\lambda\mu})_{lk}$ is the $lk$-th entry of the matrix $A_{\lambda\mu}$ and $\theta^{-1}(l)=(\alpha_1,\hdots,\alpha_d)$, $\theta^{-1}(k)=(\beta_1,\hdots,\beta_d)$.
\smallskip
Since $K$ has a diagonal power series expansion it is clear, from the equation $\eqref{eq}$, that $A_{\lambda\mu}=0$ unless $\lambda=\mu$. Moreover, in a similar way the same equation also shows that $(A_{\lambda\l})_{lk}=0$ if $l \neq k$ and
\begin{eqnarray}\label{eq1} (A_{\lambda\l})_{ll} &=& a_{\alpha_1,\hdots,\alpha_d,\lambda_1,\hdots,\lambda_{m-d}}. \end{eqnarray}
Now from the hypothesis we have that the Taylor coefficients, $a_{\alpha}$ satisfy the inequality stated in part $(b)$ of the Theorem 5.4 in \cite{GBKCD}. As a consequence, a straight forward calculation using the equation $\eqref{eq1}$ shows that the matrices $A_{\lambda\l}$ also satisfy the same inequality in \cite{GBKCD} but with matrix valued constants. Furthermore, it follows, from the equation $\eqref{eq1}$, that the coordinate functions of $\mathsf{Z}$ act on $J({\mathscr M})|_{\text{res}\mathsf{Z}}$ by weighted shift operators with weights determined by matrices $A_{\lambda\l}$. Therefore, by the same theorem in \cite{GBKCD} the Hilbert module, $J({\mathscr M})|_{\text{res}\mathsf{Z}}$, as a module over ${\mathcal A}(\mathsf{Z})$ is in $B_{N+1}(\mathsf{Z})$.
\end{proof}
We note that the above theorem provides examples of quotient modules which are in the Cowen-Douglas class. Thus, it motivates to consider the following class of Hilbert modules.
\begin{defn} Let $\Omega \subset {\mathbb C}^m$ be bounded domain and $\mathcal{Z} \subset \Omega$ be a connected complex submanifold of codimension $d$. Then we say that the pair of Hilbert modules $({\mathscr M},{\mathscr M}_q)$ over the algebra ${\mathcal A}(\Omega)$ is in $B_{r,k}(\Omega,\mathcal{Z})$ if
\begin{enumerate}
\item ${\mathscr M} \in B_r(\Omega)$;
\item there exists a resolution of the module ${\mathscr M}_q$ as in $\eqref{exact}$ where the module ${\mathscr M}$ appearing in the resolution is quasi-free of rank $r$ over the algebra ${\mathcal A}(\Omega)$;
\item for $f \in {\mathcal A}(\Omega)$, the restriction of the map $J_f$ to the submanifold defines the module action on $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ which is an isomorphic copy of ${\mathscr M}_q$; and
\item the quotient module ${\mathscr M}_q$ as a module over the algebra ${\mathcal A}(\Omega)|_{\text{res}\mathcal{Z}}$ is in $B_{(N+1)r}(\mathcal{Z})$ where $N={d+k-1\choose k-1}-1$.
\end{enumerate}
\end{defn}
\section{Jet Bundle}\label{JB}
This section is devoted to provide geometric invariants of quotient modules introduced in the previous section. Suppose, to begin with, we have the Hilbert module ${\mathscr M}$ in $B_r(\Omega)$ with the submodule ${\mathscr M}_0$ and quotient module ${\mathscr M}_q$, as introduced in Section $\ref{QM}$, satisfying the exact sequence $\eqref{exact}$. Then we have, following Remark $\ref{modvec}$, that ${\mathscr M}$ gives rise to a {\bf h} $E$ with the frame $\{K(.,\overline{w})\sigma_1,\hdots,K(.,\overline{w})\sigma_r:w \in \Omega^* \}$ on $\Omega^*$. Now to make calculations simpler let us consider the map $\mathit{c}:\Omega \rightarrow \Omega^*$ defined by $w\mapsto \overline{w}$ and pull back the bundle $E$ to a vector bundle over $\Omega$. Then we denote this new bundle with the same letter $E$ and note that $E$ is a {\bf h} over $\Omega$ with the global holomorphic frame $\textbf{\textit{s}}:=\{s_1(w),\hdots,s_r(w):w \in \Omega \}$ with $s_j(w):=K(.,\overline{w})\sigma_j$, $1 \leq j \leq r$. Correspondingly, we have $\partial^ls_j(w)=\partial^lK(.,\overline{w})\sigma_j$, $1 \leq j \leq r$, $0 \leq l \leq N$, where $N={d+k-1 \choose k-1}-1$.
\smallskip
The jet bundle construction of a line bundle relative to a hypersurface, introduced in the paper \cite{OQMAM}, involves the frame of the line bundle and directional derivatives of the frame in the normal direction to the hypersurface. We have generalized in previous section this notion of jet construction for a Hilbert module relative to a smooth irreducible complex analytic set of arbitrary codimension. In this section, we attempt to describe the same for vector bundles obtained above, that is, the jet bundle construction for such a trivial vector bundle relative to a connected complex submanifold of codimension $d\geq 1$.
\smallskip
We start with a {\bf h} $E$ over $\Omega$ corresponding to the Hilbert module ${\mathscr M} \in B_r(\Omega)$ described above and $\mathcal{Z} \subset \Omega$ is a connected complex submanifold of codimension $d$. Without loss of generality assume that $0 \in \mathcal{Z}$ and let $(U,\phi)$ be an admissible coordinate chart (Definition $\ref{ad}$) at $0$ of $\mathcal{Z}$. So pretending $U$ as $\Omega$ we have that $\phi(\Omega\cap\mathcal{Z})=\{w\in \phi(\Omega):w_1=\cdots=w_d=0\}$. Since we are interested to investigate unitary invariants of the quotient module ${\mathscr M}_q$ with $({\mathscr M},{\mathscr M}_q) \in B_{r,k}(\Omega,\mathcal{Z})$, following the Proposition $\ref{eococh}$, it is enough to consider the submanifold $\phi(\Omega \cap \mathcal{Z}) \subset \phi(\Omega)$. Therefore, pretending $\phi(\Omega)$ as $\Omega$, we consider the submanifold $\mathsf{Z}$ defined as
$$\mathsf{Z} := \{z=(z_1,\cdots,z_m) \in \Omega : z_1=\cdots=z_d=0\}.$$
\smallskip
We then define the jet bundle $J^kE$ of order $k$ of $E$ relative to the submanifold $\mathsf{Z}$ on $\Omega$ by declaring $\{\textbf{\textit{s}},\partial^1\textbf{\textit{s}},\hdots,\partial^N \textbf{\textit{s}}\}$ as a frame for $J^kE$ on $\Omega$ where the differential operators $\partial^j$, $0 \leq j \leq N$ are as introduced in $\eqref{not}$, and by $\partial^l\textbf{\textit{s}}$ we mean the ordered set of sections $\{\partial^ls_1,\hdots,\partial^ls_r\}$, $0 \leq l \leq N$. Since we have a global frame on $J^kE$ we do not need to worry about the transition rule.
\smallskip
At this point, we should note that our construction depends on the choice of the normal direction to $\mathcal{Z}$ which is, a priori, not unique. Nevertheless one way to show that our construction is essentially unambiguous is the following proposition.
\begin{prop}
Let $(U_1,\phi_1)$ and $(U_2,\phi_2)$ be two admissible coordinate charts of $\Omega$ around some point $p \in \mathcal{Z}$. Then two jet bundles $J^k_1E$ and $J^k_2E$ obtained as above with respect to $(U_1,\phi_1)$ and $(U_2,\phi_2)$, respectively, are equivalent holomorphic vector bundles over $U_1\cap U_2$.
\end{prop}
\begin{proof}
In fact, from Proposition $\ref{coch}$ it is clear, for a frame $\textbf{\textit{s}}=\{s_1,\hdots,s_r\}$ of $E$ on $U_1\cap U_2$, that on a small enough {neighbourhood} $U$ of $p$ in $U_1\cap U_2$ we have,for $i=1,2$,
$$A_{k,\phi_i}(z)\cdot
\begin{pmatrix}
s^i_{01}(\lambda) & \cdots & s^i_{0r}(\lambda)\\
s^i_{11}(\lambda) & \cdots & s^i_{1r}(\lambda)\\
\vdots & & \vdots\\
s^i_{N1}(\lambda) & \cdots & s^i_{Nr}(\lambda)\\
\end{pmatrix}=
\begin{pmatrix}
s_1(z) & \cdots & s_r(z)\\
\partial s_1(z) & \cdots & \partial s_r(z)\\
\vdots & & \vdots\\
\partial^N s_1(z) & \cdots & \partial^N s_r(z)\\
\end{pmatrix},\text{ for }z\in U\text{ and }\lambda_i \in \phi_i(U)$$ where $\lambda_i=(\lambda_{i1},\hdots,\lambda_{id})$, $(\alpha_1,\cdots,\alpha_d)=\theta^{-1}(l)$, and $s^i_{lj}=\frac{\partial^{|\alpha|}}{\partial\lambda_{i1}^{\alpha_1}\cdots\partial\lambda_{id}^{\alpha_d}}(s_j\circ\phi_i^{-1})$, for $1 \leq j \leq r$. Since $A_{k,\phi_i}(z)$, for $i=1,2$ and $z\in U$, are invertible (Proposition $\ref{coch}$) we can see that $(A_{k,\phi_1}(z)\circ A_{k,\phi_2}(z)^{-1})\otimes I_r$ is the desired bundle map where $I_r$ is the identity matrix of order $r$.
\end{proof}
\smallskip
Now in course of completing our construction to make the jet bundle $J^kE$ a Hermitian holomorphic vector bundle we need to put a Hermitian metric on $J^kE$ compatible with the metric on $E$. To this extent, if $H(w)=((\langle s_i(w),s_j(w)\rangle_{E}))_{i,j=1}^r$ is the metric on $E$ over $\Omega$ then the Hermitian metric on $J^kE$ with respect to the frame $\{\textbf{\textit{s}},\partial\textbf{\textit{s}},\hdots,\partial^N \textbf{\textit{s}}\}$ is given by the Grammian $JH:=((JH_{lt}))_{l,t=0}^{N}$ with $r\times r$ blocks $$JH_{lt}(w):=((\langle\partial^l s_i(w), \partial^t s_j(w)\rangle))_{i,j=1}^r\,\,\,\text{for}\,\,0 \leq l,t \leq N, w \in \Omega.$$
This completes our construction of the jet bundle.
\begin{rem}\label{rjbi}
Note that, for the Hilbert module ${\mathscr M}$ over ${\mathcal A}(\Omega)$ with the corresponding {\bf h} $E$ over $\Omega$, the {\bf h} $\mathscr{E}$ obtained from $J({\mathscr M})$ is equivalent to the jet bundle $J^kE|_{\text{res}\mathsf{Z}}\rightarrow \mathsf{Z}$ of $E$ relative to $\mathsf{Z}$. To facilitate, let ${\mathscr M}$ be a reproducing kernel Hilbert module over ${\mathcal A}(\Omega)$ which is in $B_r(\Omega)$. Let $K=((K_{ij}))_{i,j=1}^r$ be the reproducing kernel of ${\mathscr M}$. Then from the preceding construction we have that the metric for the jet bundle is given by the formula
$$\langle\partial^l K(.,\overline{w})\sigma_i,\partial^t K(.,\overline{w})\sigma_j\rangle=\partial^l\bar\partial^t K_{ij}(\overline{w},\overline{w})\,\,\,\text{for}\,\,w\in\Omega,~0 \leq l,t \leq N,~1 \leq i,j \leq r.$$
On the other hand, the jet construction presented in Section $\ref{QM}$ gives rise to the Hilbert module $J({\mathscr M})$ where $J$ is the unitary module map $J:{\mathscr M} \rightarrow J({\mathscr M})$. Therefore, the vector bundle $\mathscr{E}$ is unitarily equivalent to $J^kE$.
\smallskip
Note that the action of the algebra ${\mathcal A}(\Omega)$ on the module $J({\mathscr M})$ defines, for every $f \in {\mathcal A}(\Omega)$, a holomorphic bundle map $\Psi_f:J^kE\rightarrow J^kE$ whose matrix representation with respect to the frame $J(\textbf{\textit{s}}):=\{\sum_{l=0}^N\partial^l s^1\otimes equivalent_l,\hdots,\sum_{l=0}^N\partial^l s_r\otimes equivalent_l\}$ is the matrix $\mathcal J(f)\otimes I_r$ where $\mathcal J(f)$ is as in $\eqref{modac}$ and $I_r$ is the identity matrix of order $r$. Thus, $\Psi_f$ induces an action of ${\mathcal A}(\Omega)$ on the holomorphic sections of the jet bundle $J^kE$ defined by \begin{eqnarray}\label{modacsec}f\cdot \sigma(w) := \Psi_f(\sigma)(w),\end{eqnarray} for $f \in {\mathcal A}(\Omega)$, $w \in \Omega$ and $\sigma$ is a holomorphic section of $J^kE$.
\smallskip
Therefore, we observe that the question of determining the equivalence classes of modules $J({\mathscr M})$ is same as understanding the equivalence classes of the jet bundles $J^kE$ with an additional assumption that the equivalence bundle map is also a module map on holomorphic sections over ${\mathcal A}(\Omega)$. Hence it is natural to give the following definition (Definition 4.2, \cite{EQHMII}).
\end{rem}
\begin{defn}\label{jbi}
Two jet bundles are said to be equivalent if there is an isometric holomorphic bundle map which induces a module isomorphism of the class of holomorphic sections.
\end{defn}
\subsection{Main results from Jet bundle}
In order to find geometric invariants of quotient modules we first investigate the simple case, $d=k=2$. We show here that the curvature is the complete set of unitary invariants of the quotient module ${\mathscr M}_q$ for a quasi-free Hilbert module ${\mathscr M}$ of rank $1$. For this case, we give a computational proof to depict the actual picture behind the general result which we will prove later in this subsection. Although the line of idea of the proof for $k=2$ essentially is same as in \cite{EQHMII}, in our case calculations become more complicated as here we have to deal with more than one transversal directions to $\mathcal{Z}$. Thus, our results extend most of the results of the paper \cite{OQMAM}, \cite{EQHMII} as well as those from a recent paper \cite{ALOTCD}.
\medskip
Without loss of generality, under some suitable change of coordinates, we can assume that $0 \in \Omega$ and $U$ is a {neighbourhood} of $0$ such that $U \cap \mathcal{Z} = \{(z_1,\hdots,z_m)\in \Omega:z_1=z_2=0\}$. Consequently, $(0,0,z_3,\hdots,z_m)$ is the coordinates of $\mathcal{Z}$ in $U$. Now let us begin with a line bundle $E$ over $U^*$ with the real analytic metric $G$ which possesses the following power series expansion
\begin{eqnarray} G(z',z'')=\sum_{\alpha,\beta=0}^{\infty}G_{\alpha\beta}(z''){z'}^{\alpha}\overline{z'}^{\beta}\end{eqnarray} where $(z',z'')\in U^*$, $\alpha,\beta$ are multi-indices, ${z'}^{\alpha}={z_1}^{\alpha_1}{z_2}^{\alpha_2}$, $\overline{z'}^{\beta}={\overline{z}_1}^{\beta_1}{\overline{z}_2}^{\beta_2}$ and $z''=(z_3,\hdots,z_m)$.
\smallskip
\begin{lem}\label{trial}
Let $\Omega \subset {\mathbb C}^m$ be a bounded domain and $\mathcal{Z}$ be a complex connected submanifold of $\Omega$ of codimension $2$. Suppose that $\mathcal K$ and $\tilde{\mathcal K}$ are the curvature tensors of line bundles $E$ and $\tilde{E}$ with respect to the Hermitian metric $\rho$ and $\tilde{\rho}$ of $E$ and $\tilde{E}$, respectively. Then $\mathcal K$ and $\tilde{\mathcal K}$ are equal on $\mathcal{Z}$ if and only if there exists holomorphic functions $\psi_{00},\psi_{10},\psi_{01}$ on $\mathcal{Z}$ such that
\begin{eqnarray} \label{em}((\tilde{\rho}_{\theta^{-1}(i)\theta^{-1}(j)}))_{i,j=0}^2=\Psi\cdot(({\rho}_{\theta^{-1}(i)\theta^{-1}(j)}))_{i,j=0}^2\cdot{\Psi}^*\end{eqnarray} on $\mathcal{Z}$ where $\theta$ is as in $\eqref{bij}$ and $\Psi$ is the $3 \times 3$ matrix
\begin{eqnarray}\label{msi}
\Psi = \begin{pmatrix}
\psi_{00} & 0 & 0\\
\psi_{10} & \psi_{00} & 0\\
\psi_{01} & 0 & \psi_{00}\\
\end{pmatrix}.
\end{eqnarray}
\end{lem}
Before going into the proof of the lemma let us give an application of it as follows.
\begin{thm}
Suppose that ${\mathscr M}$ and $\tilde{{\mathscr M}}$ are pair of quasi-free Hilbert modules of rank $1$ over ${\mathcal A}(\Omega)$ and that $E$ and $\tilde{E}$ are the line bundles corresponding to ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively. Let ${\mathscr M}_q={\mathscr M}\ominus{\mathscr M}_0$ and $\tilde{{\mathscr M}}_q=\tilde{{\mathscr M}}\ominus\tilde{{\mathscr M}}_0$ be a pair of quotient modules of Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively, over ${\mathcal A}(\Omega)$. Assume that $({\mathscr M},{\mathscr M}_q)$ and $(\tilde{{\mathscr M}},\tilde{{\mathscr M}}_q)$ are in $B_{1,2}(\Omega,\mathcal{Z})$. Then the quotient modules ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are isomorphic if and only if the corresponding curvature tensors $\mathcal K$ and $\tilde{\mathcal K}$ of the line bundles $E$ and $\tilde{E}$, respectively, are equal on $\mathcal{Z}$.
\end{thm}
\begin{proof}
In fact, Theorem $\ref{qm}$ provides that equivalence of ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ is same as the equivalence of $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ and $J(\tilde{{\mathscr M}})|_{\text{res}\mathcal{Z}}$. So let us begin with an isometric module map $\Psi : J({\mathscr M})|_{\text{res}\mathcal{Z}} \rightarrow J(\tilde{{\mathscr M}})|_{\text{res}\mathcal{Z}}$. Since $\Psi$ intertwines the module action $\Psi$ is of the form given in $\eqref{msi}$. Moreover, being an isometry, $\Psi$ satisfies \begin{eqnarray}\label{ek} JK|_{\mathcal{Z}} = \Psi\cdot J\tilde{K}|_{\mathcal{Z}}\cdot \Psi^*\end{eqnarray} which is equivalent to saying that $\Psi$ satisfies the identity $\eqref{em}$ on $\mathcal{Z}$ as, for $z \in \mathcal{Z}$, $\rho(z)$ is nothing but $K(z,z)$. Then the Lemma $\ref{trial}$ proves the necessity part.
\smallskip
Conversely, following the Lemma $\ref{trial}$ the equality of curvature tensors $\mathcal K$ and $\tilde{\mathcal K}$ on $\mathcal{Z}$ implies that $\Psi$ is of the form given in $\eqref{msi}$ and satisfies $\eqref{ek}$ which in turn yields that $\Psi$ is an isometry from $J({\mathscr M})|_{\text{res}\mathcal{Z}}$ onto $J(\tilde{{\mathscr M}})|_{\text{res}\mathcal{Z}}$ and intertwines the module action. Hence this completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma $\ref{trial}$]
Let us begin with the assumption that there exists holomorphic functions $\psi_{00},\psi_{10},\psi_{01}$ on $\mathcal{Z}$ such that $\eqref{em}$ holds on $\mathcal{Z}$. Then we wish to show that $\mathcal K$ and $\tilde{\mathcal K}$ are equal restricted to $\mathcal{Z}$. We have, by the local expression of the curvature $\eqref{locK}$, that ${\mathcal K}_{i\overline{j}}=\partial_i\bar\partial_j\log\rho$ and $\tilde{\mathcal K}_{i\overline{j}}=\partial_i\bar\partial_j\log\tilde{\rho}$, for $i,j=1,\hdots,m$. So let us calculate $\tilde{\mathcal K}_{i\overline{j}}(z)$ for any point $z \in \mathcal{Z}$ and $i,j=1,\hdots,m$.
\begin{eqnarray*}
\tilde{\mathcal K}_{1\overline{1}}(z)
&=& [respectively\partial_1\bar\partial_1respectively - \bar\partial_1respectively\partial_1respectively]{{respectively}^{-2}}|_z\\
&=& [\rho_{(0,0)(0,0)}\rho_{(1,0)(1,0)}-\rho_{(0,0)(1,0)}
\rho_{(1,0)(0,0)}]\rho_{(0,0)(0,0)}^{-2}\\
&=& {\mathcal K}_{1\overline{1}}(z)
\end{eqnarray*}
We also find that $\tilde{\mathcal K}_{2\overline{2}}(z)=\mathcal K_{2\overline{2}}(z)$ for $z\in\mathcal{Z}$ by doing a similar calculation as in the case of $\mathcal K_{1\overline{1}}(z)$. Now we calculate $\tilde{\mathcal K}_{1\overline{2}}(z)$.
\begin{eqnarray*}
\tilde{\mathcal K}_{1\overline{2}}(z)
&=& [respectively\partial_1\bar\partial_2respectively-\bar\partial_2respectively\partial_1respectively]{{respectively}^{-2}}|_z\\
&=& [\rho_{(0,0)(0,0)}\rho_{(1,0)(0,1)}-\rho_{(1,0)(0,0)}\rho_{(0,0)(0,1)}]\rho_{(0,0)(0,1)}^{-2}\\
&=& {\mathcal K}_{1\overline{2}}(z)
\end{eqnarray*}
Since $\tilde{\mathcal K}_{1\overline{2}}(z)=\overline{\tilde{\mathcal K}}_{2\overline{1}}$ we have $\tilde{\mathcal K}_{2\overline{1}}(z)=\mathcal K_{2\overline{1}}(z)$ for $z\in\mathcal{Z}$. Finally, let us calculate $\tilde{\mathcal K}_{i\overline{1}}(z)$, for $z \in \mathcal{Z}$ and $2 < i \leq m$.
\begin{eqnarray*}
\tilde{\mathcal K}_{i\overline{1}}(z)
&=& [\tilde{\rho}\partial_i\bar\partial_1\tilde{\rho}-\bar\partial_1\tilde{\rho}\partial_i\tilde{\rho}]
\tilde{\rho}^{-2}|_z\\
&=& [\rho_{(0,0)(0,0)}\partial_i\rho_{(0,0)(1,0)}-\rho_{(0,0)(1,0)}\partial_i\rho_{(0,0)(0,0)}]\rho_{(0,0)(0,0)}^{-2}\\
&=& \mathcal K_{i1}(z)
\end{eqnarray*}
Similarly one can show that $\tilde{\mathcal K}_{i\overline{2}}(z)=\mathcal K_{i\overline{2}}(z)$ for $z\in\mathcal{Z}$ and $1\leq i\leq m$. Thus, we are done with the converse part using the skew symmetry property of the matrix $((\mathcal K_{i\overline{j}}(z)))_{i,j=1}^m$.
Now let us prove the forward direction, namely, assuming that $\mathcal K$ and $\tilde{\mathcal K}$ are equal along $\mathcal{Z}$ we want to find $\psi_{00},\psi_{10},\psi_{01}$ holomorphic on $\mathcal{Z}$ such that
$\eqref{em}$ holds.
\smallskip
Let $respectively=r\cdot\rho$, and $\Gamma=\log r$. Then $\Gamma$ is real analytic function on $\Omega$. We can, therefore, expand $\Gamma$ in power series, that is,
\begin{eqnarray}\label{GPE}\Gamma(z',z'')=\sum_{\alpha,\beta=0}^{\infty}\Gamma_{\alpha\beta}(z''){z'}^{\alpha}\overline{z'}^{\beta} \end{eqnarray}
where $\alpha,\beta$ are multi-indices, ${z'}^{\alpha}={z_1}^{\alpha_1}{z_2}^{\alpha_2}$, $\overline{z'}^{\beta}={\overline{z_1}}^{\beta_1}{\overline{z_2}}^{\beta_2}$ and $z''=(z_3,\hdots,z_m)$
\smallskip
We have, from our assumption, that $\mathcal K$ and $\tilde{\mathcal K}$ are equal along $\mathcal{Z}$ which is equivalent to the fact that $\partial_i\bar\partial_j\Gamma=0$, for $1 \leq i,j \leq m$, along $\mathcal{Z}$. We separate out this into following three different cases.
\medskip
\textbf{Case I:} $\partial_i\bar\partial_j\Gamma=0$ along $\mathcal{Z}$, for $i=1,2$, $j=3,\hdots,m$.
\smallskip
For $i=1$ we have $\partial_1\bar\partial_j\Gamma|_{\mathcal{Z}}=0$, for $j=3,\hdots,m$, which is, from $\eqref{GPE}$, equivalent to $\bar\partial_j\Gamma_{(1,0)(0,0)}=0$ on $\mathcal{Z}$. In other words, $\Gamma_{(1,0)(0,0)}$ is holomorphic on $\mathcal{Z}$. Similarly considering the case with $i=2$, we get $\Gamma_{(0,1)(0,0)}$ is also holomorphic on $\mathcal{Z}$.
\medskip
\textbf{Case II:} $\partial_i\bar\partial_j \Gamma =0$ along $\mathcal{Z}$, for $i,j=1,2$.
\smallskip
In this case, we are considering the equation $\partial_i\bar\partial_j {\Gamma}|_{\mathcal{Z}} =0 $, for $i,j=1,2$. For $i=j=1$, we have $\partial_1\bar\partial_1 \Gamma|_{\mathcal{Z}} =0 $, that is, $\Gamma_{(1,0)(1,0)}=0$ on $\mathcal{Z}$ and, for $i=j=2$, $\Gamma_{(0,1)(0,1)}=0$ along $\mathcal{Z}$. Finally, for $i=1,j=2$, it is easy to verify from the equation $\eqref{GPE}$ that $\Gamma_{(1,0)(0,1)}=0$ on $\mathcal{Z}$ and doing the same calculation with $i$ and $j$ interchanged we have $\Gamma_{(0,1)(1,0)}=0$ on the submanifold $\mathcal{Z}$.
\medskip
\textbf{Case III:} $\partial_i\bar\partial_j\Gamma=0$ along $\mathcal{Z}$, for $i,j=3,\hdots,m$.
\smallskip
In this last case, we have $\partial_i\bar\partial_j\Gamma|_{\mathcal{Z}}=0$, $i,j=3,\hdots,m$ which together with power series expansion of $\Gamma$ yield that $\partial_i\bar\partial_j\Gamma_{(0,0)(0,0)}=0$, for $i,j=3,\hdots,m$, on $\mathcal{Z}$. Since $\mathcal{Z}$ is a complex submanifold with coordinates $z=(0,0,z_3,\hdots,z_m)\in\mathcal{Z}$ the above equations together imply that $\Gamma_{(0,0)(0,0)}(z'')=\psi_1(z'')+\overline{\psi_2}(z'')$, for $z''\in\mathcal{Z}$ and some holomorphic functions $\psi_1,\psi_2$ on $\mathcal{Z}$.
\medskip
Now, substituting the above coefficients in the equation $\eqref{GPE}$ and noting that $\Gamma$ is real valued, we have
$$
\Gamma(z',z'')= \psi_1+\beta_1 z_1+\eta_1 z_2+\overline{\psi_2}+\overline{\beta_2 z_1}+\overline{\eta_2 z_2}+\text{ (terms of degree}\geq 3)
$$
where $\psi_i,\beta_i,\eta_i$, $i=1,2$, are holomorphic functions on $\mathcal{Z}$. Since $\Gamma$ is a real valued function $\Gamma=\frac{\Gamma+\overline{\Gamma}}{2}$ and hence we have
\begin{eqnarray}
\Gamma(z',z'')= \psi+\beta z_1+\eta z_2+\overline{\psi}+\overline{\beta z_1}+\overline{\eta z_2}+\text{ (terms of degree}\geq 3)
\end{eqnarray}
where $\psi=\frac{\psi_1+\psi_2}{2}$, $\beta=\frac{\beta_1+\beta_2}{2}$ and $\eta=\frac{\eta_1+\eta_2}{2}$. So from the definition of $\Gamma$ we can write
\begin{eqnarray*}
r &=& \exp\Gamma\\
&=& |\exp\psi|^2\cdot|(1+\beta z_1+\overline{\beta z_1}+|\beta|^2 z_1\overline{z}_1+\cdots)|^2\cdot(|1+\eta z_2+\overline{\eta z_2}+|\eta|^2 z_2\overline{z}_2+\cdots)|^2\cdots\\
&=& |\exp\psi|^2\cdot(1+\beta z_1+\eta z_2+\overline{\beta z_1}+\overline{\eta z_2}+|\beta|^2 z_1\overline{z}_1+\beta\overline{\eta}z_1\overline{z}_2+\overline{\beta}\eta\overline{z}_1 z_2+|\eta|^2 z_2\overline{z}_2+\cdots)\\
\end{eqnarray*}
Thus, putting the above expression of $r$ in $respectively=r\cdot\rho$ and equating the coefficients of $respectively$ and $\rho$ we see that $\psi_{00},\psi_{10}$ and $\psi_{01}$ with $$\psi_{00}=\exp\psi,\psi_{10}=\exp\psi\beta,\psi_{01}=\exp\psi\eta $$ yield our desired matrix in $\eqref{msi}$.
\end{proof}
\medskip
It would be nice if one could carry forward the arguments used in the proof of Lemma $\ref{trial}$ in order to achieve similar results in the case of arbitrary order of vanishing of vector valued functions. However, for general $k$, it would be cumbersome to continue the calculation done in the above Lemma. On the other hand, application of normalized frames makes the calculations simpler and enables us to get a conceptual proof in the general case as well. We adopt the idea of using normalized frame from the paper \cite{ALOTCD} in our case to provide the geometric invariants for quotient modules using jet bundle construction relative to a smooth complex submanifold of codimenssion $d$. To this extent the following theorem provides the required dictionary between the analytic theory and geometric theory for quotient modules obtained from submodules consisting of vector valued holomorphic functions on $\Omega$ vanishing along a smooth complex submanifold of codimension $d$.
\begin{thm}\label{anvsgeo}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$ and $\mathsf{Z}$ be the complex submanifold in $\Omega$ of codimension $d$. Suppose that $({\mathscr M},{\mathscr M}_q)$ and $(\tilde{{\mathscr M}},\tilde{{\mathscr M}}_q)$ are in $B_{r,k}(\Omega,\mathsf{Z})$. Then the quotient modules ${{\mathscr M}}_q$ and $\tilde{{\mathscr M}}_q$ are equivalent as modules over ${\mathcal A}(\Omega)$ if and only if the jet bundles $J^kE|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ are equivalent where $E$ and $\tilde{E}$ are the {\bf h s} over $\Omega$ corresponding to Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively.
\end{thm}
\begin{proof}
Proof follows from Theorem $\ref{qm}$ and Remark $\ref{rjbi}$.
\end{proof}
Thanks to Theorem $\ref{anvsgeo}$ we are now prepared to determine geometric invariants of quotient modules ${\mathscr M}_q$ by studying the geometry of the jet bundles $J^lE|_{\text{res}\mathsf{Z}}$, for $0 \leq l \leq k$. Before proceeding further, let us recall a fact from complex analysis.
\begin{lem}\label{lem1}
Let $\Omega \subset {\mathbb C}^m$ be a domain and $f(z,w)$ be a function on $\Omega \times \Omega$ which is holomorphic in $z$ and anti-holomorphic in $w$. If $f(z,z)=0$ for all $z \in \Omega$, then $f(z,w)=0$ identically on $\Omega$.
\end{lem}
Since this lemma is well known \cite[Proposition 1]{DATOBS} we omit the proof. We use the lemma several times in the proof of following theorems.
\smallskip
Note also that a {\bf h} can not have holomorphic orthonormal frame in general. Instead one can have (Lemma 2.4 of \cite{CGOT}) a holomorphic frame on a {neighbourhood} of a point which is orthonormal at that point. Then using the technique of the proof of Lemma 2.4 in \cite{CGOT} in a similar way, we have the following existence of normalized frame of a {\bf h} over $\Omega$ along a submanifold of codimension at least $d$ in $\Omega$. In the following proposition we use the notation $z=(z',z'')$ where $z'=(z_1,\cdots,z_d)$ and $z''=(z_{d+1},\hdots,z_m)$.
\begin{prop}\label{prop1}
Let $E$ be a {\bf h} of rank $r$ over a bounded domain $\Omega\subset {\mathbb C}^m$. Assume that $0 \in \Omega$ and $\mathsf{Z} \subset \Omega$ is the submanifold defined by $z_1=\cdots=z_d=0$. Then there is a holomorphic frame $\textbf{\textit{s}}(z',z'')=\{s_1(z',z''),\hdots,s_r(z',z'')\}$ on a {neighbourhood} of the origin in $\Omega$ such that $((\<\partial^l s_i(0,z''),s_j(0,0)\>))_{i,j=1}^r$ is the zero matrix for any integer $l$ and $((\<s_i(0,z''),s_j(0,0)\>))_{i,j=1}^r$ is the identity matrix on $\mathsf{Z}$.
\end{prop}
We say a frame is \textbf{\textit{normalized at origin}} if it satisfies the properties in the above proposition.
\begin{thm}\label{jean}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$ and $\mathsf{Z}$ be the complex submanifold in $\Omega$ of codimension $d$ defined by $z_1=\cdots=z_d=0$. Assume that pair of Hilbert modules $({\mathscr M},{\mathscr M}_q)$ and $(\tilde{{\mathscr M}},\tilde{{\mathscr M}}_q)$ are in $B_{1,k}(\Omega,\mathsf{Z})$. Then ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are unitarily equivalent as modules over ${\mathcal A}(\Omega)$ if and only if $\partial^l\bar\partial^j\norm{\tilde{s}}^2=\partial^l\bar\partial^j\norm{s}^2 $ on $\mathsf{Z}$ for all $0 \leq l,j \leq N$ where $\{s(z)\}$ and $\{\tilde{s}(z)\}$ are frames of the line bundles $E$ and $\tilde{E}$ on $\Omega$ associated to the Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively, normalized at origin.
\end{thm}
\begin{proof}
We begin with the observation, following Theorem $\ref{anvsgeo}$, that the quotient modules ${{\mathscr M}}_q$ and $\tilde{{\mathscr M}}_q$ are equivalent as modules over ${\mathcal A}(\Omega)$ if and only if the jet bundles $J^kE|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ are equivalent where $E$ and $\tilde{E}$ are the {\bf h s} over $\Omega$ corresponding to Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively. So it is enough to prove that there exists a jet bundle isomorphism $\Phi:J^kE|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ if and only if $\partial^l\bar\partial^j\norm{\tilde{s}}^2=\partial^l\bar\partial^j\norm{s}^2 $ on $\mathsf{Z}$ for all $0 \leq l,j \leq N$.
\smallskip
We start with the necessity. Let $\Phi:J^k E|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ be a jet bundle isomorphism. Consequently, by Definition $\ref{jbi}$, $\Phi$ intertwines the module actions on the space of holomorphic sections and preserves the Hermitian metrics. The isomorphism $\Phi$ can be represented by an $(N+1) \times (N+1)$ complex matrix $((\phi_{ij}))_{i,j=0}^N$ with respect to the frames $\{s(0,z''),\partial s(0,z''),\hdots,\linebreak \partial^N s(0,z'')\}$ and $\{\tilde{s}(0,z''),\partial\tilde{s}(0,z''),\hdots,\partial^N\tilde{s}(0,z'')\}$ where $\phi_{ij}$ are holomorphic functions on $\mathsf{Z}$. Then in terms of matrices the fact that $\Phi$ is an isomorphism of two jet bundles $J^k E|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ translates to the following two matrix equations on $\mathsf{Z}$:
\begin{equation}\label{jme}
((\langle\partial^l s,\partial^j s\rangle))_{l,j=0}^N = ((\phi_{ij}))_{i,j=0}^N((\langle\partial^l\tilde{s},\partial^j\tilde{ s}\rangle))_{l,j=0}^N(((\phi_{ij}))_{i,j=0}^N)^*
\end{equation}
\begin{equation}\label{me}
((\phi_{ij}))_{i,j=0}^N((\mathcal J(f)_{lk}))_{l,k=0}^N=((\mathcal J(f)_{lk}))_{l,k=0}^N((\phi_{ij}))_{i,j=0}^N.
\end{equation}
Then the proof of the forward direction easily follows from the following claims.
\medskip
\textbf{Claim 1.}
\begin{enumerate}
\item[(a)] For $0 \leq k \leq N$ and $z=(0,z'') \in \mathsf{Z}$, $\phi_{kk}(0,z'')=\phi_{00}(0,z'')$.
\item[(b)] Let $1 \leq i,j \leq N$, $\alpha=(\alpha_1,\hdots,\alpha_d)=\theta^{-1}(i),\beta=(\beta_1,\hdots,\beta_d)=\theta^{-1}(j)$. Then for $z \in \mathsf{Z}$ we have
\begin{equation}
\phi_{ij}(0,z'')=\left\{
\begin{array}{@{}ll@{}}
{\alpha \choose \alpha-\beta}\phi_{\theta(\alpha-\beta)0}(0,z'') & \text{if}\,\,\, \alpha_t \geq \beta_t \,\,\,\forall\,\, t=1,\cdots,d, \\
0 & \text{otherwise}.
\end{array}\right.
\end{equation}
\end{enumerate}
Before going into the proof of Claim 1, let us make some observations about the matrix $\mathcal J(f)=((\mathcal J(f)_{lk}))_{l,k=0}^N$ which will be used in the proof. So to begin with, let $\gamma=(\gamma_1,\hdots,\gamma_d)$ with $\theta(\gamma)=l$, $0 \leq l \leq N$, and $\gamma_j \in \mathbb{N} \cup \{0\}$, $1 \leq j \leq d$. We also denote the subdiagonals of the matrix $\mathcal J(f)$ as $S_0,\hdots,S_N$, that is, $S_j$ is the set
\begin{eqnarray*} S_j &:=& \{\mathcal J(f)_{j0},\hdots,\mathcal J(f)_{j+kk+1},\hdots,\mathcal J(f)_{NN-j+1}\}\\
&=& \{\mathcal J(f)_{lt}:j \leq l \leq N, 0 \leq t \leq N-j+1, l-t=j-1\}\end{eqnarray*}. Thus, $S_0$ consists of all diagonal entries and $S_N$ is the singleton set $\{\mathcal J(f)_{N0}\}$. Then a simple calculation using the definition $\eqref{modac}$ of $\mathcal J(f)$ yields the following property.
\begin{enumerate}
\item[(\textbf{P1})] For $f(z',z'')=z_1^{\gamma_1} \cdots z_d^{\gamma_d}$ with $\theta(\gamma_1,\hdots,\gamma_d)=l$, the matrix $\mathcal J(f)|_{\mathsf{Z}}$ has only non-zero entries along the subdiagonals, $S_j$, for $l \leq j \leq N$.
\end{enumerate}
Now we note, for $l \leq i \leq N$, that common entries of $i$-th row of $\mathcal J(f)$ with subdiagonals $S_l,\hdots,S_N$ are $\mathcal J(f)_{i0},\hdots,\mathcal J(f)_{ii-l+1}$, respectively. Therefore, for $l \leq i \leq N$ with $\theta^{-1}(i)=\alpha=(\alpha_1,\hdots,\alpha_d)$ and $f(z',z'')=z_1^{\gamma_1} \cdots z_d^{\gamma_d}$, the above property, $(\textbf{P1})$, shows that non-zero entries of $i$-th row of the matrix $\mathcal J(f)|_{\mathsf{Z}}$ must live in the set $\{\mathcal J(f)_{ij}(0,z''):0 \leq j \leq i-l+1\}$. On the other hand, for $0 \leq j \leq i-l+1$ with $\theta^{-1}(j)=\beta=(\beta_1,\hdots,\beta_d)$,
$$\mathcal J(z_1^{\gamma_1} \cdots z_d^{\gamma_d})_{ij}|_{\mathsf{Z}} \neq 0 \iff \partial^{\alpha-\beta}(z_1^{\gamma_1} \cdots z_d^{\gamma_d})|_{\mathsf{Z}} \neq 0 \iff \beta =\alpha-\gamma.$$
Thus, we have the following property of $\mathcal J(z_1^{\gamma_1} \cdots z_d^{\gamma_d})$.
\begin{enumerate}
\item[(\textbf{P2})] For $\alpha,\gamma$, $i,l$ as above and $f(z',z'')=z_1^{\gamma_1} \cdots z_d^{\gamma_d}$, $\mathcal J(f)_{i\theta(\alpha-\gamma)}(0,z'')$ is the only non-zero entry of $i$-th row of $\mathcal J(f)|_{\mathsf{Z}}$. In particular, we observe that $\mathcal J(f)_{ij}(0,z'')=0$, for any $j$ with $0 \leq j \leq N$ whenever $\alpha_t < \gamma_t$ for some $t \in \{1,\hdots,d\}$.
\end{enumerate}
\smallskip
Since $\Phi$ is a jet bundle isomorphism between $J^kE|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$, by the definition (Definition $\ref{jec}$) $\Phi$ commutes with the module action of ${\mathcal A}(\Omega)$ on the sections of the above jet bundles, namely, from $\eqref{modacsec}$ we have the equation $\eqref{me}$ holds on $\mathsf{Z}$ for all $f \in {\mathcal A}(\Omega)$. Let $g(z',z'')={z_1}^{\eta_1}\cdots{z_d}^{\eta_d}$, for given $0 \leq k \leq N$ with $\theta^{-1}(k)=(\eta_1,\hdots,\eta_d)$. Then from $(\textbf{P2})$ we have that $\mathcal J(g)_{k0}(0,z'')$ is the only non-zero entry of $k$-th row of $\mathcal J(g)|_{\mathsf{Z}}$. Therefore, equating $k0$-th entry of matrices in $\eqref{me}$ with $f=g$ we obtain, for $z \in \mathsf{Z}$, that $$\phi_{kk}(0,z'')\mathcal J(g)_{k0}(0,z'')=\mathcal J(g)_{k0}(0,z'')\phi_{00}(0,z'')$$ which proves (a) of the claim above as $\mathcal J(g)_{k0}=\eta!$.
\smallskip
To prove the part (b) let $0 \leq i,j \leq N$, $\theta^{-1}(i)=(\alpha_1,\hdots,\alpha_d)$, $\theta^{-1}(j)=(\beta_1,\hdots,\beta_d)$ and $g(z',z'')=z_1^{\beta_1}\cdots z_d^{\beta_d}$. We also assume that $i < j$. Then, using $(\textbf{P2})$ with $f=g$ and $l=j$, we see that $\mathcal J(g)_{j0}(0,z'')$ is the only non-zero entry of the matrix $\mathcal J(g)(0,z'')$ for $(0,z'') \in \mathsf{Z}$. Therefore, only $i0$-th entry of the matrix in left hand side of $\eqref{me}$ contains $\phi_{ij}$. On the other hand, the $i0$-th entry of the matrix in other side in $\eqref{me}$ is $0$ as $i < j$, thanks to the property $(\textbf{P1})$. Thus, comparing the $i0$-th entry of matrices in $\eqref{me}$ we conclude that $\phi_{ij}(0,z'')=0$ on $\mathsf{Z}$, for $1 \leq i < j \leq N$.
\smallskip
Now we assume that $i > j$. Since only non-zero entry of $j$-th row of the matrix $\mathcal J(g)|_{\mathsf{Z}}$ is $\mathcal J(g)_{j0}(0,z'')$ it is clear that only $i0$-th entry of the matrix in left hand side of $\eqref{me}$ contains $\phi_{ij}$. The $i0$-th entry of this matrix is $\beta!\phi_{ij}(0,z'')$ as $\mathcal J(g)_{j0}(0,z'')=\beta!$, for $(0,z'') \in \mathsf{Z}$. So, as before in order to calculate $\phi_{ij}$, we need to compare $i0$-th entry of the matrices in $\eqref{me}$. From $(\textbf{P2})$ we have that only non-zero entry of $i$-th row of $\mathcal J(g)|_{\mathsf{Z}}$ is $\mathcal J(g)_{i\theta(\alpha-\beta)}(0,z'')$ which, by definition $\eqref{modac}$ of $\mathcal J(g)$ with $g(z',z'')=z_1^{\beta_1}\cdots z_d^{\beta_d}$, is
$${\alpha \choose \alpha-\beta}\partial^{\alpha-(\alpha-\beta)}g(z',z'')|_{\mathsf{Z}}={\alpha \choose \alpha-\beta}\partial^{\beta}g(z',z'')|_{\mathsf{Z}}={\alpha \choose \alpha-\beta}\beta!,$$ provided $\alpha_t \geq \beta_t$, for all $t=1,\hdots,d$. Furthermore, if $\alpha_t < \beta_t$ for some $t \in \{1,\hdots,d\}$, it follows from $(\textbf{P2})$ that every entry of $i$-th row is zero. Therefore, equating $i0$-th entry of matrices in $\eqref{me}$ we get
\begin{equation}
\phi_{ij}(0,z'')=\left\{
\begin{array}{@{}ll@{}}
{\alpha \choose \alpha-\beta}\phi_{\theta(\alpha-\beta)0}(0,z'') & \text{for}\,\,\, \alpha_t \geq \beta_t\,\,\,\forall\,\,t=1,\hdots,d, \\
0 & \text{otherwise} \\
\end{array}\right.
\end{equation}
which completes the proof of Claim 1.
\medskip
Thus, Claim 1 shows that the matrix $((\phi_{ij}(0,z'')))_{i,j=0}^N$ is a lower triangular matrix. Consequently, we have that $\Phi$ induces bundle morphisms $\Phi|_{J^lE|_{\text{res}\mathsf{Z}}}:J^lE|_{\text{res}\mathsf{Z}}\rightarrow J^l\tilde{E}|_{\text{res}\mathsf{Z}}$, for $0 \leq l \leq k$.
\smallskip
\textbf{Claim 2.} $\phi_{00}$ is a constant function and $\phi_{ii}=\phi_{00}$, for $i=0,\hdots,N$, on $\mathsf{Z}$.
\medskip
Note that it is enough to show that $\phi_{00}$ is a constant function on $\mathsf{Z}$ thanks to Claim 1. In fact, from the equation $\eqref{jme}$ we have
$$\<s(0,z''), s(0,z'')\>=\phi_{00}(0,z'')\<\tilde{s}(0,z''),\tilde{s}(0,z'')\>\overline{\phi_{00}(0,z'')}.$$
Consequently, Lemma $\ref{lem1}$ and Proposition $\ref{prop1}$ together yield that $$\phi_{00}(0,z'')\overline{\phi_{00}(0,0)}=1.$$ Hence we are done with Claim 2.
\medskip
\textbf{Claim 3.} $((\phi_{ij}(0,z'')))_{i,j=0}^N$ is a constant diagonal matrix with diagonal entries $\phi_{ii}=\phi_{00}$, for $0 \leq i \leq N$, that is, $((\phi_{ij}(0,z'')))_{i,j=0}^N = \phi_{00}\cdot I$ where $I$ is the $(N+1)\times (N+1)$ identity matrix.
\medskip
In view of Claim 1 and Claim 2, it is enough to show that $\phi_{l0}=0$, for $0< l \leq N$, on $\mathsf{Z}$. So calculating $l0$-th entry of the matrices in the equation $\eqref{jme}$ and using the Lemma $\ref{lem1}$ we have
$$\langle\partial^l s(0,z''), s(0,w'')\rangle=\left(\sum_{j=0}^l \phi_{lj}(0,z'')\langle\partial^j\tilde{s}(0,z''), \tilde{s}(0,w'')\rangle\right)\overline{\phi}_{00}$$ and consequently, after putting $w''=0$ and applying the Proposition $\ref{prop1}$ to the frames $\{s\}$ and $\{\tilde{s}\}$ at origin we get $\phi_{l0}(0,z'')=0$ on $\mathsf{Z}$.
\smallskip
Thus, Claim 1, Claim 2, Claim 3 and the equation $\eqref{jme}$ together yield that
\begin{eqnarray}\label{jec} \partial^l\bar\partial^j\norm{s(0,z'')}^2=\phi_{00}\partial^l\bar\partial^j\norm{\tilde{s}(0,z'')}^2 \overline{\phi}_{00}=\partial^l\bar\partial^j\norm{\tilde{s}(0,z'')}^2\end{eqnarray} on $\mathsf{Z}$, for $0 \leq l,j \leq N$.
\smallskip
The converse statement is easy to see. Indeed, if the equation $\eqref{jec}$ happens to be true then the desired jet bundle isomorphism $\Phi$ is given by the constant matrix $I$ with respect to the frames $\{s, \partial s,\hdots, \partial^N s\}$ and $\{\tilde{s}, \partial \tilde{s},\hdots, \partial^N \tilde{s}\}$ where $I$ is the identity matrix of order $N+1$.
\end{proof}
\begin{thm}\label{jean1}
Let $\Omega$ be a bounded domain in ${\mathbb C}^m$ and $\mathsf{Z}$ be the complex submanifold in $\Omega$ of codimension $d$ defined by $z_1=\cdots=z_d=0$. Assume that $({\mathscr M},{\mathscr M}_q)$ and $(\tilde{{\mathscr M}},\tilde{{\mathscr M}}_q)$ are pair of Hilbert modules in $B_{r,k}(\Omega,\mathsf{Z})$. Then ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are unitarily equivalent as modules over ${\mathcal A}(\Omega)$ if and only if there exists a constant unitary matrix $D$ such that $\partial^l\bar\partial^j H = D(\partial^l\bar\partial^j\tilde{H})D^* $ on $\mathsf{Z}$, for all $0 \leq l,j \leq N$ where $H(z)$ and $\tilde{H}(z)$ are the Grammian matrices for the holomorphic frames $\textbf{s}$ and $\tilde{\textbf{s}}$ of the {\bf h s} $E$ and $\tilde{E}$ on $\Omega$ associated to the Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively, normalized at origin.
\end{thm}
\begin{proof}
We note, as in Theorem $\ref{jean}$, that it is enough to show that the jet bundles $J^kE|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ are equivalent according to the Definition $\ref{jbi}$ if and only if there exists a constant unitary matrix $D$ such that $\partial^l\bar\partial^j H = D(\partial^l\bar\partial^j\tilde{H})D^* $ on $\mathsf{Z}$, for all $0 \leq l,j \leq N$, where $H(z)$ and $\tilde{H}(z)$ are as above.
\medskip
So, to begin with, let $\Phi:J^k E|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ be a jet bundle isomorphism. Then the isomorphism $\Phi$ can be represented by an $(N+1) \times (N+1)$ block matrix $((\Phi_{lt}))_{l,t=0}^N$ with respect to the frames $\{\textbf{\textit{s}}, \partial \textbf{\textit{s}}(0,z''),\hdots,\partial^N \textbf{\textit{s}}(0,z'')\}$ and $\{\tilde{\textbf{\textit{s}}},\partial\tilde{\textbf{\textit{s}}}(0,z''),\hdots,\partial^N\tilde{\textbf{\textit{s}}}(0,z'')\}$ where $\Phi_{lt}$ are holomorphic $r \times r$ matrix valued functions on $\mathsf{Z}$. Then in terms of matrices the fact that $\Phi$ is an isometry of two jet bundles $J^k E|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$ translates to the following matrix equation on $\mathsf{Z}$:
\begin{equation}\label{jme1}
((\partial^l\bar\partial^t H))_{l,t=0}^N = ((\Phi_{lt}))_{l,t=0}^N((\partial^l\bar\partial^t\tilde{ H}))_{l,t=0}^N(((\Phi_{lt}))_{l,t=0}^N)^*.
\end{equation}
\smallskip
Let $E_i|_{\text{res}\mathsf{Z}}$ and $\tilde{E}_i|_{\text{res}\mathsf{Z}}$ be line bundles determined by the frames $\{s_i\}$ and $\{\tilde{s}_i\}$, respectively, on $\mathsf{Z}$, for $1 \leq i \leq r$. Then we can have the decomposition $E|_{\text{res}\mathsf{Z}}=\oplus_{i=1}^r E_i|_{\text{res}\mathsf{Z}}$ and $J^kE|_{\text{res}\mathsf{Z}}=\oplus_{i=1}^rJ^kE_i|_{\text{res}\mathsf{Z}}$ with $\{s_i,\partial s_i,\hdots,\partial^N s_i\}$ as a frame on $\mathsf{Z}$. Further, let $P_i:J^kE|_{\text{res}\mathsf{Z}}\rightarrow J^kE_i|_{\text{res}\mathsf{Z}}$ and $\tilde{P}_i:J^k\tilde{E}|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}_i|_{\text{res}\mathsf{Z}}$ be projection morphisms where the frame $\{\tilde{s}_i,\partial \tilde{s}_i,\hdots,\partial^N \tilde{s}_i\}$ defines the jet bundle $J^k\tilde{E}_i|_{\text{res}\mathsf{Z}}$. Then note that the matrix of $\Phi$ with respect to the frames $J(\textbf{\textit{s}})=\{\sum_{l=0}^N\partial^l s_1\otimes equivalent_l,\hdots,\sum_{l=0}^N\partial^l s_r\otimes equivalent_l\}$ and $J(\tilde{\textbf{\textit{s}}})=\{\sum_{l=0}^N\partial^l \tilde{s}_1\otimes equivalent_l,\hdots,\sum_{l=0}^N\partial^l \tilde{s}_r\otimes equivalent_l\}$ is $(([P_{ij}]))_{i,j=1}^r$ where $[P_{ij}]$ represents the matrix of $\tilde{P}_i\Phi P^*_j$ with respect to the frames $\{s_j,\partial s_j,\hdots,\partial^N s_j\}$ and $\{\tilde{s}_i,\partial \tilde{s}_i,\hdots,\partial^N \tilde{s}_i\}$. Since $\Phi$ is a jet bundle isomorphism (Definition $\ref{jbi}$) it intertwines the module action on the class of holomorphic sections of $J^k E|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$. As a consequence, we have $$(([P_{ij}]))_{i,j=1}^r(\mathcal J(f)\otimes I_r) = (\mathcal J(f)\otimes I_r)(([P_{ij}]))_{i,j=1}^r,$$ for $f \in {\mathcal A}(\Omega)$, which is equivalent to the fact that the bundle morphisms $\tilde{P}_i\Phi P^*_j$ intertwine the module action $\eqref{modacsec}$ on holomorphic sections of $J^k E_j|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}_i|_{\text{res}\mathsf{Z}}$. Thus, $\tilde{P}_i\Phi P^*_j$ defines a jet bundle morphism from $J^kE_j|_{\text{res}\mathsf{Z}}$ onto $J^k\tilde{E}_i|_{\text{res}\mathsf{Z}}$.
\smallskip
We, therefore, can apply Claim 1 in Theorem $\ref{jean1}$ to $\tilde{P}_i\Phi P^*_j$ to conclude, for $0 \leq l,t \leq N$, that
\begin{eqnarray*}
[P_{ij}]_{lt} &=& {\alpha \choose \alpha-\beta}[P_{ij}]_{\theta(\alpha-\beta)0}(0,z''), ~(\alpha - \beta)\in (\mathbb{N} \cup \{0\})^d\\
&=& {\alpha \choose \alpha-\beta}(\Phi_{\theta(\alpha-\beta)0}(0,z''))_{ij}, ~(\alpha - \beta)\in (\mathbb{N} \cup \{0\})^d,
\end{eqnarray*} otherwise, $[P_{ij}]_{lt}$ is the zero matrix.
Then it follows that the matrix of $\Phi(0,z'')$ with respect to the frames $\{\textbf{\textit{s}},\partial\textbf{\textit{s}},\hdots,\partial^N\textbf{\textit{s}}\}$ and $\{\tilde{\textbf{\textit{s}}},\partial\tilde{\textbf{\textit{s}}},\hdots,\partial^N\tilde{\textbf{\textit{s}}}\}$ is a lower triangular block matrix with $\Phi_{ll}(0,z'')=\Phi_{00}(0,z'')$ for $0 \leq l \leq N$ and, for $0 \leq l,t \leq N$, $\alpha=(\alpha_1,\hdots,\alpha_d)=\theta^{-1}(l),\beta=(\beta_1,\hdots,\beta_d)=\theta^{-1}(t)$ and $1 \leq i,j \leq r$,
\begin{eqnarray}\label{mb}
(\Phi_{lt}(0,z''))_{ij}={\alpha \choose \alpha-\beta}(\Phi_{\theta(\alpha-\beta)0}(0,z''))_{ij} \text{ if } (\alpha - \beta)\in (\mathbb{N} \cup \{0\})^d,
\end{eqnarray}
and is zero, otherwise, on $\mathsf{Z}$. Now a similar proof as in Claim 2 in Theorem $\ref{jean}$ with matrix valued holomorphic functions $H, \tilde{H}$ and $\Phi_{00}$ on $\mathsf{Z}$ yields that $\Phi_{00}$ is a constant unitary matrix. Thus, the proof will be done once we prove that $\Phi_{l0}=0$, for $0< l \leq N$, on $\mathsf{Z}$. So calculating the $l0$-th block of the matrices in the equation $\eqref{jme1}$ and using the Lemma $\ref{lem1}$ we get
$$((\langle\partial^l s_i(0,z''), s_j(0,w'')\rangle))_{i,j=1}^r=\left(\sum_{t=0}^l \Phi_{lt}(0,z'')((\langle\partial^t\tilde{s}_i(0,z''), \tilde{s}_j(0,w'')\rangle))_{i,j=1}^r\right)\Phi^*_{00}$$ and consequently, after putting $w''=0$ and applying the Proposition $\ref{prop1}$ to the frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ at origin we get $\Phi_{l0}(0,z'')=0$ on $\mathsf{Z}$. Thereby from $\eqref{jme1}$ we have
\begin{eqnarray}\label{jec1}\partial^l\bar\partial^j H(0,z'') = D(\partial^l\bar\partial^j \tilde{H}(0,z''))D^*\end{eqnarray} on $\mathsf{Z}$ for all $0 \leq l,j \leq N$ where $D=\Phi_{00}$.
\smallskip
For the converse direction, note that the equation $\eqref{jec1}$ canonically gives rise to the jet bundle isomorphism $\Phi$ by prescribing the matrix of $\Phi$ as $D\otimes I$ with respect to the frames $J(\textbf{\textit{s}})$ and $J(\tilde{\textbf{\textit{s}}})$ where $I$ is the identity matrix of order $N+1$.
\end{proof}
\begin{cor}
Let $\textbf{T}=(T_1,\hdots,T_m)$ and $\tilde{\textbf{T}}=(\tilde{T}_1,\hdots,\tilde{T}_m)$ be two operator tuples in $B_1(\Omega)$. Then $\textbf{T}$ and $\tilde{\textbf{T}}$ are unitarily equivalent if and only if there are jet bundle isomorphisms $\Phi_k:J^k E|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}|_{\text{res}\mathsf{Z}}$, for every $k \in \mathbb{N} \cup \{0\}$ where $\mathsf{Z}$ is any singleton set $\{p\}$, for $p \in \Omega$.
\end{cor}
\begin{proof}
The necessity part is trivial and so we only show that $\textbf{T}$ and $\tilde{\textbf{T}}$ are unitarily equivalent assuming that there are jet bundle isomorphisms $\Phi_k:J^k E|_{\text{res}\mathsf{Z}}\rightarrow J^k\tilde{E}|_{\text{res}\mathsf{Z}}$, for every $k \in \mathbb{N} \cup \{0\}$.
\smallskip
Let $E$ and $\tilde{E}$ be vector bundles over $\Omega$ corresponding to operator tuples $\textbf{T}$ and $\tilde{\textbf{T}}$, respectively, and $\mathsf{Z}=\{0\}$. Thus, the codimension of $\mathsf{Z}$ is $m$. We now wish to apply the previous theorem for each non-negative integer $k$. So let us start with frames $s$ and $\tilde{s}$ for $E$ and $\tilde{E}$, respectively, normalized at origin. Then by Theorem $\ref{jean}$ we have, for every $k \in \mathbb{N} \cup \{0\}$, $\partial^l\bar\partial^j\norm{\tilde{s}}^2=\partial^l\bar\partial^j\norm{s}^2$ at $0$ for all $0 \leq l,j \leq N(k)$ where $N(k)= {d+k-1 \choose k-1}-1$. In other words, translating the notations used in the above equation we get
\begin{eqnarray}\label{tc}\partial^{\alpha_1}_1\cdots\partial^{\alpha_m}_m\bar\partial^{\beta_1}_1\cdots\bar\partial^{\beta_m}_m\norm{s(0)}^2 &=&
\partial^{\alpha_1}_1\cdots\partial^{\alpha_m}_m\bar\partial^{\beta_1}_1\cdots\bar\partial^{\beta_m}_m\norm{\tilde{s}(0)}^2\end{eqnarray}
for all $\alpha,\beta \in (\mathbb{N}\cup\{0\})^m$.
Now since $s$ and $\tilde{s}$ both are holomorphic on their domains of definition $\norm{s}^2$ and $\norm{\tilde{s}}^2$ are real analytic there. Consequently, using the power series expansion of $\norm{s}^2$ and $\norm{\tilde{s}}^2$ together with the equation $\eqref{tc}$ we obtain that $$\norm{s(z)}^2=\norm{\tilde{s}(z)}^2$$ on some open {neighbourhood}, say $\Omega_0$, of the origin in $\Omega$. Thus, the bundle map $\Phi:E\rightarrow \tilde{E}$ determined by the formula $\Phi(s(z))=\tilde{s}(z)$ defines an isometric bundle isomorphism between $E$ and $\tilde{E}$ over $\Omega_0$. Then our result is a direct consequence of the Rigidity theorem in \cite{CGOT}.
\end{proof}
\begin{rem}
Note that the above theorem shows that the unitary equivalence of local operators (1.5 in \cite{CGOT}) $N^{(k)}_{\omega_0}$ and $\tilde{N}^{(k)}_{\omega_0}$ corresponding to $\textbf{T}$ and $\tilde{\textbf{T}}$, respectively, for all $k\geq 0$ but at a fixed point $\omega_0 \in \Omega$ implies the unitary equivalence of $\textbf{T}$ and $\tilde{\textbf{T}}$. In other words, any tuples of operators $\textbf{T}\in B_1(\Omega)$ enjoy the "Taylor series expansion" property. Moreover, following the technique used in Theorem 18 in \cite{ALOTCD}, it is seen that the same property is also enjoyed by any $\textbf{T}\in B_r(\Omega)$, $r \geq 1$.
\end{rem}
The following theorem is one of the main results in this article which generalizes the study of quotient modules done in the paper \cite{EQHMII} to the case of arbitrary codimension. For the definition of bundle maps used in the following theorem we refer the readers to $\eqref{covd1}$.
\begin{thm}\label{mthm}
Let $\Omega \subset {\mathbb C}^m$ be a bounded domain and $\mathsf{Z} \subset \Omega$ be the complex manifold of codimension $d$ defined by $z_1=\cdots=z_d=0$. Suppose that pair of Hilbert modules $({\mathscr M},{\mathscr M}_q)$ and $(\tilde{{\mathscr M}},\tilde{{\mathscr M}}_q)$ are in $B_{r,k}(\Omega,\mathsf{Z})$. Then ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are isomorphic as modules over ${\mathcal A}(\Omega)$ if and only if following conditions hold:
\begin{enumerate}
\item[(i)] There exists holomorphic isometric bundle map $\Phi:E|_{\text{res}\mathsf{Z}}\rightarrow \tilde{E}|_{\text{res}\mathsf{Z}}$ where $E$ and $\tilde{E}$ are {\bf h s} over $\Omega$ corresponding to the Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$ over ${\mathcal A}(\Omega)$.
\item[(ii)] The transverse curvature of $E$ and $\tilde{E}$ as well as their covariant derivatives of order at most $k-2$, along the transverse directions to $\mathsf{Z}$, are intertwined by $\Phi$ on $\mathsf{Z}$.
\item[(iii)] The bundle map $\Phi$ intertwines the bundle maps $\mathcal J_i^l(H):=\bar\partial_i(H^{-1}\partial^l H)$ and $\mathcal J_i^l(\tilde{H}):=\bar\partial_i(\tilde{H}^{-1}\partial^l \tilde{H})$, $d+1 \leq i \leq m$, holds on $\mathsf{Z}$ where $\textbf{s}=\{s_1,\hdots,s_r\}$ and $\tilde{\textbf{s}}=\{\tilde{s}_1,\hdots,\tilde{s}_r\}$ are frames of $E$ and $\tilde{E}$, respectively, for $0 \leq l \leq N$, and $H$ and $\tilde{H}$ are Grammians of $\textbf{s}$ and $\tilde{\textbf{s}}$, respectively.
\end{enumerate}
\end{thm}
\medskip
\begin{rem}
At this point although it seems that the condition (iii) in the above theorem depends on the choice of a frame it is not the case. For instance, if $\textbf{\textit{t}}$ is another frame normalized at origin we have $\textbf{\textit{t}}=\textbf{\textit{s}}A$ for some holomorphic function $A:\mathsf{Z}\rightarrow GL_r({\mathbb C})$. Now since both $\textbf{\textit{s}}$ and $\textbf{\textit{t}}$ are normalized at origin the same proof as in Claim 2 in Theorem $\eqref{jean}$ with matrix valued holomorphic functions shows that $A$ is a constant unitary matrix. Thus, we have $H=AHA^*$ and hence it follows that $$\mathcal J_i^l(G)=A\mathcal J_i^l(H)A^{-1}, d+1 \leq i \leq m,$$ where $G$ is the Gramm matrix of the frame $\textbf{\textit{t}}$.
\end{rem}
\begin{proof}
Let $\Omega \subset {\mathbb C}^m$ and $\mathsf{Z} \subset \Omega$ be as given. Suppose that ${\mathscr M}_q$ and $\tilde{{\mathscr M}}_q$ are equivalent as modules over ${\mathcal A}(\Omega)$. Then by Theorem $\ref{jean1}$ there exists a constant unitary matrix $D$ such that \begin{eqnarray}\label{dmeq}\partial^l\bar\partial^j H(0,z'')=D(\partial^l\bar\partial^j\tilde{H}(0,z''))D^*,\text{ for }(0,z'')\in \mathsf{Z} \text{ and }0 \leq l,j \leq N,\end{eqnarray} where $H(z)$ and $\tilde{H}(z)$ are the Grammian matrices for holomorphic frames $\textbf{\textit{s}}=\{s_1,\hdots,s_r\}$ and $\tilde{\textbf{\textit{s}}}=\{\tilde{s}_1,\hdots,\tilde{s}_r\}$ of the Hermitian holomorphic vector bundles $E$ and $\tilde{E}$ on $\Omega$ associated to Hilbert modules ${\mathscr M}$ and $\tilde{{\mathscr M}}$, respectively, normalized at origin. In particular, for $l=j=0$, $\eqref{dmeq}$ becomes $$H(0,z'')=D\tilde{H}(0,z'')D^*, \text{ for }(0,z'')\in \mathsf{Z}.$$
\smallskip
Let $\Phi:E|_{\text{res}\mathsf{Z}}\rightarrow\tilde{E}|_{\text{res}\mathsf{Z}}$ be the bundle morphism whose matrix representation with respect to the frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ is $D$. Then $\Phi$ is the desired isometric bundle map in (i). Further, the equation $\eqref{dmeq}$ together with (i) of Lemma $\ref{cume}$ yields (ii), and since $D$ is a constant unitary matrix on $\mathsf{Z}$, (iii) is an easy consequence of $\eqref{dmeq}$ with $j=0$.
\smallskip
Now let us prove the converse direction. To do so we show that the condition (i), (ii), (iii) in the statement together imply the condition of the Theorem $\ref{jean1}$, that is, we need to show that there exists a constant unitary matrix $D$ on $\mathsf{Z}$ such that the equation $\eqref{meq}$ holds on the submanifold $\mathsf{Z}$ for $0 \leq l,t \leq N$ and frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ of $E$ and $\tilde{E}$, respectively, normalized at origin.
\smallskip
We first extend the holomorphic isometric bundle map $\Phi: E|_{\text{res}\mathsf{Z}} \rightarrow \tilde{E}|_{\text{res}\mathsf{Z}}$, obtained from condition (i), to a family of linear isometries $\hat{\Phi}_{z_0}: J^kE|_{z_0} \rightarrow J^k\tilde{E}|_{z_0}$ for every $z_0 \in \mathsf{Z}$. Then we show that this extension is actually a jet bundle isomorphism providing our desired matrix. So let us begin with frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ for $E$ and $\tilde{E}$, respectively, normalized at $z_0 \in \mathsf{Z}$, for an arbitrary $z_0 \in \mathsf{Z}$. Condition (i) then yields an isometric holomorphic bundle map $\Phi: E|_{\text{res}\mathsf{Z}} \rightarrow \tilde{E}|_{\text{res}\mathsf{Z}}$ and consequently, we have a holomorphic $r \times r$ matrix valued function $\phi$ on $\mathsf{Z}$ such that \begin{eqnarray}\label{meq} H(0,z'') &=& \phi(0,z'')\tilde{H}(0,z'')\phi(0,z'')^*\end{eqnarray} where $\phi$ represents $\Phi$ with respect to frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$. Since both $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ are normalized at $z_0$, the above equation $\eqref{meq}$ shows that $\phi(0,z''_0)$ is a unitary matrix. Furthermore, from condition (ii) of our hypothesis along with second statement of Lemma $\ref{cume}$ we have, for $0 \leq \alpha_1+\cdots+\alpha_d \leq k-1,0 \leq \beta_1+\cdots+\beta_d \leq k-1$, \begin{eqnarray}\label{jmeq}\partial_1^{\alpha_1}\cdots\partial_d^{\alpha_d}\bar\partial_1^{\beta_1}\cdots\bar\partial_d^{\beta_d}
H(0,z''_0) &=& \phi(0,z''_0)\partial_1^{\alpha_1}\cdots\partial_d^{\alpha_d}\bar\partial_1^{\beta_1}\cdots
\bar\partial_d^{\beta_d}\tilde{H}(0,z''_0)\phi(0,z''_0)^*\end{eqnarray} as $\partial_1^{\alpha_1}\cdots\partial_d^{\alpha_d}H(0,z''_0)$ (respectively, $\partial_1^{\alpha_1}\cdots\partial_d^{\alpha_d}\tilde{H}(0,z''_0)$) and $\bar\partial_1^{\beta_1}\cdots
\bar\partial_d^{\beta_d}H(0,z''_0)$ (respectively, $\bar\partial_1^{\beta_1}\cdots
\bar\partial_d^{\beta_d}\tilde{H}(0,z''_0)$) are zero matrices for any $\alpha_i, \beta_i \geq 0$, $i=1,\hdots,d$. Thus, the equations above $(\ref{meq},~\ref{jmeq})$ lead to the following natural isometric extension, $\hat{\Phi}_{z_0}: J^kE|_{z_0} \rightarrow J^k\tilde{E}|_{z_0}$ defined by
\begin{eqnarray}\label{ext} \hat{\Phi}_{z_0}(\partial^ls_j(0,z''_0))=\sum_{i=1}^r\phi_{ji}(0,z''_0)\partial^l\tilde{s}_i(0,z''_0), \text{ } 0 \leq l \leq N, \text{ } 1 \leq j \leq r.\end{eqnarray}
Then we note, for $0 \leq l \leq N$, $ 1 \leq j \leq r$, $z_0=(0,z''_0)\in \mathsf{Z}$ and $f \in {\mathcal A}(\Omega)$, that
$$\hat{\Phi}_{z_0}\mathcal J(f)(z_0)(\partial^l s_j(0,z''_0)) = \hat{\Phi}_{z_0}(\sum_{p=0}^l\partial^{l-p}f(z_0)\partial^p s_j(0,z''_0)) = \mathcal J(f)(z_0)(\hat{\Phi}_{z_0}(\partial^l s_j(0,z''_0)).$$
Thus, the above extension $\eqref{ext}$ intertwines the module action $\eqref{modacsec}$ on the sections of $J^k E$ and $J^k \tilde{E}$ over $\mathsf{Z}$. From now on, in the rest of the proof, we denote this extension by $\hat{\Phi}$. We also note that $\hat{\Phi}$ satisfies the equation $\eqref{jme1}$.
\smallskip
Let us now work with frames $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$ of $E$ and $\tilde{E}$, respectively, normalized at origin, and $D(0,z''):=((\hat{\Phi}_{ij}(0,z'')))_{i,j=0}^N$ be the matrix of $\hat{\Phi}$ with respect to the frames $\{\textbf{\textit{s}},\partial \textbf{\textit{s}}(0,z''),\linebreak\hdots,\partial^N \textbf{\textit{s}}(0,z'')\}$ and $\{\tilde{\textbf{\textit{s}}},\partial\tilde{\textbf{\textit{s}}}(0,z''),\hdots,\partial^N\tilde{\textbf{\textit{s}}}(0,z'')\}$. Then we wish to show that $D$ is a constant matrix on $\mathsf{Z}$.
\smallskip
We point out that the $(1,1)$-th block of $D(0,z'')$, namely, $\hat{\Phi}_{00}(0,z'')$ is the matrix representation of $\Phi: E|_{\text{res}\mathsf{Z}} \rightarrow \tilde{E}|_{\text{res}\mathsf{Z}}$ with respect to the above frames, $\textbf{\textit{s}}$ and $\tilde{\textbf{\textit{s}}}$, and hence $\hat{\Phi}_{00}(0,z'')$ is holomorphic on $\mathsf{Z}$. So recalling the proof of Claim 2 in Theorem $\ref{jean}$ with matrix valued holomorphic functions, we conclude that $\hat{\Phi}_{00}(0,z'')$ is a constant unitary matrix, say, $\hat{\Phi}_{00}$ on $\mathsf{Z}$.
\smallskip
We also note, from the construction of $\hat{\Phi}$ above, that $\hat{\Phi}(J^lE|_{\text{res}\mathsf{Z}}) \subset J^l\tilde{E}|_{\text{res}\mathsf{Z}}$, for $0 \leq l \leq k$. Consequently, $D(0,z'')$ is a lower triangular matrix. Moreover, since $\hat{\Phi}$ commutes with the module action on the sections of $J^k E|_{\text{res}\mathsf{Z}}$ and $J^k\tilde{E}|_{\text{res}\mathsf{Z}}$, the same proof as in Theorem $\ref{jean1}$ shows that entries of the matrix satisfy the properties stated in $\eqref{mb}$. So it is enough to show that $\hat{\Phi}_{l0}(0,z'')=0$ on $\mathsf{Z}$ for $0 < l \leq N$. In order to show this, we first need to establish that $\hat{\Phi}_{l0}(0,z'')$ is a holomorphic function on $\mathsf{Z}$ so that we can invoke the argument used in the Theorem $\ref{jean1}$ to show $\hat{\Phi}_{l0}(0,z'')=0$ on $\mathsf{Z}$ using the Lemma $\ref{lem1}$ and Proposition $\ref{prop1}$.
\smallskip
Here we prove our claim by using mathematical induction. For the base case, let us calculate $\hat{\Phi}_{10}(0,z'')$ from the equation $\eqref{jme1}$ by equating $10$-th block and we have
\begin{eqnarray*}
\hat{\Phi}_{10}(0,z'') &=& (\partial_1 H(0,z'')\hat{\Phi}_{00}-\hat{\Phi}_{00}\partial_1 \tilde{H}(0,z''))\tilde{H}^{-1}(0,z'').
\end{eqnarray*}
For $d+1 \leq i \leq m$, differentiating both sides of the above equation with respect to $\overline{z}_i$ we obtain
\begin{eqnarray*}
\bar\partial_i\hat{\Phi}_{10}(0,z'') &=& \bar\partial_i[\partial_1 H(0,z'')\hat{\Phi}_{00}\tilde{H}^{-1}(0,z'')]-\hat{\Phi}_{00}\bar\partial_i[\partial_1 \tilde{H}(0,z'')\tilde{H}^{-1}(0,z'')]\\
&=& \bar\partial_i[\partial_1 H(0,z'')H^{-1}(0,z'')]\hat{\Phi}_{00}-\hat{\Phi}_{00}\bar\partial_i[\partial_1 \tilde{H}(0,z'')\tilde{H}^{-1}(0,z'')]
\end{eqnarray*}
where the second inequality holds as $\hat{\Phi}_{00}$ is a constant unitary matrix satisfying the equation $\eqref{jme1}$. Then using condition (iii) with $l=1$ we have $\bar\partial_i\hat{\Phi}_{10}(0,z'')=0$ on $\mathsf{Z}$, for $d+1 \leq i \leq m$, which completes the proof of the base case. Now let $\bar\partial_i\hat{\Phi}_{j0}(0,z'')=0$ on $\mathsf{Z}$, for $0 < j \leq l$, $d+1 \leq i \leq m$ and for some $1 \leq l \leq N$. Since $\hat{\Phi}_{j0}$ is holomorphic on $\mathsf{Z}$, for $0 \leq j \leq l$, recalling the equation $\eqref{mb}$, and using Lemma $\ref{lem1}$ and Proposition $\ref{prop1}$ as in the proof of Claim 2 in Theorem $\ref{jean}$ with matrix valued holomorphic functions, we conclude that the $(l+1)$-th row of $D$ contains only two non-zero blocks, namely, $\hat{\Phi}_{l+10}$ and $\hat{\Phi}_{l+1l+1}$. Therefore, from the equation $\eqref{jme1}$ by equating $l+10$-th block we have \begin{eqnarray*}
\hat{\Phi}_{l+10}(0,z'') &=& (\partial^{l+1} H(0,z'')\hat{\Phi}_{l+1l+1}(0,z'')-\hat{\Phi}_{l+1l+1}(0,z'')\partial^{l+1} \tilde{H}(0,z''))\tilde{H}^{-1}(0,z'').
\end{eqnarray*}
Now as before, for $d+1 \leq i \leq m$, applying the differential operator $\bar\partial_i$ both sides of the equation above and recalling that $\hat{\Phi}_{l+1l+1}(0,z'')=\hat{\Phi}_{00}$, for $(0,z'')\in \mathsf{Z}$, we get
\begin{eqnarray*}
\bar\partial_i\hat{\Phi}_{l+10}(0,z'') &=& \bar\partial_i[\partial^{l+1} H(0,z'')\hat{\Phi}_{00}\tilde{H}^{-1}(0,z'')]-\hat{\Phi}_{00}\bar\partial_i[\partial^{l+1} \tilde{H}(0,z'')\tilde{H}^{-1}(0,z'')]\\
&=& \bar\partial_i[\partial^{l+1} H(0,z'')H^{-1}(0,z'')]\hat{\Phi}_{00}-\hat{\Phi}_{00}\bar\partial_i[\partial^{l+1} \tilde{H}(0,z'')\tilde{H}^{-1}(0,z'')]
\end{eqnarray*}
Then by condition (iii) we conclude that $\bar\partial_i\hat{\Phi}_{l+10}(0,z'')=0$ on $\mathsf{Z}$ for $d+1 \leq i \leq m$. So by mathematical induction $\hat{\Phi}_{l0}(0,z'')$ is holomorphic on $\mathsf{Z}$ for $0 \leq l \leq N$. Now, as in the proof of Theorem $\ref{jean1}$, Proposition $\ref{prop1}$ and Lemma $\ref{lem1}$ together yield our desired conclusion.
\end{proof}
\begin{rem}
From the proof of Theorem $\ref{mthm}$, it is clear that, for $r=1$ and $d=k=2$, condition (i), (ii) and (iii) of Theorem $\ref{mthm}$ together yield that the curvatures of the bundles $E|_{\text{res}\mathsf{Z}}$ and $\tilde{E}|_{\text{res}\mathsf{Z}}$ are equal. Further, the matrix in $\eqref{msi}$ turns out to be the diagonal matrix $\psi_{00}I$ with respect to a normalized frame at origin where $I$ is the identity matrix of order $3$. Moreover, following the proof of Claim 2 in Theorem $\ref{jean}$ we see that $\psi_{00}$ is a constant function on $\mathcal{Z}$ with $|\psi_{00}|=1$. Thus, Theorem $\ref{mthm}$ is exact generalization of Lemma $\ref{trial}$.
\smallskip
We also note that three conditions $(i),(ii)$ and $(iii)$ listed in the theorem above correspond to the condition that the metric of $E$ and $\tilde{E}$ are equivalent to order $k$, in the sense of the paper \cite{EQHMII}, on $\mathsf{Z}$ while the codimension of $\mathsf{Z}$ is $1$. Consequently, following \cite[Remark 6.1]{EQHMII}, we see that the condition $(iii)$ in the above theorem corresponds to equality of the second fundamental forms for the inclusion $E|_{\text{res}\mathsf{Z}} \subset J^2E|_{\text{res}\mathsf{Z}}$ and $\tilde{E}|_{\text{res}\mathsf{Z}}\subset J^2\tilde{E}|_{\text{res}\mathsf{Z}}$, for $k=2$.
\end{rem}
\section{Examples and Application}\label{exap}
For $\lambda \geq 0$, let ${\mathcal H}^{(\lambda)}$ be the Hilbert space of holomorphic functions on ${\mathbb D}$ with the reproducing kernel $K^{(\lambda)}(z,w)=(1-z \overline{w})^{-\lambda}$ for $z,w \in {\mathbb D}$. It is then evident that the set $\{e^{(\lambda)}_n(z):=c_n^{-\frac{1}{2}}z^n:n \geq 0 \}$ forms a complete orthonormal set in ${\mathcal H}^{(\lambda)}$ where $c_n$ are the $n$-th coefficient of the power series expansion of $(1-|z|^2)^{- \lambda}$, in other words, $$ c_n = {-{\lambda} \choose n} = \frac{\lambda(\lambda+1)(\lambda+2)\cdots (\lambda+n-1)}{n!}.$$ Let us recall that for $\lambda \geq 0$, the natural action of polynomial ring ${\mathbb C}[z]$ on each Hilbert space ${\mathcal H}^{(\lambda)}$ makes it into a Hilbert module over ${\mathbb C}[z]$. We also point out that, for $\lambda > 1$, ${\mathcal H}^{(\lambda)}$ becomes a Hilbert module over the disc algebra ${\mathcal A}({\mathbb D})$.
\medskip
It is well known that product of two reproducing kernels is also a reproducing kernel \cite[8]{TRK}. So, for $\alpha=({\alpha}_1,\hdots,{\alpha}_m)$ with ${\alpha}_i\geq 0$, $i=1,\hdots,m$, let us consider the Hilbert space ${\mathcal H}^{(\alpha)}:= {\mathcal H}^{({\alpha}_1)}\otimes \cdots \otimes {\mathcal H}^{({\alpha}_m)}$ with the natural choice of complete orthonormal set $\{e^{({\alpha}_1)}_{i_1}(z)\otimes \cdots \otimes e^{({\alpha}_m)}_{i_m}(z):i_j \geq 0, j=1,\hdots,m\}$. Then under the identification of the functions $z_1^{i_1}\cdots z_m^{i_m}$ on ${\mathbb D}^m:={\mathbb D}\times\cdots\times{\mathbb D}$, ${\mathcal H}^{(\alpha)}$ naturally possesses an obvious reproducing kernel $$K^{(\alpha)}(z,w):=(1-z_1\overline{w}_1)^{-{\alpha}_1}\cdots(1-z_m\overline{w}_m)^{-{\alpha}_m}$$ on ${\mathbb D}^m$. Furthermore, the natural action of ${\mathbb C}[\textbf{z}]$ on ${\mathcal H}^{(\alpha)}$ makes it a Hilbert module over ${\mathbb C}[\textbf{z}]$, for ${\alpha}_i \geq 0$, $i=1,\hdots,m$, where by ${\mathbb C}[\textbf{z}]$ we mean ${\mathbb C}[z_1,\hdots,z_m]$.
\medskip
Let us now consider the subspace ${\mathcal H}^{(\alpha)}_0$ consisting of holomorphic functions in ${\mathcal H}^{(\alpha)}$ which vanish to order $2$ along the diagonal $\Delta:=\{(z_1,\hdots,z_m)\in {\mathbb D}^m:z_1=\cdots=z_m\}$, that is, following the definition given in Section $\ref{SM}$, $${\mathcal H}^{(\alpha)}_0=\{f \in {\mathcal H}^{(\alpha)}:f=\partial_1 f= \cdots =\partial_m f = 0\,\,\text{on}\,\,\Delta\}.$$
We are now interested in describing the quotient space ${\mathcal H}^{(\alpha,\beta,\gamma)}_q := {\mathcal H}^{(\alpha,\beta,\gamma)} \ominus {\mathcal H}^{(\alpha,\beta,\gamma)}_0$ in case of $m=3$, where $\alpha,\beta,\gamma\geq 0$.
\smallskip
In order to achieve our goal we first compute an orthonormal basis for ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ with the help of which we find the desired expression of the reproducing kernel $K^{(\alpha,\beta,\gamma)}_q$ of ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ obtained from the Theorem $\ref{qm}$. We then present an application of the Theorem $\ref{mthm}$ showing that the unitary equivalence classes of weighted Hardy modules are precisely determined by those of the quotient modules obtained from the submodules of functions vanishing to order at least $2$ along the diagonal set $\Delta:=\{(z_1,z_2,z_3)\in{\mathbb D}^3:z_1=z_2=z_3\}$.
\subsection{Examples}
We start with the submodule ${\mathcal H}^{(\alpha,\beta,\gamma)}_0$ which is the closure of the ideal \linebreak $I:=<(z_1-z_2)^2,(z_1-z_2)(z_1-z_3),(z_1-z_3)^2>$ in the Hilbert space ${\mathcal H}^{(\alpha,\beta,\gamma)}$. It then follows that $B:=\{z_1^i z_2^j z_3^k(z_1-z_2)^2,z_1^i z_2^j z_3^k(z_1-z_2)(z_1-z_3),z_1^i z_2^j z_3^k(z_1-z_3)^2: i,j,k \in \mathbb{N} \cup \{0\}\}$ is a spanning set for the submodule ${\mathcal H}^{(\alpha,\beta,\gamma)}_0$. So to calculate an orthonormal basis for ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ it is enough to find an orthonormal basis for the orthogonal complement of $B$. It is easily verified that $\{g^{(p)}_1,g^{(p)}_2,g^{(p)}_3:p \in \mathbb{N} \cup \{0\}\}$ forms a basis for $B^{\perp}$ where
$$ g^{(p)}_1 := \sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{z^{p-l-k}_1 z^{k}_2}{\norm {z^{p-l-k}_1}^2\norm {z^k_2}^2}\right)\frac{z^l_3}{{\norm {z^l_3}}^2},~
g^{(p)}_2 := \sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{l z^{p-l-k}_1 z^{k}_3}{\norm {z^{p-l-k}_1}^2\norm {z^k_3}^2}\right)\frac{z^l_2}{{\norm {z^l_2}}^2}$$
$$ g^{(p)}_3 := \sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{l z^{p-l-k}_1 z^{k}_2}{\norm {z^{p-l-k}_1}^2\norm {z^k_2}^2}\right)\frac{z^l_3}{{\norm {z^l_3}}^2}.$$
Then note that the corresponding orthogonal basis for $B^{\perp}$ is $\{f^{(p)}_1,f^{(p)}_2,f^{(p)}_3:p \in \mathbb{N} \cup \{0\}\}$ with
$$ f^{(p)}_1 := g^{(p)}_1,~ f^{(p)}_2 := \left \< g^{(p)}_1,g^{(p)}_2\right \> g^{(p)}_1 - \norm{g^{(p)}_1}^2 g^{(p)}_2,~ f^{(p)}_3 := \left \< \tilde{f}^{(p)}_3,f^{(p)}_2\right \> f^{(p)}_2 - \norm{f^{(p)}_2}^2 \tilde{f}^{(p)}_3 $$
where $\tilde{f}^{(p)}_3$ is orthogonal to $g^{(p)}_1$ and is given by the following formula $$\tilde{f}^{(p)}_3 = \left\<g^{(p)}_1,g^{(p)}_3\right\>g^{(p)}_1-\norm{g^{(p)}_1}^2 g^{(p)}_3.$$
Thus, our required orthonormal set of vectors in the quotient module ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ is $$\mathscr{B}:=\left\{e^{(p)}_1=\frac{f^{(p)}_1}{\norm{f^{(p)}_1}},e^{(p)}_2=\frac{f^{(p)}_2}{\norm{f^{(p)}_2}},e^{(p)}_3=\frac{f^{(p)}_3}{\norm{f^{(p)}_3}}\right\}_{p=0}^{\infty}.$$
Following the Theorem $\ref{qm}$ to calculate the reproducing kernel we need to describe the unitary map
\begin{eqnarray}\label{jac}
h \mapsto \textbf{h}|_{\Delta}:= {\sum_{l=0}^N \partial^l h \otimes equivalent_l}|_{\Delta}
\end{eqnarray}
for $h \in {\mathcal H}^{(\alpha,\beta,\gamma)}$ where $N=\frac{k(k-1)}{2}$. In our calculation, for $k=2$, it is enough to determine the action of this map on the orthonormal basis $\mathscr{B}$. In this context the following calculations provide us the necessary ingredients to compute the action of this unitary map.
\medskip
We first note that
\begin{eqnarray*}
(1-|z_1|^2)^{-(\alpha + \beta + \gamma)} &=& (1-|z_1|^2)^{-\alpha}(1-|z_2|^2)^{-\beta}(1-|z_3|^2)^{-\gamma}|_{\Delta}\\
&=& \sum_{p=0}^{\infty}\left[\sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{1}{\norm {z^{p-l-k}_1}^2\norm {z^k_2}^2}\right)\frac{1}{{\norm {z^l_3}}^2}\right]\cdot|z_1|^{2p}\\
\end{eqnarray*}
which implies that $\norm{f^{(p)}_1}^2$ is the $p$-th coefficient of the power series expansion of $(1-|z_1|^2)^{-(\alpha + \beta + \gamma)}$. Thus, we have $ \norm{f^{(p)}_1}^2= {-(\alpha+\beta+\gamma) \choose p}$. Further we point out that
\begin{eqnarray*}
\beta(1-|z_1|^2)^{-(\alpha + \beta + \gamma +1)} &=& (1-|z_1|^2)^{-\alpha}(1-|z_3|^2)^{-\gamma}\frac{d}{d|z_2|^2}(1-|z_2|^2)^{-\beta}|_{\Delta}\\
&=& \sum_{p=0}^{\infty}\left[\sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{1}{\norm {z^{p-l-k}_1}^2\norm {z^k_3}^2}\right)\frac{l}{{\norm {z^l_2}}^2}\right]\cdot|z_1|^{2(p-1)}\\
\end{eqnarray*}
which, as before, together with a similar calculation for $\gamma(1-|z_1|^2)^{-(\alpha + \beta + \gamma +1)}$ ensure that \begin{eqnarray*} \left\<g^{(p)}_1,g^{(p)}_2\right\>=\beta {-(\alpha+\beta+\gamma+1) \choose (p-1)}\,\,\,\text{and}\,\,\,\left\<g^{(p)}_1 ,g^{(p)}_3 \right\>=\gamma{ -(\alpha+\beta+\gamma+1) \choose (p-1)}.\end{eqnarray*}
\smallskip
Now to calculate the inner product of $g^{(p)}_2$ and $g^{(p)}_3$ we consider the following power series
\begin{eqnarray*}
\beta\gamma(1-|z_1|^2)^{-(\alpha + \beta + \gamma +2)} &=& (1-|z_1|^2)^{-\alpha}\frac{d}{d|z_2|^2}(1-|z_2|^2)^{-\beta}\frac{d}{d|z_3|^2}(1-|z_3|^2)^{-\gamma}|_{\Delta}\\
&=& \sum_{p=0}^{\infty}\left[\sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{k(p-l-k)}{\norm {z^{p-l-k}_2}^2\norm {z^k_3}^2}\right)\frac{1}{{\norm {z^l_1}}^2}\right]\cdot|z_1|^{2(p-2)}\\
\end{eqnarray*}
which shows that $\left\<g^{(p)}_2,g^{(p)}_3\right\>=\beta\gamma{-(\alpha+\beta+\gamma+2)\choose (p-2)}$. Furthermore, we have
\begin{eqnarray*}
\beta(1+\beta |z_1|^2)(1-|z_1|^2)^{-(\alpha+\beta+\gamma+2)} &=& (1-|z_1|^2)^{-\alpha}(1-|z_3|^2)^{-\gamma}\left(\frac{d}{d|z_2|^2}(|z_2|^2\frac{d}{d|z_2|^2}(1-|z_2|^2)^{-\beta})\right)\rvert_{\Delta}\\
&=& \sum_{p=0}^{\infty}\left[\sum_{l=0}^{p}\left(\sum_{k=0}^{p-l}\frac{1}{\norm {z^{p-l-k}_1}^2\norm {z^k_3}^2}\right)\frac{l^2}{{\norm {z^l_2}}^2}\right]\cdot|z_1|^{2(p-1)}\\
\end{eqnarray*}
and hence with the help of a similar calculation we find that
$$\norm{g^{(p)}_2}^2=\beta\left({-(\alpha+\beta+\gamma+2)\choose (p-1)}+\beta{-(\alpha+\beta+\gamma+2)\choose (p-2)}\right)~ \text{and} $$
$$\norm{g^{(p)}_3}^2=\gamma\left({-(\alpha+\beta+\gamma+2)\choose (p-1)}+\gamma{-(\alpha+\beta+\gamma+2)\choose (p-2)}\right).$$
\smallskip
Thus, the above calculations lead us to compute the norms of the vectors $\{f^{(p)}_1,f^{(p)}_2,f^{(p)}_3:p \in \mathbb{N} \cup \{0\}\}$. We have already showed that $\norm{f^{(p)}_1}^2$ is $-(\alpha+\beta+\gamma)\choose p$ and recalling the definition of $f^{(p)}_2$ and $f^{(p)}_3$ it is easily seen that
\begin{eqnarray*}
\norm{f^{(p)}_2}^2 &=& \norm{g^{(p)}_1}^2\left(\norm{g^{(p)}_1}^2\norm{g^{(p)}_2}^2-|\left\<g^{(p)}_1,g^{(p)}_2\right\>|^2\right)\\
&=& \frac{\beta(\alpha+\gamma)}{(\alpha+\beta+\gamma)}{-(\alpha+\beta+\gamma)\choose p}^2{-(\alpha+\beta+\gamma+2)\choose (p-1)}\\
\end{eqnarray*}
and a similarly one can find, from the definition of $f^{(p)}_3$, that
$$ \norm{f^{(p)}_3}^2 = \frac{\alpha{\beta}^2 \gamma(\alpha+\gamma)}{(\alpha+\beta+\gamma)^2}{-(\alpha+\beta+\gamma)\choose p}^6{-(\alpha+\beta+\gamma+2)\choose (p-1)}^3.$$
We also point out that similar computations as above give rise to following identities:
$$ \partial_1 g^{(p)}_1 = \alpha {-(\alpha+\beta+\gamma+1) \choose (p-1)},\,\,\,\partial_2 g^{(p)}_1=\beta{ -(\alpha+\beta+\gamma+1) \choose (p-1)},$$
$$\partial_1 g^{(p)}_2 = \alpha\beta {-(\alpha+\beta+\gamma+2) \choose (p-2)},\,\,\,
\partial_2 g^{(p)}_2 = \beta\left({-(\alpha+\beta+\gamma+2)\choose (p-1)}+\beta{ -(\alpha+\beta+\gamma+2) \choose (p-2)}\right),$$
$$\partial_1 g^{(p)}_3 = \alpha\gamma {-(\alpha+\beta+\gamma+2) \choose (p-2)}\,\,\,\text{and}\,\,\,\partial_2 g^{(p)}_3=\beta\gamma{ -(\alpha+\beta+\gamma+2) \choose (p-2)}.$$
\medskip
So we are now in position to calculate the orthonormal basis for the quotient module ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ and their derivatives along $z_1$ and $z_2$ direction restricted to the diagonal set $\Delta$ which we exactly require to compute the reproducing kernel of the quotient module. Let us begin, from $\eqref{jac}$, by pointing out that
$$
e^{(p)}_1 \mapsto
\left(
\begin{array}{c}
{-(\alpha+\beta+\gamma)\choose p}^{\frac{1}{2}}z_1^p\\
\alpha\sqrt{\frac{p}{\alpha+\beta+\gamma}}{-(\alpha+\beta+\gamma+1)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
\beta\sqrt{\frac{p}{\alpha+\beta+\gamma}}{-(\alpha+\beta+\gamma+1)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
\end{array}
\right),
~e^{(p)}_2 \mapsto
\left(
\begin{array}{c}
0\\
\frac{\alpha\beta}{\sqrt{\beta(\alpha+\gamma)}}\frac{1}{\sqrt{\alpha+\beta+\gamma}}{-(\alpha+\beta+\gamma+2)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
\frac{\beta\gamma}{\sqrt{\beta(\alpha+\gamma)}}\frac{1}{\sqrt{\alpha+\beta+\gamma}}{-(\alpha+\beta+\gamma+2)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
\end{array}
\right)$$
$$\text{and}~~
e^{(p)}_3 \mapsto
\left(
\begin{array}{c}
0\\
\sqrt{\frac{\alpha\gamma}{\alpha+\gamma}}{-(\alpha+\beta+\gamma+2)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
-\sqrt{\frac{\alpha\gamma}{\alpha+\gamma}}{-(\alpha+\beta+\gamma+2)\choose (p-1)}^{\frac{1}{2}}z^{p-1}_1\\
\end{array}
\right).\\
$$
\smallskip
This allows us to compute the reproducing kernel of the quotient module ${\mathcal H}^{(\alpha,\beta,\gamma)}_q$ as follows
$$K_q(\textbf{z},\textbf{w})=\sum_{p=0}^{\infty}e^{(p)}_1(\textbf{z})\cdot e^{(p)}_1(\textbf{w})^* + e^{(p)}_2(\textbf{z})\cdot e^{(p)}_2(\textbf{w})^* + e^{(p)}_3(\textbf{z})\cdot e^{(p)}_3(\textbf{w})^*,\,\,\textbf{z},\textbf{w}\in {\mathbb D}^3$$
which is a $3\times 3$ matrix valued function $((K_q(\textbf{z},\textbf{z})_{ij}))_{i,j=1}^3$ on ${\mathbb D}^3$ as expected. To compute the kernel $K_q(\textbf{z},\textbf{z})$ for $\textbf{z} \in \Delta$ we note, for $\textbf{z}=(z_1,z_1,z_1)$ in $\Delta$, that
$$
K_q(\textbf{z},\textbf{z})_{11} = K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z}),~
K_q(\textbf{z},\textbf{z})_{12} = \partial_1K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z}),~
K_q(\textbf{z},\textbf{z})_{13} = \partial_2K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z}),
$$
\begin{eqnarray*}
K_q(\textbf{z},\textbf{z})_{23} &=& \frac{\alpha\beta}{\alpha+\beta+\gamma}\frac{d}{d|z_1|^2}\left({|z_1|^2(1-|z_1|^2)^{-(\alpha+\beta+\gamma+1)}}\right)\\ &+& \left(\frac{\alpha\beta\gamma}{(\alpha+\gamma)(\alpha+\beta+\gamma)}-\frac{\alpha\gamma}{\alpha+\gamma}\right)(1-|z_1|^2)^{-(\alpha+\beta+\gamma+2)}\\
&=& \alpha\beta |z_1|^2 (1-|z_1|^2)^{-(\alpha+\beta+\gamma+2)}\\ &=& \bar\partial_1\partial_2K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z}),\\
K_q(\textbf{z},\textbf{z})_{22} &=& \frac{{\alpha}^2}{\alpha+\beta+\gamma}\frac{d}{d|z_1|^2}\left({|z_1|^2(1-|z_1|^2)^{-(\alpha+\beta+\gamma+1)}}\right)\\ &+& \left(\frac{{\alpha}^2\beta}{(\alpha+\gamma)(\alpha+\beta+\gamma)}+\frac{\alpha\gamma}{\alpha+\gamma}\right)(1-|z_1|^2)^{-(\alpha+\beta+\gamma+2)}\\
&=& [\alpha+{\alpha}^2 |z_1|^2] (1-|z_1|^2)^{-(\alpha+\beta+\gamma+2)}\\ &=& \partial_1\bar\partial_1K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z}),\\
\end{eqnarray*}
and similar calculations also yield that $K_q(\textbf{z},\textbf{z})_{21}=\bar\partial_1K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z})$, $K_q(\textbf{z},\textbf{z})_{31}=\bar\partial_2 K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z})$, $K_q(\textbf{z},\textbf{z})_{32}=\bar\partial_2\partial_1K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z})$, and $K_q(\textbf{z},\textbf{z})_{33}=\partial_2\bar\partial_2 K^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z})$. Thus, we have
$$K_q(\textbf{z},\textbf{z})|_{\Delta}=JK^{(\alpha,\beta,\gamma)}(\textbf{z},\textbf{z})|_{\Delta}$$
which verifies the Theorem $\ref{qm}$.
\medskip
\subsection{Application}
Let us consider the family of Hilbert modules $\textit{Mod}({\mathbb D}^m):=\{{\mathcal H}^{(\alpha)}:\alpha=(\alpha_1,\hdots,\alpha_m)\geq 0\}$ over the polydisc ${\mathbb D}^m$ in ${\mathbb C}^m$. In this subsection we prove that for any pair of tuples $\alpha=(\alpha_1,\hdots,\alpha_m)$ and $\alpha'=(\alpha_1',\hdots,\alpha_m')$, the unitary equivalence of two quotient modules ${\mathcal H}^{(\alpha)}_q$ and ${\mathcal H}^{(\alpha')}_q$, obtained from the submodules of functions vanishing of order $2$ along the diagonal set $\Delta$, implies the equality of the Hilbert modules ${\mathcal H}^{(\alpha)}$ and ${\mathcal H}^{(\alpha')}$. In other words, the restriction of the curvature of the jet bundle $J^2E^{(\alpha)}$ to the diagonal $\Delta$ is a complete unitary invariant for the class $\textit{Mod}({\mathbb D}^m)$ where the jet bundle $J^2E^{(\alpha)}$ is defined by the global frame $\{K^{(\alpha)}(.,\overline{w}),\partial_1 K^{(\lambda)}(.,\overline{w}),\hdots,\partial_m K^{(\lambda)}(.,\overline{w})\}$ where $\partial_j$ are the differential operators with respect to the variable $z_j$, for $j=1,\hdots,m$
\begin{thm}\label{WHM}
For $\alpha=(\alpha_1,\hdots,\alpha_n)$ and ${\alpha}'=({\alpha}'_1,\hdots,{\alpha}'_n)$ with $\alpha_i,{\alpha}'_i \geq 0$, for all $i=1,\hdots,n$, the quotient modules ${\mathcal H}^{(\alpha)}_q$ and ${\mathcal H}^{({\alpha}')}_q$ are unitarily equivalent if and only if $\alpha_i={\alpha}'_i$, for all $i=1,\hdots,n$.
\end{thm}
\begin{proof}
The proof of sufficiency is trivial. So we only prove the necessity. Let us begin by pointing out that the diagonal set $\Delta$ in ${\mathbb D}^m$ can be described as the zero set of the ideal $I:=<z_1-z_2,\hdots,z_i-z_{i+1},\hdots,z_{m-1}-z_m>$. Then it is easy to verify that $\phi: U \rightarrow {\mathbb C}^m$ defined by $$\phi(z_1,\hdots,z_m)=(z_1-z_2,\hdots,z_i-z_{i+1},\hdots,z_{m-1}-z_m,z_m)$$ yields an admissible coordinate system (Definition $\ref{ad}$) around the origin. We choose $U$ small enough so that $\phi(U)\subset {\mathbb D}^m$. A simple calculation then shows that $\phi^{-1}:\phi(U)\rightarrow U$ takes the form $$\phi^{-1}(u_1,\hdots,u_m)=(\sum_{j=1}^m u_j, \hdots,\sum_{j=i}^m u_j,\hdots,u_{m-1}+u_m, u_m).$$
For rest of the proof we pretend $U$ to be ${\mathbb D}^m$ thanks to the Remark $\ref{remres}$. Now recalling the Proposition $\ref{eococh}$ it is enough to prove that ${\alpha}_i={\alpha}'_i$, $i=1,\hdots,m$, provided $\phi^*{\mathcal H}^{(\alpha)}_q$ is uitarily equivalent to $\phi^*{\mathcal H}^{({\alpha}')}_q$ where $\phi^*{\mathcal H}^{(\alpha)}_q$ and $\phi^*{\mathcal H}^{({\alpha}')}_q$ are the quotient modules obtained from the submodules $\phi^*{\mathcal H}^{(\alpha)}_0$ and $\phi^*{\mathcal H}^{({\alpha}')}_0$ of the Hilbert modules $\phi^*{\mathcal H}^{(\alpha)}$ and $\phi^*{\mathcal H}^{({\alpha}')}$, respectively.
\smallskip
We now note that $\phi^*{\mathcal H}^{(\alpha)}$ and $\phi^*{\mathcal H}^{({\alpha}')}$ are reproducing kernel Hilbert modules with reproducing kernels
$$\mathsf{K}(\textbf{u})=\prod_{i=1}^m\left(1-|{\sum_{j=1}^mu_j}|^2\right)^{-{\alpha}_i}\,\,\,\text{and}\,\,\,\mathsf{K'}(\textbf{u})=\prod_{i=1}^m\left(1-|{\sum_{j=1}^mu_j}|^2\right)^{-{\alpha}'_i},$$
respectively, where $\textbf{u}=(u_1,\hdots,u_m)\in \phi(U)$. We also pint out that the submodules $\phi^*{\mathcal H}^{(\alpha)}_0$ and $\phi^*{\mathcal H}^{({\alpha}')}_0$ consists of functions in $\phi^*{\mathcal H}^{(\alpha)}$ and $\phi^*{\mathcal H}^{({\alpha}')}$, respectively, vanishing along the submanifold $\mathsf{Z}:=\{(0,\hdots,0,u_m):u_m \in {\mathbb D}\}\cap \phi(U)$ of order $2$.
\smallskip
Since $\phi^*{\mathcal H}^{(\alpha)}_q$ and $\phi^*{\mathcal H}^{({\alpha}')}_q$ are unitarily equivalent recalling the Theorem $\ref{mthm}$ we conclude that \begin{eqnarray}\label{eqn3}\mathcal{K}|_{\mathsf{Z}}&=&\mathcal{K}'|_{\mathsf{Z}}\end{eqnarray} where $\mathcal{K}$ and $\mathcal{K}'$ are the curvature matrix for the vector bundles $E$ and $E'$ over $\phi(U)$ obtained from the Hilbert modules $\phi^*{\mathcal H}^{(\alpha)}$ and $\phi^*{\mathcal H}^{({\alpha}')}$, respectively. Now we have, by definition, $\mathcal{K}(\textbf{u})=((\mathcal{K}_{ij}(\textbf{u})))_{i,j=1}^m$ where
$$\mathcal{K}_{ij}(\textbf{u})=\frac{\partial^2}{\partial u_i\partial \overline{u}_j}\log\mathsf{K}(\textbf{u},\textbf{u}),$$ for $\textbf{u}=(u_1,\hdots,u_m)\in \phi(U)$. Thus, for $1 \leq i \leq m$ and $\textbf{u}\in \phi(U)$,
$$
\mathcal{K}_{ii}(\textbf{u}) = \frac{\partial^2}{\partial u_i\partial \overline{u}_i}\log\mathsf{K}(\textbf{u},\textbf{u}) = \sum_{l=1}^i {\alpha}_l\left(1-|{\sum_{j=l}^m u_j}|^2\right)^{-1}.
$$
A similar computation also yields, for $i=1,\hdots,m$ and $\textbf{u} \in \phi(U)$, that
$$\mathcal{K}'_{ii}(\textbf{u}) = \sum_{l=1}^i {\alpha}'_l\left(1-|{\sum_{j=l}^m u_j}|^2\right)^{-1}.$$
We now note that, for $\textbf{u}\in \mathsf{Z}$, $$\mathcal{K}_{ii}(\textbf{u})=\frac{\sum_{l=1}^i {\alpha}_l}{(1-|u_m|^2)}\,\,\,\text{and}\,\,\,\mathcal{K}'_{ii}(\textbf{u})=\frac{\sum_{l=1}^i {\alpha}'_l}{(1-|u_m|^2)}.$$
Thus, by using the equality in equation $\eqref{eqn3}$ it is not hard to see that ${\alpha}_i={\alpha}'_i$, $i=1,\hdots,m$.
\end{proof}
\medskip
\textbf{Acknowledgement.} The author thanks to Dr. Shibananda Biswas and Prof. Gadadhar Misra for their support and valuable suggestions in preparation of this paper.
\medskip
\bibliographystyle{plain}
|
2,869,038,154,909 | arxiv | \section{Introduction}
Recently, results from several high-statistics measurements of $\eebar$ annihilation into multipion
states have been published \cite{FOCUS,BABAR,BABAR2,CMD,CMD2}. One of the striking features in the
data is a pronounced structure in the vicinity of the antinucleon-nucleon ($\NNbar$) threshold
that appears either as a sharp drop for $e^+e^- \to 3(\pi^+\pi^-)$ \cite{BABAR,CMD} or as a dip
for $e^+e^- \to 2(\pi^+\pi^-\pi^0)$ \cite{BABAR,CMD2}, $e^+e^- \to \omega\pi^+\pi^-\pi^0$ \cite{BABAR},
and for $e^+e^- \to 2(\pi^+\pi^-)\pi^0$ \cite{BABAR2} in the reaction cross section.
Earlier measurements with lower statistics had already suggested the presence of such a dip,
cf. the review \cite{Whalley}.
Phenomenological fits to the $\eebar$ data locate the structure at $1.91$ GeV \cite{FOCUS1}
or at $1.86$--$1.88$ GeV \cite{BABAR} while the $\ppbar$ threshold is at $1.8765$ GeV.
Naturally, this very proximity of the $\bar NN$ threshold suggests that the $\bar NN$ channel
should have something to do with the appearance of that structure in the multipion production cross
sections. A common speculation is that it could be a signal for a $\bar pp$ bound state
\cite{FOCUS1,CMD} or a subtreshold $\bar pp$ resonance \cite{CMD2}.
Such a conjecture seems to be also in line with experimental findings
in a related reaction, namely $\eebar \to \ppbar$, where a near-threshold enhancement
seen in the cross section \cite{Aubert,Lees} is likewise associated with
a possible $\bar pp$ bound state, cf. also Ref.~\cite{Antonelli:1996}.
For a discussion of other structures seen in
$\eebar \to \ppbar$, see e.g. Ref.~\cite{Lorenz:2015pba}.
Given the complexity of the reaction mechanism one cannot expect a microscopic calculation
of those multipion production cross sections soon. Indeed, so far there is not even a
calculation that quantifies the impact of the opening of the $\bar NN$ channel on
those reactions.
The idea that $e^+e^- \to 6\pi$ and $\eebar \to \ppbar$ could be closely related is
picked up in Ref.~\cite{Russian} for interpreting the drop/dip.
Aspects related to the question of an $\bar NN$ bound state are discussed in
Ref.~\cite{Rosner} where it is emphasized that ordinary threshold effects like
cusps could also explain the structures seen in the multipion channels.
In the present paper we investigate the significance of the $\bar NN$ channel for the
$e^+e^- \to 5\pi$ and $e^+e^- \to 6\pi$ reactions. Specifically, we aim at a reliable
estimation of the influence of the $\bar NN$ channel on those multipion production cross
sections around the $\bar NN$ threshold. The calculation builds on earlier works
published in Refs.~\cite{NNbar,NPA}.
This concerns (i) an $\NNbar$ potential constructed in the framework of
chiral effective field theory (EFT), that reproduces the amplitudes determined
in a partial-wave analysis of $\ppbar$ scattering data \cite{Timmermans}
from the $\NNbar$ threshold up to laboratory energies of $T_{lab} \approx 200-250$ MeV \cite{NNbar}
and (ii) an analysis of the reaction $\eebar \to \ppbar$ (and $\ppbar \to \eebar$, respectively)
where the effect of the interaction in the $\NNbar$ system was taken into account
rigorously \cite{NPA} and where the experimentally observed near-threshold enhancement in
the cross section and the associated steep rise of the electromagnetic form factors in the
time-like region is explained solely in terms of the $\ppbar$ interaction.
Note that the employed $\NNbar$ interaction is also able to describe the large near-threshold
enhancement observed in the reaction $J/\psi \to \gamma\ppbar$ \cite{Jpsi}.
\section{Formalism}
The estimation of the influence of the $\bar NN$ channel is done in the same
framework as the studies mentioned above \cite{NNbar,NPA} and consistently with them.
It amounts to solving the following formal set of coupled equations:
\begin{eqnarray}
\nonumber
T_{\bar NN\to \bar NN} &=& V_{\bar NN\to \bar NN} + V_{\bar NN\to \bar NN} G_0 T_{\bar NN\to \bar NN}, \\
\nonumber
T_{e^+e^- \to \bar NN} &=& V_{e^+e^- \to \bar NN} + V_{e^+e^- \to \bar NN} G_0 T_{\bar NN\to \bar NN}, \\
\label{eq:LS0}
T_{\bar NN\to \nu } &=& V_{\bar NN\to \nu } + T_{\bar NN\to \bar NN} G_0 V_{\bar NN\to \nu }, \\
\nonumber
& & \\ \nonumber
T_{e^+e^- \to \nu } &=& A_{e^+e^- \to \nu} + V_{e^+e^- \to \bar NN} G_0 T_{\bar NN\to \nu } \\
&=& A_{e^+e^- \to \nu} + T_{e^+e^- \to \bar NN} G_0 V_{\bar NN\to \nu },
\label{eq:LS}
\end{eqnarray}
with $G_0$ the free $\NNbar$ propagator and $\nu =3(\pi^+\pi^-)$, etc.
Here the first one is the Lippmann-Schwinger equation from which the $\NNbar$ scattering amplitude
is obtained, see Ref.~\cite{NNbar} for details.
The second equation provides the $e^+e^- \to \bar NN$ transition amplitude, which was calculated in
distorted-wave Born approximation in Ref.~\cite{NPA}. The third equation defines the amplitude
for $\NNbar$ annihilation into the (various) $5 \pi$ and $6 \pi$ channels. These amplitudes will be
established in the present work. Luckily there is experimental information for all considered multipion
channels, i.e. for $\ppbar \to 3(\pi^+\pi^-)$, $\ppbar \to 2(\pi^+\pi^-\pi^0)$, $\ppbar \to 2(\pi^+\pi^-)\pi^0$,
and $\ppbar \to \omega \pi^+\pi^-\pi^0$ \cite{Klempt,McCrady,Sai}, so that the corresponding
transition potentials $V_{\bar NN\to\nu }$ can be constrained by a fit to data.
Once $T_{e^+e^- \to \bar NN}$ ($V_{e^+e^- \to \bar NN}$) and $T_{e^+e^- \to \nu }$ ($V_{e^+e^- \to \nu }$)
are fixed the contributions to the $e^+e^- \to \nu$ reactions that proceed via an
intermediate $\NNbar$ state, cf. the second terms on the right hand side of Eq.~(\ref{eq:LS}),
are likewise fixed. Note that the two lines in Eq.~(\ref{eq:LS}) are equivalent.
The only unknown quantity in the equations above is $A_{e^+e^- \to\nu }$.
It stands for all other contributions to $e^+e^- \to \nu$, i.e. practically speaking it represents
the background to the loop contribution due to two-step $e^+e^- \to \NNbar \to \nu$ transition.
Of course, it is impossible to take into account the full complexity of the
$\eebar \to 5 \pi $, $\eebar \to 6 \pi $ and $\NNbar \to 5 \pi $, $\NNbar \to 6 \pi $ reactions.
Thus, in the following we want to describe the simplifications and approximations made in our study.
First of all this concerns the reaction dynamics. Following the strategy in Ref.~\cite{NNbar}, the
elementary transition (annihilation) potential for $\NNbar \to\nu$ is parameterized by
\begin{equation}
V_{\NNbar \to \nu}(q)=\tilde C_\nu + C_\nu q^2,
\label{eq:VNN}
\end{equation}
i.e. by two contact terms analogous to those that arise up to next-to-next-to-leading order (NNLO)
in the treatment of the $\NNbar$ or $NN$ interaction within chiral EFT. The quantity $q$ in
Eq.~(\ref{eq:VNN}) is the center-of mass (c.m.) momentum in the $\NNbar$ system. Since the threshold
for the production of $5$ or $6$ pions lies significantly below
the one for $\NNbar$ the pions carry - on average - already fairly high momenta. Thus,
the dependence of the annihilation potential on those momenta should be small for energies around
the $\NNbar$ threshold and it is, therefore, neglected.
The constants $\tilde C_\nu$ and $C_\nu$ are determined by a fit to the $\NNbar \to \nu$
cross section (and/or branching ratio) for each annihilation channel $\nu$.
The term $A_{\eebar \to\nu}$ is likewise parameterized in the form~(\ref{eq:VNN}), but as a
function of the $\eebar$ c.m. momentum. The arguments for this simplification are the same as
above and they are valid again, of course, only for energies
around the $\NNbar$ threshold. However, since in the $\eebar$ case this term does represent
actually a background amplitude and not a transition potential we allow the corresponding
constants to be complex numbers which are fixed by a fit to the $\eebar \to\nu$ cross sections.
In our study of the electromagnetic form factors in the time-like region \cite{NPA}
we adopted the standard one-photon approximation. In this case there are only two partial
waves that can contribute to the $\eebar \to\NNbar$ transition, namely the
(tensor) coupled ${^3S_1} - {}^3D_1$ partial waves. We make the same assumption in the
present work. For $\NNbar \to 5 \pi, 6\pi$ there are no general limitations on the
partial waves. However, since we restrict ourselves to energies close to the
$\NNbar$ threshold and we expect the annihilation operator to be of rather short range
any $\NNbar$ partial waves besides the ${^3S_1}$ and the ${^1S_0}$ should play a minor role.
In the actual calculation we use only the ${^3S_1}$ partial wave. Thus, the corresponding
transition potential $V_{\NNbar \to\nu} $ might actually overestimate the true contribution
of this partial wave and, therefore, the resulting amplitude for
the two step process $\eebar \to \NNbar \to\nu$ has to be considered as an upper
limit. Judging from available branching ratios for $\ppbar \to \omega\omega$ \cite{Klempt},
where near threshold only the ${^1S_0}$ can contribute, its contribution (to the $2(\pi^+\pi^-\pi^0)$
channel) could be in the order of 20\% of the one from the ${^3S_1}$ partial wave.
Note that there are selection rules for the $\NNbar \to 5 \pi, 6\pi$ transitions, because $G$-parity is preserved.
For $n$ pions the $G$-parity is defined by $G=(-1)^n$ while for $\bar NN$ it is given by $G=(-1)^{L+S+I}$,
where $L$, $S$, $I$ denote the orbital angular momentum, and the total spin and isospin, respectively.
Thus, the $G$-parity for the six pion final states (i.e. also for $\omega \pi^+\pi^-\pi^0$) is positive
which confines the isospin for the $\bar NN$ pair to $I=1$ in the ${^3S_1} - {}^3D_1$ partial wave.
Conversely, the five-pion decay mode can occur only from the $I=0$ ${^3S_1} - {}^3D_1$ $\NNbar$ partial wave.
The explicit form of Eq.~(\ref{eq:LS}) reads
\begin{eqnarray}\label{eq:ee6piN}
&& T_{\nu, e^+e^-}(Q,q_e;E)= A_{\nu,e^+e^-}(Q,q_e) + \sum_{\NNbar} \int_0^\infty \frac{dq q^2}{(2\pi)^3} \nl
&&\qquad \times V_{\nu,\NNbar}(Q,q) \frac{1}{E-2E_q+i0^+}T_{\NNbar,\eebar}(q,q_e;E), \nl
\end{eqnarray}
written here in matrix notation. The sum refers to $\ppbar$ and $\nnbar$ intermediate states.
The corresponding expression for $\ppbar \to \nu$ can be obtained by substituting $\eebar$ by $\ppbar$
in Eq.~(\ref{eq:ee6piN}), those for the other amplitudes in Eq.~(\ref{eq:LS0}) are given in Refs.~\cite{NNbar,NPA}.
The quantity $Q$ stands here symbolically for the momenta in the $5 \pi$ and $6 \pi$ channels.
But since we assumed that the transition potentials do not depend on the pion momenta,
cf. Eq.~(\ref{eq:VNN}), $Q$ does not enter anywhere
into the actual calculation and we do not need to specify this quantity. All amplitudes (and the potentials)
can be written and evaluated as functions of the c.m. momenta in the $\NNbar$ ($q_p$) and $\eebar$ ($q_e$)
systems and of the total energy $E=2 \sqrt{m_p^2 + q^2_p} = 2 \sqrt{m_e^2 + q^2_e} $.
The quantity $E_q$ in Eq.~(\ref{eq:ee6piN}) is given by $E_q= \sqrt{m_p^2 + q^2}$.
Since the amplitudes do not depend on $Q$ the integration over the multipion phase space can
be done separately when the cross sections are calculated. In practice, it amounts only to a
multiplicative factor and, moreover, to factors that are the same for $\eebar \to \nu$ and
$\NNbar \to \nu$ at the same total energy $E$. We performed this phase space integration
numerically at the initial stage of the present work but it became clear that we can get
more or less equivalent results if we simulate that multipion phase space by
effective two-body channels with a threshold that coincides with the ones of the
multipion systems. In effect the differences in the phase space can be simply absorbed
into the constants in the transition potentials, see Eq.~(\ref{eq:VNN}) --
which anyway have to be fitted to the data. All results presented in this manuscript
are based on an effective two-body phase space.
Of course, this simulation via effective two-body channels works only for energies around
the $\NNbar$ threshold. We cannot extend our calculation down to the threshold of the multipion
channels. However, one has to keep in mind that also the validity of our $\NNbar$ interaction
is limited to energies not too far away from the $\NNbar$ threshold. Thus, we have to
restrict our study to that small region around the threshold anyway.
With the definitions of the $T$-matrices above, the cross section is obtained via
\begin{equation}
\sigma_{\eebar\to\nu} (E)=\frac{3 E^2 \beta}{2^{10}\pi^3}\,|T_{\nu,\eebar}(E)|^2,
\end{equation}
and similarly for $\ppbar\to\nu$. The quantity $\beta$ denotes the phase space factor for
an effective two-body system with equal masses $M$, $\beta=\sqrt{(E^2-4M^2)}/\sqrt{(E^2-4m_e^2)}$,
with $2M=6m_\pi$, $5m_\pi$, or $m_\omega3m_\pi$. For $\ppbar\to\nu$ the
electron mass ($m_e$) has to be replaced by the one of the proton.
\section{The $\NNbar \to 5\pi, 6\pi$ reactions}
First we need to fix the constants $\tilde C_\nu$ and $C_\nu$ in the $\NNbar \to \nu$
transition potentials. We do this by considering available branching ratios
of $\ppbar$ annihilation at rest for the $3(\pi^+\pi^-)$, $2(\pi^+\pi^-\pi^0)$,
$\omega \pi^+\pi^-\pi^0 $, and $2(\pi^+\pi^-)\pi^0$ channels \cite{Klempt,McCrady}.
For the annihilation into $3(\pi^+\pi^-)$ and $2(\pi^+\pi^-)\pi^0$ there are, in addition,
in flight measurements for energies not too far from the $\NNbar$ threshold
\cite{Sai}. Following Ref.~\cite{Juelich_NNbar} we evaluate the relative cross sections
at low energy and compare those with the measured branching ratios. Specifically, we
calculate the cross sections at $p_{\rm lab} = 106$ MeV/c ($T_{\rm lab} \approx 5$ MeV)
because at this energy the total annihilation cross section is known from
experiment \cite{BRrest} and it can be used to calibrate the cross sections for the
individual $5 \pi$ and $6\pi$ channels based on the branching ratios.
Our results for $\ppbar\to 3(\pi^+\pi^-)$ and $\bar pp\to 2(\pi^+\pi^-)\pi^0$
are shown in Fig.~\ref{fig:ppbar6pi}. Note that the lowest ``data'' point is not
from a measurement but deduced from the branching ratios \cite{Klempt} and
the total annihilation cross section \cite{BRrest} as discussed in the preceding
paragraph.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=110mm,clip]{ppbarpions.eps}
\caption{(Color online) Cross section for (a) $\bar pp\to 2(\pi^+\pi^-)\pi^0$
and (b) $\ppbar\to 3(\pi^+\pi^-)$.
The solid curves represent our result. Data are taken from Ref.~\cite{Sai} (open circles).
The ``data'' points at $106$~MeV/c (filled circles) are deduced from
information on the branching
ratios of $\bar pp$ annihilation at rest, see text.
}
\label{fig:ppbar6pi}
\end{center}
\end{figure}
There are no in flight data for $\ppbar \to 2(\pi^+\pi^-\pi^0)$ and
$\ppbar \to \omega \pi^+\pi^-\pi^0$. Here we fit to the central value of the branching
ratios, 17.7\% \cite{Klempt} and 16.1\% \cite{McCrady}, respectively, and assume
that the energy dependence is the same as for the $3(\pi^+\pi^-)$ channel. The resulting
cross sections at $p_{\rm lab} = 106$ MeV/c are $63.2$ mb for the $2(\pi^+\pi^-\pi^0)$
case and $57.5$ mb for $\omega 3\pi$.
Note that the uncertainty in the energy dependence is not too critical.
Important is first and foremost the absolute value of those cross sections close to
the $\NNbar$ threshold because that value is decisive for the magnitude of the
$\eebar \to \NNbar \to \nu$ two-step contribution and, in turn, for the relevance
of the $\NNbar$ intermediate state for the $\eebar \to \nu$ reaction.
The results above are based on the NNLO EFT $\NNbar$ interaction with the cutoff
combination $\{\Lambda, \tilde\Lambda\} = \{450,500\}$ MeV, cf. Ref.~\cite{NNbar}
for details. Exploratory calculations for the other cutoff combinations considered
in Ref.~\cite{NNbar} turned out to be very similar. Like for $\NNbar$
scattering itself, much of the cutoff dependence is absorbed by the contact terms
($\tilde C_\nu$ and $C_\nu$ in Eq.~(\ref{eq:VNN})) that are fitted to the data so that the
variation of the results for energies of, say, $\pm 50$ MeV around the $\NNbar$
threshold is rather small. For consistency the momentum dependent regulator
function as given in Eq.~(2.21) in Ref.~\cite{NNbar} is also attached to all
momentum dependent quantities here, for example to the transition potential in
Eq.~(\ref{eq:VNN}).
Because of the coupled nature of the $^3S_1$-$^3D_1$ $\NNbar$ partial wave, in principle,
the $D$ wave should be also included in Eq.~(\ref{eq:ee6piN}) and, consequently, also in
Eq.~(\ref{eq:VNN}). Then there would be an additional contact term of the form
$D_\nu q^2$ \cite{NNbar}, representing the $\NNbar \to \nu$ transition potential from
the $\NNbar$ $^3D_1$ state, and a summation over the intermediate $\NNbar$ $^3D_1$
state arises, in addition to the integration over the intermediate
momentum in Eq.~(\ref{eq:ee6piN}).
We ignore these complications here because transitions starting from the
$\NNbar$ $D$ wave are strongly suppressed for energies around the $\NNbar$ threshold
and the contribution from the loop can be anyway effectively included in the contact
terms of the transition from the $\NNbar$ $S$-wave state.
\section{Results for $\eebar \to 5\pi, 6\pi$}
Once the contact terms in $V_{\NNbar \to \nu}$ are fixed from a fit to the pertinent
data the corresponding part of the ${\eebar \to \nu}$ amplitude that comes from the
transition via an intermediate $\NNbar$ state is also completely fixed, cf. Eq.~(\ref{eq:LS}).
We then add $A_{\eebar\to\nu}$. This term is assumed to be of the same functional
form as Eq.~(\ref{eq:VNN}), however, it can no longer be
identified with a transition potential (like for ${\NNbar \to \nu}$) but rather has to
account for all other contributions to ${\eebar \to \nu}$, besides the one that includes
the intermediate $\NNbar$ state. Specifically, this term can have a relative phase as
compared to the contribution from the $\NNbar$ loop. Therefore, in this case the
parameters can and should be complex. Since this background amplitude simulates a possibly
very large set of transition
processes it should have a weak dependence on the total energy in the region of the $\NNbar$
threshold and this feature is implemented by the ansatz~(\ref{eq:VNN}) with $q$ being interpreted
as the c.m. momentum in the ${\eebar}$ system. The two complex constants in the
analogous Eq.~(\ref{eq:VNN}) for $A_{\eebar\to\nu}$
are adjusted in a fit to the cross sections of each of the four ${\eebar \to \nu}$ reactions
studied in the present investigation. For the fit we considered data in the range
$1750 \ \rm{MeV} \le E \le 1950$~MeV, i.e. in a region that spans more or less symmetrically the $\NNbar$ threshold.
In principle, the ${\eebar \to \ppbar}$ amplitude that enters the loop contribution in
Eq.~(\ref{eq:ee6piN}) can be taken straight from Ref.~\cite{NPA} where it was fixed
in a fit to the ${\eebar \to \ppbar}$ cross section. The results in that work
were obtained by using a $\ppbar$ amplitude which is the sum of the isospin $I=0$ and
$I=1$ amplitudes, i.e. $T_{\ppbar} = (T^{I=1}+T^{I=0})/2$. However, it was found
that employing other combinations of $T^{1}$ and $T^{0}$ lead to very similar
results and in all cases an excellent agreement with the energy dependence exhibited
by the data could be achieved. Thus, since isospin is not conserved in the reaction
${\eebar \to \ppbar}$ the actual isospin content of the produced $\ppbar$ could not be fixed.
The mentioned selection rules for $\NNbar \to n\,\pi$ imply that the $6\pi$ final state
can only be reached from an $I=1$ $^3S_1$ $\NNbar$ state while $5$ pions have to come
from the corresponding $I=0$ state. Thus, the magnitude of the $\NNbar$ loop contribution
to ${\eebar \to \nu}$ depends decisively on the isospin content of the intermediate
$\NNbar$ state. We did calculations for ${\eebar \to \nu}$ with the combination as used in Ref.~\cite{NPA}
but it turned out that a slightly larger $I=1$ admixture, namely $T_{\bar pp}\approx 0.7\,T^1+0.3\,T^0$,
is preferable and leads to a somewhat better overall agreement with the experiments and, therefore,
we adopt this combination here.
The cross section for ${\eebar \to \nnbar}$ is also known experimentally \cite{Antonelli,Xsec_nnbar},
though with somewhat less accuracy. It agrees with the one for ${\eebar \to \ppbar}$ within the error bars
\cite{Xsec_nnbar}.
Therefore, we simply put $T_{\eebar \to \nnbar}=T_{\eebar \to \ppbar}$ in the sum in Eq.~(\ref{eq:ee6piN}),
which is certainly justified as can be seen from the actual ${\eebar \to \nnbar}$ result presented in
Fig.~\ref{fig:nnbar}.
\begin{figure}
\begin{center}
\includegraphics[height=60mm,clip]{eennbar.eps}
\caption{(Color online) Cross section for $\eebar\to\nnbar$. The solid line represents our result.
Data are taken from Refs.~\cite{Antonelli} (circles) and \cite{Xsec_nnbar} (squares).
}
\label{fig:nnbar}
\end{center}
\end{figure}
As discussed above, $D$-wave contributions were ignored in case of $\ppbar\to \nu$. However, for
$\eebar\to \nu$ around the $\NNbar$ threshold the momentum in the incoming system is no longer
small and the $\eebar$ $^3D_1$ component cannot be neglected. However, it can be easily included
because the $\eebar \to \ppbar$ transition amplitudes from the $^3S_1$ and $^3D_1$ $\eebar$
states are proportional to each other, see Eq.~(6) of Ref.~\cite{NPA}. Thus, for including
the $D$-wave contribution we simply have to multiply the $S$-wave cross section by a
factor $1.5$.
\begin{figure*}[htbp]
\begin{center}
\includegraphics[height=120mm,clip]{eepions.eps}
\caption{(Color online) Cross section for (a) $e^+e^-\to 3(\pi^+\pi^-)$,
(b) $\omega\pi^+\pi^-\pi^0$,
(c) $2(\pi^+\pi^-\pi^0)$, and (d) $2(\pi^+\pi^-)\pi^0$.
The solid (red) curves represent our full result, including the $\NNbar$ intermediate state,
while the dash-dotted (black) curves are based on the background term alone.
The vertical lines indicate the $\bar pp$ threshold.
The dashed (red) curve in (a) corresponds to amplifying deliberately the $\bar NN$ loop
contribution by a factor of four.
Data are taken from Refs.~\cite{BABAR,BABAR2} (circles) and \cite{CMD,CMD2} (squares).
}
\label{fig:ee6pi}
\end{center}
\end{figure*}
Our results are shown in Fig.~\ref{fig:ee6pi}.
Obviously, in three of the four considered reactions the contribution from the two-step
process $\eebar\to \NNbar\to\nu$ is large enough to be of relevance and, moreover, together
with a suitably adjusted background a rather good description of the cross sections around
the $\NNbar$ threshold can be achieved (solid curves). The cross section due to the background
alone is indicated by the dash-dotted curves. It is practically constant and does not exhibit
any structure. The contribution involving the intermediate $\NNbar$ state generates a distinct
structure at the $\NNbar$ threshold and is responsible for the fact that the full result is
indeed in line with the behaviour suggested by the measurements. Of course,
in case of the $2(\pi^+\pi^-\pi^0)$ channel the data could hint at a minimum at an
energy slightly above the threshold. However, there is also some variation between the
two experiments. Further measurements with improved statistics and also with a better
momentum resolution would be quite helpful. This applies certainly also to the other channels.
No satisfactory result could be achieved for the reaction $e^+e^-\to 3(\pi^+\pi^-)$. Here the
amplitude due to the intermediate $\NNbar$ state would have to be roughly a factor four
larger in order to explain the data, see the dashed line. We emphasize that this
curve is shown only for illustrative purposes! At the moment we do not have any physical
arguments why that particular amplitude should be increased by a factor four. Indeed, we have
examined and explored various uncertainties that could be used to motivate an
amplification of the amplitude but without success.
For example, assuming that the $\eebar\to \ppbar$ reaction is given
by the isospin $1$ alone changes the result only marginally and the same is the case if
we take into account that the $\eebar\to \nnbar$ cross section could by slightly larger than the
one for $\eebar\to \ppbar$ as indicated by the data in Ref.~\cite{Antonelli}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=110mm,clip]{loop_bkg1.eps}
\caption{(Color online) Real (dotted/blue curve) and imaginary part (dashed/green), and
modulus (solid/red line) of the $\bar NN$ loop contribution, see Eq.~\eqref{eq:ee6piN}.
The modulus of the background term is shown by the dash-dotted (black) curves.
}
\label{fig:ppbar6piA}
\end{center}
\end{figure}
Finally, let us come to the key question, namely are those structures seen in the
experiment a signal for a $\NNbar$ bound state? As discussed in Ref.~\cite{NNbar},
we did not find any near-threshold poles for our EFT $\NNbar$ interaction in the
$^{3}S_1$--$^{3}D_1$ partial wave with $I=1$. However, there is a pole in
the $I=0$ case and this pole corresponds to a ``binding'' energy of
$E_B= (+4.8-{\rm i}\,68.2.9)$ MeV for the NNLO interaction employed
in the present study \cite{NNbar}.
The positive sign of the real part of $E_B$ indicates that the pole
we found is actually located above the $\bar NN$ threshold (in the energy plane).
As discussed in Ref.~\cite{NNbar}, the pole moves below the threshold when we switch
off the imaginary part of the potential and that is the reason why we refer to it as bound state.
There is a distinct difference in the $\eebar\to\nu$ amplitudes due to the $\NNbar$
loop contribution for the two isospin channels, see Fig.~\ref{fig:ppbar6piA}, and
the modulus exhibits indeed the features one expects in case of the absence/presence
of a bound state, namely a genuine cusp or a rounded step and a maximum below the
threshold.
However, the structure in the cross section is strongly influenced and
modified by the interference with the (complex) background amplitude as
testified by the results in Fig.~\ref{fig:ee6pi}. Thus, at this
stage we do not see a convincing evidence for the presence of an $\NNbar$
bound state in the data for $2(\pi^+\pi^-)\pi^0$ and for the opposite
in case of $\omega\pi^+\pi^-\pi^0$, say. However, high accuracy data around
the $\NNbar$ threshold together with a better theoretical understanding of the background
could certainly change the perspective for more reliable conclusions in the future.
In any case, our results corroborate that one should see an effect of the opening
of the $\NNbar$ channel in the cross sections of the considered $\eebar\to\nu$
reactions. Thus, the observation of a dip or a cusp-like structure in that
energy region is not really something unusual or exotic.
As argued above, our calculation provides a fairly reliable estimate for the
amplitude that results from two-step processes with an intermediate $\NNbar$ state.
Though all the reactions considered in the present study are obviously dominated by
processes that are not related to the $\NNbar$ interaction, cf.
Fig.~\ref{fig:ppbar6piA}, the amplitude due to the
coupling to the $\NNbar$ system is large enough so that it can produce sizeable
interference effects. In three of the four reactions investigated those
interference effects are indeed sufficient to explain the behavior of the measured
cross sections in the region around the $\NNbar$ threshold.
Should one expect similar structures also in other annihilation channels such as
$\eebar \to \pi^+\pi^-$, $\eebar \to 2(\pi^+\pi^-)$, etc.? An educated guess can be
made based on the relative magnitude of the branching ratios for the pertinent $\NNbar$
annihilation channels compared to annihilation cross sections from the $\eebar$ state.
Judging from the branching ratios summarized in Ref.~\cite{Klempt}
and the compilation of $\eebar$ induced cross sections in Refs.~\cite{Druzhinin,Bevan}
the most promising case is certainly the $\pi^+\pi^-\pi^0 $ channel, see Table~\ref{tab:N1}.
In fact, the available data \cite{Aubert:2004,Logashenko} could hint at an anomaly around the
$\NNbar$ threshold. In case of the 4$\pi$ channels there is a kink at the $\NNbar$ threshold,
see, e.g. Ref.~\cite{Logashenko}. On the other hand, for the $\pi^+\pi^-$ case the
branching ratio is very small, see Table~\ref{tab:N1}, so that we do not expect any
noticeable effects there. Indeed, the data for $\eebar \to \pi^+\pi^-$ \cite{Aubert:2009}
support this conjecture.
\begin{table}
\caption{Branching ratios for $\ppbar$ annihilation at rest \cite{Klempt}
and ${\eebar}$ annihilation cross sections around the $\NNbar$ threshold \cite{Druzhinin,Bevan}.
}
\renewcommand{\arraystretch}{1.3}
\label{tab:N1}
\vspace{0.2cm}
\centering
\begin{tabular}{|c||c|c|}
\hline
$\nu$ & \ BR for $\ppbar \to \nu$ [\%] \ & \ $\sigma_{\eebar \to \nu}$ [nb] \ \\
\hline
$\pi^+\pi^- $ & 0.314$\pm$0.012 & $\approx$\,1 \\
$\pi^+\pi^-\pi^0 $ & 6.7$\pm$1.0 & $\approx$\,1 \\
$2(\pi^+\pi^-) $ & 5.6$\pm$0.9 & $\approx$\,6 \\
$\pi^+\pi^-2\pi^0 $ & 12.2$\pm$1.8 & $\approx$\,9 \\
$2(\pi^+\pi^-)\pi^0$ & 21.0$\pm$3.2 & $\approx$\,2 \\
$\,2(\pi^+\pi^-\pi^0)\,$ & 17.7$\pm$2.7 & $\approx$\,4 \\
$3(\pi^+\pi^-) $ & 2.1$\pm$0.25 & $\approx$\,1 \\
\hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\end{table}
\section{Summary}
We analyzed the origin of the structure observed in the reactions $e^+e^-\to 3(\pi^+\pi^-)$, $2(\pi^+\pi^-\pi^0)$,
$\omega\pi^+\pi^-\pi^0$, and $e^+e^-\to 2(\pi^+\pi^-)\pi^0$ around the
$\bar pp$ threshold in recent BaBar and CMD measurements. Specifically,
we evaluated the contribution of the two-step process $e^+e^-\to \bar NN \to$ multipions to the
total reaction amplitude. The amplitude for $e^+e^-\to \bar NN$ was constrained from near-threshold
data on the $e^+e^-\to \bar pp$ cross section and the one for $\bar NN \to$ multipions was fixed from
available experimental information, for all those $5 \pi$ and $6\pi$ states.
The resulting amplitude turned out to be large enough to play a role for the considered $e^+e^-$ annihilation
channels and, in three of the four reactions, even allowed us to reproduce the data quantitatively near the
$\bar NN$ threshold once the interference with a background amplitude was taken into account. The
latter simulates other transition processes that do not involve an $\bar NN$ intermediate state.
In case of the reaction $e^+e^-\to 3(\pi^+\pi^-)$ there is also a visible effect from the $\bar NN$
channel, however, overall the magnitude of the pertinent amplitude is too small.
In our study the structures seen in the experiments emerge as a threshold effect due to the opening
of the $\bar NN$ channel. The question of a $\bar NN$ bound state is discussed, however, no firm
conclusion can be made. But it is certainly safe to say that the near-threshold behavior of
the $e^+e^-\to \bar pp$ cross section and the structures seen in $e^+e^-\to 3(\pi^+\pi^-)$, etc.
have the same origin.
\section*{Acknowledgements}
We would like to thank Eberhard Klempt for help and comments regarding the branching
ratio measurements in $\ppbar$ annihilation and Evgeni P. Solodov for sending
us the BaBar data for $\eebar \to \omega\pi^+\pi^-\pi^0$.
This work is supported in part by the DFG and the NSFC through
funds provided to the Sino-German CRC 110 ``Symmetries and
the Emergence of Structure in QCD''.
|
2,869,038,154,910 | arxiv |
\section{Acknowledgments}
We thank the anonymous PKC 2022 reviewers for their valuable comments.
Rohit Chatterjee and Xiao Liang are supported in part by Omkant Pandey's DARPA SIEVE Award HR00112020026 and NSF grants 1907908 and 2028920. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government, DARPA, or NSF.
Kai-Min Chung is supported by Ministry of Science and Technology, Taiwan, under Grant No. MOST 109-2223-E-001-001-MY3.
Giulio Malavolta is supported by the German Federal Ministry of Education and Research BMBF (grant 16K15K042, project 6GEM).
\bibliographystyle{alpha}
\section{Additional Preliminaries}
\label{sec:add-prelim}
\subsection{Preliminaries for Lattice}
\label{sec:add-prelim:lattice}
{Throughout the current paper, we denote the Gram-Schmidt ordered orthogonalization of a matrix $\vb{A} \in \mathbb{Z}^{m\times m}$ by $\tilde{\vb{A}}$}.
\subsubsection{Lattices}
We define the notion of a lattice and integer lattice.
\begin{definition}[Lattice]
Let $\mathbf{B} = [~\mathbf{b}_1~|~\dots~|~\mathbf{b}_m~]$ be a basis of linearly independent vectors $\mathbf{b}_i \in \mathbb{R}^m, \ i \in [m]$. The {\em lattice} generated by $\mathbf{B}$ is defined as $\Lambda = \Set{\mathbf{y} \in \mathbb{R}^m: \exists s_i\in \mathbb{Z}, \mathbf{y} = \sum_1^ms_i\mathbf{b}_i}$. The {\em dual lattice} $\Lambda^*$ of $\Lambda$ is defined as $\Lambda^* = \Set{\mathbf{z} \in \mathbb{R}^m: \forall y \in \Lambda,\langle\mathbf{z,y} \rangle \in \mathbb{Z}}$
\end{definition}
\begin{definition}[Integer Lattice]
For a prime $q$, a modular matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ and vector $\mathbf{u} \in \mathbb{Z}_q^n$, we define the m-dimensional (full rank) integer lattice $\Lambda_q^\perp(\mathbf{A}) = \Set{\mathbf{e} \in \mathbb{Z}^m:\mathbf{Ae}=0\ (\mod q)}$, and the `shifted' lattice as the coset $\Lambda_q^\mathbf{u}(\mathbf{A}) = \Set{\mathbf{e} \in \mathbb{Z}^m:\mathbf{Ae}=\mathbf{u}\ (\mod q)}$
\end{definition}
\subsubsection{Lattice Trapdoors, Discrete Gaussians}
The works \cite{STOC:Ajtai96,EC:MicPei12} show how to sample close to uniform matrices $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ along with a matrix trapdoor $\mathbf{T_A}$ that consists of a basis of low norm vectors for the associated lattice $\Lambda_q^\perp(\mathbf{A})$. We call this sampling procedure $\algo{TrapGen}$.
\begin{lemma}[Trapdoor Matrices]\label{lemma:trapgen}
There is a PPT algorithm $\algo{TrapGen}$ that given as input integers $n \geq 1$, $q \geq 2$, and (sufficiently large) $m = O(n\log q)$, outputs a matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ and a trapdoor matrix $\mathbf{T_A} \in \mathbb{Z}_q^{m\times m}$, such that $\mathbf{AT_A} = 0$, the distribution of $\mathbf{A}$ is statistically close to uniform over $\mathbb{Z}_q^{n\times m}$, and $||\widetilde{\vb{T}}_{\vb{A}}|| = O(\sqrt{n\log q})$.
\end{lemma}
We now define the notion of discrete Gaussian distributions.
\begin{definition}[Discrete Gaussians]\label{def:DiscGaussian}
Let $m \in \mathbb{Z}_{>0}$, $\Lambda \subset \mathbb{Z}^m$. For any vector $\mathbf{c} \in \mathbb{R}^m$, and positive real $\sigma \in \mathbb{R}_{>0}$, define the Gaussian function $\rho_{\sigma,\mathbf{c}}(\mathbf{x}) = \mathsf{exp}(-\pi||\mathbf{x-c}||^2/\sigma^2)$ over $\mathbb{R}^m$ with center $\mathbf{c}$ and width $\sigma$. Define the discrete Gaussian distribution over $\Lambda$ with center $\mathbf{c}$ and width $\sigma$ as $\mathcal{D}_{\Lambda,\sigma,\mathbf{c}} = \rho_{\sigma,\mathbf{c}}/\rho_{\sigma}(\Lambda)$ where $\rho_{\sigma}(\Lambda) = \sum_{x \in Lambda} \rho_{\sigma,\mathbf{c}}$. For convenience, we use the shorthand $\rho_\sigma$ and $\mathcal{D}_{\Lambda,\sigma}$ for $\rho_{\sigma,\mathbf{0}}$ and $\mathcal{D}_{\Lambda,\sigma,\mathbf{0}}$ respectively.
\end{definition}
The following lemma is a very useful concentration bound on the norm of discrete guassian samples, depending on the basis they were sampled using.
\begin{lemma}[Discrete Gaussian Concentration \cite{FOCS:MicReg04}]\label{lemma:discgaussconc}
For any lattice $\Lambda$ of integer dimension $m$ with basis $\mathbf{T}$, $\mathbf{c} \in \mathbb{R}^m$, and Gaussian width parameter $\sigma \geq ||\widetilde{\mathbf{T}}||\cdot\omega\big(\sqrt{\log m}\big)$, we have $$\Pr[\mathbf{x}\gets\mathcal{D}_{\Lambda,\sigma, \vb{c}}: ||\mathbf{x-c}||>\sigma\sqrt{m}] \leq \negl(n)$$
\end{lemma}
\subsubsection{The Gadget Matrix}
The gadget matrix $\mathbf{G}$ was defined in \cite{EC:MicPei12}. We use the following two properties of $\mathbf{G}$ in particular:
\begin{lemma}[{\cite[Theorem 1]{EC:MicPei12}}]\label{lemma:gadgettrapdoor}
Let $q$ be a prime, and $n,m$ be integers with $m=n\log q$. There is a fixed full-rank matrix $\mathbf{G} \in \mathbb{Z}_q^{n\times m}$ such that the lattice $\Lambda_q^\perp(\mathbf{G})$ has a publicly known trapdoor matrix $\mathbf{T_G} \in \mathbb{Z}^{n\times m}$ with $||\widetilde{\vb{T}}_{\vb{G}}|| \leq \sqrt{5}$.
\end{lemma}
\begin{lemma}[{\cite[Lemma 2.1]{EC:BGGHNS14}}]\label{lemma:gadgetinverse}
There is a deterministic algorithm, denoted by $\mathbf{G}^{-1}(\cdot):\mathbb{Z}_q^{n\times m} \rightarrow \mathbb{Z}^{m\times m}$ that takes a matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ as input, and outputs a `preimage' $\mathbf{G}^{-1}(\mathbf{A})$ of $\mathbf{A}$ such that $\mathbf{G} \cdot \mathbf{G}^{-1}(\mathbf{A})=\mathbf{A}$ $(\mathrm{mod}~q)$ and $||\mathbf{G}^{-1}(\mathbf{A})||\leq m$.
\end{lemma}
\subsubsection{Hardness Assumptions}
We recall the LWE and SIS problems, and their hardness based on worst case lattice problems.
For a positive integer dimension $n$ and modulus~$q$, and an error
distribution~$\chi$ over~$\mathbb{Z}$, the LWE distribution and decision
problem are defined as follows. For an $\mathbf{s} \in \mathbb{Z}^{n}$, the LWE
distribution $A_{\mathbf{s},\chi}$ is sampled by choosing a uniformly
random $\mathbf{a} \gets \mathbb{Z}_q^{n}$ and an error term $e \gets \chi$, and
outputting $(\mathbf{a}, b = \langle\mathbf{s}, \mathbf{a}\rangle + e) \in \mathbb{Z}_q^{n+1}$.
\begin{definition}
\label{def:lwe}
The decision-LWE$_{n,q,\chi}$ problem is to distinguish, with
non-negligible advantage, between any desired (but polynomially
bounded) number of independent samples drawn from $A_{\mathbf{s},\chi}$
for a single $\mathbf{s} \gets \mathbb{Z}_q^{n}$, and the same number of
\emph{uniformly random} and independent samples over
$\mathbb{Z}_q^{n+1}$.
\end{definition}
A standard
instantiation of LWE is to let~$\chi$ be a \emph{discrete Gaussian}
distribution over~$\mathbb{Z}$ with parameter $r = 2\sqrt{n}$. A sample drawn
from this distribution has magnitude bounded by, say,
$r\sqrt{n} = \Theta(n)$ except with probability at most $2^{-n}$, and
hence this tail of the distribution can be entirely removed. For this
parameterization, it is known that LWE is at least as hard as
\emph{quantumly} approximating certain ``short vector'' problems on
$n$-dimensional lattices, in the worst case, to within
$\widetilde{O}(q\sqrt{n})$
factors~\cite{STOC:Regev05,STOC:PeiRegSte17}.
Classical reductions are also known for different
parameterizations~\cite{STOC:Peikert09,STOC:BLPRS13}.
\begin{definition}\label{def:sis}
The $\mathbf{SIS}_{q,\beta,n,m}$ problem is: given an uniformly random matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$, find a nonzero integral vector $\mathbf{z} \in \mathbb{Z}^m$ such that $\mathbf{Az}=\vb{0}\mod q$, and $||\mathbf{z}|| \leq \beta$.
\end{definition}
When $q \geq \beta\cdot \widetilde{O}(\sqrt{n})$, solving $\mathbf{SIS}_{q,\beta,n,m}$ is at least as hard as approximating certain worst-case lattice problems (namely, SIVP) to within a $\beta\cdot \widetilde{O}(\sqrt{n})$ factor \cite{FOCS:MicReg04}.
\subsection{Random Sampling Related}
\label{sec:additional-prelims:sampling-related}
We recall the following generalization of the leftover hash lemma.
\begin{lemma}[{\cite[Lemma 4]{EC:AgrBonBoy10}}]\label{lemma:lhl}
Suppose that $m > (n+1)\log_2q+\omega(\log n)$ and that $q > 2$ is a prime. Let $\mathbf{R}$ be an $m\times k$ matrix chosen uniformly from $\Set{-1,1}^{m\times k}\mod q$ where $k=k(n)$ is polynomial in $n$. Let $\mathbf{A}$ and $\mathbf{B}$ be matrices chosen uniformly in $\mathbb{Z}_q^{n\times m}$ and $\mathbb{Z}_q^{n\times k}$ respectively. Then for all vectors $\mathbf{w} \in \mathbb{Z}_q^m$, the distribution $(\mathbf{A},\mathbf{AR},\mathbf{R}^\top\mathbf{w})$ is statistically close to the distribution $(\mathbf{A},\mathbf{B},\mathbf{R}^\top\mathbf{w})$.
\end{lemma}
We will give an argument to show how \Cref{lemma:ext-lhl} follows from this. This goes as follows: assume we start with $(\mathbf{A}',\mathbf{A}'\vb{R},\mathbf{R}^\top\mathbf{w})$. This is statistically close to $(\mathbf{A},\mathbf{A}\vb{R},\mathbf{R}^\top\mathbf{w})$ since $\mathbf{A}$ is sampled uniformly, and $\vb{A}'\sind\vb{A}$.
By \Cref{lemma:lhl} above, $(\mathbf{A},\mathbf{AR},\mathbf{R}^\top\mathbf{w}) \sind (\mathbf{A},\mathbf{B},\mathbf{R}^\top\mathbf{w})$. The latter is in turn statistically close to $(\mathbf{A}',\mathbf{B},\mathbf{R}^\top\mathbf{w})$. Therefore, we have $(\mathbf{A}',\mathbf{B},\mathbf{R}^\top\mathbf{w}) \sind (\mathbf{A}',\mathbf{A}'\vb{R},\mathbf{R}^\top\mathbf{w})$, concluding the proof for \Cref{lemma:ext-lhl}.
We also recall the following concentration bound on the operator norm for the matrices $\mathbf{R}$.
\begin{lemma}[{\cite[Lemma 5]{EC:AgrBonBoy10}}]\label{lemma:operatorbd}
Let $\mathbf{R}$ be an uniformly random chosen matrix from $\Set{-1,1}^{m\times m}$, then $\Pr[||\mathbf{R}||_2 > 12\sqrt{2m}] < e^{-m}$.
\end{lemma}
\subsection{Key-Homomorphic Evaluation Algorithms}
\label{sec:additional-prelims:key-homo-eval}
We recall the matrix key-homomorphic evaluation algorithm from \cite{C:GenSahWat13,EC:BGGHNS14,ITCS:BraVai14} more fully. This was developed in the context of fully homorphic encryption and attribute-based encryption. This template works generally as follows: given a Boolean {\sf NAND} circuit $C:\Set{0,1}^\ell \rightarrow \Set{0,1}$ with fan-in 2, $\ell$ matrices $\Set{\mathbf{A}_i = \mathbf{AR}_i + x_i\mathbf{G} \in \mathbb{Z}_q^{n\times m}}_{i\in [\ell]}$ which correspond to each input wire of $C$ where $\mathbf{A}\pick \mathbb{Z}_q^{n\times m}$, $\mathbf{R}_i \pick \Set{-1,1}^{m\times m}$, $x_i \in \Set{0,1}$ and $\mathbf{G} \in \mathbb{Z}_q^{n\times m}$ is the gadget matrix, the key-homomorphic evaluation algorithm deterministically computes $\mathbf{A}_C = \mathbf{AR}_C + C(x_1,\dots,x_\ell)\mathbf{G} \in \mathbb{Z}_q^{n\times m}$ where $\mathbf{R}_C \in \Set{-1,1}^{m\times m}$ has low norm and $C(x_1,\dots,x_\ell) \in \Set{0,1}$ is the output bit of $C$ on the arguments $x_1,\dots,x_\ell$. This is done by inductively evaluating each {\sf NAND} gate. For a {\sf NAND} gate $g(u,v;w)$ with input wires $u,v$ and output wire $w$, we have (inductively) matrices $\mathbf{A}_u = \mathbf{AR}_u + x_u\mathbf{G}$, and $\mathbf{A}_v = \mathbf{AR}_v + x_v\mathbf{G}$ where $x_u$ and $x_v$ are the input bits of $u$ and $v$ respectively, and the evaluation algorithm computes
\begin{equation}
\begin{split}
\mathbf{A}_w & = \mathbf{G} - \mathbf{A_u}\cdot\mathbf{G}^{-1}(\mathbf{A}_v) \\
& = \mathbf{G} - (\mathbf{AR}_u+x_u\mathbf{G})\cdot\mathbf{G}^{-1}(\mathbf{AR}_v + x_v\mathbf{G}) \\
& = \mathbf{AR}_g + (1-x_ux_v)\mathbf{G}
\end{split}
\end{equation}
where $1-x_ux_v \coloneqq \algo{NAND}(x_u,x_v)$, and $\mathbf{R}_g = -\mathbf{R}_u\cdot\mathbf{G}^{-1}(\mathbf{A}_v)-x_u\mathbf{R}_v$ has low norm if both $\mathbf{R}_u$ and $\mathbf{R}_v$ have low norm.
In \cite{ITCS:BraVai14}, Brakerski and Vaikuntanathan observed that the norm of $\mathbf{R}_C$ in the outlined evaluation procedure grows asymmetrically (in the $\mathbf{R}$s corresponding to the input wires). They exploited this observation to design a special evaluation algorithm that evaluates circuits in $\mathsf{NC^1}$ with moderate blowup in the norm of $\mathbf{R}_C$. Specifically, the observation is that any circuit with depth $d$ can be simulated by a length $4^d$ and width 5 branching program by Barrington's theorem, recalled below:
\begin{theorem}[Barrington's Theorem]\label{theorem:barrington}
Every Boolean {\sf NAND} circuit $C$ that acts on $\ell$ inputs and has depth $d$ can be computed by a width 5 permutation branching program of length $4^d$. Given the description of the circuit $C$, the description of the corresponding branching program can be computed in $\poly(\ell,4^d)$ time.
\end{theorem}
Such a branching program can then be computed by multiplying $4^d$ many $5\times 5$ permutation matrices. It is shown in \cite{ITCS:BraVai14} that homomorphically evaluating the multiplication of permutation matrices using the above procedure and the asymmetric noise growth feature only increases the noise by a polynomial factor, and thus allows us to use an LWE or SIS modulus that is polynomial in the security parameter. In our constructions, we will use this particular evaluation method just as in \cite{ITCS:BraVai14} and denote it by $\algo{Eval}_{BV}$.
We will use a claim regarding the noise growth properties of $\algo{Eval}_{BV}$. It can be obtained from Claim 3.4.2 and Lemma 3.6 of \cite{ITCS:BraVai14} and Barrington's Theorem.
\begin{lemma}\label{lemma:rnorm}
Let $C:\Set{0,1}^\ell \rightarrow \Set{0,1}$ be a {\sf NAND} Boolean circuit. Let $\Set{\mathbf{A}_i = \mathbf{AR}_i + x_i\mathbf{G} \in \mathbb{Z}_q^{n\times m}}_{i\in [\ell]}$ be $\ell$ distinct matrices corresponding to the input wires of $C$, where $\mathbf{A}\pick \mathbb{Z}_q^{n\times m}$, $\mathbf{R}_i \pick \Set{-1,1}^{m\times m}$, $x_i \in \Set{0,1}$ and $\mathbf{G} \in \mathbb{Z}_q^{n\times m}$ is the gadget matrix. There is an efficient deterministic algorithm $\algo{Eval}_{BV}$ that takes as input $C$ and $\Set{\mathbf{A}_i}_{i\in [\ell]}$ and outputs a matrix $\mathbf{A}_C = \mathbf{AR}_C + C(x_1,\dots,x_\ell)\mathbf{G} = \algo{Eval}_{BV}(C,\mathbf{A}_1,\dots,\mathbf{A}_\ell)$ where $\mathbf{R}_C \in \mathbb{Z}^{m\times m}$ and $C(x_1,\dots,x_\ell)$ is the output of $C$ on the arguments $x_1,\dots,x_\ell$. $\algo{Eval}_{BV}$ runs in time $\poly(4^d,\ell,n,\log q)$.
Let $||\mathbf{R}_{max}||_2 = max\Set{||\mathbf{R}_i||_2}_{i \in [\ell]}$, the norm of $\mathbf{R}_C$ in $\mathbf{A}_C$ output by $\algo{Eval}_{BV}$ can be bounded with overwhelming probability by
\begin{equation}
\begin{split}
||\mathbf{R}_C||_2 & \leq O(L\cdot ||\mathbf{R}_{max}||_2.m)\\
& \leq O(L\cdot 12\sqrt{2m}\cdot m) \\
& \leq O(4^dm^{3/2})
\end{split}
\end{equation}
where $L$ is the length of the width 5 branching program which simulates $C$ and we have used \Cref{lemma:operatorbd} to obtain $||\mathbf{R}_i||_2 \leq 12\sqrt{2m}$ for each $i$ with overwhelming probability.
In particular, if $C$ is in $\mathsf{NC^1}$ and has depth $d=c\log l$ for a constant $c$, then $L=4^d=\ell^{2c}$ and $\leq O(\ell^{2c}\cdot m^{3/2})$
\end{lemma}
\section{Blind-Unforgeable Signatures in the Plain Model}
\subsection{Notation and Building Blocks}
We assume familiarity with standard lattice-based cryptographic notions and procedures. Here we will recall certain techniques and properties to be directly used in our plain model construction.
We recall standard lattice-related concepts (e.g., parameters, hardness, trapdoors) in \Cref{sec:add-prelim:lattice}.
For a vector $\mathbf{u}$, we let $||\mathbf{u}||$ denote its $\ell_2$ norm. For a matrix $\mathbf{R} \in \mathbb{Z}^{k\times m}$, we define two matrix norms:
\begin{itemize}
\item
$||\mathbf{R}||$ denotes the $\ell_2$ norm of the largest column of $\mathbf{R}$;
\item
$||\mathbf{R}||_2$ denotes the operator norm of $\mathbf{R}$, defined as $||\mathbf{R}||_2 = \mathsf{sup}_{\vb{x}\in \mathbb{R}^{m+1}}||\mathbf{R}\cdot\mathbf{x}||$.
\end{itemize}
We denote the Gram-Schmidt ordered orthogonalization of a matrix $\vb{A} \in \mathbb{Z}^{m\times m}$ by $\tilde{\vb{A}}$. For a prime $q$, a modular matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ and vector $\mathbf{u} \in \mathbb{Z}_q^n$, we define the m-dimensional (full rank) lattice $\Lambda_q^{\vb{u}}(\mathbf{A}) = \Set{\mathbf{e} \in \mathbb{Z}^m:\mathbf{Ae}=\vb{u}~(\mathrm{mod}~q)}$. In particular, $\Lambda_q^\perp(\mathbf{A})$ denotes the lattice $\Lambda_q^{\vb{0}}(\mathbf{A})$.
\para{Lattice Sampling Algorithms.} Our construction uses the `left-right trapdoors' framework introduced in \cite{EC:AgrBonBoy10,PKC:Boyen10}, which uses two sampling algorithms $\algo{SampleLeft}$ and $\algo{SampleRight}$.
\subpara{$\algo{SampleLeft}$:} The algorithm $\algo{SampleLeft}$ works as follows:
\begin{itemize}
\item {\em Inputs:} A full-rank matrix $\mathbf{A} \in \mathbb{Z}_q^{n\times m}$ and a short basis $\mathbf{T_A}$ of $\Lambda_q^\perp(\mathbf{A})$, along with a matrix $\mathbf{B} \in \mathbb{Z}_q^{n\times m_1}$, a vector $\mathbf{u} \in \mathbb{Z}_q^{n}$, and a Gaussian parameter $s$.
\item {\em Output:} Let $\mathbf{F} = [\mathbf{A}~|~\mathbf{B}]$. $\algo{SampleLeft}$ outputs a vector $\mathbf{d} \in \mathbb{Z}^{m+m_1}$ in $\Lambda_q^\mathbf{u}(\mathbf{F})$.
\end{itemize}
\begin{theorem}[$\algo{SampleLeft}$ Closeness \cite{EC:AgrBonBoy10,EC:CHKP10}]\label{theorem:sampleleft}
Let $q > 2$, $m > n$ and $s > ||\widetilde{\vb{T}}_{\vb{A}}||\cdot\omega\big(\sqrt{\log (m+m_1)}\big)$. Then, the $\algo{SampleLeft}(\mathbf{A},\mathbf{B},\mathbf{T_A},\mathbf{u},s)$ (as defined above) outputs $\mathbf{d} \in \mathbb{Z}^{m+m_1}$ distributed statistically close to $\mathcal{D}_{\Lambda_q^\mathbf{u}(\mathbf{F}),s}$.
\end{theorem}
\subpara{$\algo{SampleRight}$:} The algorithm $\algo{SampleRight}$ works as follows:
\begin{itemize}
\item {\em Inputs:} Matrices $\mathbf{A} \in \mathbb{Z}_q^{n\times k}$ and $\mathbf{R} \in \mathbb{Z}_q^{k\times m}$, a full-rank matrix $\mathbf{B} \in \mathbb{Z}_q^{n\times m}$, a short basis $\mathbf{T_B}$ of $\Lambda_q^\perp(\mathbf{B})$, a vector $\mathbf{u} \in \mathbb{Z}_q^{n}$, and a Gaussian parameter $s$.
\item {\em Output:} Let $\mathbf{F} = [\mathbf{A}~|~\mathbf{AR}+\mathbf{B}]$. It outputs a vector $\mathbf{d} \in \mathbb{Z}^{m+m_1}$ in the set $\Lambda_q^\mathbf{u}(\mathbf{F})$.
\end{itemize}
\begin{theorem}[$\algo{SampleRight}$ Closeness \cite{EC:AgrBonBoy10}]\label{theorem:sampleright}
Let $q > 2$, $m > n$ and $s > ||\widetilde{\vb{T}}_{\vb{B}}||\cdot ||\vb{R}||_2 \cdot \omega(\sqrt{\log m})$. Then $\algo{SampleRight}(\mathbf{A},\mathbf{B},\mathbf{R},\mathbf{T_B},\mathbf{u},s)$ (as defined above) outputs $\mathbf{d} \in \mathbb{Z}^{m+k}$ distributed statistically close to $\mathcal{D}_{\Lambda_q^\mathbf{u}(\mathbf{F}),s}$.
\end{theorem}
\para{Random Sampling Related.}
The following is a simple corollary of \cite[Lemma 4]{EC:AgrBonBoy10} (see \Cref{sec:additional-prelims:sampling-related} for details).
\begin{corollary}\label{lemma:ext-lhl}
Suppose that $m > (n+1)\log_2q+\omega(\log n)$ and that $q > 2$ is a prime. Let $\mathbf{R}$ be an $m\times k$ matrix chosen uniformly from $\Set{-1,1}^{m\times k}\mod q$ where $k=k(n)$ is polynomial in $n$. Let $\mathbf{A}' \in \mathbb{Z}_q^{n\times m}$ be sampled from a distribution statistically close to uniform over $\mathbb{Z}_q^{n\times m}$. Let $\mathbf{R}$ be an $m\times k$ matrix chosen uniformly from $\Set{-1,1}^{m\times k}\mod q$ where $k=k(n)$ is polynomial in $n$. Let $\mathbf{B}$ be chosen uniformly in $\mathbb{Z}_q^{n\times k}$. Then for all vectors $\mathbf{w} \in \mathbb{Z}_q^m$, the distributions $(\mathbf{A}',\mathbf{A}'\vb{R},\mathbf{R}^\top\mathbf{w})$ and $(\mathbf{A}',\mathbf{B},\mathbf{R}^\top\mathbf{w})$ are statistically close.
\end{corollary}
\para{Key-Homomorphic Evaluation.}
We briefly recall the matrix key-homomorphic evaluation algorithm, as found in \cite{C:GenSahWat13,EC:BGGHNS14,ITCS:BraVai14} (see \Cref{sec:additional-prelims:key-homo-eval} for details). This template evaluates NAND circuits, gate by gate, in a homomorphic manner. For a {\sf NAND} gate $g(u,v;w)$ with input wires $u,v$ and output wire $w$, we have (inductively) matrices $\mathbf{A}_u = \mathbf{AR}_u + x_u\mathbf{G}$, and $\mathbf{A}_v = \mathbf{AR}_v + x_v\mathbf{G}$ where $x_u$ and $x_v$ are the input bits of $u$ and $v$, and the evaluation algorithm computes:
$$\mathbf{A}_w
= \mathbf{G} - \mathbf{A_u}\cdot\mathbf{G}^{-1}(\mathbf{A}_v)
= \mathbf{G} - (\mathbf{AR}_u+x_u\mathbf{G})\cdot\mathbf{G}^{-1}(\mathbf{AR}_v + x_v\mathbf{G})
= \mathbf{AR}_g + (1-x_ux_v)\mathbf{G},
$$
where $1-x_ux_v \coloneqq \algo{NAND}(x_u,x_v)$, and $\mathbf{R}_g = -\mathbf{R}_u\cdot\mathbf{G}^{-1}(\mathbf{A}_v)-x_u\mathbf{R}_v$ has low norm if both $\mathbf{R}_u$ and $\mathbf{R}_v$ have low norm.
\para{Biased Bit-QPRF}. We will need a single-bit-output QPRF that output 1 with a customizable probability $\epsilon$. Moreover, we need it to be implementable in $\mathsf{NC}^1$. Such a QPRF can be built using the PRF constructed in \cite{EC:BanPeiRos12} assuming QLWE with super-polynomial modulus. See \Cref{sec:additional-prelims:biased-QPRF} for more details and the formal definition.
\subsection{Our Construction}
Our signature scheme uses a biased bit QPRF $\PRF$ whose input space $\mathcal{X}$ corresponds to our message space $\mathcal{M}$, and the algorithms $\algo{SampleLeft}$, $\algo{SampleRight}$ given as in \Cref{theorem:sampleleft} and \Cref{theorem:sampleright} respectively, and $\algo{TrapGen}$ that can sample matrices in $\mathbb{Z}_q^{n\times m}$ statistically close to uniform, along with a corresponding `short' or `trapdoor' basis for the associated lattice. This is formally defined in \Cref{lemma:trapgen}. The construction is as follows:
\begin{ConstructionBox}[label={constr:BU:plain}]{Blind-Unforgeable Signatures in the Plain Model}
{\bf Paramters:} Set message length $t(\secpar)$ and row size $n(\secpar)$ as free parameters (polynomial in $\secpar$). PRF key size is set as $k(\secpar)$, and the depth for $\mathsf{C}_{\algo{PRF}}$ is given by $d(\secpar)$. We set $m=n^{1+\eta}$ for proper running of $\algo{TrapGen}$, and $\mathsf{sigsize}_\secpar = s\sqrt{2m}$ for the validity of $\algo{SampleLeft}$ output (to ensure completeness). Set $s = O(4^dm^{3/2})\cdot\omega(\sqrt{\log m})$ to ensure statistical closeness of $\algo{SampleLeft}$ and $\algo{SampleRight}$, and correspondingly set {$\beta = O(16^dm^{7/2})\cdot\omega(\sqrt{\log m})$} and {$q = O(16^dm^4)\cdot \big(\omega(\sqrt{\log m})\big)^2$} to have an overall reduction to an appropriately hard instance of SIS. For further details about these choices, see \Cref{sec:appendix:parameters-plain-BU-sig}.
\subpara{$\Gen(1^\SecPar)$:}
\begin{enumerate}
\item
Sample a matrix $\mathbf{A}$ along with a `trapdoor' basis $\mathbf{T_A}$ for $\Lambda^\perp_q(\mathbf{A})$ using $\algo{TrapGen}$.
\item
Sample a matrix $\mathbf{A}'$, `PRF key' matrices $\mathbf{B}_1,\dots,\mathbf{B}_k$, and `PRF input' matrices $\mathbf{C}_0,\mathbf{C}_1$ uniformly from $\mathbb{Z}_q^{n \times m}$ ($k$ is the PRF key length).
\item
Fix the Gaussian width parameter $s$ as given in parameter selection.
\item Fix a Boolean circuit description $\mathsf{C}_{\algo{PRF}}$ of the algorithm $\algo{PRF}_{(\cdot)}(\cdot)$.
\item Output $vk=(\mathbf{A},\mathbf{A}',\Set{\mathbf{B}_i}_{i=1}^k, \{\mathbf{C}_0,\mathbf{C}_1\}, \algo{PRF},s,\mathbf{C}_{\PRF})$ and $sk=\mathbf{T_A}$.
\end{enumerate}
\subpara{$\Sign(sk,vk,M)$:} let $(M_1,\dots,M_t) \in \bits^t$ be the bit-wise representation of $M$.
\begin{enumerate}
\item Run the \cite{ITCS:BraVai14} evaluation algorithm $\Eval_\textsc{bv}$ to homomorphically evaluate the circuit $\mathbf{C}_{\PRF}$ using the `encoded' PRF key bits $\Set{\mathbf{B}_i}_{i\in[k]}$ and message bits $\Set{\mathbf{C}_{M_j}}_{j \in[t]}$. This yields
$$\mathbf{A}_\textsc{prf,m} \coloneqq \Eval_\textsc{bv}(\mathbf{C}_{\PRF},\Set{\mathbf{B}_i}_{i\in[k]},\Set{\mathbf{C}_{M_j}}_{j \in [t]}) \in \mathbb{Z}_q^{n \times m}.$$
\item
Set $\mathbf{F}_\textsc{m} \coloneqq [\mathbf{A}~|~\mathbf{A}' - \mathbf{A}_\textsc{prf,m}]$.
\item
Use $\algo{SampleLeft}$ to obtain $\mathbf{d}_\textsc{m}$ with distribution statistically close to $\gets \mathcal{D}_{\Lambda_q^\perp(\mathbf{F}_\textsc{m}),s}$ (see \Cref{theorem:sampleleft}).
\item
Output $\sigma = \mathbf{d}_\textsc{m} \in \mathbb{Z}_q^{2m}$.
\end{enumerate}
\subpara{$\Verify(vk,M,\sigma)$:}
\begin{enumerate}
\item
Compute $\mathbf{A}_\textsc{prf,m}$, $\mathbf{F}_\textsc{m}$ as before.
\item
Check that $\sigma \in \mathbb{Z}_q^{2m}$, $\sigma\neq 0$, and $||\sigma|| \leq \mathsf{sigsize}_\secpar$. If it fails, output $0$.
\item
If $\mathbf{F}_\textsc{m} \cdot \sigma = 0 \mod q$, output $1$, otherwise output $0$.
\end{enumerate}
\end{ConstructionBox}
\subsection{Parameter Selection for \Cref{constr:BU:plain}}
\label{sec:appendix:parameters-plain-BU-sig}
The security parameter $\secpar$ is represented as before. We set message length $t(\secpar)$ and row size $n(\secpar)$ as free parameters (polynomial in $\secpar$). PRF key size is set as $k(\secpar)$, and the depth for $\mathsf{C}_{\algo{PRF}}$ is set to be $d(\secpar)$. We must set the parameters properly to ensure that the following conditions are satisfied:
\begin{enumerate}
\item We must have $m=n^{1+\eta}$, with $\eta$ being given by $n^\eta > O(\log q)$. This is to ensure that $\algo{TrapGen}$ can run properly, by \Cref{lemma:trapgen}.
\item We require that $s > ||\widetilde{\mathbf{T}}_{\vb{G}}||\cdot||\mathbf{R}||_2\cdot\omega(\sqrt{\log m})$, where $\mathbf{R} = \mathbf{R_{A'}} - \mathbf{R}_{\mathbf{A}_{\mathrm{PRF},\mathrm{M}}}$ (the latter will be defined in the course of the proof), for the statistical similarity of $\algo{SampleLeft}$ and $\algo{SampleRight}$, as per \Cref{theorem:sampleleft,theorem:sampleright}.
\item Since signatures are vectors of length $2m$ over $\mathbb{Z}_q$ sampled from (statistically close to) $\mathcal{D}_{\Lambda_q^\perp(\mathbf{F}_m),s}$, for most honestly generated signatures to be valid, it is necessary to set $\mathsf{sigsize}_\secpar \geq s\sqrt{2m}$, in accordance with \Cref{lemma:discgaussconc}.
\item For hardness of the SIS instance, we require that the width parameter {$\beta$ satisfy $\beta \geq O(4^d\cdot m^{3/2}\cdot s\sqrt{2m})$}.
\item Finally, for standard average-to-worst case hardness reductions to apply for SIS, we require that {$q \geq \beta \cdot\omega(\sqrt{n\log n})$}.
\end{enumerate}
Accordingly, we set the remaining parameters as follows:
\begin{itemize}
\item We set $m=n^{1+\eta},\mathsf{sigsize}_\secpar=s\sqrt{2m}$ just as indicated.
\item By the bounds on $\mathbf{R}_{\mathbf{A}_{\mathrm{PRF},\mathrm{M}}}$ implied by \Cref{lemma:rnorm}, it suffices to set $s=O(4^dm^{3/2})\cdot\omega(\sqrt{\log m})$ to satisfy constraint 2 above.
\item Using the above $s$ and so as to just about satisfy constraint 4, we set {$\beta = O(16^dm^{7/2})\cdot\omega(\sqrt{\log m})$}.
\item We set $q$ based on $\beta$ above so as to just satisfy the final constraint, namely {$q = O(16^dm^4)\cdot\omega(\sqrt{\log m})^2$}.
\end{itemize}
Since we consider PRFs in $\mathsf{NC}^1$, we can write $d = c \log \ell$ (for some constant $c$) where $\ell = t+k$ is the input length for the PRF. This yields {$\beta = O(\ell^{4c}m^{7/2})\cdot\omega(\sqrt{\log m})$ and $q = O(\ell^{4c}m^4)\cdot\omega(\sqrt{\log m})^2$}.
\subsection{Proof of Security}
Completeness follows straightforwardly from the correctness of $\algo{SampleLeft}$ (\Cref{theorem:sampleleft}) for $\mathcal{D}_{\Lambda_q^\bot(\mathbf{F}),s}$. In the following, we prove BU-security.
\begin{theorem}\label{thm:plain-BU-sig:security}
Let $\secpar$ denote the security parameter, and $\algo{PRF}$ be a biased bit QPRF as defined in \Cref{def:biased-bit-PRF} above. If the parameters $n,m,q,\beta,s,d$ are picked as discussed above, and the $\mathbf{SIS}_{q,\beta,n,m}$ problem is hard for QPT adversaries, then our signature scheme $\Sig$ constructed as above, with the indicated parameters, satisfies Blind-Unforgeability as in \Cref{def:pq:blind-unforgeability}.
\end{theorem}
\begin{proof}
Consider a QPT $\Adv$ that is able to produce forgeries w.r.t.\ $\Sig$ in the blind-unforgeability challenge. Our proof proceeds using a series of hybrid experiments. In the final hybrid we show a reduction from an adversary producing succesful forgeries to the hardness of $\mathbf{SIS}_{q,\beta,n,m}$. The hybrids are as follows:
\para{Hybrid $H_0$:} This is the blind-unforgeability game (\Cref{expr:BU:ordinary-signature}). Namely, for an adversary-specified $\epsilon$, the challenger manually samples an $\epsilon$-weight set $B_\epsilon$ over messages, and does not answer queries in $B_\epsilon$. Signing and verification keys are chosen just as in the ordinary signing procedure.
\para{Hybrid $H_1$:} This hybrid is identical to the previous one, except that we change the ordinary key generation into the following:
\begin{enumerate}
\item
Sample $\mathbf{A}$ with a `trapdoor' basis $\mathbf{T_A}$ for $\Lambda^\perp_q(\mathbf{A})$ using $\algo{TrapGen}$ as before.
\item
Sample `low-norm' matrices: $\mathbf{R_{A}'},\Set{\mathbf{R}_{\vb{B}_i}}_{i=1}^k,\mathbf{R_C}_0,\mathbf{R_C}_1 \pick \Set{-1,1}^{m\times m}$.
\item
Let $\algo{PRF}$ and $\mathsf{C}_{\algo{PRF}}$ be as before.
\item \label[Step]{item:PQSig:plain:h-1:4}
Sample a PRF key $k_\epsilon \gets \algo{PRF.Gen}(1^\secpar,\epsilon)$, where $k_\epsilon = s_1,\dots,s_k$ (i.e. has length $k$).
\item
Set $\mathbf{A}' = \mathbf{AR}_{\vb{A}'} + \mathbf{G}$, where $\vb{G}$ the gadget matrix $\mathbf{G}$, which has a publicly-known trapdoor $\widetilde{\mathbf{T}}_{\mathbf{G}}$ (as described in \Cref{lemma:gadgettrapdoor}).
\item \label[Step]{item:PQSig:plain:h-1:6}
Set $\mathbf{C}_b = \mathbf{AR}_{\vb{C}_b} + b\mathbf{G}$ for $b \in \Set{0,1}$, and sample $\mathbf{B}_i \pick \mathbb{Z}_q^{n \times m}$ for every $i \in [k]$.
\item
Fix the Gaussian width parameter $s$ as before.
\item
Output $vk=(\mathbf{A},\mathbf{A}',\Set{\mathbf{B}_i}_{i=1}^k, \{\mathbf{C}_0,\mathbf{C}_1\}, s, \algo{PRF},\mathbf{C}_{\PRF})$, and $sk=(\mathbf{T_A},k_\epsilon)$.
\end{enumerate}
Note that while this hybrid generates a key $k_\epsilon$, it never uses it.
\subpara{$H_0 \sind H_1$:} The only thing that changes (w.r.t.\ $\Adv$) is the distribution of the various components $(\mathbf{A'},\mathbf{C}_0,\mathbf{C}_1)$ of the verification key handed out by the challenger. However, by \Cref{lemma:ext-lhl} these distributions are all statistically close to the corresponding distributions in $H_0$. Note that the verification key is picked at the start of the challenge and provided to $\Adv$, so there is no scope for $\Adv$ to have quantum access to these component distributions. Thus the outputs in these hybrids are statistically close.
\para{Hybrid $H_2$:} This hybrid is identical to the previous one, except that we change how the challenger picks the blindset---Instead of manually sampling $B_\epsilon$ as a random $\epsilon$-weight set, it now sets $B_\epsilon$ to be the set of messages $M$ where $\algo{PRF}_{k_\epsilon}(M)$ is 1 (note that the challenger now possesses $k_\epsilon$ as part of $sk$, and can compute $\algo{PRF}_{k_\epsilon}(\cdot)$). Observe that the challenger in this hybrid is now efficient.
\subpara{$H_1 \cind H_2$:} Note that setup and key generation in $H_2$ is identical to that in $H_1$---In particular, the adversary learns {\em no} information about the key $k_\epsilon$. The indistinguishability between $H_1$ and $H_2$ then follows immediately from the security of the biased bit-QPRF (\Cref{def:biased-bit-PRF}).
\para{Hybrid $H_3$:} This hybrid is identical to the previous one, except that we change how the matrices $\vb{B}_i$'s (in \Cref{item:PQSig:plain:h-1:6}) are generated. Namely, we now set $$\forall i \in [k], ~~\mathbf{B}_i \coloneqq \mathbf{AR}_{\vb{B}_i} + s_i \cdot\mathbf{G}.$$
(Recall that $s_i$ is the $i$-th bit of the $k_\epsilon$ generated in \Cref{item:PQSig:plain:h-1:4}.)
\subpara{$H_2 \sind H_3$:} The only things that change between these hybrids are the matrices $\Set{\vb{B}_i}_{i \in [k]}$. Again, using \Cref{lemma:ext-lhl} the distributions for $\vb{B}_i$ for each $i \in [k]$ are all statistically close to the corresponding distributions in $H_2$, and just as in the similarity argument between $H_2$ and $H_3$, we can conclude that these hybrids too have indistinguishable outputs.
\iffalse
\para{Hybrid $H_1$:} Here we change the ordinary key generation into the following:
\begin{enumerate}
\item
Sample $\mathbf{A}$ with a `trapdoor' basis $\mathbf{T_A}$ for $\Lambda^\perp_q(\mathbf{A})$ using $\algo{TrapGen}$ as before.
\item
Pick `low-norm' matrices. $\mathbf{R_{A}'},\Set{\mathbf{R}_{\vb{B}_i}}_{i=1}^k,\mathbf{R_C}_0,\mathbf{R_C}_1 \gets \Set{-1,1}^{m\times m}$
\item
Let $\algo{PRF}$ and $\mathsf{C}_{\algo{PRF}}$ be as before.
\item
Sample a PRF key $k_\epsilon \gets \algo{PRF.Gen}(1^\secpar,\epsilon)$, where $k_\epsilon = s_1,\dots,s_k$.
\item
Set $\mathbf{A}' = \mathbf{AR}_{\vb{A}'} + \mathbf{G}$, where $\vb{G}$ the gadget matrix $\mathbf{G}$, which has a publicly-known trapdoor $\widetilde{\mathbf{T}}_{\mathbf{G}}$ ( as described in \Cref{lemma:gadgettrapdoor}).
\item
Set $\mathbf{C}_b = \mathbf{AR}_{\vb{C}_b} + b\mathbf{G}$ for $b \in \Set{0,1}$, $\mathbf{B}_i = \mathbf{AR}_{\vb{B}_i} + s_i\mathbf{G}$ for every $i \in [k]$.
\item
Fix the Gaussian width parameter $s$ as before.
\item
Output $vk=(\mathbf{A},\mathbf{A}',\Set{\mathbf{B}_i}_{i=1}^k, \{\mathbf{C}_0,\mathbf{C}_1\}, s, \algo{PRF},\mathbf{C}_{\PRF})$, and $sk=(\mathbf{T_A},k_\epsilon)$.
\end{enumerate}
\xiao{remark that $k_\epsilon$ is not used in this hybrid, though we generate it.}
\subpara{$H_0 \sind H_1$:} The only thing that changes (w.r.t.\ $\Adv$) is the distribution of the various components $(\mathbf{A'},\Set{\mathbf{B}_i}_{i=1}^k,\mathbf{C}_0,\mathbf{C}_1)$ of the verification key handed out by the challenger. However, by \Cref{lemma:ext-lhl} these distributions are all statistically close to the corresponding distributions in $H_0$. Note that the verification key is picked at the start of the challenge and provided to $\Adv$, so there is no scope for $\Adv$ to have quantum access to these component distributions. Thus the outputs in these hybrids are statistically close.
\para{Hybrid $H_2$:} In this hybrid, we change how the challenger picks the blindset. Instead of manually sampling $B_\epsilon$ as a random $\epsilon$-weight set, it now sets $B_\epsilon$ to be the set of messages $M$ where $\algo{PRF}_{k_\epsilon}(M)$ is 1 (note that the challenger now possesses $k_\epsilon$ as part of $sk$, and can compute $\algo{PRF}_{k_\epsilon}(\cdot)$). Observe that the challenger in this hybrid is now efficient.
\subpara{$H_1 \cind H_2$:} follows immediately from the security of the biased bit-QPRF (\Cref{def:biased-bit-PRF}).
\xiao{A reviewer equstion this step: we give the adversary $\vb{B}_i$ which contain info about PRF key. Talk with Rohit to add a remark for this.}
\fi
\para{Hybrid $H_4$:} Observe that, starting from $H_1$, we have:
\begin{align*}
\mathbf{F}_\textsc{m}
& = [\vb{A} ~|~ \vb{A}' - \vb{A}_\textsc{prf,m}] =
\big[\vb{A} ~|~ \vb{A}' - \Eval_\textsc{bv}(\mathbf{C}_{\PRF},\Set{\mathbf{B}_i}_{i\in[k]},\Set{\mathbf{C}_{M_j}}_{j \in [t]}) \big]\\
& = \big[ \vb{A} ~|~ \vb{A}' - \big( \vb{A}\vb{R}_\textsc{prf,m} + \PRF_{k_\epsilon}(M) \cdot\vb{G} \big) \big]\\
& = \big[\mathbf{A}~|~\mathbf{A}(\mathbf{R_{A'}}-\mathbf{R}_\textsc{prf,m}) + \big(1-\algo{PRF}_{k_\epsilon}(M)\big)\cdot\mathbf{G}\big].
\end{align*}
In this hybrid, we switch to using $\algo{SampleRight}$ to answer signing queries, instead of using $\algo{SampleLeft}$. That is, we run $\algo{SampleRight}$ using $\mathbf{T_G}$, the publicly available trapdoor for $\mathbf{G}$. Note this means that now the challenger cannot answer queries where the `right half' of $\mathbf{F}_\textsc{m}$ does not include $\mathbf{G}$, i.e., $\algo{PRF}_{k_\epsilon}(M) =1$. But due to the way $H_2$ generate the blindset, such a query is anyway answered with ``$\bot$''.
\subpara{$H_3 \cind H_4$:} We first show that these two hybrids answer signature queries for any {\em classical} query $M$ in a {\em statistically} indistinguishable manner. For any query $M$, there are two cases: (1) if $\algo{PRF}_{k_\epsilon}(M) =1$, the challengers in both $H_3$ and $H_4$ return $\bot$. In this case, these distributions are identical. (2) Else, we have $\algo{PRF}_{k_\epsilon}(M) =0$. Since $\mathbf{F}_\textsc{M}$ is computed identically in both hybrids, and by \Cref{theorem:sampleleft,theorem:sampleright} both $\algo{SampleLeft}$ and $\algo{SampleRight}$ sample from distributions statistically close to $\mathcal{D}_{\Lambda_q^\perp(\mathbf{F}_\textsc{m}),s}$, i.e., they are also statistically close to each other. Thus overall the distributions of signatures returned in $H_3$ and $H_4$ are statistically close to each other, say with less than distance $\Delta(\SecPar)$ (which is negligible in $\SecPar$). Now since \Adv is a quantum machine making at most polynomially (say $q(\secpar)$) many quantum queries. Then, we can use \Cref{lemma:oracleindist} to conclude that \Adv distinguishes between $H_3$ and $H_4$ with probability at most $\sqrt{8C_0q^3\Delta}$, which is negligible.
\para{Hybrid $H_5$:} In this hybrid, the challenger no longer samples $\mathbf{A}$ using $\algo{TrapGen}$. Instead, it samples $\mathbf{A}$ uniformly from $\mathbb{Z}_q^{n \times m}$.
\subpara{$H_4 \sind H_5$:} This follows immediately from \Cref{lemma:trapgen}.
\para{Reduction to QSIS.} We can now describe our reduction $\mathcal{R}$ in this hybrid:
\begin{enumerate}
\item Asks for and recieves a uniform matrix in $\mathbb{Z}_q^{n \times m}$ as the $\mathbf{SIS}_{q,\beta,n,m}$ challenge.
\item Sets $\mathbf{A}$ to be this matrix (instead of sampling $\mathbf{A}$ by itself).
\item When the adversary returns a forgery $(M^*,\sigma^*)$, $\mathcal{R}$ checks if this is valid, i.e., that (i) $M^* \in B_\epsilon$, (ii) $\sigma^* \in \mathbb{Z}_q^{2m}$, (iii) $\sigma^*\neq 0$, (iv) $\mathbf{F}_{\textsc{m}^*} \cdot \sigma^* = 0 \mod q$ and (v) $||\sigma|| \leq \mathsf{sigsize}_\secpar$. If any of these checks fail, it aborts.
\item Represent $\sigma^*$ as $[\mathbf{d}^\top_1 |~ \mathbf{d}^\top_2]^\top$, with $\mathbf{d}_1, \vb{d}_2 \in \mathbb{Z}_q^m$. $\mathcal{R}$ computes $\mathbf{e} = \mathbf{d}_1 + \mathbf{R}\mathbf{d}_2$ where $\mathbf{R} = \mathbf{R}_{\vb{A}'} - \vb{R}_\textsc{prf,m}$ (we will use this shorthand going forward), and presents $\mathbf{e}$ as its solution to the SIS challenge $\mathbf{A}$.
\end{enumerate}
Now we can prove that $\vb{e}$ is indeed an SIS solution with non-negligible probability by an argument very similar as in the final reduction for \cite[Theorem 3.1]{AC:BoyLi16}. We present the final reduction in the following.
\subpara{The Final Reduction.}
Before showing that the reduction's output $\mathbf{e}$ indeed breaks the given SIS challenge, we must first examine the possibility of a `related message' attack. Namely, we want to avoid a situation where the adversary can directly use signatures for one message to get signatures on another since this would render our reduction moot. We show that this is not the case by showing that an adversary cannot come up with two messages $M,M'$ such that $\mathbf{F}_\textsc{m} = \mathbf{F}_\textsc{m}'$. The following lemma accomplishes this task.
\begin{lemma}
If a QPT adversary produces two distinct messages $M,M'$ such that $\mathbf{A}_\textsc{prf,m} = \mathbf{A}_{\textsc{prf},\textsc{m}'}$ with non-negligible probability, then we can break the $\mathbf{SIS}_{q,\beta,n,m}$ challenge.
\end{lemma}
\begin{proof}
With the verification key in $H_5$ picked just as in $H_2$, if $\mathbf{A}_\textsc{prf,m} = \mathbf{A}_{\textsc{prf},\textsc{m}'}$, then we have $$\mathbf{AR}_\textsc{prf,m} + \algo{PRF}_{k_\epsilon}(M)\mathbf{G} = \mathbf{AR}_{\textsc{prf}, \textsc{m}'} + \algo{PRF}_{k_\epsilon}(M')\mathbf{G}.$$
Note that we have $\algo{PRF}_{k_\epsilon}(M) \neq \algo{PRF}_{k_\epsilon}(M')$ with probability $2\epsilon \cdot (1-\epsilon)$, which is a constant. If this holds, we have $\mathbf{A}(\mathbf{R}_\textsc{prf,m} - \mathbf{R}_{\textsc{prf}, \textsc{m}'}) \pm \mathbf{G} = 0 \mod q$. Now by \Cref{lemma:gadgettrapdoor} and using $\algo{SampleRight}$ we can find a low-norm vector $\mathbf{d} \in \mathbb{Z}_q^{m \times m}$ such that $\mathbf{Gd} = 0 \mod q$, $\mathbf{d} \neq 0$ and $||\mathbf{d}|| \leq s'\sqrt{2m}$ (for some $s' \geq \sqrt{5}\omega(\sqrt{\log m})$). Then $[\mathbf{A}(\mathbf{R}_\textsc{prf,m} - \mathbf{R}_{\textsc{prf}, \textsc{m}'}) \pm \mathbf{G}]\mathbf{d} = 0 \mod q$, yielding $\mathbf{A}(\mathbf{R}_\textsc{prf,m} - \mathbf{R}_{\textsc{prf}, \textsc{m}'})\mathbf{d} = 0 \mod q$. By our choice of parameters, $(\mathbf{R}_\textsc{prf,m} - \mathbf{R}_{\textsc{prf}, \textsc{m}'})$ has low enough norm and so $(\mathbf{R}_\textsc{prf,m} - \mathbf{R}_{\textsc{prf}, \textsc{m}'})\mathbf{d}$ is a valid SIS solution for $\mathbf{A}$. This happens with non-negligible probability using our starting assumption, and thus we break $\mathbf{SIS}_{q,\beta,n,m}$ as claimed.
\end{proof}
Now we can turn to validating our reduction. It is straightforward to verify that if $\sigma^*$ is a valid signature, then $\mathbf{e}$ is a valid integer solution to $\mathbf{A}$. Indeed, we have $\mathbf{F}_{\textsc{m}^*}\cdot \sigma^* = 0 \mod q$, which from the above boils down to
$$[\mathbf{A}~|~\mathbf{A}(\mathbf{R_{A'}}-\mathbf{R}_\textsc{prf,m}) + \big(1-\algo{PRF}_{k_\epsilon}(M)\big)\mathbf{G}] \cdot \sigma^* = 0 \mod q,$$
which can be rewritten as $\mathbf{A}(\mathbf{d}_1 + \mathbf{R}\mathbf{d}_2) = 0 \mod q$, proving our claim.
It remains to verify that $\mathbf{e}$ is (i) short and (ii) nonzero. Let us begin with shortness. Since $||\sigma^*|| \leq s\sqrt{2m}$, we have $\mathbf{||d_1||,||d_2||} \leq s\sqrt{2m}$. We then have $||\mathbf{e}|| \leq ||\mathbf{d}_1|| + ||\mathbf{d}_2||\cdot||\mathbf{R}||_2$. By our parameter choices, and using \Cref{lemma:rnorm}, this latter term is again at most $O(4^d m^{3/2})s\sqrt{2m}$. By our choice of parameters, this is less that $\beta \ge O(4^d\cdot m^{3/2}) \cdot s\sqrt{m}$, and so $\mathbf{e}$ is indeed a valid solution.
Next let us show that $\mathbf{e}$ is nonzero with overwhelming probability. Note that by assumption, $\sigma^*$ is nonzero so at least one of $\mathbf{d}_1$ or $\mathbf{d}_2$ must be so. If $\mathbf{d}_2$ is zero, then we have that $\mathbf{e}$ is directly is nonzero, so let us focus on the case that $\mathbf{d}_2$ is nonzero. Expressing $\mathbf{d}_2$ as $(d_1,\dots,d_m)^\top$, we must have that at least one of the coordinates of $\mathbf{d}_2$ is nonzero. Let $d_j$ be such a coordinate. Expressing $\mathbf{R}$ as $(\mathbf{r}_1,\dots,\mathbf{r}_m)$, we have that $$\mathbf{R}\cdot\mathbf{d}_2 = \mathbf{r}_jd_j + \sum\limits_{i=1,i\neq j}^{m}\mathbf{r}_id_i.$$
Now we note that for the (fixed) $M^*$ for which $\Adv$ makes its forgery, $\mathbf{R}$ (and in turn $\mathbf{r}_j$) depends only on the low-norm matrices $\mathbf{R_{A'}},\Set{\mathbf{R_{B_i}}}_{i \in [k]},\mathbf{R_{C_0}},\mathbf{R_{C_1}}$ and $k_\epsilon$. Now the {\em only} information about $\mathbf{R}$ (in turn, $\mathbf{r}_j$) $\Adv$ has is derived from the components of $vk$, namely, $\mathbf{A}',\Set{\mathbf{B}_i}_{i \in [k]},\mathbf{C}_0,\mathbf{C}_1$. This implies that {\em any} $\mathbf{r}'_j \in \Set{-1,1}^m$ such that $\mathbf{Ar}_j = \mathbf{Ar}'_j$ is in fact a valid vector in the sense that replacing $\mathbf{r}_j$ with $\mathbf{r}'_j$ is completely indistinguishable to the adversary. By the pigeonhole principle, there are exponentially many such distinct $\mathbf{r}'_j$'s so that $\mathbf{Ar}_j = \mathbf{Ar}'_j$, and for such an admissible $\mathbf{r}'_j$, the probability that $\mathbf{r}'_j\cdot d_j$ hits a fixed value in $\mathbb{Z}^m_q$ is exponentially small. It is straightforward to see that this implies that $\mathbf{e}$ is zero with at most negligible probability (since the chance that $\mathbf{r}'_j\cdot d_j$ equals the exact value needed to cancel out the other terms in $\mathbf{e}$ is negligible).
Finally, it is straightforward to verify that $H_4$ runs in polynomial time, and in turn that the reduction $\mathcal{R}$ is efficient. If $\Adv$ produces a valid forgery within $B_\epsilon$ with probability $\nu(\secpar)$, $\mathcal{R}$ breaks the $\mathbf{SIS}_{q,\beta,n,m}$ challenge with probability $\nu(\secpar)-\negl(\secpar)$. We conclude that $\Adv$ wins the blind-unforgeability experiment with at most negligible probability.
\end{proof}
\section{Blind-Unforgeable Signatures in the QROM}
\label{sec:BU:sig:QROM}
We show here that the signature scheme in \cite{STOC:GenPeiVai08} is a blind-unforgeable signature in the quantum random oracle model. This construction relies on the notion of {\em preimage sampleable functions}.
\begin{definition}[Preimage Sampleable Functions \cite{STOC:GenPeiVai08}]
\label{def:PSF}
A {\em preimage sampleable function} family \PSF consists of the following PPT algorithms:
\begin{itemize}
\item $\Gen(1^\SecPar)$ samples a public/secret key pair $(pk,sk)$.
\end{itemize}
For any $(pk, sk)$ in the range of $\Gen(1^\SecPar)$:
\begin{itemize}
\item $\F(pk,\cdot)$ computes a function from set $\mathcal{X}_\SecPar$ to set $\mathcal{Y}_\SecPar$.
\item $\Samp(1^\SecPar)$ samples an $x$ from some (possibly non-uniform) distribution $\mathcal{X}_\SecPar$ such that $\F(pk,x)$ is distributed uniformly over $\mathcal{Y}_\SecPar$.
\item $\InvF(sk,y)$ takes as input any $y \in \mathcal{Y}_\SecPar$ and outputs a preimage $x \in \mathcal{X}_\SecPar$ such that $\F(pk,x)=y$, {\em and} $x$ is distributed statistically close to $\Samp(1^\SecPar)$ conditioned on $\F(pk,x)=y$.
\end{itemize}
These algorithms satisfy the following properties:
\begin{enumerate}
\item \label{item:def:psf:preimage-min-entropy}
{\bf Preimage Min-entropy:} For each $y \in \mathcal{Y}_\SecPar$, the conditional min-entropy of $x \leftarrow \Samp(1^\SecPar)$ given $\F(pk,x)=y$ is at least $\omega(\log n)$.
\item \label{item:def:psf:cr}
{\bf Collision Resistance:} For any QPT algorithm \Adv, the probability that $\Adv(1^\SecPar,pk)$ outputs distinct $x,x' \in \mathcal{X}_\SecPar$ such that $F(pk,x) = \F(pk,x')$ is negligible in $\SecPar$.
$$\Pr[
\begin{array}{l}
(pk, sk)\gets \Gen(1^\secpar);\\ (x, x')\gets \Adv(1^\SecPar,pk)
\end{array}: x\ne x' ~\wedge~ \F(pk,x) = \F(pk,x')] = \negl(\secpar).$$
\end{enumerate}
\end{definition}
\cite{STOC:GenPeiVai08} constructs such PSFs based on the hardness of the SIS problem. They also give a signature scheme using PSFs, a hash function modeled as a random oracle, and a pseudorandom function. In the following, we first recall their signature scheme in \Cref{prot:GPVSig}, and then prove in \Cref{thm:GPV:security-proof} that
this construction satisfies \Cref{def:classical-BU-sig} in the QROM if the PRF is a QPRF.
\begin{ConstructionBox}[label={prot:GPVSig}]{The GPV Signature \cite{STOC:GenPeiVai08}}
Let $\PSF$ be a preimage sampleable function. Let \PRF be a pseudorandom function, and $H$ be a hash function. The signature scheme \Sig is defined as follows:
\subpara{$\Gen(\SecPar)$:}
\begin{enumerate}
\item Generate $(sk',pk') \leftarrow \PSF.\Gen(\SecPar)$;
\item Sample a PRF key $k\gets \PRF.\Gen(1^\secpar)$;
\item Output $sk=(sk',k)$ and $pk=pk'$.
\end{enumerate}
\subpara{$\Sign(sk,m)$:}
\begin{enumerate}
\item Compute $r \leftarrow \PRF(k,m)$ and $h=H(m)$;
\item Compute $\sigma = \InvF(sk',h;r)$;
\item Output $\sigma$.
\end{enumerate}
\subpara{$\Verify(pk,m,\sigma)$:}
\begin{enumerate}
\item Compute $h=H(m)$ and $h'=\F(pk',\sigma)$;
\item If $h=h'$ output $1$; otherwise, output $0$.
\end{enumerate}
\end{ConstructionBox}
\begin{theorem}\label{thm:GPV:security-proof}
Assume that \PSF be a preimage sampleable function, PRF is a quantum secure pseudorandom function, and $H$ realizes a random oracle. Then \Cref{prot:GPVSig} is a short blind-unforgeable signature in the quantum random oracle model.
\end{theorem}
\begin{proof}
Completeness and shortness of \Cref{prot:GPVSig} are straightforward. In the following, we prove blind-unforgeability.
\para{The Joint Oracle.} First notice that in the blind-unforgeability game in the ORAM, the adversaries has quantum oracle access to {\em two} oracles: $H$ and $B_\varepsilon\Sign$. As we will show later (in particular, in hybrids $H_1$ and $H_2$ below), we need to change the way these two oracles are sampled, without being noticed by the adversary. We will need to argue the indistinguishability of this switch by invoking \Cref{lemma:oracleindist}; but \Cref{lemma:oracleindist} is for adversaries that have access to a {\em single} oracle ($\mathcal{O}$ or $\mathcal{O}'$). Therefore, we start by slightly changing our oracle interface.
Instead of maintaining separate random and signing oracles (i.e., $H$ and $B_\varepsilon\Sign$ respectively), we will maintain a single {\em joint oracle} $\mathcal{O}$ that can answer both types of queries `jointly'. In more detail, we ask the adversary to include a {\em flag} bit $c \in \bits$ in each query. If $c = 0$, $\mathcal{O}$ will respond as $H$; if $c = 1$, it will respond as $B_\epsilon\Sign$. Formally, $\mathcal{O}$ implements the following mapping:
$$\sum_{c,m,t}\Psi_{c,m,t} \ket{c,m,t} \mapsto \sum_{c,m,t}\Psi_{c,m,t} \ket{c,m,t \xor G(c, m)},$$
where $G(c,m) =
\begin{cases}
H(m) & c = 0 \\
B_\epsilon \Sign(m) & c =1
\end{cases}$.
This transformation is without loss of generality---indeed, any adversary $\Adv'$ that wins the original blind-unforgeability game can be transformed into an adversary \Adv that breaks blind-unforgeability w.r.t.\ the joint oracle $\mathcal{O}$. \Adv need only forward queries from $A'$ to $\mathcal{O}$ and corresponding responses back to $\Adv'$ by setting a proper (classical) bit $c$ .
It is straightforward to see that $\Adv'$ gets identical responses whether interacting with \Adv or in the QROM challenge. We therefore conclude that \Adv has the same success probability as $\Adv'$, and this validates our single joint oracle interface. For the rest of this proof, we presuppose an adversary \Adv that directly interacts with the joint oracle.
We will use \Adv to obtain a contradiction, by employing hybrid arguments. For ease of exposition we will also use the following shorthand: denoting $\F(pk,\cdot)$ by $f(\cdot)$, and $\InvF(sk,\cdot)$ by $f^{-1}(\cdot)$. Consider the following hybrid experiments:
\para{Hybrid $H_0$:} This is simply the normal blind-unforgeability challenge with the joint oracle. In particular, for each $m$ in the superpostion of the adversary's quantum query $\Sigma_m \Psi_m \ket{m}$, the hash $h$ is computed (implicitly) as the random oracle output $H(m)$, and the signature $\sigma_m$ is computed according to $\sigma_m = f^{-1}(h;r)$ where $r=\PRF_k(m)$ (note that, in accordance with the challenge, the signing algorithm will be invoked only if $m \notin B_\varepsilon$).
\para{Hybrid $H_1$:} In this hybrid we change how $r$ is generated. Instead of computing $r=\PRF_k(m)$, we set $r=J(m)$ where $J(\cdot)$ is a {\em random} function over the range and domain of $\PRF$.\footnote{Note that the adversary's query is quantum: $\Sigma_m \Psi_m \ket{m}$. To keep the notation succinct, it suffices to describe the computation for each $m$ in the superposition. We stick to this convention for later hybrids in this proof.}
\subpara{$\Output(H_0) \cind \Output(H_1)$:} This follows directly from the quantum security of the PRF. Any adversary that has distinguishable outputs in these two hybrids is easily converted into a QPT algorithm that can distinguish between the QPRF and a uniformly random function given quantum oracle access to them in turn.
\para{Hybrid $H_2$:} In this hybrid we change both components of the oracle. The sining oracle is now re-defined as $\sigma_m = \Sig.\Sign(m) \coloneqq \Samp\big(1^\SecPar;J(m)\big)$. Further, the hash oracle query is now answered by $h=f(\sigma_m)$. That is, when the adversary asks for $H(m)$, the hybrid first computes $\sigma_m = \Samp\big(1^\SecPar; J(m)\big)$, and then returns $h = f(\sigma_m)$ to the adversary.
\subpara{$\Output(H_1) \sind \Output(H_2)$:} Recall that we view the random oracle $H$ and the signing oracle together as a joint oracle. The only thing that changes in $H_2$ is the computation of parts of the joint oracle response. We go through these carefully. For every $m$ (in the superposition of $\Adv$'s quantum query), the response changes from
\begin{itemize}
\item
$\Big(H(m),~f^{-1}\big(H(m);J(m)\big)\Big)$, i.e., the joint oracle in $H_1$, {\bf to}
\item
$\Big(f\big(\Samp(\SecPar;J(m))\big),~\Samp\big(\SecPar;J(m)\big)\Big)$, i.e, the joint oracle in $H_2$.
\end{itemize}
Note that both $H(m)$ and $J(m)$ are uniformly random. Therefore, by the properties of $\Samp$ and $f^{-1}$ (\Cref{def:PSF}), the above two distributions are statistically close to each other. Denoting the joint oracle in $H_1$ by $\mathcal{O}_1$, and that in $H_2$ by $\mathcal{O}_2$, we conclude that for any (classical) query point $m$, the distributions of the responses returned by $\mathcal{O}_1$ and $\mathcal{O}_2$ conditioned on $m$ are statistically close, say less than distance $\Delta(\SecPar)$ (which is negligible in $\SecPar$). Now since \Adv is a quantum machine making at most polynomially (say $q(\secpar)$) many quantum queries. Then, we can use \Cref{lemma:oracleindist} to conclude that \Adv distinguishes between $\mathcal{O}_1$ and $\mathcal{O}_2$ with probability at most $\sqrt{8C_0q^3\Delta}$, which is negligible in $\SecPar$.
\para{Hybrid $H_3$:} Observe that the hybrid $H_2$ is {\em not} efficiently implementable as it needs to sample a random function $J(\cdot)$. In the classical setting, $H_2$ can be made efficient by lazy-sampling $J(\cdot)$; however, here we cannot resort to lazy-sampling as the adversary has {\em quantum} access to the oracle. Thus, we take the following alternative approach to have an efficient hybrid. Assume $q$ is the upper-bound of the number of quantum queries made by the adversary. In hybrid $H_3$, we sample a $2q$-wise independent hash function $J'(\cdot)$, and replace the random function $J(\cdot)$ with $J'(\cdot)$. Everything else remains the same as in $H_2$.
\subpara{$\Output(H_2) \idind \Output(H_3)$}: This follows immediately from \Cref{lemma:2qwise}.
\para{Reducing to the security of PSF.} Now consider the eventual forgery output by \Adv in $H_3$, $(m^*,\sigma_m^*)$. If this is valid, it follows from \Cref{def:classical-BU-sig} that $m^*\in B_\epsilon$ and $\Sig.\Verify(vk, m^*, \sigma_m^*) = 1$, which means $H(m^*)=f(\sigma_m^*)$.
Let $\sigma_m' \coloneqq \Samp(1^\SecPar;J'(m^*))$. Note that this is exactly $\Sig.\Sign$'s output on $m^*$ in $H_3$. Due to the way $H_3$ implements the oracles, this presents only two possibilities:
\begin{enumerate}
\item
Either we have $\sigma_m' = \sigma_m^*$, in which \Adv is able to pick out the value $\sigma_m'$ among all possible preimages of $h= f(\sigma_m^*)$. Observe that $m^* \in B_\epsilon$, which means that $\Adv$ never saw that value $\sigma_m'$ before as $\Sig.\Sign$ returns $\top$ for messages in $B_\epsilon$.
Therefore, by the preimage min-entropy property (\Cref{item:def:psf:preimage-min-entropy}) of PSF, the set of allowed pre-images for $h$ has conditional minentropy at least $\omega(\log \SecPar)$, which means that \Adv can only predict this value with at most negligible probability, so this case has a negligible chance of occurrence.
\item
Else, we must have $\sigma_m' \neq \sigma_m^*$ in which case \Adv has obtained colliding preimages, i.e. $\sigma_m' \neq \sigma_m^*$ such that $f(\sigma_m') = f(\sigma_m^*)$. This contradicts the collision resistance of PSF (note that this is the reason why we need $H_3$ to be efficiently implementable), so this case can also only arise with at most negligible probability.
\end{enumerate}
We conclude that a successful forgery occurs only with negligible probability if the PSF satisfies the aforementioned properties.
\end{proof}
\subsection{Discussion on Compactness}
\label{sec:discussion}
Our construction of post-quantum ring signatures (i.e., \Cref{constr:pq:ring-sig}) is currently only of theoretical interest. It is not efficient, and it does not enjoy compactness (i.e., the signatures size is independent of, or even poly-logarithmic on, the ring size). It is an interesting problem for future research to construction practical or compact ring signatures that satisfies our notion of post-quantum security. In the following, we briefly discuss why this seems non-trivial.
\para{Efficiency.} Almost all known efficient ring signatures are in the random oracle model, following the Fiat-Shamir paradigm (e.g., \cite{AFRICACRYPT:ABBFG13,EC:LLNW16,torres2018post,DBLP:conf/icics/BaumLO18,DBLP:journals/ijhpcn/WangZZ18,CCS:EZSLL19,AC:BeuKatPin20,DBLP:conf/crypto/LyubashevskyNS21}). Although these constructions are based on post-quantum hardness assumptions, their security proofs can only handle adversaries making {\em classical} random oracle queries. Making these constructions secure in the QROM requires a quantum version of the {\em forking lemma} \cite{EC:PoiSte96,CCS:BelNev06}, which seems hard to prove. Indeed, this problem is still open even for (ordinary) signatures in the post-quantum setting (e.g., see the discussion in \cite{AC:Unruh17}). (As a side note, our construction in \Cref{sec:BU:sig:QROM} does not face this problem as it follows the hash-and-sign paradigm, instead of Fiat-Shamir.)
\para{Compactness.} The original construction in \cite{C:CGHKLMPS21} does achieve compactness. Although based on their work, our construction in \Cref{sec:PQ-ring-sig:def} gives up compactness by using the underlying $\Sig$ to sign $\Ring \| m$ together\footnote{Note that our construction will not become compact even if we use a compact $\Sig$. This is because the size of the $\ZAP$ proof for the validity of the signature for $\Ring\|m$ also depends on the size of the rings.}; in contrast, \cite{C:CGHKLMPS21} only uses $\Sig$ to sign $m$. Our choice is critical to achieving BU: when proving BU for our $\RS$, we need to reduce to the BU of $\Sig$. The $\RS$ game will ``blind'' $(\Ring, m)$ pairs, while the $\Sig$ game only blinds messages $m$. If we do not use $\Ring\|m$ as the message for $\Sig$ to sign, the reduction will not be able to create the blindset in a consistent manner. This problem cannot be resolved by applying some type of ``hash'' function on $\Ring\|m$ and asking $\Sig$ to sign the short digest. Indeed, blinding $(\Ring, m)$ pairs with probability $\epsilon$ is different from blinding the hash result of $\Ring\|m$, unless the ``hash'' has pseudo-random output. Replacing the ``hash'' with a PRF does not work either, as the verifier also needs to evaluate the ``hash'' to verify the signature. We leave it as an open question to construct {\em compact} ring signatures achieving our post-quantum security notion.
\section{Introduction}
Recent advances in quantum computing have uncovered several new threats to the existing body of cryptographic work. As demonstrated several times in the literature (e.g., \cite{STOC:Watrous06,AC:BDFLSZ11,FOCS:Zhandry12,DBLP:conf/eurocrypt/AgarwalBGKM21}), building quantum-secure primitives requires more than taking existing constructions and replacing the underlying assumptions with post-quantum ones. It usually requires new techniques and analysis. Moreover, for specific primitives, even giving a meaningful security notion against quantum adversaries is a non-trivial task (e.g., \cite{EC:BonZha13,C:BonZha13,DBLP:journals/qic/Zhandry15,EC:Unruh16,EC:AMRS20}). This work focuses on {\em post-quantum security} of digital signature schemes, namely, classical signatures schemes for which we want to protect against quantum adversaries.
\para{Post-Quantum Unforgeable Signatures.} To build post-quantum secure signature schemes, the first step is to have a notion of unforgeability that protects against adversaries with quantum power. Probably the most natural attempt is to take the standard existential unforgeability (EUF) game, but require unforgeability against all {\em quantum polynomial-time} (QPT) adversaries (instead of all {\em probabilistic polynomial-time} (PPT) adversaries). We emphasize that the communication between the EUF challenger and the QPT adversary is still classical. Namely, the adversary is not allowed to query the challenger's circuit in a quantum manner. Herein, we refer to this notion as PQ-EUF. Usually, PQ-EUF can be achieved by existing constructions in the classical setting via replacing the underlying hardness assumptions with quantum-hard ones (e.g., hard problems on lattice or isogeny-based assumptions).
\subpara{The (Quantum) Random Oracle Model.} In the classical setting, the random oracle model (ROM) \cite{CCS:BelRog93} has been accepted as a useful paradigm to obtain efficient signature schemes. When considering the above PQ-EUF notion in the ROM, two choices arise---one can either allow the adversary {\em classical} access to the RO (as in the classical setting)\footnote{ To avoid confusion, we henceforth denote this model as CROM (``C'' for ``classical'').}, or {\em quantum} access to the RO. The latter was first formalized as the {\em quantum random oracle model} (QROM) by Boneh et al.\ \cite{AC:BDFLSZ11}, who showed that new techniques are necessary to achieve unforgeability against QPT adversaries in this model. Then, a large body of literature has since investigated the PQ-EUF in QROM \cite{FOCS:AmbRosUnr14,AC:Unruh17,EC:KilLyuSch18,C:DFMS19,C:LiuZha19,C:DonFehMaj20,EPRINT:GHHM20}.
\subpara{One-More Unforgeability vs Bind Unforgeability.} Starting from \cite{FOCS:Zhandry12}, people realize that the definitional approach taken by the above PQ-EUF may not be sufficient to protect against quantum adversaries. The reason is that quantum adversaries may try to attack the concerned protocol/primitive by executing it {\em quantumly}, even if the protocol/primitive by design is only meant to be executed classically. As argued in existing literature (e.g., \cite{DBLP:conf/icits/DamgardFNS13,C:GagHulSch16}), such an attack could possibly occur in a situation where the computer executing the classical protocol is a quantum machine, and an adversary somehow manages to observe the communication before measurement. Other examples include adversaries managing to trick a classical device (e.g., a smart card reader) into showing full or partial quantum behavior by, for example, cooling it down and shielding it from any external electromagnetic or thermal interference. Moreover, this concern may also arise in the security reduction (even) w.r.t.\ classical security games but against QPT adversaries. For example, some constructions may allow the adversary to obtain an {\em indistinguishability obfuscation} of, say, a PRF; the QPT adversary can then implement it as a quantum circuit to conduct superposition attacks. Recently, this issue has received an increasing amount of attention \cite{EC:BonZha13,C:BonZha13,DBLP:journals/qic/Zhandry15,EC:Unruh16,C:GagHulSch16,C:SonYun17,AC:HosYas18,AC:HosIwa19,C:CzaHulSch19,EC:AMRS20,DBLP:journals/iacr/CarstensETU20,DBLP:journals/iacr/ChevalierEV20,alagic2021impossibility,hosoyamada2021tight,DBLP:conf/crypto/HosoyamadaS21,cryptoeprint:2021:421}.
To address the aforementioned security threats to digital signatures, it is reasonable to give the QPT adversary $\Adv$ {\em quantum access} to the signing oracle in the EUF game. This raises an immediate question---How should the game decide if $\Adv$'s final forgery is valid? Recall that in the classical setting (or the PQ-EUF above), the game records all the signing queries made by $\Adv$; to decide if $\Adv$ wins, it needs to make sure that $\Adv$'s final forgery message-signature pair is different from the ones $\Adv$ learned from the signing oracle. However, this approach does not fit into the quantum setting, since it is unclear how to record $\Adv$'s {\em quantum} queries without irreversibly disturbing them.
Boneh and Zhandry \cite{C:BonZha13} proposed the notion of {\em one-more unforgeability}. This requires that the adversary cannot produce $\mathsf{sq} + 1$ valid message-signature pairs with only $\mathsf{sq}$ signing queries (an approach previously taken to define blind signatures \cite{AC:PoiSte96}). When restricted to the classical setting, this definition is equivalent to the standard unforgeability of ordinary signatures, by a simple application of the pigeonhole principle. \cite{C:BonZha13} shows how to convert any PQ-EUF signatures to one-more unforgeable ones using a {\em chameleon hash function} \cite{DBLP:conf/ndss/KrawczykR00}; it also proves that the PQ-EUF signature scheme by Gentry, Peikert, and Vaikuntanathan \cite{STOC:GenPeiVai08} (henceforth, GPV) is one-more unforgeable in the QROM, assuming the PRF in that construction is quantum secure (i.e., being a QPRF \cite{FOCS:Zhandry12}).
However, as argued in \cite{C:GarYueZha17,EC:AMRS20}, one-more unforgeability does not seem to capture all that we can expect from quantum unforgeability. For example, an adversary may produce a forgery for a message in a subset $A$ of the message space, while making queries to the signing oracle supported on a disjoint subset $B$. Also, an adversary may make multiple quantum signing queries, but then must consume, say, all of the answers in order to make a single valid forgery. This forgery might be for a message that is different from all the messages in all the superpositions of previous queries. This clearly violates what we intuitively expect for unforgeability, but the one-more unforgeability definition may never rule this out.
To address these problems, Alagic at el.\ \cite{EC:AMRS20} propose {\em blind-unforgeability} (BU). Roughly, the blind-unforgeability game modifies the (quantum-accessible) signing oracle by asking it to always return ``$\bot$'' for messages in a ``blinded'' subset of the message space. The adversary's forgery is considered valid only if it lies in the blinded subset. In this way, the adversary is forced to forge a signature for a message she has not seen a signature before, consistent with our intuition for unforgeability. \cite{EC:AMRS20} shows that blind-unforgeability, when restricted to the classical setting, is also equivalent to PQ-EUF; Moreover, it does not suffer from the above problems for one-more unforgeability\footnote{Although \cite{EC:AMRS20} claimed that blind-unforgeability implies one-more unforgeability, their proof was flawed \cite{BU:vs:OneMore}. The relation between these two notions is still an open problem.}.
In terms of constructions, \cite{EC:AMRS20} show that Lamport's one-time signature \cite{lamport1979constructing} is BU in the QROM, assuming the OWF is modeled as a (quantum-accessible) random oracle. Later, \cite{majenz2021quantum} show that the Winternitz one-time signature \cite{C:Merkle89a} is BU in the QROM, assuming the underlying hash function is modeled as a (quantum-accessible) random oracle. To the best of our knowledge, these are the only schemes known to achieve BU. This gives rise to the following question:
\begin{quote}
{\bf Question 1:}
{\em Is it possible to build (multi-time) signature schemes achieving blind-unforgeability, either in the QROM or the plain model?}
\end{quote}
{\bf Post-Quantum Secure Ring Signatures.} In a {\em ring signature} scheme \cite{AC:RivShaTau01,TCC:BenKatMor06}, a user can sign a message with respect to a {\em ring} of public keys, with the knowledge of a signing key corresponding to any public key in the ring. It should satisfy two properties:
\begin{enumerate}
\item
{\em Anonymity} requires that no user can tell which user in the ring actually produced a given signature;
\item
{\em Unforgeability} requires that no user outside the specified ring can produce valid signatures on behalf of this ring.
\end{enumerate}
In contrast to its notional predecessor, {\em group signatures} \cite{EC:ChaVan91}, no central coordination is required for producing and verfying ring signatures. Due to these features, ring signatures (and their variants) have found natural applications related to whistleblowing, authenticating leaked information, and more recently to cryptocurrencies \cite{torres2018post,EPRINT:Noether15}, and thus have received extensive attention (see, e.g., \cite{C:CGHKLMPS21} and related work therein).
For ring signatures from {\em latticed-based} assumptions,
there exist several constructions in the CROM \cite{AFRICACRYPT:ABBFG13,EC:LLNW16,torres2018post,DBLP:conf/icics/BaumLO18,DBLP:journals/ijhpcn/WangZZ18,CCS:EZSLL19,AC:BeuKatPin20,DBLP:conf/crypto/LyubashevskyNS21}, but only two schemes are known in the plain model \cite{DBLP:journals/iacr/BrakerskiK10,C:CGHKLMPS21}. The authors of \cite{C:CGHKLMPS21} also initiate the study of quantum security for ring signatures. They propose a definition where the QPT adversary is allowed quantum access to the signing oracle in both the anonymity and unforgeability game, where the latter is a straightforward adaption of the aforementioned one-more unforgebility for ordinary signatures. As noted in their work, this approach suffers from two disadvantages:
\begin{enumerate}
\item
Their unforgeability definition seems weak in the sense that, when restricted to the classical setting, it is unclear if their unforgeability is equivalent to the standard one (see \Cref{sec:tech-overview:PQ-ring-sig}). This is in contrast to ordinary signatures, for which one-more unforgeability is equivalent to the standard existential unforgeability;
\item
Their construction only partially achieves (even) this seemingly weak definition. In more detail, their security proof only allows the adversary to conduct superposition attacks on the messages, but not on the rings. As remarked by the authors, this is not a definitional issue, but rather a limitation of their technique. Indeed, \cite{C:CGHKLMPS21} left it as an open question to have a construction protecting against superposition attacks on both the messages and the rings.
\end{enumerate}
The outlined gap begs the following natural question:
\begin{quote}
{\bf Question 2:} {\em Can we have a proper unforgeability notion for ring signatures that does not suffer from the above disadvantage? If so, can we have a construction achieving such a notion?}
\end{quote}
\para{Our Results.} In this work, we resolve the aforementioned questions:
\begin{enumerate}
\item
We show that the GPV signature, which relies on the quantum hardness of SIS (QSIS), can be proven BU-secure in the QROM. Since our adversary has quantum access to the signing oracle, we also need to replace the PRF in the original GPV scheme with a QPRF, which is also known from QSIS. As will be discussed later in \Cref{sec:tech-overview:BU:QROM}, our security proof is almost identical to the proof in \cite{C:BonZha13} for the one-more unforgeability of GPV, except how the desired contradiction is derived in the last hybrid. Interestingly, our proof for BU turns out to be simpler than that in \cite{C:BonZha13} (for one-more unforgeability). We remark that the GPV scheme is {\em short} (i.e., the signature size only depends on the security parameter, but not the message size).
\item
We also construct a BU-secure signature {\em in the plain model}, assuming quantum hardness of Learning with Errors (QLWE) with super-polynomial modulus. Our construction is inspired by the signature (and adaptive IBE) scheme by Boyen and Li \cite{AC:BoyLi16}. This signature scheme is also short.
\item \label{item:contribution:ring-sig-def}
We present a new definition of post-quantum security for ring signatures, by extending blind-unforgeability from \cite{EC:AMRS20}. We show that this definition, when restricted to the classical setting, is equivalent to the standard security requirements for ring signatures.
\item We build a ring signature satisfying the above definition. Our construction is a compiler that converts any BU (ordinary) signature to a ring signature achieving the definition in \Cref{item:contribution:ring-sig-def}, assuming QLWE.
\end{enumerate}
\section{Post-Quantum Ring Signatures}
\subsection{Definition}\label{sec:PQ-ring-sig:def}
\subsubsection{Classical Ring Signatures}
We start by recalling the classical definition of ring signatures \cite{TCC:BenKatMor06,EC:BDHKS19}.
\begin{definition}[Ring Signature]
\label{def:classical:ring-signature}
A ring signature scheme \algo{RS} is described by a triple of PPT algorithms \algo{(Gen,Sign,Verify)} such that:
\begin{itemize}
\item {\bf $\Gen(1^\lambda,N)$:} on input a security parameter $1^\lambda$ and a super-polynomial\footnote{The $N$ has to be super-polynomial to support rings of {\em arbitrary} polynomial size.} $N$ (e.g., $N = 2^{\log^2\secpar}$) specifying the maximum number of members in a ring, output a verification and signing key pair $\algo{(VK,SK)}$.
\item {\bf $\algo{Sign}(\SK,\Ring, m)$:} given a secret key \SK, a message $m \in \mathcal{M}_\lambda$, and a list of verification keys (interpreted as a ring) $\algo{R = (VK_1,\cdots,VK_\ell)}$ as input, and outputs a signature $\Sigma$.
\item {\bf $\algo{Verify}(\Ring,m,\Sigma)$:} given a ring $\algo{R = (VK_1,\dots,VK_\ell)}$, message $m \in \mathcal{M}_\lambda$ and a signature $\Sigma$ as input, outputs either 0 (rejecting) or 1 (accepting).
\end{itemize}
These algorithms satisfy the following requirements:
\begin{enumerate}
\item {\bf Completeness:} for all $\lambda \in \Naturals$, $\ell \le N$, $i^* \in [\ell]$, and $m \in \mathcal{M}_\lambda$, it holds that $\forall i \in [\ell]$ $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N)$ and $\Sigma \gets \algo{Sign}(\SK_{i^*},\Ring,m)$ where $\algo{R = (VK_1,\dots,VK_\ell)}$, we have
$$\Pr[\algo{RS.Verify}(\Ring,m,\Sigma) = 1] = 1,$$
where the probability is taken over the random coins used by $\Gen$ and $\algo{Sign}$.
\item {\bf Anonymity:} For any $Q = \poly(\secpar)$ and any PPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:classical-anonymity} that
$$\algo{Adv}_\textsc{Anon}^{\secpar, Q}(\Adv) \coloneqq \big|\Pr\big[\algo{Exp}_\textsc{Anon}^{\secpar, Q}(\Adv) = 1\big] - 1/2\big| \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:classical-anonymity}]{Classical Anonymity \textnormal{$\algo{Exp}_\textsc{Anon}^{\secpar, Q}(\Adv)$}}
\begin{enumerate}[label={\arabic*.},leftmargin=*,itemsep=0em]
\item
For each $i \in [Q]$, the challenger generates key pairs $(\VK_i,\SK_i) \gets \Gen(1^\lambda, N;r_i)$. It sends $\Set{(\VK_i, \SK_i, r_i)}_{i\in [Q]}$ to $\Adv$;
\item
$\Adv$ sends a challenge to the challenger of the form $(i_0,i_1,\Ring,m)$.\footnote{We stress that $\Ring$ might contain keys that are not generated by the challenger in the previous step. In particular, it might contain maliciously generated keys.} The challenger checks if $\VK_{i_0} \in \Ring$ and $\VK_{i_1} \in \Ring$. If so, it samples a uniform bit $b$, computes $\Sigma \gets \algo{Sign}(\SK_{i_b},\Ring,m)$, and sends $\Sigma$ to $\Adv$.
\item
$\Adv$ outputs a guess $b'$. If $b' = b$, the experiment outputs 1, otherwise 0.
\end{enumerate}
\end{ExperimentBox}
\item {\bf Unforgeability:} for any $Q=\poly(\lambda)$ and any PPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:classical:unforgeability} that
$$\algo{Adv}_\textsc{Unf}^{\secpar, Q}(\Adv) \coloneqq \Pr\big[\algo{Exp}_\textsc{Unf}^{\secpar, Q}(\Adv) = 1\big] \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:classical:unforgeability}]{Classical Unforgeability \textnormal{$\algo{Exp}_\textsc{Unf}^{\secpar, Q}(\Adv)$}}
\begin{enumerate}[label={\arabic*.},leftmargin=*,itemsep=0em]
\item For each $i \in [Q]$, the challenger generates $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N;r_i)$, and stores these key pairs along with their corresponding randomness. It then sets $\mathcal{VK} = \Set{\VK_1,\dots,\VK_Q}$ and initializes a set $\mathcal{C} = \emptyset$.
\item The challenger sends $\mathcal{VK}$ to $\Adv$.
\item $\Adv$ can make polynomially-many queries of the following two types:
\begin{itemize}[leftmargin=*,itemsep=0em,topsep=0em]
\item {\bf Corruption query $(\mathsf{corrupt},i)$:} The challenger adds $\VK_i$ to the set $\mathcal{C}$ and returns the randomness $r_i$ to \Adv.
\item {\bf Signing query $(\algo{sign},i,\Ring,m)$:} The challenger first checks if $\VK_i \in \Ring$. If so, it computes $\Sigma \gets \algo{Sign}(\SK_i,\Ring,m)$ and returns $\Sigma$ to \Adv. It also keeps a list of all such queries made by \Adv.
\end{itemize}
\item Finally, $\Adv$ outputs a tuple $(\Ring^*, m^*, \Sigma^*)$. The challenger checks if:
\begin{enumerate}
\item
$\Ring^* \subseteq \mathcal{VK \setminus C}$,
\item
$\Adv$ never made a signing query of the form $(\algo{sign},\cdot,\Ring^*, m^*)$, {\bf and}
\item
$\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$.
\end{enumerate}
If so, the experiment outputs 1; otherwise, 0.
\end{enumerate}
\end{ExperimentBox}
\end{enumerate}
\end{definition}
We mention that the unforgeability and anonymity properties defined in \autoref{def:classical:ring-signature} correspond respectively to the notions of \emph{unforgeability with insider corruption} and \emph{anonymity with respect to full key exposure} presented in \cite{TCC:BenKatMor06}.
\subsubsection{Defining Post-Quantum Security}\label{sec:ring-sig:pq-definition}
We aim to build a classical ring signature that is secure against adversaries making superposition queries to the signing oracle. Formalizing the security requirements in this scenario is non-trivial. An initial step toward this direction has been taken in \cite{C:CGHKLMPS21}. But their definition has certain restrictions (discussed below). In the following, we develop a new definition building on ideas from \cite{C:CGHKLMPS21}.
\para{Post-Quantum Anonymity.} Recall that in the classical anonymity game (\Cref{expr:classical-anonymity}), the adversary's challenge is a quadruple $(i_0,i_1,\Ring,m)$. To define post-quantum anonymity, a natural attempt is to allow the adversary to send a superposition over components of quadruple, and to let the challenger respond using the following unitary mapping\footnote{Of course, the challenger also needs to check if $\VK_{i_0} \in \Ring$ and $\VK_{i_1} \in \Ring$. But we can safely ignore this for our current discussion.}:
$$\sum_{i_0,i_1,\Ring,m,t} \psi_{i_0, i_1, \Ring, m, t}\ket{i_0, i_1, \Ring, m, t} \mapsto \sum_{i_0,i_1,\Ring,m,t} \psi_{i_0, i_1, \Ring, m, t}\ket{i_0, i_1, \Ring, m, t \oplus \Sign(\SK_{i_b}, m, \Ring;r)}.$$
However, as observed in \cite{C:CGHKLMPS21}, this will lead to an unsatisfiable definition due to an attack from \cite{C:BonZha13}. Roughly speaking, the adversary could use classical values for $\Ring$, $m$, and $i_1$, but she puts a uniform superposition of all valid identities in the register for $i_0$. After the challenger's signing operation, observe that if $b = 0$, the last register will contain signatures in superposition (as $i_0$ is in superposition); if $b = 1$, it will contain a classical signature (as $i_1$ is classical). These two cases can be efficiently distinguished by means of a Fourier transform on the $i_0$'s register followed by a measurement. Therefore, to obtain an achievable notion, we should not allow superpositions over $(i_0, i_1)$.
Now, $\Adv$ only has the choice to put superpositions over $\Ring$ and $m$. The definition in \cite{C:CGHKLMPS21} further forbids $\Adv$ from putting superpositions over $\Ring$. But this is only because they fail to prove security if superposition attacks on $\Ring$ is allowed. Indeed, they leave open the problem to construct a scheme that protects against superposition attacks on $\Ring$. In this work, we solve this problem: our definition allows superposition attacks on both $\Ring$ and $m$.
\begin{definition}[Post-Quantum Anonymity]
\label{def:pq:anonymity}
Consider a triple of PPT algorithms $\RS = (\algo{Gen}, \Sign, \Verify)$ that satisfies the same syntax as in \Cref{def:classical:ring-signature}. $\RS$ achieves post-quantum anonymity if for any $Q=\poly(\lambda)$ and any QPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:pq:anonymity} that
$$\algo{PQAdv}_\textsc{Anon}^{\secpar,Q}(\Adv) \coloneqq \big|\Pr\big[\algo{PQExp}_\textsc{Anon}^{\secpar,Q}(\Adv) = 1\big] - 1/2\big| \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:pq:anonymity}]{Post-Quantum Anonymity \textnormal{$\algo{PQExp}_\textsc{Anon}^{\secpar,Q}(\Adv)$}}
\begin{enumerate}
\item
For each $i \in [Q]$, the challenger generates key pairs $(\VK_i,\SK_i) \leftarrow \RS.\algo{Gen}(1^\SecPar, N;r_i)$. The challenger sends $\Set{(\VK_i, \SK_i, r_i)}_{i\in [Q]}$ to $\Adv$;
\item
$\Adv$ sends $(i_0, i_1)$ to the challenger, where both $i_0$ and $i_1$ are in $[Q]$;
\item
$\Adv$'s challenge query is allowed to be a superposition of rings {\em and} messages. The challenger picks a random bit $b$ and a random string $r$. It signs the message using $\SK_{i_b}$ and randomness $r$, while making sure that $\VK_{i_0}$ and $\VK_{i_1}$ are indeed in the ring specified by $\Adv$. Formally, the challenger implements the following mapping:
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{ \Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor f(\Ring, m)},$$
where
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i_b}, R, m;r) & \text{if}~\VK_{i_0}, \VK_{i_1} \in R \\
\bot & \text{otherwise}
\end{cases}
$.
\item
$\Adv$ outputs a guess $b'$. If $b' = b$, the experiment outputs 1, otherwise 0.
\end{enumerate}
\end{ExperimentBox}
\end{definition}
\para{Post-Quantum Unforgeability.} In the classical unforgeability game (\Cref{expr:classical:unforgeability}), $\Adv$ can make both {\sf corrupt} and {\sf sign} queries. As discussed in \Cref{sec:tech-overview:PQ-ring-sig}, we do not consider quantum {\sf corrupt} queries, or superposition attacks over the identity in $\Adv$'s {\sf sign} queries.
We also remark that in the unforgeability game, \cite{C:CGHKLMPS21} does not allow superpositions over the ring. Instead of a definitional issue, this is again only because they are unable to prove the security of their scheme if superposition attacks on the ring is allowed. In contrast, our construction can be proven secure against such attacks; thus, this restriction is removed from our definition.
To define quantum unforgeability, \cite{C:CGHKLMPS21} adapts one-more unforgeability \cite{C:BonZha13} to the ring setting: they require that, with $\mathsf{sq}$ quantum signing queries, the adversary cannot produce $\mathsf{sq} + 1$ signatures, where all the rings are subsets of $\mathcal{VK}\setminus \mathcal{C}$. This definition, {\em when restricted to the classical setting}, seems to be weaker than the standard unforgeability in \Cref{def:classical:ring-signature}
That is, in the classical setting, any $\RS$ satisfying the unforgeability in \Cref{def:classical:ring-signature} is also one-more unforgeable; but the reverse direction is unclear (we discuss this in \Cref{sec:one-more:PQ-EUF:ring-sig}). Instead, our definition extends the blind-unforgeability for ordinary signatures (\Cref{def:classical-BU-sig}) to the ring setting. We present this definition in \Cref{def:pq:blind-unforgeability}.
\begin{definition}[Post-Quantum Blind-Unforgeability]
\label{def:pq:blind-unforgeability}
Consider a triple of PPT algorithms $\RS = (\algo{Gen}, \Sign, \Verify)$ that satisfies the same syntax as in \Cref{def:classical:ring-signature}. For any security parameter $\secpar$, let $\mathcal{R}_\secpar$ and $\mathcal{M}_\secpar$ denote the ring space and message space, respectively. $\RS$ achieves blind-unforgeability if for any $Q=\poly(\lambda)$ and any QPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:pq:blind-unforgeability} that
$$\algo{PQAdv}_\textsc{bu}^{\secpar,Q}(\Adv) \coloneqq \Pr\big[\algo{PQExp}_\textsc{bu}^{\secpar,Q}(\Adv) = 1\big] \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:pq:blind-unforgeability}]{Post-Quantum Blind-Unforgeability \textnormal{$\algo{PQExp}_\textsc{bu}^{\secpar,Q}(\Adv)$}}
\begin{enumerate}
\item
$\Adv$ sends a constant $0\le \epsilon \le 1$ to the challenger;
\item
For each $i \in [Q]$, the challenger generates $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N;r_i)$, and stores these key pairs along with their corresponding randomness. It then sets $\mathcal{VK} = \Set{\VK_1,\dots,\VK_Q}$ and initializes a set $\mathcal{C} = \emptyset$; The challenger sends $\mathcal{VK}$ to $\Adv$;
\item
The challenger defines a {\em blindset} $B^\RS_\varepsilon \subseteq 2^{\mathcal{R}_\secpar} \times \mathcal{M}_\SecPar$: every pair $(\Ring, m) \in 2^{\mathcal{R}_\secpar} \times \mathcal{M}_\SecPar$ is put in $B^\RS_\varepsilon$ with probability $\varepsilon$;
\item
$\Adv$ can make polynomially-many queries of the following two types:
\begin{itemize}
\item
{\bf Classical corruption query $(\mathsf{corrupt},i)$:} The challenger adds $\VK_i$ to the set $\mathcal{C}$ and returns the randomness $r_i$ to \Adv.
\item
{\bf Quantum Signing query $(\algo{sign},i, \sum \psi_{\Ring, m, t}\ket{\Ring, m, t})$:} That is, $\Adv$ is allowed to query the signing oracle on some classical identity $i$ and superpositions over rings and messages. The challenger samples a random string $r$ and performs
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor B^\RS_\epsilon f(\Ring, m)},$$
where $B^\RS_\epsilon f(\Ring, m) \coloneqq
\begin{cases}
\bot & \text{if}~(\Ring, m) \in B^\RS_\epsilon \\
f(\Ring, m) & \text{otherwise}
\end{cases}
$, and
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i}, m, \Ring;r) & \text{if}~\VK_{i} \in \Ring \\
\bot & \text{otherwise}
\end{cases}
$.
\end{itemize}
\item Finally, $\Adv$ outputs $(\Ring^*, m^*, \Sigma^*)$. The challenger checks if:
\begin{enumerate}
\item
$\Ring^* \subseteq \mathcal{VK \setminus C}$,
\item
$\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$, {\bf and}
\item
$(\Ring^*,m^*) \in B^\RS_\epsilon$.
\end{enumerate}
If so, it outputs 1; otherwise, it outputs 0.
\end{enumerate}
\end{ExperimentBox}
\end{definition}
In contrast to the ``one-more'' unforgeability, we show in \Cref{lem:euf-ring-sig:bu-ring-sig} that, when restricted to the classical setting, \Cref{def:pq:blind-unforgeability} (for ring signatures) is indeed equivalent to the standard existential unforgeability in \Cref{def:classical:ring-signature}. Its proof is almost identical to that of \cite[Proposition 2]{EC:AMRS20}.
\begin{lemma}\label{lem:euf-ring-sig:bu-ring-sig}
Restricted to (classical) QPT adversaries, a ring signature $\RS$ scheme is blind-unforgeable (\Cref{def:pq:blind-unforgeability}) if and only if it satisfies the unforgeability requirement in \Cref{def:classical:ring-signature}.
\end{lemma}
\begin{proof}
We show necessity and sufficiency in turn. In the following, by ``\Cref{expr:pq:blind-unforgeability}'', we refer to the classical version of \Cref{expr:pq:blind-unforgeability}, where the signing query is of the form $(\algo{sign}, i, \Ring, m)$ (i.e., $(\Ring, m)$ is classical), and is answered as $B^\RS_\epsilon f(\Ring, m)$.
\subpara{Necessity ($\Leftarrow$).} Let us first show how blind-unforgeability implies standard unforgeability (for classical settings). Assume we have an adversary $\Adv_\textsc{euf}$ that breaks standard unforgeability of $\RS$ as per \Cref{def:classical:ring-signature}, i.e., in \Cref{expr:classical:unforgeability} it produces a forgery $(m^*,\Ring^*,\Sigma^*)$ that is valid with non-negligible probability $\nu(\SecPar)$. We show that this is easily converted into an adversary $\Adv_\textsc{bu}$ that wins \Cref{expr:pq:blind-unforgeability} with non-negligible probability as well. $\Adv_\textsc{bu}$ first sets $\epsilon(\SecPar)$ equal to $1/p(\SecPar)$, where $p(\secpar)$ denotes the (polynomial) running time of $\Adv_\textsc{euf}$ (the reasoning behind this choice will become clear very soon). It then simply forwards all the queries from $\Adv_\textsc{euf}$ to the blind-unforgeability challenger and the responses back to $\Adv_\textsc{euf}$. It also outputs whatever eventual forgery $\Adv_\textsc{euf}$ does.
Let us consider the success probability of $\Adv_\textsc{bu}$. To start with, note that $\Adv_\textsc{euf}$ makes at most $p(\SecPar)$ many queries of the signing oracle. In each such query, we know that the
$(\Ring, m)$ pair is in the blind set independently with probability $\epsilon$. Thus it is not in the blind set with probability $1-\epsilon$, and if so the query is answered properly. In turn, the probability that all the queries made are answered properly is then at least $(1-\epsilon)^{p(\SecPar)} \approx 1/e$ (this uses independence and $\epsilon = 1/p$), and so the probability that the forgery $(\Ring^*,m^*,\Sigma^*)$ is valid
is then at least $(1-\epsilon)^{p(\SecPar)}\cdot\nu(\SecPar)$.
Finally, the forgery, even if successful, might lie in the blind set with probability $\epsilon$. So, the total probability that $\Adv_\textsc{euf}$ outputs a valid forgery {\em for the blind unforgeability game} is $(1-\epsilon)^{p(\SecPar)+1} \cdot \nu(\SecPar) \approx (1-\epsilon) \cdot \nu \cdot 1/e$, which is non-negligible since $\nu$ is non-negligible by assumption. Thus if $\Adv_\textsc{euf}$ violates standard ring signature unforgeability according to \Cref{def:classical:ring-signature}, then $\Adv_\textsc{bu}$ violates blind unforgeability for ring signatures according to \Cref{def:pq:blind-unforgeability}, as claimed.
\subpara{Sufficiency ($\Rightarrow$).} Let us now turn to the other direction of the equivalence. Assume now that there exists an adversary $\Adv_\textsc{bu}$ that can break blind unforgeability of $\RS$, i.e., win \Cref{expr:pq:blind-unforgeability} with non-negligible probability $\nu(\secpar)$. We show an adversary $\Adv_\textsc{euf}$
that can win \Cref{expr:classical:unforgeability} with non-negligible probability. $\Adv_\textsc{euf}$ simply simulates \Cref{expr:pq:blind-unforgeability} for $\Adv_\textsc{bu}$ by answering oracle queries according to a locally-simulated version of $B^\RS_\epsilon f(\Ring,m)$. Concretely, the adversary $\Adv_\textsc{euf}$ proceeds by drawing a subset $B^\RS_\epsilon$ in the same manner as the challenger in \Cref{expr:pq:blind-unforgeability} and answering queries made by $\Adv_\textsc{bu}$ according to $B^\RS_\epsilon f(\Ring,m)$. Two remarks are in order:
\begin{enumerate}
\item \label{item:euf-ring-sig:bu-ring-sig:remark1}
when $(\Ring, m) \in B^\RS_\secpar$, no signature needs to be done. That is, this query can be answered by $\Adv_\textsc{euf}$ without calling its own signing oracle;
\item
$\Adv_\textsc{euf}$ can construct the set $B^\RS_\epsilon$ by ``lazy sampling'', i.e., when a particular query $(\algo{sign}, i, \Ring, m)$ is made by $\Adv_\textsc{bu}$,
whether $(\Ring, m) \in B^\RS_\epsilon$ and ``remembering'' this information in case the query is asked again.
\end{enumerate}
By assumption, $\Adv_\textsc{bu}$ produces a valid forgery. And it follows from \Cref{item:euf-ring-sig:bu-ring-sig:remark1} that this forgery must be on a point which was not queried by $\Adv_\textsc{euf}$, thus, also serving as a valid forgery for $\Adv_\textsc{euf}$'s game.
\iffalse
At this point we will consider the specific kind of ring signature scheme $\RS$ being used: specifically, whether this scheme has deterministic signatures or not. The overall idea is that $\Adv_\textsc{euf}$ will {\em locally} maintain and administer a blind set of size $\epsilon$ in its interation with $\Adv_\textsc{bu}$; but the exact manner on implementation will depend on what kind of scheme we use. The reason for this distinction will become clear from the following arguments.
Let us first consider the case of a deterministic ring signature $\RS$. In this case, we use essentially the same reduction as in the proof for \cite[Proposition 2]{EC:AMRS20} (they consider such an equivalence only for weak unforgeability - and this is precisely why their reduction works for the deterministic case). Namely, $\Adv_\textsc{euf}$ behaves as follows: it maintains a blind set with respect to messages and rings only. This is equivalent to maintaining a blind set over message, ring and signature triples for deterministic signatures. Whenever $\Adv_\textsc{bu}$ makes a query, $\Adv_\textsc{euf}$ adds this to the blind set with probability $\epsilon$. If the query gets added, $\Adv_\textsc{euf}$ simply replies with $\bot$ and does not forward the query to its own challenger. If not, it forwards the query and sends the response back to $\Adv_\textsc{bu}$ as per usual. Finally, $\Adv_\textsc{bu}$ outputs a forgery, say $(m^*,\Ring^*,\Sigma^*)$. With probability $\nu$, this is good enough to win the blind unforgeability challenge, implying that at the very least this forgery is not with respect to a message to a {\em successful} query. Indeed, nothing prevents $\Adv_\textsc{bu}$ from producing a forgery with respect to any message in $\Adv_\textsc{euf}$'s local blind set, including messages that it queried and recieved no response for. Crucially however, for these messages $\Adv_\textsc{euf}$ never queries its own oracle so even a forgery with respect to such messages suffices to win the standard unforgeability challenge. Thus $\Adv_\textsc{euf}$ can simply use the forgery output by $\Adv_\textsc{bu}$ in its own challenge, and win with probability at least $\nu(\SecPar)$.
Now we turn to the case of randomized ring signatures. For any such scheme $\RS$, since the signing algorithm $\RS.\Sign$ is randomized, we have the property that for any particular message $m$, ring $R$ and signing key $sk$ there are multiple signatures that are possible (depending on the randomness used by the signing algorithm). Define the guessing probability $\eta_{m,R,sk}$ to be the inverse of the number of possible signatures that can be generated by $\RS.\Sign$ given a choice of $(m,R,sk)$ (for most schemes, the number of signatures is usually independent of these choices and depends only on \SecPar). Further define $\eta_\RS = \mathsf{min}_{m,R,sk}\{\eta_{m,R,sk}\}$. By definition, we have that $\eta_\RS \leq 1/2$ for any randomized ring signature scheme $\RS$. We now describe the function of $\Adv_\textsc{euf}$ for this case. It forwards queries from $\Adv_\textsc{bu}$ and responses back to it, as before. Additionally, it maintains a local blind set (consisting of message, ring and signature triples). In contrast to the previous case, $\Adv_\textsc{euf}$ now queries its own (standard unforgeability) challenger on {\em all} queries $\Adv_\textsc{bu}$ makes. Upon obtaining the signature from the challenger, it adds the resulting triple to the blindset with probability $\epsilon$, and only returns $\bot$ to $\Adv_\textsc{bu}$ for such a query. Ultimately $\Adv_\textsc{bu}$ outputs a forgery $(m^*,\Ring^*,\sigma^*)$. This is guaranteed to be a valid forgery for the blinded unforgeability challenge (i.e., i.e., that $\Ring^*$ has no corrupted members, and that $\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$) with probability at least $\nu$.
Let us consider two possibilities. Either the forgery $(m^*,\Ring^*,\sigma^*)$ was not queried by $\Adv_\textsc{euf}$ before, in which case it can simply use this forgery and win the standard unforgeability game. Otherwise, $\Adv_\textsc{bu}$ indeed asked for a signature with respect to $(m^*,\Ring^*)$ and obtained no response. In this case, $\Adv_\textsc{bu}$ may have output a signature $\Sigma^*$, and correspondingly $\Adv_\textsc{euf}$ recieved some signature $\Sigma'$ earlier from its challenger. Under our assumptions so far, both of these are valid signatures for $(m^*,\Ring^*)$. Now if $\Sigma^* = \Sigma'$ then $\Adv_\textsc{euf}$ cannot use $\Sigma^*$ as its forgery, since it has already seen it as a query response. However, the probability that $\Sigma^* = \Sigma'$ is at least $\eta_\RS$ from our earlier arguments (recall that $\Adv_\textsc{bu}$ never sees $\Sigma'$) and so we have that in this case the forgery $(m^*,\Ring^*,\Sigma^*)$ can still be used by $\Adv_\textsc{euf}$ with probability $1-\eta_\RS$. It is easy to see that overall, if $\Adv_\textsc{bu}$ succeeds with probability $\nu$, then $\Adv_\textsc{euf}$ still succeeds with probability at least $(1-\eta_\RS)\nu$. This is still non-negligible ($\eta_\RS$ is at most a constant), and so our claim follows.
\fi
\end{proof}
\noindent{To conclude, we present the complete definition for quantum ring signatures.}
\begin{definition}[Post-Quantum Secure Ring Signatures]
\label{def:pq:ring-signatures}
A post-quantum secure ring signature scheme $\RS$ is described by a triple of PPT algorithms $(\Gen,\Sign,\Verify)$ that share the same syntax as in \Cref{def:classical:ring-signature}. Moreover, these algorithms also satisfy the {\em completeness} requirement as per \Cref{def:classical:ring-signature}, the {\em post-quantum anonymity} requirement as per \Cref{def:pq:anonymity}, and the {\em post-quantum blind-unforgeability} requirement as per \Cref{def:pq:blind-unforgeability}.
\end{definition}
\subsection{Building Blocks}
\subsubsection{Lossy PKEs with Special Properties}
We need the following lossy PKE.
\begin{definition}[Special Lossy PKE]\label{def:special-LE}
For any security parameter $\secpar \in \Naturals$, let $\mathcal{M}_\secpar$ denote the message space.
A special lossy public-key encryption scheme $\algo{LE}$ consists of the following PPT algorithms:
\begin{itemize}
\item
$\algo{MSKGen}(1^\secpar, Q)$, on input a number $Q \in \Naturals$, outputs $\big(\Set{\pk_i}_{i \in [Q]}, \msk \big)$. We call $\pk_i$'s the {\em injective public keys}, and $\msk$ the {\em master secret key}.
\item
$\algo{MSKExt}(\msk, \pk)$, on input a master secret key $\msk$ and an injective public key $\pk$, outputs a secret key $\sk$.
\item
$\KSam^\ls(1^\secpar)$ outputs key $\pk_\ls$, which we call {\em lossy public key}.
\item
$\algo{Valid}(\pk,\sk)$, on input a public $\pk$ and a secret key $\sk$, outputs either $1$ (accepting) or $0$ (rejecting).
\item
$\RndExt(\pk)$ outputs a $r$ which we call {\em extracted randomness}.
\item
$\algo{Enc}(\pk, m)$, on input a public key $\pk$, and a message $m \in \mathcal{M}_\secpar$, outputs $\ct$.
\item
$\algo{Dec}(\sk, \ct)$, on input a secret key $\sk$ and a ciphertext $ct$, outputs $m$.
\end{itemize}
These algorithms satisfy the following properties:
\begin{enumerate}
\item \label{item:def:LE:completeness}
{\bf Completeness.} For any $\secpar\in\Naturals$, any $(\pk, \sk)$ s.t.\ $\Valid(\pk, \sk) =1$, and any $m \in \mathcal{M}_\secpar$, it holds that
$$\Pr[\Dec\big(\sk, \Enc(\pk, m)\big)= m] = 1.$$
\item \label{def:LE:property:lossyness}
{\bf Lossiness of lossy keys.} For any $\pk_\ls$ in the range of $\KSam^\ls(1^\secpar)$ and any $m_0, m_1 \in \mathcal{M}_\secpar$, it holds that
$$\big\{ \Enc(\pk_\ls, m_0) \big\}_{\secpar\in\Naturals} \sind \big\{ \Enc(\pk_\ls, m_1) \big\}_{\secpar\in\Naturals}.$$
\item \label{def:LE:property:GenWithSK:completeness}
{\bf Completeness of Master Secret Keys:} for any $Q = \poly(\secpar)$, it holds that
$$\Pr[
\big(\Set{\pk_i}_{i \in [Q]}, \algo{msk} \big) \gets \algo{MSKGen}(1^\secpar, Q):
\begin{array}{l}
\forall i \in [Q], \Valid(\pk_i,\sk_i\big) =1, \\
\textrm{where}~\sk_i \coloneqq \algo{MSKExt}\big(\algo{msk}, \pk_i)
\end{array}
] \ge 1 - \negl(\secpar).$$
\item \label{def:LE:property:IND:GenWithSK-KSam}
{\bf IND of $\algo{MSKGen}$/$\KSam^\ls$ mode:} For any $Q=\poly(\secpar)$, the following two distributions are computationally indistinguishable:
\begin{itemize}
\item
$\forall i\in [Q]$, sample $\pk_i \gets \KSam^\ls(1^\secpar; r_i)$, then output $\Set{\pk_i, r_i}_{i\in [Q]}$;
\item
Sample $\big(\Set{\pk_i}_{i \in [Q]}, \algo{msk} \big) \gets \algo{MSKGen}(1^\secpar, Q)$ and output $\big\{\pk_i, \algo{RndExt}(\pk_i)\big\}_{i\in [Q]}$.
\end{itemize}
\item \label{item:def:LE:computationally-unique-sk}
{\bf Almost-Unique Secret Key:} For any $Q = \poly(\secpar)$, it holds that
$$
\Pr[
\big(\Set{\pk_i}_{i \in [Q]}, \algo{msk} \big) \gets \algo{MSKGen}(1^\secpar, Q):
\begin{array}{l}
\text{There exist}~ i \in [Q] ~\text{and}~ \sk'_i~\text{such that} \\
\sk'_i \ne \algo{MSKExt}(\algo{msk},\pk_i) ~\wedge~ \Valid(\pk_i, \sk'_i) = 1
\end{array}
] = \negl(\secpar).$$
\end{enumerate}
\end{definition}
We propose an instantiation of such a lossy PKE using dual mode LWE commitments \cite{STOC:GorVaiWic15}. In lossy (statistically hiding) mode, the public key consists of a uniformly sampled matrix $\mathbf{A}$ and a message $m$ is encrypted by computing $\mathbf{A}\mathbf{R} + m\mathbf{G}$, where $\mathbf{R}$ is a low-norm matrix and $\mathbf{G}$ is the gadget matrix. Note that the random coins used to sample $\mathbf{A}$ simply consists of the matrix $\mathbf{A}$ itself. Furthermore, we can switch $\mathbf{A}$ to be an LWE-matrix (using some secret vector $\mathbf{s}$) to make the encryption scheme injective. Such a modification is computationally indistinguishable by an invocation of the LWE assumption. Note that this is true also in the presence of the output of $\algo{RndExt}(\mathbf{A})$, since the algorithm simply returns $\mathbf{A}$. Furthermore, by setting the dimensions appropriately, the secret $\mathbf{s}$ is uniquely determined by $\mathbf{A}$ with overwhelming probability. Finally, we note that we can define a master secret key for all keys in injective mode using a simple trick: sample a PRF key $k$ and sample the $i$-th key pair using $\mathsf{PRF}(k,i)$ as the random coins. It is not hard to see that the distribution of public/secret keys is computationally indistinguishable by the pseudorandomness of $\mathsf{PRF}$. Furthermore, given $k$ one can extract the $i$-th secret key simply by recomputing it.
\subsubsection{ZAPs for Super-Complement Languages}
As mentioned in \Cref{sec:tech-overview:PQ-ring-sig}, \cite{C:CGHKLMPS21} uses a ZAP (for $\NP\cap\coNP$) to prove a statement that the (ring) signature contains a ciphertext of a valid signature w.r.t.\ the building-block signature scheme. Let us denote this language as $L$. In the security proof, they need to argue that the adversary cannot prove a false statement $x^*\notin L$. However, this $L$ is not necessarily in $\coNP$; thus, there may not exist a non-witness $\tilde{w}$ for the fact that $x^*\notin L$. Therefore, it is unclear how to use a ZAP for $\NP\cap \coNP$ here. To address this issue, the authors of \cite{C:CGHKLMPS21} propose the notion of {\em super-complement languages}. This notion considers a pair of $\NP$ languages $(L, \tilde{L})$ such that $(x\in \tilde{L}) \Rightarrow (x \notin L)$. Their ZAP achieves soundness such that the cheating prover cannot prove $x \in L$ (except with negligible probability) once there exists a ``non-witness'' $\tilde{w}$ s.t.\ $(x,\tilde{w}) \in R_{\tilde{L}}$. The $\tilde{L}$ is set to a special language that captures some {\em necessary conditions} for any forged signatures to be valid. Thus, a winning adversary will break the soundness of the ZAP, leading to a contradiction.
In the following, we present the original definition of super-complement languages. But we will only need a special case of it (see \Cref{rmk:super-complement-language:special-case}).
\begin{definition}[Super-Complement \cite{C:CGHKLMPS21}]\label{def:super-complement-language}
Let $(L, \widetilde{L})$ be two $\NP$ languages where the elements of $\widetilde{L}$ are represented as pairs of bit strings. We say $\widetilde{L}$ is a \emph{super-complement} of $L$, if
$\widetilde{L} \subseteq (\bits^*\setminus L) \times \bits^*$.
I.e., $\widetilde{L}$ is a super complement of $L$ if for any $x= (x_1,x_2)$, $x \in \widetilde{L} \Rightarrow x_1 \not \in L$.
\end{definition}
Notice that, while the complement of $L$ might not be in $\NP$, it must hold that $\widetilde{L} \in \NP$. The language $\widetilde{L}$ is used to define the soundness property. Namely, producing a proof for a statement $x = (x_1, x_2) \in \widetilde{L}$, should be hard. We also use the fact that $\widetilde{L} \in \NP$ to mildly strengthen the soundness property. In more detail, instead of having selective soundness where the statement $x \in \widetilde{L}$ is fixed in advance, we now fix a non-witness $\widetilde{w}$ and let the statement $x$ be adaptively chosen by the malicious prover from all statements which have $\widetilde{w}$ as a witness to their membership in $\widetilde{L}$.
\begin{remark}\label{rmk:super-complement-language:special-case}
Our application only needs a special case of the general form given in \Cref{def:super-complement-language}---we will only focus on $\tilde{L}$ where the $x_2$ part is an empty string. Formally, we consider the special case where $\tilde{L} \subseteq \bits^* \setminus L $ (i.e., $x \in \tilde{L} \Rightarrow x \notin L$).
\end{remark}
We now define ZAPs for super-complement languages. We remark that the original definition (and construction) in \cite{C:CGHKLMPS21} captures the general $(L, \tilde{L})$ pairs defined in \Cref{def:super-complement-language}. Since we only need the special case in \Cref{rmk:super-complement-language:special-case}, we will define the ZAP only for this case.
\begin{definition}[ZAPs for Special Super-Complement Languages]
\label{def:generalizedzap-interface}
Let $L,\widetilde{L} \in \NP$ be the special super-complement language in \Cref{rmk:super-complement-language:special-case}. Let $R$ and $\widetilde{R}$ denote the $\NP$ relations corresponding to $L$ and $\widetilde{L}$ respectively. Let $\{C_{n,\ell}\}_{n,\ell}$ and $\{\widetilde{C}_{{n},\tilde{\ell}}\}_{{n},\tilde{\ell} }$ be the $\NP$ verification circuits for $L$ and $\widetilde{L}$ respectively. Let $\widetilde{d} = \widetilde{d}({n},\tilde{\ell})$ be the depth of $\widetilde{C}_{n,\tilde{\ell}}$. A {\em ZAP} for $(L,\widetilde{L})$
is a tuple of PPT algorithms
$(\algo{V}, \algo{P}, \algo{Verify})$ having the following
interfaces (where $1^{n}, 1^{\lambda}$ are implicit inputs to
$\algo{P}$, $\algo{Verify}$):
\begin{itemize}
\item
{\bf $\algo{V}(1^{\lambda}, 1^n,1^{\widetilde{\ell}}, 1^{\widetilde{D}})$:}
On input a security parameter $\lambda$,
statement length~$n$ for $L$,
witness length $\widetilde{\ell}$ for $\widetilde{L}$,
and $\NP$ verifier circuit depth upper-bound $\widetilde{D}$ for $\widetilde{L}$, output a first message
$\rho$.
\item
{\bf $\algo{P}\big(\rho, x, w\big)$:} On input a string~$\rho$, a statement $x \in \bits^{n}$, and a witness $w$ such that $(x, w) \in R$, output a proof $\pi$.
\item
{\bf $\algo{Verify}\big(\rho, x,\pi\big)$:} On input a string $\rho$, a
statement $x$, and a proof $\pi$, output either 1 (accepting) or 0 (rejecting).
\end{itemize}
The following requirements are satisfied:
\begin{enumerate}
\item {\bf Completeness:} For every $x \in L$,
every $\widetilde{\ell} \in \Naturals$,
every $\widetilde{D} \ge \widetilde{d}(\abs{x},\widetilde{\ell})$,
and every
$\lambda \in \Naturals$, it holds that
$$\Pr[ \rho \gets \algo{V}(1^{\lambda}, 1^{\abs{x}},1^{\widetilde{\ell}}, 1^{\widetilde{D}});\pi \gets \algo{P}(\rho,x,w) : \algo{Verify}\big(\rho,x ,\pi\big)=1 ] = 1.$$
\item {\bf Public coin:}
$\algo{V}(1^{\lambda}, 1^n,1^{\widetilde{\ell}}, 1^{\widetilde{D}})$ simply outputs a uniformly
random string.
\item \label{item:def:zap:soundness}
{\bf Selective non-witness adaptive-statement soundness:}
For any non-uniform QPT machine $P^*_{\lambda}$, any $n,\widetilde{D} \in \Naturals$, and any non-witness $\widetilde{w} \in \bits^{*}$,
\begin{equation*}
\Pr[
\begin{array}{l}
\rho \gets \algo{V}(1^{\lambda}, 1^{n},1^{\abs{\widetilde{w}}}, 1^{\widetilde{D}});\\
\big(x,\pi^*\big) \gets P^*_{\lambda}(\rho)
\end{array}:
\begin{array}{l}
\algo{Verify}(\rho,x,\pi^*) = 1 ~\wedge\\
\widetilde{D} \ge \widetilde{d}(\abs{x}, \abs{\widetilde{w}}) ~\wedge~ (x,\widetilde{w}) \in \widetilde{R}
\end{array} ] \le \negl(\SecPar).
\end{equation*}
\item {\bf Statistical witness indistinguishability:} For every (possibly unbounded) ``cheating''
verifier $V^{*}=(V^{*}_{1}, V^{*}_{2})$ and every $n,\widetilde{\ell},\widetilde{D} \in \Naturals$, the
probabilities
$$\Pr[V^{*}_{2}(\rho,x,\pi,\zeta)=1 ~\wedge~ (x,w) \in \mathcal{R} ~\wedge~ (x,w') \in \mathcal{R} ]$$
in the
following two experiments differ only by $\negl(\lambda)$:
\begin{itemize}
\item {\em Experiment 1:}
$(\rho,x,w,w',\zeta) \gets
V^{*}_{1}(1^{\lambda}, 1^{n},1^{\widetilde{\ell}}, 1^{\widetilde{D}})$, $\pi \gets \algo{P}(\rho,x,w)$;
\item {\em Experiment 2:}
$(\rho,x,w,w',\zeta) \gets
V^{*}_{1}(1^{\lambda}, 1^{n},1^{\widetilde{\ell}}, 1^{\widetilde{D}}),$ $\pi \gets \algo{P}(\rho,x,w')$.
\end{itemize}
\end{enumerate}
\end{definition}
\begin{lemma}[\cite{C:CGHKLMPS21}]\label{lem:ZAP:from:QLWE}
Assuming QLWE, there exist ZAPs as per \Cref{def:generalizedzap-interface} for any super-complement language as per \Cref{def:super-complement-language}.
\end{lemma}
\subsection{Construction}
\label{sec:ring-sig:construction}
Our construction $\RS$, shown in \Cref{constr:pq:ring-sig}, relies on the following building blocks:
\begin{enumerate}
\item
Pair-wise independent hash functions;
\item
A blind-unforgeable signature scheme $\algo{Sig}$ satisfying \Cref{def:classical-BU-sig};
\item
A lossy PKE scheme $\LE$ satisfying \Cref{def:special-LE};
\item
A ZAP for special super-complement languages $\algo{ZAP}$ satisfying \Cref{def:generalizedzap-interface}.
\end{enumerate}
We remark that the $\RS.\Sign$ algorithm runs $\ZAP$ on a special super-complement language $(L, \tilde{L})$, whose definition will appear after the construction in \Cref{sec:building-blocks:language}. This arrangement is because we believe that the language $(L, \tilde{L})$ will become easier to understand once the reader has slight familiarity with \Cref{constr:pq:ring-sig}.
\begin{ConstructionBox}[label={constr:pq:ring-sig}]{Post-Quantum Ring Signatures}
Let $\widetilde{D} = \widetilde{D}(\lambda, N)$
be the maximum depth of the $\algo{NP}$ verifier circuit for language $\widetilde{L}$ restricted to statements where the the ring has at most $N$ members, and the security parameter for \Sig and $\algo{LE}$ is $\lambda$. Let $n = n(\lambda,\log N)$ denote the maximum size of the statements of language $L$ where the ring has at most $N$ members and the security parameter is $\lambda$. Recall that for security parameter $\lambda$, secret keys in $\LE$ have size $\widetilde{\ell} = \ell_{\sk}(\lambda)$. We now describe our ring signature construction:
\subpara{Key Generation Algorithm $\algo{Gen}(1^\lambda,N)$:}
\begin{itemize}
\item sample signing/verification key pair: $(vk,sk) \gets \algo{Sig.Gen}(1^\lambda)$;
\item sample obliviously an injective public key of $\algo{LE}$: $pk \gets \LE.\KSam^\ls(1^\secpar)$;
\item compute the first message $\rho \leftarrow \algo{ZAP.V}(1^\lambda,1^n,1^{\widetilde{\ell}},1^{\widetilde{D}})$ for \ZAP;
\item output the verification key $\VK\coloneqq (vk,pk,\rho)$ and signing key $\SK\coloneqq (sk,vk,pk,\rho)$.
\end{itemize}
\subpara{Signing Algorithm $\algo{Sign}(\SK,\Ring, m)$:}
\begin{itemize}
\item parse $\Ring = (\VK_1,\dots,\VK_{\ell})$; and parse $\SK=(sk,vk,pk,\rho)$;
\item compute $\sigma \gets \algo{Sig.Sign}(sk,\Ring\|m)$;
\item let $\VK \coloneqq \VK_i \in R$ be the verification key corresponding to $\SK$;
\item
sample two pairwise-independent hash functions $\PI_1$ and $\PI_2$, and compute
$$r_{c_1} = \PI_1(\Ring\|m), ~~r_{c_2} = \PI_2(\Ring\|m).$$
\item compute $c_1 \gets \algo{LE.Enc}(pk,(\sigma,vk);r_{c_1})$ and $c_2 \gets \algo{LE.Enc}(pk,0^{|\sigma|+|vk|};r_{c_2})$;
\item
let $\VK_1 = (vk_1,pk_1,\rho_1)$ denote the lexicographically smallest member of $\Ring$ (as a string; note that this is necessarily unique);
\item
fix statement $x=(\Ring, m, c_1,c_2)$ and witness $w=(vk,pk, \sigma,r_{c_1})$. We remark that this statement and witness correspond to a super-complement language $(L, \tilde{L})$ that will be defined in \Cref{sec:building-blocks:language}. Looking ahead, $x$ with witness $w$ is a statement in the $L$ defined in \Cref{eq:def:language:L}; $x$ constitutes a statement that is {\em not} in the $\tilde{L}$ defined in \Cref{eq:def:language:L-tilde}.
\item sample another pairwise-independent hash function $\PI_3$ and compute $r_\pi = \PI_3(\Ring\|m)$;
\item compute $\pi \gets \algo{ZAP.P}(\rho_1,x,w; r_\pi)$;
\item output $\Sigma = (c_1,c_2,\pi)$.
\end{itemize}
\subpara{Verification Algorithm $\algo{Verify}(\Ring,m, \Sigma)$:}
\begin{itemize}
\item identify the lexicographically smallest verification key $\VK_1$ in $\Ring$;
\item fix $x=(\Ring, m, c_1, c_2)$; read $\rho_1$ from $\VK_1$;
\item compute and output $\algo{ZAP}.\algo{Verify}(\rho_1,x,\pi)$.
\end{itemize}
\end{ConstructionBox}
\subsubsection{The Super-Complement Language Proven by the ZAP}
\label{sec:building-blocks:language}
We now define the super-complement language $(L, \widetilde{L})$ used in \Cref{constr:pq:ring-sig}. This deviates from the $(L, \tilde{L})$ defined in \cite[Section 5]{C:CGHKLMPS21}, to accommodate \Cref{constr:pq:ring-sig}.
For a statement of the form $x_1 = (\Ring, m,c)$ and witness $w = \big(\mathsf{VK}=(vk,pk,\rho),\sigma,r_c \big)$, define relations $R_1$, $R_2$, and $R_3$ as follows:
\begin{align*}
(x_1,w) \in R_1 & ~\Leftrightarrow~ \VK \in \Ring, \\
(x_1,w) \in R_2 & ~\Leftrightarrow~ \algo{LE.Enc}\big(pk,(\sigma,vk);r_c\big) = c,\\
(x_1,w) \in R_3 & ~\Leftrightarrow~ \algo{Sig.Verify}(vk, \Ring\|m,\sigma)=1.
\end{align*}
Next, define the relation $R'$ as $R'\coloneqq R_1\cap R_2\cap R_3$. Let $L'$ be the language corresponding to $R'$. Define language $L$ as
\begin{equation}\label{eq:def:language:L}
L \coloneqq \big\{x=(\Ring, m,c_1,c_2) ~\big|~
(\Ring, m, c_1) \in L' ~\vee~
(\Ring, m, c_2) \in L'
\big\}.
\end{equation}
Now, we define another language $\widetilde{L}$ and prove that it is a super-complement of $L$ in \Cref{lemma:ring-sig:complement}.
Let $x_1 = (\Ring, m,c)$ as above, but let $\widetilde{w} \coloneqq msk$. Define the following relations:
\begin{align}
(x_1,\widetilde{w})\in R_4 & ~\Leftrightarrow~ \forall j \in [\ell]: \algo{LE.Valid}\big(pk_j, \algo{LE.MSKExt}(msk, pk_j)\big)=1
\label{def:relation:R4}\\
(x_1,\widetilde{w})\in R_5 & ~\Leftrightarrow~
\left\{
\begin{array}{l}
\exists\algo{VK} \in \mathsf{R}: \algo{VK} = (vk,pk,\rho) ~\text{such that:}\\
\algo{LE.Valid}\big(pk, \algo{LE.MSKExt}(msk, pk)\big)=1 ~\wedge\\
\algo{LE.Dec}\big(\algo{LE.MSKExt}(msk, pk), c \big) = (\sigma,vk) ~\wedge\\
\algo{Sig.Verify}(vk, \Ring\|m, \sigma)=1
\end{array}
\right. \label{def:relation:R5}
\end{align}
where, for each $j \in [\ell]$, $\algo{VK}_j = (vk_j, pk_j, \rho_j)$ is the $j$-th member in $\Ring$. Let $L_4$ and $L_5$ be the languages corresponding to $R_4$ and $R_5$, respectively. Define further the relation $\widehat{R}$ according to $\widehat{R} \coloneqq R_4 \setminus R_5$, and let $\widehat{L}$ be the corresponding language. Define $\tilde{L}$ as follows:
\begin{equation}\label{eq:def:language:L-tilde}
\widetilde{L} \coloneqq \big\{x=(\Ring, m,c_1,c_2) ~\big|~
(\Ring, m,c_1) \in \widehat{L} ~\wedge~
(\Ring, m,c_2) \in \widehat{L}
\big\}.
\end{equation}
Following a similar proof as for \cite[Lemma 5.1]{C:CGHKLMPS21}, we can show that $\tilde{L}$ is indeed a super-complement of $L$.
\begin{MyClaim}
\label{lemma:ring-sig:complement}
If $\LE$ satisfies the completeness defined in \Cref{item:def:LE:completeness} of \Cref{def:special-LE}, then the language $\widetilde{L}$ defined in \Cref{eq:def:language:L-tilde} is a super-complement (as per \Cref{def:super-complement-language}) of the language $L$ defined in \Cref{eq:def:language:L}.
\end{MyClaim}
\begin{proof}
To prove this claim, we need to show that for any statement $x$ of the following form
\begin{equation}\label{eq:ring-sig:complement:statement:x}
x = (\Ring, m, c_1, c_2),
\end{equation}
it holds that $x \in \tilde{L} \Rightarrow x \notin L$ (see \Cref{rmk:super-complement-language:special-case}). In the following, we finish the proof by showing the contrapositive: $x \in L \Rightarrow x\notin \tilde{L}$.
For any $x$ as in \Cref{eq:ring-sig:complement:statement:x}, we define
$$x_1 \coloneqq (\Ring, m,c_1) ~\text{and}~ x_2 \coloneqq (\Ring, m,c_2).$$
To prove ``$x \in L \Rightarrow x\notin \tilde{L}$'', it suffices to show that the following \Cref{eq:ring-sig:complement:expression1,eq:ring-sig:complement:expression2} hold for every $w= (\algo{VK}= (vk,pk,\rho),\sigma,r_c)$ and every $\widetilde{w} = msk$:
\begin{align}
&(x_1,w) \in R' \wedge \big(x_1,\widetilde{w}\big) \in R_4 ~\Rightarrow~ \big(x_1, \widetilde{w} \big) \in R_5 \label[Expression]{eq:ring-sig:complement:expression1}\\
&(x_2,w) \in R' \land \big(x_2,\widetilde{w}\big) \in R_4 ~\Rightarrow~ \big(x_2, \widetilde{w} \big) \in R_5. \label[Expression]{eq:ring-sig:complement:expression2}
\end{align}
We first prove \Cref{eq:ring-sig:complement:expression1}. If $(x_1,\tilde{w}) \in R_4$, then for all $\algo{VK} = (vk, pk, \rho) \in \algo{R}$, we know that $\algo{LE.MSKExt}(msk, pk)$ is a valid secret key for $pk$.
This means that:
\begin{itemize}
\item[] {\bf Fact:} any ciphertext w.r.t.\ any $pk$ (contained in any $\VK$) in $\Ring$ can be decrypted correctly by $\algo{LE.MSKExt}(msk, pk)$.
\end{itemize}
Also, observe that $(x_1,\tilde{w}) \in R'$ means $(x_1,w) \in R_1 \cap R_2 \cap R_3$, which says $c_1$ is a valid ciphertext of a signature for $\Ring\|m$, encrypted by some $pk$ in the ring $\Ring$. Then, by the above {\bf Fact}, we must have $(x_1, \tilde{w}) \in R_5$.
\Cref{eq:ring-sig:complement:expression2} can be proven similarly. This finish the proof of \Cref{lemma:ring-sig:complement}.
\end{proof}
\subsection{Proof of Security}
We now prove that \Cref{constr:pq:ring-sig} is a post-quantum secure post-quantum ring signature satisfying \Cref{def:pq:ring-signatures}. Its completeness follows straightforwardly from the completeness of $\ZAP$ and $\algo{Sig}$. We next prove post-quantum anonymity and blind-unforgeability in \Cref{sec:prove:pq:anonymity} and \Cref{sec:prove:pq:blind-unforgeability}, respectively.
\subsubsection{Proving Post-Quantum Anonymity}
\label{sec:prove:pq:anonymity}
In this section, we prove the following \Cref{lem:ring-sig:pq:anonymity}, which establishes post-quantum anonymity for \Cref{constr:pq:ring-sig}.
\begin{lemma}
\label{lem:ring-sig:pq:anonymity}
Assume $\algo{LE}$ satisfies the lossiness (\Cref{def:LE:property:lossyness}) described in \Cref{def:special-LE} and $\algo{ZAP}$ is statistically witness indistinguishable. Then, \Cref{constr:pq:ring-sig} satisfies the post-quantum anonymity described in \Cref{def:pq:anonymity}.
\end{lemma}
Let $\Adv$ be a QPT adversary participating in \Cref{expr:pq:anonymity}. Recall that the classical identities specified by $\Adv$ is $(i_0, i_1)$ and the quantum query is $\sum_{\Ring, m, t} \psi_{\Ring, m, t} \ket{\Ring, m, t}$. We will show a sequence of hybrids where the challenger switches from signing using $i_0$ to signing using $i_1$. It is easy to see that the scheme is post-quantum anonymous if $\Adv$ cannot tell the difference between each pair of adjacent hybrids.
\para{Hybrid $H_0$:} This hybrid simply runs the anonymity game with $b=0$. That is, $\Adv$'s query is answered as follows:
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{ \Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor f(\Ring, m)},$$
where
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i_0}, \Ring\| m;r) & \text{if}~\VK_{i_0}, \VK_{i_1} \in R \\
\bot & \text{otherwise}
\end{cases}
$. We remark that $f(\Ring,m)$ is performed quantumly for each $(\Ring, m)$ pair in the superposition. We say that $\Adv$ {\em wins} if it outputs $b' = b$ $(= 0)$.
It is worth noting that although $\RS.\Sign$ is a randomized algorithm, it uses only a single random tape $r$ for all the $(\Ring, m)$ pairs in the superposition (See \Cref{rmk:single-randomness}). In \Cref{constr:pq:ring-sig}, this means that the pair-wise independent hash functions $\PI_1, \PI_2, \PI_3)$ are sampled only once (i.e., they remain the same for all the $(\Ring, m)$ pairs in the superposition).
\para{Hybrid $H_1$:} In this hybrid, for each signing query from $\Adv$, instead of sampling a pair-wise indepedent function $\PI_2(\cdot)$ and compute $r_{c_2} = \PI_2(\Ring\|m)$, we compute $r_{c_2} = P_2(\Ring\|m)$, where $P_2(\cdot)$ is a {\em random} function. In effect, $r_{c_2}$ is now randomly sampled for each $(\Ring, m)$ pairs.
\subpara{$H_0 \idind H_1$:}
This follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_2$:} Here we switch $c_2$ from an encryption of a zero string to $c_2 \gets \algo{LE.Enc}(pk_{i_1},(\sigma',vk_{i_1});r_{c_2})$, where $\sigma' \gets \Sig.\Sign(sk_{i_1}, \Ring\|m)$. In this hybrid, it is worth noting that the previous ``dummy ciphertext'' $c_2$ becomes a valid one, i.e., it encrypts a valid signature for $\Ring\|m$ using identity $i_1$.
\subpara{$H_1 \sind H_2$:} In both $H_1$ and $H_2$, $r_{c_2}$ is sampled (effectively) uniformly at random for each $(\Ring, m)$ pair in the superposition in each signing query. Consider an oracle $\mathcal{O}$ that takes $(\Ring, m)$ as input and returns $(c_1,c_2,\pi)$ just as in $H_1$, and an analogous oracle $\mathcal{O}'$ that takes the same input and returns $(c_1,c_2,\pi)$ computed just as in $H_2$. Note that the only difference between the outputs of $\mathcal{O}$ and $\mathcal{O}'$ is in $c_2$, which encrypts $0^{|\sigma|+|vk|}$ in $H_1$ and $(\sigma',vk_{i_1})$ in $H_2$. Recall that $pk_{i_1}$ is produced using $\LE.\KSam^\ls$ and therefore, by lossiness (\Cref{def:LE:property:lossyness}), we have that the distributions of $c_2$ in $H_1$ and $H_2$ are statistically indistinguishable, implying that the outputs of $\mathcal{O}$ and $\mathcal{O}'$ are statistically close for every input $(\Ring,m)$, say less than distance $\Delta$ (which is negligible in \secpar). Then, by \Cref{lemma:oracleindist}, the probability that $\Adv$ distinguishes these two oracles even with $q = \poly(\secpar)$ quantum queries is at most $\sqrt{8C_0q^3\Delta}$, which is negligible since $\Delta$ is negligible. Similarity of these hybrids is immediate.
\para{Hybrid $H_3$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_2$ to compute $r_{c_2}$, instead of using a truly random function. Effectively we are undoing the change made in $H_1$.
\subpara{$H_2 \idind H_3$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_4$:} Here, we compute $r_\pi$ as the output of a {\em random} function $r_\pi = P_3(\Ring\|m)$, instead of being computed using $\PI_3$ as before. In effect, $r_\pi$ is now uniformly random.
\subpara{$H_3 \idind H_4$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_5$:} As mentioned in $H_2$, the ``block'' $(\Ring, m, c_2)$ is valid. Recall that in previous hybrids, $\ZAP$ uses the witness $w$ corresponding to the block $(\Ring, m, c_1)$. In this hybrid, we switch the witness used by $\ZAP$ from $w=(vk_{i_0},pk_{i_0}, \sigma, r_{c_1})$ to $w'=(vk_{i_1},pk_{i_1}, \sigma', r_{c_2})$, i.e., the witness corresponding to the $(\Ring, m, c_2)$ block.
\subpara{$H_4 \sind H_5$:} In both $H_4$ and $H_5$, $r_\pi$ is sampled (effectively) uniformly at random for each $(\Ring, m)$ pairs in the superposition for each query.
Consider an oracle $\mathcal{O}$ that takes $(\Ring,m)$ as input and returns $(c_1,c_2,\pi)$ just as in $H_4$, and an analogous oracle $\mathcal{O}'$ that takes the same input and returns $(c_1,c_2,\pi)$ computed just as in $H_5$. Note that the only difference between the outputs of $\mathcal{O}$ and $\mathcal{O}'$ is in $\pi$, which is generated using $w$ in $H_4$ and using $w'$ in $H_5$. Since both $w$ and $w'$ are valid witnesses, by the {\em statistical} witness indistinguishability of $\algo{ZAP}$, we have that the distributions of $\pi$ in $H_4$ and $H_5$ are statistically indistinguishable for every $(\Ring, m)$ pair (aka the input to the $\mathcal{O}$ or $\mathcal{O}'$). In other words, the outputs of $\mathcal{O}$ and $\mathcal{O}'$ are statistically close for every input $(\Ring,m)$, say less than distance $\Delta$ (which is negligible in \secpar). Then, the statistical indistinguishability follows from \Cref{lemma:oracleindist}.
\para{Hybrid $H_6$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_3$ to compute $r_\pi$, instead of using a truly random function. Effectively we are undoing the change made in $H_4$.
\subpara{$H_5 \idind H_6$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_7$:} In this hybrid, instead of sampling $r_{c_1} = \PI_1(\Ring\|m)$, we instead compute $r_{c_1}$ as the output of a {\em random} function $r_{c_1} = P_1(\Ring\|m)$. In effect, $r_{c_1}$ is now randomly sampled.
\subpara{$H_6 \idind H_7$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_8$:} In this hybrid, we switch $c_1$ from an encryption of $(\sigma, vk_{i_0})$ to one of $(\sigma', vk_{i_1})$.
\subpara{$H_7 \sind H_8$:} This follows from the same argument for $H_1\sind H_2$.
\para{Hybrid $H_9$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_1$ to compute $r_{c_1}$, instead of using a truly random function. Effectively we are undoing the change made in $H_7$.
\subpara{$H_8 \idind H_9$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{10}$:} In this hybrid, we switch to computing $r_\pi$ as the output of a {\em random} function $r_\pi = P_3(\Ring\|m)$, instead of being computed using $\PI_3$.
\subpara{$H_9 \idind H_{10}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{11}$:} In this hybrid, we again switch the witness used to generate $\pi$, from $w'$ to $w''=(vk_{i_1},pk_{i_1}, \sigma',r_{c_1})$.
\subpara{$H_{10} \sind H_{11}$:} This follows from the same argument for $H_4 \sind H_5$.
\para{Hybrid $H_{12}$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_3$ to compute $r_\pi$, instead of using a truly random function.
\subpara{$H_{11} \idind H_{12}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{13}$:} Here, we switch to computing $r_{c_2}$ as the output of a {\em random} function $r_{c_2} = P_2(\Ring\|m)$.
\subpara{$H_{12} \idind H_{13}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{14}$:} In this hybrid, we switch $c_2$ to an encryption of zeroes, namely $c_2 = \LE.\Enc(pk, 0^{|\sigma|+|vk|}; r_{c_2})$, instead of an encryption of $(\sigma',vk_{i_1})$.
\subpara{$H_{13} \sind H_{14}$:} This argument is identical to that for simlarity between $H_1 \sind H_2$.
\para{Hybrid $H_{15}$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_2$ to compute $r_{c_2}$, instead of using a truly random function.
\subpara{$H_{14} \idind H_{15}$:} This again follows from \Cref{lemma:2qwise}.
\vspace{1em}
Observe that $H_{15}$ corresponds to sign using identity $i_1$ in \Cref{expr:pq:anonymity}. This finishes the proof of \Cref{lem:ring-sig:pq:anonymity}.
\subsubsection{Proving Post-Quantum Blind-Unforgeability}
\label{sec:prove:pq:blind-unforgeability}
In this section, we prove the following \Cref{lem:ring-sig:pq:blind-unforgeability}, which establishes post-quantum blind-unforgeability for \Cref{constr:pq:ring-sig}.
\begin{lemma}
\label{lem:ring-sig:pq:blind-unforgeability}
Assume $\Sig$ is blind-unforgeable as per \Cref{def:classical-BU-sig}, $\algo{LE}$ satisfies the completeness of master secret keys property (\Cref{def:LE:property:GenWithSK:completeness}) and the almost-unique secret key property (\Cref{item:def:LE:computationally-unique-sk}), and $\algo{ZAP}$ has the selective non-witness adaptive-statement soundness (\Cref{item:def:zap:soundness}). Then, \Cref{constr:pq:ring-sig} is blind-unforgeable as per \Cref{def:pq:blind-unforgeability}.
\end{lemma}
Consider a QPT adversary $\Adv_\RS$ participating in \Cref{expr:pq:blind-unforgeability}. We proceed using a sequence of hybrids to set up our reduction to the blind-unforgeability of \algo{Sig}.
\para{Hybrid $H_0$:} This is just the post-quantum blind-sunforgeability game (\Cref{expr:pq:blind-unforgeability}) for our construction. In particular, for all $i\in [Q]$, the encryption key $pk_i$ is generated as $pk_i\gets \LE.\KSam^\ls(1^\secpar;r_i)$. Recall that we are in the full key exposure setting, so both the public keys and random coins $\Set{pk_i, r_i}_{i \in [Q]}$ are given to $\Adv$.
\para{Hybrid $H_1$:} In this experiment, the only difference is that, the challenger generates the $\Set{pk_i}_{i \in [Q]}$ by running $\big( \Set{pk_i}_{i\in [Q]}, \msk \big)\gets \algo{LE.MSKGen}(1^\secpar, Q)$. The challenger keeps $\msk$ to itself, and sends $\big\{pk_i, \algo{LE.RndExt}(pk_i)\big\}_{i\in [Q]}$ to $\Adv$.
\subpara{$H_0 \cind H_1$:} This follows immediately from the IND of $\algo{MSKGen}$/$\KSam^\ls$ property (\Cref{def:LE:property:IND:GenWithSK-KSam}) of $\LE$ as specified in \Cref{def:special-LE}. It is worth noting that $\Adv_\RS$'s quantum access to the signing algorithm does not affect this proof at all, since the $pk_i$'s (contained in $\VK_i$'s) are sampled classically by the challenger before $\Adv_\RS$ makes any quantum {\sf sign} queries.
\para{Reduction to the BU of $\Sig$.} We proceed to show that post-quantum blind-unforgeability holds in $H_1$. Consider the adversary's forgery attempt
$$\big(\algo{R}^*, m^*, \Sigma^* = (c_1^*, c_2^*, \pi^*)\big)~\text{satisfying}~(\Ring^*, m^*) \in B^\RS_\epsilon.$$
Let $x^* \coloneqq (\Ring^*, m^*, c^*_1, c^*_2)$. Let $\algo{VK}_1^*$ = ($vk_1^*,pk_1^*,\rho_1^*$) be the lexicographically smallest verification key in $\algo{R}^*$.
Observe that for the $x^*$ defined above, one of the following two cases must happen: $x^* \in \tilde{L}$ or $x^* \notin \tilde{L}$. (Recall that $\tilde{L}$ is the super-complement of $L$ defined in \Cref{eq:def:language:L-tilde}.) In the following, we show two claims. \Cref{lemma:unfroge:inltilde} says that it cannot be the case that $x^* \in \tilde{L}$, unless the $\ZAP$ verification rejects (thus, the forgery is invalid). \Cref{lemma:unforge:sig} says that $x^* \notin \tilde{L}$ cannot happen either. Therefore, \Cref{lemma:unfroge:inltilde,lemma:unforge:sig} together show that any QPT adversary has negligible chance of winning the blind-unforgeability game for $\RS$ in $H_1$. Note that winning the post-quantum blind-unforgeability game for $\RS$ is an event that can be efficiently tested. Thus, by $H_0 \cind H_1$, no QPT adversaries can win the post-quantum blind-unforgeability game for $\RS$ in hybrid $H_0$. This concludes the proof of \Cref{lem:ring-sig:pq:blind-unforgeability}.
Now, the only thing left is to state and prove \Cref{lemma:unfroge:inltilde,lemma:unforge:sig}, which is done in the following.
\begin{MyClaim}
\label{lemma:unfroge:inltilde}
In $H_1$, assume that $\algo{ZAP}$ satisfies selective non-witness adaptive statement soundness (\Cref{item:def:zap:soundness}).
Then, the following holds:
$$\Pr[x^* \in \widetilde{L} ~\wedge~ \algo{\algo{ZAP}.Verify}(\rho_1^*,x^*,\pi^*)=1] = \negl(\lambda).$$
\end{MyClaim}
\begin{proof}
First, notice that by definition, the $\Ring^*$ in $\Adv_\RS$'s forgery contains only $\VK$'s from the set $\mathcal{VK} = \Set{\VK_j}_{j\in [Q]}$ generated by the challenger. Therefore, it suffices to show that for each $j \in [Q]$,
\begin{equation}\label{eq:lemma:unfroge:inltilde:real-goal}
\Pr[x^* \in \widetilde{L} ~\wedge~ \algo{ZAP.Verify}(\rho_j,x^*,\pi^*)=1] = \negl(\lambda),
\end{equation}
where $\rho_j$ denotes the first-round message of \ZAP corresponding to the $j$-th verification key $\algo{VK}_j$ generated in the game.
Let $\Adv_\RS$ be an adversary attempting to output a forgery such that
$$x^* \in \widetilde{L}~~\text{and}~~\algo{ZAP}.\Verify(\rho_j,x^*,\pi^*) =1.$$
We build an adversary $\Adv_\ZAP$ against the selective non-witness adaptive-statement soundness of $\algo{ZAP}$ for $(L, \widetilde{L})$ (defined in \Cref{eq:def:language:L,eq:def:language:L-tilde} respectively).
The algorithm $\Adv_\ZAP$ proceeds as follows:
\begin{itemize}
\item On input the 1st \algo{ZAP} message $\widehat{\rho}$, it sets $\rho_{j} = \widehat{\rho}$ and proceeds exactly as $H_1$.
\item Upon receiving the forgery attempt $\big(\Ring^*, m^*, \Sigma^* =(c^*_1, c^*_2, \pi^*)\big)$ from $\Adv$, it outputs $$\big(x^* \coloneqq (\Ring^*, m^*, c^*_1, c^*_2),~\pi^*\big).$$
\end{itemize}
We remark that $H_1$ is quantum. So, $\Adv_\ZAP$ needs to be a quantum machine to simulate $H_1$ for $\Adv_\RS$. This is fine since we assume that the soundness (\Cref{item:def:zap:soundness}) of ZAP in \Cref{def:generalizedzap-interface} holds against QPT adversaries.
To finish the proof, notice that $x^* \in \widetilde{L}$ means that there exists a ``non-witness'' $\tilde{w}^*$ such that $(x^*, \tilde{w}^*) \in \tilde{R}$. Therefore, if \Cref{eq:lemma:unfroge:inltilde:real-goal} does not hold, $(\rho_j, x^*, \pi^*)$ will break the soundness (\Cref{item:def:zap:soundness}) w.r.t.\ the non-witness $\tilde{w}$.
\end{proof}
\begin{MyClaim}
\label{lemma:unforge:sig}
In $H_1$, assume that $\algo{Sig}$ satisfies the blind-unforgeability as per \Cref{def:classical-BU-sig}, $\algo{LE}$ satisfies the completeness of master secret keys propoerty (\Cref{def:LE:property:GenWithSK:completeness}) and the almost-unique secret key property (\Cref{item:def:LE:computationally-unique-sk}). Then,
$\Pr\big[x^* \not \in \widetilde{L}\big] = \negl(\lambda)$.
\end{MyClaim}
\begin{proof}
Let $\Adv_\RS$ be a QPT adversary attempting to output a forgery w.r.t.\ our $\RS$ scheme such that $x^* \not \in \widetilde{L}$. We build an algorithm $\Adv_\Sig$ against the blind-unforgeability of $\algo{Sig}$. The algorithm $\Adv_\Sig$ (playing the blind-unforgeability game \Cref{expr:BU:ordinary-signature} for $\Sig$) proceeds as follows:
\begin{enumerate}
\item
invoke $\Adv_\RS$ to obtain the $\epsilon$ for the blind-unforgeability game of $\RS$; give this $\epsilon$ to $\Adv_\Sig$'s own challenger for the blind-unforgeability game of $\Sig$;
\item
receive $\widehat{vk}$ from its own challenger; pick an index $j \pick [Q]$ uniformly at random; set $vk_{j} \coloneqq \widehat{vk}$; then, proceeds as in $H_1$ to prepare the rest of the verification keys and continue the execution with $\Adv_\RS$.
\item \label[Step]{step:unforge:sig:signing-query}
when $\Adv_\RS$ sends a quantum signing query $(\algo{sign},i, \sum \psi_{\Ring, m, t}\ket{\Ring, m, t})$, if the specified identity $i$ is not equal to $j$, it proceeds as in $H_1$; otherwise, it uses the blind-unforgeability (for $\Sig$) game's signing oracle $\Sig.\Sign$ to obtain a $\Sig$ signature for the $j$-th party and then continues exactly as in $H_1$; (See the paragraph right after the description of $\Adv_\Sig$.)
\item
if $\Adv_\RS$ tries to corrupt the $j$-th party, $\Adv_\Sig$ aborts; (It is worth noting that the identities are classical. So, $\Adv_\RS$'s quantum power does not affect this step.)
\item
upon receiving the forgery attempt $\Sigma^*$ from $\Adv_\RS$, $\Adv_\Sig$ decrypts $c_1^*$ using $\msk$ to recover $\sigma_1^*$. (Recall that, the secret key for $pk_j$ can be obtained as $\algo{LE.MSKExt}(\msk, pk_j)$). If $$\algo{Sig.Verify}(vk_j, \Ring^*\|m^*, \sigma_1^*)=1,$$ it sets $\widehat{\sigma} := \sigma_1^*$. Otherwise, it decrypts $c_2^*$ with $\msk$ to recover $\sigma_2^*$, and sets $\widehat{\sigma} \coloneqq \sigma_2^*$. It outputs $(\Ring^*\|m^*,\widehat{\sigma})$.
\end{enumerate}
We first remark that, up to (inclusively) \Cref{step:unforge:sig:signing-query}, $\Adv_\RS$'s view is identical to that in $H_1$. Recall that in $H_1$, the challenger maintains a blindset $B^\RS_\epsilon$ such that any $(\Ring, m) \in B^\RS_\epsilon$ will not be answered (this is inherited from $H_0$, which is exactly \Cref{expr:pq:blind-unforgeability}). In contrast, in the execution of $\Adv_\Sig$ described above, $\Adv_\Sig$ first forwards the $\sum \psi_{\Ring, m, t}\ket{\Ring, m, t}$ part of $\Adv_\RS$'s query to its $\Sig.\Sign$ oracle to obtain
$\sum_{\Ring, m, t} \psi_{\Ring, m, t} \ket{\Ring, m, t\xor B^\Sig_\epsilon\Sig.\Sign(sk_j, \Ring\|m)}$ (note that $sk_j = \widehat{sk}$), and then performs the remaining computation exactly as in $H_1$. Note that the $B^\Sig_\epsilon$ is the blindset maintained by the $\Sig$ signing algorithm. Importantly, since the ``messages'' singed by $\Sig$ are of the form $\Ring\|m$, $B^\Sig_\epsilon$ is actually generated identically to $B^\RS_\epsilon$---that is, both of them are generated by including each $(\Ring, m)$ pair in with (the same) probability $\epsilon$.
To finish the proof, we show that
$(\Ring^*\|m^*,\widehat{\sigma})$ is a valid forgery against $\Sig$'s blind-unforgeability game with probability at least $\frac{1}{Q}\big(\Pr\big[x^* \not \in \widetilde{L}\big]- \negl(\lambda)\big)$.
Recall that we are focusing on the case $x^* \notin \tilde{L}$, where $\tilde{L}$ is defined in \Cref{eq:def:language:L-tilde}; without loss of generality, assume that $(\Ring^*, m^*, c_1^*) \not \in \widehat{L}$. Then, observe that due to the way $H_1$ generates the public keys (more acurately, \Cref{def:LE:property:GenWithSK:completeness}) and that $\Ring^* \subseteq \mathcal{VK}\setminus \mathcal{C}$ (in particular, $\Ring^*\subseteq \mathcal{VK}$), we have
\begin{equation}\label[Expression]{eq:unforge:sig:proof:L4}
\big((\Ring^*, m^*,c_1^*), \msk\big) \in R_4 ~~\text{(recall that $R_4$ is defined in \Cref{def:relation:R4})}.
\end{equation}
Since we assume that $(\Ring^*, m^*,c_1^*) \notin \widehat{L}$, \Cref{eq:unforge:sig:proof:L4} and the definition of $\widehat{L}$ imply the existence of a string $\widetilde{w}$ such that
\begin{equation}\label[Expression]{eq:unforge:sig:proof:L5}
\big((\Ring^*, m^*,c_1^*),\widetilde{w}\big) \in R_5 ~~\text{(recall that $R_5$ is defined in \Cref{def:relation:R5})}.
\end{equation}
We remark that the $\tilde{w}$ may not equal $\msk$. However, note that $R_5$ (\Cref{def:relation:R5}) tests if
$$\algo{LE.Valid}\big(pk, \algo{LE.MSKExt}(\tilde{w},pk)\big) = 1$$ with respect to the $pk$ contained in some $\VK$ in the ring. If this test passes, by \Cref{eq:unforge:sig:proof:L4} and the almost-unique secret key property (\Cref{item:def:LE:computationally-unique-sk}) of $\LE$, it must hold for this $pk$ that $$\algo{LE.MSKExt}(\tilde{w}, pk) = \algo{LE.MSKExt}(\msk, pk),$$ except with negligible probability.
To summarize, the above argument implies the following facts:
\begin{enumerate}
\item
by our assumption, $(\Ring^*, m^*) \in B^\RS_\secpar$; this also implies $(\Ring^*, m^*) \in B^\Sig_\secpar$ because $B^\Sig_\epsilon = B^\RS_\epsilon$ as argued earlier;
\item
by \Cref{eq:unforge:sig:proof:L5}, for some $\VK = (vk^*, pk^*, \rho^*) \in \Ring^*$, it must hold that
$$\LE.\Dec\big(\algo{LE.MSKExt}(\tilde{w},pk), c_1^*\big) = (\sigma^*,vk^*)~~\text{and}~~\algo{Sig.Verify}(vk^*, \Ring^*\|m^*,\sigma^*)=1.$$ Also, as mentioned earlier, $\algo{LE.MSKExt}(\tilde{w}, pk^*) = \algo{LE.MSKExt}(\msk, pk^*)$ for this $pk^*$.
\end{enumerate}
The above means that the $\Adv_\RS$ uses a
$\algo{VK^*}=(vk^*, pk^*, \rho^*) \in \mathsf{R^*} \subseteq \mathcal{VK}\setminus\mathcal{C}$ such that $c_1^*$ encrypts (among other things) a signature $\sigma^*$ that is valid for the forgery message $\Ring^*\|m^*$ w.r.t. key $vk^*$ (for the blind-unforgeability game of $\Sig$). Moreover, $\Adv_\Sig$ can extract this forgery message efficiently by decrypting $c^*$ using $\algo{LE.MSKExt}(\msk, pk^*)$!
Finally, observe that index $j$ is sampled uniformly.
Therefore, we have that $(\widehat{vk} =)$ $vk_j = vk^*$ with probability $1/Q$.
\end{proof}
\section{One-More Unforeagibility vs PQ-EUF for Ring Signatures}
\label{sec:one-more:PQ-EUF:ring-sig}
The ring-signature analog of the one-more unforgeability by Boneh and Zhandry \cite{C:BonZha13}, {\em when restricted to the classical setting}, seems to be weaker than the standard unforgeability in \Cref{def:classical:ring-signature}.\footnote{This is in contrast to the case of ordinary signatures, where one-more unforgeability is equivalent to the standard existential unforgeability \cite{C:BonZha13}.} That is, in the classical setting, any $\RS$ satisfying the unforgeability in \Cref{def:classical:ring-signature} is also one-more unforgeable; but the reverse direction is unclear. We provide discussion in the following.
To argue that one-more unforgeability is no weaker than \Cref{def:classical:ring-signature}, one needs to show how to convert a forger $\Adv_\textsc{euf}$ winning in \Cref{expr:classical:unforgeability} to another forger $\Adv_\textsc{om}$ winning in the (classical version of) ``one-more forgery'' game. Conceivably, $\Adv_\textsc{om}$ will run $\Adv_\textsc{euf}$ internally; thus, $\Adv_\textsc{om}$ will make no less {\sf sign} queries than $\Adv_\textsc{euf}$. Recall that $\Adv_\textsc{om}$ needs to forge one more signature than the total number of its queries.
Also, crucially, all the ring signatures presented by $\Adv_\textsc{om}$ at the end must have {\em no} corrupted members in the accompanying ring. Now ideally one might imagine that we can simply use the queries made by $\Adv_\textsc{om}$ (which are really queries by $\Adv_\textsc{euf}$) to meet the ``one-more'' challenge; however, this is thwarted immediately due to the fact that $\Adv_\textsc{euf}$ has absolutely no obligation to make queries meeting this requirement, so even if the final forgery produced by $\Adv_\textsc{euf}$ is valid, our attempted reduction does not have any means to provide $\Adv_\textsc{om}$ with all the signatures it needs to win the ``one-more'' challenge (since not all of the queries can be reused). Indeed, it is not hard to find attacks that use this definitional gap to violate standard unforgeability, while being ruled out as a valid attack against one-more ring unforgeability. Contrast this with a comparison in the other direction: an adversary $\Adv_\textsc{om}$ for the one-more unforgeability experiment is easily converted into a standard $\Adv_\textsc{euf}$ adversary since not all of the signatures output by $\Adv_\textsc{om}$ at the end can be previous queries (by the pigeonhole principle); $\Adv_\textsc{euf}$ simple outputs the one that is not.
We remark however that this definitional gap between standard ring signature unforgeability and the ``one-more'' version may not be inherent; rather, we just do not know how to meet this gap. Our arguments here should not be interpreted as a proof showing that the former notion is strictly stronger than the latter. We leave it as an open question to either demonstrate a separation, or prove that the two are actually equivalent.
\subsubsection{Post-Quantum Anonymity}
\label{sec:prove:pq:anonymity}
We establish the post-quantum anonymity by proving the following lemma.
\begin{lemma}
\label{lem:ring-sig:pq:anonymity}
Assume $\algo{LE}$ satisfies the lossiness (\Cref{def:LE:property:lossyness}) described in \Cref{def:special-LE}, $\algo{SPB}$ is index hiding, and $\algo{ZAP}$ is statistically witness indistinguishable. Then, \Cref{constr:pq:ring-sig} satisfies the post-quantum anonymity described in \Cref{def:pq:anonymity}.
\end{lemma}
\xiao{give a sketch of this proof, and move the full proof to the appendix.}
Let $\Adv$ be a QPT adversary participating in \Cref{expr:pq:anonymity}. Recall that the classical identities specified by $\Adv$ is $(i_0, i_1)$ and the quantum query is $\sum_{\Ring, m, t} \psi_{\Ring, m, t} \ket{\Ring, m, t}$. We will show a sequence of hybrids where the challenger switches from signing using $i_0$ to signing using $i_1$. It is easy to see that the scheme is post-quantum anonymous if $\Adv$ cannot tell the difference between each pair of adjacent hybrids.
\para{Hybrid $H_0$:} This hybrid simply runs the anonymity game with $b=0$. That is, $\Adv$'s query is answered as follows:
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{ \Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor f(\Ring, m)},$$
where
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i_0}, \Ring, m;r) & \text{if}~\VK_{i_0}, \VK_{i_1} \in R \\
\bot & \text{otherwise}
\end{cases}
$. We remark that $f(\Ring,m)$ is performed quantumly for each $(\Ring, m)$ pair in the superposition. We say that $\Adv$ {\em wins} if it outputs $b' = b$ $(= 0)$.
It is worth noting that although $\RS.\Sign$ is a randomized algorithm, it uses only a single random tape $r$ for all the $(\Ring, m)$ pairs in the superposition (See \Cref{rmk:single-randomness}). In \Cref{constr:pq:ring-sig}, this means that the $(hk_1, shk_1, hk_2, shk_2, \PI_1, \PI_2, \PI_3)$ values are only sampled once (i.e., they remain the same for all the $(\Ring, m)$ pairs in the superposition).
\para{Hybrid $H_1$:} Recall that in $H_0$, the 2nd (classical) \SPB key pair $(hk_2, shk_2)$ is generated using identity $i_0$. In this hybrid, we generate them using identity $i_1$:
$$(hk_2, shk_2) \gets \SPB.\Gen(1^\secpar, M, i_1).$$
Everything else in $H_1$ remains identical to $H_0$.
\subpara{$H_0 \cind H_1$:} This follows immediately from the index hiding property (against QPT adversaries) of \SPB defined in \Cref{def:SPB-hash}. Note that $\Adv$'s quantum access to the signing algorithm does not affect this proof at all. Because, as mentioned above in $H_0$, the \SPB key generation happens (classically) only once for all the rings and messages in superposition.
\para{Hybrid $H_2$:} In this hybrid, for each signing query from $\Adv$, instead of sampling $r_{c_2} \gets \PI_2(\Ring\|m)$, we instead compute $r_{c_2}$ as the output of a {\em random} function $r_{c_2} \gets P_2(\Ring\|m)$. In effect, $r_{c_2}$ is now randomly sampled for each $(\Ring, m)$ pairs.
\subpara{$H_1 \idind H_2$:}
This follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_3$:} Here we switch $c_2$ from an encryption of a zero string to $c_2 \gets \algo{LE.Enc}(pk_{i_1},(\sigma',vk_{i_1});r_{c_2})$, where $\sigma' \gets \Sig.\Sign(sk_{i_1}, m)$. In this hybrid, it is worth noting that the previous ``dummy block'' $(c_2, hk_2, h_2)$ becomes a valid one, where $c_2$ encrypting a valid signature for $m$ using identity $i_1$, $hk_2$ is a $\SPB$ hashing key binding identity $i_1$, and $h_2$ is a hash of $\Ring$ using $hk_2$.
\subpara{$H_2 \sind H_3$:} In both $H_2$ and $H_3$, $r_{c_2}$ is sampled (effectively) uniformly at random for each $(\Ring, m)$ pair in the superposition in each signing query. Consider an oracle $\mathcal{O}$ that takes $(\Ring, m)$ as input and returns $(c_1,c_2,\pi)$ just as in $H_2$, and an analogous oracle $\mathcal{O}'$ that takes the same input and returns $(c_1,c_2,\pi)$ computed just as in $H_3$. Note that the only difference between the outputs of $\mathcal{O}$ and $\mathcal{O}'$ is in $c_2$, which encrypts $0^{|\sigma|+|vk|}$ in $H_2$ and $\sigma',vk_{i_1}$ in $H_3$. Recall that $pk_{i_1}$ is produced using $\LE.\KSam^\ls$ and therefore by lossiness (\Cref{def:LE:property:lossyness}), we have that the distributions of $c_2$ in $H_2$ and $H_3$ are statistically indistinguishable, implying that the outputs of $\mathcal{O}$ and $\mathcal{O}'$ are statistically close for every input $(m,\Ring)$, say less than distance $\Delta$ (which is negligible in \secpar). Then, by \Cref{lemma:oracleindist}, the probability that $\Adv$ distinguishes these two oracles even with $q$ quantum queries is at most $\sqrt{8C_0q^3\Delta}$, which is negligible since $\Delta$ is negligible and $q$ is polynomial. Similarity of these hybrids is immediate.
\para{Hybrid $H_4$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_2$ to compute $r_{c_2}$, instead of using a truly random function. Effectively we are undoing the change made in $H_2$.
\subpara{$H_3 \idind H_4$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_5$:} Here, we compute $r_\pi$ as the output of a {\em random} function $r_\pi \gets P_3(\Ring\|m)$, instead of being computed using $\PI_3$ as before. In effect, $r_\pi$ is now uniformly random.
\subpara{$H_4 \idind H_5$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_6$:} As mentioned in $H_3$, the ``block'' $(c_2, hk_2, h_2)$ is valid. Recall that in previous hybrids, $\ZAP$ uses the witness $w$ corresponding to the block $(c_1, hk_1, h_1)$. In this hybrid, we switch the witness used by $\ZAP$ from $w$ to $w'=(vk_{i_1},pk_{i_1},i_1,\tau_2, \sigma', r_{c_2})$ (i.e., the witness corresponding to the $(c_2, hk_2, h_2)$ block).
\subpara{$H_5 \sind H_6$:} In both $H_5$ and $H_6$, $r_\pi$ is sampled (effectively) uniformly at random for each $(\Ring, m)$ pairs in the superposition for each query.
Consider an oracle $\mathcal{O}$ that takes $(\Ring,m)$ as input and returns $(c_1,c_2,\pi)$ just as in $H_5$, and an analogous oracle $\mathcal{O}'$ that takes the same input and returns $(c_1,c_2,\pi)$ computed just as in $H_6$. Note that the only difference between the outputs of $\mathcal{O}$ and $\mathcal{O}'$ is in $\pi$, which is generated using $w$ in $H_5$ and using $w'$ in $H_6$. Since both $w$ and $w'$ are valid witnesses, by the statistical witness indistinguishability of $\algo{ZAP}$, we have that the distributions of $\pi$ in $H_5$ and $H_6$ are statistically indistinguishable for every $(\Ring, m)$ pair (aka the input to the $\mathcal{O}$ or $\mathcal{O}'$). In other words, the outputs of $\mathcal{O}$ and $\mathcal{O}'$ are statistically close for every input $(m,\Ring)$, say less than distance $\Delta$ (which is negligible in \secpar). Then, the statistical indistinguishability follows from \Cref{lemma:oracleindist}.
\para{Hybrid $H_7$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_3$ to compute $r_\pi$, instead of using a truly random function. Effectively we are undoing the change made in $H_5$.
\subpara{$H_6 \idind H_7$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_8$:} In this hybrid, instead of sampling $r_{c_1} \gets \PI_1(\Ring\|m)$, we instead compute $r_{c_1}$ as the output of a {\em random} function $r_{c_1} \gets P_1(\Ring\|m)$. In effect, $r_{c_1}$ is now randomly sampled.
\subpara{$H_7 \idind H_8$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{9}$:} In this hybrid, we now switch $c_1$ from an encryption of $(\sigma, vk_{i_0})$ to one of $(\sigma', vk_{i_1})$.
\subpara{$H_8 \sind H_{9}$:} This follows from the same argument for $H_2\sind H_3$.
\para{Hybrid $H_{10}$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_1$ to compute $r_{c_1}$, instead of using a truly random function. Effectively we are undoing the change made in $H_8$.
\subpara{$H_{9} \idind H_{10}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{11}$:} In this hybrid, the \SPB key pair $(hk_1, shk_1)$ is now generated using identity $i_1$, instead of $i_0$. Namely, we compute $(hk_1, shk_1) \gets \SPB.\Gen(1^\secpar, M, i_1)$.
\subpara{$H_{10} \cind H_{11}$:} This follows from the same argument for $H_0 \cind H_1$.
\para{Hybrid $H_{12}$:} In this hybrid, we switch to computing $r_\pi$ as the output of a {\em random} function $r_\pi \gets P_3(\Ring\|m)$, instead of being computed using $\PI_3$.
\subpara{$H_{11} \idind H_{12}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{13}$:}In this hybrid, we again switch the witness used to generate $\pi$, from $w'$ to $w''=(vk_{i_1},pk_{i_1},i_1,\tau_1,\sigma',r_{c_1})$.
\subpara{$H_{12} \sind H_{13}$:} This follows from the same argument for $H_5 \sind H_6$.
\para{Hybrid $H_{14}$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_3$ to compute $r_\pi$, instead of using a truly random function.
\subpara{$H_{13} \idind H_{14}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{15}$:} Here, we switch to computing $r_{c_2}$ as the output of a {\em random} function $r_{c_2} \gets P_2(\Ring\|m)$.
\subpara{$H_{14} \idind H_{15}$:} This again follows from \Cref{lemma:2qwise}.
\para{Hybrid $H_{16}$:} In this hybrid, we switch $c_2$ to an encryption of zeroes, namely $c_2 \gets \LE.\Enc(pk, 0^{|\sigma|+|vk|}; r_{c_2})$, instead of an encryption of $(\sigma',vk_{i_1})$.
\subpara{$H_{15} \sind H_{16}$:} This argument is identical to that for simlarity between $H_2\sind H_3$.
\para{Hybrid $H_{17}$:} In this hybrid, we switch back to using the pairwise independent hash function $\PI_2$ to compute $r_{c_2}$, instead of using a truly random function.
\subpara{$H_{16} \idind H_{17}$:} This again follows from \Cref{lemma:2qwise}.
\vspace{1em}
Observe that $H_{17}$ corresponds to sign using identity $i_1$ in \Cref{expr:pq:anonymity}. This finishes the proof of \Cref{lem:ring-sig:pq:anonymity}.
\subsubsection{Post-Quantum Blind-Unforgeability}
\label{sec:prove:pq:blind-unforgeability}
In this section, we prove the following lemma.
\begin{lemma}
\label{lem:ring-sig:pq:blind-unforgeability}
Assume $\Sig$ is blind-unforgeable as per \Cref{def:classical-BU-sig}, $\algo{LE}$ satisfies the almost-unique secret key property (\Cref{item:def:LE:computationally-unique-sk}), $\algo{SPB}$ is somewhere perfectly binding, and $\algo{ZAP}$ has the selective non-witness adaptive-statement soundness (\Cref{item:def:zap:soundness}). Then, \Cref{constr:pq:ring-sig} is blind-unforgeable as per \Cref{def:pq:blind-unforgeability}.
\end{lemma}
Consider a QPT adversary $\Adv_\RS$ participating in \Cref{expr:pq:blind-unforgeability}. We proceed with a sequence of hybrids to set up our reduction to the blind-unforgeability of \algo{Sig}.
\para{Hybrid $H_0$:} This is just the post-quantum blind-sunforgeability game (\Cref{expr:pq:blind-unforgeability}) for our construction. In particular, for all $i\in [Q]$, the encryption key $pk_i$ is generated as $pk_i\gets \LE.\KSam^\ls(1^\secpar)$.
\para{Hybrid $H_1$:} In this experiment, the only difference is that, the challenger first picks a uniformly random secret key for $\LE$ as $sk_\LE \pick \mathcal{SK}_\secpar$, and then generates the corresponding public keys for the adversary using this, namely $pk_i \gets \algo{LE.GenWithSK}(sk_{\LE})$ for all $i \in [Q]$. The challenger now stores $sk_{\LE}$.
\subpara{$H_0 \cind H_1$:} This follows immediately from the IND of $\GenWithSK$/$\KSam^\ls$ property (\Cref{def:LE:property:IND:GenWithSK-KSam}) of $\LE$ as specified in \Cref{def:special-LE}. It is worth noting that $\Adv_\RS$'s quantum access to the signing algorithm does not affect this proof at all. Because the $pk_i$'s (contained in $\VK_i$'s) are sampled classically by the challenger before $\Adv_\RS$ makes any quantum {\sf sign} queries.
\para{Reduction to the BU of $\Sig$.} We proceed to show that post-quantum blind-unforgeability holds in $H_1$. Consider the adversary's forgery attempt
$$\big(\algo{R}^*, m^*, \Sigma^* = (c_1^*, hk_1^*,c_2^*,hk_2^*,\pi^*)\big)~\text{satisfying}~(\Ring^*, m^*) \in B^\RS_\epsilon.$$
Let $x_1^*$ be the statement corresponding to $\Sigma^*$: $x_1^* \coloneqq (m^*,c_1^*,c_2^*,hk_1^*,hk_2^*,h_1^*,h_2^*)$, where $h_1^* = \algo{SPB.Hash}(hk_1^*,\mathsf{R^*})$ and $h_2^* = \algo{SPB.Hash}(hk_2^*,\mathsf{R^*})$. Let $x^*_2 \coloneqq \Ring^*$ and $x^* \coloneqq (x^*_1, x^*_2)$. Let $\algo{VK}_1^*$ = ($vk_1^*,pk_1^*,\rho_1^*$) be the lexicographically smallest verification key in $\algo{R}^*$.
Observe that for the $x^*$ defined above, one of the following two cases must happen: $x^* \in \tilde{L}$ or $x^* \notin \tilde{L}$. (Recall that $\tilde{L}$ is the super-complement of $L$ defined in \Cref{eq:def:language:L-tilde}.) In the following, we show two claims. \Cref{lemma:unfroge:inltilde} says that it cannot be the case that $x^* \in \tilde{L}$, unless the $\ZAP$ verification rejects (thus, the forgery is invalid). \Cref{lemma:unforge:sig} says that $x^* \notin \tilde{L}$ cannot happen either. Therefore, \Cref{lemma:unfroge:inltilde,lemma:unforge:sig} together show that any QPT adversary has negligible chance of winning the blind-unforgeability game for $\RS$ in $H_1$. Note that winning the post-quantum blind-unforgeability game for $\RS$ is an event that can be efficiently tested. Thus, by $H_0 \cind H_1$, no QPT adversaries can win the post-quantum blind-unforgeability game for $\RS$ in hybrid $H_0$. This eventually finishes the proof of \Cref{lem:ring-sig:pq:blind-unforgeability}.
Now, the only thing left is to state and prove \Cref{lemma:unfroge:inltilde,lemma:unforge:sig}, which is done in the following.
\begin{MyClaim}
\label{lemma:unfroge:inltilde}
In $H_1$, assume that $\algo{ZAP}$ satisfies selective non-witness adaptive statement soundness (\Cref{item:def:zap:soundness}).
Then, the following holds:
$$\Pr[x^* \in \widetilde{L} ~\wedge~ \algo{\algo{ZAP}.Verify}(\rho_1^*,x^*,\pi^*)=1] = \negl(\lambda).$$
\end{MyClaim}
\begin{proof}
First, notice that by definition, the $\Ring^*$ in $\Adv_\RS$'s forgery contains only $\VK$'s from the set $\mathcal{VK} = \Set{\VK_j}_{j\in [Q]}$ generated by the challenger. Therefore, it suffices to show that for each $j \in [Q]$,
\begin{equation}\label{eq:lemma:unfroge:inltilde:real-goal}
\Pr[x^* \in \widetilde{L} ~\wedge~ \algo{ZAP.Verify}(\rho_j,x^*,\pi^*)=1] = \negl(\lambda),
\end{equation}
where $\rho_j$ denotes the first-round message of \ZAP corresponding to the $j$-th verification key $\algo{VK}_j$ generated in the game.
Let $\Adv_\RS$ be an adversary attempting to output a forgery such that $x^* \in \widetilde{L}$ {\em and} $\algo{ZAP}.\Verify(\rho_j,x^*,\pi^*) =1$. We build an adversary $\Adv_\ZAP$ against the selective non-witness adaptive-statement soundness of $\algo{ZAP}$ for $(L, \widetilde{L})$ (defined in \Cref{eq:def:language:L,eq:def:language:L-tilde} respectively).
The algorithm $\Adv_\ZAP$ proceeds as follows:
\begin{itemize}
\item On input the first \algo{ZAP} message $\widehat{\rho}$, it sets $\rho_{j} = \widehat{\rho}$ and then proceeds exactly as $H_1$.
\item Upon receiving the forgery attempt $(\Ring^*, m^*, \Sigma^*)$ from $\Adv$, it constructs the corresponding $x^*$ and $\pi^*$, and outputs $(x^*,\pi^*)$.
\end{itemize}
We remark that $H_1$ is quantum. So, $\Adv_\ZAP$ needs to be a quantum machine to simulate $H_1$ for $\Adv_\RS$. This is fine since we assume that the soundness (\Cref{item:def:zap:soundness}) of ZAP in \Cref{def:generalizedzap-interface} holds against QPT adversaries.
To finish the proof, notice that $x^* \in \widetilde{L}$ means that there exists a non-witness $\tilde{w}^*$ such that $(x^*, \tilde{w}^*) \in \tilde{R}$. Therefore, if \Cref{eq:lemma:unfroge:inltilde:real-goal} does not hold, $(\rho_j, x^*, \pi^*)$ will break the soundness (\Cref{item:def:zap:soundness}) w.r.t.\ the non-witness $\tilde{w}$.
\end{proof}
\begin{MyClaim}
\label{lemma:unforge:sig}
In $H_1$, assume that $\algo{Sig}$ satisfies the blind-unforgeability as per \Cref{def:classical-BU-sig}, $\LE.\GenWithSK$ has the almost-unique secret key property defined in \Cref{item:def:LE:computationally-unique-sk},
and $\algo{SPB}$ is somewhere perfectly binding. Then,
$\Pr\big[x^* \not \in \widetilde{L}\big] = \negl(\lambda)$.
\end{MyClaim}
\begin{proof}
Let $\Adv_\RS$ be a QPT adversary attempting to output a forgery w.r.t.\ our $\RS$ scheme such that $x^* \not \in \widetilde{L}$. We build an algorithm $\Adv_\Sig$ against the blind-unforgeability of $\algo{Sig}$. The algorithm $\Adv_\Sig$ (playing the blind-unforgeability game \Cref{expr:BU:ordinary-signature} for $\Sig$) proceeds as follows:
\begin{enumerate}
\item
invoke $\Adv_\RS$ to obtain the $\epsilon$ for the blind-unforgeability game of $\RS$; give this $\epsilon$ to $\Adv_\Sig$'s own challenger for the blind-unforgeability game of $\Sig$;
\item
receive $\widehat{vk}$ from its own challenger; pick an index $j \pick [Q]$ uniformly at random; set $vk_{j} \coloneqq \widehat{vk}$; then, proceeds as in $H_1$ to prepare the rest of the verification keys and continue the execution with $\Adv_\RS$.
\item \label[Step]{step:unforge:sig:signing-query}
when $\Adv_\RS$ sends a quantum signing query $(\algo{sign},i, \sum \psi_{\Ring, m, t}\ket{\Ring, m, t})$, if the specified identity $i$ is not equal to $j$, it proceeds as in $H_1$; otherwise, it uses the blind-unforgeability (for $\Sig$) game's signing oracle $\Sig.\Sign$ to obtain a $\Sig$ signature for the $j$-th party and then continues exactly as in $H_1$; (See the paragraph right after the description of $\Adv_\Sig$.)
\item
if $\Adv_\RS$ tries to corrupt the $j$-th party, $\Adv_\Sig$ aborts; (It is worth noting that the identities are classical. So, $\Adv_\RS$'s quantum access to the signing oracle does not affect this step.)
\item
upon receiving the forgery attempt $\Sigma^*$ from $\Adv_\RS$, $\Adv_\Sig$ decrypts $c_1^*$ using $sk_\LE$ to recover $\sigma_1^*$. If $\algo{Sig.Verify}(vk_j,m^*, \sigma_1^*) \text{ accepts}$, it sets $\widehat{\sigma} := \sigma_1^*$. Otherwise, it decrypts $c_2^*$ with $sk_\LE$ to recover $\sigma_2^*$, and sets $\widehat{\sigma} \coloneqq \sigma_2^*$. It outputs $(m^*,\widehat{\sigma})$.
\end{enumerate}
We first remark that, up to (inclusively) \Cref{step:unforge:sig:signing-query}, $\Adv_\RS$'s view is identical to that in $H_1$. Recall that in $H_1$, the challenger maintains a blindset $B_\epsilon$ such that any $(\Ring, m) \in B^\RS_\epsilon$ will not be answered (this is inherited from $H_0$, which is exactly \Cref{expr:pq:blind-unforgeability}). In contrast, in the execution of $\Adv_\Sig$ described above, $\Adv_\Sig$ first forwards the $\sum \psi_{\Ring, m, t}\ket{\Ring, m, t}$ part of $\Adv_\RS$'s query to its $\Sig.\Sign$ oracle to obtain
$$\sum_{\Ring, m, t} \psi_{\Ring, m, t} \ket{\Ring, m, t\xor \Sig.\Sign(sk_j, \Ring\|m)}, \text{ (note that $sk_j = \widehat{sk}$)}$$ and then performs the remaining computation exactly as in $H_1$. Note that the $\Sig$ signing oracle also maintains a blindset $B^\Sig_\epsilon$ for $\Sig$'s message sapce; and importantly, since the ``messages'' singed by $\Sig$ are of the form $\Ring\|m$, $B^\Sig_\epsilon$ is actually generated identically to $B^\RS_\epsilon$---that is, both of them are generated by including each $(\Ring, m)$ pair in with (the same) probability $\epsilon$.
To finish the proof, we show that
$(\Ring^*\|m^*,\widehat{\sigma})$ is a valid forgery against $\Sig$'s blind-unforgeability game with probability at least $\frac{1}{Q}\big(\Pr\big[x^* \not \in \widetilde{L}\big]- \negl(\lambda)\big)$.
Recall that we are focusing on the case $x^* \notin \tilde{L}$, where $\tilde{L}$ is defined in \Cref{eq:def:language:L-tilde}; without loss of generality, assume that $(m^*, c_1^*, hk_1^*, h_1^*, \algo{R^*}) \not \in \widehat{L}$. Then, observe that due to the way $H_1$ generates the public keys and also the definition of $h_1^*$, we know that
\begin{equation}\label[Expression]{eq:unforge:sig:proof:L4}
\big((m^*,c_1^*,hk_1^*,h_1^*,\algo{R^*}), sk_\LE\big) \in R_4 ~~\text{(recall that $R_4$ is defined in \Cref{def:relation:R4})}.
\end{equation}
Since we assume that $(m^*,c_1^*,hk_1^*,h_1^*,\algo{R^*}) \notin \widehat{L}$, \Cref{eq:unforge:sig:proof:L4} and the definition of $\widehat{L}$ imply the existence of a string $\widetilde{w}$ such that
\begin{equation}\label[Expression]{eq:unforge:sig:proof:L5}
\big((m^*,c_1^*,hk_1^*,h_1^*,\algo{R^*}),\widetilde{w}\big) \in R_5 ~~\text{(recall that $R_5$ is defined in \Cref{def:relation:R5})}.
\end{equation}
By the almost-unique secret key property of $\LE.\GenWithSK$ (\Cref{item:def:LE:computationally-unique-sk}), it follows that $\widetilde{w} = sk_\LE$ (except with negligible probability). Consequently, the following hold:
\begin{enumerate}
\item
by our assumption, $(\Ring^*, m^*) \in B^\RS_\secpar$; this also implies $(\Ring^*, m^*) \in B^\Sig_\secpar$ because $B^\Sig_\epsilon = B^\RS_\epsilon$ as argued earlier;
\item
by \Cref{eq:unforge:sig:proof:L4} and the somewhere perfectly binding property of \algo{SPB}, there exists $\algo{VK^*} = (vk^*,pk^*,\rho^*)$ such that
$$\algo{VK}^* \in \algo{R^*} ~~\text{and}~~ \LE.\Valid(pk^*, sk_\LE) = 1;$$
\item
by \Cref{eq:unforge:sig:proof:L5}, $\LE.\Dec\big(\tilde{w} ~(= sk_\LE),c_1^*\big) = (\sigma^*,vk^*)$ for some $\sigma^*$ and $vk^*$ and $\algo{Sig.Verify}(vk^*, \Ring^*\|m^*,\sigma^*)=1$.
\end{enumerate}
The above means that the $\Adv_\Sig$ uses a verification key $\algo{VK^*} \in \mathsf{R^*}$ and that $c_1^*$ encrypts (among other things) a signature $\sigma^*$ that is valid for the forgery message $\Ring^*\|m^*$ w.r.t. key $vk^*$ for the blind-unforgeability game of $\Sig$. Moreover, $\Adv_\Sig$ can extract this forgery message efficiently by decrypting $c^*$ using $sk_\LE$!
Finally, observe that index $j$ is sampled uniformly.
Therefore, we have that $(\widehat{vk} =)$ $vk_j = vk^*$ with probability $1/Q$.
\end{proof}
\section{Preliminaries}
{\bf Notation.} For a set $\mathcal{X}$, let $2^{\mathcal{X}}$ denote the power set of $\mathcal{X}$ (i.e., the set of all subsets of $\mathcal{X}$. Let $\secpar \in \Naturals$ denote the security parameter.
A non-uniform QPT adversary is defined by $\Set{\mathsf{QC}_\secpar, \rho_\secpar}_{\secpar \in \Naturals}$, where $\Set{\mathsf{QC}_\secpar}_\secpar$ is a sequence of polynomial-size non-uniform quantum circuits, and $\Set{\rho_\secpar}_\secpar$ is some polynomial-size sequence of mixed quantum states. For any function $F: \bits^n \rightarrow \bits^m$, ``quantum access'' will mean that each oracle
call to $F$ grants an invocation of the $(n + m)$-qubit unitary gate $\ket{x,t} \mapsto \ket{x, t \xor F(x)}$; we stipulate that for any $t \in \bits^*$, we have $t\xor\bot = \bot$. Symbols $\cind$, $\sind$ and $\idind$ are used to denote computational, statistical, and perfect indistinguishability respectively. Computational indistinguishability in this work is by default w.r.t.\ non-uniform QPT adversaries.
We provide more preliminaries on lattices and the \cite{ITCS:BraVai14} key-homomorphic evaluation method in \Cref{sec:add-prelim}.
\subsection{Quantum Oracle Indistinguishability}
We will need the following lemmata.
\begin{lemma}[\cite{C:Zhandry12}]\label{lemma:2qwise}
Let $H$ be an oracle drawn from a 2q-wise independent distribution. Then, the advantage of any quantum algorithm making at most $q$ queries to $H$ has in distinguishing $H$ from a truly random function is 0.
\end{lemma}
\begin{lemma}[\cite{C:BonZha13}]\label{lemma:oracleindist}
Let $\mathcal{X}$ and $\mathcal{Y}$ be sets, and for each $x \in \mathcal{X}$, let $D_x$ and $D'_x$ be distributions on $\mathcal{Y}$ such that $|D_x-D'_x| \leq \epsilon$ for some value $\epsilon$ that is independent of $x$. Let $O: \mathcal{X} \rightarrow \mathcal{Y}$ be a function where, for each $x$, $O(x)$ is drawn from $D_x$, and let $O'(x)$ be a function where, for each $x$, $O'(x)$ is drawn from $D'(x)$. Then any quantum algorithm making at most $q$ queries to either $O$ or $O'$ cannot distinguish the two, except with probability at most $\sqrt{8C_0q^3\epsilon}$.
\end{lemma}
\subsection{Blind-Unforgeable Signatures}
We recall in \Cref{def:classical-BU-sig} the definition for blind unforgeable signature schemes in \cite{EC:AMRS20}. The authors there provide a formal definition for MACs. We extend it in the natural way to the signature setting.
\begin{definition}[Blind-Unforgeable Signatures]\label{def:classical-BU-sig}
For any security parameter $\secpar \in \Naturals$, let $\mathcal{M}_\secpar$ denote the message space and $\mathcal{T}_\secpar$ denote the signature space.
A {\em blind-unforgeable} signature scheme \Sig consists of the following PPT algorithms:
\begin{itemize}
\item
$\Gen(1^\secpar)$ outputs a verification and signing key pair $(vk,sk)$.
\item
$\Sign(sk, m;r)$ takes as input a signing key $sk$, a message $m \in \mathcal{M}_\SecPar$, and a randomness $r$ (which we avoid specifying unless pertinent). It outputs a signature $\sigma \in \mathcal{T}_\secpar$.
\item
$\Verify( vk, m, \sigma)$ takes as input a verification key $vk$, a message $m \in \mathcal{M}_\SecPar$ and a signature $\sigma \in \mathcal{T}_\SecPar$. It outputs a bit signifying accept (1) or reject (0).
\end{itemize}
These algorithms satisfy the following requirements:
\begin{enumerate}
\item
{\bf Completeness:} For any $\secpar \in \Naturals$, any $(vk, sk)$ in the range of $\Gen(1^\secpar)$, and any $m\in \mathcal{M}_\secpar$, it holds that $$\Pr[\Verify\big(vk,m,\Sign(sk,m)\big)=1]= 1-\negl(\secpar).$$
\item
{\bf Blind-Unforgeability:} For any non-uniform QPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:BU:ordinary-signature} that
$$\algo{PQAdv}_\textsc{bu}^{\secpar}(\Adv) \coloneqq \Pr\big[\algo{PQExp}_\textsc{bu}^{\secpar}(\Adv) = 1\big] \le \negl(\secpar).$$
\begin{ExperimentBox}[
label={expr:BU:ordinary-signature},
]{Blind-Unforgeability Game \textnormal{$\algo{PQExp}^{\secpar}_\textsc{bu}(\Adv)$}}
\begin{enumerate}[label={\arabic*.}
\item
$\Adv$ sends a constant $0\le \epsilon \le 1$ to the challenger;
\item
The challenger generates $(vk, sk) \leftarrow \Gen(1^\SecPar)$ and provides $vk$ to \Adv.
\item
The challenger defines a {\em blindset} $B^\Sig_\varepsilon \subseteq \mathcal{M}_\SecPar$ as follows: every $m \in \mathcal{M}_\SecPar$ is put in $B^\Sig_\varepsilon$ independently with probability $\varepsilon$.
\item
$\Adv$ is allowed to make $\poly(\secpar)$ quantum queries. For each query, the challenger samples a (classical) random string $r$ and performs the following mapping:
$$\sum_{m,t} \psi_{m,t}| m,t \rangle \mapsto \sum_{m,t} \psi_{m,t}| m,t \xor B^\Sig_\varepsilon\Sign(sk,m;r)\rangle,$$
where $B^\Sig_\varepsilon\Sign(sk,m;r) =
\begin{cases}
\bot & \text{if}~m \in B^\Sig_\varepsilon \\
\Sign(sk,m;r) & \text{otherwise}
\end{cases}
$.
\item
Finally, $\Adv$ outputs $(m^*, \sigma^*)$; the challenger checks if:
\begin{enumerate}
\item
$m^* \in B^\Sig_\epsilon$; {\bf and}
\item
$\Verify(vk, m^*, \sigma^*) = 1$.
\end{enumerate}
If so, the experiment outputs 1; otherwise, it outputs 0.
\end{enumerate}
\end{ExperimentBox}
\item {\bf Shortness (Optional):}The signature scheme is short if the signature size is at most a polynomial on the security parameter {\em and} the logarithm of the message size.
\end{enumerate}
\end{definition}
\begin{remark}[One randomness to rule them all\footnote{Inspired by J.\ R.\ R.\ Tolkien. Indeed, this is a ``ring'' signature paper.}]\label{rmk:single-randomness}
The signing algorithm in our definition samples signing randomness once per every query, as opposed to sampling signing randomness for every classical message in the superposition. This was established as a reasonable definitional choice in \cite{C:BonZha13}, where they observed that one could ``de-randomize'' the signing procedure by simply using a quantum PRF to generate randomness for each possible message in superposition, and use this for signing.
We stick with this convention when defining post-quantum security for both ordinary signatures (\Cref{def:classical-BU-sig}) and ring signatures (\Cref{def:pq:anonymity,def:pq:blind-unforgeability}).
\end{remark}
\begin{remark}
We let the adversary choose $\varepsilon$. This is equivalent to quantifying over all values of $\varepsilon$ as in the definition in \cite{EC:AMRS20}.
\end{remark}
\subsection{Quantum-Access Secure Biased Bit-PRF}
\label{sec:additional-prelims:biased-QPRF}
We will need a {\em quantum-access secure} PRF having a {\em biased single-bit} output. It should also be implementable by $\mathsf{NC}^1$ circuits. Let us first present the definition.
\begin{definition}[Biased Bit-QPRFs]\label{def:biased-bit-PRF}
A biased bit-QPRF on domain $\bits^{n(\secpar)}$ consists of:
\begin{itemize}
\item
$\Gen(1^\secpar, \epsilon)$: takes as input a constant $\epsilon \in [0,1]$, outputs a key $k_\epsilon$;
\item
$\PRF_{k_\epsilon}(x)$: takes as input $x \in \bits^{n(\secpar)}$, outputs a bit $b \in \bits$,
\end{itemize}
such that for any $\epsilon\in[0,1]$ and any QPT $\Adv$ having {\em quantum access} to its oracle,
$$\big| \Pr\big[k_\epsilon \gets \Gen(1^\secpar, \epsilon): \Adv^{\PRF_{k_\epsilon}(\cdot)} = 1\big] - \Pr\big[F \pick \mathcal{F}\big(n(\secpar),\epsilon\big): \Adv^{F(\cdot)} = 1\big]\big|\le \negl(\secpar),$$
where $\mathcal{F}\big(n(\secpar),\epsilon\big)$ is the collection of all functions from $\bits^{n(\secpar)}$ to $\bits$ that output $1$ with probability $\epsilon$.
\end{definition}
It is known that the $\mathsf{NC}^1$ PRF from \cite{EC:BanPeiRos12} is quantum-access secure (i.e., a QPRF) \cite{FOCS:Zhandry12}. It can be made biased by standard techniques (e.g., using the standard QPRF to ``de-randomize'' a $\epsilon$-biased coin-tossing circuit). Note that the \cite{EC:BanPeiRos12} PRF relies on the quantum hardness of LWE with {\em super-polynomial} modulus. It is worthing noticing that such an LWE hardness assumption is stronger than the SIS assumption with
polynomial modulus (see \Cref{def:sis}).
\section{Post-Quantum (Compact) Ring Signatures}
\subsection{Definitions}\label{sec:PQ-ring-sig:def}
\subsubsection{Classical Ring Signatures}
We start by recalling the classical definition of ring signatures \cite{TCC:BenKatMor06,EC:BDHKS19}.
\begin{definition}[Ring Signature]
\label{def:classical:ring-signature}
A ring signature scheme \algo{RS} is described by a triple of PPT algorithms \algo{(Gen,Sign,Verify)} such that:
\begin{itemize}
\item {\bf $\Gen(1^\lambda,N)$:} on input a security parameter $1^\lambda$ and a super-polynomial\footnote{The $N$ has to be super-polynomial to support rings of {\em arbitrary} polynomial size.} $N$ (e.g., $N = 2^{\log^2\secpar}$) specifying the maximum number of members in a ring, output a verification and signing key pair $\algo{(VK,SK)}$.
\item {\bf $\algo{Sign}(\SK,\Ring, m)$:} given a secret key \SK, a message $m \in \mathcal{M}_\lambda$, and a list of verification keys (interpreted as a ring) $\algo{R = (VK_1,\cdots,VK_\ell)}$ as input, and outputs a signature $\Sigma$.
\item {\bf $\algo{Verify}(\Ring,m,\Sigma)$:} given a ring $\algo{R = (VK_1,\dots,VK_\ell)}$, message $m \in \mathcal{M}_\lambda$ and a signature $\Sigma$ as input, outputs either 0 (rejecting) or 1 (accepting).
\end{itemize}
These algorithms satisfy the following requirements:
\begin{enumerate}
\item {\bf Completeness:} for all $\lambda \in \Naturals$, $\ell \le N$, $i^* \in [\ell]$, and $m \in \mathcal{M}_\lambda$, it holds that $\forall i \in [\ell]$ $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N)$ and $\Sigma \gets \algo{Sign}(\SK_{i^*},\Ring,m)$ where $\algo{R = (VK_1,\dots,VK_\ell)}$, we have $\Pr[\algo{RS.Verify}(\Ring,m,\Sigma) = 1] = 1$,
where the probability is taken over the random coins used by $\Gen$ and $\algo{Sign}$.
\item {\bf Anonymity:} For any $Q = \poly(\secpar)$ and any PPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:classical-anonymity} that
$\algo{Adv}_\textsc{Anon}^{\secpar, Q}(\Adv) \coloneqq \big|\Pr\big[\algo{Exp}_\textsc{Anon}^{\secpar, Q}(\Adv) = 1\big] - 1/2\big| \le \negl(\secpar)$.
\begin{ExperimentBox}[label={expr:classical-anonymity}]{Classical Anonymity \textnormal{$\algo{Exp}_\textsc{Anon}^{\secpar, Q}(\Adv)$}}
\begin{enumerate}[label={\arabic*.},leftmargin=*,itemsep=0em]
\item
For each $i \in [Q]$, the challenger generates key pairs $(\VK_i,\SK_i) \gets \Gen(1^\lambda, N;r_i)$. It sends $\Set{(\VK_i, \SK_i, r_i)}_{i\in [Q]}$ to $\Adv$;
\item
$\Adv$ sends a challenge to the challenger of the form $(i_0,i_1,\Ring,m)$.\footnote{We stress that $\Ring$ might contain keys that are not generated by the challenger in the previous step. In particular, it might contain maliciously generated keys.} The challenger checks if $\VK_{i_0} \in \Ring$ and $\VK_{i_1} \in \Ring$. If so, it samples a uniform bit $b$, computes $\Sigma \gets \algo{Sign}(\SK_{i_b},\Ring,m)$, and sends $\Sigma$ to $\Adv$.
\item
$\Adv$ outputs a guess $b'$. If $b' = b$, the experiment outputs 1, otherwise 0.
\end{enumerate}
\end{ExperimentBox}
\item {\bf Unforgeability:} for any $Q=\poly(\lambda)$ and any PPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:classical:unforgeability} that
$\algo{Adv}_\textsc{Unf}^{\secpar, Q}(\Adv) \coloneqq \Pr\big[\algo{Exp}_\textsc{Unf}^{\secpar, Q}(\Adv) = 1\big] \le \negl(\secpar)$.
\begin{ExperimentBox}[label={expr:classical:unforgeability}]{Classical Unforgeability \textnormal{$\algo{Exp}_\textsc{Unf}^{\secpar, Q}(\Adv)$}}
\begin{enumerate}[label={\arabic*.},leftmargin=*,itemsep=0em]
\item For each $i \in [Q]$, the challenger generates $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N;r_i)$, and stores these key pairs along with their corresponding randomness. It then sets $\mathcal{VK} = \Set{\VK_1,\dots,\VK_Q}$ and initializes a set $\mathcal{C} = \emptyset$.
\item The challenger sends $\mathcal{VK}$ to $\Adv$.
\item $\Adv$ can make polynomially-many queries of the following tow types:
\begin{itemize}[leftmargin=*,itemsep=0em,topsep=0em]
\item {\bf Corruption query $(\mathsf{corrupt},i)$:} The challenger adds $\VK_i$ to the set $\mathcal{C}$ and returns the randomness $r_i$ to \Adv.
\item {\bf Signing query $(\algo{sign},i,\Ring,m)$:} The challenger first checks if $\VK_i \in \Ring$. If so, it computes $\Sigma \gets \algo{Sign}(\SK_i,\Ring,m)$ and returns $\Sigma$ to \Adv. It also keeps a list of all such queries made by \Adv.
\end{itemize}
\item Finally, $\Adv$ outputs a tuple $(\Ring^*, m^*, \Sigma^*)$. The challenger checks if:
\begin{itemize}[leftmargin=*,itemsep=0em,topsep=0em]
\item $\Ring^* \subseteq \mathcal{VK \setminus C}$,
\item $\Adv$ never made a signing query of the form $(\algo{sign},\cdot,\Ring^*, m^*)$, {\bf and}
\item $\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$.
\end{itemize}
If so, the experiment outputs 1; otherwise, it outputs 0.
\end{enumerate}
\end{ExperimentBox}
\item {\bf Compactness (Optional):}
the scheme is said to be {\em compact} if the size of a signature is upper-bounded by a polynomial in $\lambda$ and $\log N$.
\end{enumerate}
\end{definition}
We mention that the unforgeability and anonymity properties defined in \autoref{def:classical:ring-signature} correspond respectively to the notions of \emph{unforgeability with insider corruption} and \emph{anonymity with respect to full key exposure} presented in \cite{TCC:BenKatMor06}.
\subsubsection{Defining Post-Quantum Security}
We aim to build a classical ring signature scheme that can protect against adversaries making superposition queries to the singing oracle. Formalizing the security requirements in this scenario is non-trivial. An initial step toward this direction has been taken in \cite{C:CGHKLMPS21}. But their definition has certain restrictions (discussed below). In the following, we develop a new definition built on top of that from \cite{C:CGHKLMPS21}.
\para{Post-Quantum Anonymity.} Recall that in the classical anonymity game (\Cref{expr:classical-anonymity}), the adversary's query is of the quadruple $(i_0,i_1,\Ring,m)$. To define post-quantum anonymity, a natural attempt is to allow the adversary to send a superposition of this quadruple, and to let the challenger respond using the following unitary mapping\footnote{Of course, the challenger also needs to check if $\VK_{i_0} \in \Ring$ and $\VK_{i_1} \in \Ring$. But we can safely ignore this for our current discussion.}:
$$\sum_{i_0,i_1,\Ring,m,t} \psi_{i_0, i_1, \Ring, m, t}\ket{i_0, i_1, \Ring, m, t} \mapsto \sum_{i_0,i_1,\Ring,m,t} \psi_{i_0, i_1, \Ring, m, t}\ket{i_0, i_1, \Ring, m, t \oplus \Sign(\SK_{i_b}, m, \Ring;r)}.$$
However, as observed in \cite{C:CGHKLMPS21}, this will lead to a unsatisfiable definition due to an attack from \cite{C:BonZha13}. Roughly speaking, the adversary could use classical values for $\Ring$, $m$, and $i_1$, but she puts a uniform superposition of all valid identities in the register for $i_0$. After the challenger's signing operation, observe that if $b = 0$, the last register will contain signatures in superposition (as $i_0$ is in superposition); if $b = 1$, it will contain a classical signature (as $i_1$ is classical). These two cases can be efficiently distinguished by means of a Fourier transform on the $i_0$'s register followed by a measurement. Therefore, to obtain an achievable notion, we should not allow superpositions over $(i_0, i_1)$.
Now, $\Adv$ only has the choice to put superpositions over $\Ring$ and $m$. The definition in \cite{C:CGHKLMPS21} further forbids $\Adv$ from putting superposition over $\Ring$. But this is only because they did not know how to prove security if superposition attacks on $\Ring$ is allowed. Indeed, they left it as an open problem to construct a scheme that protects against superposition attacks on $\Ring$. We in this work solve this problem. So, our definition allows superposition attacks on both $\Ring$ and $m$.
\begin{definition}[Post-Quantum Anonymity]
\label{def:pq:anonymity}
Consider a triple of PPT algorithms $\RS = (\algo{Gen}, \Sign, \Verify)$ that satisfies the same syntax as in \Cref{def:classical:ring-signature}. $\RS$ achieves post-quantum anonymity if for any $Q=\poly(\lambda)$ and any QPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:pq:anonymity} that
$$\algo{PQAdv}_\textsc{Anon}^{\secpar,Q}(\Adv) \coloneqq \big|\Pr\big[\algo{PQExp}_\textsc{Anon}^{\secpar,Q}(\Adv) = 1\big] - 1/2\big| \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:pq:anonymity}]{Post-Quantum Anonymity \textnormal{$\algo{PQExp}_\textsc{Anon}^{\secpar,Q}(\Adv)$}}
\begin{enumerate}[leftmargin=*,itemsep=0em]
\item
For each $i \in [Q]$, the challenger generates key pairs $(\VK_i,\SK_i) \leftarrow \RS.\algo{Gen}(1^\SecPar, N;r_i)$. The challenger sends $\Set{(\VK_i, \SK_i, r_i)}_{i\in [Q]}$ to $\Adv$;
\item
$\Adv$ sends $(i_0, i_1)$ to the challenger, where both $i_0$ and $i_1$ are in $[Q]$;
\item
$\Adv$'s challenge query is allowed to be a superposition of rings {\em and} messages. The challenger picks a random bit $b$ and a random string $r$. It signs the message using $\SK_{i_b}$ and randomness $r$, while making sure that $\VK_{i_0}$ and $\VK_{i_1}$ are indeed in the ring specified by $\Adv$. Formally, the challenger implements the following mapping:
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{ \Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor f(\Ring, m)},$$
where
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i_b}, R, m;r) & \text{if}~\VK_{i_0}, \VK_{i_1} \in R \\
\bot & \text{otherwise}
\end{cases}
$.
\item
$\Adv$ outputs a guess $b'$. If $b' = b$, the experiment outputs 1, otherwise 0.
\end{enumerate}
\end{ExperimentBox}
\end{definition}
\para{Post-Quantum Unforgeability.} In the classical unforgeability game (\Cref{expr:classical:unforgeability}), $\Adv$ can make both {\sf corrupt} queries and {\sf sign} queries. \xiao{As mentioned in the intro/tech-overview,} we do not consider quantum {\sf corrupt} queries, because it is unclear what it means to ``corrupt a superposition of all the ring members''. For the same reason, we do not consider superposition attacks over the identity in $\Adv$'s {\sf sign} queries, and only allows the ring and the message to be in superposition.
\begin{xiaoenv}{Superpositions for Unforgeability Game}
We need to argue the requirement of classical identity properly in the intro. Also, for {\sf sign} queries, mention the setting that a single user is corrupted and enforced into quantum behaviors. However, this user only knows his own key; even if he is quantum now, he can only sign with his own key.
\end{xiaoenv}
We remark that in the unforgeability game, \cite{C:CGHKLMPS21} does not allow superpositions over the ring. Instead of a definitional issue, this is again only because they does not know how to prove the security of their construction if superposition attacks on the ring is allowed. In contrast, our construction can be proven secure against such attacks; thus, this restriction is removed from our definition.
In the post-quantum unforgeability game, extra caution is needed to define ``valid'' forgeries. \cite{C:CGHKLMPS21} follows the ``one-more forgery'' definition \xiao{as mentioned in intro/tech-overview/prelim?}. Concretely, it is required that the adversary cannot produce $(\mathsf{sq} + 1)$ valid signatures by making only $\mathsf{sq}$ quantum {\sf sign} queries.
However, in contrast to ordinary signatures, there is one caveat unique to the ring setting. Recall in \Cref{expr:classical:unforgeability} that the adversary's forgery contains a ring $\Ring^*$. This forgery will be considered as valid only if $\Ring^*$ consists of the uncorrupted members (i.e.\ $\Ring^* \subseteq \mathcal{VK}\setminus \mathcal{C}$). A natural generalization of the ``one-more forgery'' approach here is to require that, with $\mathsf{sq}$ quantum signing queries, adversary cannot produce $\mathsf{sq} + 1$ signatures, where all the rings are subsets of $\mathcal{VK}\setminus \mathcal{C}$. But this definition, {\em when restricted to the classical setting}, seems to be weaker than the standard unforgeability in \Cref{def:classical:ring-signature}.\footnote{This is in contrast to the case of ordinary signatures, where ``one-more'' unforgeability is equivalent to the standard existential unforgeability \cite{C:BonZha13}.} That is, in the classical setting, any $\RS$ satisfying the unforgeability in \Cref{def:classical:ring-signature} is also unforgeable w.r.t.\ the ``one-more'' definition; but the reverse direction is unclear. The reason is as follows. \xiao{check Rothit's writing..} To argue that ``one-more'' unforgeability is no weaker than \Cref{def:classical:ring-signature}, one needs to show how to convert a forger $\Adv_\textsc{euf}$ winning in \Cref{expr:classical:unforgeability} to another forger $\Adv_\textsc{om}$ winning in the (classical version of) ``one-more forgery'' game. Conceivably, $\Adv_\textsc{om}$ will run $\Adv_\textsc{euf}$ internally; thus, $\Adv_\textsc{om}$ will make no less {\sf sign} queries than $\Adv_\textsc{euf}$. Recall that $\Adv_\textsc{om}$ needs to forge one more signature than the total number of its queries.
Also, crucially, all the ring signatures presented by $\Adv_\textsc{om}$ at the end must have {\em no} corrupted members in the accompanying ring. Now ideally one might imagine that we can simply use the queries made by $\Adv_\textsc{om}$ (which are really queries by $\Adv_\textsc{euf}$) to meet the ``one-more'' challenge; however, this is thwarted immediately due to the fact that $\Adv_\textsc{euf}$ has absolutely no obligation to make queries meeting this requirement, so even if the final forgery produced by $\Adv_\textsc{euf}$ is valid, our attempted reduction does not have any means to provide $\Adv_\textsc{om}$ with all the signatures it needs to win the ``one-more'' challenge (since not all of the queries can be reused). Indeed, it is not hard to find attacks that use this definitional gap to violate standard unforgeability, while being ruled out as a valid attack against ``one-more'' ring unforgeability. Contrast this with a comparison in the other direction: an adversary $\Adv_\textsc{om}$ for the ``one-more'' security definition is easily converted into a standard $\Adv_\textsc{euf}$ adversary since not all of the signatures output by $\Adv_\textsc{om}$ at the end can be queries (by the pigeonhole principle); $\Adv_\textsc{euf}$ simple outputs the one that is not.
We remark however that this definitional gap between standard ring signature unforgeability and the ``one-more'' version may not be inherent; rather, we just do not know how to meet this gap. Our arguments here should not be interpreted as a proof showing that the former notion is strictly stronger than the latter. We leave it as an open question to either demonstrate a separation, or prove that the two are actually equivalent.
In this work, we present an alternative post-quantum unforgeability definition for ring signatures. Our idea is to extend the blind-unforgeability for ordinary signatures (\Cref{def:classical-BU-sig}) to the ring setting. We present this version in \Cref{def:pq:blind-unforgeability}. In contrast to the ``one-more'' unforgeability, we will show in \Cref{lem:euf-ring-sig:bu-ring-sig} that, when restricted to the classical setting, this blind-unforgeability for ring signatures is indeed equivalent to the standard existential unforgeability in \Cref{def:classical:ring-signature}.
\begin{definition}[Post-Quantum Blind-Unforgeability]
\label{def:pq:blind-unforgeability}
Consider a triple of PPT algorithms $\RS = (\algo{Gen}, \Sign, \Verify)$ that satisfies the same syntax as in \Cref{def:classical:ring-signature}. For any security parameter $\secpar$, let $\mathcal{R}_\secpar$ and $\mathcal{M}_\secpar$ denote the ring space and message space, respectively. $\RS$ achieves blind-unforgeability if for any $Q=\poly(\lambda)$ and any QPT adversary $\Adv$, it holds w.r.t.\ \Cref{expr:pq:blind-unforgeability} that
$$\algo{PQAdv}_\textsc{bu}^{\secpar,Q}(\Adv) \coloneqq \Pr\big[\algo{PQExp}_\textsc{bu}^{\secpar,Q}(\Adv) = 1\big] \le \negl(\secpar).$$
\begin{ExperimentBox}[label={expr:pq:blind-unforgeability}]{Post-Quantum Blind-Unforgeability \textnormal{$\algo{PQExp}_\textsc{bu}^{\secpar,Q}(\Adv)$}}
\begin{enumerate}[leftmargin=*,itemsep=0em]
\item
$\Adv$ sends a constant $0\le \epsilon \le 1$ to the challenger;
\item
For each $i \in [Q]$, the challenger generates $(\VK_i,\SK_i) \gets \Gen(1^\lambda,N;r_i)$, and stores these key pairs along with their corresponding randomness. It then sets $\mathcal{VK} = \Set{\VK_1,\dots,\VK_Q}$ and initializes a set $\mathcal{C} = \emptyset$; The challenger sends $\mathcal{VK}$ to $\Adv$;
\item
The challenger defines a {\em blindset} $B^\RS_\varepsilon \subseteq 2^{\mathcal{R}_\secpar} \times \mathcal{M}_\SecPar$: every pair $(\Ring, m) \in 2^{\mathcal{R}_\secpar} \times \mathcal{M}_\SecPar$ is put in $B^\RS_\varepsilon$ with probability $\varepsilon$;
\item
$\Adv$ can make polynomially-many queries of the following two types:
\begin{itemize}[leftmargin=*,itemsep=0em,topsep=0em]
\item
{\bf Classical corruption query $(\mathsf{corrupt},i)$:} The challenger adds $\VK_i$ to the set $\mathcal{C}$ and returns the randomness $r_i$ to \Adv.
\item
{\bf Quantum Signing query $(\algo{sign},i, \sum \psi_{\Ring, m, t}\ket{\Ring, m, t})$:} That is, $\Adv$ is allowed to query the signing oracle on some classical identity $i$ and superpositions over rings and messages. The challenger samples a random string $r$ and performs the following mapping:
$$\sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t} \mapsto \sum_{\Ring, m, t}\psi_{\Ring, m, t} \ket{\Ring, m, t\xor B^\RS_\epsilon f(\Ring, m)},$$
where $B^\RS_\epsilon f(\Ring, m) \coloneqq
\begin{cases}
\bot & \text{if}~(\Ring, m) \in B^\RS_\epsilon \\
f(\Ring, m) & \text{otherwise}
\end{cases}
$, and
$f(\Ring, m) \coloneqq
\begin{cases}
\RS.\Sign(\SK_{i}, m, \Ring;r) & \text{if}~\VK_{i} \in \Ring \\
\bot & \text{otherwise}
\end{cases}
$.
\end{itemize}
\item Finally, $\Adv$ outputs a tuple $(\Ring^*, m^*, \Sigma^*)$. The challenger checks if:
\begin{itemize}[leftmargin=*,itemsep=0em,topsep=0em]
\item $\Ring^* \subseteq \mathcal{VK \setminus C}$;
\item $\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$; {\bf and}
\item $(\Ring^*,m^*) \in B^\RS_\epsilon$.
\end{itemize}
If so, the experiment outputs 1; otherwise, it outputs 0.
\end{enumerate}
\end{ExperimentBox}
\end{definition}
\begin{lemma}\label{lem:euf-ring-sig:bu-ring-sig}
Restricted to (classical) QPT adversaries, a ring signature $\RS$ scheme is blind-unforgeable (\Cref{def:pq:blind-unforgeability}) if and only if it satisfies the unforgeability requirement in \Cref{def:classical:ring-signature}.
\end{lemma}
\begin{proof}
\xiao{This argument is very similar to \cite[Proposition 2]{EC:AMRS20}. It can go to the appendix.}
We show both sides of this equivalence in turn. In the following, by ``\Cref{expr:pq:blind-unforgeability}'', we refer to the classical version of \Cref{expr:pq:blind-unforgeability}, where the signing query is of the form $(\algo{sign}, i, \Ring, m)$ (i.e., $(\Ring, m)$ is classical), and is answered as $B^\RS_\epsilon f(\Ring, m)$.
\subpara{Necessity ($\Leftarrow$).} Let us first show how blind-unforgeability implies standard unforgeability (for classical settings). Assume we have an adversary $\Adv_\textsc{euf}$ that breaks standard unforgeability of $\RS$ as per \Cref{def:classical:ring-signature}, i.e., in \Cref{expr:classical:unforgeability} it produces a forgery $(m^*,\Ring^*,\Sigma^*)$ that is valid with non-negligible probability $\nu(\SecPar)$. We show that this is easily converted into an adversary $\Adv_\textsc{bu}$ that wins \Cref{expr:pq:blind-unforgeability} with non-negligible probability as well. $\Adv_\textsc{bu}$ first sets $\epsilon(\SecPar)$ equal to $1/p(\SecPar)$, where $p(\secpar)$ denotes the (polynomial) running time of $\Adv_\textsc{euf}$ (the reasoning behind this choice will become clear very soon). It then simply forwards all the queries from $\Adv_\textsc{euf}$ to the blind-unforgeability challenger and the responses back to $\Adv_\textsc{euf}$. It also outputs whatever eventual forgery $\Adv_\textsc{euf}$ does.
Let us consider the success probability of $\Adv_\textsc{bu}$. To start with, note that $\Adv_\textsc{euf}$ makes at most $p(\SecPar)$ many queries of the signing oracle. In each such query, we know that the
$(\Ring, m)$ pair is in the blind set independently with probability $\epsilon$. Thus it is not in the blind set with probability $1-\epsilon$, and if so the query is answered properly. In turn, the probability that all the queries made are answered properly is then at least $(1-\epsilon)^{p(\SecPar)} \approx 1/e$ (this uses independence and $\epsilon = 1/p$), and so the probability that the forgery $(\Ring^*,m^*,\Sigma^*)$ is valid
is then at least $(1-\epsilon)^{p(\SecPar)}\cdot\nu(\SecPar)$.
Finally, the forgery, even if successful, might lie in the blind set with probability $\epsilon$. So, the total probability that $\Adv_\textsc{euf}$ outputs a valid forgery {\em for the blind unforgeability game} is $(1-\epsilon)^{p(\SecPar)+1} \cdot \nu(\SecPar) \approx (1-\epsilon) \cdot \nu \cdot 1/e$, which is non-negligible since $\nu$ is non-negligible by assumption. Thus if $\Adv_\textsc{euf}$ violates standard ring signature unforgeability according to \Cref{def:classical:ring-signature}, then $\Adv_\textsc{bu}$ violates blind unforgeability for ring signatures according to \Cref{def:pq:blind-unforgeability}, as claimed.
\subpara{Sufficiency ($\Rightarrow$).} Let us now turn to the other direction of the equivalence. Assume now that there exists an adversary $\Adv_\textsc{bu}$ that can break blind unforgeability of $\RS$, i.e., win \Cref{expr:pq:blind-unforgeability} with non-negligible probability $\nu(\secpar)$. We show an adversary $\Adv_\textsc{euf}$
that can win \Cref{expr:classical:unforgeability} with non-negligible probability. $\Adv_\textsc{euf}$ simply simulates \Cref{expr:pq:blind-unforgeability} for $\Adv_\textsc{bu}$ by answering oracle queries according to a locally-simulated version of $B^\RS_\epsilon f(\Ring,m)$. Concretely, the adversary $\Adv_\textsc{euf}$ proceeds by drawing a subset $B^\RS_\epsilon$ in the same manner as the challenger in \Cref{expr:pq:blind-unforgeability} and answering queries made by $\Adv_\textsc{bu}$ according to $B^\RS_\epsilon f(\Ring,m)$. Two remarks are in order:
\begin{enumerate}
\item \label{item:euf-ring-sig:bu-ring-sig:remark1}
when $(\Ring, m) \in B^\RS_\secpar$, no signature needs to be done. That is, this query can be answered by $\Adv_\textsc{euf}$ without calling its own signing oracle;
\item
$\Adv_\textsc{euf}$ can construct the set $B^\RS_\epsilon$ by ``lazy sampling'', i.e., when a particular query $(\algo{sign}, i, \Ring, m)$ is made by $\Adv_\textsc{bu}$,
whether $(\Ring, m) \in B^\RS_\epsilon$ and ``remembering'' this information in case the query is asked again.
\end{enumerate}
By assumption, $\Adv_\textsc{bu}$ produces a valid forgery. And it follows from \Cref{item:euf-ring-sig:bu-ring-sig:remark1} that this forgery must be on a point which was not queried by $\Adv_\textsc{euf}$, thus, also serving as a valid forgery for $\Adv_\textsc{euf}$'s game.
\iffalse
At this point we will consider the specific kind of ring signature scheme $\RS$ being used: specifically, whether this scheme has deterministic signatures or not. The overall idea is that $\Adv_\textsc{euf}$ will {\em locally} maintain and administer a blind set of size $\epsilon$ in its interation with $\Adv_\textsc{bu}$; but the exact manner on implementation will depend on what kind of scheme we use. The reason for this distinction will become clear from the following arguments.
Let us first consider the case of a deterministic ring signature $\RS$. In this case, we use essentially the same reduction as in the proof for \cite[Proposition 2]{EC:AMRS20} (they consider such an equivalence only for weak unforgeability - and this is precisely why their reduction works for the deterministic case). Namely, $\Adv_\textsc{euf}$ behaves as follows: it maintains a blind set with respect to messages and rings only. This is equivalent to maintaining a blind set over message, ring and signature triples for deterministic signatures. Whenever $\Adv_\textsc{bu}$ makes a query, $\Adv_\textsc{euf}$ adds this to the blind set with probability $\epsilon$. If the query gets added, $\Adv_\textsc{euf}$ simply replies with $\bot$ and does not forward the query to its own challenger. If not, it forwards the query and sends the response back to $\Adv_\textsc{bu}$ as per usual. Finally, $\Adv_\textsc{bu}$ outputs a forgery, say $(m^*,\Ring^*,\Sigma^*)$. With probability $\nu$, this is good enough to win the blind unforgeability challenge, implying that at the very least this forgery is not with respect to a message to a {\em successful} query. Indeed, nothing prevents $\Adv_\textsc{bu}$ from producing a forgery with respect to any message in $\Adv_\textsc{euf}$'s local blind set, including messages that it queried and recieved no response for. Crucially however, for these messages $\Adv_\textsc{euf}$ never queries its own oracle so even a forgery with respect to such messages suffices to win the standard unforgeability challenge. Thus $\Adv_\textsc{euf}$ can simply use the forgery output by $\Adv_\textsc{bu}$ in its own challenge, and win with probability at least $\nu(\SecPar)$.
Now we turn to the case of randomized ring signatures. For any such scheme $\RS$, since the signing algorithm $\RS.\Sign$ is randomized, we have the property that for any particular message $m$, ring $R$ and signing key $sk$ there are multiple signatures that are possible (depending on the randomness used by the signing algorithm). Define the guessing probability $\eta_{m,R,sk}$ to be the inverse of the number of possible signatures that can be generated by $\RS.\Sign$ given a choice of $(m,R,sk)$ (for most schemes, the number of signatures is usually independent of these choices and depends only on \SecPar). Further define $\eta_\RS = \mathsf{min}_{m,R,sk}\{\eta_{m,R,sk}\}$. By definition, we have that $\eta_\RS \leq 1/2$ for any randomized ring signature scheme $\RS$. We now describe the function of $\Adv_\textsc{euf}$ for this case. It forwards queries from $\Adv_\textsc{bu}$ and responses back to it, as before. Additionally, it maintains a local blind set (consisting of message, ring and signature triples). In contrast to the previous case, $\Adv_\textsc{euf}$ now queries its own (standard unforgeability) challenger on {\em all} queries $\Adv_\textsc{bu}$ makes. Upon obtaining the signature from the challenger, it adds the resulting triple to the blindset with probability $\epsilon$, and only returns $\bot$ to $\Adv_\textsc{bu}$ for such a query. Ultimately $\Adv_\textsc{bu}$ outputs a forgery $(m^*,\Ring^*,\sigma^*)$. This is guaranteed to be a valid forgery for the blinded unforgeability challenge (i.e., i.e., that $\Ring^*$ has no corrupted members, and that $\algo{Verify}(\Ring^*,m^*,\Sigma^*) = 1$) with probability at least $\nu$.
Let us consider two possibilities. Either the forgery $(m^*,\Ring^*,\sigma^*)$ was not queried by $\Adv_\textsc{euf}$ before, in which case it can simply use this forgery and win the standard unforgeability game. Otherwise, $\Adv_\textsc{bu}$ indeed asked for a signature with respect to $(m^*,\Ring^*)$ and obtained no response. In this case, $\Adv_\textsc{bu}$ may have output a signature $\Sigma^*$, and correspondingly $\Adv_\textsc{euf}$ recieved some signature $\Sigma'$ earlier from its challenger. Under our assumptions so far, both of these are valid signatures for $(m^*,\Ring^*)$. Now if $\Sigma^* = \Sigma'$ then $\Adv_\textsc{euf}$ cannot use $\Sigma^*$ as its forgery, since it has already seen it as a query response. However, the probability that $\Sigma^* = \Sigma'$ is at least $\eta_\RS$ from our earlier arguments (recall that $\Adv_\textsc{bu}$ never sees $\Sigma'$) and so we have that in this case the forgery $(m^*,\Ring^*,\Sigma^*)$ can still be used by $\Adv_\textsc{euf}$ with probability $1-\eta_\RS$. It is easy to see that overall, if $\Adv_\textsc{bu}$ succeeds with probability $\nu$, then $\Adv_\textsc{euf}$ still succeeds with probability at least $(1-\eta_\RS)\nu$. This is still non-negligible ($\eta_\RS$ is at most a constant), and so our claim follows.
\fi
\end{proof}
\subsection{Building Blocks}
\subsubsection{Somewhere Perfect Binding Hash}
As in \cite{EC:BDHKS19,C:CGHKLMPS21}, we will only define and employ somewhere perfectly binding hash functions {\em with private local openings}. We present the definition from \cite{C:CGHKLMPS21} with following modification. In \cite{C:CGHKLMPS21} (inherited from \cite{ITCS:HubWic15,EC:BDHKS19}), the $\SPB.\Gen$ algorithm needs a parameter $n$ specifying the size of the database that will be hashed later. Due to a technical reason in our application, we need to make $\SPB.\Gen$ depend only on the {\em upper bound} of supported databases' size. That is, running $\SPB.\Gen$ on an integer $M$ will yield a key that can be used to hash any database of size $\le M$. Fortunately, existing constructions achieve this property by setting $M$ to be super-polynomial on $\secpar$ and padding any database of size $\le M$ with some special symbol (e.g., ``$\bot$'') before hashing. This will not blow up the running time of $\algo{SPB.Hash}$ because the subtree with only ``$\bot$'' leaves is dummy, thus $\algo{SPB.Hash}$ does not really to compute it.
Also, the standard efficiency property of SPB usually requires that the $\algo{SPB.Open}$ and $\algo{SPB.Verify}$ algorithms run in logarithmic time on the actual database size. In contrast, our definition relaxes this to be logarithmic on the upper bound parameter $M$. This already suffices for our application (see \Cref{rmk:SPB:setting-M}).
\begin{remark}[On efficiency]\label{rmk:SPB:setting-M}
In applications (e.g., our \Cref{constr:pq:ring-sig}), one can set, say, $M = 2^{\log^2\secpar}$ such that the SPB scheme can be used to hash any polynomial-size database while still achieving compactness---According to \Cref{item:def:SPB:efficiency} below, if $M = 2^{\log^2\secpar}$, then the efficiency will be $\log(M)\poly(\secpar)=\log^2(\secpar) \poly(\secpar)$, which is effectively independent of the actual size of the database.
\end{remark}
The definition is given as follows. \cite{EC:BDHKS19} builds such a scheme assuming quantum hardness of LWE.
\begin{definition}[SPB Hash \cite{C:CGHKLMPS21}]\label{def:SPB-hash}
A {\em somewhere perfectly binding (SPB) hash} scheme with private local openings consists of a tuple of probabilistic polynomial time algorithms $(\algo{Gen,Hash,Open,Verify})$ with the following syntax:
\begin{itemize}
\item {$\algo{Gen}(1^\lambda, M, \mathsf{ind})$} takes as input an integer $M \le 2^\secpar$ indicating the upper bound of the database size and an index $\mathsf{ind}$; it outputs a key pair $(hk,shk)$.
\item {$\algo{Hash(hk,db)}$,} given a hash key $hk$ and database \algo{db} as input, outputs a hash value $h$.
\item {$\algo{Open}(hk,shk,\algo{db,ind})$,} given a hash key $hk$, secret key $shk$, database \algo{db} and index \algo{ind} as input, outputs a witness $\tau$.
\item {$\algo{Verify}(hk,h,\mathsf{ind},x,\tau)$,} given as input a hash key $hk$, a hash value $h$, an index \algo{ind}, a value $x$ and a witness $\tau$, outputs either 1 (accepting) or 0 (rejecting).
\end{itemize}
To maintain clarity, we will not explicitly specify the block size of databases as input to $\algo{Gen}$, but assume that this is clear from the specific usage and hardwired into the algorithm. Also, for any index $\algo{ind}$ greater than the actual size of a database $\algo{db}$, we use a special symbol ``$\bot$'' to denote the value $\algo{db_{ind}}$. The SPB scheme should satisfy the following properties:
\begin{enumerate}
\item {\bf Correctness:} for any $\lambda \in \Naturals$, any $M\le 2^\secpar$, any database \algo{db} of size $\le M$, and any index $\algo{ind} \in [M]$, we have,
$$
\Pr[
\begin{array}{l}
(hk,shk) \gets \algo{Gen}(1^\lambda, M,\algo{ind});\\
h \gets \algo{Hash}(hk,\algo{db}); \\
\tau \gets \algo{Open}(hk,shk,\algo{db,ind})
\end{array}:
\algo{Verify}(hk,h,\algo{ind}, \algo{db_{ind}},\tau) = 1
] = 1.
$$
\item \label{item:def:SPB:efficiency}
{\bf Efficiency:} all hash keys $\algo{hk}$ generated by $ \Gen(1^\secpar, M, \algo{ind})$ and all witnesses $\tau$ generated by $\algo{Open}(hk,shk, \algo{db}, \algo{ind})$ where $|\algo{db}| \le M$ are of size $\log(M)\poly(\lambda)$. Further, for databases of size $\le M$, $\algo{Verify}$ can be computed by a circuit of size $\log(M)\poly(\lambda)$.
\item {\bf Somewhere perfectly binding:} for all $\lambda\in \Naturals$, all $M \le 2^\secpar$, all databases $\algo{db}$ of size $\le M$, all indices $\algo{ind} \in [M]$, all purported hashing keys $hk$, all purported witnesses $\tau$, and all $h$ in the support of $\algo{Hash}(hk,\algo{db})$, it holds that
$$\algo{Verify}(hk,h,\algo{ind},x,\tau)=1 ~\Rightarrow~ x=\mathsf{db_{ind}}.$$
\item {\bf Index hiding:} for any $M\le 2^\secpar$ and any $\algo{ind}_0,\algo{ind}_1 \in [M]$,
\[ \big\{(hk,shk) \gets \algo{Gen}(1^\lambda, M, \algo{ind}_0): hk \big\}_{\secpar \in \Naturals} \cind \big\{(hk,shk) \gets \algo{Gen}(1^\lambda,M,\algo{ind}_1) : hk \big\}_{\secpar \in \Naturals} \]
\end{enumerate}
\end{definition}
\subsubsection{Lossy PKE with Oblivious Key Generation}
\xiao{@Giulio: can you cite the correct work that build a lossy encryption (probably with slight modifications) satisfying \Cref{def:special-LE}?}
\begin{definition}[Special Lossy PKE]\label{def:special-LE}
For any security parameter $\secpar \in \Naturals$, let $\mathcal{M}_\secpar$ denote the message space, and $\mathcal{SK}_\secpar$ denote the {\em PPT-samplable} secret key space.
A special lossy public-key encryption scheme $\algo{LE}$ consists of the following PPT algorithms:
\begin{itemize}
\item
$\algo{GenWithKey}(1^\secpar, \sk)$, on input a secret key $\sk \in \mathcal{SK}_{\lambda}$, outputs $\pk$, which we call {\em injective public key}.
\item
$\KSam^\ls(1^\secpar)$ outputs key $\pk_\ls$, which we call {\em lossy public key}.
\item
$\algo{Valid}(\pk,\sk)$, on input a public $\pk$ and a secret key $\sk$, outputs either $1$ (accepting) or $0$ (rejecting).
\item
$\RndExt(\pk)$ outputs a $r$ which we call {\em extracted randomness}.
\item
$\algo{Enc}(\pk, m)$, on input a public key $\pk$, and a message $m \in \mathcal{M}_\secpar$, outputs $\ct$.
\item
$\algo{Dec}(\sk, \ct)$, on input a secret key $\sk$ and a ciphertext $ct$, outputs $m$.
\end{itemize}
These algorithms satisfy the following properties:
\begin{enumerate}
\item \label{item:def:LE:completeness}
{\bf Completeness.} For any $\secpar\in\Naturals$, any $(\pk, \sk)$ s.t.\ $\Valid(\pk, \sk) =1$, and any $m \in \mathcal{M}_\secpar$, it holds that $\Pr[\Dec\big(\sk, \Enc(\pk, m)\big)= m] = 1$.
\item \label{def:LE:property:lossyness}
{\bf Lossiness of lossy keys.} For any $\pk_\ls$ in the range of $\KSam^\ls(1^\secpar)$ and any $m_0, m_1 \in \mathcal{M}_\secpar$,
$$\big\{ \Enc(\pk_\ls, m_0) \big\}_{\secpar\in\Naturals} \sind \big\{ \Enc(\pk_\ls, m_1) \big\}_{\secpar\in\Naturals}.$$
\item \label{def:LE:property:GenWithSK:completeness}
{\bf Completeness of $\GenWithSK$:} It holds that
$$\Pr[\sk\pick \mathcal{SK}_\secpar; \pk \gets \GenWithSK(\sk): \Valid(\pk,\sk) =1 ] \ge 1 - \negl(\secpar).$$
\item \label{def:LE:property:IND:GenWithSK-KSam}
{\bf IND of $\GenWithSK$/$\KSam^\ls$ mode:} For any $Q=\poly(\secpar)$, the following two distributions are computationally indistinguishable:
\begin{itemize}
\item
$\forall i\in [Q]$, sample $\pk_i \gets \KSam^\ls(1^\secpar; r_i)$, then output $\Set{\pk_i, r_i}_{i\in [Q]}$;
\item
Sample $sk \pick \mathcal{SK}_\secpar$; $\forall i\in [Q]$, sample independently $pk_i \gets \GenWithSK(1^\secpar, sk)$; then, output $\Set{\pk_i, \algo{RndExt}(\pk_i)}_{i\in [Q]}$.
\end{itemize}
\item \label{item:def:LE:computationally-unique-sk}
{\bf Almost-Unique Secret Key:} It holds that
$$
\Pr[
\begin{array}{l}
sk \pick \mathcal{SK}_\secpar;\\
pk \gets \GenWithSK(1^\secpar, sk);
\end{array}:
\begin{array}{l}
\text{There exists an}~sk'~\text{such that} \\
sk' \ne sk ~\wedge~ \Valid(pk, sk') = 1
\end{array}
] = \negl(\secpar).
$$
\end{enumerate}
\end{definition}
\subsubsection{Compact ZAPs for Super-Complement Languages}
\para{The Role of ZAPs.} \cite{C:CGHKLMPS21} uses a (special) ZAP for security. It plays a similar role as that of NIZK in the Naor-Yung paradigm \cite{STOC:NaoYun90}: the signing algorithm generates two ciphertexts $(c_1, c_2)$, where $c_1$ encrypts a real signature using the ring and identity specified by the adversary, and $c_2$ encrypts a dummy signature. To establish anonymity, $c_1$ will be switched to an encryption of a signature using another identity. This can be done relying on the WI property of ZAP in a similar manner as in \cite{STOC:NaoYun90}. To establish unforgeability, the hybrid will use the secret key to extract the forgery signature from $c_1$ or $c_2$ output by the adversary, which is used to break the unforgeability of the building-block (ordinary) signature scheme $\Sig$. The soundness of ZAP plays an important role in this argument to ensure that at least one of $(c_1, c_2)$ encrypts a valid signature.
However, ZAPs from \NP are currently not known from LWE. \cite{C:CGHKLMPS21} bypasses this obstacle by constructing a ZAP for $\NP \cap\coNP$ assuming LWE, and shows that it suffices for the ring signature application. Roughly, this is because that if the forgery does not contain a valid signature w.r.t.\ the building-block $\Sig$, then the secret key serves as a non-witness. That is, the hybrid can decrypt both $(c_1, c_2)$ and tell that neither of the plaintexts are a valid $\Sig$ signature. This means that the statement proven by the ZAP really lies in $\NP\cap \coNP$.
\para{The Super-Complement Languages.} To obtain a compact construction, the signature size should be sub-polynomial on the ring size. In particular, the size of the ZAP proof should be sub-polynomial on the ring size. To capture this property, \cite{C:CGHKLMPS21} abstracts out the following notion of {\em super-complement languages}, which can be interpreted as a generalization of $\NP\cap\coNP$ where an (augmented) complement langauge is specified explictly. As it will become clear in \Cref{sec:building-blocks:language}, this helps to split the ring and the other sub-polynomial-size part in a signature. In the following, we follow the \cite{C:CGHKLMPS21} framework to present the definition of super-complement languages and ZAPs for them. Then, We will instantiate in \Cref{sec:building-blocks:language} a concrete super-complement language used in our ring signature construction.
\begin{definition}[Super-Complement \cite{C:CGHKLMPS21}]\label{def:super-complement-language}
Let $(L, \widetilde{L})$ be two $\NP$ languages where the elements of $\widetilde{L}$ are represented as pairs of bit strings. We say $\widetilde{L}$ is a \emph{super-complement} of $L$, if
$\widetilde{L} \subseteq (\bits^*\setminus L) \times \bits^*$.
I.e., $\widetilde{L}$ is a super complement of $L$ if for any $x= (x_1,x_2)$, $x \in \widetilde{L} \Rightarrow x_1 \not \in L$.
\end{definition}
We now define ZAPs for super-complement languages consisting of pairs of the form $(L,\widetilde{L})$. Notice that, while the complement of $L$ might not be in $\NP$, it must hold that $\widetilde{L} \in \NP$. The language $\widetilde{L}$ is used to define the soundness property. Namely, producing a proof for a statement $x = (x_1, x_2) \in \widetilde{L}$, should be hard. We also use the fact that $\widetilde{L} \in \NP$ to mildly strengthen the soundness property. In more detail, instead of having selective soundness where the statement $x \in \widetilde{L}$ is fixed in advance, we now fix a non-witness $\widetilde{w}$ and let the statement $x$ be adaptively chosen by the malicious prover from all statements which have $\widetilde{w}$ as a witness to their membership in $\widetilde{L}$.
\begin{definition}[ZAPs for Super-Complement Languages \cite{C:CGHKLMPS21}]
\label{def:generalizedzap-interface}
Let $L,\widetilde{L} \in \NP$ s.t.\ $\widetilde{L}$ is a super-complement of $L$. Let $R$ and $\widetilde{R}$ denote the $\NP$ relations corresponding to $L$ and $\widetilde{L}$ respectively. Let $\{C_{n,\ell}\}_{n,\ell}$ and $\{\widetilde{C}_{\tilde{n},\tilde{\ell}}\}_{\tilde{n},\tilde{\ell} }$ be the $\NP$ verification circuits for $L$ and $\widetilde{L}$ respectively. Let $\widetilde{d} = \widetilde{d}(\tilde{n},\tilde{\ell})$ be the depth of $\widetilde{C}_{n,\ell}$. A \emph{compact relaxed ZAP} for~$L,\widetilde{L}$
is a tuple of PPT algorithms
$(\algo{V}, \algo{P}, \algo{Verify})$ having the following
interfaces (where $1^{n}, 1^{\lambda}$ are implicit inputs to
$\algo{P}$, $\algo{Verify}$):
\begin{itemize}
\item
{\bf $\algo{V}(1^{\lambda}, 1^n,1^{\widetilde{\ell}}, 1^{\widetilde{D}})$:}
On input a security parameter $\lambda$,
statement length~$n$ for $L$,
witness length $\widetilde{\ell}$ for $\widetilde{L}$,
and $\NP$ verifier circuit depth upper-bound $\widetilde{D}$ for $\widetilde{L}$, output a first message
$\rho$.
\item
{\bf $\algo{P}\big(\rho,x = (x_1,x_2),w\big)$:} On input a string~$\rho$, a statement $(x_1 \in \bits^{n},x_2)$, and a witness $w$ such that $(x_1,w) \in R$, output a proof~$\pi$.
\item
{\bf $\algo{Verify}\big(\rho,x = (x_1,x_2),\pi\big)$:} On input a string $\rho$, a
statement $x$, and a proof $\pi$, output either 1 (accepting) or 0 (rejecting).
\end{itemize}
The following requirements are satisfied:
\begin{enumerate}
\item {\bf Completeness:} For every $(x_1,w) \in L$, every $x_2 \in \bits^*$,
every $\widetilde{\ell} \in \Naturals$,
every $\widetilde{D} \ge \widetilde{d}(\abs{x_1} + \abs{x_2},\widetilde{\ell})$,
and every
$\lambda \in \Naturals$, \[\Pr[\algo{Verify}\big(\rho,x = (x_1,x_2),\pi\big)=1 ] = 1, \]
where
$\rho \gets \algo{V}(1^{\lambda}, 1^{\abs{x_1}},1^{\widetilde{\ell}}, 1^{\widetilde{D}})$ and
$\pi \gets \algo{P}(\rho,x,w)$.
\item {\bf Public coin:}
$\algo{V}(1^{\lambda}, 1^n,1^{\widetilde{\ell}}, 1^{\widetilde{D}})$ simply outputs a uniformly
random string.
\item \label{item:def:zap:soundness}
{\bf Selective non-witness adaptive-statement soundness:}
For any non-uniform QPT machine $P^*_{\lambda}$, any $n,\widetilde{D} \in \Naturals$, and any non-witness $\widetilde{w} \in \bits^{*}$, the following holds
\begin{equation*}
\Pr[
\begin{array}{l}
\rho \gets \algo{V}(1^{\lambda}, 1^{n},1^{\abs{\widetilde{w}}}, 1^{\widetilde{D}});\\
\big(x = (x_1,x_2),\pi^*\big) \gets P^*_{\lambda}(\rho)
\end{array}:
\begin{array}{l}
\algo{Verify}(\rho,x,\pi^*) = 1 ~\wedge\\
\widetilde{D} \ge \widetilde{d}(\abs{x}, \abs{\widetilde{w}}) ~\wedge~ (x,\widetilde{w}) \in \widetilde{R}
\end{array} ] \le \negl(\SecPar).
\end{equation*}
\item {\bf Statistical witness indistinguishability:} For every (possibly unbounded) ``cheating''
verifier $V^{*}=(V^{*}_{1}, V^{*}_{2})$ and every $n,\widetilde{\ell},\widetilde{D} \in \Naturals$, the
probabilities
\[ \Pr[V^{*}_{2}(\rho,x,\pi,\zeta)=1 ~\wedge~ (x,w) \in \mathcal{R} ~\wedge~ (x,w') \in \mathcal{R} ] \] in the
following two experiments differ only by $\negl(\lambda)$:
\begin{itemize}
\item in experiment 1,
$(\rho,x,w,w',\zeta) \gets
V^{*}_{1}(1^{\lambda}, 1^{n},1^{\widetilde{\ell}}, 1^{\widetilde{D}}), \pi \gets \algo{P}(\rho,x,w)$;
\item in experiment 2,
$(\rho,x,w,w',\zeta) \gets
V^{*}_{1}(1^{\lambda}, 1^{n},1^{\widetilde{\ell}}, 1^{\widetilde{D}}), \pi \gets \algo{P}(\rho,x,w')$.
\end{itemize}
\item {\bf Compactness (Optional):} The ZAP is {\em compact} if the size of the proof $\pi$ (output by $\algo{P}$) is a fixed polynomial in $n$, $\widetilde{\ell}$, $\widetilde{D}$, $|C|$, and $\lambda$. In particular, it is independent of the size of $\widetilde{C}$ and $x_2$.
\end{enumerate}
\end{definition}
\cite{C:CGHKLMPS21} constructs such an ZAP assuming LWE. We remark that \cite{C:CGHKLMPS21} only claims soundness (\Cref{item:def:zap:soundness}) against non-uniform PPT adversaries. But it is straightforward to see that their proof for soundness also works against non-uniform QPT adversaries assuming the quantum hardness of LWE.
\begin{lemma}[\cite{C:CGHKLMPS21}]\label{lem:ZAP:from:QLWE}
Assuming QLWE, there exist compact ZAPs (as per \Cref{def:generalizedzap-interface}) for any super-complement language (as per \Cref{def:super-complement-language}).
\end{lemma}
\subsection{Construction}
Our construction, shown in \Cref{constr:pq:ring-sig}, relies on the following building blocks. Note that except for \Cref{item:RS:building-block}, which is a information-theoretical primitive, all of the others can be constructed assuming QLWE.
\begin{enumerate}
\item \label{item:RS:building-block}
Pair-wise independent functions;
\item
An blind-unforgeable signature scheme $\algo{Sig}$ satisfying \Cref{def:classical-BU-sig};
\item
A lossy public key encryption scheme $\LE$ satisfying \Cref{def:special-LE};
\item
A somewhere perfectly binding hash function $\algo{SPB}$ satisfying \Cref{def:SPB-hash};
\item
A compact ZAP scheme $\algo{ZAP}$ satisfying \Cref{def:generalizedzap-interface}.
\end{enumerate}
We remark that the $\RS.\Sign$ algorithm runs $\ZAP$ on a special super-complement language $(L, \tilde{L})$, whose definition will appear after the construction in \Cref{sec:building-blocks:language}. Such an arrangement is because we find that the language $(L, \tilde{L})$ will become easier to understand once the reader has the big picture of \Cref{constr:pq:ring-sig}.
\begin{xiaoenv}{Explain the difference with \cite{C:CGHKLMPS21}:}
Explain the difference with \cite{C:CGHKLMPS21}:
\begin{itemize}
\item
we replace PKE with $\LE$ to obtain post-quantum anonymity.
\item
the ZAP must be statistical WI, while \cite{C:CGHKLMPS21} only needs computational WI\footnote{Although \cite{C:CGHKLMPS21} builds and employs a ZAP with statistical WI in their ring signature, computational WI already suffices for their security proof.}.
\item
We need pair-wise independent hash function to invoke Zhandry's lemma.
\item
Need to make the $\SPB.\Gen(1^\secpar, M, i)$ use a universal upper bound for database size, independent of $|\Ring|$. Thus, the \SPB keys are generated only once, instead of one time for each $\Ring$'s in superposition. I think we can use $M$ as the universal upper bound. But we need to make sure that a $\SPB.\Gen(1^\secpar, M, i)$ can be used to hash $\Ring$ with $|\Ring| \le M$. I think this could be done by padding $\Ring$ with dummies to size $M$. Note that the signature size scales with $\log(M)$.
\item
The $\algo{Sig}$ in our construction should be BU secure.
\item
I need to sign it as $\algo{Sig}.\Sign(sk, \Ring \| m)$, where $\Sig$ is a short signature scheme.
\end{itemize}
\end{xiaoenv}
\begin{construction}
\label{constr:pq:ring-sig}
Let $\widetilde{D} = \widetilde{D}(\lambda, N)$
be the maximum depth of the $\algo{NP}$ verifier circuit for language $\widetilde{L}$ restricted to statements where the the ring has at most $N$ members, and the security parameter for $\algo{SPB}$, \Sig, and $\algo{LE}$ is $\lambda$. Let $n = n(\lambda,\log N)$ denote the maximum size of the statements of language $L$ where the ring has at most $N$ members and the security parameter is $\lambda$. Recall that for security parameter $\lambda$, secret keys in $\LE$ have size $\widetilde{\ell} = \ell_{\sk}(\lambda)$. We now describe our ring signature construction:
\subpara{Key Generation Algorithm $\algo{Gen}(1^\lambda,N)$:}
\begin{itemize}[topsep=0.2em,itemsep=0em]
\item sample signing/verification key pair: $(vk,sk) \gets \algo{Sig.Gen}(1^\lambda)$;
\item sample obliviously an injective public key of $\algo{LE}$: $pk \gets \LE.\KSam^\ls(1^\secpar)$;
\item compute the first message $\rho \leftarrow \algo{ZAP.V}(1^\lambda,1^n,1^{\widetilde{\ell}},1^{\widetilde{D}})$ for the relaxed ZAP scheme;
\item output the verification key $\VK\coloneqq (vk,pk,\rho)$ and signing key $\SK\coloneqq (sk,vk,pk,\rho)$.
\end{itemize}
\subpara{Signing Algorithm $\algo{Sign}(\SK,\Ring, m)$:}
\begin{itemize}[topsep=0.2em,itemsep=0em]
\item parse $\Ring = (\VK_1,\dots,\VK_{\ell})$; and parse $\SK=(sk,vk,pk,\rho)$;
\item compute $\sigma \gets \algo{Sig.Sign}(sk,\Ring\|m)$; \xiao{this could lead a problem to define $L$ and $\tilde{L}$.}
\item let $\VK \coloneqq \VK_i \in R$ be the verification key corresponding to $\SK$;
\item let $M = 2^{\log^2\secpar}$ (see \Cref{rmk:SPB:setting-M}); sample independently two pairs of hash keys:\vspace{-0.7em}
$$(hk_1,shk_1) \gets \algo{SPB.Gen}(1^\lambda,M,i),~(hk_2,shk_2) \gets \algo{SPB.Gen}(1^\lambda,M,i),\vspace{-0.7em}$$ and compute $h_1 \gets \algo{SPB.Hash}(hk_1,\Ring)$ and $h_2 \gets \algo{SPB.Hash}(hk_2,\Ring)$;
\item compute the opening $\tau_1 \gets \algo{SPB.Open}(hk_1,shk_1,\Ring,i)$ to position $i$;
\item
sample tow pairwise-independent functions $\PI_1$ and $\PI_2$, and compute
\vspace{-0.8em}$$r_{c_1} = \PI_1(\Ring\|m), ~~r_{c_2} = \PI_2(\Ring\|m).\vspace{-0.8em}$$
\item compute $c_1 \gets \algo{LE.Enc}(pk,(\sigma,vk);r_{c_1})$ and $c_2 \gets \algo{LE.Enc}(pk,0^{|\sigma|+|vk|};r_{c_2})$;
\item
let $\VK_1 = (vk_1,pk_1,\rho_1)$ denote the lexicographically smallest member of $\Ring$ (as a string; note that this is necessarily unique);
\item
fix statement $x_1=(m,c_1,c_2,hk_1,hk_2,h_1,h_2)$, witness $w=(vk,pk,i,\tau_1,\sigma,r_{c_1})$, and statement $x_2 = \Ring$. We remark that these statements and witness correspond to a super-complement language $(L, \tilde{L})$ that will be defined in \Cref{sec:building-blocks:language}. Looking ahead, $x_1$ with witness $w$ is a statement in the $L$ defined in \Cref{eq:def:language:L}; $(x_1, x_2)$ constitute a statement that is {\em not} in the $\tilde{L}$ defined in \Cref{eq:def:language:L-tilde}.
\item sample another pairwise-independent function $\PI_3$ and compute $r_\pi = \PI_3(\Ring\|m)$;
\item compute $\pi \gets \algo{ZAP.P}(\rho_1,x=(x_1,x_2),w; r_\pi)$;
\item output $\Sigma = (c_1,hk_1,c_2,hk_2,\pi)$.
\end{itemize}
\subpara{Verification Algorithm $\algo{Verify}(\Ring,m, \Sigma)$:}
\begin{itemize}[topsep=0.2em,itemsep=0em]
\item identify the lexicographically smallest verification key $\VK_1$ in $\Ring$;
\item compute $h'_1 = \algo{SPB.Hash}(hk_1,\Ring)$ and $h'_2 = \algo{SPB.Hash}(hk_2,\Ring)$;
\item fix $x_1=(m,c_1,c_2,hk_1,hk_2,h'_1,h'_2)$ and $x_2 = \Ring$; read $\rho_1$ from $\VK_1$;
\item compute and output $\algo{ZAP}.\algo{Verify}(\rho_1,x,\pi)$.
\end{itemize}
\end{construction}
\subsubsection{The Super-Complement Language Proven by the ZAP}
\label{sec:building-blocks:language}
We now define the super-complement language $(L, \widetilde{L})$ used in \Cref{constr:pq:ring-sig}. It is similar to the $(L, \tilde{L})$ defined in \cite[Section 5]{C:CGHKLMPS21}, but with necessary modifications to accommodate \Cref{constr:pq:ring-sig}.
For a statement of the form $y_1 = (m,c,hk,h)$ and witness $w = \big(\mathsf{VK}=(vk,pk,\rho),i,\tau,\sigma,r_c \big)$, define relations $R_1,R_2$ and $R_3$ as follows:
\begin{align*}
(y_1,w) \in R_1 & ~\Leftrightarrow~ \algo{SPB.Verify}(hk,h,i,\mathsf{VK},\tau)=1\\
(y_1,w) \in R_2 & ~\Leftrightarrow~ \algo{LE.Enc}\big(pk,(\sigma,vk);r_c\big) = c\\
(y_1,w) \in R_3 & ~\Leftrightarrow~ \algo{Sig.Verify}(vk,m,\sigma)=1 ~~~~\red{(\Ring\|m)???}
\end{align*}
Next, define the relation $R'$ as $R'\coloneqq R_1\cap R_2 \cap R_3$. Let $L'$ be the language corresponding to $R'$. Define language $L$ as
\begin{equation}\label{eq:def:language:L}
L \coloneqq \bigg\{(m,c_1,c_2,hk_1,hk_2,h_1,h_2) ~\bigg|~
\begin{array}{l}
(m,c_1,hk_1,h_1) \in L' ~\vee \\
(m,c_2,hk_2,h_2) \in L'
\end{array}\bigg\}.
\end{equation}
Now, we define another language $\widetilde{L}$ and prove that it is a super-complement of $L$ in \Cref{lemma:ring-sig:complement}. Let $x_2 \coloneqq \algo{R}$ where $\Ring = (\algo{VK_1,\dots,VK_\ell})$. Let $y \coloneqq (y_1,x_2)$, and $\widetilde{w} \coloneqq s$. Define the following relations:
\begin{align}
(y,\widetilde{w})\in R_4 & ~\Leftrightarrow~ \big(\forall j \in [\ell]: \algo{LE.Valid}(pk_j,s)=1\big) \wedge \big(h = \algo{SPB.Hash}(hk,\algo{R})\big) \label{def:relation:R4}\\
(y,\widetilde{w})\in R_5 & ~\Leftrightarrow~
\left\{
\begin{array}{l}
\algo{LE.Dec}(s,c) = (\sigma,vk) \wedge \algo{Sig.Verify}(vk, m, \sigma)=1 ~\wedge \\
\exists\algo{VK} \in \mathsf{R}: \algo{VK} = (vk,pk,\rho) \text{ for some } pk \text{ and } \rho
\end{array}
\right. \label{def:relation:R5}
\end{align}
where, for each $j \in [\ell]$, $\algo{VK}_j = (vk_j, pk_j, \rho_j)$. Let $L_4$ and $L_5$ be the languages corresponding to $R_4$ and $R_5$, respectively. Define further the relation $\widehat{R}$ according to $\widehat{R} \coloneqq R_4 \setminus R_5$, and let $\widehat{L}$ be the corresponding language. Define $\tilde{L}$ as follows:
\begin{equation}\label{eq:def:language:L-tilde}
\widetilde{L} \coloneqq \bigg\{(
\underbrace{m,c_1,c_2,hk_1,hk_2,h_1,h_2}_{\coloneqq x_1},\underbrace{\mathsf{R}{\color{white})}}_{\coloneqq x_2}) ~\bigg|~
\begin{array}{l}
(m,c_1,hk_1,h_1,\algo{R}) \in \widehat{L} ~\wedge \\
(m,c_2,hk_2,h_2,\algo{R}) \in \widehat{L}
\end{array}
\bigg\}.
\end{equation}
The next claim shows that $\tilde{L}$ is indeed the super-complement of $L$.
\begin{MyClaim}
\label{lemma:ring-sig:complement}
If $\algo{SPB}$ is somewhere perfectly binding and $\LE$ is complete (\Cref{item:def:LE:completeness}), then the $\widetilde{L}$ (\Cref{eq:def:language:L-tilde}) is a super-complement of $L$ (\Cref{eq:def:language:L}).
\end{MyClaim}
\begin{proof}
It is straightforward to see that $L, \tilde{L} \in \NP$. To prove this lemma, we need to show that for any statement $x$ of the following form
\begin{equation}\label{eq:ring-sig:complement:statement:x}
x = \big(x_1 =(m,c_1,c_2,hk_1,hk_2,h_1,h_2),x_2 = \algo{R}\big),
\end{equation}
it holds that $x \in \tilde{L} \Rightarrow x_1 \notin L$ (see \Cref{def:super-complement-language}). In the following, we finish the proof by showing the contrapositive: $x_1 \in L \Rightarrow x\notin \tilde{L}$.
For any $x$ as in \Cref{eq:ring-sig:complement:statement:x}, we define
$$y_1 \coloneqq (m,c_1,hk_1,h_1) ~\text{and}~ y'_1 \coloneqq (m,c_2,hk_2,h_2).$$
To prove ``$x_1 \in L \Rightarrow x\notin \tilde{L}$'', it suffices to show that the following \Cref{eq:ring-sig:complement:expression1,eq:ring-sig:complement:expression2} hold for every $w= (\algo{VK}= (vk,pk,\rho),i,\tau,\sigma,r_c)$ and every $\widetilde{w} = s$:
\begin{align}
&(y_1,w) \in R' \wedge \big((y_1,x_2),\widetilde{w}\big) \in R_4 ~\Rightarrow~ \big((y_1,x_2), \widetilde{w} \big) \in R_5 \label[Expression]{eq:ring-sig:complement:expression1}\\
&(y_1',w) \in R' \land \big((y_1',x_2),\widetilde{w}\big) \in R_4 ~\Rightarrow~ \big((y_1',x_2), \widetilde{w} \big) \in R_5. \label[Expression]{eq:ring-sig:complement:expression2}
\end{align}
We first prove \Cref{eq:ring-sig:complement:expression1}. If $(y_1,w) \in R_1$ and $\big((y_1,x_2),\widetilde{w}\big) \in R_4$, then, by the somewhere perfectly binding of $\algo{SPB}$, we have $\algo{VK} \in \algo{R}$. Now, notice that by the completeness property (\Cref{item:def:LE:completeness}) of $\algo{LE}$, since $\algo{LE.Valid}(pk,s) = 1$, if $(y_1,w) \in R_2 \cap R_3$, then $(y_1,\widetilde{w}) \in R_5$.
\Cref{eq:ring-sig:complement:expression2} can be proven similarly. This finish the proof of \Cref{lemma:ring-sig:complement}.
\end{proof}
\subsection{Proving Post-Quantum Security}
We now prove that \Cref{constr:pq:ring-sig} is a post-quantum secure post-quantum ring signature satisfying \Cref{def:pq:ring-signatures}. Its completeness follows straightforwardly from the completeness of $\algo{SPB}$, $\ZAP$, and $\algo{Sig}$. For compactness, notice that the only steps in \Cref{constr:pq:ring-sig} that could possibly depend on the ring size is the \ZAP proof and the $\Sig.\Sign(sk, \Ring\| m)$.
, the compactness of \Cref{constr:pq:ring-sig} follows from the compactness of $\ZAP$. Next, we prove the post-quantum anonymity and unforgeability in \Cref{sec:prove:pq:anonymity} and \Cref{sec:prove:pq:blind-unforgeability}, respectively.
\section{Technical Overview}
\red{tech overview goes here...}
\section{Technical Overview}
\subsection{BU Signatures in the QROM}\label{sec:tech-overview:BU:QROM}
We show that the GPV signature scheme from \cite{STOC:GenPeiVai08} is BU-secure in the QROM.
The GPV signature scheme follows the hash-and-sign paradigm and relies crucially on the notion of {\em preimage sampleable functions} (PSFs). As the name indicates, these functions can be efficiently inverted given a secret inverting key in addition to being efficiently computable. Further, the joint distribution of image-preimage pairs is statistically close, no matter whether the image or the preimage is sampled first. PSFs also provide collision resistance, as well as {\em pre-image min-entropy}: given any image, the set of possible preimages has $\omega(\log \secpar)$ bits of min-entropy, meaning that a specific preimage can only be predicted with negligible chance.
The GPV scheme uses a hash function $H$ modeled as a random oracle. It first hashes the message $m$ using $H$ to obtain a digest $h$. The signing key includes the PSF secret key, and the signature is a preimage of $h$ (the signing randomness is generated using a quantum secure PRF over the message).
To verify a signature, one simply computes its image under the PSF and compares it with the digest.
Notice that in the proof of (post-quantum) blind-unforgeability, the adversary has quantum access to both $H$ and the signing algorithm. To show blind-unforgeability, we will move to a hybrid experiment where the $H$ and the signing algorithm $\Sign$ are constructed differently, but their {\em joint distribution} is statistically close to that in the real execution. To do so, the hybrid will set the signature for a message $m$ to a random preimage from the domain of the PSF (note that this procedure is ``de-randomized'' using the aforementioned PRF). To answer a $H$-oracle query on $m$, the hybrid will first compute its signature (i.e., the PSF preimage corresponding to $m$), and then return the PSF evaluation on this signature (aka preimage) as the output of $H(m)$. Observe that, in this hybrid, the $(H, \Sign)$ oracles are constructed by first sampling preimages for the PSF, and then evaluating the PSF in the ``forward'' direction; in contrast, in the real game, the $(H, \Sign)$ oracles can be interpreted as sampling a image for PSF first, and then evaluating the PSF in the ``reverse'' direction using the inverting key. From the property of PSFs given above, these two approaches induce statistically-close joint distributions of $(H, \Sign)$ on each (classical) query. A lemma from \cite{C:BonZha13} then shows that these are also indistinguishable to adversaries making polynomially-many {\em quantum} queries.
So far, our proof is identical to that of \cite{C:BonZha13}, where GPV is shown to be one-more unforgeable. This final part is where we differ. In the final hybrid, if the adversary produces a successful forgery for a message in the blind set, only two possibilities arise. Since the image of the signature under the PSF must equal the digest, the signature must either (i) provide a second preimage for $h$ to the one computed by the challenger, creating a collision for the PSF, or (ii) equal the one the challenger itself computes, compromising preimage min-entropy of the PSF. This latter claim requires special attention in \cite{C:BonZha13}. A reduction to the min-entropy condition is not immediate, since it is unclear if the earlier quantum queries of $\Adv$ already allow $\Adv$ information about the preimages for the $q+1$ forgeries it outputs. To handle this, \cite{C:BonZha13} prove a lemma (\cite[Lemma 2.6]{C:BonZha13}) showing $q$ quantum queries will not allow $\Adv$ to predict $q+1$ preimages, given the min-entropy condition. In contrast, this last argument is superfluous in our case, since the blind unforgeability game {\em automatically} prevents any information for queries in the blindset from reaching the adversary. We can therefore directly appeal to the min-entropy condition for case (ii) above.
We present the formal construction and the corresponding proof in \Cref{sec:BU:sig:QROM}.
\subsection{BU Signatures in the Plain Model}
To construct a BU signature {\em in the plain model}, we make use of the signature template introduced in \cite{AC:BoyLi16}, which in turn relies on key-homomorphic techniques as used in \cite{ITCS:BraVai14}. We will refer to the \cite{ITCS:BraVai14} homomorphic evaluation procedure as $\algo{Eval}_{\textsc{bv}}$. The \cite{AC:BoyLi16} scheme uses the `left-right trapdoor' paradigm. Namely, the verification key contains a matrix $\vb{A}$ sampled with a `trapdoor' basis $\vb{T}_{\vb{A}}$, and $\vb{A}_0,\vb{C}_0,\vb{A}_1,\vb{C}_1$, which can be interpreted as BV encodings of 0 and 1 respectively, as well as similar encodings $\Set{\vb{B}_i}_{i \in [|k|]}$ of the bits of a key $k$ for a bit-PRF (the use of this PRF is the key innovation in \cite{AC:BoyLi16}). The corresponding signing key contains $\vb{T}_{\vb{A}}$. To sign, one computes BV encodings $\mathbf{C}_{M_1},\dots,\mathbf{C}_{M_t}$ of a $t$-bit message $M$, then computes $\vb{A}_{\textsc{prf,m}} = \algo{Eval}_{\textsc{bv}}(\Set{\vb{B}_i}_{i \in [|k|]},\Set{\vb{C}_j}_{j \in [t]},\algo{PRF})$. Two signing matrices $\mathbf{F}_{M,b} = [\mathbf{A}~|~\mathbf{A}_b - \mathbf{A}_\textsc{prf,m}]$ ($\forall b \in \Set{0,1}$) are then generated (crucially, the adversary cannot tell these apart because of the PRF). A signature is a {\em short non-zero} vector $\sigma \in \mathbb{Z}^{2m}$ satisfying $\mathbf{F}_{M,b}\cdot\sigma= 0$ for any one of the $\mathbf{F}_{M,b}$'s. As pointed out, $\vb{T}_{\vb{A}}$ allows the signer to produce a short vector for either $\mathbf{F}_{M,b}$.
To show unforgeability, one constructs a reduction that
\begin{enumerate}
\item
replaces the left matrix with an SIS challenge (thus losing $\vb{T}_{\vb{A}}$), {\bf and}
\item
replaces the other matrices used to generate the right half with their `puncturable' versions (e.g., $\mathbf{A}_b$ now becomes $\mathbf{AR}_b + \mathbf{G}$, where $\mathbf{R}_b$ is an uniform low-norm matrix and $\mathbf{G}$ is the gadget matrix), with the end result being that the matrix $\vb{A}_{\textsc{prf,m}}$ becomes $\vb{A}\vb{R'} + \vb{G}$ and $\mathbf{F}_{M,b}$ now looks like $[\mathbf{A}~|~\mathbf{AR}+(b-\algo{PRF}_k(M))\mathbf{G}]$ (with $\vb{R},\vb{R'}$ being suitable low-norm matrices).
\end{enumerate}
The crucial point is this: having sacrificed $\vb{T}_{\vb{A}}$, the reduction cannot sign like a normal signer. However it still retains a trapdoor for the gadget matrix $\mathbf{G}$, and for {\em exactly one} of the $\mathbf{F}_{M,b}$, a term in $G$ survives in the right half. This suffices to obtain a `right trapdoor', and in turn, valid signatures for any $M$. On the other hand, a forging adversary lacks the PRF key and so it cannot tell apart $\mathbf{F}_{M,0}$ from $\mathbf{F}_{M,1}$. Thus the forgery must correspond to $\mathbf{F}_{M,\algo{PRF}_k(M)}$ with probability around $1/2$, and the reduction can use this solution to obtain a short solution for the challenge $\mathbf{A}$.
However, the blind-unforgeability setting differs in several meaningful ways. Here, we no longer expect a forgery for any possible message, so the additional machinery to have two signing matrices for every message becomes superfluous. Indeed, for us the challenge is to disallow signing queries in the blindset (even if they are made as part of a query superposition) and to prevent forgeries in the blindset. Accordingly, we interpret the function of the PRF in a different manner. We simply have the bit-PRF act as the characteristic function for the blindset. Then we can extend the approach above to the blind-unforgeability setting very easily: we use a single signing matrix $\mathbf{F}_\textsc{m} = [\mathbf{A}~|~\mathbf{A'} - \mathbf{A}_\textsc{prf,m}]$ (where $\mathbf{A'}$ `encodes 1'). In the reduction, after making changes just as before, we obtain that $\mathbf{F}_\textsc{m} = \big[\mathbf{A}~|~\mathbf{AR} - \big(1-\algo{PRF}_k(M)\big)\mathbf{G}\big]$. For messages where the PRF is not 1, we can answer signing queries using the trapdoor for $\vb{G}$; For messages where it is 1, we cannot, and further we can use a forgery for such a message to break the underlying SIS challenge. In effect, the reduction enforces the requisite blindset behavior naturally.
A caveat is that the bit-PRF based approach may not correctly model a blindset, which is a random $\epsilon$-weight set of messages. Indeed, we require a slight modification of a normal bit-PRF to allow us the necessary latitude in approximating sets of any weight $\epsilon \in [0,1]$. Moreover, due to the adversary's quantum access to the signing oracle, this PRF must be quantum-access secure; and to allow the BV homomorphic evaluation, the PRF must have $\mathsf{NC}^1$ implementation. Fortunately, such a {\em biased} bit-PRF can be built by slightly modifying the PRF from \cite{EC:BanPeiRos12}, assuming QLWE with super-polynomial modulus.
\subsection{Post-Quantum Secure Ring Signatures}
\label{sec:tech-overview:PQ-ring-sig}
{\bf Defining Post-Quantum Security.} To reflect the {\em quantum power} of an QPT adversary $\Adv$, one needs to give $\Adv$ quantum access to the signing oracle in the security game. While this is rather straightforward for anonymity, the challenge here is to find a proper notion for unforgeability (thus, here we only focus on the latter). Let us first recall the {\em classical} unforgeability game for a ring signature. In this game, $\Adv$ learns a ring $\mathcal{R}$ from the challenger, and then can make two types of queries:
\begin{itemize}
\item
by a {\em corruption query} $(\algo{corrupt}, i)$, $\Adv$ can corrupt a member in $\mathcal{R}$ to learn its secret key;
\item
by a {\em signing query} $(\algo{sign}, i, \Ring^*, m)$, $\Adv$ can create a ring $\Ring^*$, specify a member $i$ that is contained in both $\mathcal{R}$ and $\Ring^*$, and ask the challenger to sign a message $m$ w.r.t.\ $\Ring^*$ using the signing keys of member $i$.
\end{itemize}
Notice that $\Ring^*$ may contain (potentially malicious) keys created by $\Adv$; but as long as the member $i$ is in both $\Ring^*$ and $\mathcal{R}$, the challenger is able to sign $m$ w.r.t.\ $\Ring^*$. The challenger also maintains a set $\mathcal{C}$, which records all the members in $\mathcal{R}$ that are corrupted by $\Adv$. To win the game, $\Adv$ needs to output a forgery $(\Ring^*, m^*, \Sigma^*)$ satisfying the following 3 requirements:
\begin{enumerate}
\item
$\Ring^* \subseteq \mathcal{R}\setminus \mathcal{C}$,
\item
$\RS.\Verify(\Ring^*,m^*, \Sigma^*) = 1$, {\bf and}
\item
$\Adv$ never made a signing query of the form $(\algo{sign}, \cdot, \mathcal{R}^*, m^*)$.
\end{enumerate}
To consider quantum attacks, we first require that corruption queries should remain classical. In practice, corruption queries translate to the attack where a ring member is totally taken over by $\Adv$. Since ring signatures are a de-centralized primitive, corrupting a specific party should not affect other parties in the system. This situation arguably does not change with $\Adv$'s quantum power. One could of course consider ``corrupting a group of users in superposition'', but the motivation and practical implications of such corruptions is unclear, and thus we defer it to future research. In this work, we restrict ourselves to classical ring member corruptions.
We will allow $\Adv$ to conduct superposition attacks over the ring and message. That is, a QPT $\Adv$ can send singing queries of the form $(\algo{sign}, i, \sum_{\Ring,m} \psi_{\Ring,m}\ket{\Ring, m})$, where the identity $i$ is classical for the same reason above. Given the argument above, one may wonder why we allow superpositions over $\Ring$ in the signing query. The reason is that unlike for corruption queries, each signing query specifies a specific member $i$ to run the signing algorithm for. No matter what $\Ring$ is, this member will only sign using her own signing key (and this is the only signing key that she knows), and this has nothing to do with other parties in the system\footnote{Indeed, $\Ring$ may even contain ``illegitimate'' or ``non-existent'' members faked by $\Adv$. Note that we do not require $\Ring\subseteq\mathcal{R}$.}. Therefore, superposition attacks over $\Ring$ can be validated just as superposition attacks over $m$, thus should be allowed.
The next step is to determine the winning condition for QPT adversaries in the above quantum unforgeability game. The approach taken by \cite{C:CGHKLMPS21} is to extend the one-more unforgeability from \cite{C:BonZha13} to the ring setting. Concretely, it is required that the adversary cannot produce $(\mathsf{sq} + 1)$ valid signatures by making only $\mathsf{sq}$ quantum {\sf sign} queries. However, there is a caveat. Recall that the $\Ring^*$ in $\Adv$'s forgery should be a subset of uncorrupted ring members (i.e., $\mathcal{R}\setminus\mathcal{C}$). A natural generalization of the ``one-more forgery'' approach here is to require that, with $\mathsf{sq}$ quantum signing queries, the adversary cannot produce $\mathsf{sq} + 1$ forgery signatures, where {\em all} the rings contained are subsets of $\mathcal{R}\setminus \mathcal{C}$. This requirement turns out to be so strict that, when restricted to the classical setting, this one-more unforgeability seems to be weaker than the standard unforgeability for ring signatures (more details in \Cref{sec:ring-sig:pq-definition} and \Cref{sec:one-more:PQ-EUF:ring-sig}).
Our idea is to extend the blind-unforgeability definition to our setting. Specifically, the challenger will create a blind set $B^\RS_\epsilon$ by including in each ring-message pair $(\Ring, m)$ with probability $\epsilon$. It will then blind the signing algorithm such that it always returns ``$\bot$'' for $(\Ring, m) \in B^\RS_\epsilon$. In contrast to one-more unforgeability, we will show that this definition, when restricted to the classical setting, is indeed equivalent to the standard unforgeability notion for ring signatures.
\para{Our Construction.} Our starting point is the LWE-based construction by Chatterjee et al.\ \cite{C:CGHKLMPS21}.
We first recall their construction: the public key consists of a public key for a public-key encryption scheme $\PKE$ and a verification key for a standard signature scheme $\Sig$, as well as the first round message of a (bespoke) ZAP argument. To sign a message, one first computes an ordinary signature $\sigma$ and then encrypts this along with a hash key $hk$ for a specific hash function (i.e., {\em somewhere perfectly-binding} hash). Two such encryptions $(c_1,c_2)$ are produced, along with the second-round message $\pi$ of the ZAP proving that one of these encryptions is properly computed using a public key that is part of the presented ring. The hash key is extraneous to our concerns here; suffice it to say that it helps encode a ``hash'' of the ring into the signature and is a key feature in establishing compactness of their scheme.
To show anonymity, one starts with a signature for $i_0$, then switches the ciphertexts $c_1$ and $c_2$ in turn to be computed using the public key for $i_1$ while changing the ZAP accordingly. Semantic security ensures that ciphertexts with respect to different public keys are indistinguishable, and WI of the ZAP allows us to switch whichever ciphertext is not being used to prove $\pi$, and also to switch a proof for a ciphertext corresponding to $i_0$ to one corresponding to $i_1$.
Unforgeability in \cite{C:CGHKLMPS21} follows from a reduction to the unforgeability of $\Sig$. Even though their construction uses a custom ZAP that only offers soundness for (effectively) $\NP \cap \coNP$, they develop techniques in this regard to show that even with this ZAP, one can ensure that if an adversary produces a forgery with non-negligible probability, then it also encrypts a valid signature for $\Sig$ in one of $c_1$ or $c_2$ with non-negligible probability. The reduction can extract this using a corresponding decryption key (which it can obtain during key generation for the experiment) and use this as a forgery for $\Sig$.
The \cite{C:CGHKLMPS21} construction can thus be seen as a compiler from ordinary to ring signatures assuming LWE. We use their template as a starting point, but there are significant differences between security notions for standard (classical) ring signatures, and our (quantum) blind-unforgeability setting. We discuss these and how to accomodate them next. The very first change that we require here is to use a blind-unforgeable signature scheme in lieu of $\Sig$, since we reduce unforgeability to that of $\Sig$.
Next, let us discuss post-quantum anonymity. Here, the adversary can make a challenge query that contains a {\em superposition} over rings and messages.
We would like to use the same approach as above, but of course computational indistiguishability is compromised against superposition queries. Two clear strengthenings are needed compared to the classical scheme: first, we need to use pairwise-independent hashing to generate signing randomness (to apply quantum oracle similarity techniques from \cite{C:BonZha13}). Second, we want to ensure statistical similarity of the components $c_1,c_2,\pi$ (in order to use an aforementioned lemma from \cite{C:BonZha13} which says that pointwise statistically close oracles are indistiguishable even with quantum queries). In particular, the $\PKE$ needs to be statistically close on different plaintexts, and the WI guarantee for the ZAP needs to be statistical. Fortunately, we can use lossy encryption for the constraint on ciphertexts, and the ZAP from \cite{C:CGHKLMPS21} is already statistical WI.
Finally, we turn to blind-unforgeability. Here, the things that change are that firstly, we need to switch to injective public keys (instead of lossy ones) to carry over the reduction from the classical case. Further, we forego using SPB hashing, because our techniques require that we sign the message along with the ring, i.e. $\algo{Sig}.\Sign(sk, \Ring \| m)$. Thus we end up compromising compactness and using an SPB would serve no purpose. The reason that we need to sign the ring too has to do with how we define the blindset and how the challenger must maintain it in the course of the unforgeability game; this turns out to be more delicate than expected (see related discussion in \Cref{sec:discussion}).
With the modifications above, we can eventually reduce the blind-unforgeability to that of $\Sig$.
|
2,869,038,154,911 | arxiv |
\section{Introduction}
\label{Intro}
Blazars are jetted active galaxies with the jet aligned along our line of sight. Due to the beamed relativistic motion of material in the jet, this component dominates the observable spectrum from radio to $\gamma$-rays. The majority of the observed emission is thought to originate from a specific volume near the base of the jet. The jet environment is generally opaque to radio emission, with specific wavelengths released depending on the distance from the BH. Thus, radio emission is not usually included in blazar models.
Multiwavelength blazar spectra are usually composed of two broad bumps with a dip in-between. The low-frequency bump is generally recognized as electron synchrotron emission, as confirmed by optical polarization measurements. Various emission mechanisms can contribute to the high-frequency bump, depending on the types of particles being modeled and the target photon fields with which they interact.
BL Lacertae (BLL) type blazars are noted for their lack of optical/UV lines, which is often thought to indicate that the broad emission line region is absent or sufficiently dim that it can be neglected. BLL electron synchrotron emission spectra tend to peak at higher energies (UV to X-rays) and are associated with less powerful jets. In the leptonic picture, the high-frequency bump of a BLL is caused by synchrotron self-Compton (SSC). In the hadronic picture, the high-frequency bump can be caused by proton synchrotron.
Flat-spectrum radio quasars (FSRQs) have broad emission lines in their optical/UV spectrum, and the electron synchrotron emission tends to peak near the optical band, on average at lower frequencies than BLLs. FSRQs are identified with the higher-powered radio jets of Fanaroff-Riley Type II jets at other orientations. In the leptonic picture, the high-frequency bump of FSRQs is usually attributed to inverse-Compton reprocessing of external photon fields that impinge on the emitting region of the jet, where they interact with the electron(positron) population(s), or external Compton (EC). External photon fields that might impinge on the jet can come from the accretion disk (EC/disk), dust torus (EC/dust), or broad line region (EC/BLR), although in practice, EC/disk is not usually significant. The cosmic microwave background is not expected to contribute significantly to EC components from a blazar primary emitting region that is near the base of the jet. In the hadronic picture, the high-frequency bump of FSRQs may be explained by proton synchrotron, either independently or in combination with synchrotron emission from p$\gamma$ cascade particle populations.
Neutrinos are the product of hadronic processes. For example, a p$\gamma$ cascade will produce a neutrino spectrum, and this is one of the more likely avenues to produce observable neutrinos in the blazar environment. A purely leptonic jet cannot produce neutrinos, and a lepto-hadronic jet in which the protons are not sufficiently accelerated to produce observable radiation would be very unlikely to produce an observable neutrino spectrum. As such, definitively identifying neutrinos with a blazar source is the smoking gun for the presence of significant proton acceleration in blazar jets. This is an important observational marker because models that track disk-jet energetics tend to disfavor hadronic and leptohadronic models because they require more energy from the accretion disk than leptonic models, in some cases to the point of seeming unphysical given current accretion models (which is notably an independent area of active study). However, if some blazars definitely produce neutrinos, then we may have to re-examine our understanding of the disk and disk-jet energetics.
When modeling the emission region in blazar jets, it is important to consider all of the physical processes self-consistently. Particles that are accelerated in one location will tend to emit in the same location, thus we consider the acceleration and cooling regions of the blazar jet to be co-spatial. The minimum variability timescale of a blazar is often understood to indicate the light-crossing timescale of the cooling region that produced the emission. Under this interpretation, it is not unreasonable to model a single homogeneous blob of a size defined by the minimum variability timescale of the high-energy photons. The peak of a flare might be interpreted as the point in time wherein the acceleration and cooling processes could be described by a(n unstable) steady-state. Thus, it follows that a homogeneous, one-zone, steady-state model could represent the physics of the particles in the primary emitting region of a blazar jet.
The 2014-2015 neutrino flare showed no evidence of multiwavelength flaring activity, and the neutrino flux was $\sim$5 times higher than the average $\gamma$-ray flux, likely implying a strong absorption of GeV photons by an intense X-ray radiation field. This poses a significant difficulty for interpreting the neutrino flares in terms of conventional one-zone models \citep[e.g.][and references therein]{2021ApJ...906...51X}. To overcome this, numerous models that go beyond the framework of the conventional one-zone model have been proposed \citep[e.g.][]{2019PhRvD..99f3008L,2019ApJ...886...23X,2020ApJ...889..118Z,2021ApJ...906...51X}. In this work we focus on the recent two-zone model from \citet{2021ApJ...906...51X}, since it predicts a substantial MeV flux that can be tested with future MeV telescopes such as AMEGO-X.
\citet{2021ApJ...906...51X} consider a two-zone radiation model with an inner and outer blob, where the inner blob is close to the SMBH (within the hot corona), and the outer blob is further away. The two blobs are assumed to be spherical plasmoids of different radii, moving with the same bulk Lorentz factor along the jet axis, and they are filled with uniformly entangled magnetic fields ($\mathrm{B_{in} > B_{out}}$), as well as relativistic electrons and protons. The electrons and protons are assumed to be accelerated primarily by Fermi-type acceleration \citep{2007Ap&SS.309..119R} at the base of the jet, and injected into the blobs. In the inner blob, the X-ray corona provides target photons for efficient neutrino production and strong GeV $\gamma$-ray absorption (which is reprocessed down to the MeV band) resulting from $p \gamma$ interactions. The outer blob is assumed to be away from the broad line region, where it is assumed that relativistic electrons radiate mainly through synchrotron radiation and synchrotron self-Compton scattering. These dissipation processes in the outer blob are responsible for the multiwavelength emission.
Although the inner-outer blob model from \citet{2021ApJ...906...51X} can account for the TXS 0506+056 neutrino flares, it was also shown that the probability of forming a blob within the X-ray corona is likely very small. More generally, however, the corona may still be a promising site for neutrino production via other mechanisms \citep[e.g.][]{2019ApJ...880...40I,2020PhRvL.125a1101M}.
\color{black}
\section{Modeling Methodology}
\label{Motive}
\begin{comment}
Define Cosmological factors and beaming treatment
The particle calculations are done in the blob frame
\end{comment}
Blazar jets form near the supermassive black hole (SMBH) in the core of active galactic nuclei via a process that is likely related to magnetohydrodynamical winds produced by accretion processes.
Some of the magnetic field lines from the disk pass through the ergosphere of the BH and bend up (and down) to contain the jet(s) above (below) the plane of the accretion disk. Thus, material originates from the accretion disk and is funneled into the base of the jet with an energy distribution that is characterized by a thermal distribution with tail.
As the material moves away from the BH, it has a characteristic bulk Lorentz factor $\Gamma = (1-\beta^2)^{-1/2}$, which is determined by the bulk relativistic speed $v = \beta c$, where $c$ is the speed of light. Since the material in a blazar jet axis makes a small angle, $\theta$, with the line-of-sight of the observer, the relativistic Doppler beaming factor $\delta_{\rm D} = [\Gamma(1-\beta \cos \theta)]^{-1}$ can be approximated as $\delta_{\rm D} \sim \Gamma$. The observed variability timescale, $t_{\rm var}$, constrains the size of the comoving blob via a causality argument regarding the light-crossing timescale. Thus, the radius of the blob in the comoving frame is given by $R'_b \lesssim c\delta_{\rm D} t_{\rm var}/(1+z)$, for a cosmological redshift $z$.
\begin{comment}
Introduce the transport equations and define each set of coefficients and their physical meaning, referring the bulk of the discussion to theory papers.
\end{comment}
In the
comoving frame, the electron energy distribution, $N_e(\gamma)$, is the solution to the steady-state Fokker-Planck equation \citep{lewis18, lewis19}
\begin{align}
\frac{\partial N_e}{\partial t} &= 0 = \frac{\partial^2}{\partial \gamma^2}\left(\frac{1}{2} \frac{d \sigma^2}{d t} N_e \right) - \frac{\partial}{\partial \gamma} \left(\left< \frac{d\gamma}{dt} \right> N_e \right) \nonumber\\
& - \frac{N_e}{t_{\rm esc}} + \dot{N}_{e,{\rm inj}} \delta(\gamma-\gamma_{\rm inj}) \ ,
\label{eq-elecTransport}
\end{align}
where $\gamma \equiv E/(m_ec^2)$ is the electron Lorentz factor, describing the particle energy. The fourth term on the right-hand side describes the rate of particle injection $\dot{N}_{e,{\rm inj}}$ into the blob with an injection Lorentz factor $\gamma_{\rm inj}$, which we constrain to be near 1. The third term on the right-hand side describes the energy-dependent (Bohm) diffusive particle escape on timescales given by
\begin{equation}
t_{\rm esc}(\gamma) = \frac{\tau}{D_0 \gamma} = \frac{R^{\prime 2}_b q B D_0}{m_e c^3} \ ,
\end{equation}
where $\tau$ is a dimensionless escape parameter that depends on the comoving radius of the blob, the fundamental charge, the jet magnetic field, the electron mass, and the stochastic diffusion parameter $D_0 \propto s^{-1}$ \citep{park95}.
Equation (\ref{eq-elecTransport}) includes a broadening coefficient term,
\begin{equation}
\frac{1}{2} \frac{d \sigma^2}{d t} = D_0 \gamma^2\ ,
\end{equation}
which encodes a formulaic description of second-order Fermi (stochastic) acceleration and includes the first term of the drift coefficient. We assume the hard-sphere approximation for characterizing the stochastic acceleration. The drift coefficient is given by
\begin{equation}
\left< \frac{d\gamma}{dt} \right> = D_0\left[ 4\gamma + a\gamma - b_{\rm syn}\gamma^2 - \gamma^2 \sum_{j=1}^J b_C^{(j)} H(\gamma\epsilon_{\rm ph}^{(j)})\right] \ ,
\end{equation}
and includes terms (from left to right) for first-order (shock) acceleration, adiabatic expansion with parameter $a$, synchrotron cooling with parameter $b_{\rm syn}$, and Compton cooling with parameter $b_{\rm C}^{(j)}H(\gamma\epsilon_{\rm ph}^{(j)})$. The Compton cooling term includes contributions from all of the relevant photon fields denoted by $(j)$ with Klein-Nishina effects encoded in the $H(\gamma\epsilon_{\rm ph}^{(j)})$ term determined by the characteristic energy of the photon field $\epsilon_{\rm ph}^{(j)}$ \citep[for additional details see][]{lewis18}.
The evolution of the proton energy distribution is described by a similar Fokker-Planck equation to that for the electrons, except with an additional term,
\begin{align}
\frac{\partial N_p}{\partial t} &= 0 = \frac{\partial^2}{\partial \gamma^2}\left(\frac{1}{2} \frac{d \sigma^2}{d t} N_p \right) - \frac{\partial}{\partial \gamma} \left(\left< \frac{d\gamma}{dt} \right> N_p \right) \nonumber\\
& - \frac{N_p}{t_{\rm esc}} - D_0 f_{\rm cascade}(\gamma) N_p + \dot{N}_{p,{\rm inj}} \delta(\gamma-\gamma_{\rm inj}) \ .
\label{eq-protTransport}
\end{align}
The additional $f_{\rm cascade}(\gamma)$ term
represents the protons that leave the system because they become other particles arising from hadronic interactions. The loss of protons due to these interactions is
energy dependent and calculated in PYTHIA 8~\citep{PYTHIA8p2}.\footnote{Additional details to be provided in an upcoming publication in preparation.}
The escape and broadening coefficient terms in the proton Fokker-Planck equation are the same as those in the electron Fokker-Planck equation, except for mass corrections where applicable. The proton drift coefficient is similar to the electron drift coefficient,
\begin{equation}
\left< \frac{d\gamma}{dt} \right> = D_0\left[ 4\gamma + a\gamma - b_{\rm syn}\gamma^2 - b_{p\gamma}(\epsilon^*) \gamma \right] \ ,
\end{equation}
except that we have excluded the term related to cooling via Compton scattering, which is not expected to be relevant for the proton population, and included a cooling term due to the $p\gamma$ interactions.
The $p\gamma$ cooling coefficient is defined in \citet{boett13} as
\begin{equation}
b_{p\gamma}(\epsilon^*) \equiv \frac{c \left<\sigma_{p\gamma} f_{p\gamma}\right>}{D_0} n_{\rm ph}(\epsilon^*) \epsilon^* \ ,
\end{equation}
where $\epsilon^* \sim 0.95 [{\rm erg}]/(E_p[{\rm erg}])$ is a dimensionless energy that depends on the proton energy $E_p$ in ergs, and $\left<\sigma_{p\gamma} f_{p\gamma}\right> \approx 10^{-28}$ cm is the elasticity-weighted interaction cross-section for the $\Delta$ resonance, which is expected to be the most relevant hadronic interaction process for blazars \citep{mucke00}.
\begin{comment}
State the spectral components calculated for each case
Mention gg-absorption and how it's included
\end{comment}
The spectral emission components include direct emission from the accretion disk \citep{shakura73}
and dust torus, in addition to emission from the jet due to synchrotron,
SSC, and EC of dust torus and BLR photons in the leptonic case and electron synchrotron, SSC and EC/dust, in addition to proton synchrotron and cascade emission in the leptohadronic case. The blazar TXS 0506+056 has
redshift $z=0.3365$ giving it a luminosity distance
$d_L=5.5\times10^{27}\ \mathrm{cm}$ in a cosmology where $(h, \Omega_m,
\Omega_\Lambda) = (0.7, 0.3, 0.7)$.
The $\nu F_\nu$ disk flux is approximated as
\begin{flalign}
f^{\rm disk}_{\epsilon_{\rm obs}} = \frac{1.12}{4\pi d_L^2}\
\left( \frac{ \epsilon}{ \epsilon_{\rm max}} \right)^{4/3} {\rm e}^{-\epsilon / \epsilon_{\rm max}}
\end{flalign}
\citep{dermer14} where $\epsilon=\epsilon_{\rm obs}(1+z)$ and $m_e c^2 \epsilon_{\rm max} = 10$ eV.
The $\nu F_\nu$ dust torus flux is approximated as a blackbody,
\begin{flalign}
f^{\rm dust}_{\epsilon_{\rm obs}} =
\frac{15 L^{\rm dust}}{4\pi^5d_L^2} \frac{(\epsilon/\Theta)^4}{\exp(\epsilon/\Theta) - 1} \ ,
\end{flalign}
where $T_{\rm dust}$ is the dust temperature, $\Theta = k_{\rm B}T_{\rm dust}/(m_ec^2)$, and $k_B$ is the Boltzmann constant.
The primary emission region $\nu F_\nu$ flux is computed using the particle distributions, which are the solutions to
the electron and proton Fokker-Planck equations (Equation [\ref{eq-elecTransport} \& \ref{eq-protTransport}]),
$N^\prime_e(\gamma^{\prime})$ \& $N^\prime_p(\gamma^{\prime})$ respectively. We now add primes to indicate the distribution is in the
frame co-moving with the blob.
The electron synchrotron flux
\begin{flalign}
f_{\epsilon_{\rm obs}}^{e,{\rm syn}} =
\frac{\sqrt{3} \epsilon' \delta_{\rm D}^4 q^3 B}{4\pi h d_{\rm L}^2}
\int^\infty_1 d\gamma^{\prime}\ N^\prime_e(\gamma^{\prime})\ R(x_e)\ ,
\label{eq-fsyn-e}
\end{flalign}
where
\begin{flalign}
x_e = \frac{4\pi \epsilon' m_e^2 c^3}{3qBh\gamma^{\prime 2}}\ ,
\label{eq-xe}
\end{flalign}
and $R(x)$ is defined by \citet{crusius86}. Similarly, the proton synchrotron flux
\begin{flalign}
f_{\epsilon_{\rm obs}}^{p,{\rm syn}} =
\frac{\sqrt{3} \epsilon' \delta_{\rm D}^4 q^3 B}{4\pi h d_{\rm L}^2}
\int^\infty_1 d\gamma^{\prime}\ N^\prime_p(\gamma^{\prime})\ R(x_p)\ ,
\label{eq-fsyn-p}
\end{flalign}
where
\begin{flalign}
x_p = \frac{4\pi \epsilon' m_p^2 c^3}{3qBh\gamma^{\prime 2}}\ ,
\label{eq-xp}
\end{flalign}
with mass corrections and different particle distributions, but notably no other parameter changes. Synchrotron self-absorption is also calculated for each case.
The SSC flux
\begin{flalign}
f_{\epsilon_s}^{\rm SSC} & = \frac{9}{16} \frac{(1+z)^2 \sigma_{\rm T} \epsilon_s^{\prime 2}}
{\pi \delta_{\rm D}^2 c^2t_{v}^2 }
\int^\infty_0\ d\epsilon^\prime_*\
\frac{f_{\epsilon_*}^{\rm syn}}{\epsilon_*^{\prime 3}}\
\nonumber \\ & \times
\int^{\infty}_{\gamma^{\prime}_{1}}\ d\gamma^{\prime}\
\frac{N^\prime_e(\gamma^{\prime})}{\gamma^{\prime2}} F_{\rm C}\left(4 \gamma^\prime \epsilon^\prime_{*}, \frac{\epsilon}{\gamma^\prime} \right) \ ,
\label{fSSC}
\end{flalign}
\citep[e.g.,][]{finke08_SSC} where $\epsilon^\prime_s= \epsilon_s(1+z)/\delta_{\rm D}$, $\epsilon^\prime_*=\epsilon_*(1+z)/\delta_{\rm D}$, and
\begin{flalign}
\gamma^{\prime}_{1} = \frac{1}{2}\epsilon^\prime_s
\left( 1+ \sqrt{1+ \frac{1}{\epsilon^\prime \epsilon^\prime_s}} \right) \ .
\end{flalign}
The function $F_{\rm C}(p,q)$ was originally derived by
\citet{jones68}, but had a mistake that was corrected by
\citet{blumen70}. The EC flux \citep[e.g.,][]{georgan01,dermer09}
\begin{flalign}
\label{eq-ECflux}
f_{\epsilon_s}^{\rm EC} & = \frac{3}{4} \frac{c\sigma_{\rm T} \epsilon_s^2}{4\pi d_L^2}\frac{u_*}{\epsilon_*^2} \delta_{\rm D}^3
\nonumber \\ & \times
\int_{\gamma_{1}}^{\gamma_{\max}} d\gamma
\frac{N^\prime_e(\gamma/\delta_{\rm D})}{\gamma^2}F_{\rm C} \left(4 \gamma \epsilon_*, \frac{\epsilon_s}{\gamma} \right) \ ,
\end{flalign}
where the lower limit is
\begin{flalign}
\gamma_{1} = \frac{1}{2}\epsilon_s \left( 1+ \sqrt{1+ \frac{1}{\epsilon \epsilon_s}} \right) \ ,
\end{flalign}
the energy density $u_*$ and dimensionless photon energy $\epsilon_*$ describe the external radiation field. The energy density of the dust torus photon field,
\begin{equation}
u_* = u_{\rm dust} = 2.2 \times 10^{-5} \bigg(\frac{ \xi_{\rm dust}}{0.1} \bigg)
\bigg( \frac{T_{\rm dust}}{1000\ {\rm K}} \bigg)^{5.2} ~ {\rm erg ~ cm^{-3}} \
\end{equation}
and the corresponding dimensionless energy
\begin{equation}
\epsilon_* = \epsilon_{\rm dust} = 5 \times 10^{-7}
\bigg( \frac{T_{\rm dust}}{1000\ {\rm K}} \bigg) \ ,
\end{equation}
are consistent with \citet{nenkova08-pt2}. We note that the dust reprocessing efficiency $\xi_{\rm dust}$ is a free parameter. The energy density of the BLR photons takes the form,
\begin{equation}
u_* = u_{\rm line} = \frac{u_{\rm line,0}}{1 + (r_{\rm blob}/r_{\rm line})^\beta} \ ,
\label{eq-uline}
\end{equation}
where $r_{\rm blob}$ is the
distance of the emitting blob from the black hole (a free parameter) and $\beta\approx 7.7$ \citep{finke16}.
The line radii $r_{\rm line}$ and intrinsic energy densities $u_{\rm line,0}$ for all broad lines used are known ratios based on composite SDSS quasar spectra \citep[e.g.][]{vanden01, finke16}. The parameters $r_{\rm
line}$ and $u_{\rm line,0}$ are determined from the disk luminosity
using relations found from reverberation mapping \citep{finke16,lewis18}
\color{black}
\color{black}
\section{Data Analysis}
\label{Data}
\begin{comment}
Discuss the origin of the data used here
historical data from SED Builder... need to cite that as well as the individual papers. I might have exported that already.
2014: Rodriguez et al. (2019) in red (remove grey points... re-ssc fit)
2017: Xue+20 with the flat top? Keivani et al. (2018) has a LP set, but I don't think we digitized it.
There will be a table(s) of physical parameters spanning the historical, 2014, and 2017 model fits to be used in respective discussions. Sample in Lewis+19
\end{comment}
\subsection{Historical Spectra}
\label{history}
Since the data available during the periods of interest is limited, especially in particular wavelengths, it is prudent to first apply the model to more complete, although non-simultaneous historical data for the source. This helps to calibrate the model and provide a baseline for the parameter space that might be specific to this source. Notably, the comparison between the model and non-simultaneous data should not be understood as an explanation of the historical activity of the source since the historical data spans many distinct states of the blazar's activity. Each state might be driven by specific physical processes, and the average of all of these observations is not necessarily related to the specifics of any physical state. On a related note, since the comparison to historical data is only for the purpose of calibration for the free parameters, it is not especially interesting to perform the comparison for all of the model types used to fit particular flares. Therefore, we used only the leptohadronic model where the SSC component is not included in the electron transport equation (the FSRQ interpretation).
The historical data was acquired from the Space Science Data Center (SSDC) SED Builder tool, which provides broadband spectral and light curve data from a selection of previously published data sets \citep{Myers03, Healey07, Jackson07, Nieppola07, Condon98, Wright94, Planck11, Gregory96, White92, Planck14, Planck15, Wright10, Bianchi11, Evans14, Voges99, Boller16, Abdo10, Nolan12, Acero15, Bartoli13}. In the radio through UV, the data spans roughly 30 years. The radio observations were made by AT, CLASSSCAT, CRATES, GBT, JVASPOL, NIEPPOCAT, NVSS, PMN, and Planck. The IR data was observed by WISE, and the Opt/UV data were observed by GALEX and Swift. The X-rays were observed by ROSAT and {\it Swift}. The $\gamma$-ray data from Fermi{\it LAT} is limited to the 3FGL and 2FHL catalogues since the current catalogue versions include all historical measurements of the source (avoiding double counting of older data), and previous versions of those catalogues tend not to include updated diffuse background models and analysis methods (making them less accurate). There is one upper limit from ARGO2LAC in the very high-energy $\gamma$-rays.
The range of models in comparison to the historical data is produced by varying the strength of the magnetic field from $0.5$ to $1.2$ G (Table \ref{tbl-freeparams}). The remainder of the free parameters were varied during the analysis, but are held constant through the demonstration of possible parameterizations of the historical data of TXS 0506+056, shown in Figure \ref{fig-TXShist}. Model 09a (blue) has the highest magnetic field value (1.2 G), which produces a synchrotron peak at lower frequencies and with a higher flux, consistent with the highest UV data. The break between the high and low frequency bumps occurs in the softer X-rays, consistent with the lower flux cluster of soft X-rays. While the combination of SSC and EC components replicates the flat $\gamma$-ray spectrum and sharp cutoff at higher frequencies, the frequency of the cutoff is not well matched.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth,trim={0 0 0 18},clip]{SED_TXS_histSamples.pdf}
\caption{A range of leptohadronic model parameterizations are compared to historical data to constrain the baseline parameter space for TXS 0506+056. It is only necessary to change a single parameter to illustrate this range of data. Model 09a is represented in blue and has a magnetic field of $B=1.2$G. Model 08 (green) has $B=0.9$G. Model 09b (pink) has $B=0.7$G. Model 09c (purple) has $B=0.5$G.}
\label{fig-TXShist}
\end{figure}
Model 09c (purple in Figure \ref{fig-TXShist}) has the smallest value for the magnetic field $B=0.5$G. Here, the electron synchrotron curve is less luminous and peaks at higher frequencies. Since the Compton emission is also at higher frequencies, the convergence of the low and high frequency bumps is at a higher frequency. In this case, the higher X-ray data would be explained as the high-frequency side of the synchrotron curve, rather than any physical component having higher flux as one might expect of a flare. Magnetic fields can suddenly decrease during magnetic reconnection. The peak of the Compton curve in this case explains the highest frequency $\gamma$-ray data, and turns over to respect the VHE upper limits.
Most of the historical data can be explained within the range of parameters considered. These models are not unique representations of the possible physics that occurred in this blazar over a 30 year period. However, the parameters provide a baseline for more specific analysis of simultaneous data with this family of transport models. Even though the models consider the acceleration and emission processes of both electrons and protons (including relevant cascades), the electrons and protons are constrained to be equal in number at the rate of injection. The natural consequence of this is that the proton synchrotron emission component is subdominant, and the leptonic emission processes form the observed spectrum at all wavelengths.
\begin{deluxetable*}{lrrrrr}
\tablecaption{Free Model Parameters \label{tbl-freeparams}}
\tablewidth{\textwidth}
\tablehead{
\colhead{Parameter (Unit)}
& \colhead{Historical}
& \colhead{FSRQ 2014}
& \colhead{FSRQ 2017}
& \colhead{BLL 2014}
& \colhead{BLL 2017}
}
\startdata
$t_{\rm var}$ (s)
&$1.0 \times 10^{4}$
&$8.0 \times 10^{3}$
&$1.0 \times 10^{3}$
&$9.0 \times 10^{2}$
&$5.0 \times 10^{3}$
\\
$B$ (G)
&$0.5, \, 0.7, \, 0.9, \, 1.2$
&$1.1$
&$1.05$
&$0.5$
&$3.0$
\\
$\delta_{\rm D}$
&$35$
&$40$
&$72$
&$100$
&$20$
\\
$r_{\rm blob}$ (cm)
&$4.0 \times 10^{17}$
&$4.0 \times 10^{17}$
&$6.2 \times 10^{17}$
&$9.7 \times 10^{17}$
&$1.0 \times 10^{18}$
\\
$\xi_{\rm dust}$
&$0.1$
&$0.1$
&$0.1$
&$0.001$
&$0.1$
\\
$T_{\rm dust}$ (K)
&$600$
&$600$
&$800$
&$300$
&$800$
\\
$L_{\rm disk}$ (erg s$^{-1}$)
&$1.0 \times 10^{45}$
&$1.0 \times 10^{45}$
&$4.0 \times 10^{45}$
&$1.0 \times 10^{45}$
&$1.0 \times 10^{46}$
\\
$D_0$ (s$^{-1}$)
&$4.8\times 10^{-6}$
&$4.8\times 10^{-6}$
&$8.0\times 10^{-6}$
&$7.0 \times 10^{-6}$
&$1.5 \times 10^{-3}$
\\
$a$
&$-3.9$
&$-3.9$
&$-3.9$
&$-3.9$
&$-3.5$
\\
$\gamma_{\rm inj}$
&$1.01$
&$1.01$
&$1.01$
&$1.01$
&$1.01$
\\
$L_{\rm inj}$ (erg s$^{-1}$)
&$6.6 \times 10^{28}$
&$3.3 \times 10^{28}$
&$9.4 \times 10^{28}$
&$7.4 \times 10^{28}$
&$8.3 \times 10^{28}$
\enddata
\end{deluxetable*}
\begin{deluxetable*}{llllllr}
\tablecaption{Calculated Parameters \label{tbl-calculated}}
\tablewidth{0pt}
\tablehead{
\colhead{Parameter (Unit)}
& \colhead{Historical}
& \colhead{FSRQ 2014}
& \colhead{FSRQ 2017}
& \colhead{BLL 2014}
& \colhead{BLL 2017}
}
\startdata
$R_{Ly\alpha}$ (cm)
&$2.7 \times 10^{16}$
&$2.7 \times 10^{16}$
&$5.7 \times 10^{16}$
&$2.7 \times 10^{16}$
&$9.4 \times 10^{16}$
\\
$R_{H\beta}$ (cm)
&$1.0 \times 10^{17}$
&$1.0 \times 10^{17}$
&$2.2 \times 10^{17}$
&$1.0 \times 10^{17}$
&$3.5 \times 10^{17}$
\\
$R^\prime_{b}$ (cm)
&$7.9 \times 10^{15}$
&$7.2 \times 10^{15}$
&$1.6 \times 10^{15}$
&$2.0 \times 10^{15}$
&$2.2 \times 10^{15}$
\\
$\phi_{j,{\rm min}}$ ($^{\circ}$)
&$0.6$
&$0.5$
&$0.07$
&$0.01$
&$0.01$
\\
\hline
\\
$P_B$ (erg s$^{-1}$)
&$1.4-8.2 \times 10^{44}$
&$7.5 \times 10^{44}$
&$1.1 \times 10^{44}$
&$7.6 \times 10^{43}$
&$1.4 \times 10^{44}$
\\
$P_{e+p}$ (erg s$^{-1}$)
&$1.3-10 \times 10^{44}$
&$7.3 \times 10^{44}$
&$1.5 \times 10^{45}$
&$1.3 \times 10^{45}$
&$3.9 \times 10^{44}$
\\
$\zeta_{e+p}^{\dagger}$
&$1.6-3.9$
&$1.0$
&$13$
&$17$
&$2.9$
\\
$\mu_{a}^{\ddagger}$
&$0.3-0.9$
&$0.6$
&$0.2$
&$0.6$
&$0.02$
\\
\hline
\\
$u_{\rm ext}$ (erg cm$^{-3}$)
&$2.1\times 10^{-5}$
&$2.1\times 10^{-5}$
&$6.2\times 10^{-5}$
&$5.9\times 10^{-8}$
&$6.4\times 10^{-5}$
\\
$u_{\rm dust}$ (erg cm$^{-3}$)
&$1.5 \times 10^{-6}$
&$1.5 \times 10^{-6}$
&$6.9 \times 10^{-6}$
&$4.2 \times 10^{-10}$
&$6.9 \times 10^{-6}$
\\
$u_{\rm BLR}$ (erg cm$^{-3}$)
&$2.0 \times 10^{-5}$
&$2.0 \times 10^{-5}$
&$5.5 \times 10^{-5}$
&$5.8 \times 10^{-8}$
&$5.7 \times 10^{-5}$
\\
\hline
\\
$A_C$
&$0.5-2.6$
&$0.7$
&$7.3$
&$0.6$
&$0.07$
\\
$L_{\rm jet}$ (erg s$^{-1}$)
&$2.5-6.3 \times 10^{43}$
&$2.9 \times 10^{43}$
&$2.4 \times 10^{43}$
&$5.9 \times 10^{42}$
&$3.6 \times 10^{44}$
\\
$\sigma_{\rm max}$
&$3.8$
&$3.5$
&$1.3$
&$1.4$
&$340$
\\
\enddata
\tablenotetext{\dagger}{ $\zeta_{e+p} = (u_e+u_{p})/u_B = (P_e + P_{p})/P_B$ is the equipartition parameter, where $\zeta_{e+p} = 1$ indicates equipartition between the particles and field.}
\tablenotetext{\ddagger}{ $\mu_a \equiv (P_B+P_e+P_p)/P_a$ is the ratio of jet to accretion power. Magnetically arrested accretion explains values of $\mu_a \lesssim$ a few. }
\end{deluxetable*}
\color{black}
\subsection{2014/15 Epoch}
\label{2014}
\begin{comment}
plot of SED showing each model in comparison to the 2014 spectrum and simulations for each model type:
\\ .$\qquad$ - FSRQ (leptohadronic)
\\ .$\qquad$ - BLL (leptohadronic with SSC)
Discuss each model with respect to the data including details about what physical processes are included, the fit limitations and how "ordinary" the flare period appears in comparison to the historical baseline for the object.
Discuss any part of the simulations that are specific to the epoch in this plot.
\end{comment}
The broadband spectral data for the 2014/2015 flare \citep{Rodrigues19} is not dissimilar to the historical range for this source. This is reflected in the model comparison with the 2014 flare in Figure~\ref{fig-flare14no}, where we use the leptohadronic model without the SSC cooling term in the electron transport equation. This model is valid where the SSC component is subdominant compared with the EC emission, and can therefore be neglected, which is the case for FSRQ. While TXS 0506+056 has historically been classified as a LSP BL Lac, some debate has arisen in recent years, and we here examine the parameters of a simulation that intentionally emphasizes the EC over the SSC component. Since this is the model we calibrated with the historical data, we should expect that a lot of the parameters will be similar, even though the available simultaneous data in this case is limited.
In Figure \ref{fig-flare14no}, the optical data is well explained by the synchrotron peak, and the X-ray upper limit is well respected. The concave nature of the $\gamma$-ray data (which may be an upward statistical fluctuation) is not possible to recreate in a model with only one dominant physical component in that regime. In particular, the highest frequency data point cannot be matched while constraining the model in this way.
In this analysis, we assume that TXS 0506+056 behaves like an FSRQ during the 2014 flare (which is not necessarily expected due to the low Compton dominance $A_C$ (Table \ref{tbl-calculated}).
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={4 0 4 18}, clip]{SED_TXS2014_noSSC.pdf}
\caption{Comparison of the FSRQ leptohadronic model to broadband multiwavelngth data from the 2014-2015 flare of TXS 0506+056 \citep[][in red]{Rodrigues19}. The individual components of the emission model are labeled with the EC components dominating in the $\gamma$-rays. The co-added total emission from the model is given by the dot dashed line.}
\label{fig-flare14no}
\end{figure}
Figure \ref{fig-flare14ssc} explores the same data set \citep{Rodrigues19}, under the assumption that TXS 0506+056 is a BLL. In this case, the electron transport equation includes SSC as a cooling term, and the parameters are tuned intentionally to emphasize the SSC component rather than EC. The optical data is well explained with electron synchrotron emission. While the synchrotron peak occurs much closer to the X-ray upper limit, there is no violation of it. The fit to the $\gamma$-ray data seems more natural in this configuration, with the SSC component following the shape of the four lowest frequency points. However the highest frequency $\gamma$-ray point remains unexplained under the paradigm of of single dominant component for the high-energy bump. In order to suppress the EC/BLR, the blob is further from the BH to decrease the energy density of the photon fields incident on the jet from the BLR. In order to suppress the EC/Dust component, it was insufficient to reduce the temperature of the dust torus, and instead we reduce the reprocessing efficiency $\xi_{\rm dust}$ by 2 dex.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={4 0 4 0}, clip]{SED_TXS2014_ssc.pdf}
\caption{Comparison of the BLL leptohadronic model to broadband multiwavelngth data from the 2014-2015 flare of TXS 0506+056 \citep[][in red]{Rodrigues19}. The individual components of the emission model are labeled with the SSC component dominating in the $\gamma$-rays. The co-added total emission from the model is given by the dot dashed line.}
\label{fig-flare14ssc}
\end{figure}
In the FSRQ case, the magnetic field $B=1.1$G is on the higher side of the range examined for the historical data, while in the BLL case it is at the lower end $B=0.5$G. The minimum variability timescale, which may be interpreted as the light crossing timescale in the emission region, $t_{\rm var} = 8 \times 10^3$s is slightly smaller than the historical case for the FSRQ and much smaller in the BLL case. From that it also follows that the size of the emitting region in the comoving frame $R'_b$ is slightly smaller or much smaller, respectively. Since the bulk Lorentz factor $\Gamma = \delta_{\rm D}$ is higher in the FSRQ case while the acceleration and emission processes occur at the same rate as the historical case, the injected particle luminosity $L_{\rm inj}$ can be lower. In the BLL case, there is more acceleration generated from stochastic scattering ($D_0$), but also more energy lost to adiabatic expansion, synchrotron radiation, and Compton emission (since the SSC component is broader). All together, this creates the need for a higher injection luminosity in the BLL case, as well as more Doppler boosting into the observer frame to explain the same data.
The Distance of the emitting region from the BH $r_{\rm blob}$ (Table \ref{tbl-freeparams}) is well under parsec-scale in all cases where the leptonic emission dominates. In the FSRQ case, the emitting region (blob) sits just outside the outer edge of the broad line region, which is roughly signified by the distance of the H$\beta$ line from the BH $R_{H\beta}$ (Table \ref{tbl-calculated}). In the BLL case, the model indicates that the emitting region is over twice the distance from the BH and well outside the BLR, which is one reason that we might observe less EC emission - the energy density of the external photon fields drop with distance. The jet opening angle $\phi_{j, {\rm min}} = 0.5 ^{\deg}$ for the FSRQ case or $\phi_{j, {\rm min}} = 0.001 ^{\deg}$ for the BLL case are consistent with a highly columnated jet if one assumes a conical shape.
In the FSRQ case, the 2014 flare is nearly in equipartition $\zeta_{e+p} = 1$ between the energy density of the particles and the field (Table \ref{tbl-calculated}), where we consider both the electron and proton energies. In the BLL case, the jet is particle dominated. It is relatively common for electron dominated emission models of blazars to suggest that the jet is slightly particle dominated, so the unusual case here is actually the FSRQ.
We cannot formally calculate the magnetization parameter in this type of model, but there is an upper limit enforced by the electron Larmor radius that will fit inside the emitting region \citep{lewis16,lewis19}. The upper limit of the magnetization parameter indicates that acceleration via reconnection is unlikely or subdominant in both FSRQ and BLL scenarios since $\sigma_{\rm max} \not\gg 1$. (Note that reconnection is not included in the model.) All of the power produced in the jet through both dominant particle types and the magnetic field, is less than the luminosity of the jet $\mu_a < 1$, indicating that ordinary accretion \citep[e.g.][]{shakura73} is sufficient to power the jet in either case.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={0 0 4 0}, clip]{ElecDist_2014ssc.pdf}
\caption{Particle distributions for the BLL case with respect to the particle Lorentz factor $\gamma$. }
\label{fig-ED14ssc}
\end{figure}
We co-solve the particle transport equations independently for each set of parameters considered. Since the transport equation contains both acceleration and emission processes, these are reflected in the shape of the particle distribution. In particular the slope of the power-law is determined in this case predominantly by the ratio between the shock acceleration/adiabatic expansion and the stochastic acceleration. Since these parameters are consistent between the electron and proton equations, as implied by the assumption that they are co-spatial, the power-law slope is the same in both distributions (Figure \ref{fig-ED14ssc}). The position of the exponential turnover in energy is determined by the rate of acceleration and the rate of cooling. Where the cooling rate becomes dominant, the particle spectrum will turn over \citep[e.g.][]{lewis19}. This occurs in different places for the electrons and the protons largely because of their mass difference, but it is also worth noting that the electrons lose energy to 3 processes (synchrotron, SSC, and EC) while the protons only cool through synchrotron in the current models. The rate of particles being injected into the jet is constrained to be equal for electrons and protons, so the total number of each particle type in the jet should be similar (although there is some conversion through $p\gamma$ cascades). There are significantly fewer protons at lower energies (where the electrons congregate) because there are a large number of protons at significantly higher energies (where there are no electrons).
\subsection{2017 Epoch}
\label{2017}
\begin{comment}
plot of SED showing each model in comparison to the 2017 spectrum and simulations for each model type:
\\ .$\qquad$ - FSRQ (leptohadronic)
\\ .$\qquad$ - BLL (leptohadronic with SSC)
Discuss each model with respect to the data including details about what physical processes are included, the fit limitations and how "ordinary" the flare period appears in comparison to the historical baseline for the object.
Discuss any part of the simulations that are specific to the epoch in this plot.
\end{comment}
The broadband spectral data for the 2017 flare \citep{Aartsen18} is distinct from the historical range in Section \ref{history} for this source in the $\gamma$-ray regime. So, it is expected that some parameters will vary more between the historical data and 2017 flare than between the historical data and 2014 flare. The \citet{Aartsen18} reduction of the {\it Fermi}-LAT $\gamma$-ray data for this period uses a 28 day window centered on the neutrino detection, and their methodology gives a flattened spectrum. Alternatively, \citet{Keivani:2018rnh}'s analysis included approximately the same temporal cuts, but resulted in a more curved spectrum. It is not clear which data reduction method should be considered correct, but it is difficult to explain a flattened spectrum with a single dominant emission component since they tend to resemble log-parabolas. Similarly to Section \ref{2014}, we employ a FSRQ inspired transport equation model where the parameters are tuned to emphasize EC for the $\gamma$-rays. Then, we examine the BLL interpretation with a transport model that treats the SSC cooling consistently for the electrons, and tune the free parameters to emphasize the SSC component in the $\gamma$-rays.
In Figure \ref{fig-flare17no}, we assume that TXS 0506+056 is a FSRQ. The optical data is well explained by the synchrotron peak. The X-ray regime is well covered and may be explained by the convergence of the synchrotron and SSC components around $\nu = 10^{18}$Hz. The high-energy $\gamma$-ray data is well explained by the EC processes. However, it is not possible to recreate the lower-energy $\gamma$-ray spectrum simultaneously in this configuration. In particular, the EC components are too narrow to explain this version of the data.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={4 0 4 0}, clip]{SED_TXS2017_clean.pdf}
\caption{Comparison of the FSRQ leptohadronic model to broadband multiwavelength data from the 2017 flare of TXS 0506+056 \citep[][in red]{Aartsen18}. The individual components of the emission model are labeled with the EC components dominating in the $\gamma$-rays. The co-added total emission from the model is given by the dot dashed line.}
\label{fig-flare17no}
\end{figure}
Figure \ref{fig-flare17ssc} explores the same data set \citep{Aartsen18}, under the assumption that TXS 0506+056 is a BLL. In this case, the electron transport equation includes SSC as a cooling term, and the parameters are tuned intentionally to emphasize the SSC component rather than EC in the $\gamma$-ray regime. Here again, the optical data is well explained with electron synchrotron emission, with or without the illustrated flare of the accretion disk. The BLL model matches the X-ray data about as well as the FSRQ model in Figure \ref{fig-flare17no}. The fit to the lower end of the $\gamma$-ray spectrum is improved in this configuration, with the SSC component being broader. However, it is not sufficiently broad to simultaneously explain the X-rays and the higher frequency $\gamma$-ray points, even though the general shape is close. In order to suppress the EC/BLR, it was not necessary to move the emitting region much further from the outer edge of the BLR or to alter the dust reprocessing efficiency. So, those parameters are similar to the FSRQ case (Tables \ref{tbl-freeparams} \& \ref{tbl-calculated}).
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={4 0 4 0}, clip]{SED_TXS2017_ssc.pdf}
\caption{Comparison of the BLL leptohadronic model to broadband multiwavelength data from the 2017 flare of TXS 0506+056 \citep[][in red]{Aartsen18}. The individual components of the emission model are labeled with the SSC components dominating in the $\gamma$-rays. The co-added total emission from the model is given by the dot dashed line.}
\label{fig-flare17ssc}
\end{figure}
The variability timescale ($t_{\rm var}$), signalling the size of the emission region ($R'_b$), for the FSRQ and BLL analyses of the 2017 flare are between the extremes that we explored in the analysis of the 2014 flare, with the BLL case being slightly larger (Table \ref{tbl-freeparams}). The jet opening angle (in a conical approximation) is especially small for both models of the 2017 flare, which may indicate that the jet is very columnated or that the jet not well approximated by a cone out past the BLR, where the emission region seems to be located in this picture. Increasing the Doppler boosting (bulk Lorentz factor) is a way to broaden the predicted spectrum without increasing the particle acceleration. It is interesting that the Doppler factor is higher in the FSRQ case, since the EC component tends to reach higher frequencies than SSC. However, the rate of particle acceleration is several orders of magnitude higher in the BLL case in order to fuel the the increased synchrotron cooling rate (higher magnetic field) and broader SSC component.
Both the FSRQ and BLL cases explored indicate the jet is particle dominated during the 2017 flare, although moreso in the FSRQ case. Both cases also seem well explained by ordinary accretion since the jet power to accretion luminosity ratio ($\mu_a < 1$). The Compton dominance $A_C$ is larger for the 2017 flare than the 2014 flare or the historical data for this source. The value for Compton dominance in Table \ref{tbl-calculated} is is more accurate in the FSRQ interpretation since the calculation is based on the ratio between the synchrotron and EC immensities. Since EC is suppressed in the BLL case, the $\gamma$-ray luminosity is not represented by the luminosity of the EC component. The total jet luminosity is higher in the BLL case because the SSC component is broader - consistent with the 2014 flare analysis. It is especially interesting that the maximum magnetization parameter $\sigma_{\rm max} \gg 1$ in the BLL case. When the magnetization parameter is near or less than 1, stochastic acceleration is likely. Whereas, when the magnetization is much greater than 1, the system is in the reconnection regime. stochastic acceleration and reconnection generally do not happen simultaneously. It is worth mentioning that reconnection is not included in this family of models and that the actual magnetization parameter may not be near the maximum allowed. However, it would be interesting to followup the analysis with a model that includes reconnection to further explore the 2017 flare in the BLL interpretation.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth, trim={0 0 4 0}, clip]{ElecDist_2017ssc.pdf}
\caption{Particle distributions for the BLL case with respect to the particle Lorentz factor $\gamma$. }
\label{fig-ED17ssc}
\end{figure}
Figure \ref{fig-ED17ssc} shows the electron and proton distributions with respect to the particle Lorentz factor for the BLL interpretation of the 2017 flare. In comparison to Figure \ref{fig-ED14ssc}, the electron distribution is qualitatively similar, but the slope of the power-law section is slightly less because the ratio between stochastic acceleration and shock acceleration/adiabatic expansion is closer to $a=-2$, which signifies horizontal \citep{lewis19}. The proton distribution power-law matches the electron distribution since the same acceleration rates apply to both populations. However, in comparing between the 2014 and 2017 particle distributions, the 2017 proton break energy is about an order of magnitude higher Lorentz factor since there is overall more acceleration available to the particles in the 2017 simulation. Since the proton population stretches to higher energies and more protons achieve higher energies, there are fewer protons that accumulate at lower energies and the electron and proton curves appear more separated.
\section{Simulations}
\label{Sims}
\color{black}
The Medium-Energy Gamma-ray Astronomy library (MEGAlib) software package\footnote{Available at \url{https://megalibtoolkit.com/home.html}} is a standard tool in MeV astronomy~\citep{2006NewAR..50..629Z}. MEGAlib was first developed for the MEGA instrument, and since then it has been further developed and successfully applied to a number of hard X-ray and $\gamma$-ray telescopes including, the Nuclear Compton Telescope (NCT), the Nuclear Spectroscopic Telescope Array (NuSTAR), the Imaging Compton Telescope (COMPTEL), and most recently, the Compton Spectrometer and Imager (COSI). Among other things, MEGAlib simulates the emission from a source, simulates the instrument response, performs the event reconstruction, and generates data for a given detector design, exposure time, and background emission (instrumental and astrophysical). Specifically, the source flux is generated from Monte Carlo simulations using \textit{cosima}, the event reconstruction is performed using \textit{revan}, and the high-level data analysis is done using \textit{mimrec}. MEGAlib is written in C++ and utilizes ROOT (v6.18.04) and Geant4 (v10.02.p03).
We use MEGAlib (v3.02.00) to simulate AMEGO-X observations of the source TXS 0506+056 during the two flare periods. The simulations employ the latest version of the AMEGO-X detector, as described in \citet{2021arXiv210802860F}. Events are simulated between 100 keV to 1 GeV, using 10 energy bins. For the first neutrino event we use a total flare time of 158 days \citep{IceCube:2018cha}, and for the second neutrino event we use a total flare time of 120 days \citep{IceCube:2018dnn}. To account for the AMEGO-X duty cycle, the exposure time is taken to be 20\% of the flare time. For simplicity, we simulate the source on axis, and perform just a single simulation. This will be updated in the next version of the draft to include 1000 simulations and the average AMEGO-X off-axis angle for a source of $\sim$30$^\circ$.
Simulating the background emission is very computationally expensive, and currently it is not practical to obtain simulated backgrounds for the full exposure time. We have therefore developed an analysis pipeline\footnote{Available at~\url{https://github.com/ckarwin/AMEGO_X_Simulations}} that enables us to simulate the source and background separately, and then combine the respective counts in order to obtain the signal-to-noise (SN) and statistical error ($\sigma$) on the flux, defined as
\begin{equation}
SN = \frac{S}{\sqrt{S+B}},
\end{equation}
\begin{equation}
\sigma = \sqrt{S+B},
\end{equation}
where $S$ is the source counts and $B$ is the background counts. The background is simulated for 1 hr and scaled up to the observation time. It includes both instrumental and astrophysical components. There are contributions from hadrons, hadronic decay, leptons, photons, and trapped hadronic decay. Note that the prompt trapped hadronic and trapped leptonic components contribute mainly during passage of the Southern Atlantic Anomaly (SAA) when the detector is expected to be off, and thus we do not include them in the overall background. It should also be noted that the current AMEGO-X background model includes the extragalactic diffuse emission from \citet{1999ApJ...520..124G}, but it does not include the Galactic diffuse component. Fig.~\ref{fig:BG} shows the total background rate, as well as the rates for the three constituent event types: untracked Compton, tracked Compton, and pair.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{BG.pdf}
\caption{AMEGO-X background rates (including both astrophysical and instrumental contributions). The background is comprised of (from left to right) untracked Compton events (blue), tracked Compton events (green), and pair events (red). The total background is the sum of the three components and is shown with the black curve.}
\label{fig:BG}
\end{figure}
The counts spectrum (C) is output from \textit{mimrec} (in units of counts/keV). This is converted to photon flux ($dN/dE$) using the effective area ($A_{\mathrm{eff}}$) and exposure time (t):
\begin{equation}
\frac{dN}{dE} = \frac{C}{A_{\mathrm{eff}} \times t},
\end{equation}
\begin{equation}
A_{\mathrm{eff}} = \frac{N_{\mathrm{obs}}}{N_{\mathrm{sim}}} \times A_{\mathrm{sph}},
\end{equation}
where $N_\mathrm{obs}$ and $N_\mathrm{sim}$ are the number of observed and simulated counts, respectively, and $A_{\mathrm{sph}}$ is the area of the surrounding sphere from which the simulated events are launched (see MEGAlib documentation for more details).
To optimize the detection sensitivity the pipeline utilizes an energy-dependent extraction region for both the source and background counts. The region is determined by the energy-dependent angular resolution, as given in~\citet{2021arXiv210802860F}, and is set self-consistently for source and background. Note that in the Compton regime the angular resolution is a measure of the full-width-at-half-maximum of the ARM (angular resolution measure), and in the pair regime the angular resolution is a measure of the 95\% containment angle. Additionally, we also optimize with respect to event type, i.e.~for each energy bin we consider the SN separately for untracked Compton, tracked Compton, and pair events, selecting the case that gives the highest SN.
Since AMEGO-X is a highly versatile instrument, covering both the Compton and pair regime, we report measurements for two distinct energy regimes, which we refer to as the low energy band (extending from 100 keV to 2.3 MeV) and the high energy band (extending from 15.2 MeV to 432.9 MeV). Note that the specific ranges of the bands were choosen to coincide with the SED binning. Below $\sim$3 MeV most of the events are from Compton interactions, whereas above $\sim$3 MeV most of the events are from pair interactions. We define the high energy band to start a bit above 3 MeV in order to avoid the transition region which has a lower sensitivity. Note that above $\sim$400 MeV the effective area quickly falls off and there are no counts in our simulations.
The simulated SED for the first event is shown in Fig.~\ref{fig:SED1}. Our representative FSRQ model is shown with the dashed purple curve, and the corresponding simulated data is shown with the peach markers. As can be seen, the emission falls below the AMEGO-X sensitivity for the entire energy range. The upper limits are plotted as the flux + 1$\sigma$ error, and thus they represent the 68\% confidence level. We also simulate the predicted flux from~\citet{2021ApJ...906...51X}. The model is shown with the tan dash-dot curve, and the corresponding data is shown with the red markers. In this case AMEGO-X is sensitive in the low energy band. In particular, the integrated flux in the low energy band is $(23.8 \pm 5.0) \times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$, with a corresponding SN=7.8. The upper limit in the high energy band is $6.5\times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$. For comparison, Fig.~\ref{fig:SED1} also shows the \textit{Fermi}-LAT data for the flare period.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{TXS_SED_first_event_with_published_LAT}
\caption{Expected AMEGO-X flux for the first flare period. The FSRQ model is shown with the purple dashed curve, and the simulated data is shown with peach markers. The model from~\citet{2021ApJ...906...51X} is shown with the tan dash-dot curve, and the simulated data is shown with red markers. For reference, the blue dash-dot curve shows the predicted flux from~\citet{2020ApJ...889..118Z}. The green markers show the LAT date during the flare period from~\citet{2019ApJ...874L..29R}.}
\label{fig:SED1}
\end{figure}
For the second flare period we simulate just the FSRQ model. Note that this event did not have efficient neutrino production, and thus an enhanced flux in the MeV band is not expected (e.g.~\citet{2021ApJ...906...51X}). In this case the high energy band is significantly detected, with a measured flux of $(8.9 \pm 2.9) \times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$ and a corresponding SN = 3.5. In the low energy band we measure an upper limit of $7.5\times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$.
Fig.~\ref{fig:LC} shows the $\gamma$-ray light curve of TXS 0506+056 over roughly a 10 year observational period. The tan bands correspond to the two flare periods, and the simulated AMEGO-X data is plotted for both the high and low energy bands, corresponding to the y-axis on the left. For the first event we show the simulated data corresponding to the model from~\citet{2021ApJ...906...51X}. The LAT data is shown with grey markers, corresponding to the y-axis on the right. To estimate the quiescent state flux we scale the FSRQ model to the LAT data for the full mission. The integrated flux is $1.2\times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$ and $3.1\times10^{-11} \mathrm{\ erg \ cm^{-2} \ s^{-1}}$ in the low energy band and high energy band, respectively. As can be seen, for a cascade scenario as described in~\citet{2021ApJ...906...51X}, AMEGO-X would have been able to significantly detect the first event (in the low energy band), which was not detected by the LAT.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{TXS_0506_AMEGO-X_LC_v2}
\caption{TXS 0506+056 $\gamma$-ray light curve covering roughly 10 years of the LAT mission. The tan bands show the periods of the two neutrino flares. The y-axis on the left gives the AMEGO-X flux, which is plotted for both the low and high energy bands. The green dash and purple dash-dot lines give an estimate of the quiescent flux. The LAT data is shown with grey (from~\citet{Petropoulou:2019zqp}) and corresponds to the y-axis on the right.}
\label{fig:LC}
\end{figure}
\section{Discussion \& Conclusions}
\label{Disc}
\begin{comment}
Drive physical interpretation, connections to the source object and signs to look for in observations.
Review previous literature interpretations.
Was anything weird in the plots or the tables that needs to be further explained?
\end{comment}
In Section \ref{Data}, we compared the effect of different assumptions about TXS 0506+056 on the fits to specific data sets and the physical parameter regimes that produce those simulated spectra. In particular, we categorically assumed that the full multiwavelength spectrum is produced in a single zone that is homogeneous, the peak of a flare may be approximated by a steady state, the acceleration and emission regions are co-spatial, and that the injection rate of protons and electrons is the same. In the case of the FSRQ interpretation, we additionally constrain that the EC emission dominates the SSC at high energies, and in turn for the BLL cases that the SSC emission dominates EC. Most of these assumptions stand in contrast to existing literature for this source in one way or another, and it is informative to explore different possible scenarios systematically. Since many of the assumptions about the physical picture disagree, it can be difficult to make direct comparisons between parameters.
The accretion disk luminosity used in this work is somewhat larger than that used in \citet{Xue21} based on extrapolations from references therein. The accretion disk emission curve is under the peak synchrotron emission curve in both sets of models, so it is not directly important to the fits, but the value can change our perception of the energy balance between the accretion disk and the jet.
\citet{Xue21} demonstrate that the presence of a corona for a 2 zone model, is able to generate the neutrino signature, while avoiding overproduction of photon emission because the same corona photon field that generates sufficient energy density for the neutrino production also creates the opacity that blocks spectral emission. Since the presence of an X-ray corona is not observationaly confirmed, it is neglected in this work. The emitting regions we consider across scenarios are between where \citet{Xue21} predict their inner and outer blobs would occur. Similarly the size of our blobs are between the sizes predicted of inner and outer blobs in \citet{Xue21}. The magnetic field ranges between the two works are similar. This work tends to find higher Doppler factors. However, from a modeling perspective Doppler brightening can produce effects on the spectrum that are in some ways similar to the effects of acceleration mechanisms. The acceleration of particles here is treated as co-spatial with the primary cooling and includes both first and second-order Fermi processes. As is common practice, \citet{Xue21} seem to use a particle injection spectrum that is pre-accelerated, and usually interpreted as shock acceleration.
In examining the 2017 flare, \citet{Keivani18} generate fits to the data using a single zone model with EC as the dominant high-energy bump, where the hadronic component that produces neutrinos is subdominant. This physical picture is more comparable to the work presented here. While their electron spectrum is qualitatively similar to that derived here, they do not treat acceleration as co-spatial with the emission region. The maximum proton energies are also similar despite the different methods, reinforcing the idea that the shape of the particle spectrum is important to the spectra produced, meaning that spectral models that match the same data set will require similar underlying particle spectra. This further implies that something about particle acceleration may be discerned from complete, simultaneous multiwavelength spectra, and the addition of neutrino spectra provides further constraints.
With regard to their EC leptonic model, \citet{Keivani18} use a BLR size similar to our finding of the position of the Lyman-$\alpha$ line in the BLR, which is often used as a proxy for the entire BLR because it has the highest intrinsic energy density of any line in the BLR. However, we know from reverberation mapping \citep[e.g.][]{kaspi05,kaspi07,finke16} that BLR regions are predictably stratified due to a photoionization gradient outside of the accretion disk. If one considers the relativistic motion of the emission region the Lyman-$\alpha$ line may not have the highest energy density in the comoving frame from the blob location. Therefore, it is beneficial to treat multiple lines when calculation EC/BLR and to use one of the expected outer lines (e.g. H$\beta$) when approximating the extent of the BLR.
\citet{Keivani18} also consider a leptohadronic modes in which cascade pair production and proton synchrotron produce the high-energy bump, a configuration not currently examined here. In those simulation, they find significantly higher magnetic field values, which is to be expected in that picture. The neutrino production in the proton dominated leptohadronic model of \citet{Keivani18} is sensitive to the Doppler boosting, and is one of the better representations of the full spectrum for the 2017 flare in the literature, suggesting that a good extension of this work would be to emphasize hadronic emission processes.
Where the models explained in this work are comparable with the extensive literature on this source, there is broad agreement. However, the breadth of approaches employed makes direct comparisons difficult.
\subsection{Simulations \& Implications}
\label{imply}
\begin{comment}
Discuss the simulations and what they mean for future observations. Spell out how observers should interpret these predictions as they plan observations and what can be expected, extrapolating to other objects and situations.
\end{comment}
Some of the challenges in constraining models of TXS 0506+056 arise from the incomplete multiwavelength spectra available. Most notably, there is currently a large sensitivity gap in the MeV band, yet this band may be critical for detecting the electromagnetic counterpart of neutrino flares. In particular, the counterparts to bright neutrino emission will not be detected by high-energy gamma-ray telescopes such as the LAT, since the radiation fields required for efficient neutrino production make the source opaque to high-energy gamma rays. Even in the case of the 2017 co-detection of neutrinos with a GeV flare, it is still not possible to definitively state whether something like a corona could contribute to the neutrino production. As shown in Section~\ref{Sims}, during the 2014-2015 TXS 0506+056 orphan neutrino flare, an MeV instrument such as AMEGO-X would have been able to detect a significant MeV flare even though there was no detectable GeV flare, for a model similar to that described in~\citet{Xue21}. Observational evidence of coronae in jetted AGN would be a significant discovery, and illuminate answers to long debated questions, including how jets form and the source of astrophysical neutrinos.
\begin{acknowledgments}
T.L. is supported by an appointment to the NASA Postdoctoral Program at NASA Goddard Space Flight Center, administered by Universities Space Research Association under contract with NASA. H.F. acknowledges support by NASA under award number 80GSFC21M0002. The authors are pleased to acknowledge conversations with Marco Ajello, as well as the AMEGO-X team.
\end{acknowledgments}
\bibliographystyle{aasjournal}
|
2,869,038,154,912 | arxiv | \section{Introduction}
Gravitational lensing has become a versatile tool to probe the cosmological model and scenarios of galaxy evolution. From the coherent distorsions, generated by the intervening matter along the line of sight, of the last scattering surface \citep[eg][]{Planck18-8} or intermediate redshift galaxies \citep{BS01,KilbingerReviewCS}, to the inner parts of massive galaxies \citep{Tre10}, lensing directly measures the fractional energy density in matter of the Universe. Since it does not rely on assumptions about the relative distribution between the galaxies and the underlying Dark Matter (DM), which drives the dynamical evolution of cosmological structures, weak lensing plays a key role in recent, ongoing or upcoming ground-based imaging surveys \citep[CFHTLenS, DES, KiDS, HSC, LSST,~][]{CFHTLens,DES,DES_CS,KIDS,HSC,LSST}. It is also at the center of the planned Euclid and WFIRST satellites \citep{Euclid,wfirst}.
The statistical power of these experiments dramatically increases and drives on its way enormous efforts for the control of systematic effects. One of them concerns the accuracy to which theoretical predictions on the statistical properties of the matter distribution when it has evolved into the non-linear regime can be made on small scale. Arguably, cosmological N-body numerical simulations have been playing a key role in solving the complex dynamical evolution of DM on scales smaller than a few Mpc \citep[eg][]{SFW06}. The upcoming Euclid or LSST missions require an extreme accuracy on the matter density power spectrum and the associated covariances which may enter a likelihood analysis of these data. The effort is currently culminating with the Flagship simulation, for instance \citep{flagship17}. But it also motivated earlier very large simulations like Horizon-4$\pi$ \citep{Teyssier++09,Pichon++10}, DEUS \citep{DEUS10} or MICE \citep{MICE1}.
It has early been envisioned to propagate light rays through such dark matter simulations in order to reproduce the deflection and distorsions of light bundles in a lumpy universe. The motivation is to derive lensing observables like convergence maps and 1-point PDFs of this field or its topological properties (peaks, voids...) or 2-point shear correlation functions \citep[eg][]{JSW00,Pichon++10,Ham01,Vale+03,H+S05,Hil++07,hil++09,Sato++09}. Since, much progress has been made on large and mildly non linear scales with the production of full sky maps with a few arc minutes angular resolution \citep[eg][]{MICE3,Giocoli+16,Takahashi++17}.
In order to make the most of the upcoming surveys, the matter distribution for Fourier modes as large as $k\sim 10 h \, {\rm Mpc}^{-1}$ must be predicted to the percent accuracy, which nowadays still represents a challenge \citep{Schneider++16}. Furthermore, at those scales, the physics of baryons can differ from the dynamics of DM and, even though, it amounts for $\sim 17\%$ of the total cosmological matter budget, it has to be taken into account \citep[OWLS simulation][]{VanDaalen2011}.
For weak lensing statistics, \citet{Semboloni++11} showed that the modelling of the 2-point shear correlation function can be significantly biased, should the baryons be simply treated like the collision-less DM. Even the number of convergence peaks itself is altered by baryons but to a
lesser extent than the power-spectrum \citep{Yang++13}.
Recently, significant progress has been made on hydrodynamical simulations which are now able to reproduce a morphological mix of galaxies in a cosmological context, by considering baryonic physics such as radiative cooling, star formation, and feedback from supernovae and Active Galactic Nuclei (AGN). Despite the tension between the high resolution needs to properly describe the galaxies formed at the center of DM halos and the necessity to simulate sizeable cosmological volumes, recent simulations, such as Horizon-AGN~\citep{Dubois++14}, Illustris/Illustris-TNG \citep{illustris,Pillepich18}, or EAGLE \citep{Eagle}, have now reached volumes of order $100 \, {\rm Mpc}$ on a side
and resolution of order $1 \, {\rm kpc}$. This opens the possibility to quantifying the effect of baryons (experiencing adiabatic pressure support, dissipative cooling, star formation, feedback...) on the total matter distribution and its impact on lensing cosmological observables \citep[see e.g.][]{VanDaalen2011, tenneti+15, hellwing+16,IllustrisTNG,Chisari++18}. Prescriptions to account for this effect \citep[eg][]{Semboloni++13,S+T15,Mead++15,R+T17} have been explored and some start to be incorporated in cosmic shear studies \citep[KIDS:][]{Hildebrandt++17}.
In this paper, we further investigate the impact of baryons on lensing observables in the Horizon-AGN~simulation. By taking advantage of the lightcone generated during the simulation run, we are able to fully account for projection effects (mixing physical scales) and small scale non-linearities occurring in the propagation of light rays (eg, Born approximation, lens-lens coupling, shear -- reduced shear corrections) which may be boosted by the steepening of the gravitational potential wells due to cooled gas sinking at the bottom of DM halos. Hence, this extends the analysis of \citet{Chisari++18} who mostly focused on the effect of baryons on the three-dimensional matter power spectrum, compared the Horizon-AGN~results with those of Illustris, OWLS, EAGLE and Illustris-TNG, found a broad qualitative agreement.
The common picture is that hot baryons which are prevented from sinking into halos like DM, induce a deficit of power inside halos (in a proportion of order $\Omega_{\rm b}/\Omega_{\rm M}$) and, at yet smaller scales ($k \gtrsim 30 h \, {\rm Mpc}^{-1}$), baryons in the form of stars (and to a lesser extent cooled gas) dramatically boost the amplitude of density fluctuations.
However, even though those results seem to converge from one simulation to another, they substantially depend on the assumptions about sub-grid physics, and in particular about AGN feedback.
Beside those encouraging successes at quantifying the nuisance of baryons on cosmological studies, hydrodynamical simulations entail a wealth of information on the relation between galaxies or galaxy properties and the halo they live in. It is, thereby, a way to understand the large scale biasing of these galaxies with respect to the overall total matter density field.
We also explore the small scale relation between galaxies and their surrounding gravitational potential sourcing the lensing deflection field. In particular, the correlation between galaxies and the tangential distortion of background sources (so-called Galaxy-Galaxy Lensing signal, GGL) has proven being a way to constrain the galaxy-mass correlation function \citep[eg][]{brainerd96,guzik01,mandelbaum06,2013MNRAS.432.1544M,Lea++12,2014MNRAS.437.2111V,2015MNRAS.447..298H,2015MNRAS.449.1352C}. In this vein, \citet{Velliscig++17} recently showed that the GGL around $z\sim 0.18$ galaxies in the EAGLE simulation is consistent with the GAMA+KiDS data \citep{Dvornik++17}.
Finally, subtle observational effects entering GGL by high redshift deflectors ($z \gtrsim 0.8)$ are investigated from the lensing information over the full past lightcone of the Horizon-AGN~simulation. The magnification bias affecting the selection of deflectors \citep{Z+H08} complicates the interpretation of GGL substantially. Currently, no such high-z lens sample has been studied because of the scarcity of even higher faint lensed sources carrying the shear signal but the situation may change with Euclid. Its slit-less grism spectroscopy will provide a large sample of H$\alpha$ emitters in the $0.9\le z\le 1.8$ redshift range. A thorough understanding of the clustering properties of this sample may be achieved with the GGL measurement of this sample by using the high-z tail of the shape catalogue obtained with the VIS imager. Some raytracing through cosmological simulations \citep{hil++09,MICE3} had briefly mentioned some aspects of the problem of magnification bias raised by \citet{Z+H08}. The Horizon-AGN~lightcone is a good opportunity to quantify those effects in order to correctly interpret upcoming GGLs.
In this paper, cosmic shear or GGL quantities are directly measured from the lensing quantities obtained by ray-tracing methods. They are not inferred from the shape of galaxies as is done in observations.
A forthcoming paper will present the generation of mock wide-field images including lensing distortions from the full view of Horizon-AGN~lightcone and the light emission predicted for the simulated stars, taking us one step closer to a full end-to-end generation of mock lensing observations.
The paper is organised as follows.
Sect.~\ref{sec:simintro} presents the Horizon-AGN~hydrodynamical simulation, the structure of its lightcone and some properties of the galaxy population, therein.
Sect.~\ref{sec:method} describes the implemented methods to generate the deflection field on thin lens planes and to propagate light rays through them.
Sect.~\ref{sec:cosmicshear}, describes the 1-point and 2-point statistics of the resulting convergence and (reduced-)shear fields. The validity of the raytracing method is quantified by comparing our results with independent methods.
Sect.~\ref{sec:GGL} measures the GGL around the galaxies in the Horizon-AGN~simulation. A comparison with observations is made for low redshift deflectors. The problem of magnification bias is investigated for future observations of high-z GGL. Sect.~\ref{sec:conclusions} wraps up.
\section{The Horizon-AGN simulation lightcone}\label{sec:simintro}
\subsection{Characteristics}\label{ssec:sim0}
The Horizon-AGN~simulation is a cosmological hydrodynamical simulation performed with {\tt RAMSES}~\citep{teyssier02}. The details of the simulations can be found in \cite{Dubois++14}. Let us first briefly summarise the main characteristics. Horizon-AGN~contains $1024^3$ dark matter particles with a mass resolution of $8 \times 10^7\hmm\msun$, in a box of comoving size $L_{\rm box} = 100\,\hmm\Mpc$ on a side. The gravity and hydrodynamics are treated in {\tt RAMSES}~with a {\it multiscale} approach with adaptive mesh refinement (AMR): starting from a uniform $1024^3$ grid, cells are then adaptively refined when the mass inside the cell exceeds $8$ times the initial mass resolution. Cells are recursively refined (or de-refined according to the refinement criterion) down to a minimum cell size of almost constant 1 proper kpc (an additional level is triggered at each expansion factor $a=0.1,0.2,0.4,0.8$). The underlying cosmology is a standard $\Lambda$CDM model consistent with the WMAP7 data \citep{komatsuetal11}, with total matter density $\Omega_{\rm m}=0.272$, dark energy density $\Omega_{\rm \Lambda}=0.728$, amplitude of the matter power spectrum $\sigma_8=0.81$, baryon density $\Omega_{\rm b}=0.045$, Hubble constant of $H_0=70.4\, \rm km\, s^{-1}\, Mpc^{-1}$, and scalar spectral index $n_{\rm s}=0.967$.
The evolution of the gas is solved on the {\tt RAMSES}~grid using a Godunov method with the approximate HLLC Riemann solver on the interpolated conservative hydrodynamical quantities, that are linearly interpolated at cell boundaries from their cell-centered values using a MinMod total variation diminishing scheme.
In addition, accurate models of unresolved sub-grid physics have been implemented.
The gas heating comes from a uniform UV background which started at the re-ionisation $z_{\rm reion} = 10$ \citep{haardt&madau96}.
The cooling function of the gas follows \citet{Sutherland1993}, from H and He collision and from the contribution of other metals.
Star formation is modelled following the Schmidt law \citep{Kennicutt1998}, with a constant star formation efficiency of $2\%$ per free fall time. It occurs when the density of the gas exceeds the threshold $0.1 \ \rm{H\, cm^{-3}}$.
The temperature at gas densities larger than $0.1 \ \rm{H\, cm^{-3}}$ is modified by a polytropic equation of state with polytropic index of $4/3$ and scaling temperature of $10^4\, \rm K$ \citep{Springel2003}.
Stellar evolution is performed assuming a \citet{Salpeter1955} initial stellar mass function. The sub-grid physics also includes stellar winds and supernova feedback in the form of heating, metal enrichment of the gas, and kinetic energy transfer to the ambient gas \citep[see][ for more details]{Kaviraj++17}.
Finally, black holes (BH) are created when the gas density exceeds $0.1 \ \rm{H\, cm^{-3}}$, and when there is not other BH in the close environment.
They grow by direct accretion of gas following an Eddington-limited Bondi-Hoyle-Littleton accretion rate, and merger when BH binaries are sufficiently close.
The AGN feedback is treated by either an isotropic injection of thermal energy, or by a jet as a bipolar outflow, depending of the ratio between the Bondi and the Eddington accretion rates
\citep[see][for details]{duboisetal12agnmodel, Volonteri++16}.
The past lightcone of the simulation was created on-the-fly as the simulation was running. Its geometry is sketched in Fig.~\ref{fig:cone-shape}. The opening angle of the cone is $2.25$ deg out to redshift $z=1$ and $1$ deg all the way to $z=8$. These two values correspond to the angular size of the full simulation box at these redshifts. We can therefore safely work in the flat sky (or infinitely remote observer) approximation.
Up to $z=1$, the volume of the cone is filled with $\sim7$ replicates of the box. Between $z=0$ and $z=4$, the narrow cone contains $\sim14$ replicates of the box and the union of the two cones contains about $19$ copies. This should be kept in mind when quantifying the statistical robustness of our results.
In order to limit projection effects, a non-canonical direction is chosen for the past lightcone but, in order to preserve periodic boundary conditions between replicates, no random rotation is applied. Projection effects will still be present and induce characteristic spectral distortions on large scales which must be taken into account.
Particles and AMR cells were extracted on-the-fly at each coarse simulation time step (when all levels are synchronized in time as a factor 2 of subcycling is used between levels) of the simulation according to their proper distance to a fiducial observer located at the origin of the simulation box. The lightcone of the simulation thus consists in 22,000 portions of concentric shells. Each of them contains stellar, black hole, DM particles (with their position and velocity, mass and age) along with AMR Eulerian cells storing the gas properties (position, density, velocity, temperature, chemical composition, and cell size) and the total gravitational acceleration vector.
\begin{figure}[h]
\includegraphics[width=0.499\textwidth]{cone-shape.pdf}
\caption{2D sketch of the past lightcone around redshift $z=1$ (orange vertical line). Each mesh is a replicate of the Horizon-AGN~simulation box (bounded with cyan lines). The tiling is performed all the way up to redshift $z\sim8$. }
\label{fig:cone-shape}
\end{figure}
\subsection{Properties of galaxies and host halos}\label{ssec:catalogs0}
The \textsc{ AdaptaHOP} halo finder \citep{aubert04} is run on the lightcone to identify galaxies from the stellar particles distribution. Local stellar particle density is computed from the 20 nearest neighbours, and structures are selected with a density threshold equal to 178 times the average matter density at that redshift. Galaxies resulting in less than 50 particles ($\simeq 10^8 \,{\rm M}_\odot$) are not included in the catalogue.
Since the identification technique is redshift dependent, \textsc{AdaptaHOP} is run iteratively on thin lightcone slices. Slices are overlapping to avoid edge effects (i.e. cutting galaxies in the extraction) and duplicate are removed.
In a second step dark matter haloes have been extracted independently from the dark matter particle distribution, with a density threshold of 80 times the average matter density, and keeping only haloes with more than 100 particles. The centre of the halo is temporarily defined as the densest particle in the halo, where the density is computed from the 20 nearest neighbours. In a subsequent step, a sphere of the size of the virial radius is drawn around it and implement a shrinking sphere method (Power et al. 2003) to recursively find the centre of mass of the halo. In each iteration, the radius of the halo is reduced by 10 $\%$. The search is stopped when a sphere 3 times larger than our spatial resolution is reached. Each galaxy is matched with its closest halo.
The simulation contains about $116,000$ galaxies and halos in the simulation box at $z=0$, with a limit of order $M_* \gtrsim 2\times10^9 {\rm M}_{\odot}$.
These yields have been extensively studied in previous papers of the Horizon-AGN~series. For instance, \citet{Kaviraj++17} compared the statistical properties of the produced galaxies, showing a reasonable agreement with observed stellar mass functions all the way to $z\sim6$. The colour and star formation histories are also well recovered and so are the black hole -- bulge relations and duty-cycles of AGNs \citep{Volonteri++16}.
Following up on an earlier work \citep{Dub++13} focusing on a handful of zoomed galaxy simulations with {\tt RAMSES}, \citet{HorizonAGN} confirmed with a much greater statistical significance in Horizon-AGN, that the morphological diversity of galaxies is well reproduced (fraction of rotation- versus dispersion-supported objects, and how this dichotomy maps into the star forming versus quiescent dichotomy). Taking advantage of a parallel simulation run with the same initial conditions and in which the AGN feedback is turned off (Horizon-noAGN), the key role of the latter in shaping the galaxy morphology was emphasised. Furthermore, \citet{Peirani++17} studied the effect of AGN feedback on the innermost density profiles (stars, gas, DM, total) and found a good agreement of the density profile, size-mass relation and dark matter fraction inside the effective radius of galaxies with observations. In particular, \citet{Peirani++18} showed that the innermost parts of Horizon-AGN~galaxies are consistent with strong lensing observations of \citet{Sonnenfeld++13} and \citet{New++13,New++15}.
Populating the lightcone yields a volume limited sample of $1.73\times 10^6$ galaxies in the narrow 1 deg cone. However, a large fraction of the low mass high redshift galaxies would not be of much practical use in a flux limited survey as shown in Fig.~\ref{fig:gals} which plots the redshift dependent limit in stellar mass attained with several $i$-band apparent limiting magnitudes. This was obtained using the COSMOS2015 photometric catalogue of \citet{Laigle16}.
\begin{figure}[h]
\includegraphics[width=0.499\textwidth]{galaxy_distrib.pdf}
\caption{Distribution in the redshift -- stellar mass plane of the $1.7$ million galaxies in the Horizon-AGN~lightcone. For guidance the stellar mass limit for completeness is shown as well as fiducial cuts in mass one would obtain with a flux limited survey of various $i$ band limiting magnitudes.}
\label{fig:gals}
\end{figure}
\section{Raytracing through the lightcone}\label{sec:method}
After briefly describing the basics of the propagation of light rays in a clumpy universe and the numerical transcription of this formalism,
let us now describe the the ray-tracing computation in the Horizon-AGN~lightcone. Our implementation of the multiple lens plane (but also the Born approximation) builds on similar past efforts \citep{Hil++08,GLAMER1,GLAMER2,Barreira++16}. It has been tailored for the post-treatment of the Horizon-AGN~past lightcone, but, provided the flat sky approximation holds, our implementation could readily be applied to any other {\tt RAMSES}~lightcone output \citep{Teyssier++09}.
As detailed below, two methods are investigated to infer deflection angles from either the distribution of various particle-like matter components or the total gravitational acceleration stored by {\tt RAMSES}.
The light rays are then propagated plane by plane (both within and beyond the Born approximation), for these two different estimates of the deflection field.
\subsection{The thin lens plane}
Let us define $\vec{\beta}$ the (un-perturbed and unobservable) source plane angular position and $\vec{\theta}$ the observed angular position of a light ray. Considering a unique, thin, lens plane, the relation between the angular position of the source $\vec{\beta}$, the deflection angle $\vec{\alpha}$ and the image $\vec{\theta}$ is simply:
\begin{equation}\label{eq:lenseq}
\vec{\beta} = \vec{\theta} -\frac{D_{\rm ls}}{D_{\rm s}} \vec{\alpha}(\vec{\theta})\,,
\end{equation}
where $D_{\rm ls}$ and $D_{\rm s}$, are the angular diameter distance between the source and the lens, and between the observer and the source, respectively.
The deflection angle $\vec{\alpha}(\vec{\theta})$ is obtained by integrating the gravitational potential $\Phi(\vec{r})$ along the line of sight (here, radial proper coordinate $x_3$)
\begin{equation}\label{eq:alphadef}
\vec{\alpha} (\vec{\theta}) = \frac{2}{c^2} \int \vec{\nabla_{\perp}} \Phi(\vec{\theta},x_3) \ {\rm d} x_3\,.
\end{equation}
Hence, across a thin lens plane, the lensing potential $\phi(\vec{\theta})$ is related to the deflection field by the Poisson equation:
\begin{equation}\label{eq:poisson}
\Delta \phi = \vec{\nabla} . \vec{\alpha} \equiv 2 \kappa \,,
\end{equation}
where the convergence $\kappa$ is the projected surface mass density $\Sigma(\vec{\theta})$ in the lens plane expressed in units of the critical density $\Sigma_{\rm crit}$
\begin{equation}
\Sigma_{\rm crit}\, \kappa(\vec{\theta}) = \Sigma(\vec{\theta}) \equiv \int \rho(\vec{\theta},z)\, {\rm d} z\,.
\end{equation}
The critical density reads:
\begin{equation}\label{eq:scrit}
\Sigma_{\rm crit} = \frac{c^2}{4 \pi G} \frac{D_{\rm s}}{D_{\rm l}D_{\rm ls}}\,,
\end{equation}
with $D_{\rm l}$, the angular diameter distance between the observer and the lens. In the above equations, all distances and transverse gradients are expressed in physical (proper) coordinates.
A Taylor expansion of the so-called lens equation \eqref{eq:lenseq} yields the Jacobian of the $\vec{\theta} \rightarrow \vec{\beta}$ mapping, which defines the magnification tensor \citep[e.g.][]{BS01}
\begin{equation}\label{eq:jacobi}
a_{ij}(\vec{\theta}) = \frac{\partial \vec{\beta}}{\partial \vec{\theta}}= \left( \delta_{ij} - \phi_{,ij}\right) \equiv \left(\begin{array}{cc} 1 - \kappa - \gamma_1 & -\gamma_2 \\ -\gamma_2 & 1 - \kappa + \gamma_1\end{array}\right)\;,
\end{equation}
where $\delta_{ij}$ is the Kronecker symbol, and the two components $\gamma_{1/2}$ of the complex spin-2 shear have been introduced. Note that subscripts following a comma denote partial derivatives along that coordinate. Both shear and convergence are first derivatives of the deflection field $\vec{\alpha}$ (or second derivatives of the lensing potential)
\begin{eqnarray}\label{eq:kg2psi}
\kappa & = &\frac{1}{2} ( \alpha_{1,1} + \alpha_{2,2}) \,,\\
\gamma_1 &= &\frac{1}{2} ( \alpha_{1,1} - \alpha_{2,2})\,,\\
\gamma_2 &= & \alpha_{1,2} = \alpha_{2,1} \,.
\end{eqnarray}
Therefore, starting from pixelised maps of the deflection field $\alpha_{1/2}(i,j)$ in a thin slice of the lightcone, one can easily derive $\gamma_{1/2}(i,j)$ and $\kappa(i,j)$ with finite differences or Fast Fourier Transforms (FFTs), even if $\alpha$ is only known on a finite aperture, without periodic boundary conditions.
Conversely, starting from a convergence map $\kappa(i,j)$, it is impossible to integrate \eqref{eq:poisson} with FFTs to get $\alpha$ (and then differentiate again to get $\gamma$) without introducing edge effects, if periodic boundary conditions are not satisfied.
Additionally, we also introduce the scalar magnification $\mu$ which is the inverse determinant of the magnification tensor $a_{i,j}$ of Eq.~\eqref{eq:jacobi}.
\subsection{Propagation of rays in a continuous lumpy Universe}
On cosmological scales, light rays cross many over/under-dense extended regions at different locations. Therefore, the thin lens approximation does not hold.
The transverse deflection induced by an infinitely thin lens plane is still given by the above equations but one needs to fully integrate the trajectory of rays along their path. Therefore, for a given source plane at comoving distance $\chi_s$, the source plane position of a ray, initially observed at position $\vec{\theta}$ is given by the continuous implicit (Voltera) integral equation \citep{Jain1997}:
\begin{equation}\label{eq:fullpropagation}
\vec{\beta}(\vec{\theta}, \chi_{\rm s}) = \vec{\theta} -\frac{2}{c^2} \int_0^{\chi_{\rm s}} {\rm d}\chi\: \frac{\chi_{\rm s}-\chi}{\chi_{\rm s} \ \chi} \vec{\nabla}_{\beta} \phi \left(\vec{\beta}(\vec{\theta},\chi), \chi\right) \,.
\end{equation}
To first order, one can evaluate the gravitational potential along an unperturbed path, so that:
\begin{equation}\label{eq:Bornpropagation}
\vec{\beta}(\vec{\theta}, \chi_{\rm s}) = \vec{\theta} -\frac{2}{c^2} \int_0^{\chi_{\rm s}} {\rm d}\chi\: \frac{\chi_{\rm s}-\chi}{\chi_{\rm s} \ \chi} \vec{\nabla}_{\theta} \phi \left(\vec{\theta},\chi\right) \,.
\end{equation}
This is known as the \textit{Born approximation}, which is common in many diffusion problems of physics.
An interesting property of the Born Approximation is that the relation between $\vec{\beta}$ and $\vec{\alpha}$ can be reduced to an effective thin lens identical to \eqref{eq:lenseq} allowing the definition of an effective convergence, which is the divergence of the effective (curl-free) deflection field: $2 \kappa_{\rm eff} = \vec{\nabla}.\vec{\alpha}_{\rm eff}$.
When the approximation does not hold, the relation between $\vec{\beta}$ and $\vec{\alpha}$ can no longer be reduced to an effective potential and some curl-component may be generated, implying that the magnification tensor is no longer symmetric but requires the addition of a rotation term and so-called B-modes in the shear field. In this more general framework, the magnification tensor should be rewritten
\begin{equation}\label{eq:jacobiROT}
a_{ij}(\vec{\theta}) = \left(\begin{array}{cc} 1 - \kappa - \gamma_1 & -\gamma_2 -\omega \\ -\gamma_2 + \omega & 1 - \kappa + \gamma_1\end{array}\right)\;.
\end{equation}
with the following definitions of the new lensing rotation term $\omega$ (and revised $\gamma_2$)
\begin{eqnarray}\label{eq:kg2aROT}
\gamma_2 & = &\frac{1}{2} ( \alpha_{1,2} + \alpha_{2,1}) \,,\\
\omega & = &\frac{1}{2} ( \alpha_{1,2} - \alpha_{2,1}) \,.
\end{eqnarray}
The image plane positions where $\omega\ne 0$ are closely related to the lines of sight along which some substantial lens-lens coupling may have occurred.
\subsection{The multiple lens planes approximation}
The numerical transcription of equation~\eqref{eq:fullpropagation} in the Horizon-AGN~past line-cone requires the slicing of the latter into a series of parallel transverse planes, which could simply be the 22,000 slabs dumped by {\tt RAMSES}~at runtime every coarse time step. These are too numerous and can safely be stacked into thicker planes by packing together 40 consecutive slabs\footnote{This number was chosen as a tradeoff between the typical number of CPU cores in the servers used to perform the calculations and the preservation of the line-of-sight native sampling of lightcone.}. Here 500 slices of varying comoving thickness are produced all the way to redshift $z=7$ to compute either the deflection field or the projected surface mass density as described below.
The discrete version of the equation of ray propagation \eqref{eq:fullpropagation} for a fiducial source plane corresponding to the distance of the plane $j+1$ reads:
\begin{equation}\label{eq:crude_multi_plane}
\vec{\beta}^{j+1} = \vec{\theta} -\sum^{j}_{i=1} \frac{D_{i;j+1}}{D_{j+1}} \vec{\alpha}^i (\vec{\beta}^i)\,,
\end{equation}
where $\vec{\alpha}^i $ is the deflection field in the lens plane $i$, $D_{j+1}$ is the angular diameter distance between the observer and the plane $j+1$, and $D_{i;j+1}$ the angular diameter distance between planes $i$ and $j+1$.
Therefore, as sketched in Fig.~\ref{fig:raytracing}, rays are recursively deflected one plane after the other, starting from unperturbed positions on a regular grid $\vec{\theta} \equiv \vec{\beta}^1$.
The practical implementation of the recursion in equation~\eqref{eq:crude_multi_plane} is computationally cumbersome and demanding in terms of memory because the computation of the source plane positions $\vec{\beta}^{j+1}$ requires holding all the $j$ previously computed source plane positions. Instead, this paper follows the approach of \citet{hil++09}, who showed that equation~\eqref{eq:crude_multi_plane} can be rewritten as a recursion over only three consecutive planes\footnote{This recursion requires the introduction of an artificial $\vec{\beta}^0\equiv\vec{\beta}^1 = \vec{\theta}$ slice in the initial setup.}
\begin{equation}\label{eq:multi_plan}
\vec{\beta}^{j+1}= \left( 1 - \frac{D_{j}}{D_{j+1}} \frac{D_{j-1;j+1}}{D_{j-1;j}} \right) \vec{\beta}^{j-1} + \frac{D_{j}}{D_{j+1}} \frac{D_{j-1;j+1}}{D_{j-1;j}} \vec{\beta}^{j} - \frac{D_{j;j+1}}{D_{j}} \vec{\alpha}^{j} (\vec{\beta}^{j} )\,.
\end{equation}
\begin{figure}[h!]
\includegraphics[width=0.49\textwidth]{raytracing}
\caption{Schematic view of the propagation of a light ray through a lightcone sliced into multiple discrete lens planes. The ray (red line) is deflected at each intersection with a thin lens plane. The deflection field is defined for each plane depending of the angular position on this plane $\vec{\alpha}^{j}(\vec{\beta}^{j})$.}
\label{fig:raytracing}
\end{figure}
Besides this thorough propagation of light rays
source plane positions and associated quantities (convergence $\kappa$, shear $\gamma$, rotation $\omega$) are additionally computed using the Born approximation, following the discrete version of equation~\eqref{eq:Bornpropagation}:
\begin{equation}\label{eq:multi_plan_born}
\vec{\beta}^{j+1} = \vec{\theta} -\sum^{j}_{i=1} \frac{D_{i;j+1}}{D_{j+1}} \vec{\alpha}^i (\vec{\theta})\,
\end{equation}
The deflection maps in each lens plane are computed on a very fine grid of pixels of constant angular size. In order to preserve the $\sim 1 \, {\rm kpc}$ spatial resolution allowed by the simulation at high redshift, $36,000\times 36,000$ deflection maps are built in the narrow 1 deg lightcone. The deflection maps in the low redshift 2.25 sq deg wide cone reaching $z=1$ are computed on a coarser $20,000\times20,000$ pixels grid since the actual physical resolution of the simulation at low redshift does justify the $0.1 \textrm{ arcsec}$ resolution of the narrow 1 sq deg field-of-view. Even though the image plane positions $\vec{\theta}=\vec{\beta}^1$ are placed on the regular pixel grid, the deflections they experience must be interpolated in between the nodes of the regular deflection map as they progress backward to a given source plane. This is done with a simple bilinear interpolation scheme.
\subsection{Total deflections from the {\tt RAMSES}~accelerations}\label{ssec:OBB}
Let us now describe how to obtain $\alpha$ to use it in Eqs.~\eqref{eq:multi_plan} and \eqref{eq:multi_plan_born}.
The first method uses the gravitational acceleration field which is registered on each (possibly-refined) grid location inside the lightcone. The very same gravitational field that was used to move particles and evolve Eulerian quantities in {\tt RAMSES}~is interpolated at every cell position and is therefore used to consistently derive the deflection field. The merits of the complex three-dimensional multi-resolution Poisson solver are therefore preserved and the transverse components of the acceleration fields can readily be used to infer the deflection field.
By integrating the transverse component of the acceleration along the light of sight, one can compute the deflection field according to equation~\eqref{eq:alphadef}.
To do so, for each light ray, gas cells which intersect the ray are considered, and the intersection length along the line-of-sight $l_i$ is computed.
Knowing the cell size $\delta_i$, and its orientation with respect to the line of sight, $l_i$ is deduced with a simple Oriented-Box-Boundary (OBB) algorithm \citep[e.g.][]{RTR3} in which it is assumed that all cells share the same orientation (flat sky approximation) and factorise out expensive dot products between normals to cell edges and the line of sight.
\begin{equation}
\vec{\alpha} (\vec{\theta})= \frac{2}{c^2} \sum_{i \in \mathcal{V}(\vec{\theta})} \vec{\nabla_{\perp}}\phi_i (\vec{\theta}) \ l_i\;,
\end{equation}
where $\mathcal{V}(\vec{\theta})$ denotes the projected vicinity of a sky position $\vec{\theta}$.
As shown in fig \ref{fig:tetris}, a fiducial light ray is drawn: at each lens plane, the deviation of the light is calculated as the direct sum of the transverse acceleration components recorded on the cells $i$, weighted by the intersection length $l_i$.
Here, the field of view is small and one can safely assume that light rays share the same orientation (flat sky approximation) and are parallel to the line-of-sight.
This method has the main advantage of preserving the gravitational force that was used when evolving the simulation. In particular, the way shot noise is smoothed out in the simulation to recover the acceleration field from a mixture of Lagrangian particles and Eulerian gas cells is faithfully respected in the raytracing.
In other word, the force felt by photons is very similar to the one felt by particles in the simulation.
Dealing with acceleration is also local, in the sense that the deflection experienced by a light ray (and related derivatives leading to e.g. shear and convergence) depends only on the acceleration of cells this ray crosses. The mass distribution outside the lightcone is therefore consistently taken into account via the acceleration field.
However, this method is sensitive to small artefacts which are present at the lightcone generation stage (i.e. simulation runtime) and which could not be corrected without a prohibitive post-processing of the lightcone outputs. When the simulation dumps two given neighbouring slabs at two consecutive time steps, problems can happen if cells on the boundary between the two slabs have been (de-)refined in the mean time. As illustrated in Fig.~\ref{fig:tetris}, such cells can be counted twice or can be missing, if they are refined (or derefined) at the next time step. Those bumps and dips in the deflection map translate into saw-tooth patterns in the convergence maps. They are however quite scarce and of very modest amplitude.
\begin{figure}[h!]
\includegraphics[width=0.49\textwidth]{tetris2.pdf}
\caption{Schematic view of the problem induced by cells at the boundary of slabs $j$ and $j+1$, which get refined between time $t$ and $t+{\rm d} t$. Missing cells (devoid of dots) or cells in excess (overlapping "dotted" cells of different colour) can end up as lightcone particles. A fiducial light ray is drawn to illustrate the intersection length $l_i$ between the ray and RAMSES cells.}
\label{fig:tetris}
\end{figure}
A 100 arcsec wide zoom into the convergence map obtained with this method is shown in the left panel of Fig.~\ref{fig:metcomp}. The source redshift is $z_{\rm s}=0.8$. A few subdominant artefacts due to missing acceleration cells are spotted.
They induce small correlations on scales smaller than a few arcsec and are otherwise completely negligible for our cosmological applications.
\begin{figure*}[h!]
\includegraphics[width=\textwidth]{kappa_comparison.png}
\caption{Comparison of $z_{\rm s}=0.8$ convergence maps obtained with the OBB method (integration of transverse accelerations in cells, {\it left}) and with the SPL method (projection of particles onto convergence planes after adaptative Gaussian smoothing, {\it right}). The latter method applies a more aggressive smoothing which better erases shot noise. Inaccuracies of long range deflections in the SPL method due to edge effects translate into a global shift for some galaxies, as compared to OBB. With this method, some missing acceleration cells produce modest artefacts on small scale, here and there.}
\label{fig:metcomp}
\end{figure*}
\subsection{Projection of smoothed particle density}\label{ssec:SPL}
The second method of computing the deflection maps in thin lens planes is more classical: it relies on the projection of particles onto surface density maps which are then turned into deflection maps. If the line-of-sight integration is performed under the Born approximation, the Fourier inversion going from the projected density to the deflection is just done once starting from the effective convergence. Otherwise, with the full propagation, many FFTs inversions on projected density maps that do not fulfil the periodic boundary condition criterion imply an accumulation of the inaccuracies in the Fourier inversion.
First of all, this method allows us to separate the contribution of each matter component to the total deflection field. One can therefore compute the contribution of stars or gas to the overall lensing near a given deflector, something that is not possible with the acceleration method since only the total acceleration is computed by the simulation.
In addition, one can project particles with an efficient and adaptive smoothing scheme. Instead of a standard nearest grid point or cloud-in-cell projection, a gaussian filter (truncated at 4 times the standard deviation $\sigma$) is used in which the width of the smoothing filter $\sigma$ is tuned to the local density, hence following the Smooth Particle Lensing (SPL) method of \citet{Aubert++07}. Since the AMR grid of {\tt RAMSES}~is adaptive, the resolution level around a given particle position from the neighbouring gas cells can be recovered. This thus bypasses the time consuming step of building a tree in the distribution of particles, which is at the heart of the SPL method.
To illustrate the merits of this method and for comparison with the previous one, let us show the same region of simulated convergence fields for a source redshift $z_{\rm s}=0.8$ in the right panel of Fig.~\ref{fig:metcomp}.
This adaptive gaussian smoothing (referred to as SPL method below) seems more efficient at smoothing the particle noise out. Between the two methods, we notice small displacements of some galaxies of a few arcsec. They are due to the long range inaccuracies generated by the Fourier inversions.
\subsection{Lensing of galaxy and halo catalogues}\label{ssec:defl-points}
In order to correlate galaxies (or halos) in the lightcone with the convergence or shear field around them and, hence, measure their GGL, one has to shift their catalogue positions $\vec{\beta}$ (which are intrinsic source plane coordinates) and infer their observed lensed image plane positions $\vec{\theta}$. These are related by the thorough lens equation \eqref{eq:fullpropagation}, or its numerical translation \eqref{eq:crude_multi_plane}. However, this equation is explicit for the $\theta\rightarrow \beta$ mapping, only. The inverse relation, which can be multivalued when strong lensing occurs, has to be solved numerically by testing for every image plane mesh $\vec{\theta}_{ij}$ whether it surrounds the coordinates $\vec{\beta}^{\rm gal}$ of the deflected galaxy when cast into the source plane $\vec{\beta}_{ij}$ \citep[e.g.][]{SEF92,keeton01soft1,bartelmann03c}. Because the method should work in the strong lensing regime, regular rectangular meshes may no longer remain convex in the source plane and, therefore, it is preferable to split each mesh into two triangles. Those triangles will map into triangles in the source plane and one can safely test whether $\vec{\beta}^{\rm gal}$ is inside them. In order to speed up the test on our large pixel grids, the image plane is partitioned into a quad-tree structure that recursively explore finer and finer meshes. The method is actually very fast and yields all the image plane antecedents of a given galaxy position $\vec{\beta}^{\rm gal}$. This provides us the updated catalogues of halos and galaxies.
Obviously, when measuring the GGL signal in the Born approximation, catalogue entries do not need to be deflected and therefore source plane and image plane coordinates are identical.
\subsection{Summary of generated deflection maps}\label{ssec:subsum}
Table \ref{tab:pros_cons} summarises the main advantages and drawbacks of the OBB and SPL methods.
Altogether, $2 \times2 $ (OBB/SPL and Born approximation/full propagation) deflection maps were generated for each of the 246 source planes all the way to $z=1$ in the wide opening angle field. Likewise, we obtained $2 \times 2 $ maps for each of the 500 source planes all the way to $z=7$ in the narrow opening angle field.
\begin{table*}{
\begin{tabular}{C{3.5cm} C{7cm} C{7cm} }
\hline
& OBB & SPL \\
\hline\hline
Deflection (per plane) & integration of transverse acceleration & particles adaptively smoothed and projected\\
&& onto density planes \\
Large scale & matter outside the lightcone is taken into account & Edge effects due to Fourier Transforms \\
Small scale & uses the multi-scale {\tt RAMSES}~potential & smoothing reduces small-scale features\\
Cells missing/in excess & produces small scale artefacts & unaffected \\
Matter component & only for the total matter & can individually consider DM, stars, and gas \\
\hline\hline
\end{tabular}
\caption{Summary of the main properties of the SPL and OBB methods ray-tracing methods.}
\label{tab:pros_cons}}
\end{table*}
\section{Cosmic shear}\label{sec:cosmicshear}
This section assesses the validity of our ray-tracing methods by measuring 1-point and 2-point statistics of the lensing quantities like convergence, and (reduced-)shear. It also compares those finding with other methods.
The focus is on the impact of baryons on small scales for multipoles $\ell\gtrsim 2000$ to check whether the baryonic component couples to other non-linear effects like the shear -- reduced shear correction and Beyond-Born treatments.
\subsection{Convergence 1-point statistics}
The most basic quantity that one can derive from the convergence field shown in the right panel of Fig.~\ref{fig:convergence_map} is the probability distribution function (PDF) of the convergence. The Fig.~\ref{fig:convergence_map} shows this quantity which is extremely non-Gaussian at the $\sim 1\arcsec$ resolution of the map. One can see the skewness of the field with a prominent high-end tail and a sharp fall off of negative convergence values.
\begin{figure*}[h]
\begin{minipage}{0.7\textwidth}
\includegraphics[width=0.8\textwidth]{zs1.pdf}
\end{minipage}
\begin{minipage}{0.67\textwidth}
\hspace*{-0.85cm} \includegraphics[width=0.4\textwidth]{zs2.pdf} \\
\hspace*{-2.cm} \includegraphics[width=0.6\textwidth]{convergence_pdf_z2.pdf}
\end{minipage}
\caption{ {\it Left panel:} Convergence map generated with a $0\farcs1$ pixel grid over a $2.25 \times 2.25$ square degrees field of view for a fiducial source plane at $z_{\rm s}\sim~1$.
{\it Right panel:} Convergence map with a field of view of $1$ sq. deg. at $z_s \sim 2$, and its corresponding convergence PDF showing the characteristic skewed distribution.
}
\label{fig:convergence_map}
\end{figure*}
\subsection{Convergence power spectrum}
In Fourier space, the statistical properties of the convergence field are commonly characterised by its angular power spectrum $P_\kappa (l)$,
\begin{equation}
\langle \hat{\kappa}(\vec{\ell}) \, \hat{\kappa}^*(\vec{\ell'}) \rangle = (2 \pi)^2\, \delta_{D}(\vec{\ell}-\vec{\ell'})\, P_{\kappa}(\ell)\,.
\end{equation}
where $\delta_{D}(\vec{\ell}$ is the Dirac delta function.
For two fiducial source redshifts ($z_{\rm s}=0.5$ and $z_{\rm s}=1$), Fig.~\ref{fig:pkappa} shows the angular power spectrum of the convergence obtained with the two ray tracing techniques: the OBB and SPL methods (respectively solid magenta and solid cyan curves). The low redshift ones are based on the $2.25\,{\rm deg}$ wide lightcone. They are thus more accurate on larger scales $\ell \lesssim 10^3$, even though the large sample variance will not permit quantitative statements.
On small scales ($\ell \sim 2\times 10^5$), the additional amount of smoothing implied by the SPL projection of particles onto the lens planes induces a deficit of power with respect to the less agressive softening of the OBB method in which shot noise has not been entirely suppressed (see Fig.~\ref{fig:metcomp}).
The middle panel of Fig.~\ref{fig:pkappa} shows the difference between power spectra inferred using the Born approximation or with the full multiple lens plane approach for the OBB method.
For angular scales $ \ell \lesssim 8 \times 10^4$, we find differences between the two propagation methods that are less than 0.5\%, or so, which is totally negligible given possible numerical errors and sampling variance limitations.
At lower angular scales $ \ell \gtrsim 10^5$, departures rise above the few percent level. Note that this scale also corresponds to scale where shot noise (from DM particles) and convergence power spectral are of equal amplitude (yellow shaded area). Below these very small scales, close to the strong lens regime, the Born approximation may start to break down \citep{Schafer2012}.
Under the Limber and Born approximations, one can express the convergence power spectrum as an integral of the three-dimensional non-linear matter power spectrum $P_\delta$ \citep{Limber,blandford91,miralda91,K92} from the observer to the source plane redshift or corresponding co-moving distance $\chi_{\rm s}$:
\begin{equation}\label{eq:limber}
P_{\kappa}(\ell) = \left( \frac{3 \Omega_{\rm m} H_0^2 }{2c^2} \right)^2 \int_0^{\chi_{\rm s}} {\rm d}\chi \left( \frac{\chi (\chi_{\rm s} - \chi)}{\chi_{\rm s} a(\chi)}\right)^2 P_{\delta}\left(\frac{\ell}{\chi},\chi \right) \,,
\end{equation}
where $a$ is the scale factor and where no spatial curvature of the Universe was assumed for conciseness and because the cosmological model in Horizon-AGN~is flat.
As a validation test of our light deflection recipes, the lensing power spectrum derived from the actual ray-tracing is compared to an integration of the three-dimensional matter power spectrum measured by \citet{Chisari++18} in the Horizon-AGN~simulation box. The red curve is the direct integration of $P_\delta(k)$ power spectra and the dashed parts of the lines corresponds to a power-law extrapolation of the $P_\delta(k)$ down to smaller scales.
In the range $3\,000 \lesssim \ell \lesssim 3\times 10^5$, an excellent agreement is found between the red curve and the spectra inferred with our two ray-tracing techniques. On larger scales, the cosmic variance (which is different in the full simulation box and the intercept of the box with the lightcone) prevents any further agreement. This is also the case for $\ell \gtrsim 3\times 10^5$ where some possibly left over shot noise in the raytracing maps and the hazardous high-$\ell$ extrapolation of the three-dimensional power spectra complicate the comparison. In addition, the low-$\ell$ oscillations of the spectrum is likely to originate from the replicates of the simulation box throughout the past lightcone.
\citet{Chisari++18} also measured matter power spectra in the Horizon-DM simulation at various redshifts. This simulation is identical to Horizon-AGN~in terms of initial conditions but has been run without any baryonic physics in it after having rescaled the mass of DM particles to conserve the same total matter density \citep{Peirani++17,Chisari++18}. The integration of this DM-only power spectrum allows to get a sense on the effect of baryons in the DM-distribution itself.
Just like the red curve was showing the result of the Limber integral in equation~\eqref{eq:limber} for Horizon-AGN, the dark blue curve shows the same integral for Horizon-DM. The latter has much less power for $\ell \gtrsim 2\times 10^4$ than either the integration of the full physics Horizon-AGN~matter power spectrum (red) or that derived directly from ray-tracing (purple or green). The boost of spectral amplitude is due to cool baryons in the form of stars at the center of halos. Moreover, we notice a deficit of power on scales $2\times 10^3 \lesssim \ell \lesssim 2\times 10^4$ for the full physics simulation. As pointed out by \citet{Semboloni++11}, the pressure acting on baryons prevents them from falling onto halos as efficiently as dark matter particles, hence reducing the depth of the potential wells, when compared to a dark-matter only run. This effect has already been investigated with more sensitivity on the three-dimensional matter power spectrum in the Horizon-AGN simulation \citep{Chisari++18}, and a clear dip in the matter density power spectrum of the full physics simulation is observed on scales $1 \lesssim k \lesssim 10 \, h \, {\rm Mpc}^{-1}$. Here, the projection somewhat smears out this dip over a larger range of scales but a $\sim 15\%$ decrease in amplitude is typically observed for $\ell=10^4$ at $z_{\rm s}=0.5$. In order to better see the changes due to the inclusion of the baryonic component, we traced rays through the lightcone by considering only the DM particles of the Horizon-AGN~run with the SPL method. For this particular integration of rays trajectories, we multiplied the mass of the dark matter particules by a factor $1+\Omega_{\rm b}/\Omega_{\rm DM}$ (where $\Omega_{\rm DM}=\Omega_{\rm m}-\Omega_{\rm b}$) to get the same overall cosmic mean matter density. The cyan curve in the upper panel shows the resulting convergence power spectrum.
The ratios between the total full physics convergence power spectrum and the rescaled dark matter contribution of this power spectrum at $z_{\rm s}=0.5, 1.0$ and $1.5$ are shown in the bottom panel and further illustrates the two different effects of baryons on intermediate and small scales.
By considering two raytracing methods to derive the convergence power spectrum, and by asserting that consistent results are obtained by integrating the three-dimensional matter power spectrum, let us now look for small scale effects involving the possible coupling between the baryonic component and shear -- reduced shear corrections.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{pk_final2.pdf
\caption{{\it Upper panel:} Convergence power spectra for source redshift $z_{\rm s}=1$ (top) and $z_{\rm s}=0.5$ (bottom) derived with the OBB (magenta) and the SPL (green) methods. The more aggressive smoothing of this latter method translates into a faster high-$\ell$ fall-off. The cyan curves (DM) only account for the dark matter component (rescaled by $1+\Omega_{\rm b}/ \Omega_{\rm M}$). The red curve corresponds to the direct integration of the 3D total matter power spectrum (Limber approximation) in the Horizon-AGN simulation (Hz-AGN). The blue curves is the direct integration of the Horizon-DM (dark matter only) matter power spectrum (Hz-DM). Dashes reflect regimes where the 3D spectra of \citet{Chisari++18} was extrapolated by a simple power law (extrapolation).
The yellow lines show the particle shot-noise contribution at two different redshifts.
{\it Middle panel:} ratio of the $z_{\rm s}=0.5$ convergence power spectra obtained with the Born approximation and the proper multiple lens plane integration showing only very small changes up to $\ell\sim 10^5$. {\it Bottom panel:} ratio of the dark-matter only to total convergence power spectra at $z_{\rm s}=0.5, \ 1.0$ and $1.5$ for the SPL method.
\label{fig:pkappa}
\end{figure}
\subsection{Shear -- reduced shear corrections to 2-point functions}
In practical situations, rather than the convergence power spectrum, which is not directly observable, wide fields surveys give access to the angular correlation of pairs of galaxy ellipticities. The complex ellipticity\footnote{$\varepsilon= (a-b)/(a+b) {\rm e}^{2i \varphi}$, with $a$ and $b$, respectively, the major and minor axis of a given galaxy, and $\varphi$ is the orientation of the major axis.} $\varepsilon$ is directly related to the shear $\gamma$. The relation between the ensemble mean ellipticity and the shear is in fact
\begin{equation}
\langle \varepsilon \rangle = g \equiv \frac{\gamma}{1-\kappa} \simeq \gamma \,,
\end{equation}
with, $g$, the so-called reduced shear. Therefore, the two point correlations of ellipticities and shear only match when the convergence $\kappa$ is small. Since the regions of large convergence are typically the centres of halos where the contribution of cooled baryons is highest, one might expect a coupling between the inclusion of baryons and the shear reduced-shear corrections needed to properly interpret the cosmological signal carried by the 2-point statistics \citep[e.g.][]{White05,Kilbinger10}
Owing to the spin-2 nature of ellipticity, one can define the angular correlation functions $\xi_\pm$
\begin{eqnarray}
\xi_\pm(\theta) &= &\langle \gamma_+(\vartheta+\theta) \gamma_+(\vartheta) \rangle_\vartheta \pm \langle \gamma_\times(\vartheta+\theta) \gamma_\times(\vartheta) \rangle_\vartheta \label{eq:xipmdef1} \,,\\
& =& 2 \pi \int {\rm d}\ell\, \ell J_{0/4}(\theta \ell) P_\kappa(\ell) \label{eq:xipmdef2} \,,
\end{eqnarray}
where $\gamma_+$ and $\gamma_\times$ are defined with respect to the separation vector between two galaxies or, here any two image plane positions at separation $\theta$. $J_0$ and $J_4$ are 0th and 4th order Bessel functions.
Instead of the shear, observers can only measure associated ellipticities $\epsilon$, which should thus replace $\gamma$ in equation~\eqref{eq:xipmdef1} in practical measurements.
The reduced shear maps were computed together with shear and convergence maps, so as to measure the modified $\xi_+$ and $\xi_-$ angular correlations to compare them with the actual correlation functions. For efficiency, the {\tt Athena} code\footnote{\url{http://www.cosmostat.org/software/athena}} was used to compute correlation functions.
The results are shown in Fig.~\ref{fig:xipm} for a fiducial source redshift $z_{\rm s}=0.5$. Here $\xi_+^g$ and $\xi_+^\gamma$ only depart from one another at the $\sim 2-3\%$ level on angular separations $\sim 1\arcmin$. The effect is slightly stronger for $\xi_-$ which is known to be more sensitive to smaller non-linear scales than $\xi_+$, but also more difficult to measure in the data because of its lower amplitude. On $1\arcmin$ scales, $\xi_-^g/\xi_-^\gamma -1 \simeq 7-8\%$.
Like for the power spectra in the previous subsection, the cyan curves represent the correlations $\xi_\pm^\gamma$ for the rescaled DM contribution. The bottom panel shows the ratio of rescaled DM over full physics reduced shear correlation functions, further illustrating the effect of baryons on small scales. Again, $\xi_-$ responds more substantially to the inclusion of baryons. The deficit of correlation amplitude when baryons are taken into account peaks at $3-4\arcmin$ and is of order 10\%. Below $1\arcmin$, the effect starts to increase but those scales are never used in practical cosmic shear applications.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{xi_spl_zs05.pdf}
\caption{{\it Upper panel:} Two-point shear correlation functions $\xi_+$ (solid lines) and $\xi_-$ (dotted lines) for a fiducial source redshift $z_{\rm s}=0.5$. We either correlate actual shear (red) or reduced shear (green) in the calculation to highlight the small scale impact of baryons on this non linear correction. {\it Middle panel:} ratio of shear correlation functions for the two cases. {\it Bottom panel:} ratio of shear correlation functions for a raytracing that only includes rescaled DM particules or all the components.}
\label{fig:xipm}
\end{figure}
As shown in the next section, those scales remain perfectly relevant for galaxy evolution studies by means of the Galaxy-Galaxy weak lensing signal.
\section{Galaxy-Galaxy lensing}\label{sec:GGL}
Focussing further into dark matter halos, let us now investigate the yields of the simulation in terms of the galaxy-galaxy weak lensing signal. The tangential alignment of background galaxies around foreground deflectors is substantially altered by the aforementioned baryonic physics, and one also expects a strong signature in this particular lensing regime.
For a circularly symmetric mass distribution $\Sigma(R)$, one can relate shear, convergence and the mean convergence enclosed inside a radius $R$ centred on a foreground galaxy or halo as:
\begin{equation}
\bar{\kappa}(<R) = \frac{2}{R^2} \int_0^R \kappa(R') R' {\rm d} R' = \kappa(R)+ \gamma(R)\;.
\end{equation}
Using the definition of the critical density \eqref{eq:scrit}, one can define the excess density
\begin{eqnarray}
\Delta \Sigma(R) & = & \frac{M(<R)}{\pi R^2} - \Sigma(R) \label{eq:massexcess1} \,,\\
& = & \Sigma_{\rm crit} \gamma(R)\;. \label{eq:massexcess2}
\end{eqnarray}
The previous section already showed that the lensing convergence or shear maps have adequate statistical properties, while Sect.~\ref{ssec:defl-points} showed how to use the associated deflection maps to map our lightcone galaxy catalogue into the image plane. In addition, galaxies should also get magnified when lensed. Future extensions of this work will include the realistic photometry of the Horizon-AGN~galaxies. One can however easily account for the magnification bias by multiplying stellar masses by the magnification $\mu$, as if luminosity or flux were a direct proxy for stellar mass. In the following, we shall refer to $M_*$ for the intrinsic and $\mu M_*$ for the magnified mass proxy.
For any given source redshift, averages of the tangential shear around galaxies of any given stellar mass $M_*$ or more realistically magnified stellar mass $\mu M_*$. This is done around deflected galaxy positions.
\subsection{Comparison with CMASS galaxies}\label{ssec:GGLlow}
Let us first make a comparison of the GGL around Horizon-AGN~galaxies with the GGL excess mass profiles obtained by \citet{Lea++17} who analysed the spectroscopic CMASS sample of massive galaxies in the footprint of the CFHTLS and CS82 imaging surveys, covering $\sim 250\,{\rm deg}^2$.
These authors paid particular attention to quantifying the stellar mass of the CMASS galaxies centred around lens redshift $z\sim0.55$. The CMASS sample is not a simple mass selection, and includes a set of colour cuts, which makes this just a broad brush comparison. These results are somewhat sensitive to the detailed distribution in stellar mass above that threshold. The sample mean mass only slightly changes with redshift but remains close to $3 \times 10^{11} {\rm M}_{\odot}$.
In order to match this lens sample, we extract from the wide low redshift lightcone the galaxies in the redshift range $0.4\le z \le 0.70$, and with a stellar mass above a threshold that is chosen to match the CMASS mean stellar mass. Even though these galaxies centred around lens redshift $z\sim 0.52$ are treated as lens galaxies, they experience a modest amount of magnification (they behave like sources behind the mass distribution at yet lower redshift, see Sect.~\ref{ssec:magnif}). We thus pick galaxies satisfying $\mu M_* > 1.7 \times 10^{11} {\rm M}_{\odot}$. At this stage, selecting on $M_*$ or $\mu M_*$ does not make any significant difference ($\lesssim 4\%$) because of the relatively low redshift of the lens sample. By doing so, we obtain the same sample mean stellar mass as the CMASS sample.
We now measure the mean tangential shear around those galaxies for a fiducial, unimportant, source redshift $z_{\rm s}=1$ and convert shear into excess density $\Delta \Sigma$.
The result can be seen in Fig.~\ref{fig:ggl-prof}. A good agreement between our predictions (OBB method, green with lighter envelope) and the observations of \citet{Lea++17} (blue dots) is found,
further suggesting that Horizon-AGN~galaxies live in the correct massive halos ($M_{\rm h} \simeq 10^{13} {\rm M}_{\odot}$), or at the very least, produce the same shear profile as CMASS galaxies around them. Note that we split the 2.25 deg field of view into 4 quadrants and used the dispersion among those areas to compute a rough estimate of model uncertainties.
On scales $R\lesssim 0.2 \hmm\Mpc$, the shear profile is 10-15\% above the observations. Answering whether the discrepancy is due to faulty subgrid baryonic physics, a missing cosmological ingredient (or not perfectly adequate cosmological parameters) or leftover systematics in the data will certainly require more GGL observations, possibly combined with yet smaller scale strong lensing and kinematical data \citep[e.g.][]{Sonnenfeld++18}. Small scale GGL is definitely a unique tool to address those issues \citep[e.g.][]{Velliscig++17}, and asserting that the galaxy-halo connection is correctly reproduced by the simulations all the way to $z\gtrsim 1$, is arguably one of the foremost goals of galaxy formation models.
Fig.~\ref{fig:ggl-prof} also shows our GGL results for the same population of lenses at the same redshift but as inferred from the SPL method (solid black) which allows to split the total lensing signal into its dark matter (blue and baryonic components (red). First of all, we do see a remarkable agreement between the two methods for the total lensing signal, except on scales $\gtrsim 2 \, {\rm Mpc}\sim 5\arcmin$ where differences start exceeding the percent level.
As already mentioned in the previous section, this is due to inaccuracies of the Fourier transforms performed with the SPL method.
We can however use this latter technic to compare the contribution of DM and baryons (stars+gas). Clearly, the total and DM profiles look very similar beyond $\sim 0.2\, {\rm Mpc}$ up to a $\sim17\%$ renormalisation of the matter density. It is only below those scales that cooled baryons (stars) start playing a substantial contribution. We predict an equal contribution of DM and stars to the total shear signal near a radius $\sim15\, {\rm kpc}$. We refer the reader to \citet{Peirani++17} for further details about the innermost density profiles around Horizon-AGN~galaxies in the context of the cusp-core problem.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{alt_ggl_prof_v2.pdf}
\caption{Comparison of the GGL tangential shear signal around $z=0.55$ Horizon-AGN ~galaxies (green curve surrounded by light-green ``ribbon'') and the GGL observations of \citet{Lea++17} (blue dots with error bars). Units are all physical (and not comoving!). Model uncertainties in the simulation past lightcone are roughly estimated by splitting the 2.25 deg wide field of view into 4 quadrants. They may be underestimated beyond $1\hmm\Mpc$. Cuts in stellar mass are expressed in units of $10^{11} {\rm M}_{\odot}$. Black, blue, and red curves show the GGL shear signal predicted with the SPL method for the total, DM, and baryonic mass distributions respectively. For clarity uncertainties are omitted. They are similar to the OBB method case (green).}
\label{fig:ggl-prof}
\end{figure}
\subsection{High redshift magnification bias}\label{ssec:magnif}
For $z_{\rm l}\gtrsim 0.6$, the lens population starts being lensed by yet nearer structures. This can lead to a magnification bias, which was studied by \citet{Z+H08}.
The spatial density of a lensed population of background sources can be enhanced or decreased by magnification as light rays travel through over- or under-dense sight-lines \citep[eg][]{moessner98b,moessner98a,menard02,scranton05}. Furthermore, the fraction of sources that are positively or negatively magnified depends on the slope of the luminosity function of the population. If it is very steep (typically the bright end of a population) one can observe a dramatic increase of the number of bright lensed objects. These deflectors appear brighter than they actually are. Fig.~\ref{fig:ggl-mubias} shows the mean magnification experienced by Horizon-AGN~lightcone galaxies above a given stellar mass threshold (mimicking a more realistic flux limit) as a function of redshift and minimum mass.
The upper panel does not take into account the effect of magnification bias whereas the lower panel does.
The ones that are consistently magnified and pass a given threshold (bottom panel) are slightly magnified on average whereas the top panel only shows a tiny constant $\mu\sim 1-3\%$ systematic residual magnification. This residual excess does not depend wether the SPL or OBB method are used, or whether we properly integrate rays or use the Born approximation. This is likely due to the replicates of the simulation box filling up the lightcone which slightly increase the probability of rays leaving an over-dense region to cross other over-dense regions on their way to the observer. This residual magnification is however tiny for sight-lines populated by galaxies and completely vanishes for rays coming for random positions.
At face value, one can see that the massive end of the galaxy stellar mass function is significantly magnification-biased. A $\sim 8\%$ effect for galaxies at $0.6\le z \le 1.2$ and $M_* \gtrsim 2 \times 10^{11} {\rm M}_{\odot}$ is typical. It can be as high at $\sim 20-50\%$ at $1.5\le z \le 2$ for $\mu M_* \gtrsim 3 \times 10^{11} {\rm M}_{\odot}$. A thorough investigation of the impact of this magnification bias when trying to put constraints on the high end of the $z\gtrsim 2$ luminosity function from observations is left for a forthcoming paper.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{ggl-mubias_off.pdf}
\includegraphics[width=0.49\textwidth]{ggl-mubias_on.pdf}
\caption{Average magnification experienced by presumably foreground deflectors accounting {\it (bottom)} or not {\it (top)} for magnification bias effect which mostly affects the rapidly declining high end of the stellar mass function. Without magnification bias, a flat nearly unity mean magnification at all redshifts is recovered to within $\sim1\%$. When the magnification bias is turned on, as expected in actual observations, no rapid rise is found ($\sim 10\%$ at $z\sim 1$ for the most massive/luminous galaxies). Cuts in stellar mass are expressed in units of $10^{11} {\rm M}_{\odot}$.}
\label{fig:ggl-mubias}
\end{figure}
Taking magnification bias into account, let us now explore three fiducial populations of massive deflectors to highlight the changes induced on projected excess density profiles. The first population consists in the aforementioned CMASS galaxies at $z=0.54$ and $\mu M_* \ge 1.7 \times 10^{11} {\rm M}_{\odot}$, the second case simply corresponds to the same lower limit on the mass but pushed to $z=0.74$. In both cases, the excess density is measured for source redshift $z_{\rm s}=0.8$. The last lens sample corresponds to the population of H$\alpha$ emitters in the $0.9\le z\le 1.8$ redshift range that will be detected by the Euclid slit-less grism spectrograph above a line flux of $\sim 2 \times 10^{-16} \, {\rm erg\, s^{-1}\, cm^{-2}}$. One expects about 2000 such sources per square degree; therefore the 2000 most massive Horizon-AGN~lightcone sources are picked in that redshift intervalle to crudely mimic an H$\alpha$ line flux selection. To account for magnification bias, the selection is made on $\mu M_*$, too, and the source redshift for this populations is set to $z_{\rm s}=2$. Results for these three populations can be seen in the top panel of Fig.~\ref{fig:ggl-prof-highz}, where we distinguish the excess density profiles accounting (dotted) or not (solid) for magnification. As anticipated, no significant change is obtained for the $z=0.54$ CMASS-like sample (green) but differences are more noticeable as lens redshift increases and on large scales ($R \gtrsim 1 \, {\rm Mpc}$), we observe a $20-50\%$ increase in $\Delta \Sigma$, consistent with the large scale linear scale-invariance bias model used by \citet{Z+H08}. Between $z=0.54$ and $z=0.74$, galaxies of the same mass seem to live in halos of the same mass (very little evolution of the $M_* - M_{\rm h}$ relation), leading to no evolution of $\Delta \Sigma$ below $\sim 200 \, {\rm kpc}$. The only difference occurs further out where the 2-halo term starts to be important in this galaxy-mass correlation function. There, galaxies of the same mass at $z=0.54$ and $z=0.74$ live in rarer excursions of the initial density field, and are thus more highly biased leading to an increase of $\Delta \Sigma$ on large scale. For the Euclid-like distant lens population, the trend is similar and the amplitude of the magnification bias effect would suggest a bias of the lens population about 30\% higher than it really is.
\begin{figure}[h]
\includegraphics[width=0.49\textwidth]{high_z_ggl_prof.pdf}
\includegraphics[width=0.49\textwidth]{high_z_ggl_prof_zs_dep.pdf}
\caption{{\it Upper panel:} Effect of magnification bias on GGL for several high-z fiducial lens samples showing an increase of excess density $\Delta \Sigma$ (or tangential shear) for $R \gtrsim 1 \, {\rm Mpc}$. Solid curves ignore the magnification whereas dotted lines account for it. {\it Lower panel:} Dependence of this effect on the source redshift. In both panels, cuts in stellar mass are expressed in units of $10^{11} {\rm M}_{\odot}$.}
\label{fig:ggl-prof-highz}
\end{figure}
The lower panel of Fig.~\ref{fig:ggl-prof-highz} shows the evolution of the magnification bias induced excess density profile with source redshift for massive deflectors at $z=0.74$. In principle, according to equation~\eqref{eq:massexcess2}, the excess density should not depend on source redshift. However, magnification bias favours the presence of over-densities in front of deflectors. The response of distance sources carrying shear to these over-densities will depend on the source redshift in a way that is not absorbed by equation~\eqref{eq:massexcess2}. Hence, a scale dependent distortion of the profiles is observed. The closer the source redshift from the deflector, the smaller the scale it kicks in. As already stressed by \citeauthor{Z+H08}, this hampers a direct application of shear-ratio tests with high redshift deflectors \citep[eg][]{J+T03}.
\section{Summary \& future prospects}\label{sec:conclusions}
Using two complementary methods to project the density or gravitational acceleration field from the Horizon-AGN~lightcone, we propagated light rays and derived various gravitational lensing observables in the simulated field of view. The simulated area is $2.25$ deg$^2$ out to $z=1$ and $1$ deg$^2$ all the way to $z=7$. The effect of baryons on the convergence angular power spectrum $P_\kappa(\ell)$ was quantified, together with the two-point shear correlations $\xi_\pm(\theta)$ and the galaxy-galaxy lensing profile around massive simulated galaxies.
For cosmic shear, the inclusion of baryons induces a deficit of power in the convergence power spectrum of order 10\% for $10^3 < \ell < 10^4$ at $z_{\rm s}=0.5$. The amplitude of the distortion is about the same at $z_{\rm s}=1$ but is slightly shifted to roughly twice as high $\ell$ multipole values. On yet higher multipoles, the cooled baryons, essentially in the form of stars, produce a dramatic boost of power, nearly a factor $2$ for $\ell \sim 10^5$. As emphasised in \citep{Chisari++18}, it is worth stressing that detailed quantitative statements on such small angular scales may still depend on the numerical implementation of baryonic processes.
For Galaxy-Galaxy lensing, the projected excess density profiles for a sample of simulated galaxies consistent with the CMASS sample at $z\sim0.52$ \citep[analysed by][]{Lea++17} were found to be in excellent agreement.
To properly analyse this signal around high redshift deflectors, the magnification bias affecting the bright end of a population of distant galaxies was carefully taken into account, showing a large scale increase of the signal as high as 30\% beyond 1 Mpc for lenses at $z\gtrsim 1$. This kind of effect is particularly pronounced for future samples of distant deflectors, such as the spectroscopic Euclid sources detected based on their H$\alpha$ line intensity.
\citet{Peirani++18} already showed that the innermost parts of Horizon-AGN~galaxies are consistent with strong lensing observations of \citet{Sonnenfeld++13} and \citet{New++13,New++15} at $z_{\rm lens}\lesssim 0.3$. We intend to make more predictions on the optical depth for strong lensing in the Horizon-AGN~lightcone with our implemented raytracing machinery. Likewise, in a forthcoming paper we will present the results of the deflection field applied to simulated images derived from the light emitted by the stars produced in the simulation, hence enabling the possibility to measure lensing quantities (shear, magnification...) in the very same way as in observations: shape measurement in the presence of noise, Point Spread Function, pixel sampling, photometric redshift determinations, realistic galaxy biasing and more generally directly predicted galaxy-mass relation, and also the intrinsic alignment of galaxies and their surrounding halos \citep{Codis2015,2015MNRAS.454.2736C,2016MNRAS.461.2702C}.
\begin{acknowledgements}
The authors would like to thank D. Aubert for making his SPL code available to us. We acknowledge fruitful discussions with K. Benabed, S. Colombi, M. Kilbinger and S. Prunet in early phases of the project. We also thank G. Lavaux, Y. Rasera and M-A Breton for stimulating interactions around this project. We are also thankful to A. Leauthaud for constructive comments about the comparison with her GGL lensing results. CL is supported by a Beecroft Fellowship.
This work was supported by the Agence Nationale de la Recherche (ANR) as part of the SPIN(E) ANR-13-BS05-0005 \href{http://cosmicorigin.org}{http://cosmicorigin.org}) ERC grant 670193, and AMALGAM ANR-12-JS05-0006 projects, and by the Centre National des Etudes Spatiales (CNES). NEC is supported by a RAS research fellowship.
This research is also funded by the European Research Council (ERC) under the Horizon 2020 research and innovation programme grant agreement of the European Union: ERC-2015-AdG 695561 (ByoPiC, https://byopic.eu).
This work has made use of the Horizon Cluster hosted by the Institut d'Astrophysique de Paris. We thank S.~Rouberol for running the cluster smoothly for us.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,154,913 | arxiv | \section{Introduction}\label{Intro}
The importance of the pair correlation has been widely
recognized for nucleon many-body systems in various circumstances,
in particular in open-shell nuclei and in neutron stars.
The pairing gap varies with
system parameters such as $N$, $Z$ and the rotational frequency
in the case of finite nuclei,
or the temperature and the density in the case of neutron stars
(cf. Refs.\cite{BM2,Shimizu89,TT93,Lombardo-Schulze,Dean03} as reviews).
The pairing correlation at low nucleon density is of special interest since
the theoretical predictions for low-density uniform matter suggest
that the pairing gap may take, at around 1/10 of the normal nuclear
density, a value which is considerably
larger than that around the normal density
\cite{TT93,Lombardo-Schulze,Dean03}.
This feature is expected to have direct relevance to the properties of neutron
stars, especially those associated with the inner crust\cite{Nstar1,Nstar2}.
The strong pairing at low density
may be relevant also to finite nuclei, if one considers
neutron-rich nuclei near the drip-line\cite{Bertsch91,DobHFB2,DD-Dob,DD-mix}.
This is because such nuclei
often accompany unsaturated low-density distribution of neutrons
(the neutron skin and/or the neutron halo) surrounding the
nuclear surface\cite{Tanihata,Tanihata-density,Ozawa}.
It is interesting to clarify how
the pair correlation in these exotic nuclei is different
from that in stable nuclei, reflecting the strong density dependence
mentioned above. In this connection we would like to ask
how the pair correlation at low nucleon density is different
from that around the normal density.
Spatial structure of the neutron Cooper pair is
focused upon as a characteristic feature of the low density nucleon pairing.
Its possible indication could be
the di-neutron correlation in
the two-neutron halo nuclei, e.g. $^{11}$Li, for which
a spatially correlated pair formed by the halo neutrons
has been predicted theoretically
\cite{Bertsch91,Hansen,Ikeda,Zhukov,Barranco01,Aoyama,Hagino05}
and debated in experimental studies\cite{Sackett,Shimoura,Zinser,Ieki}.
Further, a recent theoretical analysis\cite{Matsuo05} using
the Hartree-Fock-Bogoliubov (HFB)
method\cite{Ring-Schuck,Blaizot-Ripka,DobHFB,Bulgac}
predicts also presence of
similar di-neutron correlation in medium-mass neutron-rich nuclei
where more than two weakly-bound neutrons contribute to form
the neutron skin
in the exterior of the nuclear surface.
It is also possible to argue importance
of the spatial correlation from a more fundamental viewpoint based
on the nucleon
interaction in the $^1S$ channel.
The bare nucleon-nucleon interaction in this
channel has a virtual state around
zero energy characterized by the large
scattering length $a\approx -18$ fm, which implies a very strong
attraction between a pair of neutrons with the spin singlet configurations.
A rather general argument\cite{Leggett,Nozieres},
which applies to a dilute limit of
any Fermion systems,
indicates that the pair correlation of the Fermions interacting with a large
scattering length differs largely from what is considered in the conventional
BCS theory\cite{BCS} assuming weak coupling: it is then appropriate
to consider a crossover
between a superfluid system of the weak-coupling BCS type
and a Bose-Einstein condensate of spatially compact bound Fermion
pairs\cite{Leggett,Nozieres,Melo,Engelbrecht,Randeria}.
This BCS-BEC crossover phenomenon
was recently observed in a ultra-cold atomic gas in a trap
for which the interaction is controllable\cite{Regal}.
In the case of the nucleon pairing,
the BCS-BEC crossover has been argued mostly
for the neutron-proton pairing in the $^{3}SD_{1}$ channel,
for which the strong spatial correlation associated with the deuteron
and the BEC of the deuterons
may emerge\cite{Alm93,Stein95,Baldo95,Lombardo01a,Lombardo01b}.
Concerning the neutron pairing in the
$^{1}S$ channel, which we discuss in the present paper,
we may also expect that the strong coupling feature may lead to
the spatial di-neutron correlation
although the realization of the crossover could be marginal and depend
on the density\cite{Stein95}.
In the present paper, we would like to clarify how the
the spatial structure of the neutron Cooper pair varies with the
density. For this purpose, we shall investigate the
neutron pair correlation in symmetric nuclear matter and neutron matter
in the low density region. Uniform matter is of course a
simplification of
the actual nucleon configurations in finite nuclei and neutron stars.
However they have a great advantage as
one can solve the gap equation in this case without ambiguity for
various interactions including the bare nucleon-nucleon forces
with the repulsive core
\cite{TT93,Lombardo-Schulze,Dean03,Baldo95,Lombardo01a,Lombardo01b,
Takatsuka72,Takatsuka84,Baldo90,Oslo96,Khodel,DeBlasio97,Serra,Garrido01}
as well as effective interactions such as
the Gogny force\cite{Serra,Kucharek89,Sedrakian03},
provided that the
BCS approximation (equivalent to the HFB in finite nuclei) is assumed.
It is straightforward then to
determine the wave function of the neutron Cooper pair
from the solution of the gap equation
\cite{TT93,Baldo95,Lombardo01a,Takatsuka84,Baldo90,Oslo96,DeBlasio97,Serra}.
This provides us with a good reference frame to study
the spatial structure of the Cooper pair as a function of the
density while
we do not intend to make precise predictions on other
properties of neutron and symmetric nuclear matter.
We shall perform an analysis employing
both a bare force and the effective Gogny force\cite{Gogny},
and using a Hartree-Fock single-particle spectrum associated with the
media. Our main conclusion will be that the spatial di-neutron correlation
is strong
in a wide range of the low density $\rho/\rho_0\approx 10^{-4}-0.5$,
independently on the adopted forces.
We shall clarify the nature of the strong
spatial di-neutron correlation in terms of the BCS-BEC crossover
model.
We shall also examine a
possibility of phenomenological description of the
spatially correlated neutron Cooper pair. Here we consider
a contact force with a parametrized
density dependent interaction strength, called often
the density dependent delta interaction (DDDI)
\cite{Bertsch91,DobHFB2,DD-Dob,DD-mix,DDpair-Chas,DDpair-Tera,DDpair-Taj,Fayans96,Fayans00,Garrido}.
The parameters of the DDDI need to be determined from
some physical constraints. For example, the interaction strength has
been constrained by conditions to
reproduce
the experimentally extracted pairing gap
in finite nuclei\cite{DobHFB2,DD-Dob,DD-mix,DDpair-Tera,DDpair-Taj,Fayans96,Fayans00}, or
the density-dependence of the neutron
pairing gap in symmetric nuclear matter
and the $^1S$
scattering length \cite{Bertsch91,Garrido}.
It should be noted here that the contact force requires
a cut-off energy, which needs to be treated as an additional model parameter.
Concerning the cut-off parameter,
attentions have been paid in the previous studies
to convergence properties of the
pairing correlation energy in finite nuclei\cite{DobHFB2},
to the energy dependence of the phase shift
\cite{Esbensen97}, or to the renormalization with respect to the
pairing gap\cite{Bulren,YuBulgac}.
In the present paper we shall take a different approach to the cut-off, i.e.,
we investigate relevance
of the cut-off parameter to the spatial structure of the neutron Cooper pair.
It will be shown
that the cut-off energy plays an important role
to describe the strong spatial correlation at low density.
Considering this as a physical
constraint on the cut-off energy,
we shall derive
new parameter sets of the DDDI.
Preliminary results of this work are reported in Ref.\cite{Matsupre}.
\section{Formulation}\label{Formulation-sec}
\subsection{BCS approximation}\label{gap-eq-sec}
We describe the neutron pair correlation in uniform neutron matter
and in uniform symmetric nuclear matter
by means of the BCS approximation, which is equivalent to the Bogoliubov's generalized
mean-field approach\cite{Ring-Schuck,Blaizot-Ripka}.
One of the basic equations is the gap equation, which is written in the momentum
representation as
\begin{eqnarray}\label{gap-eq}
\Delta(p)&=& - {1 \over 2 (2\pi)^3}\int d \mbox{\boldmath $k$} \tilde{v}(\mbox{\boldmath $p$}-\mbox{\boldmath $k$})
{\Delta(k) \over E(k)}, \\
E(k)&=&\sqrt{(e(k)-\mu)^2 + \Delta(k)^2}.
\end{eqnarray}
Here
$\Delta(k)$ is the pairing gap dependent on the single-particle
momentum $k$, while
$e(k)$ and $E(k)$ are the single-particle and the quasiparticle
energies. $\tilde{v}(\mbox{\boldmath $p$}-\mbox{\boldmath $k$})$
is the matrix element of the nucleon-nucleon interaction in the $^1S$ channel.
The gap equation needs to be solved together with the number equation
\begin{equation}\label{num-eq}
\rho \equiv {k_F^3 \over 3\pi^2} = {1\over (2\pi)^3} \int d \mbox{\boldmath $k$}
\left(1 + {e(k)-\mu \over E(k)}\right),
\end{equation}
which determines the relation between the neutron density $\rho$
(the Fermi momentum $k_F$)
and the chemical potential $\mu$.
The solution of these equations defines the ground state
wave function of the BCS type
and the static pairing properties at zero temperature
for a given density $\rho$.
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig1.eps}
}
\caption{
The nucleon-nucleon potential $v(r)$ in the ${}^1S$ channel
for the G3RS and Gogny D1 forces, plotted with the solid and
dotted curves, respectively,
as a function of the relative distance $r$ between neutrons.
\label{force}}
\end{figure}
As the interaction acting in the $^1S$ channel
we shall adopt a bare nucleon-nucleon force,
the G3RS force\cite{Tamagaki},
and the effective interaction given by Gogny\cite{Gogny}.
The G3RS force is a local potential representation
of the bare nucleon-nucleon interaction which is
given by a superposition of three
Gaussian functions:
\begin{equation}\label{G3RS-eq}
v(\mbox{\boldmath $r$}) = \sum_i v_i e^{-r^2/\mu_i^2}.
\end{equation}
One component represents
a repulsive core with the height of
$v_1=2000$ MeV and the range parameter $\mu_1=0.447$ fm,
while two other Gaussians
with $v_{2,3}=-240, -5$ MeV and $\mu_{2,3}=0.942, 2.5$ fm
represent the attraction dominant for $1\m@thcombine<\sim r \m@thcombine<\sim 3$ fm (see Fig.\ref{force}).
In spite of the simple three Gaussian representation, the G3RS reproduces
rather well the $^1S$ phase shift up to about $300$ MeV in the c.m. energy
of the scattering nucleons.
The associated scattering length
$a=-17.6$ fm is in close agreement with the experimental value
$a=-18.5\pm0.4$ fm\cite{Aexp}. The G3RS has been used in some
BCS calculations for the $^{1}S$ pairing at low density
and for the $^{3}P_2$ pairing at high density\cite{TT93,Lombardo-Schulze,Dean03}.
From a practical point of view, the analytic form makes it
easy to evaluate the
matrix elements of the interaction.
The Gogny force is an effective interaction
which is designed for the HFB description
of the pairing correlation in finite nuclei while keeping
some aspects of the $G$-matrix\cite{Gogny}.
It is also a local potential represented
as a combination of two Gaussian functions in the form of
Eq.(\ref{G3RS-eq}) with the
range parameters $\mu_{1,2}=0.7, 1.2$ fm.
In the following we show mostly results obtained with the
parameter set D1\cite{Gogny}
as we find no qualitative difference in the results for the
parameter set D1S\cite{Gogny-D1S}. A common feature of the
G3RS and Gogny forces is that both are attractive
in the range $1 \m@thcombine<\sim r \m@thcombine<\sim 3$ fm while they differ
largely for $r\m@thcombine<\sim 0.5$ fm, where
the Gogny force exhibits only a
very weak repulsion
instead of the
short-range core present in the G3RS force (Fig.\ref{force}).
The interaction range of the two forces is
of the order of 3fm. Note that
the experimental effective range is $r_{e}=2.80\pm0.11$ \cite{Aexp}.
We shall also apply a zero-range
contact force $v(\mbox{\boldmath $r$}) \propto \delta(\mbox{\boldmath $r$})$.
Our treatment of this interaction will be described separately in
Section \ref{DDDI-sec}.
As the single-particle energy $e(k)$ we use, in the case of the Gogny interaction,
the Hartree-Fock single-particle
spectrum derived directly from the same interaction.
In the case of the bare force, it would be
better
from a viewpoint of the
self-consistency to use the Brueckner Hartree-Fock spectrum.
But for simplicity
we adopt in the present analysis an effective mass approximation.
Namely the single-particle energy is given by
$e(k)=k^2/2m^* $, where the effective mass
$m^*=\left(\partial^2 e(k) / \partial^2 k|_{k_F}\right)^{-1}$
is derived from the
Gogny HF spectrum\cite{Kucharek89} for the parameter set D1.
We solve the gap and number equations, (\ref{gap-eq}) and (\ref{num-eq}),
without
introducing any cut-off.
The momentum integrations in the two equations are performed
using a direct numerical method, where
the maximum momentum $k_{max}$ for the integration is
chosen large enough so that the result does not depend on the
choice of $k_{max}$.
We adopt $k_{max}=20$ fm$^{-1}$
for the G3RS, and $k_{max}=10$ fm$^{-1}$ for the Gogny interaction.
Note here that it is dangerous to
introduce a small energy window around
the chemical potential (or the Fermi energy) or a cut-off at a
small momentum in evaluating the r.h.s. of
the gap equation. Such an approximation may be justified only in the
case of the weak coupling BCS where the pairing gap is considerably
smaller than the Fermi energy, but it is not applicable to the
strong coupling case\cite{Leggett}.
Note also that
the chemical potential $\mu$
and the Fermi energy $e_F$ are not the same except in the limit of the weak
coupling. We define
the Fermi momentum $k_F$ through the
nominal relation to the density $\rho = {1 \over 3\pi^2} k_F^3$,
and the Fermi energy $e_F$ by $e_F\equiv e(k_F)$.
The pairing gap
$\Delta_F\equiv\Delta(k_F)$ at the Fermi momentum is used below
as a measure of the pair correlation.
In the following $\rho$ denotes always the neutron density.
(The total nucleon density is $\rho_{tot}=2\rho$
in the case of symmetric nuclear matter.)
We define, as a reference value, the normal neutron density by
$\rho_0={1 \over 3\pi^2} k_{F0}^3$
with $k_{F0}=1.36$ fm$^{-1}$.
In order to investigate the spatial structure of the neutron Cooper
pair, it is useful to look into
its wave function
represented as a function of the relative distance between the partner neutrons of the pair.
It is given by
\begin{eqnarray}\label{Cooper-eq}
\Psi_{pair}(r) &\equiv&
C' \left<\Phi_0\right|\psi^\dagger(\mbox{\boldmath $r$}\uparrow)
\psi^\dagger(\mbox{\boldmath $r$}'\downarrow)\left|\Phi_0 \right>
=
{C \over (2\pi)^3}
\int d \mbox{\boldmath $k$} u_kv_k e^{i\mbox{\boldmath $k$}\cdot(\mbox{\boldmath $r$}-\mbox{\boldmath $r$}')}, \\
u_kv_k&=& {\Delta(k) \over 2E(k)},
\end{eqnarray}
in terms of the $u,v$-factors except by the normalization factors $C$ and $C'$. Here
$\psi^\dagger(\mbox{\boldmath $r$} \sigma) \ \ (\sigma=\uparrow,\downarrow)$ is the creation operator of neutron and
$\left|\Phi_0\right>$ is the BCS ground state.
The Cooper pair wave function depends only on
the relative distance $r=|\mbox{\boldmath $r$}-\mbox{\boldmath $r$}'|$ between the partners
as it is an s-wave.
We evaluate the momentum integral in Eq.(\ref{Cooper-eq})
in the same way as in the gap and the number equations.
It is useful to evaluate the size of the neutron Cooper pair.
A straightforward measure is the
r.m.s. radius of the Cooper pair
\begin{equation}\label{rms-eq}
\xi_{rms}= \sqrt{\left<r^2\right>},
\end{equation}
where
\begin{equation}\label{rms2-eq}
\left<r^2\right>= \int d\mbox{\boldmath $r$} r^2 |\Psi_{pair}(r)|^2
= {\int_0^{\infty} dk k^2 \left({\partial \over \partial k}u_kv_k\right)^2 \over
\int_0^{\infty} dk k^2 \left(u_kv_k\right)^2}
\end{equation}
can be calculated directly from the Cooper pair wave function
$\Psi_{pair}(r)$ and/or from the $u,v$-factors in the momentum space.
If one assumes weak coupling, the Pippard's coherence length \cite{BCS}
\begin{equation}\label{Pippard-eq}
\xi_P ={\hbar^2 k_F \over m^{*}\pi\Delta_F}
\end{equation}
given analytically in terms of the gap and the Fermi momentum may be
used also as another estimate of the size of the Cooper pair. In the following we mostly use
$\xi_{rms}$ since this quantity itself has a solid meaning even in the
case of the strong coupling BEC case and in the crossover region
between BCS and BEC.
We shall use $\xi_{P}$ for qualitative discussions.
In the present paper, we neglect
higher order many-body effects which go beyond
the BCS approximation. In many calculations
\cite{Chen86,Chen93,Ainsworth89,
Wambach93,Schulze96,Schulze01,Lombardo01,Shen03,Schwenk,Lombardo04}
the higher order effects in low density neutron matter
are predicted to reduce the pairing gap by about a factor of two, which is
however very much dependent on the prescriptions
adopted\cite{Lombardo-Schulze,Dean03} except
for the low density limit
$\rho\rightarrow 0$\cite{Heiselberg00}.
A recent Monte Carlo study\cite{Fabrocini05}
using the realistic bare force suggests the gap close to the BCS result.
The higher order effects in symmetric nuclear matter\cite{Lombardo04}
and in finite nuclei\cite{Milan1,Milan2,Milan3}
are estimated to increase the gap. Keeping in
mind these ambiguities, we consider that
the BCS approximation provides a meaningful
zero-th order reference.
\subsection{Pairing gap and coherence length}\label{gap-sec}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig2.eps}
}
\caption{
The pairing gap $\Delta_F$ in neutron and symmetric nuclear matter
as a function of the neutron density
$\rho/\rho_0$.
The results for the G3RS force are shown with the
symbols: cross for the free single-particle spectrum,
square for neutron matter, and diamond for
nuclear matter. The results with the Gogny D1 force
are plotted with the dashed, dotted, and
solid curves for matter with the free single-particle spectrum,
for neutron matter, and for symmetric matter, respectively.
\label{gap}}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig3.eps}
}
\caption{
The r.m.s. radius $\xi_{rms}$ of the neutron Cooper pair
in uniform matter, plotted as a function of
the neutron density $\rho/\rho_0$.
The results for symmetric nuclear and neutron matter
obtained with the Gogny D1 force are shown by the solid and dotted
curves, respectively, while the results for symmetric nuclear and neutron
matter with the G3RS force are
shown by the diamond and square symbols, respectively.
The average inter-neutron
distance $d=\rho^{-1/3}$ is plotted with the dot-dashed line.
\label{rms}}
\end{figure}
Figure \ref{gap} shows
the neutron pairing gap $\Delta_F$
obtained with the G3RS and Gogny D1 forces both
for neutron matter and for symmetric matter. Results
assuming the free single-particle spectrum (equivalent to the use
of $m^*=m$) are also shown for comparison. The pairing gap
becomes maximum around $\rho/\rho_0\sim 0.1-0.3$ in all the cases.
The gap decreases gradually
with further decreasing the density. The difference between
neutron matter and symmetric nuclear matter, which originates from
the effective mass effect, becomes negligible at
low density $\rho/\rho_0 \m@thcombine<\sim 5\times 10^{-2}$.
The pairing gap obtained with the G3RS force is
very similar to those obtained with more realistic models of
the bare force
(OPEG\cite{Takatsuka72,Takatsuka84,TT93},
Reid\cite{Takatsuka84,TT93,Khodel}, Argonne\cite{Khodel,Baldo90}, Paris\cite{Baldo90},
Bonn\cite{Oslo96,Serra}, Nijmegen\cite{Lombardo-Schulze}).
This is because the gap is essentially determined by the $^{1}S$ phase shift
function\cite{phase-shift}, and the G3RS force reproduces the experimental phase shift,
though not as accurately as the modern forces.
There is small difference from them: the maximum gap
$\Delta_F \approx 2.5$ MeV around $\rho/\rho_0 \sim 0.2$
(or $k_F \sim 0.8 {\rm fm}^{-1}$) for neutron matter is
slightly smaller in the G3RS by about 5-20\%.
The difference may be due to a limitation
of the simple three-Gaussian representation, but
the quality is enough for the following discussions.
The gap obtained with the Gogny force is consistent
with those in the previous calculations\cite{Kucharek89,Serra,Sedrakian03}.
If we compare with the Gogny and G3RS results, they exhibit a
similar overall
density dependence, and
a significant difference between the two forces is seen
only at modest density
$\rho/\rho_0 \m@thcombine>\sim 5\times 10^{-2}$. Garrido {\it et al.}
\cite{Garrido} suggests that the similarity may indicate a possible
cancellation among higher order effects in the case of symmetric matter.
We find a less significant difference in the gap
at
rather low density $\rho/\rho_0 \m@thcombine<\sim 10^{-3}$,
though not visible in Fig.\ref{gap}.
This arises from the
fact that the scattering length $a=-13.5$ fm of the Gogny D1
force deviates from the G3RS value $a=-17.6$ fm.
The r.m.s. radius $\xi_{rms}$ of the neutron Cooper pair
calculated for neutron and symmetric nuclear matter with the
bare or the Gogny forces are shown in Fig.\ref{rms} and Table \ref{coherence-length}.
The calculated $\xi_{rms}$ is consistent with the r.m.s. radius (or the Pippard's
coherence length) reported in the
BCS calculations using other models of
the bare force\cite{Takatsuka84,TT93,DeBlasio97,Serra}. Here we would like
emphasize characteristic density dependence of $\xi_{rms}$. It is seen
from Fig.\ref{rms} that
$\xi_{rms}$ decreases
dramatically by nearly a factor of ten
from a large value
of the order of $\sim 50$ fm around the normal density $\rho/\rho_0\sim 1$
to considerably smaller values $\xi_{rms} \approx 4.5-6$ fm
at density around
$\rho/\rho_0 \sim 0.1$.
The size of the Cooper pair stays at small values
$\xi_{rms} \approx 5-6$ fm in the density region $\rho/\rho_0 \sim
10^{-2}-0.1$. It then turns to increase, but only gradually, at
further low density. These features are commonly seen
for both neutron and symmetric nuclear
matter, and for both the G3RS and Gogny forces.
We would like to emphasize also the smallness of the neutron Cooper pair.
This may be elucidated if we
compare the r.m.s. radius $\xi_{rms}$
with the average inter-neutron distance $d\equiv\rho^{-1/3}=3.09k_F^{-1}$.
It is seen from Fig.\ref{rms} that $\xi_{rms}$
becomes smaller than $d$ in a very wide range of density
$\rho/\rho_0 \sim 10^{-4} - 0.1$.
The ratio $\xi_{rms}/d$ can reach values as small as
$\approx 0.5$ at density around $\rho/\rho_0 \sim 10^{-2}$.
The relation $\xi_{rms} <d$, i.e.
the size of the neutron Cooper pair smaller than
the average inter-neutron distance, suggests that the neutron
Cooper pair exhibits strong spatial di-neutron correlation.
To understand this strong spatial correlation,
it is useful to consider the ratio $\Delta_F/e_F$
between the pairing gap and the Fermi energy rather than the
absolute magnitude of the gap. The pairing gap
$\Delta_F \approx 0.2$ MeV at the density $\rho/\rho_0=1/512$
for example appears small in the absolute
scale, but
the gap to Fermi-energy ratio amounts
to $\Delta_F/e_F \approx 0.4$ (see Table \ref{coherence-length}),
which is larger than the
value $\Delta_F/e_F \approx 0.25$ at $\rho/\rho_0=1/8$ where the
gap is nearly the maximum. If we use
the Pippard's coherence length $\xi_P=\hbar^2 k_F /m^{*}\pi\Delta_F$ in place
of the r.m.s. radius $\xi_{rms}$ (this may be justified at least for qualitative
discussion since the two quantities
agree within 10-25 \%, see Table \ref{coherence-length}), the ratio
between the r.m.s. radius and the average inter-neutron distance
is related to the gap to Fermi-energy ratio as
$\xi_{rms}/d \sim \xi_P/d \sim 0.2 e_F/\Delta_F$.
Consequently we can expect in the zero-th order argument that
the strong spatial correlation $\xi_{rms}/d \m@thcombine<\sim 1$
emerges when the gap to Fermi-energy
ratio is larger than $\Delta_F/e_F \m@thcombine>\sim 0.2$.
This is realized in the present calculations in the density range
$\rho/\rho_0 \sim 10^{-4}-0.1$. In the following we shall investigate in detail
the spatial correlation in the neutron Cooper pair around this density range.
We shall comment here comparison with
the $^{3}SD_{1}$ neutron-proton pairing in symmetric nuclear
matter. In this case the BCS pairing gap calculated with a
realistic bare force (the
Paris force) is of the order of
8 MeV at maximum\cite{Garrido01}. The r.m.s. radius $\xi_{rms}$
of the neutron-proton Cooper pair is quite small, reaching to
the minimum value $\xi_{rms} \sim 2$ fm at density around
$\rho/\rho_0 \sim 0.2$ \cite{Lombardo01a}. Consequently,
the r.m.s. radius $\xi_{rms}$ becomes considerably smaller
than the average inter-particle distance $d$ in the density interval
from $\rho/\rho_0 \sim 0.5$ down to the zero
density limit, where $\xi_{rms}$ becomes identical to the
r.m.s. radius of the deuteron\cite{Lombardo01a}.
In the case of the
$^{1}S$ neutron pairing, by contrast, the signature of the
strong coupling $\xi_{rms} \m@thcombine<\sim d$ is obtained in the wide but
limited range of the density $\rho/\rho_0 \sim 10^{-4}-0.1$.
Apart from this difference, it is noted that the qualitative trend of
the Cooper pair size, e.g. shrinking
with increasing the density from the zero density limit, is similar to
that discussed in the neutron-proton case\cite{Lombardo01a}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{lccccccccccc}
\hline
\hline
& $k_F$ & $\rho/\rho_0$ & $d$ & $m^*/m$ & $e_F$& $\Delta_F$ &
$\xi_{rms}$ & $\xi_P$ & $P(d)$ &$(1/k_Fa)_\xi$& $(1/k_Fa)_\Delta$\\
\hline
\multicolumn{3}{l}{symmetric matter}, Gogny D1 &&&&&&&& \\
& 1.36 & 1 & 2.27 & 0.668 &62.0& 0.64 & 46.60 & 41.76
&0.18&-2.91&-2.99\\
& 1.079 & 1/2 & 2.87 & 0.744 &33.8& 2.03 & 10.80 & 9.45
&0.48&-1.83&-1.84\\
& 0.68 & 1/8 & 4.55 & 0.891 &10.9& 2.60 & 4.81 & 3.87
&0.81&-0.97&-0.90\\
& 0.34 & 1/64 & 9.10 & 0.984 &2.45& 0.97 & 5.87 & 4.71
&0.92&-0.59&-0.52\\
& 0.17 & 1/512 & 18.20 & 0.998 &0.60& 0.22 & 12.05 & 10.30
&0.91&-0.62&-0.58\\
\hline
\multicolumn{3}{l}{neutron matter}, G3RS &&&&&&&& \\
& 1.36 & 1 & 2.27 & 0.905 &42.4& 0.14& 159.8 & 144.1 &0.09
&-3.70&-3.70\\
& 1.079 & 1/2 & 2.87 & 0.925 &26.1& 1.52& 11.61 & 10.13 &0.47
&-1.88&-1.85\\
& 0.68 & 1/8 & 4.55 & 0.969 &9.89& 2.37& 4.94 & 3.90 &0.80
&-0.99&-0.90\\
& 0.34 & 1/64 & 9.10 & 0.995 &2.41& 0.98& 5.90 & 4.61 &0.92
&-0.60&-0.50\\
& 0.17 & 1/512 & 18.20 & 0.999 &0.60& 0.24& 11.16 & 9.30 &0.92
&-0.55&-0.50\\
\hline
\hline
\end{tabular}
\end{center}
\caption{The pairing gap $\Delta_F$, the r.m.s. radius
$\xi_{rms}$, the Pippard's coherence
length $\xi_P$, and the probability $P(d)$ within the average
inter-neutron distance $d$,
associated with the neutron Cooper pair in symmetric nuclear matter
obtained with the Gogny D1 interaction and in
neutron matter with the G3RS force, at the neutron density
$\rho/\rho_0=1, 1/2, 1/8, 1/64, 1/512$, or
equivalently $k_F=1.36, 1.094, 0.68, 0.34, 0.17$ fm$^{-1}$.
The Fermi energy $e_F$, the effective mass $m^*/m$, and the
average inter-neutron distance $d$ are also shown.
The parameters $(1/k_F a)_{\xi}$ and $(1/k_F a)_{\Delta}$ of the
regularized delta interaction model are also listed (see the text).
The units for $k_F$ and $\Delta_F, e_F$ are fm$^{-1}$ and MeV,
respectively while that for $d, \xi_{rms}$ and $\xi_{P}$ is fm.
\label{coherence-length}
}
\end{table}
\section{
Spatial structure of neutron Cooper pair}\label{spatial-sec}
\subsection{Cooper pair wave function: basics}\label{Cooper-wf-sec}
In examining the spatial structure of the neutron Cooper pairs,
we shall focus mostly
on the symmetric nuclear matter case obtained with the Gogny force and
the neutron matter case with the G3RS force.
The r.m.s. radii in these two cases represent
a rough mean value of the four results plotted in Fig.\ref{rms}. Note also
that the two r.m.s. radii coincide with each other within $10\%$ in
a very wide interval of density $\rho/\rho_0 = 10^{-3}- 0.5$.
It is by accident, but this feature can be exploited to
single out influences of different interactions since the
comparison can be made with the r.m.s. radii kept the same.
\begin{figure}[htbp]
\centerline{
\includegraphics[width=8.5cm]{fig4a.eps}
\includegraphics[width=8.5cm]{fig4b.eps}
}
\caption{(a-e)
The wave function $\Psi_{pair}(r)$ of the neutron Cooper pair as a function
of the relative distance $r$ between the pair partners at
the neutron density $\rho/\rho_0=1, 1/2, 1/8, 1/64, 1/512$.
The solid curve is for the pair in symmetric nuclear matter
obtained with the Gogny D1 force, while the dotted curve is for that
in neutron matter with the G3RS.
The vertical dotted
line represents the r.m.s. radius $\xi_{rms}$ of the Cooper pair in
neutron matter with the G3RS while the dashed line for
symmetric nuclear matter with Gogny D1.
Here and also in the following figures the wave function
is normalized by $\int_0^\infty |\Psi_{pair}(r)|^2 r^2 dr=1$.
The arrow indicates the average inter-neutron distance $d$.
(f-j) The same as (a-e) but for
the probability density $r^2|\Psi_{pair}(r)|^2$.
The thin dotted line in (g) and (i) is the wave function of
the fictitious "bound state" in the free space described in the text.
\label{Cooper-wf}}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=0,width=8.5cm]{fig5.eps}
}
\caption{
The probability $P(r)$ for the partner neutrons of the Cooper pair to
be correlated
within a relative distance $r$, calculated at density $\rho/\rho_0=1/2$.
The result for symmetric nuclear matter with the Gogny D1 force
is plotted with the solid curve while the dotted curve represents
the result for neutron matter with the G3RS force.
The dashed vertical line indicates the r.m.s. radius of the
Cooper pair in the symmetric matter case. The vertical dotted line
marks the position $r=3$ fm.
\label{pairprob}}
\end{figure}
Figures \ref{Cooper-wf}(a-e) show
the wave function $\Psi_{pair}(r)$ of the neutron Cooper pair
for the representative values of density listed in Table \ref{coherence-length}.
The result for neutron matter
calculated with the G3RS force and that for symmetric
nuclear matter with the Gogny D1 force are plotted in the same figure
for the reasons mentioned just above.
The
probability density $r^2|\Psi_{pair}(r)|^2$ multiplied
by the volume element $r^2$ is plotted in Fig. \ref{Cooper-wf}(f-j).
As a quantitative measure of the spatial correlation,
we evaluate also the probability $P(r)$ for the partners of the
neutron Cooper pair to come close with each other within a
relative distance $r$. It is nothing but a partial
integration of the probability density $r^2|\Psi_{pair}(r)|^2$
up to the distance $r$:
\begin{equation}\label{prob-eq}
P(r) = {\int_0^{r} |\Psi_{pair}(r')|^2 r'^2 dr' \over
\int_0^{\infty} |\Psi_{pair}(r')|^2 r'^2 dr'}.
\end{equation}
An example of this quantity is shown in Fig.\ref{pairprob}
in the case of $\rho/\rho_0=1/2$.
Before proceeding to the main analysis we shall first
point out that the G3RS and Gogny forces provide
essentially the same spatial structure of the Cooper pair except
at very short relative distances.
In Fig.\ref{Cooper-wf}, a clear difference between the two forces
is seen at short
relative distances $r \m@thcombine<\sim 1$ fm.
(Note that the normal density case, Fig.\ref{Cooper-wf}(a,f), is
not relevant for this discussion since the gaps and the r.m.s. radii are
very different.) Apparently the suppression of the
wave function seen at $r \m@thcombine<\sim 1$ fm in the G3RS case is
caused by the strong repulsive core present in the bare force.
The Cooper pair wave function for the Gogny force does
not show this short range correlation because of the lack of the core.
On the other hand, by looking
at distances $r>1$ fm slightly larger than the core radius
we find that
the Cooper pair wave functions obtained with the two
forces agree quite well with each other.
This observation applies also to the probability density
$r^2|\Psi_{pair}(r)|^2$ and the probability $P(r)$,
for which the difference at short distances $r<1$ fm becomes barely visible
as the volume element is small at such short distances. Thus
the spatial structure of the neutron Cooper pair does not depend on
whether the interaction is the bare force or the effective Gogny force,
provided that two cases gives the same r.m.s. radius of the Cooper pair.
In the following, we shall
concentrate on behaviors which are common to the two interactions.
The Cooper pair wave function $\Psi_{pair}(r)$ in the coordinate representation
is reported in some of the previous BCS calculations adopting other models of
the bare force\cite{Takatsuka84,Baldo90,Oslo96,DeBlasio97,Serra}
and the Gogny force\cite{Serra}.
Our wave function appears consistent with those in
Refs.\cite{Takatsuka84,Baldo90,Oslo96,Serra}
although
in these references the wave function is shown
only at very limited numbers of density values and
up to not very large relative distances.
We found, however,
that the shape of the Cooper pair wave function shown in
Ref.\cite{DeBlasio97} differs largely from our results (Fig.\ref{Cooper-wf}),
especially at relative distances smaller than several fm.
\subsection{Density dependence}
If we compare in Fig.\ref{Cooper-wf}
the Cooper pair wave functions at different density values,
important features show up.
An apparent observation is that
the spatial extension or the size of the Cooper pair
varies strongly with the density, in accordance
with the strong density dependence of the r.m.s radius $\xi_{rms}$
discussed in the previous section (Fig.\ref{rms}). We emphasize here
another prominent feature.
Namely {\it the profile} of the Cooper pair wave function
also changes significantly with the density.
At the normal density $\rho/\rho_0=1$ (Fig.\ref{Cooper-wf}(a,f)), the
Cooper pair wave function is spatially extended:
the r.m.s. radius of the Cooper pair is
as large as $\xi_{rms}\m@thcombine>\sim 50$ fm.
The profile of the Cooper pair wave function in this case exhibits
an exponential fall-off convoluted
with an oscillation. This behavior is
consistent with the well known expression\cite{BCS}
$r\Psi_{pair}(r) \sim K_0(r/\pi \xi_P) \sin(k_F r)$ for the Cooper pair
wave function in the weak coupling BCS situation. Here $K_0$ is the
modified Bessel function, which behaves asymptotically as
$K_0(r/\pi \xi_P) \sim (\xi_P/r)^{1/2}\exp(-(r/\pi\xi_P))$.
The position of the first node
$r \approx \pi k_F^{-1}$ approximately
corresponds to the average inter-particle
distance $d=3.09k_F^{-1}(=2.3 {\rm fm})$.
The wave function has significant amplitude for $r>d$ since we here have
a relation $\xi_{rms,P} \gg d$ (see Table \ref{coherence-length}).
This is a typical behavior in the situation of the
weak coupling BCS.
The Cooper pair wave function at the density $\rho/\rho_0=1/8$
(Fig.\ref{Cooper-wf}(c,h)) is very
different from that at the normal density.
Apart from the considerably small spatial extension
($\xi_{rms}= 4.8-4.9$ fm), the
functional form of the wave function behaves quite differently.
We find that
amplitude of the wave function is strongly concentrated within
the average inter-neutron
distance $d$, and that the oscillating amplitude beyond $d$ is quite small.
This is consistent with the observation in the previous section that
the r.m.s. radius $\xi_{rms}$ of the Cooper pair
is smaller than the average inter-neutron distance $d$ in this case
(cf.Fig.\ref{rms} and Table \ref{coherence-length}).
The probability $P(d)$
for the partners of the Cooper pair to be correlated
within the inter-nucleon distance
$d$,
exceeds $0.8$ (Fig.\ref{prob} and Table \ref{coherence-length}),
indicating directly the strong spatial di-neutron correlation.
The neutron Cooper pair wave functions
at $\rho/\rho_0=1/64$ and 1/512, Fig.\ref{Cooper-wf}(d,e,i,j), exhibit a
behavior similar to that at $\rho/\rho_0=1/8$.
Inspecting more closely, we notice that
the concentration within $r<d$ is stronger than at $\rho/\rho_0=1/8$ while
the spatial extension
itself is slightly larger ($\xi_{rms} \approx 6-12$ fm).
We observe also
smaller oscillating amplitude
in the large distance region $r > d$ (Fig.\ref{Cooper-wf}),
larger values of $P(d)\approx 0.9$, and smaller ratio $\xi_{rms}/d$
(Table \ref{coherence-length}).
They all point to
stronger spatial di-neutron
correlation at these values of density.
The Cooper pair wave function
at $\rho/\rho_0 = 1/2$ (Fig.\ref{Cooper-wf}(b,g)) exhibits
an intermediate feature
between that at the normal density $\rho/\rho_0=1$ and those at
$\rho/\rho_0 =1/64 - 1/512$. In particular, we notice that
the spatial correlation seen at the lower density persists
to some significant extent also in this case.
For example,
the probability density
is strongly concentrated to the short distance region of
$r \m@thcombine<\sim 3$ fm (Fig.\ref{Cooper-wf}(g)). This is more apparent in
the plot of $P(r)$ shown in
Fig.\ref{pairprob}, where we find that
the probability $P(r)$ increases steeply with increasing $r$
from $r=0$, and
reaches $\sim 50\%$ already at $r=3$ fm, which
is roughly the interaction range of the nucleon force.
This strong concentration within $r<3$ fm
may be elucidated by comparing with
what could be
expected if a bound pair having
the same r.m.s. radius ($\xi_{rms}=10.8$ fm in this case) was formed
in the free space.
(We calculate this fictitious ``bound state'' wave function by increasing the
strength of the Gogny D1 potential by a numerical factor.)
It is noticed that the profile of the Cooper pair wave function
differs from the ``bound state'' wave function which is plotted
with the thin dotted line
in Fig.\ref{Cooper-wf}(g).
In this "bound state" wave function,
concentration of the probability within $r \m@thcombine<\sim 3$ fm
is not very large, i.e., $P(3{\rm fm})=0.24$,
while the probability $P(3{\rm fm})$ associated with the neutron Cooper
pair wave function is about twice this value.
This indicates
that the spatial
di-neutron correlation is also strong for the moderate low density region
$\rho/\rho_0\sim 0.5$.
A remnant of this spatial correlation is found also at the normal
density (Fig.\ref{Cooper-wf}(a,f)),
but in this case the concentration within the interaction range
is not very large ($P(3{\rm fm})=0.21$),
due to the very large Cooper pair size
( $\xi_{rms} \sim 50$ fm).
In contrast the Cooper pair wave function at the lower density,
$\rho/\rho_0=1/64$ (Fig.\ref{Cooper-wf}(i))
for example,
is much more similar to the "bound state" wave function.
Figure \ref{prob} shows the overall behavior of
$P(3{\rm fm})$ and $P(d)$ as a function of the density. It is seen
that the strong concentration
within the
interaction range, say $P(3{\rm fm})>0.5$,
is realized in
the density region $\rho/\rho_0 \approx 5\times 10^{-2}-0.5$.
The probability $P(3{\rm fm})$ reaches the maximum value
$\sim 0.7$ around $\rho/\rho_0 \sim 0.1$, where the r.m.s.
radius is the smallest. At lower density
$\rho/\rho_0 \m@thcombine<\sim 10^{-1}$, the probability
$P(3{\rm fm})$ decreases gradually in accordance with the gradual increase
of the r.m.s. radius of the Cooper pair. Note however that in this density region
($\rho/\rho_0 \sim 10^{-4}-10^{-1}$) the concentration of the probability
within
the average inter-neutron distance $d$ remains very large, i.e., $P(d) \m@thcombine>\sim 0.8$.
All the above analyses indicate that the
spatial di-neutron correlation is strong in the quite wide density interval
$\rho/\rho_0 \sim 10^{-4} - 0.5$.
It is interesting to compare our result with that in a similar analysis
of the Cooper pair wave
function for the $^3SD_1$ neutron-proton pairing. In that case
the Cooper pair wave function is found to merge smoothly into the
deuteron wave function in the low density limit\cite{Baldo95}.
Correspondingly,
the r.m.s. radius of the Cooper pair approaches to that of the deuteron,
which is much smaller than the average
inter-particle distance\cite{Lombardo01a}.
This is interpreted as a realization of
the BEC of the deuterons in the low density
region and the BCS-BEC crossover taking place with change of the
density\cite{Stein95,Baldo95,Lombardo01a,Lombardo01b}. In the
neutron pairing case, the similarity of the Cooper pair
wave function to a bound state wave function is found only
in a limited density range
$\rho/\rho_0 \sim 10^{-4}-10^{-1}$, and it never converges
to a bound state wave function (NB. there is no bound state in this
channel in the free space).
The r.m.s. radius $\xi_{rms}$ of the Cooper pair is only comparable
to the average inter-neutron distance $d$ in the same density interval.
Although the spatial correlation is strong as discussed above,
these
qualitative observations alone are not enough to assess
whether the region of the BCS-BEC crossover
is reached in the case of the neutron pairing.
We shall investigate this issue on more quantitative bases in the next section.
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig6.eps}
}
\caption{
The probability $P(d)$ for the partner neutrons to be correlated within
the average inter-neutron distance $d$ and the
probability $P(3{\rm fm})$ within $r=3$ fm. The solid and dotted curves are
for symmetric nuclear matter obtained with the Gogny D1
force, while the square and diamond symbols are for neutron matter with
the G3RS
force.
\label{prob}}
\end{figure}
\section{Relation to the BCS-BEC crossover}\label{BCS-BEC-sec}
In the previous section, we have seen the strong spatial
correlation in the neutron Cooper pair wave function at low density.
In the present section we shall elucidate its
implication by making a connection
to the BCS-BEC crossover phenomenon.
For this purpose, we shall first describe a reference Cooper pair
wave function based on a simple
solvable model of the BCS-BEC crossover, and then compare our results with that of the reference model.
As such a reference, we adopt a $^{1}S$ pairing model that applies generically to
a dilute gas limit of any Fermion systems\cite{Leggett,Melo,Engelbrecht},
for which the average inter-particle distance
$d=\rho^{1/3}$ is supposed to be much larger
than the range of interaction.
The dilute limit is equivalent to treating
the interaction matrix elements
as a constant, or assuming a contact interaction.
Using the relation between the interaction constant
and the zero-energy $T$-matrix or the scattering length $a$,
the gap equation (\ref{gap-eq}) is written in a regularized form:
\begin{equation}\label{reg-gap-eq}
{m \over 4\pi \hbar^2 a }= - {1 \over 2(2\pi)^3}\int d \mbox{\boldmath $k$}
\left({1 \over E(k)} - {1 \over e(k)}\right).
\end{equation}
The regularized gap equation (\ref{reg-gap-eq}) and the number equation (\ref{num-eq})
are now expressed
analytically in terms of some special functions,
and are easily solvable\cite{Marini,Papenbrock}.
In this model a dimensionless parameter
$1/k_F a$, characterized by the scattering length and the Fermi momentum,
is the only parameter that controls the
strength of interaction, and hence properties of the pair correlation
are determined solely by $1/k_F a$ while the length scale is given by
$k_F^{-1}$ (or the inter-particle distance $d=3.09k_F^{-1}$), and
the energy scale by the Fermi energy $e_F= \hbar^2 k_F^2 /2m$
\cite{Leggett,Melo,Randeria,Engelbrecht,Marini}.
The gap to Fermi-energy ratio $\Delta/e_F$ and the ratio $\xi_{rms}/d$
between the r.m.s. radius and the average inter-particle distance
are then monotonic functions of the
interaction parameter $1/k_F a$\cite{Engelbrecht,Marini}. The functional form
of the Cooper pair wave function, defined by Eq.(\ref{Cooper-eq}),
is also
determined only by $1/k_F a$, except for the length scale.
Since there is no available analytic expression for the wave function,
we evaluate Eq.(\ref{Cooper-eq}) by performing the momentum integral
numerically with a help of an explicit use of a smooth cut-off
function of a Gaussian form.
The cut-off scale is chosen large enough so that the results shown below
do not depend on it.
The range $1/k_F a \ll -1$ of the interaction parameter corresponds to
the situation of the weak coupling BCS, for which
the pairing gap is given by
the well known formula\cite{Leggett,Randeria,Lombardo-Schulze}
$\Delta/e_F \approx 8e^{-2}\exp\left(\pi / 2k_F a\right)$.
In the opposite range $1/k_F a \gg 1$, the
situation of the Bose-Einstein condensation (BEC) of bound
Fermion pairs (bosons) is realized.
The crossover between the weak coupling BCS and the strong coupling
BEC corresponds to the interval
$-1 \m@thcombine<\sim 1/k_F a \m@thcombine<\sim 1$, as described in Refs.
\cite{Leggett,Melo,Randeria,Engelbrecht}.
(The case with the infinite scattering
length $1/k_F a=0$ is the midway of the
crossover, called the unitarity limit.)
In the following we shall adopt $1/k_F a=\pm 1$
according to Ref.\cite{Engelbrecht}
as boundaries characterizing the crossover
although the transition is smooth in nature.
In Table \ref{BCS-BEC} we list
the values of $\xi_{rms}/d$
and $\Delta_F/e_F$ at the ``boundaries''
$1/k_Fa= \pm 1$ of the crossover domain and
at the unitarity limit $1/k_Fa=0$ \cite{Engelbrecht,Marini}.
Note that the r.m.s. radius comparable to the average inter-particle
distance $0.2 \m@thcombine<\sim \xi_{rms}/d \m@thcombine<\sim 1.1$, or the
gap comparable to the Fermi energy
$0.2 \m@thcombine<\sim \Delta_F/e_F \m@thcombine<\sim 1.3$ corresponds to
the BCS-BEC crossover domain $-1 \m@thcombine<\sim 1/k_F a \m@thcombine<\sim 1$.
We have discussed in the previous section
the probability $P(d)$ for the paired neutrons
to come closer than the average inter-neutron distance $d$
(cf. Fig.\ref{prob}). As the same quantity is easily calculated
also in the analytic model of the BCS-BEC crossover
(the result is shown in Fig.\ref{prob-delta}), this quantity
may be used also as a measure of the crossover.
The calculated boundary values corresponding to $1/k_F a = \pm1, 0$
are listed in Table \ref{BCS-BEC}.
The crossover region is specified
by $0.8 \m@thcombine<\sim P(d)\m@thcombine<\sim 1.0$ while the boundary to the strong
coupling BEC regime ($P(d) \approx 1$) is hardly visible in this measure.
In the case of nucleonic matter, the assumption of
the dilute gas limit
may be justified only at very low density $\rho/\rho_0 \m@thcombine<\sim 10^{-5}$
(or $k_F \m@thcombine<\sim 0.05$ fm$^{-1}$)\cite{Heiselberg01,Khodel,Lombardo-Schulze},
and hence we cannot apply
the above analytic model in a direct manner to the region of the
density $\rho/\rho_0=10^{-5}-1$ which we are dealing with.
In order to make the application possible, we shall stand
on a more flexible viewpoint by
regarding the interaction parameter $1/k_F a$ as a freely adjustable
variable, rather than by fixing it from the physical value of the
neutron scattering length. We shall call the model treated in this way
the regularized delta interaction model to distinguish from
the original idea of the dilute gas limit.
The interaction parameter $1/k_F a$ needs to be determined then.
We shall require the condition that the regularized delta interaction
model gives, for a given value of density, the same r.m.s.
radius $\xi_{rms}/d$
as that of the microscopically calculated neutron Cooper pair.
The parameter determined in this way may be denoted $(1/k_Fa)_{\xi}$.
We can also determine the interaction parameter to
reproduce the ratio $\Delta_F/e_F$ between the gap and the
Fermi energy, which we shall denote $(1/k_Fa)_{\Delta}$. The values of
$(1/k_Fa)_{\xi}$ and $(1/k_Fa)_{\Delta}$ thus determined are listed in
Table \ref{coherence-length}. There is no sizable difference
between $(1/k_Fa)_{\xi}$ and $(1/k_Fa)_{\Delta}$.
The Cooper pair wave
functions obtained in this reference model are shown in
Fig.\ref{Cooper-wf-delta}.
It is hard to distinguish between the two options of
$1/k_F a$. We now compare them
with the neutron Cooper pair obtained with the
Gogny force for symmetric matter at the three representative
values of density $\rho/\rho_0=1, 1/8$ and 1/512.
\begin{figure}[htbp]
\centerline{
\includegraphics[width=8.5cm]{fig7a.eps}
\includegraphics[width=8.5cm]{fig7b.eps}
}
\caption{(a-c)
The neutron Cooper pair wave function $\Psi_{pair}(r)$
in the regularized delta interaction model, plotted
with the dotted curve and the cross symbol
in the cases of
$(1/k_F a)_{\xi}$ and $(1/k_F a)_{\Delta}$, respectively.
The neutron Cooper pair wave function in
symmetric nuclear matter obtained with the Gogny D1 force
is also shown by the solid curve.
(d-f)
The same as (a-c) but for the probability density $r^2|\Psi_{pair}(r)|^2$.
\label{Cooper-wf-delta}}
\end{figure}
It is seen from Fig.\ref{Cooper-wf-delta} that
the wave function of the regularized delta interaction model and
and that of the neutron
Cooper pair behave very similarly at
distances far outside the interaction range, $r \m@thcombine>\sim 5$ fm.
In contrast, we notice a
sizable disagreement for $r \m@thcombine<\sim 3$ fm.
The disagreement is understandable
as the wave function within the interaction range $r \approx 3$ fm
of the finite range Gogny force could not be
described by the zero-range delta interaction.
The Cooper pair wave function in the regularized delta interaction model
exhibits the known divergence $\Psi_{pair}(r) \propto 1/r$ for
$r \rightarrow 0$ (cf. Fig.\ref{Cooper-wf-delta}(a-c)), and consequently
the disagreement between the Gogny model and the regularized
delta interaction model
becomes serious at very short relative distances
$r \m@thcombine<\sim 1$ fm.
Note however that the squared wave function weighted with the
volume element $r^2$ stays finite as seen Fig.\ref{Cooper-wf-delta}(d-f)
and hence there is no diverging difference in the probability
density.
These observations suggest that
the regularized delta interaction model
can account for the essential features of the spatial structure of the
neutron Cooper pair as far as the interaction strength
$1/k_F a$ is chosen appropriately.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{ccccc}
\hline
\hline
$1/k_F a$ & $\xi_{rms}/d$ & $\Delta/e_F$ & $P(d)$ & \\
\hline
-1 & 1.10 & 0.21 & 0.807 & boundary to BCS \\
0 & 0.36 & 0.69 & 0.990 & unitarity limit \\
1 & 0.19 & 1.33 & 1.000 & boundary to BEC \\
\hline
\hline
\end{tabular}
\end{center}
\caption{The reference values of
$1/k_F a$, $\xi_{rms}/d$ and $\Delta/e_F$
characterizing the
BCS-BEC crossover in the regularized delta interaction model.}
\label{BCS-BEC}
\end{table}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=5cm]{fig8.eps}
}
\caption{
The probability $P(d)$ for the partner particles to be correlated
within the average inter-particle distance $d$
for the regularized delta interaction model.
\label{prob-delta}}
\end{figure}
We thus have a reference frame, i.e.,
the regularized delta interaction model,
to which the neutron pairing is mapped. The question on possible relation
to the BCS-BEC crossover phenomena can be addressed now quantitatively.
We first look into
the ratio $\xi_{rms}/d$ between the r.m.s radius $\xi_{rms}$ of
the Cooper pair and the average inter-particle distance $d$.
The values of $\xi_{rms}/d$ for the neutron
Cooper pair obtained with the G3RS force and the Gogny D1 interaction
are compared in Fig.\ref{rmsoverd} with the reference values
defining the ``boundaries'' of the BCS-BEC crossover domain
(Table \ref{BCS-BEC}).
It is seen in
Fig.\ref{rmsoverd} that
the calculated ratio $\xi_{rms}/d$ enters the domain
of the BCS-BEC crossover, $1.10 > \xi_{rms}/d (>0.19) $,
in the density interval $\rho/\rho_0 \approx 10^{-4} - 0.1$.
Note also the calculated ratio becomes closest to
the unitarity limit $\xi_{rms}/d=0.36$ around the density
$\rho/\rho_0 \sim 10^{-2}$.
In the other way around, the weak coupling BCS regime is realized
only at very low density $\rho/\rho_0 \m@thcombine<\sim 10^{-4}$ and
around the normal density $\rho/\rho_0 \m@thcombine>\sim 0.2$.
Comparing in Fig.\ref{deltaoveref}
the gap to Fermi-energy ratio $\Delta_F/e_F$ with the
boundary values $0.21<\Delta_F/e_F<1.33$,
we have the same observation that
the density region $\rho/\rho_0 \sim 10^{-4}-0.1$ corresponds to
the domain of the BCS-BEC crossover. Comparison of
the third measure $P(d)$, performed in Fig.\ref{prob}, provides
us the same information.
It is seen also
in the values of $(1/k_Fa)_\xi$ and $(1/k_Fa)_\Delta$
listed in Table \ref{coherence-length} that
the condition of the crossover region $(1/k_Fa)_{\xi,\Delta}>-1$
is met in the cases of $\rho/\rho=1/8,1/64$ and 1/512.
On the basis of the above analysis, we conclude that the
strong spatial correlation at short relative distances
seen in the neutron Cooper pair
in the very wide density range
$\rho/\rho_0 \approx 10^{-4} - 0.1$ is the behavior associated
with the BCS-BEC crossover.
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig9.eps}
}
\caption{
The ratio $\xi_{rms}/d$ between the r.m.s. radius $\xi_{rms}$ of the neutron
Cooper pair and the average inter-nucleon distance $d$, calculated
with the Gogny D1 force for
symmetric nuclear and neutron matter
(the solid and dotted lines, respectively),
and those with the G3RS force for symmetric nuclear
and neutron matter (the diamond and square
symbols), plotted as a function of the neutron density.
The reference values
characterizing the BCS-BEC crossover listed in Table \ref{BCS-BEC}
are also shown with the horizontal
dotted and dashed lines.
\label{rmsoverd}}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig10.eps}
}
\caption{
The ratio $\Delta_F/e_F$ between the neutron pairing gap $\Delta_F$
and the neutron Fermi energy $e_F$, plotted as a function of
the neutron density. The curves and the symbol are the same as in
Fig.\ref{rmsoverd}.
The reference values
characterizing the BCS-BEC crossover listed in Table \ref{BCS-BEC}
are also shown with the horizontal
lines.
\label{deltaoveref}}
\end{figure}
\section{density-dependent delta interaction}\label{DDDI-sec}
\subsection{DDDI and the cut-off energy}\label{cutoff-sec}
The contact force whose interaction strength is chosen
as a density-dependent parameter, called often the density-dependent delta
interaction (DDDI), has been employed as a phenomenological
effective interaction describing the pairing correlation in finite nuclei,
especially unstable nuclei with large neutron excess
\cite{DobHFB2,DD-Dob,DD-mix,Matsuo05,YuBulgac,Grasso,Stoitsov03,Teran,Bender}.
In the previous section we found that
the regularized contact interaction model, which employs also the
contact force but with an analytic regularization,
describes the essential feature of the spatial structure of the
neutron Cooper pair wave function in the whole density range.
This suggests a possibility that
the phenomenological DDDI may also describe the
spatial correlation in the neutron pairing.
We would like to examine from this viewpoint in what conditions
the DDDI can be justified.
The density-dependent delta interaction has a form
\begin{equation}\label{dddi-eq}
v(\mbox{\boldmath $r$})={1-P_\sigma \over 2}V_0[\rho]\delta(\mbox{\boldmath $r$}),
\end{equation}
where $V_0[\rho]$ is the interaction strength which is supposed to be
dependent on the density. The force acts only in the
$^{1}S$ channel due to the projection operator $(1-P_\sigma)/2$.
It should be noted that in the applications of the density-dependent
delta interaction to finite nuclei,
an explicit and finite cut-off energy
needs to be introduced.
The cut-off energy in this case is regarded as an additional model parameter.
Applying the DDDI
to the neutron pairing in uniform matter,
the gap equation reads
\begin{equation}\label{gap-dddi-eq}
\Delta= - {V_0 \over 2(2\pi)^3}\int' d \mbox{\boldmath $k$}
{\Delta \over E(k)}
\end{equation}
where the pairing gap here is a momentum independent constant $\Delta$.
For the single-particle energy $e(k)$,
we adopt the effective mass approximation.
We use the effective mass derived from the Hartree-Fock spectrum
of the Gogny D1.
The momentum integral in Eq.(\ref{gap-dddi-eq}) is performed
under a sharp cut-off condition
\begin{equation}\label{ecut}
e(k) < \mu + e_{cut}
\end{equation}
where we define the cut-off energy $e_{cut}$ as relative energy
from the chemical potential $\mu$.
We shall treat $e_{cut}$ as
a common constant which is applied to all density.
We use the same cut-off in calculating the Cooper pair wave function.
We remark that our definition of the cut-off
is different from a similar one adopted in
Refs.\cite{Bertsch91,Garrido},
where a cut-off
energy is defined with respect to the
single-particle energy $e(k)$ measured from the bottom of
the spectrum $e(k=0)$, e.g., by imposing $e(k) < 60$ MeV\cite{Garrido}
independent of the density. In our case, by contrast, we fix
an energy window above the chemical potential $\mu$ to $e_{cut}$.
We think that the cut-off energy $e_{cut}$ defined in this way can be
compared with the quasiparticle energy cut-off $E_i <E_{cut}$ adopted often
in the HFB calculations for finite nuclei ($E_i$ is the quasiparticle energy
of the single-particle state $i$)
\cite{DobHFB,DobHFB2,DD-Dob,DD-mix,Matsuo05,YuBulgac,Grasso,Stoitsov03,Teran,Bender}.
Note that in finite nuclei the density varies
locally with the position coordinate while the
quasiparticle energy is defined globally. This may imply, in the
sense of the local density approximation, that a fixed density-independent
cut-off quasiparticle energy $E_{cut}$ is applied to each value of
local density.
Note also that both our $e_{cut}$ and the cut-off quasiparticle energy
$E_{cut}$ are quantities measured from the chemical potential,
which approximately coincide
if $e_{cut}$ and $E_{cut}$ are sufficiently larger than the pairing gap.
Thus we can compare directly $e_{cut}$ and $E_{cut}$, provided that
$e_{cut}$ is chosen as a density-independent constant.
At the zero-density limit with $\mu=0$,
the cut-off energy $e_{cut}$ simply defines an upper bound on the
free single-particle energy $e(k)= \hbar^2 k^2/2m$.
\subsection{Constraints on $e_{cut}$}\label{cutoffdep-sec}
Let us
investigate how the energy cut-off influences the Cooper pair wave function.
We performed several calculations using different values of
$e_{cut}=5,10,30, 50, 100, 200$ MeV for symmetric nuclear matter.
In doing so, we choose
the interaction strength $V_0$ for each value of density so that the
gap $\Delta$ calculated with a given cut-off energy
$e_{cut}$ coincides with the gap $\Delta_F$ obtained
with the Gogny D1 force. Note that the
interaction strength $V_0$ thus determined
depends on both the cut-off energy
and the density.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{cccccccc}
\hline
\hline
&\multicolumn{6}{c}{$\xi_{rms}$ [fm]} & \\
$\rho/\rho_0$ & $e_{cut}=$5 & 10 & 30 & 50 & 100 & 200 MeV & Gogny D1\\
\hline
1 & \underline{48.6} & \underline{47.3} & \underline{46.7} & \underline{46.6} & \underline{46.5} & \underline{46.5}
& 46.6 \\
1/2 & 16.9 & 13.8 & \underline{11.2} & \underline{10.9} & \underline{10.7} & \underline{10.6} &
10.8 \\
1/8 & 14.9 & 10.0 & 6.0 & \underline{5.3} & \underline{4.8} & \underline{4.6} & 4.8 \\
1/64 & 10.5 & 7.9 & \underline{6.2} & \underline{6.0} & \underline{5.7} & \underline{5.5} & 5.9
\\
1/512 & \underline{13.2} & \underline{12.5} & \underline{12.0} & \underline{11.9} & \underline{11.8} & \underline{11.7} & 12.1 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{The r.m.s. radius $\xi_{rms}$ of the neutron
Cooper pair in symmetric nuclear matter obtained with the
density dependent delta interaction
and different cut-off energies
$e_{cut}=5,10,30,50,100,200$ MeV.
In the rightmost column, the r.m.s. radius for the Gogny D1 force
is also listed.
The underline means that
the calculated number agrees with the reference Gogny D1 result within
$10\%$.}
\label{rms-dddi}
\end{table}
\begin{figure}[htbp]
\centerline{
\includegraphics[width=8.5cm]{fig11a.eps}
\includegraphics[width=8.5cm]{fig11b.eps}
}
\caption{
(a-c) The neutron Cooper pair wave function $\Psi_{pair}(r)$
in symmetric nuclear matter calculated with the DDDI
having different cut-off energies
$e_{cut}=5,10,30,50,100$ MeV.
The result obtained with the Gogny D1 interaction is shown also by
the solid curve.
(d-f) The same as (a-c) but for the probability density $r^2|\Psi_{pair}(r)|^2$.
\label{Cooper-wf-dddi}}
\end{figure}
Table \ref{rms-dddi} shows the r.m.s. radius
$\xi_{rms}$ of the neutron Cooper pair calculated with the DDDI
for different values of the cut-off energy.
Here $\xi_{rms}$ is calculated by using the
Cooper pair wave function in the coordinate representation evaluated up to
$r=500$ fm.
It is seen from Table \ref{rms-dddi} that the result apparently
depends on the cut-off energy $e_{cut}$.
In the cases of $\rho/\rho_0=1/64 - 1/2$, the
dependence of $\xi_{rms}$ on $e_{cut}$ is very strong. The values
calculated with the small cut-off energy $e_{cut}=5, 10$ MeV
largely deviate from
those obtained with the Gogny force even though the interaction strength
is chosen to reproduce the same reference pairing gap.
We consider that
the small cut-off energies $e_{cut}=5, 10$ MeV are unacceptable
since they fail to describe the small size $\xi_{rms} \sim 5$ fm
of the neutron Cooper pair at the density around
$\rho/\rho_0=10^{-2} - 0.1$. If we require that the DDDI reproduces
the r.m.s. radius of the neutron
Cooper pair within an accuracy of 10\%
in the whole density region of interest,
use of a large value of the cut-off energy satisfying $e_{cut}\m@thcombine>\sim 50$ MeV
is suggested.
To see roles of the cut-off energy in more details,
we show in Fig.\ref{Cooper-wf-dddi}
the neutron Cooper pair wave functions $\Psi_{pair}(r)$ obtained
for $e_{cut}=5,10,30,50,100$ MeV.
The plot of $\Psi_{pair}(r)$ indicates clearly
that the Cooper pair wave function depends sensitively on the
cut-off energy $e_{cut}$.
If we adopt the small cut-off energies $e_{cut}=5,10$ MeV
the wave function obtained with the DDDI
fails to produce the strong spatial correlation
at the short relative distances $r \m@thcombine<\sim 3$ fm,
which is the characteristic feature
of the neutron
Cooper pair wave function common to the Gogny and G3RS forces.
(Fig.\ref{Cooper-wf-dddi} shows only the Gogny result for comparison,
but we remind the reader of Fig.\ref{Cooper-wf} where the G3RS case
is also shown.)
The plot of the probability density
$r^2|\Psi_{pair}(r)|^2$ at the density $\rho/\rho_0=1/8$
indicates that even the wave function at
larger distances is not described well if the small cut-off energies
$e_{cut}=5, 10$ MeV are adopted. This is nothing but the
difficulty mentioned above in describing the r.m.s. radius with these
small cut-off energies.
If we use a large cut-off energy, say $e_{cut} \m@thcombine>\sim 30-50$ MeV,
the wave function at large distances converges reasonably to that
obtained with the Gogny force. Concerning the wave function at short relative
distances $r \m@thcombine<\sim 3$ fm, on the other hand, we find no convergence
with respect to the cut-off energy. The
value of the wave function $\Psi_{pair}(0)$ at zero relative distance
$r=0$ increases monotonically with increasing $e_{cut}$. (Increasing further $e_{cut}\rightarrow \infty$,
$\Psi_{pair}(r)$ will approach to the one for the
regularized delta interaction model shown in Fig.\ref{Cooper-wf-delta}, and
the value $\Psi_{pair}(0)$ at $r=0$ will diverge.)
It may be possible to regard $e_{cut}$ as a parameter which simulates
the finite range of the neutron-neutron interaction.
It is then reasonable to require that
the wave function $\Psi_{pair}(r)$ of the DDDI model with an
appropriate choice of $e_{cut}$ describes
that of the Gogny force at distances $r\m@thcombine<\sim 3$ fm (within
the interaction range) as well as at larger distances.
In the case of $\rho/\rho_0=1/8$, for example,
this requirement is approximately satisfied if we choose
$e_{cut}=30$ or $50$ MeV, see Fig.\ref{Cooper-wf-dddi}(b).
At $\rho/\rho_0=1/512$, a good description of the wave function is
obtained with $e_{cut}=30$ MeV
(Fig.\ref{Cooper-wf-dddi}(c)), and similarly we find $e_{cut}\sim 70$ MeV for the
normal density $\rho/\rho_0=1$ (Fig.\ref{Cooper-wf-dddi}(a)).
If we do not include in the comparison
the wave function at very short distances
$r \m@thcombine<\sim 1$ fm where the repulsion due to the core influences in the
case of the bare force, the constraint on the cut-off energy may be slightly
relaxed. For example,
at the density $\rho/\rho_0=1/512$,
the wave functions for $e_{cut}=30$ and $50$ MeV
differ only by about $\m@thcombine<\sim 20\%$ at distances
$ 1<r<3$ fm, and hence the
cut-off energy $e_{cut}=50$ MeV may also be accepted.
Within this tolerance we can choose a
value around $e_{cut} \sim 50$ MeV as the cut-off
energy which can be used commonly
in the whole density region of interest.
This value can be compromised with the constraint $e_{cut} \m@thcombine>\sim 50$ MeV
which we obtained from the condition on the r.m.s. radius of the neutron
Cooper pair.
It is interesting to note that
cut-off quasiparticle energies around $E_{cut}=50-70$ MeV have been
employed in many of recent HFB applications to finite nuclei
\cite{DobHFB,DobHFB2,DD-Dob,DD-mix,Matsuo05,YuBulgac,Grasso,Stoitsov03,Teran,Bender}.
These cut-off energies are consistent with the
constraint $e_{cut} \sim 50$ MeV
suggested from the above analysis.
Much smaller cut-off energies $\m@thcombine<\sim 10$ MeV adopted in early applications
of the DDDI \cite{DDpair-Taj,DDpair-Tera} are not appropriate from the view point of
the spatial structure of the neutron Cooper pair wave function.
In Ref.\cite{Bertsch91} the cut-off energy of $20$ MeV
for the single-particle energy (40 MeV in the center of mass
frame energy) was shown to describe reasonably
the scattering wave function at zero energy.
This cut-off energy is not very different from the cut-off energy $e_{cut} \sim 30$ MeV
which we find most reasonable (among the selected examples) in the lowest density case
$\rho/\rho_0=1/512$.
In the the delta interaction model adopted in Ref.\cite{Esbensen97}
the cut-off energy is examined with respect to
the low-energy scattering
phase shift in the $^1S$ channel. The cut-off value adopted
is around 5-10 MeV in the single-particle energy
($9-20$ MeV for the center of mass frame energy),
in disagreement with our value $e_{cut} \sim 30$ MeV.
The difference seems to
originate from different strategies to the delta interaction:
the momentum dependence of the interaction matrix element
is accounted for by the cut-off energy in Ref.\cite{Esbensen97}
while it is taken into account in the present approach mostly
through the density dependent interaction strength $V_0[\rho]$.
\subsection{DDDI parameters}\label{parm-sec}
It is useful to parameterize the interaction strength of the DDDI
in terms of a
simple function of the density. The following form is often assumed
\cite{Bertsch91,Garrido,DobHFB2,DD-Dob,DD-mix,DDpair-Taj,DDpair-Tera}:
\begin{eqnarray}\label{dddi-int-eq}
V_0[\rho]&=&v_0 \left(1 - \eta \left({\rho_{tot} \over \rho_c}\right)^\alpha\right), \\
\rho_c&=&0.16\ {\rm fm}^{-3}
\end{eqnarray}
where $\rho_{tot}$ is the total nucleon density.
The parameters $v_0,\eta$ and $\alpha$ need to be determined.
In the works
by Bertsch and Esbensen\cite{Bertsch91} and Garrido {\it et al.}\cite{Garrido}
the parameters are determined so that the parameterized DDDI reproduces
the pairing gap obtained with the Gogny force
in symmetric nuclear matter as well as the experimental s-wave scattering
length at zero density.
We shall follow a similar line, but we add the important constraint
that the spatial structure of the neutron Cooper pair is also
reasonably reproduced. As discussed in the subsection just above,
this can be achieved if we constrain the cut-off energy to a value around
$e_{cut} \sim 50$ MeV. Note also
our definition of the cut-off energy is different from that
in Refs.\cite{Bertsch91,Garrido},
as mentioned in Subsection \ref{cutoff-sec}.
Our procedure is as follows. We consider symmetric nuclear
matter. The cut-off energy is fixed
to $e_{cut}=50$ MeV or 60 MeV for the reasons mentioned above.
We then fix the interaction strength $V_0[0]=v_0$ at zero density
to a value $v_0$ which reproduces the scattering length $a$ in the
free space.
$v_0$ satisfying this condition is given by\cite{Bertsch91,Garrido}
\begin{eqnarray}\label{dddi-zero-eq}
v_0&=& -{ 2\pi^2\hbar^2 m^{-1} \over k_c - \pi/2a},\\
k_c &=& \sqrt{2me_{cut}}/\hbar.
\end{eqnarray}
If we use as the scattering length $a$ in Eq.(\ref{dddi-zero-eq})
the one associated with the Gogny force,
the pairing gap of the DDDI in the low density
limit $\rho/\rho_0 \rightarrow 0$
coincides with that of the Gogny force. However, since the
scattering length $a=-13.5$ fm of the Gogny D1 is slightly off
the experimental value, we adopt the experimental one $a=-18.5$ fm
for Eq.(\ref{dddi-zero-eq}).
This is equivalent to constrain the DDDI at the low density limit
by the bare nucleon force.
To determine the other parameters $\eta$ and $\alpha$ controlling
the density-dependence of the interaction strength,
we first calculate at several representative points of density
the values of $V_0$ with which the neutron gap $\Delta_F$
of the Gogny D1 force is reproduced.
We then search the parameters $\eta$ and $\alpha$
so that the simple function Eq.(\ref{dddi-int-eq}) fits well to the values of
$V_0$ thus determined.
We consider
the density interval
$\rho/\rho_0 \sim 10^{-2}- 1$ ($k_F \sim 0.3 - 1.4$ fm$^{-1}$)
in this fitting.
The obtained values of the parameters (denoted DDDI-D1) are shown in
Table \ref{dddi-parm}, where we list also the parameter set obtained
when we use $e_{cut}=60$ MeV instead of 50 MeV. Another
set of the parameters derived in the same way from the Gogny D1S force
(DDDI-D1S) is listed. We performed the same procedure also for the
G3RS force (DDDI-G3RS).
The pairing gap obtained with these parameterizations of
the DDDI are shown in Fig.\ref{dddi-gap}.
The resultant gap $\Delta$ agrees with that of the corresponding
reference gap to the accuracy of about one hundred keV
for the whole density region below $\rho/\rho_0=1$. Although
the results for $e_{cut}=60$ MeV are not shown here,
the agreement with the reference gaps is as good as in the $e_{cut}=50$ MeV
case.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{lcccc}
\hline
\hline
& \hspace{5mm} $v_0$ [MeV fm$^{-3}$] \hspace{5mm}& $\eta$ & $\alpha$ \\
\hline
DDDI-D1\hspace{20mm} &&&& \\
\hspace{5mm}$e_{cut}=50$ MeV & -499.9 & 0.627 & 0.55 \\
\hspace{5mm}$e_{cut}=60$ MeV & -458.4 & 0.603 & 0.58 \\
DDDI-D1S \hspace{20mm}&&&& \\
\hspace{5mm}$e_{cut}=50$ MeV & -499.9 & 0.652 & 0.56 \\
\hspace{5mm}$e_{cut}=60$ MeV & -458.4 & 0.630 & 0.60 \\
DDDI-G3RS \hspace{20mm}&&&& \\
\hspace{5mm}$e_{cut}=50$ MeV & -499.9 & 0.872 & 0.58 \\
\hspace{5mm}$e_{cut}=60$ MeV & -458.4 & 0.845 & 0.59 \\
\hline
\hline
\end{tabular}
\end{center}
\caption{
The parameter sets of the density-dependent delta interaction
with the cut-off energies $e_{cut}=50$ and 60 MeV, derived from
the procedure applied to the Gogny D1 and D1S, and the G3RS forces.
See text for details. }
\label{dddi-parm}
\end{table}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig12.eps}
}
\caption{The pairing gap in symmetric nuclear matter
obtained with the DDDI parameter sets shown in Table \ref{dddi-parm}
with the cut-off energy $e_{cut}=50$ MeV. The solid,
dotted and dashed curves represent the result for the
parameter sets
DDDI-D1, DDDI-D1S, and DDDI-G3RS, respectively.
The symbols represent the gap for the reference
calculations with the Gogny D1 (square) and D1S (cross)
forces, and the G3RS force (circle).
\label{dddi-gap}}
\end{figure}
\begin{figure}[htbp]
\centerline{
\includegraphics[angle=270,width=8.5cm]{fig13.eps}
}
\caption{The density-dependent interaction strength
$V_0[\rho]$ of the DDDI for the parameter sets
DDDI-D1 (solid curve) and DDDI-G3RS (dotted curve) with
$e_{cut}=60$ MeV. For comparison, $V_0[\rho]$
for the phenomenological DDDI parameters of the surface and mixed types are also
shown by the symbols. See the text for details.
\label{dddi-v0}}
\end{figure}
It is noticed in Table \ref{dddi-parm} that the parameter $\alpha$
takes a similar value $\alpha = 0.58-0.60$ (in the case of $e_{cut}=60$ MeV)
for all of DDDI-D1, DDDI-D1S, and DDDI-G3RS,
while the difference between the Gogny forces (DDDI-D1,-D1S) and
the G3RS force (DDDI-G3RS) is readily recognized
in the value of $\eta$. It is seen also that the difference
in the cut-off energy influences slightly the values of $v_0$,
$\eta$ and $\alpha$. If we compare our result with
that of Ref.\cite{Garrido},
our parameter values $\eta=0.60-0.63$ for the prefactor
and $\alpha=0.55-0.58$ for the power imply
stronger density dependence in $V_0[\rho]$ than that in
Ref.\cite{Garrido}, where
the
parameters are determined as $\eta=0.45, \alpha=0.47$ from a
similar fitting to the gap with the Gogny D1 force.
This is due to the difference in the cut-off schemes mentioned
in Subsection \ref{cutoff-sec}.
Since the chemical
potential $\mu$ increases with the density, the
energy window measured from the chemical potential $\mu$
decreases with increasing the density in the scheme of Ref.\cite{Garrido}
where the cut-off $e(k) < 60$MeV is adopted for all density
while in our cut-off
scheme the energy window
is kept constant $e_{cut}=50,60$ MeV independent of the density.
Consequently the stronger density dependence in $V_0[\rho]$
is needed in our case.
It may be interesting to compare our DDDI parameter sets with those
determined phenomenologically from the experimental
pairing gap in the ground states of finite nuclei. Such a comparison
is made in
Fig.\ref{dddi-v0}, where the density dependent interaction strength
$V_0[\rho]$ is plotted. The phenomenological DDDI's employed here are the
so called surface and mixed types, for which the prefactor parameter
$\eta$ is fixed to $\eta=1$ and $\eta=0.5$, respectively.
The power parameter
is assumed as $\alpha=1$ for the mixed type\cite{DD-mix},
while for the surface type
we choose here $\alpha=1$ or 1/2.
(Note that the power parameter in the surface
type DDDI was investigated in Ref.\cite{DD-Dob},
and $ 1/2 \m@thcombine<\sim \alpha \m@thcombine<\sim 1$
is suggested as a reasonable range of the parameter.)
The strength of the surface type DDDI used here is
$v_0=-521$ MeV fm$^{-3}$ for $\alpha=1$, and
$v_0=-781$ for $\alpha=1/2$,
taken from Ref.\cite{DD-Dob},
where the value of $v_0$ is determined by a Skyrme HFB calculation for
$^{120}$Sn to reproduce
the gap $1.25$ MeV.
The equivalent energy cut-off 60 MeV adopted in the HFB calculation
corresponds to our cut-off $e_{cut}=60$ MeV.
In the case of the mixed type DDDI, the adopted strength is
$v_0=-290$ MeV fm$^{-3}$ derived from
the same condition on $^{120}$Sn.
It is seen in Fig.\ref{dddi-v0} that the density dependence in
the present parameterizations of $V_0[\rho]$ is mild in the range
$\rho_{tot}/\rho_c \m@thcombine>\sim 0.4$, resembling more
to that of the mixed
type DDDI than to that of the surface type.
At lower density $\rho_{tot}/\rho_c \m@thcombine<\sim 0.3$ the density dependence
is stronger
than that of the mixed type DDDI. Note that
the interaction strength $V_0[\rho]$ itself is not large: it takes
the values between those of the mixed and the surface
type DDDI's in this low density region. The above comparisons suggest that
the density dependence in
the present parameterization
of $V_0[\rho]$ may not be very unrealistic for applications to
finite nuclei. It could be also suggested that the present parameterization
will be free from the problems pointed to in Ref.\cite{DD-Dob} for
strongly density dependent DDDI's such as
the surface type DDDI with small values
of the power $\alpha\m@thcombine<\sim 1/2$.
For more definite conclusions, we need to make more quantitative analyses
using
the HFB calculation performed directly for finite nuclei.
It is also interesting to compare
with a new approach to the DDDI with a microscopically
derived cut-off factors\cite{Duguet}.
This is, however, beyond the scope of this paper,
and such analyses will be pursued in future.
\section{Conclusions}\label{concl-sec}
We have analyzed the spatial structure of
the neutron Cooper pair
obtained with the BCS approximation for neutron and symmetric nuclear
matter
using a bare force and the effective Gogny interaction.
The size of the Cooper pair varies significantly
with the density: its r.m.s. radius $\xi_{rms}$ becomes
as small as $\sim 5$ fm around
$\rho/\rho_0 \approx 10^{-2}-0.2$, and $\xi_{rms}$
smaller than the average inter-neutron distance $d$ is
realized in a very wide range
$\rho/\rho_0\approx 10^{-4}-0.1$ at low density.
The analysis of the Cooper pair wave function
indicates that the probability for the spin up and down neutrons in the
pair to be correlated within the
average inter-neutron distance $d$ exceeds more than 0.8 in this density
range. The strong spatial correlation
at short relative distances is also seen for modest density
$\rho/\rho_0 \sim 0.5$, at which the concentration of the pair neutrons
within the interaction range $\sim 3$ fm reaches about 0.5.
These observations suggest that
the spatial di-neutron correlation is strong, at least in the level of the
mean-field approximation, in
low-density superfluid uniform matter
in the wide range of density $\rho/\rho_0\approx 10^{-4}-0.5$.
The essential feature does not depend on the interactions.
We have investigated the behaviors of the strong di-neutron correlation
in connection with the crossover phenomenon
between the conventional pairing of the weak coupling BCS type and
the Bose-Einstein condensation of the bound neutron pairs. Comparing with
the analytic BCS-BEC crossover model assuming a contact interaction, we found
that the density region
$\rho/\rho_0\approx 10^{-4}-10^{-1}$
corresponds to the domain of the BCS-BEC crossover.
We have examined also how the density dependent delta interaction (DDDI)
combined with a finite cut-off energy can describe the spatial
correlation of the neutron Cooper pair.
The spatial correlation at short relative distances and the
r.m.s. radius of the pair are described consistently in a wide density
region $0<\rho/\rho_0\m@thcombine<\sim 1$ provided that we adopt
a cut-off energy around $e_{cut}\sim 50$ MeV defined with respect to the
chemical potential.
We have derived a possible parametrization of the DDDI, which
satisfies this new condition on top of the constraints
on the gap in symmetric nuclear matter and on the scattering length
in the free space.
The new DDDI parameterizations may be consistent with or at least not strongly
contradictory to the phenomenological DDDI's derived from the gap
in finite nuclei.
\section*{Acknowledgments}
The author thanks P.~Schuck and M.~Hjorth-Jensen for useful discussions.
He thanks also the Yukawa Institute for Theoretical Physics at
Kyoto University and the Institute for Nuclear Theory at University
of Washington. Discussions during the YITP workshop YITP-W-05-01 on
``New Developments in Nuclear Self-Consistent Mean-Field Theories''
and the INT workshop ``Towards a universal density functional
for nuclei'' in the program INT-05-3 ``Nuclear Structure Near the Limits
of Stability'' were useful to complete this work.
Discussions with the members of the
Japan-U.S. Cooperative Science Program
``Mean-Field Approach to Collective Excitations in Unstable
Medium-Mass and Heavy Nuclei'' are also acknowledged.
This work was supported by the Grant-in-Aid for Scientific
Research (No. 17540244) from the Japan Society for the Promotion
of Science.
|
2,869,038,154,914 | arxiv | \section{Introduction}
\label{sec:0}
These lectures are intended as a pedagogical introduction to the attractor
mechanism. With this mission in mind we will seek to be explicit and,
to the extent possible, introduce the various ingredients using rather
elementary concepts. While this will come at some loss in mathematical
sophistication, it should be helpful to students who are not already
familiar with the attractor mechanism and, for the experts, it may serve
to increase transparency.
A simple and instructive setting for studying the attractor mechanism is M-theory
compactified to five dimensions on a Calabi-Yau three-fold. The resulting low energy
theory has $N=2$ supersymmetry and it is based on real special geometry.
We will focus on this setting because of the pedagogical mission of the lectures: real
special geometry is a bit simpler than complex special geometry,
underlying $N=2$ theories in four dimensions.
The simplest example where the attractor mechanism applies is that of a regular,
spherically symmetric black hole that preserves supersymmetry. In the first lecture
we develop the attractor mechanism in this context, and then verify the results by
considering the explicit black hole geometry.
In the second lecture we generalize the attractor mechanism to situations
that preserve supersymmetry, but not necessarily spherical symmetry.
Some representative examples are rotating black holes, multi-center black holes,
black strings, and black rings. Each of these examples introduce new features
that have qualitative significance for the implementation of the attractor mechanism.
The approach will follow the paper \cite{Kraus:2005gh} rather closely, with the
difference that here we include many more examples and other pedagogical material
that should be helpful when learning the subject.
In the third lecture we consider an alternative approach to the attractor
mechanism which amount to seeing the attractor behavior as a result of an
extremization procedure, rather than a supersymmetric flow. One setting that
motivates this view is applications to black holes that are extremal, but not
supersymmetric. Extremization principles makes it clear that the attractor mechanism
applies to such black holes as well.
Another reason for the interest in extremization principles is more philosophical:
we would like to understand what the attractor mechanism means in terms of
physical principles. There does not yet seem to be a satisfactory formulation
that encompasses all the different examples, but there are many interesting
hints.
The literature on the attractor mechanism is by now enormous. As general references
let us mention from the outset the original works \cite{Ferrara:1995ih} establishing the
attractor mechanism. It is also worth highlighting the review \cite{Moore:2004fg} which
considers the subject using more mathematical sophistication than we do here.
In view of the extensive literature on the subject we will not be comprehensive
when referencing. Instead we generally provide just a few references that may serve
as entry points to the literature.
\section{The Basics of the Attractor Mechanism}
\label{sec:1}
In this section we first introduce a few concepts from the geometry of
Calabi-Yau spaces and real special geometry.
We then review the compactification of eleven-dimensional
supergravity on a Calabi-Yau space and the resulting $N=2$ supergravity
Lagrangian in five dimensions. This sets up a discussion of the attractor mechanism
for spherically symmetric black holes in five dimensions. We conclude the lecture
by giving explicit formulae in the case of toroidal compactification.
\subsection{Geometrical Preliminaries}
On a complex manifold with hermitian metric $g_{\mu{\bar\nu}}$
it is useful to introduce the K\"{a}hler two-form $J$ through
\begin{equation}
J = i g_{\mu{\bar\nu}} dz^\mu \wedge dz^{\bar\nu}~.
\end{equation}
K\"{a}hler manifolds are complex manifolds with hermitian metric
such that the corresponding K\"{a}hler form is closed, $dJ=0$.
The linear space spanned by all closed $(1,1)$ forms (modulo
exact forms) is an important structure that is known as the
Dolbault cohomology and given the symbol $H^{1,1}_{\bar\partial}$.
If we denote by $J_I$ a basis of this cohomology we can expand the
closed K\"{a}hler form as
\begin{equation}
\label{aexp}
J = X^I J_I ~~~~~;~I=1,\ldots, h_{11}~.
\end{equation}
This expansion is a statement in the sense of cohomology so it should be
understood modulo exact forms.
Introducing the basis $(1,1)$-cycles $\Omega^I$ we can write the
expression
\begin{equation}
\label{XIdef}
X^I = \int_{\Omega^I} J~~~~~; ~I=1,\ldots, h_{11},
\end{equation}
for the real expansion coefficients $X^I$ in (\ref{aexp}). We see that they
can be interpreted geometrically as the volumes of $(1,1)$-cycles within the
manifold. The $X^I$ are known as K\"{a}hler moduli. In the context of
compactification the K\"{a}hler moduli become functions on spacetime and
so the $X^I$ will be interpreted as scalar fields.
One of several ways to define a Calabi-Yau space is that it is a K\"{a}hler manifold
that permits a globally defined holomorphic three-form. One consequence of this
property is that Calabi-Yau spaces do not have any $(0,2)$ and $(2,0)$ forms. For this
reason the $(1,1)$ cycles $\Omega^I$ are in fact the only two-cycles on the manifold.
The two-cycles $\Omega^I$ give rise to a dual basis of four-cycles $\Omega_I$,
$I=1,\ldots, h_{11}$, constructed such that their intersection numbers
with the two-cycles are canonical $(\Omega^I,\Omega_J)=\delta^I_J$. The volumes
of the four-cycles are measured by the K\"{a}hler form as
\begin{equation}
X_I = {1\over 2} \int_{\Omega_I} J\wedge J~.
\end{equation}
The integral can be evaluated by noting that the two-form $J_I$ covers
the space transverse to the $4$-cycle $\Omega_I$. Therefore
\begin{equation}
\label{fourc}
X_I = {1\over 2} \int_{CY} J\wedge J\wedge J_I = {1\over 2} C_{IJK}
X^J X^K~,
\end{equation}
where the integrals
\begin{equation}
C_{IJK} = \int_{CY} J_I \wedge J_J \wedge J_K~,
\end{equation}
are known as intersection numbers because they count the points
where the four-cycles $\Omega_I$, $\Omega_J$ and $\Omega_K$
all intersect.
\subsection{The Effective Theory in Five Dimensions}
\label{effthy}
We next review the compactification of M-theory on a Calabi-Yau manifold \cite{Cadavid:1995bk}.
The resulting theory in five dimensions can be approximated at large distances
by $N=2$ supergravity. In addition to the $N=2$ supergravity multiplet, the
low energy theory will include matter organized into a number of $N=2$
vector multiplets and hypermultiplets. In discussions of the attractor mechanism
the hyper-multiplets decouple and can be neglected. We therefore
focus on the gravity multiplet and the vector multiplets.
The $N=2$ supergravity multiplet in five dimensions contains the metric, a vector
field, and a gravitino (a total
of 8+8 physical bosons+fermions). Each $N=2$ vector multiplet in five dimensions
contains a vector field, a scalar field, and a gaugino (a total of 4+4 physical bosons+fermions).
It is useful to focus on the vector fields. These fields all have their origin
in the three-form in eleven dimensions which can be expanded as
\begin{equation}
\label{gaud}
{\cal A} = A^I \wedge J_I~~~~;~I=1,\ldots, h_{11}~.
\end{equation}
The $J_I$ are the elements of the basis of $(1,1)$ forms introduced
in (\ref{aexp}). Among the
$h_{11}$ gauge fields $A^I$, $I=1,\ldots, h_{11}$, the linear combination
\begin{equation}
\label{graviphoton}
A^{\rm grav} = X_I A^I~,
\end{equation}
is a component of the gravity multiplet. This linear combination is
known as the graviphoton. The remaining $n_V = h_{11} -1$ vector fields are
components of $N=2$ vector multiplets.
The scalar components of the vector multiplets are essentially the scalar fields
$X^I$ introduced in (\ref{XIdef}). The only complication is that, since one of
the vector fields does not belong to a vector multiplet, it must be that
one of the scalars $X^I$ also does not belong to a vector multiplet.
Indeed, it turns out that the overall volume of the Calabi-Yau space
\begin{equation}
\label{constraint}
{\cal V} = {1\over 3!} \int_{CY} J\wedge J\wedge J = {1\over 3!} C_{IJK} X^I X^J X^K~,
\end{equation}
is in a hyper-multiplet. As we have already mentioned, hyper-multiplets decouple and
we do not need to keep track of them. Therefore (\ref{constraint}) can be treated as a
constraint that sets a particular combination of the $X^I$'s to a constant.
The truly independent scalars obtained by solving the constraint (\ref{constraint}) are
denoted $\phi^i$, $i=1,\ldots, n_V$. These are the scalars that belong to vector multiplets.
We now have all the ingredients needed to present the Lagrangean of the theory.
The starting point is the bosonic part of eleven-dimensional supergravity
\begin{equation}
S_{11} = {1\over 2\kappa_{11}^2} \int \left[ - R ~{}^*1 - {1\over 2}{\cal F} \wedge {}^* {\cal F} - {1\over 3!}
{\cal F} \wedge {\cal F} \wedge {\cal A} \right]~,
\end{equation}
where the four-form field strength is ${\cal F}=d{\cal A}$. The coupling constant
is related to Newton's constant as $\kappa^2_D = 8\pi G_D$.
Reducing to five dimensions we find
\begin{equation}
\label{fivedact}
S_5
= {1\over 2\kappa_5^2} \int
\left[ - R ~{}^*1 - G_{IJ} dX^I \wedge {}^* dX^J
- G_{IJ} F^I \wedge {}^* F^J
- {1\over 3!} C_{IJK} F^I \wedge F^J\wedge A^K\right]~,
\end{equation}
where $F^I = dA^I$ and $\kappa_{5}^2=\kappa^2_{11}/{\cal V}$. The hodge-star is now the
five-dimensional one, although we have not introduced new notation to stress this fact.
The gauge kinetic term in (\ref{fivedact}) is governed by the metric
\begin{equation}
\label{GIJdef}
G_{IJ} = {1\over 2} \int_{CY} J_I \wedge {}^*J_J ~.
\end{equation}
It can be shown that
\begin{equation}
\label{bbaa}
G_{IJ} = - {1\over 2} \partial_I \partial_J (\ln {\cal V}) = -{1\over 2{\cal V}} (C_{IJK} X^K -
{1\over{\cal V}}X_I X_J)~,
\end{equation}
where the notation $\partial_I = {\partial\over\partial X^I}$.
Combining (\ref{fourc}) and (\ref{constraint}) we have the relation
\begin{equation}
\label{bba}
X_I X^I=3{\cal V}~,
\label{xixi}
\end{equation}
and so (\ref{bbaa}) gives
\begin{equation}
G_{IJ}X^J = {1\over 2{\cal V}} X_I~.
\label{gijlower}
\end{equation}
The metric $G_{IJ}$ (and its inverse $G^{IJ}$) thus lowers (and raises) the
indices $I,J=1,\cdots, h_{11}$. It is sometimes useful to extend this action
to the intersection numbers $C_{IJK}$ so that, {\it e.g.}, the constraint
(\ref{constraint}) can be be reorganized as
\begin{equation}
{\cal V}^2 = {1\over 3!}C^{IJK} X_I X_J X_K~,
\label{constrainttwo}
\end{equation}
where all indices were either raised or lowered.
The effective action in five dimensions
(\ref{fivedact}) was written in terms of the fields $X^I$ which include
some redundancy because the constraint (\ref{constraint}) should be imposed on them.
An alternative form of the scalar term which employs only the unconstrained scalars
$\phi^i$ is
\begin{equation}
{\cal L}_{\rm scalar} = -{1\over 2\kappa_5^2} g_{ij} d\phi^i \wedge {}^* d\phi^j~,
\end{equation}
where the metric on moduli space is
\begin{equation}
g_{ij}
= G_{IJ} \partial_i X^I \partial_j X^J~.
\label{gijdef}
\end{equation}
Here derivatives with respect to the unconstrained fields are
\begin{equation}
\partial_i X^I= {\partial X^I\over\partial\phi^i} ~.
\end{equation}
So far we have just discussed the bosonic part of the supergravity action. We will not need
the explicit form of the terms that contain fermions. However, it is important that the
full Lagrangean is invariant under the supersymmetry variations
\begin{eqnarray}
\label{susyone}
\delta \psi_\mu &=& \left[ D_\mu (\omega) + {i\over 24}
X_I (\Gamma_\mu^{~\nu\rho} - 4\delta^\nu_\mu \Gamma^\rho)F_{\nu\rho}^I \right] \epsilon~, \\
\label{susytwo}
\delta\lambda_i &=& -{1\over 2} G_{IJ} \partial_i X^I \left[ {1\over 2}
\Gamma^{\mu\nu} F_{\mu\nu}^J + i \Gamma^\mu \partial_\mu X^J \right] \epsilon~,
\end{eqnarray}
of the gravitino $\psi_\mu$ and the
gauginos $\lambda_i$, $i=1,\ldots, n_V$.
Here $\epsilon$ denotes the infinitesimal supersymmetry parameter. $D_\mu(\omega)$ is
the covariant derivative formed from the connection $\omega$ and acting on the spinor $\epsilon$.
The usual $\Gamma$-matrices in five dimensions are denoted
$\Gamma^\mu$; and their multi-index versions $\Gamma^{\mu\nu}$ and $\Gamma^{\mu\nu\rho}$
are fully anti-symmetrized products of those.
\subsection{A First Look at the Attractor Mechanism}
\label{attractor}
We have now introduced the ingredients we need for a first look at the attractor mechanism.
For now we will consider the case of supersymmetric black holes. As the terminology
indicates, such black holes preserve at least some of the supersymmetries. This means
$\delta \psi_\mu = \delta\lambda_i=0$ for some components of the supersymmetry
parameter $\epsilon$. A great deal can be learnt from these conditions by analyzing the
explicit formulae (\ref{susyone}-\ref{susytwo}).
In order to make the conditions more explicit we will make some simplifying assumptions.
First of all, we will consider only stationary solutions in these lectures.
This means we assume that
the configuration allows a time-like Killing vector. The corresponding coordinate will be
denoted $t$. All the fields are independent of this coordinate. The
supersymmetry parameter $\epsilon$ satisfies
\begin{equation}
\label{susyproj}
\Gamma^{\hat t} \epsilon = -i\epsilon~,
\end{equation}
where hatted coordinates refer to a local orthonormal basis. In order to keep
the discussion as simple and transparent as possible we will for now also
assume radial symmetry. This last assumption is very strong and will be relaxed
in the following lecture. At any rate, under these assumptions
the gaugino variation (\ref{susytwo}) reads
\begin{equation}
\label{az}
\delta\lambda_i = {i\over 2}G_{IJ} \partial_i X^I ( F^J_{m{\hat t}} - \partial_m X^J )
\Gamma^m \epsilon =0~,
\end{equation}
where $m$ is the spatial index. We exploited that, due to radial symmetry, only
the electric components $F^I_{m{\hat t}}$ of the field strength can be nonvanishing.
We next assume that the solution preserve $N=1$ supersymmetry so that (\ref{susyproj})
are the only projections imposed on the spinor $\epsilon$. Then $\Gamma^m\epsilon$
will be nonvanishing for all $m$ and the solutions to (\ref{az}) must satisfy
\begin{equation}
\label{aza}
G_{IJ} \partial_i X^I (F_{m{\hat t}}^J - \partial_m X^J ) =0~.
\end{equation}
This is a linear equation that depends on the bosonic fields alone. It
essentially states that the gradient of the scalar field (of type $J$)
is identified with the electric field (of the same type). This identification is at
the core of the attractor mechanism. Later we will take more carefully
into account the presence of the overall projection operator
$G_{IJ} \partial_i X^I$ in (\ref{aza}). This operator takes into account that fact that
no scalar field is a superpartner of the graviphoton. This restriction arises here because
the index $i=1,\ldots, n_V$ enumerating the gauginos is one short of the vector
field index $I=1,\ldots, n_V+1$.
The conditions (\ref{aza}) give rise to an important monotonicity property that
controls the attractor flow. To see this, multiply by $\partial_r\phi^i$ and sum over $i$.
After reorganization we find
\begin{equation}
\label{monot}
G_{IJ} \partial_r X^I F_{r{\hat t}}^J = G_{IJ} \partial_r X^I \partial_r X^J \geq 0~.
\end{equation}
The quantity on the right hand side of the equation is manifestly positively definite.
In order to simplify the left hand of the equation we need to analyze Gauss' law for
the flux. For spherically symmetric configurations the Chern-Simons terms in the
action (\ref{fivedact}) do not contribute so the Maxwell equation is just
\begin{equation}
d(G_{IJ} {}^* F^J ) =0~.
\end{equation}
Using the explicit form of the metric for a radially symmetric
extremal black hole in five dimensions
\begin{equation}
\label{lineelm}
ds^2 = - f^{2} dt^2 + f^{-1} (dr^2 + r^2 d\Omega_3^2)~,
\end{equation}
the component form of the corresponding Gauss' law reads
\begin{equation}
\partial_r (G_{IJ} r^3 f^{-1} F^J_{r{\hat t}}) =0~.
\end{equation}
This can be integrated to give the explicit solution
\begin{equation}
\label{gauol}
G_{IJ} F^J_{r{\hat t}} = f\cdot{1\over r^3}\cdot {\rm const} \equiv f \cdot {Q_I \over r^3}~,
\end{equation}
for the radial dependence of the electric field. Inserting this in (\ref{monot}) we find the
flow equation
\begin{equation}
\label{floweq}
\partial_r ( X^I Q_I ) = f^{-1} r^3 G_{IJ} \partial_r X^I \partial_r X^J \geq 0~.
\end{equation}
We can summarize this important result as the statement that the central charge
\begin{equation}
\label{zedef}
Z_e \equiv X^I Q_I~,
\end{equation}
depends monotonically on the radial coordinate $r$. It starts as a
maximum in the asymptotically flat space and decreases as the black
hole is approached. This is the attractor flow.
In order to analyze the behavior of (\ref{floweq}) close to the horizon it
is useful to write it as
\begin{equation}
\label{floweqn}
r\partial_r Z_e = f^{-2} r^4 ~\epsilon \geq 0~,
\end{equation}
where the energy density in the scalar field is
\begin{equation}
\epsilon = g^{rr} G_{IJ} \partial_r X^I \partial_r X^J~.
\label{enerden}
\end{equation}
According to the line element (\ref{lineelm}) an event horizon at $r=0$ is characterized
by the asymptotic behavior $f\sim r^2$. Therefore the measure factor
$f^{-2} r^4$ is finite there. Importantly, when $f\sim r^2$ the proper distance to
the horizon diverges as $\int_0 dr/r$. Since the horizon area is finite this means the proper
volume of the near horizon region diverges. This is a key property of extremal black holes.
In the present discussion the important consequence is that the energy density of the scalars
in the near horizon region must vanish, or else they would have infinite energy, and so
deform the geometry uncontrollably. We conclude that the right hand side of (\ref{floweqn})
vanishes at the horizon, {\it i.e.} the inequality is saturated there.
We therefore find the extremization condition
\begin{equation}
r\partial_r Z_e=0~,~~({\rm at~horizon})~.
\end{equation}
This is the spacetime form of the attractor formula.
There is another form of the attractor formula that is cast entirely in terms of the
moduli space. To derive it, we begin again from (\ref{aza}), simplify using Gauss'
law (\ref{gauol}), and introduce the central charge (\ref{zedef}). We can write
the result as
\begin{equation}
\partial_i Z_e = \sqrt{g_\perp} g_{ij} \partial_n \phi^j~,
\label{modext}
\end{equation}
where $\sqrt{g_\perp}= f^{-3/2} r^3$ is the area element,
$g_{ij}$ is the metric on moduli space introduced in (\ref{gijdef}),
$\partial_n = \sqrt{g^{rr}}\partial_r$ is the proper normal derivative, and the $\phi^j$ are the
unconstrained moduli. As discussed in the previous paragraph, the energy density
(\ref{enerden}) must vanish at the horizon for extremal black holes.
This means the contribution from each of the unconstrained moduli must vanish by itself,
and so the right hand side of (\ref{modext}) must vanish for all values of the index $i$.
We can therefore write the attractor formula as an extremization principle over
moduli space
\begin{equation}
\partial_i Z_e=0~,~~({\rm at~horizon})~.
\label{modattr}
\end{equation}
This form of the attractor formula determines the values $X^I_{\rm ext}$ of the scalar
fields at the horizon in terms of the charges $Q_I$.
We can solve ({\ref{modattr}) explicitly. In order to take the constraint (\ref{constraint})
on the scalars properly into account it is useful to rewrite the extremization principle
as
\begin{equation}
D_I Z_e=0~,~~({\rm at~horizon})~,
\label{modatt}
\end{equation}
where the covariant derivative is defined as
\begin{equation}
D_I Z_e = \left(\partial_I-{1\over 3}(\partial_I\ln {\cal V})\right)Z_e
= (\partial_I - {1\over 3{\cal V}}X_I)Z_e = Q_I - {1\over 3{\cal V}}X_I Z_e~.
\label{covdef}
\end{equation}
We see that $Q_I\propto X_I$ at the attractor point, with the constant of proportionality determined
by the constraint (\ref{constrainttwo}) on the scalar. We thus find the explicit result
\begin{equation}
{X_{I}^{\rm ext}\over {\cal V}^{2/3}} = {Q_I \over \left( {1\over 3!} C^{JKL} Q_J Q_K Q_L\right)^{1/3} } ~,
\label{ellattra}
\end{equation}
for the attractor values of the scalar fields in terms of the charges.
As a side product we found
\begin{equation}
{Z_e^{\rm ext}\over {\cal V}^{1/3}} = 3 \left( {1\over 3!}C^{JKL} Q_J Q_K Q_L\right)^{1/3}~,
\label{expze}
\end{equation}
for the central charge at the extremum.
\subsection{A Closer Look at the Attractor Mechanism}
\label{moredetails}
Before considering examples, we follow up on some of the important features
of the attractor mechanism that we skipped in the preceding subsection: we
introduce the black hole entropy, we discuss the interpretation of
the central charge, and we present some details on the units.
\subsubsection{Black Hole Entropy}
Having determined the scalars $X^I$ in terms of the charges we can now
express the central charge (\ref{zedef}) in terms of charges alone. It turns
out that for spherically symmetric black holes the resulting expression is
in fact related to the entropy through the simple formula
\begin{equation}
\label{bhent}
S = 2\pi\cdot{\pi\over 4G_5}\cdot\left( {1\over 3{\cal V}^{1/3}} Z_e^{\rm ext}\right)^{3/2}~.
\end{equation}
The simplest way to establish this relation is to inspect a few explicit black hole
solutions and then take advantage of near horizon symmetries to extend the
result to large orbits of black holes that are known only implicitly. The
significance of the formula (\ref{bhent}) is that it allows the determination of
the black hole entropy without actually constructing the black hole geometry.
In view of the explicit expression (\ref{expze}) for the central charge at the extremum
we find the explicit formula
\begin{equation}
\label{bhentexp}
S = 2\pi\cdot{\pi\over 4G_5}\cdot \sqrt{ {1\over 3!}C^{JKL} Q_J Q_K Q_L }~,
\end{equation}
for the black hole entropy of a spherically symmetric, supersymmetric black hole in five dimensions.
\subsubsection{Interpretation of the Central Charge}
In the preceding subsection
we introduced the central charge (\ref{zedef}) rather formally, as the
linear combination of charges that satisfies a monotonic flow. This characterization can be
supplemented with a nice physical interpretation as follows. The eleven-dimensional
origin of the gauge potential $A^I_t$ can be determined from the decomposition (\ref{gaud}).
It is a three-form with one index in the temporal direction and the other two within the
Calabi-Yau, directed along a $(1,1)$-cycle of type $I$. Such a three-form is sourced by
$M2$-branes wrapped on the corresponding $(1,1)$-cycle which we have denoted
$\Omega^I$. The volume of this cycle is precisely $X^I$, according to (\ref{XIdef}). Putting
these facts together it is seen that the central charge (\ref{zedef}) is the total volume of the
wrapped cycles, with multiple wrappings encoded in the charge $Q_I$. We can interpret
the underlying microscopics as a single $M2$-brane wrapping some complicated cycle
$\Omega$ within the Calabi-Yau which can be characterized in terms of a decomposition
\begin{equation}
\Omega = Q_I \Omega^I~,
\end{equation}
on the canonical cycles $\Omega^I$. Then the central charge is identified with the mass
of this $M2$-brane, up to an overall factor of the tension.
There is yet another interpretation of the central charge which takes as
starting point the $N=2$ supersymmetry algebra
\begin{equation}
\label{susyalge}
\{ Q^A_\alpha , Q^B_\beta \} = 2 \left( \delta^{AB} P_\mu (\Gamma^\mu )_{\alpha\beta}
+ \delta_{\alpha\beta} \epsilon^{AB} Z_e \right)~,
\end{equation}
where $A,B=1,2$ distinguish the two supercharges.
The last term on the right hand side (proportional to $Z_e$) is the
central term. It is introduced from a purely algebraic point of view,
as a term that commutes with all other generators of the algebra.
The algebra is most usefully analyzed in the restframe where
$P_\mu (\Gamma^\mu )_{\alpha\beta}=P_0 (\Gamma^0)_{\alpha\beta}$.
Consider a state that is annihilated by one or more of the supercharges $Q^A_\alpha$.
Taking expectation value on both sides with respect to this states,
and demanding positive norm of the state, we find
the famous BPS inequality
\begin{equation}
M = |P_0 | \geq Z_e~,
\end{equation}
with the inequality saturated exactly when supersymmetry is preserved by the state.
Supersymmetric black holes are BPS states and so their mass should
agree with the algebraic central charge. In the preceding paragraph
we showed that the mass agrees with the central charge introduced geometrically,
so the alternate introductions of the central charge agree.
\subsubsection{Some Comments on Units and Normalizations}
Let us conclude this subsection with a few comments on units. It is standard to
introduce the eleven dimensional Planck length through
$\kappa_{11}^2 = (2\pi)^7 l_P^9$. In this notation the five dimensional Newton's constant
is
\begin{equation}
G_5 = {\pi\over 4}\cdot {(2\pi l_P)^6\over {\cal V}}\cdot l^3_P~,
\end{equation}
and the
$M2$-brane tension is $\tau_{M2} = {1\over (2\pi)^2 \ell_P^3}$.
The relation to standard string theory units are $l_P = g^{1/3}_s\sqrt{\alpha^\prime}$
and the radius of the M-theory circle is $R_{11}=g_s\sqrt{\alpha^\prime}$. Now,
the physical charges $Q_I$ were introduced in (\ref{gauol}) as the constant of
integration from Gauss' law, following standard practice in supergravity. Such
physical charges are proportional to quantized charges $n_I$ according to
\begin{equation}
\label{chargeunit}
Q_I = ( {{\cal V}\over(2\pi l_P)^6})^{-2/3} \cdot l^2_P \cdot
n_I = \left( {\pi\over 4G_5}\right)^{-2/3} n_I~.
\end{equation}
The mass of the brane configuration is
\begin{equation}
\label{massunit}
M = \tau_{M2} X^I n_I =
{1\over l_P^3} \cdot {{\cal V}\over (2\pi l_P)^6} \cdot {X^I\over {\cal V}^{1/3}}\cdot Q_I
= {\pi\over 4G_5}\cdot {X^I\over {\cal V}^{1/3}}\cdot Q_I~.
\end{equation}
The formulae (\ref{chargeunit}-\ref{massunit}) are the precise versions of the informal
notions that the charge $Q_I$ counts the number of branes and that the central
charge $Z_e$ agrees with the mass. We see that there are awkward constants
of proportionality, which vanish in units where $G_5 = {\pi\over 4}$ and the volumes of
two-cycles are measured relative to ${\cal V}^{1/3}$. In this first lecture we will
for the most part go through
the trouble of keeping all units around, to make sure that it is clear where the various
factors go. In later lectures we will revert to the simplified units.\footnote{
In fact, the supersymmetry algebra (\ref{susyalge}) was
already simplified this way, to avoid overly heavy notation.}
If needed, one can restore units by referring back to the simpler special cases.
\subsection{An Explicit Example}
\label{example}
We conclude this introductory lecture by working out a simple example explicitly.
The example we consider is when the Calabi-Yau space is just a torus $CY=T^6$.
Strictly speaking a torus is not actually a Calabi-Yau space if by the latter we mean a
space with exactly $SU(3)$ holonomy. The issue is that for M-theory on $T^6$
the effective five-dimensional theory has $N=8$ supersymmetry rather than
$N=2$ supersymmetry as we have assumed. This means there are extra
gravitino multiplets in the theory which we have not taken into account. However,
these gravitino multiplets decouple from the black hole background and so it is consistent
to ignore them, in much the same way that we already ignore the $N=2$ hypermultiplets.
We can therefore use the formalism reviewed above without any change.
In the explicit example we will
further assume that the metric on the torus is diagonal so that the K\"{a}hler form
takes the product form
\begin{equation}
\label{tormet}
J = i \left(
X^1 dz^1\wedge d{\bar z}^1 + X^2 dz^2 \wedge d{\bar z}^2+ X^1 dz^3\wedge d{\bar z}^3 \right)~.
\end{equation}
Then the scalar fields $X^I$ with $I=1,2,3$ are just the volumes of
each $T^2$ in the decomposition $T^6 = (T^2)^3$. The only nonvanishing intersection numbers
of these two-cycles are $C_{123}=1$ (and cyclic permutations).
The constraint (\ref{constraint}) on the scalars therefore takes the simple form
\begin{equation}
\label{torusc}
X^1 X^2 X^3 ={\cal V}~.
\end{equation}
The volumes (\ref{fourc})
of four-cycles on the torus are
\begin{equation}
X_1 = X^2 X^3 = {\cal V}/X^1~~~({\rm and~cyclic~permutations})~.
\label{invvol}
\end{equation}
\subsubsection{Attractor Behavior}
The central charge (\ref{zedef}) for this example is
\begin{equation}
\label{av}
Z_e = X^1 Q_1 + X^2 Q_2 + X^3 Q_3~.
\end{equation}
According to the extremization principle we can determine the scalar fields at the horizon
by minimizing this expression over moduli space. The constraint (\ref{torusc})
can be implemented by solving in terms of one of the $X^I$'s and then
extremizing (\ref{av}) over the two remaining moduli. Alternatively, one can employ
Lagrange multipliers. Either way the result for the scalars in terms of the charges is
\begin{equation}
\label{attrpt}
{X_1^{\rm ext}\over {\cal V}^{1/3}} = \left({Q_1^2\over Q_2 Q_3}\right)^{1/3}
= { Q_1 \over \left(Q_1 Q_2 Q_3\right)^{1/3} }~~~~~({\rm and ~cyclic ~permutations})~.
\end{equation}
These are the horizon values for the scalars predicted by the attractor mechanism. They
agree with the general formula (\ref{ellattra}). Below
we confirm these values in the explicit solutions.
At the attractor point (\ref{attrpt}) the three terms in the central charge (\ref{av}) are
identical. The central charge takes the value
\begin{equation}
Z_{\rm ext} = 3(Q_1 Q_2 Q_3)^{1/3}~.
\end{equation}
The black hole entropy (\ref{bhent}) becomes
\begin{equation}
\label{bhex}
S = 2\pi\cdot {\pi\over 4G_5}\cdot (Q_1 Q_2 Q_3 )^{1/2} =
2\pi (n_1 n_2 n_3 )^{1/2}~.
\end{equation}
This is the entropy computed using the attractor formalism, {\it i.e.} without
explicit construction of the black hole geometry. At the risk of seeming heavy
handed, we wrote (\ref{bhex}) both in terms of the proper (dimensionful) charges
$Q_I$ and also in terms of the quantized charges $n_I$.
The entropy formula (\ref{bhex}) is rather famous so let us comment a little
more on the relation to other work. The $M2$-brane black hole considered here
can be identified, after duality to type IIB theory, with the $D1-D5$ black hole
considered by Strominger and Vafa \cite{Strominger:1996sh}.
In this duality frame two of the M2-brane charges become the background D-branes
and the third charge is the momentum $p$ along the $D1$-brane. Then (\ref{bhex})
coincides with Cardy's formula
\begin{equation}
S = 2\pi \sqrt{ch\over 6}~,
\end{equation}
where the central charge $c=6N_1 N_5$ for the CFT on the D-branes and $h=p$ for
the energy of the excitations. In the present lectures we are primarily interested in
macroscopic features of black holes and no further details on the microscopic
theory will be needed. For more review on this
consult {\it e.g.} \cite{David:2002wn}.
\subsubsection{Explicit Construction of the Black Holes}
We can compare the results
from the attractor computation with an explicit construction of the black hole.
The standard form of the $M2$-brane solution in eleven-dimensional supergravity
is
\begin{equation}
\label{M2sol}
ds^2_{11} = H^{-2/3} dx^2_\parallel
+ H^{1/3} dx^2_\perp~.
\end{equation}
Here the space parallel to the $M2$-brane is
\begin{equation}
dx^2_\parallel= -dt^2 + dx^2_1 + dx^2_2~,
\end{equation}
when the spatial directions of the $M2$-brane have coordinates $x_1$ and $x_2$.
The transverse space $dx^2_\perp$
is written similarly in terms of the remaining eight coordinates. The function $H$ can be
any harmonic on the transverse space; the specific one needed in our example is given
below.
The harmonic function rule states that composite solutions can be formed by
superimposing three $M2$-brane solutions of the form (\ref{M2sol}) with cyclically
permuted choices of parallel space.
The only caveat is that we must smear along all directions within the torus, {\it i.e.} the
harmonic functions can depend only on the directions transverse to all the different
branes. This procedure gives the standard intersecting $M2$-brane solution
\begin{equation}
\label{intsol}
ds^2_{11} = - f^2 dt^2 + f^{-1} (dr^2 + r^2 d\Omega^2_3)
+ \left[ \left({H_2 H_3 \over H^2_1}\right)^{1/3} (dx^2_1 + dx^2_2) + {\rm cyclic} \right]~,
\end{equation}
where
\begin{equation}
\label{fdef}
f = (H_1 H_2 H_3 )^{-1/3}~.
\end{equation}
We introduced radial coordinates in the four spatial dimensions transverse to all the branes.
The harmonic functions are
\begin{equation}
\label{harmfct}
H_I = X_{I\infty} + {Q_I\over r^2}~~~~;~~I=1,2,3~.
\end{equation}
Comparing the intersecting brane solution (\ref{intsol}) with the torus metric (\ref{tormet})
we determine the scalar fields as
\begin{equation}
\label{XIex}
{X^1\over {\cal V}^{1/3}} = \left({H_2 H_3 \over H^2_1}\right)^{1/3} ~~~~~({\rm and ~cyclic ~permutations})~.
\end{equation}
The only remaining matter fields from the five-dimensional point of view are the gauge
fields
\begin{equation}
\label{AIdef}
A^I = \partial_r H^{-1}_I dt~~~~~;~I=1,2,3~.
\end{equation}
The scalar fields $X^I$ (\ref{XIex}) depend in a non-trivial way on the radial coordinate
$r$. One can verify that the dependence is such that $Z_e=X^I Q_I$ is a monotonic
function of the radii, but we will focus on the limiting values.
The constants $X_{I\infty}$ in the harmonic functions (\ref{harmfct}) were introduced
in order to obtain the correct limit as $r\to\infty$
\begin{equation}
X^1 \to \left({X_{2\infty} X_{3\infty}\over (X_{1\infty})^2}\right)^{1/3} {\cal V}^{1/3}
= X^1_\infty~~~~~({\rm and ~cyclic ~permutations})~.
\end{equation}
We used the constraint (\ref{torusc}) in the asymptotic space and the relation (\ref{invvol})
for the volumes of four-cycles.
As the horizon ($r=0$) is approached the moduli simplify to
\begin{equation}
{X^1\over {\cal V}^{1/3}} \to { X^1_{\rm hor} \over {\cal V}^{1/3}}=
\left({Q_2 Q_3\over Q_1^2}\right)^{1/3} ~~~~~
({\rm and ~cyclic ~permutations})~.
\end{equation}
In view of (\ref{invvol}) this agrees with the values (\ref{attrpt}) predicted by the attractor
mechanism.
We can also compute the black hole entropy directly from the geometry
(\ref{intsol}). The horizon at $r=0$
corresponds to a three-sphere with finite radius $R= (Q_1 Q_2 Q_3)^{1/6}$.
Since $V_{S^3} = 2\pi^2$ for a unit three-sphere
this gives the black hole entropy
\begin{equation}
S = {A\over 4G_5} = {1\over 4G_5} \cdot 2\pi^2\cdot R^3 = 2\pi (n_1 n_2 n_3 )^{1/2}~.
\end{equation}
This explicit result for the black hole entropy is in agreement with (\ref{bhex}) computed
from the attractor mechanism.
\section{Black Ring Attractors}
\label{sec:2}
In this lecture we generalize the discussion of the attractor mechanism to a
much larger class of stationary supersymmetric black solutions to the
$N=2$ theory in five dimensions introduced in section (\ref{effthy}). By giving up spherical
symmetry and allowing for dipole charges we can discuss multi-center black holes,
rotating black holes and, especially, black rings.
\subsection{General Supersymmetric Solutions}
The most general supersymmetric metric with a time-like Killing vector is
\begin{equation}
\label{metric}
ds^2 = -f^2 (dt+\omega)^2 + f^{-1} ds^2_4 ~,
\end{equation}
where
\begin{equation}
\label{basemet}
ds^2_4 = h_{mn} dx^m dx^n~,
\end{equation}
is the metric of a four-dimensional base space and $\omega$ is a one-form on that
base space. In the simplest examples the base is just flat space, but generally it
can be any hyper-K\"{a}hler manifold in four dimensions. The matter fields needed
to support the solution are the field strengths $F^I=dA^I$ given by
\begin{equation}
\label{FIgen}
F^I = d( fX^I (dt+\omega)) + \Theta^I~,
\end{equation}
and the scalar fields $X^I$ satisfying the sourced harmonic equation
\begin{equation}
\label{XIgen}
{}^{(4)}\nabla^2(f^{-1} X_I) = {1\over 4} C_{IJK} \Theta^J \cdot \Theta^K~,
\end{equation}
on the base space. In these equations $\Theta^I$ is a closed self-dual two-form
$\Theta^I ={}^{*_4}\Theta^I $ on the base. This two-form vanishes in the
most familiar solutions but in general it must be turned on. For example,
it plays a central role for black rings. The inner product between
two-forms is defined as the contraction
\begin{equation}
\label{contdef}
\alpha\cdot\beta = {1\over 2}\alpha_{mn} \beta^{mn}~.
\end{equation}
The self-dual part of the one-form $\omega$ introduced in the metric (\ref{metric})
is sourced by $\Theta^I$ according to
\begin{equation}
\label{omegadef}
d\omega + {}^{*_4} d\omega = -f^{-1} X_I \Theta^I~.
\end{equation}
The general solution specified by equations (\ref{metric}-\ref{omegadef}) is a bit
impenetrable at first sight but things will become clearer as we study these
equations. At this point we just remark that the form of the solution given above
has reduced the full set of Einstein's equation and matter equations to a
series of equations that are linear, if solved in the right order: first specify
the hyper-K\"{a}hler base (\ref{basemet}) and choose a self-dual two
form $\Theta^I$ on that base. Then solve (\ref{XIgen}) for $f^{-1}X_I$.
Determine the conformal factor $f$ of the metric from the constraint (\ref{constraint})
and compute $\omega$ by solving
(\ref{omegadef}). Finally the field strength is given in (\ref{FIgen}) \footnote{We
need $X^I$ which can be determined from (\ref{fourc}). On a general Calabi-Yau
this is a nonlinear equation, albeit an algebraic one.}
\subsection{The Attractor Mechanism Revisited}
\label{attrmechrev}
We next want to generalize the discussion of the attractor mechanism from the
spherical case considered in section (\ref{attractor}) to the more general solutions
described above. Thus we consider the gaugino variations
\begin{eqnarray}
\delta\lambda_i &=& {i\over 2} G_{IJ} \partial_i X^I \left[
{i\over 2} F^J_{\mu\nu} \Gamma^{\mu\nu} - \partial_\mu X^J \Gamma^\mu \right] \epsilon~, \\
&=& {i\over 2} G_{IJ} \partial_i X^I \left[
F^J_{m{\hat t}} \Gamma^m +{i\over 2} F^J_{mn}\Gamma^{mn}
-\partial_m X^J \Gamma^m \right]\epsilon~.
\label{magsusy}
\end{eqnarray}
In the second equation we imposed the supersymmetry projection (\ref{susyproj})
on the spinor $\epsilon$. In contrast to the spherically symmetric case
(\ref{az}) there are in general both electric $E^I_m \equiv F^I_{m{\hat t}}$
and magnetic $B^I_{mn} \equiv F^I_{mn}$ components of the field strength.
However, as we explain below, it turns out that the magnetic field in fact
does not contribute to (\ref{magsusy}). Therefore we have
\begin{equation}
{i\over 2} G_{IJ} \partial_i X^I \left[
E^J_{m}-\partial_m X^J \right]\Gamma^m \epsilon =0~.
\end{equation}
Since this is valid for all components of $\epsilon$ we find
\begin{equation}
\label{susyprr}
G_{IJ} \partial_i X^I \left[
E^J_{m}
-\partial_m X^J \right] =0~,
\end{equation}
just like (\ref{aza}) for the spherical symmetric case. In particular, we see that the
gradient of the scalar field is related to the electric field quite generally.
Of course this can be seen already from the explicit form (\ref{FIgen})
of the field strength, which can be written in components as
\begin{eqnarray}
E^I_m & \equiv& F^I_{m{\hat t}} = f^{-1} \partial_m (fX^I) ~,
\label{elFI} \\
B^I_{mn} & \equiv& F^I_{mn} = fX^I (d\omega)_{mn} + \Theta^I_{mn}~.
\label{magnFI}
\end{eqnarray}
The point here is that we see how the relation (\ref{elFI}) between the electric field
and the gradient of scalars captures an important part of the attractor mechanism
even when spherical symmetry is given up.
The key ingredient in reaching this result was the claim that the magnetic
part (\ref{magnFI}) does not contribute to the supersymmetry variation (\ref{magsusy}).
It is worth explaining in more detail how this comes about. The first term in
(\ref{magnFI}) is of the form $F^I_{mn} \propto X^I (d\omega)_{mn}$. This term cancels
from (\ref{magsusy}) because
\begin{equation}
G_{IJ} \partial_i X^I X^J =0~,
\end{equation}
due to special geometry. Let us prove this.
Lowering the index using the metric (\ref{gijlower}) we can use (\ref{fourc}) to find
\begin{equation}
X_I \partial_i X^I = {1\over 2} C_{IJK} X^J X^K \partial_i X^I
= {1\over 3!} \partial_i ( C_{IJK} X^I X^J X^K ) =0~.
\end{equation}
due to the constraint (\ref{constraint}) on the scalars $X^I$. This is
what we wanted to show.
We still need to consider the second term in (\ref{magnFI}), the one taking the form
$F^I_{mn} \propto \Theta^I_{mn}$. This term cancels from the supersymmetry
variation (\ref{magsusy}) because the supersymmetry projection (\ref{susyproj})
combines with self-duality of $\Theta^I_{mn}$ to give
\begin{equation}
\label{suprojj}
\Theta^I_{mn} \Gamma^{mn} \epsilon =0~.
\end{equation}
In order to verify this recall that the $SO(4,1)$ spinor representation can be
constructed from the more familiar $SO(3,1)$ spinor representation by
including Lorentz generators from using the chiral matrix
$\Gamma^4\equiv \gamma^5 = -i\gamma^0 \gamma^1 \gamma^2 \gamma^3$.
All spinors that survive the supersymmetry projection (\ref{susyproj})
therefore satisfy $\Gamma^{1234}\epsilon =\epsilon$ by construction and
this means the $\Theta_{12}$ term in (\ref{suprojj}) cancels the $\Theta_{34}$ term,
etc.
After this somewhat lengthy and technical aside we return to analyzing the
conditions (\ref{susyprr}). Following the experience from the spherically
symmetric case we would like to trade the electric field for the charges,
by using Gauss' law. The Lagrangean (\ref{fivedact}) gives the Maxwell equation
\begin{equation}
d( G_{IJ} {}^* F^J) = {1\over 2}C_{IJK} F^J \wedge F^K~,
\end{equation}
with the source on the right hand side arising from the Chern-Simons term.
Considering the coefficient of the purely spatial four-form we find
Gauss' law
\begin{equation}
\label{gauslaw}
\nabla^m (f^{-1} E_{mI} ) = -{1\over 8} C_{IJK} \Theta^J \cdot \Theta^K~.
\end{equation}
In arriving at this result we must take into account off-diagonal terms in the
metric (\ref{metric}) due to the shift by $\omega$ of the usual time element $dt$.
These contributions cancel with the terms coming from the first term in the
field strength (\ref{magnFI}). Effectively this means only the term of
the form $F^I_{mn} \sim \Theta^I_{mn}$ remains and it is those terms that give rise
to the inhomogenous terms in (\ref{gauslaw}). The physical interpretation is that the
electric field is sourced by a distributed magnetic field which we may
interpret as a delocalized charge density.
We are now ready to derive the generalized flow equation.
Multiplying (\ref{susyprr}) by $\partial_n \phi^i$ and contract with the
base metric $h^{mn}$ we find
\begin{equation}
\partial^m X^I E_{mI} = G_{IJ} \partial^m X^I \partial_m X^J~,
\end{equation}
which can be reorganized as
\begin{equation}
\nabla^m(X^I f^{-1} E_{mI}) - X^I\nabla^m(f^{-1} E_{mI}) = f^{-1} G_{IJ}
\partial^m X^I \partial_m X^J~,
\end{equation}
and then Gauss' law (\ref{gauslaw}) gives
\begin{equation}
\nabla^m (X^I f^{-1} E_{mI}) = f^{-1} G_{IJ} \partial^m X^I \partial_m X^J
- {X^I\over 8} C_{IJK} \Theta^J \cdot \Theta^K~.
\label{floeqn}
\end{equation}
This is the generalized flow equation. In the case where $\Theta^I=0$ the right hand side is
positive definite and then the flow equation generalizes the monotonicity
property found in (\ref{floweq}) to many cases without radial symmetry. However, the most
general case has nonvanishing $\Theta^I$ and such general flows are more complicated.
\subsection{Charges}
In order to characterize the more general flows with precision, it is useful to be more
precise about how charges are defined.
Consider some bounded spatial region $V$. It is natural to define the electric charge
in the region by integrating the electric flux through the boundary $\partial V$ as
\begin{equation}
\label{QIdef}
Q_I (V) = {1\over 2\pi^2} \int_{\partial V} dS f^{-1} n^m E_{mI}~,
\end{equation}
where $n^m$ is an outward pointing normal on the boundary. If we consider two
nested regions $V_2\subset V_1$ we have
\begin{equation} Q_I(V_1) - Q_I(V_2) = -{1\over 16\pi^2} \int d^4 x\sqrt{h}
C_{IJK} \Theta^J \cdot \Theta^K~,
\end{equation}
where the second step used Gauss' law (\ref{gauslaw}). This means the charge is
monotonically decreasing as we move to larger volumes. The reason that it does not
have to be constant is that in general the delocalized source on the right hand side
of (\ref{gauslaw}) contributes.
The central charge is constructed from the electric charges by dressing them
with the scalar fields. It was originally introduced
in (\ref{zedef}) but, in analogy with the definition (\ref{QIdef}) of the electric
charge in a volume of space, we may dress the electric field by the scalars as well
and so introduce the central charge in a volume of space as
\begin{equation}
\label{Zdef}
Z_e(V) = {1\over 2\pi^2} \int_{\partial V} dS f^{-1} n^m X^I E_{mI}~.
\end{equation}
Considering again a nested set of regions we can use (\ref{floeqn})
to show that the central charge satisfies
\begin{equation}
Z_e(V_1) - Z_e(V_2) = {1\over 2\pi^2} \int d^4 x \sqrt{h}
\left[ f^{-1} G_{IJ} \nabla^m X^I \nabla_m X^J
- {1\over 8} C_{IJK} X^I \Theta^J \cdot \Theta^K \right]~.
\end{equation}
When $\Theta^I=0$ the central charge is monotonically increasing as we
move outwards. This generalizes the result from the spherically symmetric
case to all cases where the two-forms vanish. When the system is not
spherically symmetric there is no unique ``radius'' but this is circumvented by
the introduction of nested regions, which gives an orderly sense of
moving "outwards". Note that in general we do not force the nested
volumes to preserve topology. In particular there can be multiple singular
points and these then provide natural centers of the successive nesting.
When the two-forms $\Theta^I\neq 0$ the electric central charge (\ref{Zdef}) may
not be monotonic and the flow equation does not provide any strong constraint
on the flow.
In order to interpret the $\Theta^I$'s properly we would like to associate charges
with them as well. Since they are two-forms it is natural to integrate them over
two-spheres and so define
\begin{equation}
\label{qdef}
q^I = -{1\over 2\pi} \int_{S^2} \Theta^I~.
\end{equation}
Since the two-forms $\Theta^I$ are closed the integral is independent under
deformations of the two-cycle and, in particular, it vanishes unless the $S^2$
is non-contractible on the base space. One way such non-trivial cycles can arise
is by considering non-trivial base spaces. For our purposes the main example will
be when the base space is flat, but endowed with singularities along
one or more closed curves (including lines going off to infinity). This situation also
gives rise to noncontractible $S^2$'s because in four Euclidean dimensions a line can be
wrapped by surfaces that are topologically a two-sphere.
The charges $q^I$ defined in (\ref{qdef}) can be usefully thought as a magnetic
charges. In our main example of a flat base space with a closed curve
we may interpret the configuration concretely in terms of electric charge distributed
along the curve. Since the curve is closed there is in general no net electric charge,
but there will be a dipole charge and it is this dipole charge that we identify as the
magnetic charge (\ref{qdef}).
In keeping with the analogy between the electric and magnetic charges we would
also like to introduce a magnetic central charge. The electric central charge (\ref{Zdef})
was obtained by dressing the ordinary charge (\ref{QIdef}) by the moduli.
In analogy, we construct the magnetic central charge
\begin{equation}
\label{Zmdef}
Z_m(V) = -{1\over 2\pi} \int_{S^2} X_I \Theta^I~.
\end{equation}
In some examples this magnetic central charge will play a role analogous to that played
by the electric central charge in the attractor mechanism.
A general configuration can be described in terms of its singularities on the
base space. There may be a number of isolated point-like singularities, to which
we assign electric charges, and there may be a number of closed curves (including lines
going off to infinity), to which we assign magnetic charges.
In four dimensions electric and magnetic charges are very similar: they are related
by electric magnetic duality, which is implemented by symplectic transformations in
the complex special geometry. In five dimensions the situation is more complicated because
the Chern-Simons term makes the symmetry between point-like electric sources and
string-like magnetic sources more subtle. Therefore we will need to treat them independently.
\subsection{Near Horizon Enhancement of Supersymmetry}
\label{susyenh}
There is another aspect of attractor behavior that we have not yet developed:
the attractor leads to enhancement of supersymmetry \cite{Chamseddine:1996pi}.
This is a very strong condition that completely
determines the attractor behavior, even when dipole charges are turned on.
The enhancement of supersymmetry means the {\it entire} supersymmetry
of the theory is preserved near the horizon. To appreciate why that is such a
strong conditions, recall the origin of the attractor flow: we considered the
gaugino variation (\ref{magsusy}) and found the flow by demanding that the
various terms cancel. The enhancement of supersymmetry at the attractor
means each term vanishes by itself.
We first determine the supersymmetry constraint on the gravitino variation (\ref{susyone}).
By considering the commutator of two variations \cite{Chamseddine:1996pi}, it can be
shown that the near horizon geometry must take the form $AdS_p\times S^q$.
In five dimensions there are just two options: $AdS_3\times S^2$ or $AdS_2\times S^3$.
The near horizon geometry of the supersymmetric black hole in five dimension that
we considered in section (\ref{example}) is indeed $AdS_2\times S^3$ \cite{Strominger}
(up to global identifications).
A more stringent test is the attractor behavior of the supersymmetric rotating black hole.
One might have expected that rotation would squeeze the sphere and make it oblate
but this would not be consistent with enhancement of supersymmetry. In fact, it turns
out that, for supersymmeric black holes, the near horizon geometry indeed remains
$AdS_2\times S^3$ \cite{Cvetic:1999ja} (up to global identifications).
There are also examples of a supersymmetric configurations with near horizon
geometry $AdS_3\times S^2$. The simplest example is the black string in five
dimensions. A more general solution is the supersymmetric black ring,
which also has near horizon geometry $AdS_3\times S^2$. Indeed, the extrinsic
curvature of the ring becomes negligible in the very near horizon geometry so
there the black ring reduces to the black string. We will consider these
examples in more detail in the next section.
The pattern that emerges from these examples is that black holes correspond
to point-like singularities on the base and a near horizon geometry
$AdS_2\times S^3$ in the complete space. On the other hand, black strings
and black rings correspond to singularities on a curve in the base and a near
horizon geometry $AdS_3\times S^2$ in the complete space. The
two classes of examples are related by electric-magnetic duality which,
in five dimensions, interchanges one-form potentials with two-form potentials
and so interchanges black holes and black strings. This duality
interchanges $AdS_3\times S^2$ with $AdS_2\times S^3$.
So far we have just considered the constraints from the gravitino variation
(\ref{susyone}). The attractor behavior of the scalars is controlled by the
gaugino variation (\ref{magsusy}) which we repeat for ease of reference
\begin{equation}
\delta\lambda_i =
{i\over 2} G_{IJ} \partial_i X^I \left[
F^J_{m{\hat t}} \Gamma^{m{\hat t}} +{i\over 2} F^J_{mn}\Gamma^{mn}
-\partial_m X^J \Gamma^m \right]\epsilon~.
\label{susythree}
\end{equation}
Near horizon enhancement of supersymmetry demands that each term in
this equation vanishes by itself, since no cancellations are possible when
the spinor $\epsilon$ remains general. Let us consider the three conditions
in turn.
The vanishing of the third term $\partial_m X^J=0$ means $X^J$ is a constant in
the near horizon geometry. The attractor mechanism will determine the value of
that constant as a function of the charges.
The first term in (\ref{susythree}) reads
\begin{equation}
\partial_i X^I E_{Im}=0~,
\end{equation}
in terms of the
electric field introduced in (\ref{elFI}). In the event that there is a point-like singularity in
the base space there is an $S^3$ in the near horizon geometry. Integrating the flux
over this $S^3$ and recalling the definition (\ref{QIdef}) of the electric charge
we then find
\begin{equation}
\partial_ i Z_e =0 ~,
\label{attzea}
\end{equation}
in terms of the electric central charge (\ref{zedef}). This is the attractor formula
(\ref{modattr}), now applicable in the near any point-like singularity in base
space. We can readily determine the explicit attractor behavior as (\ref{ellattra})
near any horizon with $S^3$ topology.
We did not yet consider the condition that the second term in (\ref{susythree})
vanishes. This term was considered in some detail after (\ref{magnFI}). There we found
that the magnetic field $B^I_{mn}=F^I_{mn}$ has a term proportional to $X^I$
which cancels automatically from the supersymmetry conditions, due to special
geometry relations. However, there is also another term in $B^I_{mn}$
which is proportional to $\Theta^I_{mn}$. This term also cancels from the supersymmetry
variation, but only for the components of the supersymmetry generator
$\epsilon$ that satisfy the projection (\ref{susyproj}). However, in the near horizon region
there is enhancement of supersymmetry and so the variation must vanish for all
components $\epsilon$. This can happen if the two-forms $\Theta^I$
take the special form \begin{equation}
\Theta^I= k X^I~,
\label{nearhTheta}
\end{equation}
where $k$ is a constant ($I$-independent) two-form because then
special geometry relations will again guarantee supersymmetry. The special
form (\ref{nearhTheta}) will determine the scalars completely.
Indeed, suppose that sources are distributed along a curve in the base space. Then
we can integrate (\ref{nearhTheta}) along the $S^2$ wrapping the curve. This
gives
\begin{equation}
q^I = -{1\over 2\pi} \int_{S^2} \Theta^I = X^I_{\rm ext} \cdot {\rm constant}~,
\end{equation}
for the dipole charges in the near horizon region. The constant of proportionality is
determined by the constraint (\ref{constraint}) and so we reach the final
result\footnote{In this sections 3 and 4 we use the simplified units where ${\cal V}=1$
and $G_5={\pi\over 4}$. See section \ref{moredetails} for details on units.}
\begin{equation}
X^I_{\rm ext} = {q^I \over \left( {1\over 3!} C_{JKL} q^J q^K q^L\right)^{1/3}}~,
\label{magattr}
\end{equation}
for the scalar field in terms of the dipole charges. The result is applicable
near singularities distributed along a curve in the base space. In particular,
this is the attractor value for the scalars in the near horizon region of black
strings and of black rings.
Our result (\ref{magattr}) was determined directly in the near horizon region, by exploiting
the enhancement of supersymmetry there. In the case where $\Theta^I\neq0$
we cannot understand the entire flow as a gradient flow of the electric central charge
$Z_e$, nor are the attractor values given by extremizing $Z_e$. In fact, we can see
that the attractor values (\ref{magattr}) amount to extremization of the {\it magnetic}
central charge (\ref{Zmdef}). However, the significance of this is not so clear since it is
only the near horizon behavior that is controlled by $Z_m$, not the entire flow. It
would be interesting to find a more complete description of the entire flow in
the most general case. For now we understand the complete flow when $\Theta^I=0$,
and the attractor behavior when $\Theta^I\neq0$.
There is in fact another caveat we have not mentioned so far. Our expression
(\ref{ellattra}) for the scalars at the electric attractor breaks down when
$C^{JKL} Q_J Q_K Q_L=0$, and similarly (\ref{magattr}) for the magnetic
attractor breaks down when $C_{IJK} q^I q^J q^K = 0$. In the electric case
the issue has been much studied: the case where $C^{JKL} Q_J Q_K Q_L=0$
corresponds to black holes with area that vanishes classically. These are the small
black holes. In some cases it is understood how higher derivative corrections to the
action modify the attractor behavior such that the geometry and the attractor values
of the scalars become regular \cite{LopesCardoso:1998wt,smallBHrefs}.
The corresponding magnetic
case $C_{IJK} q^I q^J q^K = 0$ corresponds to small black rings. This case has
been studied less but it is possible that a similar picture applies in that situation.
\subsection{Explicit Examples}
In this subsection we consider a number of explicit geometries. In each example we
first determine the attractor behavior abstractly, by applying the attractor mechanism,
and then check the results by inspecting the geometry.
\subsubsection{The Rotating Supersymmetric Black Hole}
The simplest example of the attractor mechanism is the spherically symmetric black
hole discussed in detail in section (\ref{example}). The generalization of the spherically
symmetric solution to include angular momentum are the rotating supersymmetric black
hole in five-dimensions. This solution is known as the BMPV black
hole \cite{Breckenridge:1996is}.
Let us consider the attractor mechanism first. The rotating black hole is electrically charged
but there are no magnetic charges, so the two-forms $\Theta^I$ vanish in this
case. We showed in section (\ref{attrmechrev}) that then the electric central charge
$Z_e$ must be monotonic just as it was in the nonrotating case. Extremizing over moduli
space we therefore return to the values (\ref{attrpt}) of the scalars found in the
non-rotating case. Alternatively we can go immediately to the general result (\ref{ellattra})
which is written for a general Calabi-Yau three-fold. Either way, we see that the
attractor values of the scalars are independent of the black hole angular momentum.
Since the rotation deforms the black hole geometry, this result is not at all obvious.
The independence of angular momentum is a prediction of the attractor mechanism.
We can verify the result by inspecting the explicit black hole solution.
The metric takes the form (\ref{metric}) where the base space $dx^2_4$ is
just flat space $R^4$
which can be written in spherical coordinates as
\begin{equation}
dx^2_4 = dr^2 + r^2 (d\theta^2 + \cos^2\theta d\psi^2 + \sin^2\theta d\phi^2)~.
\end{equation}
Although the solution is rotating, it is almost identical to the non-rotating example
discussed in section (\ref{example}): the conformal factor $f$ is given again by (\ref{fdef})
where the harmonic functions $H_I$ are given by (\ref{harmfct}). Additionally, the matter
fields remain the scalar fields (\ref{XIex}) and the gauge fields (\ref{AIdef}). The only
effect of adding rotation is that now the one-forms $\omega$ are
\begin{equation}
\omega = -{J\over r^2}(\cos^2\theta d\phi + \sin^2\theta d\psi)~.
\end{equation}
As an aside we note that the self-dual part of $d\omega$ vanishes, as it must for solutions with
$\Theta^I=0$, but the anti-selfdual part is non-trivial: it carries the angular momentum.
Now, for the purpose of the attractor mechanism we are especially interested in the scalar fields.
As just mentioned, these take the form (\ref{XIex}) in terms of the harmonic
functions, independently of the angular momentum. This means they will in fact
approach the attractor values (\ref{attrpt}) at the horizon. In particular,
the result is independent of the angular momentum, as predicted by the
attractor mechanism.
\subsubsection{Multi-center black holes}
From the supergravity point of view, the $M2$-brane solution (\ref{M2sol})
is valid for {\it any} harmonic function $H$ on the transverse space.
Similarly, the intersecting brane solution (\ref{intsol}) (and its generalization to an arbitrary
Calabi-Yau three-fold) remains valid for more general harmonic functions $H_I$.
In particular, the standard harmonic functions (\ref{harmfct}) can be replaced by
\begin{equation}
H_I = X_{I\infty} + \sum_{i=1}^N {Q_I^{(i)}\over |\vec{r} - \vec{r}_i|^2}~.
\label{multiharm}
\end{equation}
where $\vec{r}_i$ are position vectors in the transverse space. We will assume
that all $Q_I^{(i)}>0$ so that the configuration is regular.
The interpretation of these more general solutions is that they correspond
to multi-center black holes, {\it i.e.} $N$ black holes coexisting in equilibrium,
with their gravitational attraction cancelled by repulsion of the charges.
The black hole centered at $\vec{r}_i$ has charges $\{ Q_I^{(i)} \}$.
The attractor behavior of these solutions is the obvious generalization of the
single center black holes. The attractor close to each center depends only
on the charges associated with that center, because the charge integrals (\ref{QIdef})
are defined with respect to singularities on the base manifolds. This immediately
implies that the attractor values for the scalars in a particular attractor region
are (\ref{ellattra}) in terms of the charges $\{ Q_I^{(i)} \}$ associated with this
particular region.
The explicit solutions verify this prediction of the attractor mechanism because
the harmonic functions (\ref{multiharm}) are dominated by the term
corresponding to a single center in the attractor regime corresponding to that center.
In some ways the multi-center solution is thus a rather trivial extension of
the single-center solution. The reason it is
nevertheless an interesting and important example is the following. Far from all
the black holes, the geometry of the multi-center black hole approaches that
of a single center solution with charges $\{ Q_I \} = \{ \sum_{i=1}^N Q_I^{(i)} \}$. Based
on the asymptotic data alone one might have expected an attractor flow governed
by the corresponding central charge $Z_e = X^I Q_I$, leading to the attractor values
for the scalars depending on the $Q_I$ in a unique fashion, independently of the
partition of the geometry into constituent black holes with charges $\{ Q_I^{(i)} \}$.
The multi-center black hole demonstrates that this expectation is false: the
asymptotic behavior does {\it not} uniquely specify the attractor values of the scalars,
and nor does it define the near horizon geometry and the entropy.
More structure appears when one goes beyond the focus on attractor
behavior and consider the full attractor flow of the scalars. As we discussed in section
(\ref{attrmechrev}), the flow of the scalars is a gradient flow
controlled by the electric central charge (this is when the dipoles vanish). The
central charge is interpreted as the total constituent mass. For generic values
of the scalar fields the actual mass of the configuration is smaller, {\it i.e.} the black holes
are genuine bound states. Now, in the course of the attractor flow, the values
of the scalars change. At some intermediate point it may be that the actual
mass of the black hole is identical to that of two (or more) clusters of constituents.
This is the point of marginal stability. There the attractor flow will split up, and continue
as several independent flows, each controlled by the appropriate sets of smaller
charges. This process then continues until the true attractor basins are reached.
The total flow is referred to as the split attractor flow. It has interesting
features which are beyond the scope of the present lecture. We refer the reader
to the original papers \cite{Denef:2000ar} and the review \cite{Moore:2004fg}.
\subsubsection{Supersymmetric Black Strings}
The black string is a five dimensional solution that takes the form
\begin{equation}
ds^2_5 = f^{-1} (-dt^2 +dx^2_4) + f^2(dr^2 + r^2 d\Omega^2_2)~,
\label{strmetric}
\end{equation}
where the conformal factor
\begin{equation}
f = {1\over 3!} C_{IJK} H^I H^J H^K~,
\end{equation}
in terms of the harmonic function
\begin{equation}
H^I = X^I_{\infty} + {q^I\over 2r}~.
\end{equation}
The geometry is supported by the gauge fields
\begin{equation}
A^I = - {1\over 2} q^I (1 + \cos\theta) d\phi~,
\label{strai}
\end{equation}
and the scalar fields
\begin{equation}
X^I = f^{-1} H^I~.
\label{strsoln}
\end{equation}
The black string solution is the long distance representation of an M5-brane that
wraps the four-cycle $q^I \Omega_I$ inside a Calabi-Yau threefold and has the
remaining spatial direction aligned with the coordinate $x^4$. This configuration
plays in important role in microscopic considerations of the four dimensional black hole
(see {\it e.g.} \cite{MSW}).
The gauge field (\ref{strai}) corresponds to the field strength
$F^I = - q^I \sin\theta d\theta d\phi$. This is a magnetic field,
with normalization of the charge in agreement with the one
introduced in (\ref{qdef}). The black string is therefore an example where
the two-forms $\Theta^I\neq 0$.
We should note that the metric (\ref{strmetric}) of the supersymmetric black string
differs from the form
(\ref{metric}), assumed in the analysis in this lecture. The reason
that a different form of the metric applies is that
the black string has a null Killing vector whereas (\ref{metric}) assumes a
time-like Killing vector. Nevertheless, we can think of the null case
as a limiting case of the time-like one. Concretely, if there
is a closed curve on the base-space of (\ref{metric}), the black string is the limit
where the curve is deformed such that two points are taken to infinity and
only a straight line remains ({\it i.e.} the return line is fully at infinity). This limiting
procedure is how the simple black string arises from the more complicated
black ring solution (see following example).
Let us now examine the attractor behavior of the black string. In section (\ref{susyenh})
we showed that near horizon enhancement of supersymmetry demands that, at the
attractor, the two forms simplify to $\Theta^I= k X^I$ where $k$ is a
constant ($I$ independent) two-form. This condition was then showed to imply the
expression (\ref{magattr}) for the scalars as functions of the magnetic charges.
We can verify the attractor behavior by inspection of the explicit solution.
Taking the limit $r\to 0$ on the scalars (\ref{strsoln}) we find
\begin{equation}
X^I_{\rm hor} = {q^I\over ({1\over 3!} C_{JKL} q^J q^K q^L)^{1/3}}~.
\end{equation}
This agrees with (\ref{magattr}) predicted by the attractor mechanism.
\subsubsection{Black Rings}
As the final example we consider the attractor behavior near the supersymmetric
black ring \cite{Elvang:2004ds,Bena:2004de}.
This is a much more involved example which in fact was the motivation for
the development of the formalism considered in this lecture.
The supersymmetric black ring is charged with respect to both electric charges
$Q_I$ and dipole charges $q^I$. Far from the ring the geometry is dominated
by the electric charges, which have the slowest asymptotic fall-off, and the value of the
charges can be determined using Gauss' law (\ref{QIdef}). The dipole charges
are determined according to (\ref{qdef}) where the by $S^2$ is wrapped around
the ring. Since the two-forms do not vanish they dominate the near horizon geometry
and the near horizon values of the scalar fields become (\ref{magattr}), as
they were for the black string.
We can verify the result from the attractor mechanism by inspecting the explicit
black ring solution. The metric takes the general form (\ref{metric}). The conformal
factor $f$ is given by (\ref{fdef}) in terms of functions $H_I$ which take the form:
\begin{equation}
H_I = X_{I\infty} + {Q_I - {1\over 2} C_{IJK} q^J q^K\over\Sigma}
+ {1\over 2} C_{IJK} q^J q^K {r^2\over\Sigma^2}~,
\end{equation}
where
\begin{equation}
\Sigma = \sqrt{ (r^2 - R^2)^2 + 4R^2 r^2 \cos^2\theta}~.
\end{equation}
Although $H_I$ play the same role as the harmonic functions in other examples they
are in fact not harmonic: they satisfy equations with sources. The expression for
$\Sigma$ vanishes when $r=R$, $\theta={\pi\over 2}$, and arbitrary $\psi$.
Therefore the functions $H_I$ diverge along a circle of radius $R$ in the base space.
This is the ring.
The full solution in five dimension remains regular, due to the conformal factor.
At large distances $H_I\sim X_{I\infty} + {Q_I \over r^2}$ so the black ring
has the same asymptotic behavior as the spherically symmetric black hole
considered in section (\ref{example}). This is because the dipole charges
die off asymptotically and so $H_I$ differ from that of a black hole only at order
${\cal O}({1\over r^4})$. However, the dipole charges dominate close to the horizon.
The scalar fields in the supersymmetric black ring solution take the form
\begin{equation}
X_I = {H_I \over ({1\over 3!}C^{JKL}H_J H_K H_L)^{1/3}}~.
\end{equation}
In the near horizon region where the "harmonic" functions $H_I$ diverge the
scalars approach
\begin{equation}
X^I = { q^I\over ({1\over 3!}C_{JKL} q^J q^K q^L )^{1/3}}~.
\end{equation}
This is in agreement with the prediction (\ref{magattr}) from the attractor mechanism.
In the preceding we defined just enough of the black ring geometry to consider
the attractor mechanism. For completeness, let us discuss also the remaining
features. They are most conveniently introduced in terms of the ring coordinates
\begin{equation}
h_{mn} dx^m dx^n = {R^2 \over (x-y)^2}
\left[ {dy^2 \over y^2 -1}
+ (y^2 -1) d\psi^2
+ {dx^2\over 1-x^2} + (1-x^2) d\phi^2 \right]~,
\end{equation}
on the base space. Roughly speaking, the $x$ coordinate is a polar
angle $x\sim\cos\theta$ that combines with $\phi$ to form two-spheres in the
geometry. The angle along the ring is $\psi$, and $y$ can be interpreted as
a radial direction with $y\to -\infty$ at the horizon. In terms of these coordinates
the two form sources are
\begin{equation}
\Theta^I = -{1\over 2} q^I ( dy\wedge d\psi + dx\wedge d\phi)~.
\end{equation}
Integrating the expression along the $S^2$'s we can verify that
the normalization agrees with the definition (\ref{qdef}) of magnetic charges.
The final element of the geometry is the one-form $\omega$ introduced in (\ref{metric}).
Its nonvanishing components are
\begin{eqnarray}
\omega_\psi &=& -{1\over R^2} (1-x^2) \left[
Q_I q^I - {1\over 6} C_{IJK} q^I q^J q^K (3 + x + y)\right]~,
\label{omegaphi} \\
\omega_\phi &=& {1\over 2} X_{I\infty} q^I (1+y) + \omega_\psi~.
\label{omegapsi}
\end{eqnarray}
In five dimensions there are two independent angular momenta which we can
choose as $J_\phi$ and $J_\psi$. The one form (\ref{omegaphi}-\ref{omegapsi})
gives their values as
\begin{eqnarray}
J_\phi &=& {\pi\over 8G_5} ( Q_I q^I - {1\over 6}C_{IJK} q^I q^J q^K )~,
\label{Jphi} \\
J_\psi &=& {\pi\over 8G_5} ( 2R^2 X_{I\infty} q^I
+ Q_I q^I - {1\over 6}C_{IJK} q^I q^J q^K )~.
\label{Jpsi}
\end{eqnarray}
These expressions will play a role in the discussion of the interpretation of
the attractor mechanism in the next section.
\section{Extremization Principles}
\label{sec:3}
An alternative approach to the attractor mechanism is to analyze the Lagrangian
directly, without using supersymmetry \cite{GibKal}. An advantage of this method is that
the results apply to all extremal black holes, not just the supersymmetric ones \cite{trivetal}.
A related issue is the understanding of the attractor mechanism in terms of the
extremization of various physical quantities.
\subsection{The Reduced Lagrangian}
The attractor mechanism can be analyzed without appealing to supersymetry, by starting
directly from the Lagrangian. In this subsection we exhibit the details.
We will consider just the spherically symmetric case with the metric
\begin{equation}
\label{sphmetric}
ds^2 = -f^2 dt^2 + f^{-1} (dr^2 + r^2 d\Omega^2_3)~.
\end{equation}
Having assumed spherical symmetry, it follows that the gauge field strengths take the form
(\ref{gauol}). The next step is to insert the {\it ansatz} into the Lagrangian (\ref{fivedact}).
The result will be a reduced Lagrangian that depends only on the radial variable. In order
to advantage of intuition from elementary mechanics, it is useful to trade the radial coordinate
for an auxiliary time coordinate defined by
\begin{equation}
dr = - {1\over 2}r^3 d\tau~~~~;~~~\partial_r = -{2\over r^3}\partial_\tau~.
\end{equation}
Introducing the convenient notation
\begin{equation}
f=e^{2U}~,
\end{equation}
a bit of computation gives the reduced action
\begin{equation}
Ld\tau = \left[ - 6(\partial_\tau U)^2 - G_{IJ}\partial_\tau X^I \partial_\tau X^J
+ {1\over 4}e^{4U}G^{IJ}Q_I Q_J\right]d\tau~,
\label{redlag}
\end{equation}
up to overall constants.
Imposing a specific {\it ansatz} on a dynamical system removes numerous degrees of
freedom. The corresponding equations of motion appear as constraints on the
reduced system. In the present setting the main issue is that the charges specified
by the {\it ansatz} are the momenta conjugate to the gauge fields. The correct
variational principle is then obtained by a Legendre transform which, in this simple case,
simply changes the sign of the potential in (\ref{redlag}). Thus the equations of motion
of the reduced system can be obtained in the usual way from the effective
Lagrangean
\begin{equation}
{\cal L} = \left[ - 6(\partial_\tau U)^2 - G_{IJ}\partial_\tau X^I \partial_\tau X^J
- {1\over 4}e^{4U}G^{IJ}Q_I Q_J\right]~.
\label{redham}
\end{equation}
It is instructive to rewrite the effective potential in (\ref{redlag}) and (\ref{redham}).
Using the relations (\ref{xixi}-\ref{gijlower}) we can show the identity
\begin{equation}
G^{IJ}Q_I Q_J = {2\over 3} Z^2_e
+ G^{IJ} D_I Z_e D_J Z_e~,
\label{rewpot}
\end{equation}
where we used the definition (\ref{zedef}) of the electric central charge $Z_e$ and
(\ref{covdef}) of the covariant derivative on moduli space. The Lagrangean (\ref{redham})
can be written as
\begin{eqnarray}
{\cal L} &=& - 6(\partial_\tau U)^2 - G_{IJ}\partial_\tau X^I \partial_\tau X^J
- {1\over 6}e^{4U}Z_e^2
- {1\over 4} e^{4U}G^{IJ} D_I Z_e D_J Z_e\\
&=& -6 \left( \partial_\tau U \pm {1\over 6} e^{2U} Z_e \right)^2
\label{compsq} \\
&-& G_{IJ} \left(\partial_\tau X^I \pm {1\over 2} e^{2U} G^{IK} D_K Z_e \right)
\left(\partial_\tau X^J \pm {1\over 2} e^{2U} G^{JL} D_L Z_e \right)
\pm \partial_\tau \left(e^{2U} Z_e \right)~, \nonumber
\end{eqnarray}
where we used\footnote{We can verify this by writing
$D_I Z_e = {\cal V}^{1/3} \partial_I ({\cal V}^{-1/3}Z_e )$.
This amounts to changing
into physical coordinates before taking the derivative and then changing back.}
\begin{equation}
\partial_\tau X^I D_I Z_e = \partial_\tau Z_e ~.
\end{equation}
Thus the Lagrangean can be written a sum of squares, up to a total derivative. We can
therefore find extrema of the action by solving the linear equations of motion
\begin{eqnarray}
\partial_\tau U &=& -{1\over 6} e^{2U} Z_e~, \\
\partial_\tau X^I &=& -{1\over 2} e^{2U} G^{IJ} D_J Z_e~.
\end{eqnarray}
The second equation is identical to the condition (\ref{modext}) that the gaugino variations
vanish, as one can verify by identifying variables according
to the various notations we have introduced. The first equation can be interpreted as
the corresponding condition that the gravitino variation vanish. To summarize, we
have recovered the conditions for supersymmetry by explicitly writing the
bosonic Larangean as a sum of squares, so that extrema can be found by solving certain
linear equations of motion. The analysis of these linear equations can now be
repeated from section \ref{attractor}. In particular, finite energy density at the horizon
(or enhancement of supersymmetry, as discussed in section \ref{susyenh})
implies the conditions $D_I Z_e=0$, and these in turn lead to the explicit form (\ref{expze})
for the attractor values of the scalars.
One of the advantages of this approach to the attractor mechanism is that it applies
even when supersymmetry is broken. To see this, consider solutions with constant value of
the scalar fields throughout spacetime $\partial_\tau X^I=0$. Extremizing the Lagrangian
with respect to the scalar fields can then be found by considering just the potential (\ref{rewpot}).
Upon variation we find
\begin{equation}
\left( {2\over 3} G_{IJ} Z_e + D_I D_J Z_e \right) D^J Z_e=0~.
\end{equation}
This equation is solved automatically for $D_J Z_e=0$. Such geometries are
the supersymmetric solutions that have been our focus. However, it is seen that there
can also be solutions where the scalars satisfy
\begin{equation}
{2\over 3} G_{IJ} Z_e + D_I D_J Z_e =0~.
\end{equation}
Such solutions do not preserve supersymmetry, but they do exhibit attractor
behavior.
\subsection{Discussion: Physical Extremization Principles}
In section \ref{susyenh} we found that the attractor values are determined by
extremizing one of the two central charges. For $\Theta^I=0$ they
are determined by extremizing the electric central charge (\ref{zedef}) over
moduli space $\partial_i Z_e=0$. On the other hand, for $\Theta^I\neq 0$,
we should instead extremize the magnetic central charge $\partial_i Z_m=0$.
These prescriptions are mathematically precise but they lack a clear physical
interpretation. It would be nice to reformulate the extremization principles in terms
of physical quantities.
Let us consider first the situation when $\Theta^I=0$. As discussed in
section \ref{moredetails}, the electric central charge can be interpreted as the
mass of the system. Therefore, extremization amounts to minimizing the mass.
If we think about the attractor mechanism this way, the monotonic flow of the
electric central charge amounts to a roll down a potential, with scalars
ultimately taking the value corresponding to dynamical equilibrium.
In particular, if the scalars are adjusted to their attractor values already at infinity
(these configurations are referred to as "double extreme black holes")
there is no flow because the configuration remains in its equilibirum.
A difficulty with this picture is the fact that the situation with $\Theta^I\neq 0$
works very differently even though the asymptotic configuration is in fact
independent of the dipole charges. We would like a physical extremization
principle that works for that case as well.
The case where $\Theta^I\neq 0$ is elucidated by
considering the combination
\begin{equation}
J_\psi - J_\phi = R^2 X_{I\infty} q^I = R^2 Z_m~,
\label{angdiff}
\end{equation}
of the angular momenta (\ref{Jphi}-\ref{Jpsi}). This quantity can be interpreted
as the intrinsic angular momentum of the black ring, not associated with the
surrounding fields. The interesting point is that extremizing $Z_m$ is the same
as extremizing $J_\psi - J_\phi$ with $R^2$ fixed. It may at first seem worrying
that we propose extremizing angular momenta. For a black hole these would
be quantum numbers measurable at infinity, and so they would be part of the
input that specifies solution. However, the black ring solution is different: we
can choose its independent parameters as $q^I$, $Q_I$, $R^2$ with the
understanding that then the angular momenta $J_\phi$ and $J_\psi$ that
support the black ring must be those determined by (\ref{Jphi}-\ref{Jpsi}).
The precise values of $J_\phi$ and $J_\psi$ so determined depend on
the scalars and the proposed extremization principle is that the scalars
at the horizon are such that the combination (\ref{angdiff}) is minimal.
The proposed principle is quite similar to the extremization of the mass in the
electric case of supersymmetric black holes. In fact, the combination (\ref{angdiff})
of angular momenta that we propose extremizing in the magnetic case
behaves very much like a mass: it can be interpreted as the momentum along
the effective string that appears in the near ring limit \cite{ringent,myringent}.
In order to elevate the extremization of (\ref{angdiff}) to a satisfying principle
one would need a geometric definition of the ring radius $R$ that works
independently of the explicit solution. Ideally, there should be some kind
of conserved integral, akin to those defining the electric charges,
or the more subtle ones appearing for dipole
charges \cite{Copsey:2005se}. Another issue
is that of more complicated multiple ring solutions, which are characterized by
several radii. This latter problem is completely analogous to the ambiguity with
assigning mass for multi black hole solutions: the asymptotics does not uniquely
specify the near horizon behavior. We will put these issues aside for now,
and seek an extremization principle that combines the extremization of (\ref{angdiff})
in the magnetic case with extremization of the mass in electric case, and
works in any basin of attraction, whether electric or magnetic in character.
To find such a principle, recall that the black hole entropy (\ref{bhent}) can be
written in terms of the central charge in the electric case. Accordingly, the
extremization over moduli space can be recast as\footnote{Although
(\ref{bhent}) was given in the spherically symmetric case, it can be generalized
to include angular momentum \cite{Kallosh:1996vy} (just subtract $J^2$ under the
square root. The argument given below carries
through.}
\begin{equation}
\partial_i S =0~.
\label{entext}
\end{equation}
The black ring entropy can be written compactly as
\begin{equation}
S = 2\pi \sqrt{J_4}~.
\end{equation}
For toroidal compactification\footnote{This statement has an obvious alternate
version that applies to general Calabi-Yau spaces.} $J_4$ is the quartic $E_{7(7)}$
invariant, evaluated at arguments that depend on the black ring parameters according to the
identifications
\begin{equation}
J_4 = J_4 (Q_I, q^I, J_\psi - J_\phi)~.
\end{equation}
The black ring is thus related to black holes in four dimensions \cite{ringent}.
In the present context the point is that the extremization principle
(\ref{entext}) applies to both electric and magnetic attractors. This
provides a thermodynamic interpretation of the attractor mechanism.
One obstacle to a complete symmetry between the electric and
magnetic cases is that near a magnetic attractor point one must
apply (\ref{entext}) with $Q_I$, $q^I$, and $R$ fixed, while near an
electric attractor it is $Q_I$ and $J$ that should be kept fixed. In either case
these are the parameters that define the solution.
There is one surprising feature of the proposed physical extremization principle:
the entropy is {\it minimized} at the attractor point. This may be the correct physics:
as one moves closer to the horizon, the geometry is closer to the microscopic
data. It is also in harmony with the result that, at least in some cases, extremization
over the larger moduli space that includes multi-center configurations gives split
attractor flows that correspond to independent regions that have even less
entropy \cite{Denef:2000ar}, with the end of the flow plausibly corresponding
to "atoms" that have no entropy at all \cite{Mathur:2005zp,atomization}.
We end with a summary of this subsection: we have proposed an extremization
principle (\ref{entext}) that applies to both the electric (black hole) and magnetic
(black ring) cases. A physical interpretation in terms of thermodynamics looks
promising at the present stage of development. In order to
fully establish the proposed principle one would need a more detailed
understanding of general flows, including those that have magnetic charges,
and one would also need a more general definition of charges.
\acknowledgement
I thank Stefano Bellucci for organizing a stimulating meeting and
Per Kraus for collaboration on the material presented in these
lectures. I also thank Alejandra Castro for reading the manuscript
carefully and proposing many improvements, and Josh Davis for
discussions.
\input{referenc}
\printindex
\end{document}
|
2,869,038,154,915 | arxiv | \section{Introduction}
It is clear that
the tensor \cite{Starobinsky} and scalar \cite{Mukhanov} power
spectra from primordial inflation are quantum gravitational effects
by how the approximate tree order results depend upon Planck's
constant $\hbar$ and Newton's constant $G$,
\begin{equation}
\Delta^2_{h}(k) \approx \frac{16 \hbar G H^2(t_k)}{\pi c^5} \qquad ,
\qquad \Delta^2_{\mathcal{R}}(k) \approx \frac{\hbar G H^2(t_k)}{\pi
c^5 \epsilon(t_k)} \; . \label{Hubbleform}
\end{equation}
(Here $H(t)$ is the Hubble parameter, $\epsilon(t)$
is the first slow roll parameter, and $t_k$ is the time of
first horizon crossing for the mode of wave number $k$\footnote{
The definition of $t_{k}$ is the time at which the physical
wave number $k/a(t)$ of some perturbation equals the Hubble parameter,
$k = H(t_k) a(t_k)$.}.) These effects were predicted
around 1980, and the detection of the scalar power spectrum in 1992
\cite{COBE} represents the first quantum gravitational data ever taken.
Much more has followed \cite{WMAP,PLANCK}, as have increasingly sensitive
bounds on the tensor power spectrum \cite{SPT,ACP}.
Although using this data to study quantum gravity has so far been limited
by the absence of a compelling model for inflation,
there is no objection to the revolutionary character of these events.
The tree order results (\ref{Hubbleform}) are just the first terms
in the quantum loop expansion in which each higher loop is suppressed
by an additional factor of $G H^2$.
Assuming single-scalar inflation, the best
current data bounds this loop-counting parameter to be no larger than about
$G H^2 \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 10^{-10}$ \cite{KOW}. That is a very small number, but it has
been suggested that the sensitivity to resolve one loop corrections might be
obtained by measuring the matter power spectrum out to redshifts of as high
as $z \sim 50$ \cite{21cm}. Reaching that goal would be very difficult, requiring
both a unique model of inflation to pin down the tree order contribution and
a secure understanding of the relevant astrophysics to extract the primordial
signal. However, the work is in progress \cite{first21}, and the project does not
seem hopeless.
The possibility of resolving one loop corrections to the power spectra
has motivated theorists to do intensive studies on the issue
\cite{Weinberg, others}. Because the effect will necessarily be very small,
much attention has been devoted to potentially large enhancements from factors
of $1/\epsilon$ in the $\zeta$ propagator \cite{SLS,JaSl,XGB}, and from
the formal infrared divergence \cite{IRstudies} of $\zeta$ and
graviton propagators implied by the approximate scale invariance of their
tree order power spectra (\ref{Hubbleform}). This has raised the issue of
precisely defining {\it what} is being loop corrected. The tree order results
(\ref{Hubbleform}) are consistent with the spatial Fourier transform of
2-point correlators of the graviton and $\zeta$ fields,
\begin{eqnarray}
\Delta^2_{h}(k) & \equiv & \lim_{t \gg t_k} \frac{k^3}{2 \pi^2} \int
\!\! d^3x \, e^{-i\vec{k} \cdot \vec{x}} \Bigl\langle \Omega
\Bigl\vert h_{ij}(t,\vec{x}) h_{ij}(t,\vec{0}) \Bigr\vert \Omega
\Bigr\rangle \; , \label{tenVEV} \\
\Delta^2_{\mathcal{R}}(k) & \equiv & \lim_{t \gg t_k} \frac{k^3}{2
\pi^2} \int \!\! d^3x \, e^{-i\vec{k} \cdot \vec{x}} \Bigl\langle
\Omega \Bigl\vert \zeta(t,\vec{x}) \zeta(t,\vec{0}) \Bigr\vert
\Omega \Bigr\rangle \; . \label{scalVEV}
\end{eqnarray}
There is no question that one loop corrections to these expressions
show sensitivity to the infrared cutoff \cite{GidSloth}. This
sensitivity can be canceled by re-defining the ``power spectra'' as
the expectation values of appropriately chosen operators
\cite{gauge}. However, such redefinitions tend to alter the
$\epsilon$ dependence of loop corrections, and they also introduce
new ultraviolet divergences whose renormalization is not currently
understood \cite{issues}.
The point of this paper is to consider another generalization of
what is meant by the ``primordial power spectra.'' This alternate
generalization is motivated by the relations which emerge from
expressions (\ref{tenVEV}) and (\ref{scalVEV}) when one uses the
free field mode sums for $h_{ij}$ and $\zeta$,
\begin{eqnarray}
\Delta^2_{h}(k) & = & \lim_{t \gg t_k} \frac{k^3}{2 \pi^2} \times 64
\pi G \times \vert u(t,k) \vert^2 \; , \label{tenmode} \\
\Delta^2_{\mathcal{R}}(k) & = & \lim_{t \gg t_k} \frac{k^3}{2 \pi^2}
\times 4 \pi G \times \vert v(t,k) \vert^2 \; , \label{scalmode}
\end{eqnarray}
where $u(t,k)$ and $v(t,k)$ are the plane wave mode functions\footnote
{$u(t,k)$ and $v(t,k)$ are not the one-particle-irreducible(1PI)
1-point functions of $h_{ij}(t, \vec{x})$ and $\zeta(t,\vec{x})$
respectively.} of tensor and scalar perturbations. The alternate generalization
is to simply extend the tree order relations (\ref{tenmode})
and (\ref{scalmode}) to all orders using the mode functions obtained
by solving the linearized Schwinger-Keldysh effective field equations\footnote{
A curious reader might wonder how the quantum corrected mode functions are related
to the Heisenberg operators which satisfy the standard commutation relations.
We demonstrate the relation between them using our worked-out example
in Appendix \ref{cano-mode}.}.
Even though equations (\ref{tenVEV}, \ref{scalVEV}) and
(\ref{tenmode}, \ref{scalmode}) are two different approaches of
quantum correcting primordial power spectra,
the diagram topology for quantum corrections to the mode function definition
is identical to that of quantum corrections to the correlator\footnote{
The generic diagram topology for the two definitions is derived in Appendix \ref{topology}.}.
Also, note that the two definitions would agree if the in-out formalism
had been employed. However, one must employ the Schwinger-Keldysh formalism
in cosmological scenarios. It is not so clear whether or not the two definitions agree
at one loop due to subtle differences in which of the four Schwinger-Keldysh propagators
appears. That is what we shall check.
In this paper we start with briefly reviewing single scalar inflation,
deriving the tree order results and reasoning alternate definitions.
This comprises of section 2. In section 3 we digress to sketch the
Schwinger-Keldysh formalism\footnote{It is also called the in-in or the
closed time path formalism.} and give rules to facilitate our computation.
In section 4 we use a worked-out example to demonstrate that two definitions
disagree at one loop. Finally we discuss the advantages and disadvantages
for each definition in section 5.
\section{Two alternate definitions for loop-corrected primordial power spectra}
Primordial power spectra not only allow us to understand the early Universe,
but also serve as a bridge that connects cosmology with fundamental theory. For
example, resolving the tensor power spectrum would confirm the existence
of gravitons and their quantization. Attaining the sensitivity to resolve
loop corrections to the power spectra would, along with a unique theory of inflation,
direct theorists in the construction of a renormalizable theory of quantum gravity.
Two of the many frustrations in the attempt to connect inflation with fundamental
theory are first, we lack a unique model of inflation --- which means we don't know
the time dependence of the scale factor $a(t)$ --- and
second, we do not have a solution for
the tree order mode functions for a general $a(t)$ even if we happened to know it.
This means that approximations must be used even for the tree order power spectra. It
also implies that we must approximate the propagators which occur in loop integrations
because these propagators are mode sums of products of unknown tree order mode
functions. These are all important problems, but here we wish to focus on
the issue of what theoretical quantity represents the observed power spectrum.
That is, what quantity would we like to compute, assuming we had the mode functions
and propagators necessary to make the computation? In particular, is it the spatial
Fourier transform of the 2-point correlators (\ref{tenVEV}) and (\ref{scalVEV}), or
should we instead use the norm squared of the mode functions (\ref{tenmode}) and
(\ref{scalmode})? We begin with a quick review of single scalar inflation which
is meant to pedagogically demonstrate that the two definitions coincide at tree order.
The burden is that they disagree at one loop.
The dynamical variables of single-scalar inflation are
the metric ${\rm g}_{\mu\nu}(t,\vec{x})$ and the inflaton field $\varphi(t,\vec{x})$.
Its Lagrangian density is,
\begin{equation}
\mathcal{L} = \frac1{16 \pi G} \, {\rm R} \sqrt{-{\rm g}} -\frac12
\partial_{\mu} \varphi \partial_{\nu} \varphi {\rm g}^{\mu\nu}
\sqrt{-{\rm g}} - V(\varphi) \sqrt{-{\rm g}} \; . \label{Ldef}
\end{equation}
Primordial inflation can be described by homogeneous, isotropic
and spatially flat background metric,
\begin{equation}
{\rm g}^0_{\mu\nu} dx^{\mu} dx^{\nu} = -dt^2 + a^2(t) d\vec{x}
\!\cdot\! d\vec{x} \; ,
\end{equation}
with the slow roll parameter,
\begin{equation}
\epsilon(t) \equiv-\frac{\dot{H}}{H^2} \qquad , \qquad
0<\epsilon(t)<1\; .\label{slowroll}
\end{equation}
Here $H(t)$ is the Hubble parameter defined as the first time derivative
of the scale factor $a(t)$,
\begin{equation}
H(t) \equiv \frac{\dot{a}}{a}.
\end{equation}
It indicates whether or not the Universe is expanding.
We follow the convention of Maldacena \cite{JM} and Weinberg \cite{SW}
for decomposing the spatial metric\footnote{This spatial metric $g_{ij}$
is from Arnowitt-Deser-Misner (ADM) decomposition
\cite{ADM}:
${\rm g}_{00} \equiv -N^2 \!+\! g_{ij} N^i N^j \;,\;
{\rm g}_{0i} \equiv -g_{ij} N^j \:,\: {\rm g}_{ij} \equiv g_{ij}\:.$ },
\begin{eqnarray}
&&g_{ij}(t,\vec{x}) \equiv a^2(t) e^{2 \zeta(t,\vec{x})}
\widetilde{g}_{ij}(t,\vec{x}) \; \\
&&\widetilde{g}_{ij}(t,\vec{x}) \equiv \Bigl( e^{h(t,\vec{x})}
\Bigr)_{ij} = \delta_{ij} + h_{ij} + \frac12 h_{ik} h_{kj} + \dots
\label{hdef}\;,
\end{eqnarray}
where the $\zeta(t,\vec{x})$ and $h_{ij}(t,\vec{x})$ fields
are the scalar and tensor perturbations respectively.
During the $50$ e-foldings of primordial inflation which is required
to explain the horizon problem, many modes must experience first horizon
crossing, $k=a(t_k)H(t_k)$. After that time they became almost constant
and survived to be detected today.
Therefore the tensor and scalar power spectra are defined (for $D = 4$
spacetime dimensions) as in (\ref{tenVEV}) and (\ref{scalVEV}).
Maldacena \cite{JM} and Weinberg \cite{SW} employ Arnowitt-Deser-Misner
(ADM) notation but they do not fix the gauge by specifying lapse $N(t,\vec{x})$
and shift $N^i(t,\vec{x})$ functions. They instead fix the surface of simultaneity
using the background value of the inflaton, $\varphi(t,\vec{x})=\varphi_0(t)$,
and impose the spatial transverse gauge condition, $\partial_j h_{ij}(t,\vec{x}) = 0$.
The lapse and shift functions hence\footnote{$N[\zeta,h](t,\vec{x})$ can be solved
exactly \cite {KOW} but there only exists a perturbative solution for
$N^i[\zeta,h](t,\vec{x})$.} can be determined as nonlocal functionals of graviton fields
from solving the gauged fixed constraint equations. Substituting those solutions into
the original Lagrangian, it is not so hard to obtain the quadratic part,
\begin{eqnarray}
\mathcal{L}_{h^2} & = & \frac{a^{D-1}}{64\pi G} \Biggl\{
\dot{h}_{ij} \dot{h}_{ij} - \frac1{a^2} \partial_k h_{ij} \partial_k
h_{ij} \Biggr\} \;, \label{freeh}\\
\mathcal{L}_{\zeta^2} & = & \frac{(D \!-\!2) \, \epsilon \,
a^{D-1}}{ 16\pi G} \Biggl\{ \dot{\zeta}^2 - \frac1{a^2} \partial_k
\zeta \partial_k \zeta \Biggr\} \; . \label{freezeta}
\end{eqnarray}
From expression (\ref{freeh}) we see that each of the $\frac12 (D-3) D$
graviton polarizations is $\sqrt{32 \pi G}$ times a canonically
normalized, massless, minimally coupled scalar. Its plane wave mode
function $u(t,k)$ obeys,
\begin{equation}
\ddot{u} + (D \!-\! 1) H \dot{u} + \frac{k^2}{a^2} u = 0 \qquad {\rm
with} \qquad u \dot{u}^* - \dot{u} u^* = \frac{i}{a^{D-1}} \; .
\label{hmodes}
\end{equation}
Expression (\ref{freezeta}) implies that the free field expansion for
$\zeta(t,\vec{x})$ is $\sqrt{8\pi G/(D-2)}$ times a canonically
normalized scalar whose plane wave mode functions $v(t,k)$ obey,
\begin{equation}
\ddot{v} + \Bigl[(D \!-\! 1) H \!+\!
\frac{\dot{\epsilon}}{\epsilon} \Bigr] \dot{v} +
\frac{k^2}{a^2} v = 0 \qquad {\rm with} \qquad
v\dot{v}^* - \dot{v}v^* =\frac{i}{\epsilon a^{D-1}} \; .
\label{zetamodes}
\end{equation}
To derive equations (\ref{tenmode}) and (\ref{scalmode}) (in $D=4$ spacetime dimensions)
for the primordial power spectra one substitutes the free field expansions
for $h_{ij}(t,\vec{x})$ and $\zeta(t,\vec{x})$ into equations (\ref{tenVEV})
and (\ref{scalVEV}).
From the tree order derivation for the tensor power spectrum we establish
the following relation\footnote{The relation for the scalar power spectrum
reaches the same form.},
\begin{eqnarray}
\frac{k^3}{2\pi^2}\lim_{t\geq t_k}\Biggl\{\int\!\! d^3 x
e^{-i\vec{k}\cdot\vec{x}}
\Bigl\langle\Omega\Bigl\vert h_0(t,\vec{x})h_0(t,0)
\Bigr\vert\Omega\Bigr\rangle = \sharp \Bigl\vert u(t,k)\Bigr\vert^2
\Biggr\}\;,\label{treeid}
\end{eqnarray}
here we suppress tensor indexes and $\sharp$ is a constant which depends upon
the field we consider. Each side of equation (\ref{treeid}) has a clear
generalization to higher orders,
\begin{eqnarray}
&&\hspace{-1cm}\bullet\int\!\! d^3 x e^{-i\vec{k}\cdot\vec{x}}
\Bigl\langle\Omega\Bigl\vert h_0(t,\vec{x})h_0(t,0)
\Bigr\vert\Omega\Bigr\rangle\longrightarrow
\int\!\! d^3 x e^{-i\vec{k}\cdot\vec{x}}
\Bigl\langle\Omega\Bigl\vert h(t,x)h(t,0)
\Bigr\vert\Omega\Bigr\rangle; \label{VEVdef}\\
&&\hspace{-1cm}\bullet\;\sharp \Biggl\vert u_{0}(t,k)\Biggr\vert^2
\longrightarrow
\sharp \Biggl\vert\;u(t,k)\!+\!\sum_{l=1}\!
\Delta u_{l}(t,k)\Biggr\vert^2\nonumber\\
&&\hspace{1.2cm}=\sharp \Biggl\{\Bigl\vert u(t,k)\Bigr\vert^2
\!+\!\Delta u_1(t,k)u^*(t,k)
\!+\!\Delta u^*_1(t,k)u(t,k)\!+\!\cdots\Biggr\},
\label{modedef}
\end{eqnarray}
where higher order mode functions can be solved by the linearized
Schwinger-Keldysh effective field equation\footnote{
$\Delta u_0(t',k)\equiv u(t',k)$},
\begin{eqnarray}
\mathcal{D}\Bigl[\Delta u_l(t,k)e^{i\vec{k}\cdot\vec{x}}\;\Bigr]
\!=\!\!\int\!\! d^4 x' \sum_{k=1}^{l}\Bigl\{M^2_{\scriptscriptstyle ++}(x;x')\!
+M^2_{\scriptscriptstyle +-}(x;x')\Bigr\}_{k}\Delta u_{l-k}(t',k)
e^{i\vec{k}\cdot\vec{x'}}\;.\label{SWeqn}
\end{eqnarray}
Here $\mathcal{D}$ is
the kinetic operator.
Note that
$\frac{k^3}{2\pi^2}\lim_{t\geq t_k}$ in (\ref{treeid}) is a common factor
for both definitions. To simplify later discussion we drop it without
changing the generic structure of the two definitions.
At this step it is clear that one could compute the loop-corrected power spectra
either by spatially Fourier transforming the 2-point corrector --(\ref{VEVdef})
or exploiting the mode function definition --(\ref{modedef}).
\section{Schwinger-Keldysh formalism}
The purpose of this section is to give the rules for the various
Schwinger-Keldysh vertices and propagators. We also introduce the linearized
Schwinger-Keldysh effective field equation and demonstrate that
a causal result in $\varphi^3$ theory can be obtained by exploiting these rules.
For most of the problems we encounter in elementary particle physics
we are allowed to assume that quantum fields begin in free vacuum
at asymptotically early times and end up the same way at asymptotically
late times, for example, scattering processes in flat space.
However, this is not valid for cosmological settings in which
the in vacuum doesn't evolve to the out vacuum. The use of the
in-out formalism would result in quantum correction terms dominated
by events from the infinite future!
A realistic scenario corresponding to what we measure would rather be
that the Universe is released from a prepared state at a finite time and
allowed to evolve as it will. The Schwinger-Keldysh formalism can give
a correct description of this. Employing it \cite{JS,KTM,BM,LVK,CSHY,RDJ,CH,FW}
also guarantees that the computation is both real and causal.
It is convenient to sketch the in-in formalism by employing a scalar field
$\varphi(x)$. The basic construction is to evolve fields forwards with
$\rlap{$\rfloor$}\lceil[d\varphi_+]e^{S[\varphi_+]}$ from the time $i$ to the time $f$ and
backwards with $\rlap{$\rfloor$}\lceil[d\varphi_{-}]e^{S[\varphi_{-}]}$. To avoid a lengthy
digression, we give the key relation between the canonical operator and the
functional integral \cite{Miao,FW,TOW},
\begin{eqnarray}
\lefteqn{\Bigl\langle \Psi \Bigl\vert
\overline{T}^*\Bigl(\mathcal{O}_2[ \varphi]\Bigr)
T^*\Bigl(\mathcal{O}_1[\varphi]\Bigr) \Bigr\vert \Psi \Bigr\rangle =
\rlap{$\Biggl\rfloor$}\Biggl\lceil [d\varphi_+] [d\varphi_-] \,
\delta\Bigl[\varphi_-(f) \!-\! \varphi_+(f)\Bigr] } \nonumber \\
& & \hspace{1.5cm} \times \mathcal{O}_2[\varphi_-]
\mathcal{O}_1[\varphi_+] \Psi^*[\varphi_-(i)] e^{i \int_{i}^{f} dt
\Bigl\{L[\varphi_+(t)] - L[\varphi_-(t)]\Bigr\}} \Psi[\varphi_+(i)]
\; ,\qquad \label{fund}
\end{eqnarray}
where $T^*$ stands for a time-ording symbol, except that
any derivatives are taken {\it outside} the time ordering, whereas
$\overline{T}^*$ is anti-time-ordered. Based on the same field
in (\ref{fund}) being represented by two different dummy functional
variables, $\varphi_{\pm}(x)$, several modified Feynman rules can be
inferred,
\begin{itemize}
\item{Each line has a polarity of either $+$ or $-$;}
\item{Vertices (and counterterms) are either all $+$ or all $-$;}
\item{Vertices (and counterterms) with $+$ polarity are the same as for the
usual Feynman rules and those with $-$ polarity have an extra minus sign;}
\item{External lines from the time-ordered operator carry $+$ polarity and
those from the anti-time-ordered operator carry $-$ polarity;}
\item{Propagators can be $++$, $-+$, $+-$ and $--$.}
\end{itemize}
Note also that we can directly read off the four propagators
from substituting the free Lagrangian in place of the full Lagrangian in
expression (\ref{fund}),
\begin{eqnarray}
i\Delta_{\scriptscriptstyle ++}(x;x^{\prime})\!\!\! &=&\!\! \!
\Bigl\langle \Omega\Bigl\vert T\Bigl(\varphi(x) \varphi(x^{\prime}) \Bigr)
\Bigr\vert\Omega \Bigr\rangle_0\; , \label{++}\\
i\Delta_{\scriptscriptstyle -+}(x;x^{\prime})\!\!\!& =&\!\!\!
\Bigl\langle \Omega \Bigl\vert\varphi(x) \varphi(x^{\prime})
\Bigr\vert\Omega \Bigr\rangle_0 ,\quad\label{-+} \\
i\Delta_{\scriptscriptstyle +-}(x;x^{\prime}) \!\!\! &=& \!\!\!
\Bigl\langle \Omega \Bigl\vert \varphi(x^{\prime}) \varphi(x) \Bigr\vert
\Omega \Bigr\rangle_0 , \quad \label{+-} \\
i\Delta_{\scriptscriptstyle --}(x;x^{\prime}) \!\!& =& \!\!
\Bigl\langle \Omega \Bigl\vert \overline{T}\Bigl(\varphi(x)
\varphi(x^{\prime}) \Bigr) \Bigr\vert \Omega \Bigr\rangle_0 . \label{--}
\end{eqnarray}
The subscript $0$ indicates vacuum expectation values in the free theory.
A careful reader might have noticed that the $++$ propagator is the usual
Feynman propagator and the $--$ one is its complex conjugate; the $-+$
propagator is similarly the conjugate of the $+-$ one.
We close by employing the Schwinger-Keldysh formalism to show
that a causal result is achieved in scalar field
theory with interaction $-\frac{1}{6}\lambda\varphi^3$.
To facilitate this simple computation we introduce the linearized
Schwinger-Keldysh effective field equation without
deriving it \cite{Miao,FW,TOW}\footnote{Although there are four
2-point 1PI (One particle irreducible) functions in the in-in formalism,
we only need two of them in the Schwinger-Keldysh effective equation.},
\begin{equation}
\frac{\delta \Gamma[\varphi_{\scriptscriptstyle
+},\varphi_{\scriptscriptstyle -}] }{\delta \varphi_{\scriptscriptstyle
+}(x)} \Biggl\vert_{\varphi_{\scriptscriptstyle \pm} = \varphi} \!\!\! =
\frac{\delta S[\varphi]}{\delta \varphi(x)} - \! \int \! d^4x^{\prime}
\Bigl[M^2_{\scriptscriptstyle ++}\!(x;x^{\prime}) +
M^2_{\scriptscriptstyle +-}\!(x;x^{\prime})\Bigr] \varphi(x^{\prime}).\label{SK}
\end{equation}
The two squared self-masses in $\varphi^3$ theory can be expressed as,
\begin{eqnarray}
M^2_{\scriptscriptstyle +\pm}(x;x')\!=\! \mp i\frac{\lambda^2}{2}
\Bigl[ i\Delta_{\scriptscriptstyle +\pm}(x;x') \Bigr]^2
\!=\!\mp i \frac{\lambda^2}{2}
\frac{\Gamma^2(\frac{D}{2}\!-\!1)}{16 \pi^{D}}
\Biggl[\frac{1}{\Delta x^2_{\scriptscriptstyle +\pm}(x;x')}
\Biggr]^{D-2},\label{TOT2pt}
\end{eqnarray}
and the two invariant intervals in the denominator of (\ref{TOT2pt}) are,
\begin{eqnarray}
&&\Delta x^2_{\scriptscriptstyle ++}(x;x')=\parallel\vec{x}-\vec{x}'\parallel^2
-(|t-t'|-i\delta)^2,\label{x++}\\
&&\Delta x^2_{\scriptscriptstyle +-}(x;x')=\parallel\vec{x}-\vec{x}'\parallel^2
-(t-t'+i\delta)^2.\label{x+-}
\end{eqnarray}
First of all, we notice that $\Delta x^2_{\scriptscriptstyle ++}$ equals
$\Delta x^2_{\scriptscriptstyle +-}$ while the time $t'$ is in the future of
the time $t$. A direct consequence of this is that the contribution from
$M^2_{\scriptscriptstyle ++}(x;x')$ cancels that from
$M^2_{\scriptscriptstyle +-}(x;x')$.
This implies no contributions from $t'$ in the future of the time $t$. Second,
when the time $t'$ lies in the past of the time $t$,
$\Delta x^2_{\scriptscriptstyle +-}(x;x')$ is the complex conjugate of
$\Delta x^2_{\scriptscriptstyle ++}(x;x')$, which indicates
$i\Delta_{\scriptscriptstyle +-}(x;x')=[i\Delta_{\scriptscriptstyle ++}(x;x')]^*$.
The combination of the two self-squared masses can be written as,
\begin{eqnarray}
\Bigl[M^2_{\scriptscriptstyle ++}\!+\!M^2_{\scriptscriptstyle +-}\Bigr](x;x')
\!=\!-i\frac{\lambda^2}{2} \Biggl\{\Bigl[i\Delta_{\scriptscriptstyle ++}(x;x')\Bigr]^2
\!\!-\!\Bigl([i\Delta_{\scriptscriptstyle ++}(x;x')]^*\Bigr)^2\Biggr\}
\!\!\longrightarrow \textrm{real}.\label{2M2}
\end{eqnarray}
One can infer from equation (\ref{2M2}) that all contributions from the past of
the time $t$ are real. Further, when the points $x^{\mu}$ and ${x'}^{\mu}$ are
spacelike separated the real parts of the invariant intervals are positive and the
different infinitesimal imaginary parts are irrelevant. Hence the $++$ and $+-$
contributions cancel.
In summary, we have established that the sum of $M^2_{\scriptscriptstyle ++}(x;x')$
and $M^2_{\scriptscriptstyle+-}(x;x')$ is zero except when ${x'}^{\mu}$ lies on or
within the past lightcone of $x^{\mu}$. Using the linearized Schwinger-Keldysh
effective equation (\ref{SK}) also guarantees that the result derived from it must be
real and causal.
\section{A worked-out example}
When one considers loop corrections to the scalar or tensor power spectra,
one inevitably needs higher order interaction vertices. Even though it is tedious
to obtain them from the gauge-fixed and constrained Lagrangian, several of them
have been worked out:
\begin{itemize}
\item{The $\zeta^3$ interaction by Maldacena \cite{JM};}
\item{Simple results for the $\zeta^4$ terms by Seery, Lidsey
and Sloth \cite{SLS};}
\item{The interactions of $\zeta^5$ and $\zeta^6$ discussed
by Jarnhus and Sloth \cite{JaSl};}
\item{The lowest $\zeta$--graviton interactions,
$\zeta h^2\;,\;\zeta^2 h$ and $\zeta^2 h^2$,
given by Xue, Gao and Brandenberger \cite{XGB}.}
\end{itemize}
Many diagrams are possible with these interactions but the simplest consists of
a single loop with two 3-point vertices. We lose nothing to consider a scalar
theory with a cubic interaction in flat spacetime,
\begin{eqnarray}
\mathcal{L}=-\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi g^{\mu\nu}
\!-\frac{\lambda}{3!}\varphi^3\;,\label{3scalar}
\end{eqnarray}
because the diagram topology is the same as for scalar-driven inflation but the
actual computation is vastly simpler.
In this section we use this worked-out example
to compute the one-loop correction to
the power spectrum. We employ both the mode function definition (\ref{modedef})
and the corrector definition (\ref{VEVdef}). What we show is that two definitions
disagree at one loop. The curious reader can find the explicit, and finite results
for each definition worked out in Appendix D.
\subsection{The mode function definition}
In this subsection we first give some identities to facilitate the computation.
We then use the linearized Schwinger-Keldysh effective field equation
to solve for the first order correction to the mode function. Finally the formal
expression for the corresponding power spectrum of $\varphi^3$ theory at one loop
is presented.
The correction to the power spectrum by definition (\ref{modedef})
at one loop order is,
\begin{eqnarray}
\Delta u_1(t,k)u^*(t,k)+ \Delta u^*_1(t,k)u(t,k)\;.\label{u1u0}
\end{eqnarray}
Here $u(t,k)$ is the tree order mode function.
Its relation with the free field expansion is,
\begin{eqnarray}
\varphi_0(t,\vec{x})=\!\!\int\!\!\frac{d^3k}{(2\pi)^3}\Biggl\{u(t,k)\alpha(k)
e^{i\vec{k}\cdot\vec{x}}+ u^*(t,k)\alpha^{\dagger}(k)
e^{-i\vec{k}\cdot\vec{x}}\Biggr\}\;.\label{field-0}
\end{eqnarray}
Applying (\ref{field-0}) to
(\ref{++}) - (\ref{--})
we obtain the propagators with different polarities in terms of the mode functions,
\begin{eqnarray}
&&i\Delta_{\scriptscriptstyle ++}(x;y)\!=
\!\!\int\!\!\frac{d^3k}{(2\pi)^3}e^{i\vec{k}\cdot(\vec{x}-\vec{y})}
\left\{\!\matrix{\theta(x^0\!\!-\!y^0)u(x^0,k)u^*(y^0,k)\cr
+\theta(y^0\!\!-\!x^0)u^*(x^0,k)u(y^0,k)}
\!\right\}\;,\label{++u} \\
&&i\Delta_{\scriptscriptstyle -+}(x;y)\!=
\!\!\int\!\!\frac{d^3k}{(2\pi)^3}e^{i\vec{k}\cdot(\vec{x}-\vec{y})}
u(x^0,k)u^*(y^0,k)\label{-+u}\;,\\
&&i\Delta_{\scriptscriptstyle +-}(x;y)\!=
\!\!\int\!\!\frac{d^3k}{(2\pi)^3}e^{-i\vec{k}\cdot(\vec{x}-\vec{y})}
u^*(x^0,k)u(y^0,k)=[i\Delta_{\scriptscriptstyle{-+}}(x;y)]^*
\;,\label{+-u}\\
&&i\Delta_{\scriptscriptstyle --}(x;y)=
[i\Delta_{\scriptscriptstyle{++}}(x;y)]^* \nonumber\\
&&\hspace{2cm}=\!\!\int\!\!\frac{d^3k}{(2\pi)^3}
e^{-i\vec{k}\cdot(\vec{x}-\vec{y})}
\left\{\!\matrix{\theta(x^0\!\!-\!y^0)u^*(x^0,k)u(y^0,k)\cr
+\theta(y^0\!\!-\!x^0)u(x^0,k)u^*(y^0,k)}
\!\right\}\;.\label{--u}
\end{eqnarray}
The symbol $\Delta u_1(t,k)$ in (\ref{u1u0}) denotes the first order correction to
the mode function. For convenience of later discussion we drop the
subscript of $\Delta u_1(t,k)$. It obeys,
\begin{eqnarray}
\mathcal{D}[\Delta u(t,k) e^{i\vec{k}\cdot\vec{x}}]-\!\int \!\!
d^4y [M^2_{\scriptscriptstyle ++}(x;y) +
M^2_{\scriptscriptstyle +-}(x;y)]
u(y^0,k)e^{i\vec{k}\cdot\vec{y}}=0\;,\label{u1eqn}
\end{eqnarray}
and can be solved formally,
\begin{eqnarray}
\Delta u(t,k)=\!\!\int\!\! d^4 y G_{\scriptscriptstyle{Ret}}(x;y)
\!\int\!\! d^4 y'[M^2_{\scriptscriptstyle ++}
\!+ M^2_{\scriptscriptstyle +-}](y;y')
e^{i\vec{k}\cdot(\vec{y'}-\vec{x})}u(y'^0,k)\;.\label{u1sol}
\end{eqnarray}
Here $G_{\scriptscriptstyle{Ret}}(x;y)$ is
the retarded Green's function for the operator
$\mathcal{D}$
and can be expressed in terms of the Schwinger-Keldysh propagators
(\ref{++}) - (\ref{--}),
\begin{eqnarray}
G_{\scriptscriptstyle{Ret}}(x;y)\!=\!-i[i\Delta_{\scriptscriptstyle{++}}
\!-i\Delta_{\scriptscriptstyle{+-}}](x;y)\;.\label{Gret}
\end{eqnarray}
Also note that the various $\pm$ polarities of the self-mass-squared for
$\varphi^3$ theory are,
\begin{eqnarray}
-iM^2_{\scriptscriptstyle \pm\pm}(y;y')\!=\!-\frac{\lambda^2}{2}
\Bigl[i\Delta_{\scriptscriptstyle \pm\pm}(y;y')\Bigr]^2 \;\;
,\;\;-iM^2_{\scriptscriptstyle \pm\mp}(y;y')\!=\!\frac{\lambda^2}{2}
\Bigl[i\Delta_{\scriptscriptstyle \pm\mp}(y;y')\Bigr]^2.\label{Msquare}
\end{eqnarray}
Inserting (\ref{u1sol}), (\ref{Gret}) and their complex conjugates given by
(\ref{+-u}), (\ref{--u}) to (\ref{u1u0}) we get,
\begin{eqnarray}
&&\hspace{-1.2cm}\Delta u(t,k) u^{*}(t,k) + \Delta u^{*}(t,k) u(t,k)=
\int\!\! d^4 y \!\int\!\! d^4 y'\times\nonumber\\
&&\hspace{-1.4cm}\left\{\!\!\matrix{[i\Delta_{\scriptscriptstyle ++}
\!-i\Delta_{\scriptscriptstyle +-}](x;y)
[-iM^2_{\scriptscriptstyle ++}
\!-iM^2_{\scriptscriptstyle +-}](y;y')
e^{i\vec{k}\cdot(\vec{y'}-\vec{x})}u^*(t,k)u(y'^0,k) \cr
+[i\Delta_{\scriptscriptstyle --}\!
-i\Delta_{\scriptscriptstyle -+}](x;y)
[-iM^2_{\scriptscriptstyle -+}
\!-iM^2_{\scriptscriptstyle --}](y;y')
e^{-i\vec{k}\cdot(\vec{y'}-\vec{x})}u(t,k)u^*(y'^0,k)}\!\!\right\}\!.
\label{Duu1}
\end{eqnarray}
Besides, there is no harm to shift the spatial coordinates in (\ref{Duu1}),
\begin{eqnarray}
\vec{y'}\longrightarrow \vec{y'} +\vec{x}\;\;\;;\;\;\;
\vec{y}\longrightarrow \vec{y} +\vec{x},
\end{eqnarray}
and it can be written as,
\begin{eqnarray}
&&\hspace{-1cm}\Delta u(t,k) u^{*}(t,k) + \Delta u^{*}(t,k) u(t,k)=
\int\!\! d^4 y \!\int\!\! d^4 y'e^{-i\vec{k}\cdot\vec{y'}}
\times\nonumber\\&&\hspace{-1cm}
\left\{\!\matrix{[i\Delta_{\scriptscriptstyle ++}
\!-i\Delta_{\scriptscriptstyle +-}](t,\vec{0};y)
[-iM^2_{\scriptscriptstyle ++}
\!-iM^2_{\scriptscriptstyle +-}](y;y')
u^*(t,k)u(y'^0,k) \cr
+[i\Delta_{\scriptscriptstyle --}\!
-i\Delta_{\scriptscriptstyle -+}](t,\vec{0};y)
[-iM^2_{\scriptscriptstyle -+}
\!-iM^2_{\scriptscriptstyle --}](y;y')
u(t,k)u^*(y'^0,k)}\!\!\right\}.\label{Duu2}
\end{eqnarray}
In the next step we employ the following identities\footnote{
$\Bigl<\varphi_0(x)\varphi_0(y)\Bigr>$ is the abbreviation of
$\Bigl<\Omega|\varphi_0(x)\varphi_0(y)|\Omega\Bigr>$.},
\begin{eqnarray}
&&\Bigl[i\Delta_{\scriptscriptstyle ++}\!-\!
i\Delta_{\scriptscriptstyle +-}\Bigr](x;y)\!=\!-
\Bigl[i\Delta_{\scriptscriptstyle --}\!-\!
i\Delta_{\scriptscriptstyle -+}\Bigr](x;y)\nonumber\\
&&\hspace{3cm}=\theta(x^0\!-\!y^0)\Bigl\{
\Bigl<\varphi_0(x)\varphi_0(y)\Bigr>
-\Bigl<\varphi_0(y)\varphi_0(x)\Bigr>\Bigr\},\\
&&[-iM^2_{\scriptscriptstyle ++}
\!-iM^2_{\scriptscriptstyle +-}](y;y')=
-[-iM^2_{\scriptscriptstyle -+}
\!-iM^2_{\scriptscriptstyle --}](y;y')\nonumber\\
&&\hspace{2cm}=-\frac{\lambda^2}{2}\theta(y^0-y'^0)
\Bigl\{\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
-\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2\Bigr\},\label{M2Delta2}
\end{eqnarray}
in (\ref{Duu2}) and a further simplification is,
\begin{eqnarray}
&&\hspace{-1cm}\Delta u(t,k) u^{*}(t,k)
+ \Delta u^{*}(t,k) u(t,k)=\nonumber\\
&&\hspace{-1cm}-\frac{\lambda^2}{2}\!\!\int_0^{t}\!\!dy^0
\!\!\int_0^{y^0}\!\!\!dy'^0\!\!\int\!\! d^3 y \!\!\int\!\! d^3 y'
e^{-i\vec{k}\cdot\vec{y'}}
\Bigl[ \Bigl<\varphi_{0}(t,\vec{0})\varphi_0(y)\Bigr>
-\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]\nonumber\\
&&\hspace{-1.2cm}\times\Bigl[\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
\!-\!\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2\Bigr]
\Bigl[u(t,k)u^*(y'^0,k)+u^*(t,k)u(y'^0,k)\Bigr].\label{Duu3}
\end{eqnarray}
\subsection{The 2-point correlator definition}
In this subsection we compute the first order corrections to the power
spectrum by spatially Fourier transforming the 2-point correlators.
Within the in-in formalism the external legs of 2-point correlators could
have the following polarities: $(++), (-+), (+-)$ and $(--)$. We begin with
the $(-+)$ 2-point correlator and compute the power spectrum by employing
the correlator definition. We found that the result doesn't agree with
(\ref{Duu3}). We also show that none of the other in-in correlators, nor
any linear combination of them, can resolve the disagreement.
The spatial Fourier transform of the 2-point correlators
of $\varphi^3$ theory is,
\begin{eqnarray}
\int\!\! d^3 x e^{-i\vec{k}\cdot\vec{x}} \Bigl<\Omega|
\varphi(t,\vec{x})\varphi(t,\vec{0})|\Omega\Bigr>\;.
\end{eqnarray}
We begin with the $(-+)$ 2-point correlator at one loop order.
The generic diagram topology is depicted in Fig.~1. The explicit form is,
\begin{eqnarray}
&&\int d^3x e^{-i\vec{k}\cdot\vec{x}}
\int\!\! d^4 y \!\int\!\! d^4 y'
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\},\\
&&=\int\!\! d^4 y \!\int\!\! d^4 y'
e^{-i\vec{k}\cdot\vec{y}}\times\nonumber\\
&&\left\{\!\matrix{\hspace{-1.5cm}u(t,k)u^*(y^0,k)
\left\{\!\matrix{+[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\}\cr
+\Bigl\{\theta(t\!-\!y^0)u^*(t,k)u(y^0,k)
+\theta(y^0\!-\!t)u(t,k)u^*(y^0,k)\Bigr\}\cr
\hspace{4cm}\times\left\{\!\matrix{
+[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\}}
\!\right\}.\label{4F1}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics[width=13.8cm]{fig1loop.eps}
\end{center}
\caption{One loop contribution to the $(-+)$ 2-point correlator.
We define the coordinates of the two external legs to be
$x^{\mu}=(t,\vec{x})$ and $x'^{\mu}=(t,\vec{0})$.}
\label{Figure1}
\end{figure}
After comparing (\ref{4F1}) with (\ref{Duu2}) we interchange $y$ with
$y'$ in (\ref{4F1}),
\begin{eqnarray}
&&\int\!\! d^4 y \!\int\!\! d^4 y'e^{-\vec{k}\cdot\vec{y'}}
\times\nonumber\\
&&\left\{\!\matrix{
\left\{\!\matrix{\,i\Delta_{\scriptscriptstyle ++}(t,\vec{0};y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]\cr
\,+i\Delta_{\scriptscriptstyle +-}(t,\vec{0};y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]}\!\right\}
u(t,k)u^*(y'^0,k)\cr
\hspace{-2.5cm}+\left\{\!\matrix{
\,i\Delta_{\scriptscriptstyle ++}(t,\vec{0};y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]\cr
\,+i\Delta_{\scriptscriptstyle +-}(t,\vec{0};y)
[-iM^2_{\scriptscriptstyle --}(y;y')]}\!\right\}\times\cr
\Bigl\{\theta(t\!-\!y'^0)u^*(t,k)u(y'^0,k)
+\theta(y'^0\!-\!t)u(t,k)u^*(y'^0,k)\Bigr\}
}\!\right\}.\label{4F2}
\end{eqnarray}
Here we used $i\Delta_{\pm\mp}(y;x)=i\Delta_{\mp\pm}(x;y)$
and $i\Delta_{\pm\pm}(y;x)=i\Delta_{\pm\pm}(x;y)$.
To simplify (\ref{4F2}), we combine the first line with the third
and the second line with the fourth. We then extract out the common expression
from each of the combinations. The total result in (\ref{4F2}) consists of
two parts. One of them is proportional to $\theta(t-y^0)\theta(y^0-y'^0)$
and the other to $\theta(t-y'^0)\theta(y'^0-y^0)$. We could use these
theta functions to restrict the range of temporal integrations and
give the final expressions in a more concise form denoted by $(A)$ and $(B)$,
\begin{eqnarray}
&&(A)\!=\! -\frac{\lambda^2}{2}
\!\!\int_0^{t}\!dy^0\!\!\int_0^{y^0}\!\!dy'^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!e^{-i\vec{k}\cdot\vec{y'}}
\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\!-\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]\nonumber\\
&&\times\Bigl[\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
u(t,k)u^*(y'^0,k)\!-\!\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
u^*(t,k)u(y'^0,k)\Bigr],\label{4Ftot1}\\
&&(B)\!=\!-\frac{\lambda^2}{2}
\!\!\int_0^t\!dy^0\!\!\int_{y^0}^t\!\!dy'^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!e^{-i\vec{k}\cdot\vec{y'}}
\Bigl[u(t,k)u^*(y'^0\!,k)\!-\!u^*(t,k)u(y'^0\!,k)\Bigr]\nonumber\\
&&\times\Bigl[\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr> \!-\!
\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr].\label{4Ftot2}
\end{eqnarray}
In order to compare (\ref{4Ftot1}) $+$ (\ref{4Ftot2}) with (\ref{Duu3}),
we make several reformulations of (\ref{4Ftot1}) and (\ref{4Ftot2}).
At the first step we convert mode functions to the vacuum expectation value (VEV)
of the products of two fields. Equations (\ref{4Ftot1}) and (\ref{4Ftot2}) can be
written as,
\begin{eqnarray}
&&(A)\!=\! -\frac{\lambda^2}{2}
\!\!\int_0^{t}\!dy^0\!\!\int_0^{y^0}\!\!dy'^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!\!\int\!\!d^3 x
e^{-i\vec{k}\cdot\vec{x}}\times\nonumber\\
&&\hspace{-1cm}\left\{\matrix{
\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
\Bigl<\varphi_0(t,\vec{x})\varphi_0(y')\Bigr>
\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\!-\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]\cr
\!\!+\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
\Bigl<\varphi_0(y')\varphi_0(t,\vec{x})\Bigr>
\Bigl[\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>
\!-\!\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>\Bigr]
}\!\!\right\}\!,
\label{4Ftot1a}\\
&&(B)\!=\!-\frac{\lambda^2}{2}
\!\!\int_0^t\!dy'^0\!\!\int_0^{y'^0}\!\!\!dy^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!\!\int\!\!d^3 x
e^{-i\vec{k}\cdot\vec{x}}\times\nonumber\\
&&\hspace{-1.2cm}\left\{\matrix{
\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\Bigl[\Bigl<\varphi_0(t,\vec{x})\varphi_0(y')\Bigr>
\!-\Bigl<\varphi_0(y')\varphi_0(t,\vec{x})\Bigr>\Bigr]\cr
\!\!+\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>
\Bigl[\Bigl<\varphi_0(y')\varphi_0(t,\vec{x})\Bigr>
\!-\!\Bigl<\varphi_0(t,\vec{x})\varphi_0(y')\Bigr>\Bigr]
}\!\!\right\}\!.\label{4Ftot2a}
\end{eqnarray}
Note that we have rearranged the order of the temporal integrations
in (\ref{4Ftot2a}). Before executing the second step, we introduce
two key identities,
\begin{eqnarray}
&&\Bigl<\varphi(t,\vec{x})\varphi(y^0,\vec{y})\Bigr>=
\Bigl<\varphi(t,\vec{0})\varphi(y^0,\vec{y}\!-\!\vec{x})
\Bigr>\;,\label{id1} \\
&&\Bigl<\varphi(t,\vec{x})\varphi(y^0,\vec{0})\Bigr>=
\Bigl<\varphi(t,-\vec{x})\varphi(y^0,\vec{0})\Bigr>\;.\label{id2}
\end{eqnarray}
The identity (\ref{id1}) comes from spatial translation invariance and
the identity (\ref{id2}) is the consequence of spatial rotation invariance.
At the second stage we repeatedly apply (\ref{id1}) and (\ref{id2}) to the
contribution (B) in (\ref{4Ftot2a}) and leave (\ref{4Ftot1a}) unchanged.
The first manipulation we make is,
\begin{eqnarray}
&&\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr> =
\Bigl<\varphi_0(t,\vec{0})\varphi_0(y^0,-\vec{y})\Bigr>\;,\nonumber\\
&&\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr> =
\Bigl<\varphi_0(y^0,-\vec{y})\varphi_0(t,\vec{0})\Bigr>\;,
\end{eqnarray}
and then shift the spatial coordinates for all of the terms in (\ref{4Ftot2a}),
\begin{eqnarray}
\vec{y}\longrightarrow\vec{y}+\vec{x}\;\;,\;\;
\vec{y'}\longrightarrow\vec{y'}+\vec{x}.
\end{eqnarray}
This change would not affect the range of integrations or the VEV
of $\varphi_0(y)\varphi_0(y')$. Here we only present what has been changed
by these transformations. The first part proportional to
$\!\Bigl<\!\varphi_0(y')\varphi_0(y)\!\Bigr>^2\!$ becomes,
\begin{eqnarray}
\hspace{-.5cm}\Bigl<\varphi_0(t,\vec{0})\varphi_0(y^0,-\vec{y}\!+\!\vec{x})\Bigr>
\Bigl[\Bigl<\varphi_0(t,\vec{x})\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})\Bigr>
\!-\Bigl<\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})
\varphi_0(t,\vec{x})\Bigr>\Bigr],\label{cha}
\end{eqnarray}
and the second part proportional to
$\!\Bigl<\!\varphi_0(y)\varphi_0(y')\!\Bigr>^2\!$
has been changed to,
\begin{eqnarray}
\Bigl<\varphi_0(y^0,-\vec{y}\!+\!\vec{x})\varphi_0(t,\vec{0})\Bigr>
\Bigl[\Bigl<\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})\varphi_0(t,\vec{x})\Bigr>
\!-\!\Bigl<\varphi_0(t,\vec{x})\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})
\Bigr>\Bigr].\label{chb}
\end{eqnarray}
In the next step we apply first (\ref{id2}) and then (\ref{id1})
to the first term of (\ref{cha}) and (\ref{chb}),
\begin{eqnarray}
&&\Bigl<\varphi_0(t,\vec{0})\varphi_0(y^0,-\vec{y}\!+\!\vec{x})\Bigr>
\!=\!\Bigl<\varphi_0(t,\vec{0})\varphi_0(y^0,\vec{y}\!-\!\vec{x})\Bigr>
\!=\!\Bigl<\varphi_0(t,\vec{x})\varphi_0(y^0,\vec{y})\Bigr>,\nonumber\\
&&\hspace{-.5cm}\Bigl<\varphi_0(y^0,-\vec{y}
\!+\!\vec{x})\varphi_0(t,\vec{0})\Bigr>
\!=\!\Bigl<\varphi_0(y^0,\vec{y}\!-\!\vec{x})\varphi_0(t,\vec{0})\Bigr>
\!=\!\Bigl<\varphi_0(y^0,\vec{y})\varphi_0(t,\vec{x})\Bigr>,
\end{eqnarray}
and employ spatial translation invariance (\ref{id1}) in the remaining terms of
(\ref{cha}) and (\ref{chb}). Take the final two terms of (\ref{cha}) as an
example,
\begin{eqnarray}
&&\Bigl[\Bigl<\varphi_0(t,\vec{x})\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})\Bigr>
\!-\Bigl<\varphi_0(y'^0,\vec{y'}\!+\!\vec{x})
\varphi_0(t,\vec{x})\Bigr>\Bigr]\nonumber\\
&&\longrightarrow\Bigl[\Bigl<\varphi_0(t,\vec{0})
\varphi_0(y'^0,\vec{y'})\Bigr>
\!-\Bigl<\varphi_0(y'^0,\vec{y'})\varphi_0(t,\vec{0})\Bigr>\Bigr].
\end{eqnarray}
After gathering all manipulations we made so far,
the contribution (\ref{4Ftot2a}) can be expressed as,
\begin{eqnarray}
&&(B)\!=\!-\frac{\lambda^2}{2}
\!\!\int_0^t\!dy'^0\!\!\int_0^{y'^0}\!\!\!dy^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!\!\int\!\!d^3 x
e^{-i\vec{k}\cdot\vec{x}}\times\nonumber\\
&&\hspace{-1cm}\left\{\matrix{
\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
\Bigl<\varphi_0(t,\vec{x})\varphi_0(y)\Bigr>
\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y')\Bigr>
\!-\Bigl<\varphi_0(y')\varphi_0(t,\vec{0})\Bigr>\Bigr]\cr
\!\!+\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
\Bigl<\varphi_0(y)\varphi_0(t,\vec{x})\Bigr>
\Bigl[\Bigl<\varphi_0(y')\varphi_0(t,\vec{0})\Bigr>
\!-\!\Bigl<\varphi_0(t,\vec{0})\varphi_0(y')\Bigr>\Bigr]
}\!\!\right\}\!.\label{4Ftot2b}
\end{eqnarray}
At the final step we interchange $y$ with $y'$ in (\ref{4Ftot2b}). It
turns out that the outcome is exactly the same as the one in (\ref{4Ftot1a}).
This means that the contribution $(A)$ precisely equals $(B)$.
Hence the total result could be written as $2\times(\ref{4Ftot1})$ or
$2\times(\ref{4Ftot1a})$. We choose the form which is close to
the expression derived from the mode function definition (\ref{Duu3}),
\begin{eqnarray}
&&(A)\!+\!(B)\!=\! -\frac{\lambda^2}{2}
\!\!\int_0^{t}\!dy^0\!\!\int_0^{y^0}\!\!dy'^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!e^{-i\vec{k}\cdot\vec{y'}}
\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\!-\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]\nonumber\\
&&\times\Bigl[2\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2
u(t,k)u^*(y'^0,k)\!-\!2\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2
u^*(t,k)u(y'^0,k)\Bigr].\label{4FTOT}
\end{eqnarray}
Equations (\ref{4FTOT}) and (\ref{Duu3}) both have the same
integrations and the common factor
$\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\!-\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]$
so we could just focus on the rest of the integrands.
The two integrands differ by having the factors,
\begin{eqnarray}
\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2u^*(t,k)u(y'^0,k)\;\textrm{and}\;
-\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2u(t,k)u^*(y'^0,k),
\end{eqnarray}
in equation (\ref{Duu3}) replaced with
\begin{eqnarray}
\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2 u(t,k)u^*(y'^0,k)\;\textrm{and}\;
-\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2u^*(t,k)u(y'^0,k)\;.
\end{eqnarray}
Therefore we conclude that the mode function definition disagrees with
the spatial Fourier transform of the $(-+)$ 2-point correlator
at one loop.
We close this subsection by exploring the other Schwinger-Keldysh
correlators. First of all, we summarize several key points
learned from the reduction of the $(-+)$ 2-point correlator,
\begin{itemize}
\item{The contribution (A) in (\ref{4Ftot1}) equals (B) in (\ref{4Ftot2})
implies,
\begin{eqnarray}
\lefteqn{\int\!\!d^4y\!\!\int\!\!d^4y'\Bigl[
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)\!+\!
\theta(t\!-\!y'^0)\theta(y'^0\!-\!y^0)\Bigr]}\nonumber\\
&\longrightarrow & 2\!\!\int_0^{t}\!dy^0\!\!\int_0^{y^0}\!\!\!dy'^0
\!\!\int\!\! d^3y\!\!\int\!\! d^3y'\!=2\!\int\!\!d^4y\!\!
\int\!\!d^4y'\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0).
\end{eqnarray}}
\item{The same theta functions, $\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)$
and $\theta(t\!-\!y'^0)\theta(y'^0\!-\!y^0)$,
appear as well in the $(+-)$, $(++)$ and $(--)$ correlators.}
\item{Based on these two facts, we lose nothing by imposing the
time ordering $t>y^0>y'^0$ before making any further simplification.}
\item{One can further infer,
\begin{eqnarray}
&&\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)\Bigl\{i\Delta_{\scriptscriptstyle ++}(x;y)
\!=\!i\Delta_{\scriptscriptstyle -+}(x;y)\Bigr\}\;,\label{++-+}\\
&&\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)\Bigl\{i\Delta_{\scriptscriptstyle ++}(x';y')
\!=\!i\Delta_{\scriptscriptstyle -+}(x';y')\Bigr\}\;,\label{++-+pr}\\
&&\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)\Bigl\{i\Delta_{\scriptscriptstyle --}(x;y)
\!=\!i\Delta_{\scriptscriptstyle +-}(x;y)\Bigr\}\;,\label{--+-}\\
&&\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)\Bigl\{i\Delta_{\scriptscriptstyle --}(x';y')
\!=\!i\Delta_{\scriptscriptstyle +-}(x';y')\Bigr\}\;.\label{--+-pr}
\end{eqnarray}}
\end{itemize}
Second, we employ the rules developed in the preceding paragraph.
It is convenient to display all of the distinct forms for the spatial
Fourier transform of the in-in correctors. Because each of these forms
has the same integration
$2\!\!\int\!d^3x e^{-i\vec{k}\cdot\vec{x}}\!\!\int\!\! d^4 y \!\int\!\! d^4 y'$,
it is enough to only list the integrand,\\
from the $(-+)$ 2-point correlator,
\begin{eqnarray}
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\};\label{4F-+}
\end{eqnarray}
from the $(+-)$ 2-point correlator,
\begin{eqnarray}
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle ++}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle ++}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle --}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle --}(x';y')}\!\right\};\label{4F+-}
\end{eqnarray}
from the $(++)$ 2-point correlator,
\begin{eqnarray}
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle ++}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle ++}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle ++}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\};\label{4F++}
\end{eqnarray}
from the $(--)$ 2-point correlator,
\begin{eqnarray}
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle --}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle --}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle --}(x';y')}\!\right\}.\label{4F--}
\end{eqnarray}
Applying the relations (\ref{++-+}), (\ref{++-+pr}) and (\ref{--+-pr}) to the first
two lines of (\ref{4F+-}) and the relations (\ref{++-+pr}), (\ref{--+-}) and (\ref{--+-pr})
to the bottom two lines of (\ref{4F+-}), the integrands of the $(+-)$ and $(-+)$
2-point correlators reach the same form.
The differences between (\ref{4F++}) and (\ref{4F-+}) are those 2-point
functions propagating between $x$ and $y$. They become identical after the relations
(\ref{++-+}) and (\ref{--+-}) are employed. Expression (\ref{4F--}) also differs from
(\ref{4F+-}) by the propagators between the coordinates $x$ and $y$, and they
agree with each other after the same reduction as in the previous case is employed.
What we have just observed implies that the integrands of the four Schwinger-Keldysh
correlators reach the same expression after imposing the time ordering $t>y^0>y'^0$.
An alert reader might also have noticed that enforcing relations
(\ref{++-+})-(\ref{--+-pr}) directly to each term of equations
(\ref{4F-+})-(\ref{4F--}) all gives,
\begin{eqnarray}
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\left\{\!\matrix{+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle ++}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle -+}(x;y)
[-iM^2_{\scriptscriptstyle +-}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle -+}(y;y')]
\,i\Delta_{\scriptscriptstyle -+}(x';y')\cr
+i\Delta_{\scriptscriptstyle +-}(x;y)
[-iM^2_{\scriptscriptstyle --}(y;y')]
\,i\Delta_{\scriptscriptstyle +-}(x';y')}\!\right\}.\label{4FSame}
\end{eqnarray}
Recall that equations (\ref{4F-+})-(\ref{4F--}) have the same integration,
\begin{eqnarray*}
2\!\!\int\!d^3x e^{-i\vec{k}\cdot\vec{x}}\!\!\int\!\! d^4 y \!\int\!\! d^4 y' \;.
\end{eqnarray*}
Hence spatially Fourier transforming all the 1-loop Schwinger-Keldysh correlators
gives the same answer displayed in (\ref{4FTOT}). Even making a linear combination
of them would not compensate for all the terms in (\ref{Duu3}).
Therefore we have explicitly demonstrated that the 2-point correlator definition
disagrees with the mode function definition at one loop.
One can also obtain a simple form for the difference between the
one loop correction to the 2-point correlator (\ref{4FTOT}) and
the one loop correction to the definition based on the mode function (\ref{Duu3}),
\begin{eqnarray}
&&-\frac{\lambda^2}{2}\!\!\int\!\!d^3x e^{-i\vec{k}\cdot\vec{x}}
\!\!\int\!\! d^4y\!\!\int\!\! d^4y'\!
\theta(t\!-\!y^0)\theta(y^0\!-\!y'^0)
\Bigl[\Bigl<\varphi_0(y)\varphi_0(y')\Bigr>^2\!+\!
\Bigl<\varphi_0(y')\varphi_0(y)\Bigr>^2\Bigr]\nonumber\\
&&\hspace{-.3cm}\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>\!-\!
\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]
\Bigl[\Bigl<\varphi_0(t,\vec{x})\varphi_0(y')\Bigr>\!-\!
\Bigl<\varphi_0(y')\varphi_0(t,\vec{x})\Bigr>\Bigr].\label{diff}
\end{eqnarray}
Also note that this difference involves two retarded Green's
functions\footnote{ (\ref{diff}) seems to have a similar structure as
equation (4.39) in \cite{TensorDiv} and (C18) in \cite{HRV} for the in-in correlator of
the metric perturbations including loop corrections from matter fields. It corresponds to
the contribution named as ``induced fluctuations''.},
\begin{eqnarray}
&&-i\theta(t\!-\!y^0)\Bigl[\Bigl<\varphi_0(t,\vec{0})\varphi_0(y)\Bigr>
\!-\!\Bigl<\varphi_0(y)\varphi_0(t,\vec{0})\Bigr>\Bigr]\;\nonumber\\
&&-i\theta(t\!-\!y'^0)\Bigl[\Bigl<\varphi_0(t,\vec{x})\varphi_0(y')\Bigr>\!-\!
\Bigl<\varphi_0(y')\varphi_0(t,\vec{x})\Bigr>\Bigr].\label{2Gret}
\end{eqnarray}
\section{Epilogue}
We have analyzed loop corrections to two different ways of defining the primordial
power spectra which happen to coincide at tree order (\ref{treeid}). One of these
definitions involves the norm squared of the mode functions (\ref{modedef}). It can
be generalized by using the linearized Schwinger-Keldysh effective field equation
to quantum correct the mode function. The other definition involves the spatial
Fourier transform of the 2-point correlator (\ref{VEVdef}), which is generalized by
simply computing the correlator to higher orders using the Schwinger-Keldysh formalism.
To simplify our analysis we employed $\varphi^3$ theory in flat space.
The fact that its interactions have the same topology as those of inflationary cosmology
justifies this simplification. What we have found is that
the two definitions do not agree even at one loop order.
The 2-point correlator definition has the advantage of representing each power
spectrum in terms of a single expectation value. However, it has been claimed that
the coincident limit of its spatial Fourier transform is singular in the gravitational
case \cite{TensorDiv}. That implies that this quantity suffers from a new sort of
ultraviolet divergence, beyond the usual ones which BPHZ renormalization
absorbs. No one currently understands how to remove this new divergence;
at a minimum it would require a composite operator renormalization. This means
that the tensor power spectrum, for example, cannot be based on the
expectation value of $h_{ij}(t,\vec{x}) h_{ij}(t,\vec{0})$ but rather this plus
some higher order operator with which its one loop corrections mix. Even more disturbing,
from the perspective of cosmology, this new divergence arises
from LATE TIME correlations between fluctuations of matter fields, rather than
from anything that happened during primordial inflation. We believe
these late time effects should be removed, the same way one edits out
infrared radiation from Jupiter, the galactic plane, and other known sources,
and the same way the observed spectrum --- with its acoustic oscillations ---
is fitted using the late time transfer function to infer the almost
perfectly scale invariant primordial spectrum.
It is therefore reasonable to pursue alternatives to the usual definition
of the power spectrum. The mode function definition, which can not
be expressed as a single VEV, does seem strange at first, but a closer look
reveals some advantages. First, it is free of the late time artifacts once
the appropriate 1PI 2-point function has been renormalized.
Second, it is arguably a reasonable translation of what we should be doing.
CMB photons do not acquire their redshifts at the surface of last scattering
but rather by propagating through the perturbed geometry between the surface
of last scattering and the late time observer. The original computation by Sachs
and Wolfe \cite{SaWo} expresses the temperature fluctuation as an integral of
the metric perturbations along the photon's worldline. The time evolution for
these metric perturbations comes from solving the linearized equations in
the background geometry of late times, but the initial conditions come from
primordial inflation. It seems at least as reasonable to take these
initial conditions from the quantum corrected mode functions as
from the correlator.
What we are interested in is what theoretical objects represent the
``primordial power spectra''. How we define the power spectra is still
an open question. Loop corrections to them might be observable in the far future,
assuming that theorists can find a unique model of inflation to fix
the tree order prediction, that astronomers can measure the matter power spectra
in 3 dimensions, and that astrophysicists can develop expertise needed to extract the
primordial signal from foregrounds.
\centerline{\bf Acknowledgements}
We have profited from conversations on this subject with M. Fr\"{o}b,
A. Roura and R. P. Woodard. S.P.M. was supported by NWO Veni Project
\# 680-47-406. S.P. is supported by the Eberly Research Funds of The
Pennsylvania State University. The Institute for Gravitation and the Cosmos
is supported by the Eberly College of Science and the Office of the Senior
Vice President for Research at the Pennsylvania State University.
S.P.M. is grateful for the hospitality of the University of Crete
where the final draft was prepared.
S.P. acknowledges the hospitality of the University of Utrecht
where the main computation was conducted.
\begin{appendices}
\section{The canonical relation for the mode function}\label{cano-mode}
In this section we elucidate the relation between the Heisenberg operator and
its mode function using the cubic interaction in our worked-out example. We start with
solving for the field operator perturbatively from the equation of motion and then
compute the expectation value of its commutator with the creation operator.
The section closes by giving the canonical relation for the mode function
which is solved from the Schwinger-Keldysh effective field equation.
The equation of motion for the case we consider (\ref{3scalar}) is,
\begin{eqnarray}
\partial^2\varphi=\frac{\lambda}{2}\varphi^2.
\end{eqnarray}
Solving for the field operator perturbatively means,
\begin{eqnarray}
&&\varphi=\varphi_{\scriptscriptstyle 0}+\lambda\varphi_{\scriptscriptstyle 1}
+\lambda^2\varphi_{\scriptscriptstyle 2}+O(\lambda^3),
\end{eqnarray}
and the field operators at the order of $\lambda^0$, $\lambda^1$ and $\lambda^2$ obey,
\begin{eqnarray}
\partial^2\varphi_{\scriptscriptstyle 0}=0\;\;,\;\;
\partial^2\varphi_{\scriptscriptstyle 1}=
\frac{1}{2}\varphi_{\scriptscriptstyle 0}^{2}\;\;,\;\;
\partial^2\varphi_{\scriptscriptstyle 2}=\frac{1}{2}
\Bigl[\varphi_{\scriptscriptstyle 0}\varphi_{\scriptscriptstyle 1}
\!+\!\varphi_{\scriptscriptstyle 1}\varphi_{\scriptscriptstyle 0}\Bigr].
\end{eqnarray}
Note that we consider the initial value data to be zeroth order so that
$\varphi_{\scriptscriptstyle 1}$ and $\varphi_{\scriptscriptstyle 2}$ etc.
all vanish at $t=0$, as do their first time derivatives.
Hence $\varphi_{\scriptscriptstyle 1},\varphi_{\scriptscriptstyle 2}$
can be expressed in terms of $\varphi_{\scriptscriptstyle 0}$,
\begin{eqnarray}
&&\varphi_{\scriptscriptstyle 1}(x)\!=\!\frac12\!\int\!\!d^4y G_{\scriptscriptstyle Ret}(x;y)
\varphi_{\scriptscriptstyle 0}^2(y),\nonumber\\
&&\hspace{-.4cm}\varphi_{\scriptscriptstyle 2}(x)\!=\!\frac14\!\int\!\!d^4y\!\!\int\!\!d^4y'
G_{\scriptscriptstyle Ret}(x;y)G_{\scriptscriptstyle Ret}(y;y')
\Bigl[\varphi_{\scriptscriptstyle 0}(y)\varphi^2_{\scriptscriptstyle 0}(y')
\!+\!\varphi^2_{\scriptscriptstyle 0}(y')\varphi_{\scriptscriptstyle 0}(y)\Bigr].
\end{eqnarray}
Even though this theory is not free, we can still organize
the initial values of the full field and its first time derivative
in terms of free creation and annihilation operators,
\begin{eqnarray}
\alpha(\vec{k})\!\equiv\!\frac{\dot{u}^*(0)\widetilde{\varphi}(0,\vec{k})
\!-\!u^*(0)\dot{\widetilde{\varphi}}(0,\vec{k})}{u(0)\dot{u}^*(0)-\dot{u}(0)u^*(0)}
\;;\;\alpha^{\dagger}(\vec{k})\!\equiv\!
\Biggl[\frac{\dot{u}^*(0)\widetilde{\varphi}(0,\vec{k})
\!-\!u^*(0)\dot{\widetilde{\varphi}}(0,\vec{k})}
{u(0)\dot{u}^*(0)-\dot{u}(0)u^*(0)}\Biggr]^{\dagger}\!\!.\;
\end{eqnarray}
These operators define the free ground state $\vert \Omega \rangle$,
\begin{eqnarray}
\alpha|\Omega\Bigr>=0=\Bigl<\Omega|\alpha^{\dagger}\;\;\;\;,\;\;\;\;
\Bigl<\Omega|\Omega\Bigr>=1\,.
\end{eqnarray}
The mode function is the matrix element of the full field between
the $t=0$ free vacuum and the $t=0$ free one particle state,
$\langle \Omega \vert \varphi \alpha^{\dagger} \vert \Omega \rangle$.
We first commute the $\alpha^{\dagger}$ through the $\varphi$,
\begin{eqnarray}
&&\Bigl[\varphi(t,\vec{x}), \alpha^{\dagger}\Bigr]
=\Bigl[\varphi_{\scriptscriptstyle 0}(t,\vec{x})
\!+\!\lambda\varphi_{\scriptscriptstyle 1}(t,\vec{x})\!+\!\lambda^2
\varphi_{\scriptscriptstyle 2}(t,\vec{x})
\!+\!O(\lambda^3), \alpha^{\dagger}\Bigr]\nonumber\\
&&=\Phi_{\scriptscriptstyle 0}(t,\vec{x})\!+\!\lambda\!\!\int\!\!d^4y
G_{\scriptscriptstyle Ret}(x;y)\varphi_{\scriptscriptstyle 0}(y)\Phi_{\scriptscriptstyle 0}(y)
\!+\!\frac{\lambda^2}{2}\!\!\int\!\!d^4y\!\!\int\!\!d^4y'G_{\scriptscriptstyle Ret}(x;y)
G_{\scriptscriptstyle Ret}(y;y')\nonumber\\
&&\hspace{3cm}\times\Biggl\{\varphi_{\scriptscriptstyle 0}^2(y')\Phi_{\scriptscriptstyle 0}(y)\!+\!
\Bigl[\varphi_{\scriptscriptstyle 0}(y)\varphi_{\scriptscriptstyle 0}(y')
\!+\!\varphi_{\scriptscriptstyle 0}(y')\varphi_{\scriptscriptstyle 0}(y)\Bigr]
\Phi_{\scriptscriptstyle 0}(y')\Biggr\},\label{commutator}
\end{eqnarray}
where $\Phi_{\scriptscriptstyle 0}(t,\vec{x})$ is a c-number,
\begin{eqnarray}
\Phi_{\scriptscriptstyle 0}(t,\vec{x})
=\Bigl[\varphi_{\scriptscriptstyle 0}(t,\vec{x}), \alpha^{\dagger}\Bigr]
=u(t,k)e^{i\vec{k}\cdot\vec{x}}.\label{Phi0}
\end{eqnarray}
Taking the expectation value of equation (\ref{commutator}) actually simplifies
the expression because the second term with a single integral vanishes and
the first term of the final line is properly by a vacuum shift. For our purpose
only the last two terms matter,
\begin{eqnarray}
&&\hspace{-1cm}\Bigl<\Omega|\Bigl[\lambda^2\varphi_{\scriptscriptstyle 2},
\alpha^{\dagger}\Bigr]|\Omega\Bigr>\!=\!\frac{\lambda^2}{2}\!\!\int\!\!d^4y\!\!\int\!\!d^4y'
G_{\scriptscriptstyle Ret}(x;y)G_{\scriptscriptstyle Ret}(y;y')
\Bigl[i\Delta_{\scriptscriptstyle -+}\!+\!i\Delta_{\scriptscriptstyle +-}
\Bigr](y;y')\Phi_{\scriptscriptstyle 0}(y')\nonumber\\
&&\hspace{.8cm}=\!-i\frac{\lambda^2}{2}\!\!\int\!\!d^4y\!\!\int\!\!d^4y'\theta(y^0\!\!-\!y'^0)
G_{\scriptscriptstyle Ret}(x;y)\Bigl[i\Delta^2_{\scriptscriptstyle -+}\!-\!
i\Delta^2_{\scriptscriptstyle +-}\Bigr](y;y')\Phi_{\scriptscriptstyle 0}(y')\,.
\label{cano-op}
\end{eqnarray}
Here the second equality is obtained by employing equation (\ref{Gret}).
Our quantum corrected mode function comes from solving the linearized effective field
equation (\ref{u1eqn}). For convenience, we re-write equation (\ref{u1sol}) as,
\begin{eqnarray}
\Delta u(t,k)e^{i\vec{k}\cdot\vec{x}}=\!\!\int\!\! d^4 y
G_{\scriptscriptstyle{Ret}}(x;y)\!\int\!\! d^4 y'[M^2_{\scriptscriptstyle ++}
\!+ M^2_{\scriptscriptstyle +-}](y;y')u(y'^0,k)e^{i\vec{k}\cdot\vec{y'}}\;,
\label{u1sol1a}
\end{eqnarray}
and applying (\ref{M2Delta2}) and (\ref{Phi0}) to (\ref{u1sol1a}) gives,
\begin{eqnarray}
\hspace{-.5cm}\Delta u(t,k)e^{i\vec{k}\cdot\vec{x}}\!=\!-i\frac{\lambda^2}{2}
\!\!\int\!\! d^4 y\!\int\!\! d^4 y'\theta(y^0\!\!-\!y'^0)G_{\scriptscriptstyle{Ret}}(x;y)
\Bigl[i\Delta^2_{\scriptscriptstyle -+}
\!-\! i\Delta^2_{\scriptscriptstyle +-}\Bigr](y;y')
\Phi_{\scriptscriptstyle 0}(y').\label{modeexact}
\end{eqnarray}
Comparing equation (\ref{cano-op}) with (\ref{modeexact}) demonstrates the
canonical relation for the quantum corrected mode function,
\begin{eqnarray}
\Delta u(t,k)e^{i\vec{k}\cdot\vec{x}}\!=\!
\Bigl<\Omega|\Bigl[\lambda^2\varphi_{\scriptscriptstyle 2}(t,\vec{x}),
\alpha^{\dagger}\Bigr]|\Omega\Bigr>\;.\label{modeOP}
\end{eqnarray}
\section{The diagram topology for quantum corrections to the mode function}\label{topology}
\begin{figure}
\begin{center}
\includegraphics[width=4cm]{tadpole2.eps}
\end{center}
\caption{This diagram characterizes the partial topology of the mode function (\ref{u1sol1a}).
The external leg denoted by a dashed line comes from the retarded Green's function whereas
the remaining part is the 2-point 1PI diagram.}
\label{Figure2}
\end{figure}
In the preceding section we have derived the relation (\ref{modeOP}) between the quantum-corrected
mode function and the canonical formalism. This also indicates that the expectation value of the commutator
of the field and the creation operator has the same topology as the mode function \cite{MW}.
The diagrammatic expression is depicted in Fig.~\ref{Figure2}. The main purpose of this section is to
show that the mode function definition and the 2-point correlator definition share the same
topology.
The diagram for the usual definition, the spatial Fourier transform of the 2-point correlator,
looks like,
\begin{eqnarray}
\!\int\!\!d^3x e^{i\vec{k}\cdot\vec{x}}\times\Bigl(\textrm{Fig.~1}\Bigr),
\end{eqnarray}
whereas the diagram for the $\Delta u(t,k)u^{*}(t,k)$\footnote{One can see that the phase terms
in equation (\ref{Phi0}) and (\ref{modeOP}) cancel out.}
part of the mode function definition is,
\begin{eqnarray}
\Bigl(\textrm{Fig.~2}\Bigr)\times u(y'^0,k)e^{i\vec{k}\cdot\vec{y'}}\times u^{*}(t,k).
\label{modediagram}
\end{eqnarray}
Note that the final three components in equation (\ref{modediagram}) can be re-written as,
\begin{eqnarray}
e^{i\vec{k}\cdot\vec{y'}}u(y'^0,k)u^{*}(t,k)\!=\!
\!\int\!\!d^3x' e^{i\vec{k}\cdot\vec{x'}}i\Delta_{\scriptscriptstyle -+}(y';x').
\end{eqnarray}
Actually each single propagator with different Schwinger-Keldysh polarity takes the form of
the linear combination of $uu^{*}$ and $u^{*}u$ multiplying by a distinct theta function.
Hence the mode function definition has the diagrammatic form,
\begin{eqnarray}
\!\int\!\!d^3x' e^{i\vec{k}\cdot\vec{x'}}\times \Bigl(\textrm{Fig.~1}\Bigr).\label{modeFunTopology}
\end{eqnarray}
Therefore we conclude that the generic diagram topology of the mode function definition is
identical to that of the correlator definition.
\section{K\"{a}llen Representation}
In this subsection we will point out the necessary conditions for K\"{a}llen representation
to hold through deriving it step by step. Although this familiar material has been covered
in a textbook of quantum field theory, the purpose here is to remind readers that the familiar
representation becomes nontrivial when applying it to FRW background in a theory without a
mass gap.
In an interacting theory the 2-point correlation function is,
\begin{eqnarray}
\Bigl<\Omega|\phi(x)\phi(y)|\Omega\Bigr>.\label{correlator}
\end{eqnarray}
It is free to insert a partition of unity between the two field operators,
\begin{eqnarray}
\textrm{I}=|\Omega\Bigr>\Bigl<\Omega|+\!\!\int\!\!\frac{d^3k}{(2\pi)^3}\frac{1}{2\omega}
|k\Bigr>\Bigl<k|+\!\!\int\!\!\frac{d^3k_1}{(2\pi)^3}\frac{1}{2\omega_1}
\!\!\int\!\!\frac{d^3k_2}{(2\pi)^3}\frac{1}{2\omega_2}|k_1k_2\Bigr>\Bigl<k_1k_2|+\cdots.\,
\end{eqnarray}
The 2-point correlation function (\ref{correlator}) can be written as ,
\begin{eqnarray}
&&\Bigl<\Omega|\phi(x)\phi(y)|\Omega\Bigr>=\Bigl<\Omega|\phi(x)|\Omega\Bigr>
\Bigl<\Omega|\phi(y)|\Omega\Bigr>+\!\!\int\!\!\frac{d^3k}{(2\pi)^3}\frac{1}{2\omega}
\Bigl<\Omega|\phi(x)|k\Bigr>\Bigl<k|\phi(y)|\Omega\Bigr>\nonumber\\
&&\hspace{2.5cm}+\!\!\int\!\!\frac{d^3k_1}{(2\pi)^3}\frac{1}{2\omega_1}
\!\!\int\!\!\frac{d^3k_2}{(2\pi)^3}\frac{1}{2\omega_2}
\Bigl<\Omega|\phi(x)|k_1k_2\Bigr>\Bigl<k_1k_2|\phi(y)|\Omega\Bigr>+\cdots.
\label{correlator1}
\end{eqnarray}
The first term of (\ref{correlator1}) can be subtracted from $\phi$ so the VEV of it
is zero if the quantum field has been properly defined,
\begin{eqnarray}
\Bigl<\Omega|\phi(x)|\Omega\Bigr>=\phi_0,
\end{eqnarray}
and the resulting sum begins with the contribution from 1-particle states.
Had a theory possessed poincar\'{e} symmetry, we can implement the following steps,
\begin{eqnarray}
\Bigl<\Omega|\phi(x)|k\Bigr>=e^{-ik^{\mu}x_{\mu}}\Bigl<\Omega|\phi(0)|k\Bigr>
=e^{-ik^{\mu}x_{\mu}}\Bigl<\Omega|\phi(0)|\vec{0}\Bigr>\equiv
\sqrt{Z}e^{-ik^{\mu}x_{\mu}}.\label{keyStep }
\end{eqnarray}
The first equality is from the spacetime\footnote{The metric convention here is
$+---$ rather than $-+++$ in general relativity or cosmology.} translation invariance of
the 3-momentum state $|k\Bigr>$ and the vacuum $\Bigl<\Omega|$ whereas
the second equality is due to the Lorentz boost invariance of $\phi(0)$ and
the vacuum $\Bigl<\Omega|$. This is not a problem at all for a quantum field theory
with a mass gap in a flat background. The first nonzero term of (\ref{correlator1}) is,
\begin{eqnarray}
\int\!\!\frac{d^3k}{(2\pi)^3}\frac{1}{2\omega}Ze^{-ik^{\mu}(x-y)_{\mu}}
=\!\int\!\!\frac{d^4k}{(2\pi)^4}\frac{iZe^{-ik^{\mu}(x-y)_{\mu}}}
{k^2\!-\!m^2+i\epsilon}
\end{eqnarray}
Here we have assumed $x^{0}>y^{0}$ for convenience. Therefore we reach the familiar
expression,
\begin{eqnarray}
\Bigl<\Omega|\phi(x)\phi(y)|\Omega\Bigr>=\!\int\!\!\frac{d^4k}{(2\pi)^4}
e^{-ik^{\mu}(x-y)_{\mu}}\Biggl\{\frac{iZ}{k^2\!-\!m^2+i\epsilon}
+\sum\frac{iZ(\mu)}{k^2\!-\!\mu^2+i\epsilon}\Biggr\}.\label{correlator2}
\end{eqnarray}
The first term of (\ref{correlator2}) is the 1-particle contribution and the second
term is the multi-particle continuation.
The primordial power spectrum is computed in FRW geometry which only possesses spatial
rotation and spatial translation invariance rather than poincar\'{e} symmetry. Hence
the 2-point correlation function is not able to reach the same expression as
(\ref{correlator2}). As a result, there are no exact 1-particle states, nor even
exact energy eigenstates. Nor is the VEV of the field zero, or even constant. Furthermore
the graviton in FRW geometry experiences much more severe IR divergences than in flat background
because a rapid spacetime expansion makes IR divergence much stronger\footnote{The Bloch-Nordesick
procedure for a massless theory in flat spacetime cannot entirely cure
the IR problem in cosmology.}. So it is also impossible to take the initial state back to
a distant infinity and to define the unique vacuum as we did for a theory with a mass gap
in flat spacetime.
Our toy model is in flat space background. However, because it is massless, $\varphi^3$ theory,
the vacuum decays and it doesn't have either time translation invariance or boost invariance
\cite{GV}. It would indeed have an IR divergence if the computation had been done
in 4-momentum space. However, in position space in the Schwinger-Keldysh (in-in) formalism
there is no IR divergence as long as the state is released at a finite time. What we would instead
get is a growth of the VEV of the field \cite{TW3}.
Based on the arguments being discussed in the preceding paragraphs, the mode function definition
cannot recover the 1-particle states in k\"{a}llen representation either for quantum gravity
in cosmology or for our toy model in flat spacetime. We also can see this from the diagram topology.
The 1-particle contribution has infinite corrections from summing up a series of the 2-point
correlator with more and more 1PI insertions as being shown in equation (7.43) of \cite{Peskin}.
The topology of the mode function definition has been derived in equation (\ref{modeFunTopology})
which shares the same topology as the spatial Fourier transform of the 2-point correlator.
The two definitions both do not receive infinite corrections as the 1-particle states do.
If the theory is poincar\'{e} invariant and has a mass gap, then we would get the same answer in
both the in-out and in-in formalisms by taking the initial time to minus infinity. However,
the subtle points the two definitions disagree at loop orders are that when there are particle
productions (which there is for cosmology and for massless, $\varphi^3$ even in a flat space)
and when we cannot take the initial time to minus infinity.
\section{The issues of divergences}
In this subsection we renormalize the ultraviolet divergences of the two definitions in our toy
model using a mass counterterm. For further clarification of the renormalized results we have
emphasized the distinction between infrared divergences and secular growth. Finally, we discuss
the extra, composite operator divergence which can occur in a model with derivative interactions,
owing to the fact that the two times coincide.
\subsection{The power spectrum from the mode function definition for $\varphi^3$ theory}
Before computing the lowest order correction to the power spectrum from the mode function definition
(\ref{modedef}), we need to obtain the first order correction to the mode function using (\ref{u1eqn}).
Instead of employing the formal expression (\ref{u1sol}) and (\ref{Duu3}), the best way to get
the finite result is to remove the ultraviolet divergence of the self-mass squared and then
integrate the finite part against the tree order mode function. Even though the last step to solve for
$\Delta u(t,k)$ from (\ref{u1eqn}) still requires one more integral coming from the retarded Green's function,
it is actually not so hard to perform because it only involves with a temporal integration.
Recall that the primitive part of the one loop self-mass squared in $\varphi^3$ theory is,
\begin{eqnarray}
-iM^2_{\scriptscriptstyle +\pm}(x;x')=\mp\frac{\lambda^2}{2}
\frac{\Gamma^2(\frac{D}{2}\!-\!1)}{16\pi^{D}}
\frac{1}{\Delta x_{\scriptscriptstyle +\pm}^{2D-4}(x;x')},\label{Msquare}
\end{eqnarray}
where $\Delta x^2_{\scriptscriptstyle +\pm}(x;x')$ is defined in (\ref{x++}) and (\ref{x+-}). Note that
integrating expression (\ref{Msquare}) with respect to $x^{\prime \mu}$ in $D=4$ dimensions would
produce a logarithmic divergence due to the singularity at $x^{\prime \mu} = x^{\mu}$. We can make the
expression integrable by extracting a d'Alembertian with respect to $x^{\mu}$,
\begin{eqnarray}
\int\!\!d^4x'\frac{1}{\Delta x_{\scriptscriptstyle +\pm}^{2D-4}(x;x')}
=\frac{1}{2(D\!-\!3)(D\!-\!4)}\partial^2\!\!\int\!\! d^4x'
\frac{1}{\Delta x_{\scriptscriptstyle +\pm}^{2D-6}(x;x')}.\label{extract}
\end{eqnarray}
The remaining obstacle to taking the $D \rightarrow 4$ limit is of course the explicit factor of
$1/(D-4)$ in expression (\ref{extract}).
The next step is to segregate the divergence into a local delta function by adding zero in the form,
\begin{eqnarray}
\partial^{2} \Bigl[\frac{1}{\Delta x^{D-2}_{\scriptscriptstyle ++}} \Bigr] - \frac{i 4
\pi^{\frac{D}{2}} \delta^D(x \!-\! x')}{\Gamma(\frac{D}{2}\!-\!1)} = 0 =
\partial^{2} \Bigl[\frac{1}{\Delta x^{D-2}_{\scriptscriptstyle +-}} \Bigr] \label{0}.
\end{eqnarray}
(For simplicity, we here and henceforth suppress the two arguments of the coordinate separation
$\Delta x^2(x;x')$.) We can then take the $D \rightarrow 4$ limit of the nonlocal part,
leaving the divergence restricted to the delta function. For the $++$ case the result is,
\begin{eqnarray}
\lefteqn{\frac{1}{(D\!-\!4)} \Biggl\{ \partial^2 \Bigl[\frac{1}{\Delta x^{2D-6}_{
\scriptscriptstyle ++}} - \frac{\mu^{2D-4}}{\Delta x^{D-2}_{\scriptscriptstyle ++}}
\Bigr] + \frac{\mu^{D-4} i 4 \pi^{\frac{D}{2}}}{\Gamma(\frac{D}{2}\!-\!1)}
\delta^D(x\!-\!x') \Biggr\} } \nonumber \\
& & = \frac{\mu^{D-4}}{(D\!-\!4)}\frac{i4\pi^{\frac{D}{2}}}{\Gamma(\frac{D}{2}\!-\!1)}
\delta^D(x\!-\!x') -\frac{\partial^2}{2} \Biggl[\frac{\ln(\mu^2\Delta x^2_{\scriptscriptstyle ++})}
{\Delta x^2_{\scriptscriptstyle ++}}\Biggr] + O(D \!-\! 4) . \label{segregate}
\end{eqnarray}
At this point it is clear that the divergent part of the $++$ self-mass squared is,
\begin{eqnarray}
\frac{-i\lambda^2}{2^4\pi^{\frac{D}{2}}}\frac{\Gamma(\frac{D}{2}\!-\!1)}{(D\!-\!3)}
\frac{\mu^{D-4}}{(D\!-\!4)}\delta^D(x\!-\!x'),\label{massCTT}
\end{eqnarray}
and it can be absorbed by a mass counterterm.\footnote{Recall that $\lambda$ has the dimension
of mass in $\varphi^3$ theory so mass is not multiplicatively renormalized.} Because there is no
delta function for the $+-$ term in expression (\ref{0}), the $+-$ self-mass squared has no
ultraviolet divergence. This accords with the fact that the Schwinger-Keldysh formalism has no
counterterms with mixed $\pm$ polarities \cite{CSHY,RDJ,FW}.
It is simpler to perform the integral (\ref{u1eqn}) by extracting one more d'Alembertian,
\begin{eqnarray}
\Biggl[\frac{\ln(\mu^2\Delta x^2_{\scriptscriptstyle +\pm})}{\Delta x^2_{\scriptscriptstyle +\pm}}\Biggr]
=\frac{\partial^2}{8}\Biggl\{\ln^2(\mu^2\Delta x^2_{\scriptscriptstyle +\pm})-
2\ln(\mu^2\Delta x^2_{\scriptscriptstyle +\pm})\Biggr\} . \label{extract1}
\end{eqnarray}
With the two simplifications,
\begin{eqnarray}
\ln(\mu^2\Delta x^2_{\scriptscriptstyle +\pm})=\theta(t\!-\!t')\theta(\Delta t\!-\!\Delta \overline{x})
\Biggl\{\ln[\mu^2(\Delta t^2\!-\!\Delta \overline{x}^2)]\pm i\pi\Biggr\} ,
\end{eqnarray}
the finite part can be written as,
\begin{eqnarray}
M^2_{\scriptscriptstyle ++}+M^2_{\scriptscriptstyle +-}=\frac{-\lambda^2}{2^8\pi^3}\partial^4\Biggl\{
\theta(t\!-\!t')\theta(\Delta t\!-\!\Delta \overline{x})\Biggl(
\ln[\mu^2(\Delta t^2\!-\!\Delta \overline{x}^2)]-1\Biggr)\Biggr\}.\label{massfinite}
\end{eqnarray}
Here $\Delta t^2$ and $\Delta \overline{x}^2$ are defined as $(t-t')^2$ and $||\vec{x}-\vec{x'}||^2$ respectively.
At this stage we are ready to integrate (\ref{massfinite}) against the tree order mode function,
\begin{eqnarray}
&&\int\!\!d^4x'\Bigl[M^2_{\scriptscriptstyle ++}(x;x')+M^2_{\scriptscriptstyle +-}(x;x')\Bigr]
u(t',k)e^{i\vec{k}\cdot\vec{x'}}\nonumber\\
&&\hspace{-.5cm}=\frac{-\lambda^{2}e^{i\vec{k}\cdot\vec{x}}}{2^8\pi^3}
\partial^4\!\!\int_0^t\!dt'\!\!\int_0^{\Delta t}\!\!drr^2d\Omega
\Biggl\{\ln[\mu^2(\Delta t^2\!-\!\Delta \overline{x}^2)]-1\Biggr\}
\frac{e^{-ikt'+i\vec{k}\cdot\vec{r}}}{\sqrt{2k}}.
\end{eqnarray}
Here we set $\vec{r}$ as $\vec{x'}-\vec{x}$. After executing the angular integration and change
the variable $r=|\vec{r}|=z\Delta t$, the expression can be simplified,
\begin{eqnarray}
\hspace{-1cm}\frac{-\lambda^2}{2^6\pi^2}\frac{e^{i\vec{k}\cdot\vec{x}}}{k}
(\partial_0^2\!+\!k^2)^2\!\!\int_0^t\!\!dt'\frac{e^{-ikt'}}{\sqrt{2k}}\Delta t^2\!\!
\int_0^1\!\!\!dzz\sin(kz\Delta t)
\Biggl\{2\ln(\mu\Delta t)\!+\!\ln(1\!\!-\!z^2)\!-\!1\Biggr\}\!.
\end{eqnarray}
To perform the $z$ integration we employ several special functions,
\begin{eqnarray}
&&\textrm{Si}(x)\equiv-\!\!\int_{x}^{\infty}\!\!dt\frac{\sin(t)}{t}=
-\frac{\pi}{2}+\!\!\int_0^x\!\!dt\frac{\sin(t)}{t};\nonumber\\
&&\textrm{Ci}(x)\equiv-\!\!\int_x^{\infty}\!\!dt\frac{\cos(t)}{t}=\gamma+\ln(x)
+\!\!\int_0^x\!\!dt\Biggl[\frac{\cos(t)\!-\!1}{t}\Biggr];\nonumber\\
&&\hspace{-.5cm}\xi(\alpha)\equiv\!\!\int_0^1\!dzz\sin(\alpha z)\ln(1\!-\!z^2)
=\frac{1}{\alpha^2}\Biggl\{2\sin(\alpha)\!-\!\Bigl[\cos(\alpha)\!+\!\alpha\sin(\alpha)\Bigr]
\Bigl[\textrm{Si}(2\alpha)\!+\!\frac{\pi}{2}\Bigr]\nonumber\\
&&\hspace{5cm}+\Bigl[\sin(\alpha)\!-\!\alpha\cos(\alpha)\Bigr]\Bigl[\textrm{Ci}(2\alpha)\!-\!
\gamma\!-\!\ln(\frac{\alpha}{2})\Bigr]\Biggr\}.
\end{eqnarray}
With these the renormalized result can be expressed as,
\begin{eqnarray}
\hspace{-.3cm}\frac{-\lambda^2}{2^6\pi^2}\frac{e^{i\vec{k}\cdot\vec{x}}}{k^3}
(\partial_0^2\!+\!k^2)^2\!\!\int_0^t\!\!dt'\frac{e^{-ikt'}}{\sqrt{2k}}
\Biggl\{\!\alpha^2\xi(\alpha)\!+\!\Bigl[2\ln(\frac{\mu\alpha}{k})\!-\!1\Bigr]
\Bigl[\sin(\alpha)\!-\!\alpha\cos(\alpha)\Bigr]\!\Biggr\}\!. \label{xiform}
\end{eqnarray}
Here $\alpha$ is $k\Delta t$.
Because the integrand of (\ref{xiform}) behaves like $\Delta t^3\ln(\Delta t)$ near
$t'\!=\!t$ we can pass three of four derivatives through the integral sign to simplify
the integrand. Passing the first two derivatives through gives,
\begin{eqnarray}
&&\hspace{-1.7cm}\frac{-\lambda^2}{2^5\pi^2}\frac{e^{i\vec{k}\cdot\vec{x}}}{k}
(\partial_0\!+\!ik)(\partial_0\!-\!ik)\!\!\int_0^t\!\!dt'\frac{e^{-ikt'}}{\sqrt{2k}}
\Biggl\{\!-\cos(\alpha)\!\!\int_0^{2\alpha}\!\!ds\frac{\sin(s)}{s}\nonumber\\
&&\hspace{3.7cm}+\sin(\alpha)\Biggl[\int_0^{2\alpha}\!\!ds\frac{\cos(s)\!-\!1}{s}
\!+\!2\ln(\frac{2\mu\alpha}{k})\Biggr]\Biggr\}.
\end{eqnarray}
Extracting the temporal phase factor and passing one more derivative through the
integral gives,
\begin{eqnarray}
&&\hspace{-1.3cm}\frac{-\lambda^2}{2^5\pi^2}e^{i\vec{k}\cdot\vec{x}}(\partial_0\!+\!ik)
\!\!\int_0^t\!\!dt'\frac{e^{-ikt'}}{\sqrt{2k}}e^{-ik\Delta t}\Biggl[
\!\int_0^{2\alpha}\!\!ds\frac{e^{is}-1}{s}\!+\!2\ln(2\mu\Delta t)\Biggr]\nonumber\\
&&\hspace{-.5cm}=\frac{-\lambda^2}{2^5\pi^2}\frac{e^{-ikt+i\vec{k}\cdot\vec{x}}}{\sqrt{2k}}
\partial_0\!\int_0^t\!\!d\Delta t\times 1\times\Biggl\{\Biggl[\!\int_0^{2k\Delta t}\!\!ds
\frac{e^{is}-1}{s}\Biggr]\!+\!2\ln(2\mu\Delta t)\Biggr\}.
\end{eqnarray}
Here we have used $(\partial_0+ik)e^{-ikt} = 0$. Further simplification can be accomplished
by performing the $\Delta t$ integration and acting the final derivative. The final
result is,
\begin{eqnarray}
\frac{-\lambda^2}{2^5\pi^2}\frac{e^{-ikt+i\vec{k}\cdot\vec{x}}}{\sqrt{2k}}
\Biggl\{\!\int_0^{2kt}\!\!ds\frac{e^{is}-1}{s}\!+\!2\ln(2\mu t)\Biggr\}
\equiv-S(t)e^{i\vec{k}\cdot\vec{x}}.
\end{eqnarray}
According to (\ref{u1eqn}) the $\Delta u(t,k)$ we want to solve for obeys,
\begin{eqnarray}
&&\hspace{-1.5cm}\mathcal{D}\Bigl[\Delta u(t,k)e^{i\vec{k}\cdot\vec{x}}\Bigr]\!=\!
-(\partial_0^2\!+\!k^2)\Delta u(t,k)e^{i\vec{k}\cdot\vec{x}}\!=\!
-S(t)e^{i\vec{k}\cdot\vec{x}}\,\,\,\Longrightarrow\nonumber\\
&&(\partial_0^2\!+\!k^2)\Delta u(t,k)\!=\!S(t)\,\,\Longrightarrow\,\,
\Delta u(t,k)\!=\!\!\int_0^{\infty}\!\!dt'G_r(t,t')S(t').\label{Deltau}
\end{eqnarray}
Here $G_r(t,t')=\theta(t\!-\!t')\frac{\sin(k\Delta t)}{k}$ is the retarded Green's function.
Plugging the explicit forms of $G_r(t,t')$ and $S(t')$ into (\ref{Deltau}) gives,
\begin{eqnarray}
\Delta u(t,k)\!=\!\frac{\lambda^2}{2^5\pi^2}\frac{e^{-ikt}}{i(2k)^{\frac32}}
\!\int_0^t\!\!dt'\Bigl\{e^{2ik\Delta t}\!-\!1\Bigr\}\Biggl\{\!\int_0^{2kt'}\!\!ds
\frac{e^{is}-1}{s}\!+\!2\ln(2\mu t')\Biggr\}.
\end{eqnarray}
It remains to perform the four $t'$ integrations and collect terms. The result is,
\begin{eqnarray}
&&\hspace{-2cm}\Delta u(t,k)\!=\!\frac{\lambda^2}{2^5\pi^2}\frac{e^{-ikt}}{(2k)^{\frac32}}\Biggl\{\!
\Biggl[\frac1{2k}\!+\!it\Biggr]\!\int_0^{2kt}\!\!ds\frac{e^{is}\!-\!1}{s}\!-\!\frac{e^{2ikt}}{2k}
\!\!\int_0^{2kt}\!\!ds\frac{e^{-is}\!-\!1}{s}\nonumber\\
&&\hspace{3cm}+\Biggl[\frac{1\!-\!e^{2ikt}}{k}\!+\!2it\Biggr]\ln(2\mu t)\!-\!it\!+\!
\frac{1\!-\!e^{2ikt}}{2k}\Biggr\}.\label{FnlDeltau}
\end{eqnarray}
Combining (\ref{FnlDeltau})$\times [u^*(t,k)=\frac{e^{-ikt}}{\sqrt{2k}}]$ with
its complex conjugate gives the lowest-order correction to the power spectrum,
\begin{eqnarray}
&&\hspace{-1.5cm}\Delta u(t,k)u^*(t,k)\!+\!\textrm{c.c.}\!=\!\frac{\lambda^2}{2^7\pi^2}
\frac1{k^3}\Biggl\{\Bigl[1\!-\!\cos(2kt)\Bigr]\Bigl[1\!-\!\gamma\!+\!\textrm{Ci}(2kt)
\!+\!\ln(\frac{2\mu^2t}{k})\Bigr]\nonumber\\
&&\hspace{5cm}-\Bigl[\sin(2kt)\!+\!2kt\Bigr]\Bigl[\frac{\pi}{2}
\!+\!\textrm{Si}(2kt)\Bigr]\Biggr\}.\label{reMoSpectra}
\end{eqnarray}
\subsection{The power spectrum from the correlator definition for $\varphi^3$ theory}
We begin by removing the ultraviolet divergence of the self-mass squared embedded
in the 2-point correlator using the same procedure prescribed in the previous subsection.
We also perform two partial integrations and carry out the spatial Fourier transforms.
Even though the 2-point correlator carries various polarities on the self-mass squared
and the external legs (Fig.\ref{Figure1}), we suppress the polarities and the relative
signs\footnote{We used the convention for the usual, in-out diagram.} in order to
investigate the generic pattern ,
\begin{eqnarray}
&&\hspace{-1.5cm}\frac{-\lambda^2}{2}\Biggl[\frac{\Gamma(\frac{D}{2}\!-\!1)}{4\pi^{\frac{D}{2}}}\Biggr]^4
\!\!\int\!\!d^{D}y\frac{1}{\Delta x^{\scriptscriptstyle D-2}(x;y)}\!\!\int\!\!d^{D}y'
\frac{1}{\Delta x^{\scriptscriptstyle 2D-4}(y;y')}\frac{1}
{\Delta x^{\scriptscriptstyle D-2}(x';y')},\nonumber\\
&&\hspace{-1.5cm}\Longrightarrow\frac{-\lambda^2}{2}\frac{1}{2^8\pi^8}\!\!\int\!\!d^{4}y
\frac{1}{\Delta x^{\scriptscriptstyle 2}(x;y)}\!\!\times\!\!\frac{-1}{4}\!\!\int\!\!d^{4}y'
\partial_y^2\Biggl[\frac{\ln[\mu^{\scriptscriptstyle 2}\Delta x^{\scriptscriptstyle 2}(y;y')]}
{\Delta x^{\scriptscriptstyle 2}(y;y')}\Biggr]\frac{1}{\Delta x^{\scriptscriptstyle 2}(x';y')}.
\end{eqnarray}
To reach the second line, we have employed (\ref{extract}) to make the function integrable
with respect to $y^{\prime \mu}$ in $D=4$ dimensions. We then segregated the ultraviolet
divergence into a local delta function using (\ref{0}) and absorbed it with a mass counterterm,
just as in the previous subsection.
Because the derivative only acts on a function of the coordinate separation, we can replace
$\partial^2_y$ with $\partial^2_{y'}$ and then partially integrate to reach the form,
\begin{eqnarray}
\frac{\ln[\mu^{\scriptscriptstyle 2}\Delta x^{\scriptscriptstyle 2}(y;y')]}
{\Delta x^{\scriptscriptstyle 2}(y;y')}\Biggl[\partial_y'^2
\frac{1}{\Delta x^{\scriptscriptstyle 2}(x';y')}\Biggr]\!+\!\textrm{two surface terms}.
\end{eqnarray}
After dropping the surface terms, the $y'$ integration can be carried out using (\ref{0}),
\begin{eqnarray}
\frac{i\lambda^2}{2^9\pi^6}\!\!\int\!\!d^4y\frac{1}{\Delta x^{\scriptscriptstyle 2}(x;y)}
\frac{\ln[\mu^{\scriptscriptstyle 2}\Delta x^{\scriptscriptstyle 2}(y;x')]}
{\Delta x^{\scriptscriptstyle 2}(y;x')}.
\end{eqnarray}
Note that only the first and third diagrams of Fig.\ref{Figure1} survive because their
right external legs carry the same polarity.
Further reduction can be accomplished by extracting another d'Alembertian using (\ref{extract1}),
performing a partial integration and then carrying out the $y$ integration,
\begin{eqnarray}
\frac{\lambda^2}{2^{10}\pi^4}\Biggl\{\ln^2[\mu^{\scriptscriptstyle 2}
\Delta x^{\scriptscriptstyle 2}_{\scriptscriptstyle -+}(x;x')]
\!-\!2\ln(\mu^{\scriptscriptstyle 2}
\Delta x^{\scriptscriptstyle 2}_{\scriptscriptstyle -+}(x;x')]
\Biggr\}\!+\!\textrm{two surface terms}. \label{re2pt}
\end{eqnarray}
It is clear that the final survival term is from the third diagram of Fig.\ref{Figure1}
because its two external legs carry the same polarity. Because $x^{\mu} \!=\!(t,\vec{x})$
and $x^{\prime \mu} \!=\!(t,\vec{0})$ have the same time components, the coordinate separation
$\Delta x^2(x;x')$ in (\ref{re2pt}) is purely spatial. The spatial Fourier transform of
(\ref{re2pt}) can be easily performed to give,
\begin{eqnarray}
&&\hspace{-1cm}\frac{\lambda^2}{2^{10}\pi^4}\!\!\int\!\!d^3x e^{-i\vec{k\cdot\vec{x}}}
\Biggl\{\!\ln^2[\mu^{\scriptscriptstyle 2}|\vec{x}|^{\scriptscriptstyle 2}]
\!-\!2\ln[\mu^{\scriptscriptstyle 2}|\vec{x}|^{\scriptscriptstyle 2}]\!\Biggr\}
\!=\!\frac{\lambda^2}{2^{10}\pi^4}\frac{4^2\pi}{k}\!\!\int_0^{\infty}\!\!drr\sin(kr)
\Big\{\!\ln^2[\mu r]\!-\!\ln[\mu r]\!\Bigr\}\nonumber\\
&&\hspace{-.7cm}=\frac{\lambda^2k^{-1}}{2^{6}\pi^3}\frac{-\partial}{\partial k}\!\!\int_0^{\infty}\!\!
dr\cos(kr)\Biggl\{\ln^2[\mu r]\!-\!\ln[\mu r]\Biggr\}\!=\!\frac{\lambda^2k^{-1}}{2^6\pi^3}
\frac{-\partial}{\partial k}\Biggl\{\frac{\pi}{k}\Bigl[\frac12\!+\!\gamma\!-\!
\ln(\frac{\mu}{k})\Bigr]\Biggr\}.
\end{eqnarray}
Two special integrals \cite{integrals} have been employed in the last equality,
\begin{eqnarray}
\int_0^{\infty}\!\!dz\frac{\sin(z)}{z}\!=\!\frac{\pi}{2}\,\,\,;\,\,\,\!\!\int_0^{\infty}\!\!
dz\frac{\sin(z)}{z}\ln(z)\!=\!\frac{\pi}{2}\gamma,\,\,\,\,\,\gamma\equiv \textrm{Euler's constant}.
\end{eqnarray}
The lowest correction to the power spectrum by the correlator definition is therefore,
\begin{eqnarray}
\int\!\!d^3x e^{-i\vec{k\cdot\vec{x}}}
\Bigl<\Omega|\varphi(t,\vec{x})\varphi(t,\vec{0})|\Omega\Bigr>_{\scriptstyle 1\,loop}
\!=\!\frac{\lambda^2}{2^6\pi^2}\frac{1}{k^3}\Bigl\{\frac12\!+\!\gamma\!-\!
\ln(\frac{\mu}{k})\Bigr\}.\label{2ptSpectra}
\end{eqnarray}
\subsection{Discussions of infrared and ultraviolet divergences}
There is an unfortunate tendency in the literature to employ the term ``infrared divergence''
to describe perfectly finite, temporally growing effects such as (\ref{reMoSpectra}). Of
course a true infrared divergence is an infinite constant, with no spacetime dependence.
That neither definition for the power spectrum can give rise to an infrared divergence is a
simple consequence of the Schwinger-Keldysh formalism with the initial states being released
at finite times. In order to produce a true infrared divergence, interactions must contribute
from arbitrarily large spatial distances, and this is precluded by causality as long as the
initial state is released at any finite time.
Ultraviolet divergences can and do occur in the Schwinger-Keldysh formalism, just as they
do in the in-out formalism. In both formalisms it is important to distinguish between the
ultraviolet divergences of non-coincident 1PI functions, which are eliminated by
conventional BPHZ renormalization, and the new divergences which can occur when one or
more of the coordinates are related. These new divergences require an extra, composite
operator renormalization. The case of the power spectrum is especially tricky because
only the time components of the two spacetime points are made to coincide. For the case
of our toy $\varphi^3$ model, this produces no extra divergence. The vertices of quantum
gravity contain derivatives, which increases the tendency for divergences. Frob, Roura
and Verdaguer have claimed that this is enough to cause the one loop correction to the
correlator definition of the power spectrum to harbor a new, composite operator
divergence. One of our points is that the mode function definition is free from this
new divergence.
\end{appendices}
|
2,869,038,154,916 | arxiv | \section{Introduction}
Let $M_m$ denote $\mathbb R^2$ equipped with a
smooth, complete, rotationally symmetric Riemannian metric
given in polar coordinates as $g_m:=dr^2+m^2(r)d\theta^2$; let $o$ denote
the origin in $\mathbb R^2$.
We say that $M_m$ is a {\it von Mangoldt plane\,} if its
sectional curvature $G_m:=-\frac{m^{\prime\prime}}{m}$ is a non-increasing
function of $r$.
The Toponogov comparison theorem was extended in~\cite{IMS-topon-vm}
to open complete manifolds with radial sectional curvature
bounded below by the curvature of a von Mangoldt plane,
leading to various applications in~\cite{ShiTan-mathZ-2002, KonOht-2007, KonTan-I}
and generalizations in~\cite{MasShi, KonTan-II, Mac}.
A point $q$ in a Riemannian manifold is called a {\it critical point
of infinity\,} if each unit tangent vector at $q$
makes angle $\le\frac{\pi}{2}$ with a ray that starts at $q$.
Let ${\mathfrak C}_m$ denote the set of critical points of infinity of $M_m$; clearly
${\mathfrak C}_m$ is a closed, rotationally symmetric subset that
contains every pole of $M_m$, so that $o\in{\mathfrak C}_m$.
One reason for studying ${\mathfrak C}_m$ is the following consequence of
the generalized Toponogov theorem of~\cite{IMS-topon-vm}.
\begin{lem}
\label{lem-intro: crit}
Let $\hat M$ be a complete noncompact Riemannian manifold
with radial curvature bounded below by
the curvature of a von Mangoldt plane $M_m$, and let $\hat r$, $r$
denote the distance functions to the basepoints $\hat o$, $o$ of $\hat M$, $M_m$,
respectively.
If $\hat q$ is a critical point of $\hat r$, then
$\hat r(\hat q)$ is contained in $r({\mathfrak C}_m)$.
\end{lem}
Combined with the critical point theory of distance
functions~\cite{Grove-crit-pt-1993},
\cite[Lemma 3.1]{Gre-surv}, \cite[Section 11.1]{Pet-book}, Lemma~\ref{lem-intro: crit}
implies the following.
\begin{cor}\label{cor-intro: crit pt theory}
In the setting of Lemma~\textup{\ref{lem-intro: crit}},
for any $c$ in $[a,b]\subset r(M_m\mbox{--}\,{\mathfrak C}_m)$,
\vspace{-10pt}
\newline
$\bullet$
the ${\hat r}^{-1}$-preimage of $[a,b]$ is homeomorphic to
${\hat r}^{-1}(a)\times [a,b]$, and
the \newline \phantom{$\bullet$}
${\hat r}^{-1}$-preimages of points in $[a,b]$
are all homeomorphic;\vspace{3pt} \newline
$\bullet$
the $\hat r^{-1}$-preimage of $[0,c]$ is homeomorphic to a compact
smooth manifold \newline\phantom{$\bullet$} with boundary, and the homeomorphism maps
$\hat r^{-1}(c)$ onto the boundary; \vspace{3pt}\newline
$\bullet$
if $K\subset\hat M$ is a compact
smooth submanifold, possibly with boundary, such
\, \phantom{$\bullet$} that
$\hat r(K)\supset r({\mathfrak C}_m)$, then $\hat M$ is diffeomorphic to
the normal bundle of $K$.
\end{cor}
If $M_m$ is von Mangoldt and $G_m(0)\le 0$,
then $G_m\le 0$ everywhere, so every point is a pole and hence
${\mathfrak C}_m=M_m$ so that Lemma~\ref{lem-intro: crit} yields no information
about the critical points of $\hat r$. Of course, there are other ways
to get this information as illustrated by classical Gromov's
estimate: if $M_m$ is the standard $\mathbb R^2$,
then the set of critical points of $\hat r$ is compact;
see e.g.~\cite[page 109]{Gre-surv}.
The following theorem determines ${\mathfrak C}_m$ when $G_m\ge 0$ everywhere;
note that the plane $M_m$ in (i)-(iii) need not be von Mangoldt.
\begin{thm}
\label{thm-intro: nonneg curv}
If $G_m\ge 0$, then
\begin{enumerate}\vspace{-5pt}
\item[\textup{(i)}] ${\mathfrak C}_m$ is the closed $R_m$-ball centered at $o$ for some
$R_m\in [0,\infty]$.
\item[\textup{(ii)}] $R_m$ is positive if and only if
$\int_1^\infty m^{-2}$ is finite.
\item[\textup{(iii)}] $R_m$ is finite if and only if $m^\prime(\infty)<\frac{1}{2}$.
\item[\textup{(iv)}]
If $M_m$ is von Mangoldt and $R_m$ is finite,
then the equation $m^\prime(r)=\frac{1}{2}$ has
a unique solution $\rho_m$, and the solution satisfies
$\rho_m> R_m$ and $G_m(r_m)>0$.
\item[\textup{(v)}]
If $M_m$ is von Mangoldt and
$R_m$ is finite and positive, then
$R_m$ is the unique solution of the integral equation
$\int_x^\infty\frac{m(x)\, dr}{m(r)\,\sqrt{m^2(r)-m^2(x)}}=\pi$.
\end{enumerate}
\end{thm}
Here is a sample application of part (iv) of
Theorem~\ref{thm-intro: nonneg curv}
and Corollary~\ref{cor-intro: crit pt theory}:
\begin{cor}
\label{cor-intro: homeo to .5-ball}
Let $\hat M$ be a complete noncompact Riemannian manifold
with radial curvature from the basepoint $\hat o$
bounded below by the curvature of a von Mangoldt plane $M_m$.
If $G_m\ge 0$ and $m^\prime(\infty)<\frac{1}{2}$, then $\hat M$ is homeomorphic
to the metric $\rho_m$-ball centered at $\hat o$, where $\rho_m$ is
the unique solution of $m^\prime(r)=\frac{1}{2}$.
\end{cor}
Theorem~\ref{thm-intro: nonneg curv}
should be compared with the following results of Tanaka:
\begin{itemize}
\item
the set of poles in any $M_m$ is a closed metric ball centered at $o$
of some radius $R_p$ in $[0,\infty]$~\cite[Lemma 1.1]{Tan-cut}.\vspace{3pt}
\item
$R_p>0$ if and only if
$\int_1^\infty m^{-2}$ is finite and
$\underset{r\to\infty}{\liminf}\, m(r)>0$~\cite{Tan-ball-poles-ex}. \vspace{3pt}
\item
if $M_m$ is von Mangoldt, then $R_p$
is a unique solution of an explicit integral
equation~\cite[Theorem 2.1]{Tan-ball-poles-ex}.
\end{itemize}
It is natural to wonder when
the set of poles equals ${\mathfrak C}_m$, and we answer the question
when $M_m$ is von Mangoldt.
\begin{thm}
\label{intro-thm: poles}
If $M_m$ is a von Mangoldt plane, then
\begin{enumerate}\vspace{-5pt}
\item[\textup{(a)}]
If $R_p$ is finite and positive, then the set of poles
is a proper subset of the component of ${\mathfrak C}_m$
that contains $o$.
\item[\textup{(b)}]
$R_p=0$ if and only if ${\mathfrak C}_m=\{o\}$.
\end{enumerate}
\end{thm}
Of course $R_p=\infty$ implies ${\mathfrak C}_m=M_m$, but
the converse is not true:
Theorem~\ref{thm-intro: slope realized} ensures
the existence of a von Mangoldt plane with
$m^\prime(\infty)=\frac{1}{2}$ and $G_m\ge 0$, and
for this plane ${\mathfrak C}_m=M_m$ by
Theorem~\ref{thm-intro: nonneg curv}, while $R_p$ is finite by
Remark~\ref{rmk: poles when m'=.5}.
We say that a ray $\gamma$ in $M_m$ {\it points away from infinity\,}
if $\gamma$ and the segment $[\gamma(0), o\,]$ make an angle $<\frac{\pi}{2}$
at $\gamma(0)$.
Define $A_m\subset M_m\,\mbox{--}\,\{o\}$ as follows: $q\in A_m$
if and only if there is a ray that starts at $q$ and points
away from infinity; by symmetry, $A_m\subset{\mathfrak C}_m$.
\begin{thm}
\label{thm-intro: away}
If $M_m$ is a von Mangoldt plane, then $A_m$ is open in $M_m$.
\end{thm}
Any plane $M_m$ with $G_m\ge 0$ has another distinguished subset, namely
the set of souls, i.e. points produced via the soul construction of
Cheeger-Gromoll.
\begin{thm}
\label{thm-intro: crit is soul}
If $G_m\ge 0$, then
${\mathfrak C}_m$ is equal to the set of souls of $M_m$.
\end{thm}
Recall that the soul construction takes as input
a basepoint in an open complete manifold $N$ of nonnegative sectional
curvature and produces a compact totally convex submanifold $S$
without boundary, called a {\it soul}, such that $N$ is diffeomorphic
to the normal bundle to $S$. Thus if $N$ is contractible,
as happens for $M_m$, then $S$ is a point.
The soul construction also gives a
continuous family of compact totally convex subsets
that starts with $S$ and ends with $N$, and
according to~\cite[Proposition 3.7]{Men}
$q\in N$ is a critical point of infinity if and only if
there is a soul construction such that
the associated continuous family of totally convex sets
drops in dimension at $q$.
In particular, any point of $S$ is a critical point of infinity,
which can also be seen directly; see the proof
of~\cite[Lemma 1]{Mae-2rays}. In Theorem~\ref{thm-intro: crit is soul}
we prove conversely that every point of ${\mathfrak C}_m$ is a soul;
for this $M_m$ need not be von Mangoldt.
In regard to part (iii) of Theorem~\ref{thm-intro: nonneg curv},
it is worth mentioning $G_m\ge 0$ implies that $m^\prime$
is non-increasing, so $m^\prime(\infty)$ exists,
and moreover, $m^\prime(\infty)\in [0,1]$ because $m\ge 0$.
As we note in Remark~\ref{rmk: app, tot curv}
for any von Mangoldt plane $M_m$, the limit
$m^\prime(\infty)$ exists as a number in $[0,\infty]$.
It follows that any $M_m$ with $G_m\ge 0$, and
any von Mangoldt plane $M_m$ admits
total curvature, which equals $2\pi(1-m^\prime(\infty))$
and hence takes values in $[-\infty, 2\pi]$; thus
$m^\prime(\infty)=\frac{1}{2}$ if and only if $M_m$
has total curvature $\pi$.
Standard examples of von Mangoldt planes of positive curvature
are the one-parametric family of paraboloids,
all satisfying $m^\prime(\infty)=0$~\cite[Example 2.1.4]{SST},
and the one-parametric family of two-sheeted hyperboloids
parametrized by $m^\prime(\infty)$,
which takes every value in $(0,1)$~\cite[Example 2.1.4]{SST}.
A property of von Mangoldt planes, discovered
in~\cite{Ele, Tan-cut} and crucial to this paper, is
that the cut locus of any $q\in M_m\,\mbox{--}\, \{o\}$ is a ray
that lies on the meridian opposite $q$.
(If $M_m$ is not von Mangoldt, its cut locus is not fully
understood, but
it definitely can be disconnected~\cite[page 266]{Tan-ball-poles-ex},
and known examples of cut loci of compact
surfaces of revolution~\cite{GluSin, SinTan-experim} suggest that it
could be complicated).
As we note in Lemma~\ref{lem: crit point and ray tangent to parallel},
if $M_m$ is a von Mangoldt plane, and if $q\neq o$,
then $q\in{\mathfrak C}_m$ if and only if the geodesic tangent
to the parallel through $q$ is a ray.
Combined with Clairaut's relation
this gives the following
``choking'' obstruction for a point
$q$ to belong to ${\mathfrak C}_m$
(see Lemma~\ref{lem: basic choke}):
\begin{prop}
\label{prop-intro: choking}
If $M_m$ is von Mangoldt and $q\in {\mathfrak C}_m$, then
$m^\prime(r_q)> 0$ and $m(r)>m(r_q)$ for $r>r_q$,
where $r_q$ is the $r$-coordinate of $q$.
\end{prop}
The above proposition is immediate from Lemmas~\ref{lem: basic choke}
and \ref{lem: crit point and ray tangent to parallel}.
We also show in Lemma~\ref{lem: m growth} that if
$M_m$ is von Mangoldt and ${\mathfrak C}_m\neq o$, then
there is $\rho$ such that $m(r)$ is increasing and
unbounded on $[\rho,\infty)$.
The following theorem collects most of what we know about ${\mathfrak C}_m$ for
a von Mangoldt plane $M_m$ with some negative curvature, where
the case $\underset{r\to\infty}{\liminf}\, m(r)=0$
is excluded because then ${\mathfrak C}_m=\{o\}$ by
Proposition~\ref{prop-intro: choking}.
\begin{thm}
\label{thm-intro: neg curv}
If $M_m$ is a von Mangoldt plane with a point where $G_m<0$
and such that $\underset{r\to\infty}{\liminf}\, m(r)>0$, then
\begin{enumerate}\vspace{-5pt}
\item[\textup{(1)}]
$M_m$ contains a line and has total curvature $-\infty$;
\item[\textup{(2)}]
if $m^\prime$ has a zero, then neither $A_m$ nor ${\mathfrak C}_m$ is connected;
\item[\textup{(3)}]
$M_m\,\mbox{--}\, A_m$ is a bounded subset of $M_m$;
\item[\textup{(4)}]
the ball of poles of $M_m$ has positive radius.
\end{enumerate}
\end{thm}
In Example~\ref{ex: m' vanishes} we construct a von Mangoldt plane
$M_m$ to which part (2) of Theorem~\ref{thm-intro: neg curv} applies.
In Example~\ref{ex: m' positive but non-connected} we produce
a von Mangoldt plane $M_m$ such that neither $A_m$ nor ${\mathfrak C}_m$ is connected
while $m^\prime>0$ everywhere.
We do not know whether there is a von Mangoldt plane
such that ${\mathfrak C}_m$ has more than two connected components.
Because of Lemma~\ref{lem-intro: crit} and
Corollary~\ref{cor-intro: crit pt theory}, one is
interested in subintervals of $(0, \infty)$
that are disjoint from $r({\mathfrak C}_m)$,
as e.g. happens for any interval on which $m^\prime\le 0$,
or for the interval $(R_m, \infty)$ in Theorem~\ref{thm-intro: nonneg curv}.
To this end we prove the following result, which
is a consequence of Theorem~\ref{thm: neck}.
\begin{thm}\label{intro-thm: neck nneg}
Let $M_n$ be a von Mangoldt plane with $G_n\ge 0$,
$n(\infty)=\infty$, and such that $n^\prime(x)<\frac{1}{2}$ for some $x$.
Then for any $z>x$ there exists $y>z$ such that
if $M_m$ is a von Mangoldt plane with
$n=m$ on $[0, y]$, then $r({\mathfrak C}_m)$ and $[x,z]$ are disjoint.
\end{thm}
In general, if $M_m$, $M_n$ are von Mangoldt planes with
$n=m$ on $[0, y]$, then the sets ${\mathfrak C}_m$, ${\mathfrak C}_n$ could be quite
different. For instance, if $M_n$ is a paraboloid, then ${\mathfrak C}_n=\{o\}$,
but by Example~\ref{ex: m' positive but non-connected}
for any $y>0$ there is a von Mangoldt $M_m$ with some negative curvature
such that $m=n$ on $[0,y]$, and by Theorem~\ref{thm-intro: neg curv}
the set $M_m\,\mbox{--}\, {\mathfrak C}_m$ is bounded
and ${\mathfrak C}_m$ contains the ball of poles of positive radius.
Basic properties of von Mangoldt planes are described in
Appendix~\ref{sec: vm planes}, in particular,
in order to construct a von Mangoldt plane with prescribed $G_m$
it suffices to check that $0$ is the only zero of
the solution of the Jacobi initial value problem (\ref{form: ode})
with $K=G_m$, where $G_m$ is smooth on $[0,\infty)$.
Prescribing values of $m^\prime$ is harder.
It is straightforward to see that if $M_m$ is a von Mangoldt plane
such that $m^\prime$ is constant near infinity, then
$G_m\ge 0$ everywhere and $m^\prime(\infty)\in [0,1]$.
We do not know whether there is
a von Mangoldt plane with $m^\prime=0$ near infinity,
but all the other values in $(0,1]$ can be prescribed:
\begin{thm}
\label{thm-intro: slope realized}
For every $s\in (0,1]$ there is $\rho>0$ and a von Mangoldt plane $M_m$
such that $m^\prime=s$ on $[\rho,\infty)$.
\end{thm}
Thus each cone in $\mathbb R^3$ can be smoothed
to a von Mangoldt plane, but we do not know how to construct
a (smooth) capped cylinder that is von Mangoldt.
{\bf Structure of the paper.}
We collect notations and conventions in Section~\ref{sec: notations}.
Properties of von Mangoldt planes are reviewed in
Appendix~\ref{sec: vm planes}, while
Appendix~\ref{sec: calc lem} contains a calculus lemma
relevant to continuity and smoothness of the turn angle.
Section~\ref{sec: turn angle} contains various results
on rays in von Mangoldt planes, including the proof of
Theorem~\ref{thm-intro: away} and
Proposition~\ref{prop-intro: choking}.
Planes of nonnegative curvature are discussed in
Section~\ref{sec: nneg}, where
Theorems~\ref{thm-intro: nonneg curv} and \ref{thm-intro: crit is soul}
are proved. A proof of
Theorem~\ref{thm-intro: slope realized}
is in Section~\ref{sec: smoothed cones}, and the other results
stated in the introduction are proved in Section~\ref{sec: other proofs}.
{\bf Acknowledgments.}
Belegradek is grateful to NSF for support (DMS-0804038).
This paper will be a part of Choi's Ph.D. thesis at Emory University.
\begin{comment}
We greatly appreciate comments by the referee,
who pointed out
a number of inaccuracies, suggested
Lemmas~\ref{lem: crit is o},
\ref{lem: approx by segments}, \ref{lem: max deviat}, and showed us
how to strengthen parts (i)-(iii) of
Theorem~\ref{thm-intro: nonneg curv},
which we originally proved only when $M_m$ was von Mangoldt.
\end{comment}
\section{Notations and conventions.}
\label{sec: notations}
All geodesics are parametrized by arclength.
Minimizing geodesics are called {\it segments}.
Let $\partial_r$, $\partial_\theta$ denote the vector fields dual
to $dr$, $d\theta$ on $\mathbb R^2$.
Given $q\neq o$, denote its polar coordinates by $\theta_q$, $r_q$.
Let $\gamma_q$, $\mu_q$, $\tau_q$ denote the geodesics defined
on $[0,\infty)$ that start at $q$
in the direction of $\partial_\theta$, $\partial_r$, $-\partial_r$, respectively.
We refer to $\tau_{q}\vert_{(r_q,\infty)}$ as the
{\it meridian opposite $q$}; note that $\tau_{q}(r_q)=o$.
Also set $\kappa_{\gamma(s)}:=\angle(\dot\gamma(s),\partial_r)$.
We write $\dot{r}$, $\dot\theta$, $\dot\gamma$, $\dot\kappa$ for the derivatives of
$r_{\gamma(s)}$, $\theta_{\gamma(s)}$, $\gamma(s)$, $\kappa_{\gamma(s)}$ by $s$, and
write $m^\prime$ for $\frac{dm}{dr}$; similar notations are used for higher derivatives.
Let $\hat\kappa(r_q)$ denote the maximum of the angles
formed by $\mu_q$ and rays emanating from $q\neq o$;
let $\xi_q$ denote the ray with $\xi_q(0)=q$
for which the maximum is attained, i.e.
such that $\kappa_{\xi_q(0)}=\hat\kappa(r_{q})$.
A geodesic $\gamma$ in $M_m\,\mbox{--}\, \{o\}$ is called
{\it counterclockwise\,} if $\dot\theta>0$ and
{\it clockwise\,} if $\dot\theta<0$.
A geodesic in $M_m$ is clockwise, counterclockwise,
or can be extended to a geodesic through $o$.
If $\gamma$ is clockwise, then it can be mapped to a
counterclockwise geodesic by an isometric involution of $M_m$.
{\bf Convention:} {\it unless stated otherwise,
any geodesic in $M_m$ that we consider is either
tangent to a meridian or counterclockwise}.
Due to this convention the Clairaut constant and the turn angle
defined below are nonnegative, which will simplify notations.
\section{Turn angle and rays in $M_m$}
\label{sec: turn angle}
This section collects what we know about rays in $M_m$
with emphasis on the cases when $G_m\ge 0$ or $G_m^\prime\le 0$.
Let $\gamma$ be a geodesic in $M_m$ that does not pass through $o$, so that
$\gamma$ is a solution of the geodesic equations
\begin{equation}\label{form: geod eq}
{\ddot r}=m\, m^\prime\, {\dot\theta}^2
\qquad\qquad
{\dot\theta}\, m^2=c
\end{equation}
where $c$ is called {\it Clairaut's constant of $\gamma$}.
The equation $\dot{\theta}\, m^2=c$ is the so called
{\it Clairaut's relation\,}, which since $\gamma$ is assumed counterclockwise,
can be written as $c=m(r_{\gamma(s)})\sin\kappa_{\gamma(s)}$.
Thus $0\le c\le m(r_\gamma(s))$ where $c= m(r_\gamma(s))$ only
at points where $\gamma$ is tangent to a parallel, and
$c=0$ when $\gamma$ is tangent to a meridian.
A geodesic is called {\it escaping} if its image is unbounded, e.g.
any ray is escaping.
\begin{fact}
\label{fact: on escaping geodesics}
\rm\hspace{-10pt}
\begin{enumerate}
\item
A parallel through $q$ is a geodesic in $M_m$ if and only if
$m^\prime(r_q)=0$~\cite[Lemma 7.1.4]{SST}.
\item
A geodesic $\gamma$ in $M_m$ is tangent to a parallel at $\gamma(s_0)$
if and only if ${\dot r}_{\gamma(s_0)}=0$.
\item
\label{fact: escaping geodesics are tangent to parallels only once}
If $\gamma$ is a geodesic in $M_m$ and
${\dot r}_{\gamma(s)}$ vanishes more than once,
then $\gamma$ is invariant under a rotation of $M_m$ about
$o$~\cite[Lemma 7.1.6]{SST} and hence not escaping.
\end{enumerate}
\end{fact}
\begin{lem}
\label{lem: basic choke}
If $\gamma_q$ is escaping,
then $m(r)>m(r_q)$ for $r>r_q$, and
$m^\prime(r_q)> 0$.
\end{lem}
\begin{proof}
Since $\gamma_q$ is escaping, the image of $s\to r_{\gamma_q}(s)$
contains $[r_q,\infty)$, and $q$ is the only point where
$\gamma_q$ is tangent to a parallel.
The Clairaut constant of $\gamma_q$ is $c=m(r_q)$, hence $m(r)>m(r_q)$
for all $r>r_q$. It follows that $m^\prime(r_q)\ge 0$. Finally,
$m^\prime(r_q)\neq 0$ else $\gamma_q$ would equal the parallel through
$q$.
\end{proof}
\begin{lem}
\label{lem: no crossing}
If $\gamma$ is escaping geodesic that is tangent to the parallel $P_q$
through $q$, then $\gamma\setminus\{q\}$ lies
in the unbounded component of $M_m\setminus P_q$.
\end{lem}
\begin{proof}
By reflectional symmetry and uniqueness of geodesics,
$\gamma$ locally stays on the same side of the parallel
$P_q$ through $q$, i.e. $\gamma$ is the union of $\gamma_q$ and
its image under the reflecting fixing $\mu_q\cup\tau_q$.
If $\gamma $ could
cross to the other side of $P_q$ at some
point $\gamma(s)$, then
$|r_{\gamma (s)}-r_q|$ would attain a maximum between
$\gamma(s)$ and $q$, and at the maximum point
$\gamma $ would be tangent to a parallel. Since $\gamma$ is escaping,
it cannot be tangent to parallels more than once, hence
$\gamma$ stays on the same side of $P_q$ at all times, and
since $\gamma$ is escaping, it stays in the unbounded component of
$M_m\setminus P_q$.
\end{proof}
For a geodesic $\gamma\colon\thinspace (s_1, s_2)\to M_m$ that does
not pass through $o$,
we define the {\it turn angle $T_\gamma$ of $\gamma$} as
\[T_\gamma:=\int_\gamma d\theta=
\int_{s_1}^{s_2}{\dot\theta}_{\gamma(s)} ds=\theta_{\gamma(s_2)}-\theta_{\gamma(s_1)}.\]
The Clairaut's relation reads ${\dot\theta}=c/m^2\ge 0$
so the above integral $T_\gamma$ converges to a number in $[0,\infty]$.
Since $\gamma$ is unit speed, we have $({\dot r})^2+m^2{\dot\theta}^2=1$.
Combining this with ${\dot\theta}=c/m^2$ gives
${\dot r}=\mathrm{sign}({\dot r})\sqrt{1-\frac{c^2}{m^2}}$, which
yields a useful formula for the turn angle:
if $\gamma$ is not tangent to a meridian or a parallel on $(s_1, s_2)$,
so that $\mathrm{sign}({\dot r}_{\gamma(s)})$ is a nonzero constant, then
\begin{equation}
\label{form: defn F_c}
\frac{d\theta}{dr}=
\frac{{\dot\theta}}{{\dot r}}=
\mathrm{sign}({\dot r}_{\gamma(s)})\ \! F_c(r)\ \ \ \text{where}\
\ \ F_c:=\frac{c}
{m\sqrt{m^2-c^2}},
\end{equation}
and thus if $r_i:=r_{\gamma(s_i)}$, then
\begin{equation}
\label{form: general turn angle}
T_\gamma=
\mathrm{sign}({\dot r})\int_{r_1}^{r_2} F_c(r) dr.
\end{equation}
Since $c^2\le m^2$, this integral is finite except possibly when
some $r_i$ is in the set $\{m^{-1}(c),\, \infty\}$.
The integral (\ref{form: general turn angle})
converges at $r_i=m^{-1}(c)$ if and only if
$m^\prime(r_i)\neq 0$.
Convergence of (\ref{form: general turn angle})
at $r_i=\infty$ implies convergence of $\int_1^\infty m^{-2} dr$,
and the converse holds under the assumption
$\underset{r\to\infty}{\liminf}\, m(r)>c$;
this assumption is true when $G_m\ge 0$ or $G_m^\prime\le 0$,
as follows from Lemma~\ref{lem: m growth} below.
\begin{ex}
\label{ex: ray turn angle}
If $\gamma$ is a ray in $M_m$ that does not pass through $o$,
then $T_\gamma\le \pi$ else there is $s$ with
$|\theta_{\gamma(s)}-\theta_{\gamma(0)}|=\pi$, and by symmetry
the points $\gamma(s)$, $\gamma(0)$ are joined by two segments, so $\gamma$
would not be a ray.
\end{ex}
\begin{ex}
\label{ex: ray exists implies properties of m}
If $T_{\gamma_q}$ is finite, then $m^\prime(r_q)\neq 0$ and
$m^{-2}$ is integrable on $[1,\infty)$, as follows immediately from
the discussion preceding Example~\ref{ex: ray turn angle}.
\end{ex}
\begin{lem}
\label{lem: turn angle escaping}
If $\gamma\colon\thinspace [0,\infty)\to M_m$ is a geodesic
with finite turn angle, then $\gamma$ is escaping.
\end{lem}
\begin{proof} Note that $\gamma$ is tangent to
parallels in at most two points, for otherwise
$\gamma$ is invariant under a rotation about $o$, and hence
its turn angle is infinite. Thus after cutting off a portion
of $\gamma$ we may assume it is never tangent to a parallel,
so that $r_{\gamma(s)}$ is monotone.
By assumption $\theta_{\gamma(s)}$ is bounded and increasing.
By Clairaut's relation
$m(r_{\gamma(s)})$ is bounded below, so that
$m(0)=0$ implies that $r_{\gamma(s)}$ is bounded below.
If $\gamma$ were not escaping, then $r_{\gamma(s)}$ would also be
bounded above, so there would exist a limit of $(r_{\gamma(s)},\theta_{\gamma(s)})$
and hence the limit of $\gamma(s)$ as $s\to\infty$,
contradicting the fact that $\gamma$ has infinite length.
\end{proof}
\begin{lem}
\label{lem: m growth}
If $m^{-2}$ is integrable on $[1,\infty)$, then
\begin{enumerate}\vspace{-5pt}
\item[\textup{(1)}]
the function $(r\log r)^{^{-\frac{1}{2}}}m(r)$ is unbounded;
\item[\textup{(2)}]
if $G_m\ge 0$, then $m^\prime>0$ for all $r$;
\item[\textup{(3)}]
if $M_m$ is von Mangoldt, then $m^\prime>0$ for all large $r$;
\item[\textup{(4)}]
if either $G_m\ge 0$ or $G_m^\prime\le 0$, then $m(\infty)=\infty$.
\end{enumerate}
\end{lem}
\begin{proof}
Since $m^{-2}$ is integrable,
the function $(r\log r)^{^{-\frac{1}{2}}}m(r)$ is unbounded,
and in particular, $m$ is unbounded.
If $G_m\ge 0$ everywhere, then $m^\prime$ is non-increasing
with $m^\prime(0)=1$, and the fact that $m$ is unbounded
implies that $m^\prime>0$ for all $r$.
If $M_m$ is von Mangoldt, and $G_m(\rho_0)< 0$, then $G_m< 0$ for $r\ge \rho_0$,
i.e. $m^\prime$ is non-decreasing on $[\rho_0, \infty)$.
Since $m$ is unbounded, there is $\rho>\rho_0$ with $m(\rho)>m(\rho_0)$
so that $\int_{\rho_0}^{\rho} m^\prime=m(\rho)-m(\rho_0)>0$.
Hence $m^\prime$ is positive somewhere
on $(\rho_0, \rho)$, and therefore on $[\rho, \infty)$.
Finally, since $m$ is an unbounded increasing function for
large $r$, the limit
$\displaystyle{\lim_{r\to\infty}\, m(r)}=m(\infty)$ exists and
equals $\infty$.
\end{proof}
\begin{lem}
If $\gamma_q$ is escaping, then
$\displaystyle{\liminf_{r\to\infty}\, m(r)}>m(r_q)$
if and only if there is a neighborhood $U$ of $q$
such that $\gamma_u$ is escaping for each $u\in U$.
\end{lem}
\begin{proof}
First, recall that $m(r)>m(r_q)$ for $r>r_q$
and $m^\prime(r_q)>0$ by Lemma~\ref{lem: basic choke}.
We shall prove the contrapositive:
$\displaystyle{\liminf_{r\to\infty}\, m(r)}=m(r_q)$
if and only if there is a sequence $u_i\to q$ such that
$\gamma_{u_i}$ is not escaping.
If there is a sequence $z_i\in M_m$ with
$r_{z_i}\to\infty$ and $m(r_{z_i})\to m(r_q)$,
then there are points $u_i\to q$ on $\mu_q$ with
$m(r_{u_i})=m(r_{z_i})$.
If $\gamma_{u_i}$ is escaping, then it meets the parallel
through $z_i$, so
Clairaut's relation implies that $\gamma_{u_i}$ is tangent
to the parallels through $u_i$ and $z_i$,
which cannot happen for an escaping geodesic.
Conversely,
suppose there are $u_i\to q$ such that $\gamma_i:=\gamma_{u_i}$ is not escaping.
Let $R_i$ be the radius of the smallest ball about $o$
that contains $\gamma_i$, and let $P_i$ be its boundary
parallel. Note that $R_i\to \infty$
as $\gamma_i$ converges to $\gamma_q$ on compact sets
and $\gamma_q$ is escaping, and hence
$\displaystyle{\liminf_{r\to\infty}\, m(r)}=
\displaystyle{\lim_{r\to\infty}\, m(R_i)}$.
For each $i$
there is a sequence $s_{i,j}$ such that the $r$-coordinates of
$\gamma_i(s_{i,j})$ converge to $R_i$, which implies
$\kappa_{\gamma_i(s_{i,j})}\to \frac{\pi}{2}$
as $j\to\infty$ and $i$ is fixed.
(Note that if $\gamma_i$ is tangent to $P_i$, then
$s_{i,j}$ is independent of $j$, namely,
$\gamma(s_{i,j})$ is the point of tangency).
By Clairaut's relation,
$m(R_i)=m(r_{u_i})$, hence
$\displaystyle{\liminf_{r\to\infty}\, m(r)}=m(r_q)$.
\end{proof}
\begin{lem} \label{lem: ray iff at most pi}
If $M_m$ is von Mangoldt, then
a geodesic $\gamma\colon\thinspace [0,\infty)\to M_m\setminus\{o\}$ is
a ray if and only if $T_\gamma\le \pi$.
\end{lem}
\begin{proof} The ``only if'' direction holds
even when $M_m$ is not von Mangoldt by
Example~\ref{ex: ray turn angle}.
Conversely, if $\gamma$ is not a ray, then $\gamma$ meets the cut locus
of $q$, which by~\cite{Tan-cut} is a subset of the
opposite meridian $\tau_{\gamma(0)}\vert_{(r_{\gamma(0)},\infty)}$.
Thus $T_\gamma>\pi$.
\end{proof}
\begin{lem}\label{lem: above rays are rays}
If $\gamma$ is a ray in a von Mangoldt plane,
and if $\sigma$ is a geodesic with $\sigma(0)=\gamma(0)$ and
$\kappa_{\gamma(0)}>\kappa_{\sigma(0)}$, then $\sigma$ is a ray and $T_\sigma\le T_\gamma$.
\end{lem}
\begin{proof} Set $q=\gamma(0)$.
If $\kappa_{\gamma(0)}=\pi$, then $\gamma=\tau_q$, so $\tau_q$ is a ray,
which in a von Mangoldt plane implies that $q$ is a
pole~\cite[Lemma 7.3.1]{SST}, so that $\sigma$ is also a ray.
If $\kappa_{\gamma(0)}<\pi$ and $\sigma$ is not a ray, then $\sigma$ is minimizing
until it crosses the opposite meridian
$\tau_q\vert_{(r_q,\infty)}$~\cite{Tan-cut}.
Near $q$ the geodesic $\sigma$ lies in the
region of $M_m$ bounded by $\gamma$ and $\mu_{q}$
hence before crossing the opposite meridian
$\sigma$ must intersect $\gamma$ or $\mu_{q}$, so they would not be rays.
Finally, $T_\sigma\le T_\gamma$ holds as $\sigma$ lies in the sector between
$\gamma$ and $\mu_{q}$.
\end{proof}
\begin{lem}\label{lem: crit point and ray tangent to parallel}
If $M_m$ is von Mangoldt and $q\neq o$, then $\gamma_q$ is a ray
if and only if $q\in{\mathfrak C}_m$.
\end{lem}
\begin{proof}
If $\gamma_q$ is a ray, then $q\in{\mathfrak C}_m$ by symmetry.
If $q\in{\mathfrak C}_m$, then either $q$ is a pole and
there is a ray in any direction, or
$q$ is not a pole. In the latter case
$\tau_q$ is not a ray~\cite[Lemma 7.3.1]{SST}, hence by the
definition of ${\mathfrak C}_m$ there is a ray $\gamma$
with $\kappa_{\gamma(0)}\ge \frac{\pi}{2}$, so $\gamma_q$ is a ray
by Lemma~\ref{lem: above rays are rays}.
\end{proof}
Recall that $\hat\kappa(r_q)$ is the maximum of the angles
formed by $\mu_q$ and rays emanating from $q\neq o$,
and $\xi_q$ is the ray for which the maximum is attained.
It is immediate from definitions that
$q\in{\mathfrak C}_m$ if and only if
$\hat\kappa(r_q)\ge \frac{\pi}{2}$.
Lemmas~\ref{lem: crit is o},
\ref{lem: approx by segments}, \ref{lem: max deviat} below
were suggested by the referee.
\begin{lem}
\label{lem: crit is o}
${\mathfrak C}_m\neq\{o\}$ if and only if\,
$\displaystyle{\liminf_{r\to\infty}\, m}>0$
and $\int_1^\infty m^{-2}$ is finite.
\end{lem}
\begin{proof}
The ``if'' direction holds because by the main result
of~\cite{Tan-ball-poles-ex}
the assumptions imply that the ball of poles has a positive radius.
Conversely, if $q\in{\mathfrak C}_m\,\mbox{--}\,\{o\}$,
then $\xi_q$ is a ray different from $\mu_q$.
By~\cite[Lemma 1.3, Proposition 1.7]{Tan-ball-poles-ex}
if either $\displaystyle{\liminf_{r\to\infty}\, m}=0$ or
$\int_1^\infty m^{-2}=\infty$, then $\mu_q$
is the only ray emanating from $q$.
\end{proof}
\begin{lem}\label{lem: approx by segments}
$\xi_q$ is the limit
of the segments $[q,\tau_q(s)]$ as $s\to\infty$.
\end{lem}
\begin{proof}
The segments $[q,\tau_q(s)]$ subconverge to a ray $\sigma$ that starts
at $q$. Since $\xi_q$ is a ray, it
cannot cross the opposite meridian $\tau_q\vert_{(r_q,\infty)}$.
As $[q,\tau_q(s)]$ and $\xi_q$ are minimizing,
they only intersect at $q$,
and hence the angle formed by $\mu_q$ and $[q,\tau_q(s)]$
is $\ge\hat\kappa(r_q)$. It follows that $\kappa_{\sigma(0)}\ge\hat\kappa(r_{q})$,
which must be an equality as $\hat\kappa(r_q)$ is a maximum,
so $\sigma=\xi_q$.
\end{proof}
\begin{lem}
\label{lem: max deviat} The function
$r\to\hat\kappa(r)$ is left continuous and
upper semicontinuous. In particular,
the set $\{q: \hat\kappa(r_q)<\alpha\}$ is open for every $\alpha$.
\end{lem}
\begin{proof}
If $\hat\kappa$ is not left continuous at $r_q$, then
there exists $\varepsilon>0$ and a sequence of points $q_i$ on $\mu_q$
such that $r_{q_i}\to r_q-$ and either
$\hat\kappa(r_{q_i})-\hat\kappa(r_q)>\varepsilon$ or $\hat\kappa(r_q)-\hat\kappa(r_{q_i})>\varepsilon$.
In the former case $\xi_{q_i}$ subconverge to a ray that makes larger angle with
$\mu_q$ that $\xi_q$, contradicting maximality of $\hat\kappa(r_q)$.
In the latter case, $\xi_{q_i}$ intersects $\xi_q$ for some $i$.
Therefore, by Lemma~\ref{lem: approx by segments}
the segment $[q_i, \tau_q(s)]$ intersects
$[q, \tau_q(s)]$ for large enough $s$ at
a point $z\neq\tau_q(s)$, so $\tau_q(s)$ is a cut point of $z$
which cannot happen for a segment. This proves
that $\hat\kappa$ is left continuous.
A similar argument shows that
$\displaystyle{\limsup_{r_{q_i}\to r_q+}\hat\kappa(r_{q_i})}\le \hat\kappa(r_q)$,
so that $\hat\kappa$ is upper semicontinuous, which implies that
$\{q: \hat\kappa(r_q)<\alpha\}$ is open for every $\alpha$.
\end{proof}
Lemmas~\ref{lem: ray iff at most pi},
\ref{lem: crit point and ray tangent to parallel}
imply that on a von Mangoldt plane
$\hat\kappa(r_q)\ge\frac{\pi}{2}$
if and only if $T_{\gamma_q}\le\pi$; the equivalence is sharpened
in Theorem~\ref{thm: max devia vs turn angle}, whose proof occupies
the rest of this section.
\begin{lem}
\label{lem: rays and parallels}
If $\sigma$ is escaping and $0<\kappa_{\sigma(0)}\le \frac{\pi}{2}$, then
$T_\sigma=\int_{r_{q}}^\infty F_{c}(r) dr$; moreover,
if $\kappa_{\sigma(0)}=\frac{\pi}{2}$, then $c=m(r_{q})$.
\end{lem}
\begin{proof}
This formula for $T_\sigma$ is immediate from
(\ref{form: general turn angle}) once it is shown that
$\sigma\vert_{(0,\infty)}$ is not tangent to a meridian
or a parallel. If $\sigma\vert_{(0,\infty)}$ were tangent to a meridian,
$\kappa_{\sigma(0)}$ would be $0$ or $\pi$, which is not the case.
Since $\sigma$ is escaping,
Fact~\ref{fact: on escaping geodesics} implies that
$\sigma$ is tangent to parallels at most once.
If $\kappa_{\sigma(0)}=\frac{\pi}{2}$, then $\sigma$ is tangent to
the parallel through $\sigma(0)$, and so $\sigma\vert_{(0,\infty)}$
is not tangent to a parallel.
Finally, if $\kappa_{\sigma(0)}< \frac{\pi}{2}$, then
$\sigma$ is not tangent to a parallel,
else it would be tangent to a parallel through $u$ with
$r_u>r_q$, which would imply $r_{\sigma(s)}\le r_u$ for all $s$
by Lemma~\ref{lem: no crossing}, which cannot happen
for an escaping geodesic.
\end{proof}
To better understand
the relationship between $\hat\kappa(r_q)$ and $T_{\gamma_q}$,
we study how $T_\sigma$ depends on $\sigma$, or equivalently
on $\sigma(0)$ and $\kappa_{\sigma(0)}$, when $\sigma$ varies
in a neighborhood of a ray $\gamma_q$.
\begin{lem}
\label{lem: turn angle of gamma_q is continuous}
If $G_m\ge 0$ or $G_m^\prime\le 0$,
then the function $u\to T_{\gamma_u}$ is continuous at each
point $u$ where $T_{\gamma_u}$ is finite.
\end{lem}
\begin{proof}
If $T_{\gamma_u}$ is finite, then $\gamma_u$ is escaping by
Lemma~\ref{lem: turn angle escaping}, and hence
$T_{\gamma_u}=\int_{r_{u}}^\infty F_{m(r_u)}$
by Lemma~\ref{lem: rays and parallels}.
We need to show that this integral depends continuously
on $r_u$.
By Lemma~\ref{lem: basic choke},
Lemma~\ref{lem: m growth}, and
the discussion preceding Example~\ref{ex: ray turn angle},
the assumptions on $G_m$ and finiteness of $T_{\gamma_u}$
imply that $m(r)>m(r_u)$ for $r>r_u$, $m^{-2}$ is integrable,
$m^\prime(r_u)> 0$, and $m(\infty)=\infty$.
Hence there exists $\delta>r_u$ with
$m^\prime\vert_{[r_u,\delta]}>0$, and
$m(r)>m(\delta)$ for $r>\delta$; it is clear
that small changes in $u$ do not affect $\delta$.
Write $\int_{r_{u}}^\infty F_{m(r_u)}=
\int_{r_{u}}^\delta F_{m(r_u)}+\int_{\delta}^\infty F_{m(r_u)}$.
On $[r_u,\delta]$ we can write
$F_{m(r_u)}=h(r,r_u)(r-r_u)^{-\frac{1}{2}}$ for some
smooth function $h$. Since
$(r-r_u)^{-\frac{1}{2}}$ is the derivative
of $2(r-r_u)^{\frac{1}{2}}$,
one can integrate $F_{m(r_u)}$ by parts which easily
implies continuous dependence of
$\int_{r_{u}}^\delta F_{m(r_u)}$ on $r_u$.
Continuous dependence of $\int_{\delta}^\infty F_{m(r_u)}$ on
$r_u$ follows because $F_{m(r_u)}$ is continuous in $r_u$,
and is dominated by $Km^{-2}$ where $K$ is a positive constant
independent of small changes of $r_u$.
\end{proof}
Next we focus on the case when
$\sigma(0)$ is fixed, while $\kappa_{\sigma(0)}$ varies
near $\frac{\pi}{2}$. To get an explicit formula for $T_\sigma$
we need the following.
\begin{lem}
\label{lem: geod just below ray}
If $M_m$ is von Mangoldt, and $\gamma_q$ is a ray, then
there is $\varepsilon>0$ such that
every geodesic $\sigma\colon\thinspace [0,\infty)\to M_m$
with $\sigma(0)=q$ and
$\kappa_{\sigma(0)}\in [\frac{\pi}{2}, \frac{\pi}{2}+\varepsilon]$
is tangent to a parallel exactly once, and if $u$ is the
point where $\sigma$ is tangent to a parallel, then
$m^\prime>0$ on $[r_u, r_q]$.
\end{lem}
\begin{proof}
If $\kappa_{\sigma(0)}=\frac{\pi}{2}$, then $\sigma=\gamma_q$, so
it is tangent to a parallel only at $q$,
as rays are escaping.
If $\kappa_{\sigma(0)}>\frac{\pi}{2}$, then
$\sigma$ converges to $\gamma_q$ on compact subsets as $\varepsilon\to 0$,
so for a sufficiently small $\varepsilon$ the geodesic $\sigma$
crosses the parallel through $q$ at some point $\sigma(s)$
such that $\kappa_{\sigma(s)}<\frac{\pi}{2}$. Since $\gamma_q$ is a ray,
rotational symmetry and
Lemma~\ref{lem: above rays are rays} imply that
$\sigma\vert_{[s,\infty)}$ is a ray, so $\sigma$ is escaping.
Thus $\sigma$ is tangent to a parallel at a point $u$
where $r_{\sigma(s)}$ attains a minimum, and is not
tangent to a parallel at any other point by
Fact~\ref{fact: on escaping geodesics}.
Finally, $r_u=\lim_{\varepsilon\to 0}r_q$, and since $m^\prime(r_q)>0$
by Proposition~\ref{prop-intro: choking},
we get $m^\prime>0$ on $[r_u, r_q]$ for small $\varepsilon$.
\end{proof}
Under the assumptions of Lemma~\ref{lem: geod just below ray}
the Clairaut constant $c$ of $\sigma$ equals
$m(r_u)=m(r_q)\sin \kappa_{\sigma(0)}$,
and the turn angle of $\sigma$ is given by
\begin{equation}
\label{form: upward turn angle via c}
T_\sigma=\int_{r_q}^\infty F_{m(r_q)}(r) dr
\quad\text{if}\quad \kappa_{\sigma(0)}=\frac{\pi}{2}\quad\text{and}
\end{equation}
\begin{equation}
\label{form: turn angle via c}
T_\sigma=\int_{r_u}^\infty F_{c}(r) dr -
\int_{r_q}^{r_u} F_{c}(r) dr =
\int_{r_q}^\infty F_{c}(r) dr +
2\int_{r_u}^{r_q} F_{c}(r) dr
\end{equation}
if $\frac{\pi}{2}<\kappa_{\sigma(0)}<\frac{\pi}{2}+\varepsilon$.
These integrals converge, i.e. $T_\sigma$ is finite, as follows from
Example~\ref{ex: ray exists implies properties of m},
and Lemmas~\ref{lem: m growth}, \ref{lem: geod just below ray}.
Since any geodesic $\sigma$ with $\sigma(0)=q$ and
$\kappa_{\sigma(0)}\in [0, \frac{\pi}{2}+\varepsilon]$
has finite turn angle,
one can think of $T_\sigma$ as a function of $\kappa_{\sigma(0)}$
where $\sigma$ varies over geodesics with $\sigma(0)=q$
and $\kappa_{\sigma(0)}\in [\,0, \frac{\pi}{2}+\varepsilon]$.
\begin{lem}
\label{lem: turn angle cont and diff}
If $M_m$ is von Mangoldt, and $\gamma_q$ is a ray, then
there is $\delta>\frac{\pi}{2}$ such that the function
$\kappa_{\sigma(0)}\to T_\sigma$ is continuous
and strictly increasing on $[\frac{\pi}{2}, \delta]$, and
continuously differentiable on $(\frac{\pi}{2}, \delta]$;
moreover, the derivative of $T_\sigma$ is infinite at $\frac{\pi}{2}$.
\end{lem}
\begin{proof} The Clairaut constant $c$ of $\sigma$ equals
$m(r_u)=m(r_q)\sin \kappa_{\sigma(0)}$, so
the assertion is immediate from (elementary but nontrivial)
Lemma~\ref{lem: calc} about continuity and differentiability
of the integrals
(\ref{form: upward turn angle via c})-(\ref{form: turn angle via c}).
\end{proof}
\begin{thm}\label{thm: max devia vs turn angle}
If $M_m$ is von Mangoldt and $q\neq o$, then
\begin{enumerate}\vspace{-5pt}
\item[\textup{(1)}]
$\hat\kappa(r_q)>\frac{\pi}{2}$ if and only if $T_{\gamma_q}<\pi$.
\vspace{2pt}
\item[\textup{(2)}]
$\hat\kappa(r_q)=\frac{\pi}{2}$ if and only if $T_{\gamma_q}=\pi$.
\vspace{-5pt}
\end{enumerate}
\end{thm}
\begin{proof} (1)
If $\hat\kappa(r_q)>\frac{\pi}{2}$,
then any geodesic $\sigma$ with $\sigma(0)=q$ and
$\kappa_{\sigma(0)}\in [\frac{\pi}{2}, \hat\kappa(r_q)]$
is a ray, and so has turn angle $\le \pi$.
By Lemma~\ref{lem: turn angle cont and diff}
the turn angle is increasing at $\frac{\pi}{2}$, so $T_{\gamma_q}<\pi$.
Conversely, if $T_{\gamma_q}<\pi$, then by
Lemma~\ref{lem: turn angle cont and diff}
the turn angle is continuous at $\frac{\pi}{2}$, so
any geodesic $\sigma$ with
$\sigma(0)=q$ and $\kappa_{\sigma(0)}$ near $\frac{\pi}{2}$ has
turn angle $<\pi$, and is therefore a ray, so
$\hat\kappa(r_q)>\frac{\pi}{2}$.
(2) follows from (1) and the fact
that $\hat\kappa(r_q)\ge\frac{\pi}{2}$ if and only if $T_{\gamma_q}\le\pi$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-intro: away}]
By Theorem~\ref{thm: max devia vs turn angle} we know that
$q\in A_m$ if and only if $T_{\gamma_q}<\pi$,
and by Lemma~\ref{lem: turn angle of gamma_q is continuous} the map
$u\to T_{\gamma_u}$ is continuous at $q$, so the set
$\{u\in M_m\,|\, T_{\gamma_u}<\pi\}$
is open, and hence so is $A_m$.
\end{proof}
\begin{proof}[Another proof of Theorem~\ref{thm-intro: away}]
Fix $q\in A_m$ so that $T_{\gamma_q}<\pi$ by
Theorem~\ref{thm: max devia vs turn angle}.
Fix $\varepsilon>0$ such that $\varepsilon+T_{\gamma_q}<\pi$.
Let $P_q$ be the parallel through $q$.
Then there is a ray $\gamma$ with $\gamma(0)=q$ and $\kappa_{\gamma(0)}>\frac{\pi}{2}$
such that $\gamma$ intersects $P_q$ at points
$q$, $\gamma(t)$, and the turn angle of $\gamma\vert_{(0,t)}$
is $<\varepsilon$.
For an arbitrary sequence $q_i\to q$ we need to show that
$q_i\in A_m$ for all large $i$.
Let $\gamma_i\colon\thinspace [0,\infty)\to M_m$ be the geodesic with
$\gamma_i(0)=q_i$ and $\kappa_{\gamma_i(0)}=\kappa_{\gamma(0)}$.
Since $\gamma_i$ converge to $\gamma$ on compact sets, for large $i$
there are $t_i>0$ such that $\gamma_i(t_i)\in P_q$ and $t_i\to t$.
The angle formed by $\gamma$ and $\mu_{\gamma(t)}$
is $<\frac{\pi}{2}$. Rotational symmetry and
Lemma~\ref{lem: above rays are rays} imply that
if $i$ is large, then $\gamma_i\vert_{[t_i,\infty)}$
is a ray whose turn angle is $\le T_{\gamma_q}$.
The turn angles of $\gamma_i\vert_{(0,t_i)}$ converge to the turn angle
of $\gamma\vert_{(0,t)}$, which is $<\varepsilon$. Thus $T_{\gamma_i}<T_{\gamma_q}+\varepsilon<\pi$
for large $i$, so that $\gamma_i$ is a ray by
Lemma~\ref{lem: ray iff at most pi}, and hence $q_i\in A_m$.
\end{proof}
\section{Planes of nonnegative curvature}
\label{sec: nneg}
A key consequence of $G_m\ge 0$ is
monotonicity of the turn angle and of $\hat\kappa$.
\begin{prop}\label{prop: G>0 and monotonicity}
Suppose that $M_m$ has $G_m\ge 0$.
If $0<r_u<r_v$ and $\gamma_u$ has finite turn angle,
then $T_{\gamma_u}\le T_{\gamma_v}$ with equality if and only if $G_m$ vanishes
on $[r_u, \infty]$.
\end{prop}
\begin{proof} The result is trivial when $G$ is everywhere zero.
Since $\gamma_u$ has finite turn angle, $m^{-2}$
is integrable, and hence
$m$ is a concave function with $m^\prime>0$ and $m(\infty)=\infty$
by Lemmas~\ref{lem: m growth}.
Set $x:=r_q$, so that the turn angle of $\gamma_q$
is $\int_x^\infty F_{m(x)}$. As $m^\prime>0$, we can
change variables by $t:=m(r)/m(x)$ or $r=m^{-1}(tm(x))$ so that
\[
\int_x^\infty F_{m(x)}(r)\,dr=
\int_1^{\frac{m(\infty)}{m(x)}}\frac{dt}{l(t,x)\,t\,\sqrt{t^2-1}}=
\int_1^{\infty}\frac{dt}{l(t,x)\,t\,\sqrt{t^2-1}}
\]
where $l(t,x):=m^\prime(r)$. Computing
\begin{comment}
\[
\frac{\partial}{\partial x}\frac{1}{l(t,x)}=
-\frac{\frac{\partial l(t,x)}{\partial x}}{(l(t,x))^2}=
\frac{m^{\prime\prime}(r)}{(m^\prime(r))^2}\,\frac{\partial r}{\partial x}=
\frac{m^{\prime\prime}(r)\,t\,m^\prime(x)}{(m^\prime(r))^3}>0
\]
\end{comment}
\[
\frac{\partial l(t,x)}{\partial x}=m^{\prime\prime}(r)\,\frac{\partial r}{\partial x}=
\frac{m^{\prime\prime}(r)\,t\,m^\prime(x)}{m^\prime(r)}=
-G(r)\,\frac{t\,m^\prime(x)}{m^\prime(r)}\le 0
\]
we see that $l(t,x)$ is non-increasing in $x$.
Thus if $r_u<r_v$, then $l(t,r_u)\ge l(t,r_v)$ for all $t$ implying
$T_{\gamma_u}\le T_{\gamma_v}$. The equality occurs precisely when
$l(t,x)$ is constant on $[1,\infty)\times [r_u, r_v]$, or equivalently,
when $G(m^{-1}(tm(x)))$ vanishes on $[1,\infty)\times [r_u, r_v]$,
which in turn is equivalent to $G=0$ on $[r_u,\infty)$, because
$tm(x)$ takes all values in $(m(r_u), \infty)$ so
$m^{-1}(tm(x))$ takes all values in $(r_u, \infty)$.
\end{proof}
\begin{lem}
\label{lem: gauss bonnet nonneg}
If $G_m\ge 0$, then $\hat{\kappa}$ is non-increasing in $r$.
\end{lem}
\begin{proof}
Let $u_1, u_2, v$ be points on $\mu_v$
with $0<r_{u_1}<r_{u_2}<r_{v}$. By Lemma~\ref{lem: approx by segments}
the ray $\xi_{u_i}$ is the limit of geodesics segments that join
$u_i$ with points $\tau_{v}(s)$ as $s\to\infty$.
The segments $[u_1,\tau_{v}(s)]$, $[u_2,\tau_{v}(s)]$
only intersect at the endpoint $\tau_{v}(s)$ for if they
intersect at a point $z$, then $z$ is a cut point for
$\tau_v(s)$, so $[\tau_{v}(s), u_i]$ cannot be minimizing.
Hence the geodesic triangle
with vertices $u_1$, $v$, $\tau_{v}(s)$ contains
the geodesic triangle with vertices
$u_2$, $v$, $\tau_{v}(s)$.
Since $G_m\ge 0$, the former triangle has larger
total curvature, which is finite as $M_m$ has finite
total curvature.
As $m$ only vanishes at $0$, concavity of $m$ implies that
$m$ is non-decreasing.
If $m$ is unbounded, Clairaut's relation
implies that the angles at $\tau_{v}(s)$ tend to zero as $s\to\infty$.
By the Gauss-Bonnet theorem $\kappa_{\xi_1(0)}-\kappa_{\xi_2(0)}$
equals the total curvature of the ``ideal'' triangle
with sides $\xi_1$, $\xi_2$, $[u_1,u_2]$. Thus
$\hat\kappa(r_{u_1})\ge\hat\kappa(r_{u_2})$ with equality if and only if
$G_m$ vanishes on $[r_{u_1}, \infty)$.
If $m$ is bounded, then $\int_1^\infty m^{-2}=\infty$, so
by~\cite[Proposition 1.7]{Tan-ball-poles-ex}
the only ray emanating from $q$ is $\mu_q$ so that
$\hat\kappa=0$ on $M_m\setminus\{o\}$.
For future use note that in this case the angle formed by
$\mu_q=\xi_q$ and $[q,\tau_q(s)]$ tends to zero as $s\to\infty$, so
Clairaut's relation together with boundedness of $m$
imply that the angle at $\tau_q(s)$ in the bigon with sides
$[q,\tau_q(s)]$ and $\tau_q$
also tends to zero as $s\to\infty$.
\end{proof}
\begin{rmk} By the above proof if $G_m\ge 0$ and
$m^{-2}$ is integrable on $[1, \infty)$, then
$\hat\kappa(r_{1})=\hat\kappa(r_{2})$ for some $r_2>r_1$ if and only if
$G_m$ vanishes on
$[r_1, \infty)$.
\end{rmk}
\begin{proof}[Proof of Theorem~\ref{thm-intro: nonneg curv}]
(i)
Since rays converge to rays,
${\mathfrak C}_m$ is closed.
As $o\in{\mathfrak C}_m$, rotational symmetry and
Lemma~\ref{lem: gauss bonnet nonneg}
implies that ${\mathfrak C}_m$ is a closed ball.
(ii)
Since $m$ is concave and positive, it is non-decreasing, so
$\displaystyle{\liminf_{r\to\infty} m}>0$, and the claim follows from
Lemma~\ref{lem: crit is o}.
(iii)
We prove the contrapositive that $M_m={\mathfrak C}_m$ if and only if
$m^\prime(\infty)\ge \frac{1}{2}$.
Note that the latter
is equivalent to $c(M_m)\le\pi$, where $c(Z)$ denotes the
total curvature of a subset $Z\subseteq M_m$ which varies in $[0,2\pi]$.
Suppose $c(M_m)\le\pi$. Fix $q\neq o$, and consider the segments
$[q,\tau_q(s)]$ that by Lemma~\ref{lem: approx by segments}
converge to $\xi_q$ as $s\to\infty$. Consider the bigon
bounded by $[q,\tau_q(s)]$ and its symmetric image under the
reflection that fixes $\tau_q\cup\mu_q$.
As in the proof of Lemma~\ref{lem: gauss bonnet nonneg}
we see that the angle at $\tau_q(s)$ goes to zero as
$s\to\infty$, so the sum of angles
in the bigon tends to $2(\pi-\hat{\kappa}(r_q))$, which
by the Gauss-Bonnet theorem cannot
exceed $c(M_m)\le\pi$. We conclude that
$\hat{\kappa}(r_q)\ge\frac{\pi}{2}$, so $q\in{\mathfrak C}_m$.
Conversely, suppose that ${\mathfrak C}_m=M_m$. Given $\varepsilon>0$
find a compact rotationally symmetric
subset $K\subset M_m$ with $c(K)>c(M_m)-\varepsilon$.
Fix $q\neq o$ and consider the rays
$\xi_{\mu_q(s)}$ as $s\to\infty$. If all these rays intersect
$K$, then they subconverge to a line~\cite[Lemma 6.1.1]{SST},
so by the splitting theorem $M_m$ is the standard $\mathbb R^2$,
and $c(M_m)=0<\pi$. Thus we can assume that there is
$v$ on the ray $\mu_q$ such that $\xi_{v}$
is disjoint from $K$. Therefore, if $s$ is large enough, then $K$ lies inside the bigon bounded by $[v,\tau_v(s)]$ and its symmetric
image under the reflection that fixes $\tau_q\cup\mu_q$.
The sum of angles
in the bigon tends to $2(\pi-\hat{\kappa}(r_{v}))$,
and by the Gauss-Bonnet theorem it is bounded below by $c(K)$.
Since $v\in{\mathfrak C}_m$, we have $\hat{\kappa}(r_{v})\ge \frac{\pi}{2}$,
and hence $c(K)\le\pi$. Thus $c(M_m)<\pi+\varepsilon$, and since $\varepsilon$
is arbitrary, we get $c(M_m)\le\pi$, which completes the
proof of (iii).
(iv) Since $R_m$ is finite, $m^\prime(\infty)<\frac{1}{2}$ by part (iii).
As $m^\prime(0)=1$, the equation
$m^\prime(x)=\frac{1}{2}$ has a solution $\rho_m$.
As $G_m\ge 0$, the function $m^\prime$ is non-increasing, so
uniqueness of the solution is equivalent to positivity of $G_m(\rho_m)$.
Since $M_m$ is von Mangoldt, $G_m(\rho_m)>0$ for otherwise
$G_m$ would have to vanish for $r\ge \rho_m$, implying
$m^\prime(\infty)=m^\prime(\rho_m)=\frac{1}{2}$,
so $R_m$ would be infinite.
Now we show that $\rho_m> R_m$. This is clear
if $R_m=0$ because $\rho_m\ge 0$ and
$m^\prime(0)=1\neq\frac{1}{2}=m^\prime(\rho_m)$.
Suppose $R_m>0$.
Then $m^{-2}$ is integrable by Lemma~\ref{lem: crit is o},
so $m^\prime>0$ everywhere by the proof of Lemma~\ref{lem: m growth}.
Hence for any $r_v\ge \rho_m$ we have
$m(r_v)\ge m(\rho_m)$, which implies $t\,m(r_v)> m(\rho_m)$
for all $t>1$.
Thus $m^{-1}(t\,m(r_v))> m^{-1}(m(\rho_m))=\rho_m$.
Applying $m^\prime$ to the inequality, we get in notations
of Proposition~\ref{prop: G>0 and monotonicity} that
$l(t,r_v)<m^\prime (\rho_m)=\frac{1}{2}$,
where the inequality is strict because $G_m(r_m)>0$ by part (iv).
Now (\ref{form: zero curv turn angle}) below implies
\[
T_{\gamma_v}=\int_1^{\infty}\frac{dt}{l(t,r_v)\,t\,\sqrt{t^2-1}}>
\int_1^\infty\!\!\frac{2\,dt}{t\,\sqrt{t^2-1}}=
\pi.
\]
Since $M_m$ is von Mangoldt, $v\notin{\mathfrak C}_m$
by Lemma~\ref{lem: crit point and ray tangent to parallel}.
In summary, if $r_v\ge \rho_m$, then $v\notin{\mathfrak C}_m$, so
$\rho_m> R_m$.
(v) Since $R_m$ is positive and finite, and $M_m$ is von Mangoldt,
there are geodesics
tangent to parallels whose turn angles are $\le \pi$, and $>\pi$,
respectively. By Proposition~\ref{prop: G>0 and monotonicity}
the turn angle is monotone with respect to $r$, so let $r_q$ be the (finite) supremum
of all $x$ such that $\int_x^\infty F_{m(x)}<\pi$. Since ${\mathfrak C}_m$ is closed,
$q\in{\mathfrak C}_m$ so that $T_{\gamma_q}\le \pi$. In fact,
$T_{\gamma_q}=\pi$ for if $T_{\gamma_q}<\pi$, then $r_q$ is not maximal
because by
Theorems~\ref{thm-intro: away} and \ref{thm: max devia vs turn angle}
the set of points $q$ with $T_{\gamma_q}<\pi$ is open in $M_m$.
If $G_m(r_q)>0$, then by monotonicity $r_q$ is a unique solution of
$T_{\gamma_q}=\pi$. If $G_m(r_q)=0$, then
$G_m\vert_{[r_q,\infty)}=0$ as $M_m$ is von Mangoldt, so
(\ref{form: zero curv turn angle})
implies that the turn angle of each $\gamma_v$ with $r_v\ge r_q$ equals
$\frac{\pi}{2m^\prime(r_q)}$. So $m^\prime(r_q)=\frac{1}{2}$
but this case cannot happen as $R_m$ is infinite by (iii).
\end{proof}
In preparation for a proof of
Theorem~\ref{thm-intro: crit is soul} we
recall that the Cheeger-Gromoll soul construction with basepoint $q$,
described e.g. in~\cite[Theorem V.3.4]{Sak-book},
starts by deleting the horoballs associated with
all rays emanating from $q$,
which results in a compact totally convex subset. The next step
is to consider the points of this subset which are at maximal distance
from its boundary, and these points in turn form a compact totally
convex subset, and after finitely many iterations the process
terminates in a subset with empty boundary, called a soul.
As we shall see below, if $G_m\ge 0$, then the soul construction
with basepoint $q\in {\mathfrak C}_m\setminus\{o\}$ takes
no more than two steps; more precisely, deleting
the horoballs for rays emanating from $q$
results either in $\{q\}$ or in a segment with $q$
as an endpoint. In the latter case the soul is the midpoint
of the segment.
In what follows we let $B_\sigma$ denote the (open)
horoball for a ray $\sigma$ with $\sigma(0)=q$, i.e.
the union over $t\in [0,\infty)$
of the metric balls of radius $t$ centered at $\sigma(t)$.
Let $H_\sigma$ denote the complement of $B_\sigma$ in the
ambient complete Riemannian manifold.
\begin{lem}
\label{lem: horoball} Let $\sigma$ be a ray in a complete
Riemannian manifold $M$, and let $q=\sigma(0)$.
Then for any nonzero $v\in T_q M$ that makes an acute angle with
$\sigma$, the point $\exp_q(tv)$ lies in the horoball
$B_\sigma$ for all small $t>0$.
\end{lem}
\begin{proof} This follows from
the definition of a horoball for if
$\Upsilon$ denotes the image of $t\to \exp_q(tv)$, then
$\displaystyle{\lim_{s\to +0}\frac{d(\sigma(s), \Upsilon)}{d(\sigma(s), q)}}=
\sin \angle(\upsilon^\prime(0), \sigma^\prime(0)) < 1$,
so $B_\sigma$ contains a subsegment of $\Upsilon\,\mbox{--}\,\{q\}$
that approaches $q$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-intro: crit is soul}]
For $q\in{\mathfrak C}_m$, let $C_q$ denote the complement in $M_m$ of the union of
the horoballs for rays that start at $q$; note that $C_q$ is compact and totally convex. If $C_q$ equals $\{q\}$, then $q$ is a soul. Otherwise,
$C_q$ has positive dimension and $q\in\partial C_q$.
Set $\gamma:=\xi_q$; thus $\gamma$ is a ray.
{\bf Case 1.} Suppose $\frac{\pi}{2}<\hat{\kappa}(r_q)<\pi$.
Let $\bar\gamma$ be the clockwise ray that is mapped to $\gamma$
by the isometry fixing the meridian through $q$.
We next show that $q$ is the intersection of the complements of the
horoballs for rays $\mu_q$, $\gamma$, $\bar\gamma$, implying that $q$ is a soul
for the soul construction that starts at $q$.
As $\kappa_{\gamma(0)}>\frac{\pi}{2}$, any nonzero $v\in T_qM_m$ forms
angle $<\frac{\pi}{2}$ with one of
$\mu^\prime(0)$, $\gamma^\prime(0)$, $\bar{\gamma}^\prime(0)$,
so $\exp_q(tv)$ cannot lie in the intersection of
$H_{\mu_q}$, $H_\gamma$, $H_{\bar\gamma}$ for small $t$, and
since the intersection is totally convex, it is $\{q\}$.
{\bf Case 2.}
Suppose $\hat{\kappa}(r_q)=\frac{\pi}{2}$, so that $\gamma=\gamma_q$,
and suppose that $G_m$ does not vanish along $\gamma$.
By symmetry and Lemma~\ref{lem: horoball},
it suffices to show that every point of the segment $[o,q)$ near $q$
lies in $B_\gamma$. Let $\alpha$ be the ray from $o$ passing through $q$.
The geodesic $\gamma$ is orthogonal to $\alpha$, and it suffices to show
that there is a focal point of $\alpha$ along $\gamma$
for if $\gamma(t_0)$ is the first focal point, and $t>t_0$
is close to $t_0$, then there is a variation of $\gamma\vert_{[0, t]}$
through curves shorter than $\gamma\vert_{[0, t]}$
that join $\gamma(t)$ to points of $\alpha-\{q\}$ near
$q$~\cite[Lemma III.2.11]{Sak-book}
so these points lie in $B(\gamma(t), t)\subset B_\gamma$.
Any $\alpha$-Jacobi field along $\gamma$ is of the form $jn$ where
$n$ is a parallel nonzero normal vector field along $\gamma$
and $j$ solves $j^{\prime\prime}(t)+G_m(r_{\gamma(t)})j(t)=0$,
$j(0)=1$, $j^\prime(0)=0$. Since $G_m\ge 0$, the function $j$
is concave, so due to its initial values, $j$ must vanish
unless it is constant. The point where $j$ vanishes is focal.
If $j$ is constant, then $G_m=0$ along $\gamma$, which is ruled out by assumption.
{\bf Case 3.}
Suppose $\hat{\kappa}(r_q)=\pi$, i.e. $\gamma=\tau_q$.
For any vector $v\in T_q M_m$ pointing inside $C_q$,
for small $t$ the point $\exp_q(tv)$
is not in the horoballs for $\mu_q$ and $\tau_q$, and hence $v$
is tangent to a parallel, i.e. $C_q$ is a subsegment of
the geodesic $\alpha$ tangent to the parallel through $q$.
As $C_q$ lies outside the horoballs for $\mu_q$ and $\tau_q$,
these rays there cannot contain focal points of $\alpha$, implying that
$G_m$ vanishes along $\mu_q$ and $\tau_q$, and hence everywhere,
by rotational symmetry, so that $M_m$ is the standard $\mathbb R^2$,
and $q$ is a soul.
{\bf Case 4.} Suppose $\hat{\kappa}(r_q)=\frac{\pi}{2}$,
so that $\gamma=\gamma_q$,
and suppose that $G_m$ vanishes along $\gamma$.
By rotational symmetry
$G_m(r)=0$ for $r\ge r_q$, so $m(r)=a(r-r_q)+m(r_q)$ for $r\ge r_q$
where $a>0$, as $m$ only vanishes at $0$.
The turn angle of $\gamma$ can be computed explicitly as
\begin{equation}
\label{form: zero curv turn angle}
\int_{x}^\infty\!\!\!\frac{dr}
{
m(r)\,
\sqrt{\frac{m(r)^2}{m(x)^2}-1}
}=
\int_1^\infty\!\!\frac{dt}{a\, t\,\sqrt{t^2-1}}=
-\frac{1}{a}\,\mathrm{arccot}(\sqrt{t^2-1})\Big{|}_1^\infty=
\frac{\pi}{2a}
\end{equation}
where $x:=r_q$. Since $\gamma$ is a ray, we deduce that $a\ge \frac{1}{2}$.
Let $z\le x$ be the smallest number such that
$m^\prime\vert_{[z,\infty)}=a$;
thus there is no neighborhood of $z$ in $(0,\infty)$
on which $G_m$ is identically zero.
Note that $m(r)=a(r-z)+m(z)$ for $r\ge z$, so
the surface $M_m\,\mbox{--}\, B(o,z)$ is isometric
to $C\,\mbox{--}\, B\left(\bar o, \frac{m(z)}{a}\right)$ where $C$ is the cone with apex $\bar o$
such that cutting $C$ along the meridian from $\bar o$
gives a sector in $\mathbb R^2$ of angle $2\pi a$
with the portion inside the radius
$\frac{m(z)}{a}$ removed.
Since $\gamma_q$ is a ray, Lemma~\ref{lem: horoball} implies the
existence of a neighborhood $U_q$ of $q$ such that
each point in $U_p\,\mbox{--}\, [o,q]$ lies in a horoball
for a ray from $q$.
We now check that $o$ lies in the horoball of $\gamma_q$.
Concavity of $m$ implies that the graph of $m$
lies below its tangent line at $z$, so evaluating the tangent line
at $r=0$ and using $m(0)=0$ gives $\frac{m(z)}{a}>z$. The Pythagorean
theorem in the sector in $\mathbb R^2$ of angle $2\pi a$ implies that
\[
d_{M_m}(\gamma_q(s), o)=\sqrt{s^2+\left(x-z+\frac{m(z)}{a}\right)^2}+
z-\frac{m(z)}{a}
\]
which is $<s$ for large $s$, implying that $o$
is in the horoball of $\gamma_q$.
To realize $q$ as a soul, we need to look at
the soul construction with arbitrary basepoint $v$, which
starts by considering the complement in $M_m$ of the union of the horoballs for all rays from $v$, which by the above is either $v$ or
a segment $[u,v]$ contained in $(o,v]$, where $u$
is uniquely determined by $v$.
It will be convenient to allow for degenerate segments for which $u=v$;
with this convention the soul is the midpoint of $[u,v]$.
Since $z$ is the smallest such that $G_m\vert_{[z,\infty)}=0$,
the focal point argument of Case $2$
shows that $u=v$ when $0<r_v<z$.
Set $y:=r_v$, and let $e(y):=r_u$; note that $0<e(y)\le y$, and the midpoint of $[u,v]$ has $r$-coordinate $h(y):=\frac{y+e(y)}{2}$.
To realize each point of $M_m$ as a soul, it suffices to show that
each positive number is in the image of $h$.
Since $h$ approaches zero as $y\to 0$ and approaches infinity as
$y\to\infty$, it is enough to show that $h$ is continuous and then
apply the intermediate value theorem.
Since $e(y)=y$ when $0<y<z$, we only need to verify continuity of $e$
when $y\ge z$.
Let $v_i$ be an arbitrary sequence of points on $\alpha$ converging to $v$,
where as before $\alpha$ is the ray from $o$ passing through $q$.
Set $v_i:=r_{v_i}$. Arguing by contradiction suppose that $e(y_i)$
does not converge to $e(y)$.
Since $0<e(y_i)\le y_i$ and $y_i\to y$,
we may pass to a subsequence such that $e(y_i)\to e_\infty\in [0,y]$.
Pick any $w$ such that $r_w$ lies between $e_\infty$ and $e(y)$.
Thus there is $i_0$ such that
either $e(y_i)<r_w<e(y)$ for all $i>i_0$, or
$e(y)<r_w<e(y_i)$ for all $i>i_0$.
As $y\ge z$, we know that
$G_m$ vanishes along $\gamma_v$, so every $\alpha$-Jacobi field along $\gamma_v$
is constant. Therefore,
the rays $\gamma_{v_i}$ converge uniformly (!) to $\gamma_v$, as $v_i\to v$,
and hence their Busemann functions $b_i$, $b$ converge pointwise.
Thus $b_i(w)\to b(w)$, but we have chosen $w$
so that $b(w)$, $b_i(w)$ are all nonzero, and
$\mathrm{sign}(b(w))=-\mathrm{sign}(b_i(w))$, which gives a contradiction
proving the theorem.
\end{proof}
\begin{rmk}
In Cases $1, 2, 3$ the soul construction terminates in one step,
namely, if $q\in {\mathfrak C}_m$, then $\{q\}$
is the result of removing the horoballs for all rays
that start at $q$. We do not know whether the same
is true in Case $4$ because the basepoint $v$
needed to produce the soul $q$ is found implicitly, via an intermediate
value theorem, and it is unclear how $v$ depends on $q$,
and whether $v=q$.
\end{rmk}
\begin{rmk}\label{rmk: poles when m'=.5}
Let $M_m$ be as in Case $4$ with
$m^\prime\vert_{[z,\infty)}=\frac{1}{2}$.
If $M_m$ is von Mangoldt, then no point
$q$ with $r_q\ge z$ is a pole because by
(\ref{form: zero curv turn angle}) the turn angle of $\gamma_q$ is $\pi$,
which by Theorem~\ref{thm: max devia vs turn angle} cannot happen for a pole.
\end{rmk}
\section{Smoothed cones made von Mangoldt}
\label{sec: smoothed cones}
\begin{proof}[Proof of Theorem~\ref{thm-intro: slope realized}]
It is of course easy to find a von Mangoldt plane $g_{m_x}$ that has zero curvature
near infinity, but prescribing the slope of $m^\prime$ there takes more effort.
We exclude the trivial case $x=1$ in which $m(r)=r$ works.
For $u\in [0,\frac{1}{4}]$ set $K_u(r)=\frac{1}{4(r+1)^2}-u$, and let
$m_u$ be the unique solution of (\ref{form: ode}) with $K=K_u$. Then
$g_{m_u}$ is von Mangoldt. For $u>0$ let $z_u\in [0,\infty)$ be the unique zero
of $K_u$; note that $z_u$ is the global
minimum of $m_u^\prime$, and $z_u\to\infty$ as $u\to 0$.
\begin{lem}
\label{lem: in prescribed slope}
The function $u\to m^\prime_u(z_u)$
takes every value in $(0,1)$ as $u$ varies in $(0,\frac{1}{4})$.
\end{lem}
\begin{proof}
One verifies that $m_0(r)=\ln(r+1)\,\sqrt{r+1}$, i.e. the right hand side
solves (\ref{form: ode}) with $K=K_0$.
Then $m_0^\prime=\frac{2+\ln(r+1)}{2\sqrt{r+1}}$ is a positive function
converging to zero as $r\to \infty$.
By Sturm comparison $m_u\ge m_0>0$
and $m_u^\prime\ge m_0^\prime>0$.
We now show that $m_u^\prime(z_u)\to 0$ as $u\to +0$.
To this end fix an arbitrary $\varepsilon>0$.
Fix $t_\varepsilon$ such that $m_0^\prime(t_\varepsilon)<\varepsilon$.
By continuous dependence on parameters
$(m_u, m^\prime_u)$ converges to $(m_0, m_0^\prime)$ uniformly
on compact sets as $u\to 0$. So for all small $u$
we have $m_u^\prime(t_\varepsilon)<\varepsilon$ and also $t_\varepsilon<z_u$.
Since $m^\prime_u$ decreases on $(0,z_u)$,
we conclude that $0<m_u^\prime(z_u)<m_u^\prime(t_\varepsilon)<\varepsilon$,
proving that $m_u^\prime(z_u)\to 0$ as $u\to +0$.
On the other hand, $m_\frac{1}{4}^\prime(z_\frac{1}{4})=1$
because $z_\frac{1}{4}=0$ and by the initial condition
$m_\frac{1}{4}^\prime(0)=1$.
Finally, the assertion of the lemma follows from
continuity of the map $u\to m_u^\prime(z_u)$, because then
it takes every value within $(0,1)$ as $u$ varies in $(0,\frac{1}{4})$.
(To check continuity of the map fix $u_*$, take an arbitrary $u\to u_*$
and note that $z_u\to z_{u_*}$, so since $m^\prime_u$ converges
to $m_{u_*}^\prime$ on compact subsets, it does so on a neighborhood of $z_{u_*}$,
so $m_u^\prime(z_u)$ converges to $m_{u_*}^\prime(z_{u_*})$).
\end{proof}
Continuing the proof of the theorem,
fix an arbitrary $u>0$.
The continuous function $\max(K_u, 0)$ is decreasing and smooth on $[0,z_u]$
and equal to zero on $[z_u,\infty)$. So
there is a family of non-increasing smooth functions $G_{u,\varepsilon}$
depending on small parameter $\varepsilon$
such that $G_{u,\varepsilon}=\max(K_u, 0)$ outside the $\varepsilon$-neighborhood of $z_u$.
Let $m_{u,\varepsilon}$ be the unique solution of (\ref{form: ode}) with $K=G_{u,\varepsilon}$;
thus $m^\prime_{u,\varepsilon}(r)=m^\prime_{u,\varepsilon}(z_u+\varepsilon)$ for all $r\ge z_u+\varepsilon$.
If $\varepsilon$ is small enough, then $G_{u,\varepsilon}\le K_0$, so $m_{u,\varepsilon}\ge m_0>0$ and
$m^\prime_{u,\varepsilon}\ge m^\prime_0>0$.
By continuous dependence on parameters,
the function $(u,\varepsilon)\to m^\prime_{u,\varepsilon}$ is continuous, and moreover
$m^\prime_{u,\varepsilon}(z_u+\varepsilon)\to m^\prime_{u}(z_u)$ as $\varepsilon\to 0$, and $u$
is fixed.
Fix $x\in (0,1)$. By Lemma~\ref{lem: in prescribed slope}
there are positive $v_1$, $v_2$
such that
$m_{v_1}^\prime(z_{v_1})<x<m_{v_2}^\prime(z_{v_2})$.
Letting $u$ of the previous paragraph to be $v_1$, $v_2$, we find $\varepsilon$
such that $m^\prime_{v_1,\varepsilon}(z_{v_1}+\varepsilon)<x<m^\prime_{v_2,\varepsilon}(z_{v_2}+\varepsilon)$,
so by the intermediate value theorem there is $u$ with
$m^\prime_{u,\varepsilon}(z_u+\varepsilon)=x$. Then the metric $g_{m_{u,\varepsilon}}$ has
the asserted properties for $\rho=z_u+\varepsilon$.
\end{proof}
\section{Other applications}
\label{sec: other proofs}
\begin{proof}[Proof of Lemma~\ref{lem-intro: crit}]
Assuming $\hat r(\hat q)\notin r({\mathfrak C}_m)$ we will show that $\hat q$
is not a critical point of $\hat r$.
Since $\hat M$ is complete and noncompact, there is a ray $\hat\gamma$ emanating
from $\hat q$.
Consider the comparison triangle
$\Delta(o, q, q_i)$ in $M_m$ for any geodesic triangle with vertices
$\hat o$, $\hat q$, $\hat\gamma(i)$. Passing to a subsequence, arrange
so that the segments $[q, q_i]$ subconverge to a ray, which we denote by $\gamma$.
Since $q\notin {\mathfrak C}_m$, the angle formed by $\gamma$ and $[q,o]$ is $>\frac{\pi}{2}$,
and hence for large $i$
the same is true for the angles formed by $[q,q_i]$ and $[q,o]$.
By comparison, $\hat\gamma$ forms angle $>\frac{\pi}{2}$ with
any segment joining $\hat q$ to $\hat o$, i.e. $\hat q$ is not
a critical point of $\hat r$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{intro-thm: poles}]
(a)
Let $P_m$ denote the set of poles;
it is a closed metric ball~\cite[Lemma 1.1]{Tan-cut}.
Moreover, $P_m$ clearly lies in the connected component
$A_m^o$ of $A_m\cup\{o\}$ that contains $o$, and hence in the component
of ${\mathfrak C}_m$ that contains $o$.
By Theorem~\ref{thm-intro: away}
$A_m$ is open in $M_m$, so $A_m\cup\{o\}$ is locally path-connected,
and hence $A_m^o$ is open in $M_m$.
If $P_m$ were equal to $A_m^o$, the latter would be closed,
implying $A_m^o=M_m$, which is impossible as the ball has finite radius.
(b) The "if'' direction is trivial as $P_m\subset{\mathfrak C}_m$. Conversely,
if ${\mathfrak C}_m\neq\{o\}$, then by Lemma~\ref{lem: crit is o}
$m^{-2}$ is integrable and
$\underset{r\to\infty}{\liminf}\, m(r)>0$, so
$R_p>0$~\cite{Tan-ball-poles-ex}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm-intro: neg curv}]
By assumption there is a point of negative curvature, and since
the curvature is non-increasing, outside a compact subset the curvature is
bounded above by a negative constant. As $\displaystyle{\liminf_{r\to\infty} m(r)}>0$,
$m$ is bounded below by a positive constant outside any neighborhood of $0$,
so $\int_0^\infty m=\infty$.
Hence the total curvature
$2\pi\int_0^\infty G_m(r)\,m(r)\,dr$ is $-\infty$.
Hence there is a metric ball $B$ of finite positive
radius centered at $o$ such that the total curvature of $B$
is negative, and such that no point of $G_m\ge 0$ lies outside $B$.
By~\cite[Theorem 6.1.1, page 190]{SST}, for any $q\in M_m$
the total curvature of the set
obtained from $M_m$ by removing all rays that start at $q$ is in $[0,2\pi]$.
So for any $q$ there is a ray that starts at $q$ and intersects $B$.
If $q$ is not in $B$, then
the ray points away from infinity, so $q\in A_m$ and
any point on this ray is in ${\mathfrak C}_m$.
Thus $M_m\,\mbox{--}\, A_m$ lies in $B$.
Since ${\mathfrak C}_m\neq\{o\}$, Theorem~\ref{intro-thm: poles} implies that $R_p>0$.
Letting $q$ run to infinity the rays subconverge to a line that intersects $B$
(see e.g.~\cite[Lemma 6.1.1, page 187]{SST}.
If $m^\prime(r_p)=0$, then the parallel through $p$ is a geodesic but not a
ray, so Lemma~\ref{lem: crit point and ray tangent to parallel}
implies that no point on the parallel through $p$ is in ${\mathfrak C}_m$. Since
${\mathfrak C}_m$ contains $o$ and all points outside a compact set,
${\mathfrak C}_m$ is not connected; the same argument proves that
$A_m$ is not connected.
\end{proof}
\begin{ex}
\label{ex: m' vanishes}
Here we modify~\cite[Example 4]{Tan-cut} to construct a von Mangoldt
plane $M_m$ such that $m^\prime$ has a zero,
and neither $A_m$ nor ${\mathfrak C}_m$ is connected.
Given $a\in (\frac{\pi}{2}, \pi)$
let $m_0(r)=\sin r$ for $r\in [0,a]$, and define $m_0$ for $r\ge a$ so that $m_0$
is smooth, positive, and $\inf\lim_{r\to\infty}\,m_0>0$.
Thus $K_0:=-\frac{m_0^{\prime\prime}}{m_0}$ equals $1$ on $[0,a]$.
Let $K$ be any smooth non-increasing function with $K\le K_0$ and $K\vert_{[0,a]}=1$.
Let $m$ be the solution of (\ref{form: ode});
note that $m(r)=\sin(r)$ for $r\in [0,a]$ so that $m^\prime$ vanishes
at $\frac{\pi}{2}$.
By Sturm comparison $m\ge m_0>0$, and hence $M_m$ is a von Mangoldt plane.
Since $m^\prime(a)<0$ and $m>0$ for all $r>0$,
the function $m$ cannot be concave, so $K=G_m$ eventually becomes negative,
and Theorem~~\ref{thm-intro: neg curv} implies that $A_m$ and ${\mathfrak C}_m$
are not connected.
\end{ex}
\begin{ex}
\label{ex: m' positive but non-connected}
Here we construct a von Mangoldt plane such that $m^\prime>0$ everywhere
but $A_m$ and ${\mathfrak C}_m$ are not connected.
Let $M_{n}$ be a von Mangoldt plane such that $G_n\ge 0$ and
$n^\prime>0$ everywhere, and $R_n$ is finite
(where $R_n$ is the radius of the ball ${\mathfrak C}_n$).
This happens e.g. for any paraboloid, any
two-sheeted hyperboloid with $n^\prime(\infty)<\frac{1}{2}$, or
any plane constructed in Theorem~\ref{thm-intro: slope realized}
with $n^\prime(\infty)<\frac{1}{2}$.
Fix $q\notin {\mathfrak C}_n$. Then $\gamma_q$ has turn angle $>\pi$, so there is
$R>r_q$ such that $\int_{r_q}^R F_{n(r_q)}>\pi$. Let $G$ be any smooth
non-increasing function such that $G=G_n$ on $[0,R]$ and $G(z)<0$ for
some $z>R$. Let $m$ be the solution of (\ref{form: ode}) with $K=G$.
By Sturm comparison $m\ge n>0$ and $m^\prime\ge n^\prime>0$
everywhere; see Remark~\ref{rmk: sturm derivative}.
Since $m=n$ on $[0,R]$, on this interval
we have $F_{m(r_q)}=F_{n(r_q)}$, so in the von Mangoldt plane $M_m$
the geodesic $\gamma_q$ has turn angle $>\pi$, which implies that no
point on the parallel through $q$ is in ${\mathfrak C}_m$.
Now parts (3)-(4) of Theorem~~\ref{thm-intro: neg curv}
imply that $A_m$ and ${\mathfrak C}_m$
are not connected.
\end{ex}
\begin{thm}
\label{thm: neck}
Let $M_m$ be a von Mangoldt plane such that
$m^\prime\vert_{[0,y]}>0$ and $m^\prime\vert_{[x,y]}<\frac{1}{2}$.
Set $f_{m,x}(y):=m^{-1}\left(\cos(\pi b)\, m(y)\right)$,
where $b$ is the maximum of $m^\prime$ on $[x,y]$.
If $x\le f_{m,x}(y)$, then $r({\mathfrak C}_m)$ and $[x, f_{m,x}(y)]$ are disjoint.
\end{thm}
\begin{proof} Set $f:=f_{m,x}$.
Arguing by contradiction assume there is
$q\in {\mathfrak C}_m$ with $r_q\in [x, f(y)]$. Then $\gamma_q$ has turn angle
$\le \pi$, so if $c:=m(r_q)$, then
\[
\pi\ge\int_{r_q}^{\infty}\frac{c\,dr}{m\,\sqrt{m^2-c^2}}>
\int_{r_q}^{y}\frac{c\,dr}{m\,\sqrt{m^2-c^2}}=
\int_c^{m(y)}\!\!\!\! \frac{c\,dm}{m^\prime(r)\,m\,\sqrt{m^2-c^2}}
\ge
\]
\[
\int_c^{m(y)}\!\!\!\! \frac{c\,dm}{b\,m\,\sqrt{m^2-c^2}}=
\frac{1}{b}\,\mathrm{arccos}\left(\frac{c}{m(y)}\right)
\]
so that $\pi b>\mathrm{arccos}\left(\frac{c}{m(y)}\right)$,
which is equivalent to
$\cos(\pi b)\, m(y)< m(r_q)$.
On the other hand, $m(f(y))$ is in the interval
$[0,m(y)]$ on which $m^{-1}$ is increasing, so $f(y)<y$,
and therefore $m$ is increasing on $[x, f(y)]$.
Hence $r_q<f (y)$ implies $m(r_q)<m(f(y))=\cos(\pi b)\, m(y)$,
which is a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{intro-thm: neck nneg}]
We use notation of Theorem~\ref{thm: neck}.
The assumptions on $n$ imply $n^\prime>0$,
$n^\prime\vert_{[x,\infty)}<\frac{1}{2}$, and
$b=n^\prime(x)$. Hence $f_{n,x}$ is an increasing
smooth function of $y$ with $f_{n,x}(\infty)=\infty$.
In particular, if $y$ is large enough, then $f_{n,x}(y)>z>x$;
fix $y$ that satisfies the inequality.
Now if $M_m$ is any von Mangoldt plane with $m=n$ on $[0,y]$,
then $f_{m,x}(y)=f_{n,x}(y)$, so $M_m$ satisfies the assumptions
of Theorem~\ref{thm: neck}, so $[x,z]$ and $r({\mathfrak C}_m)$ are disjoint.
\end{proof}
|
2,869,038,154,917 | arxiv | \section*{Introduction}
In addition to its unique electronic properties and optical transparency, that render it potential applications in spintronics and photovoltaic devices\cite{Zhang2014a}, graphene has been recognized as the ultimate mono-atomic protective membrane of metal surfaces against corrosion\cite{Dedkov2008a,Dedkov2008,Bohm2014,Bunch2008}. The chemical vapor deposition (CVD) of graphene has by now been established as a scalable method for depositing graphene albeit with inevitable surface defects due to the non-epitaxial nature of the growth that proceeds at multiple points of nucleation\cite{Hofmann2015}. So, covering weak oxidizing metals (e.g. Ni, Co) with graphene can protect their surfaces over long periods to atmospheric exposure\cite{Weatherup2015}, because the metal-oxides formed at defects protect against further oxidation. On the other hand, for strong oxidizers (e.g., Fe or Eu) in atmospheric environment, corrosion through graphene-defects or other protective layers gradually spreads over the whole surface and even penetrates the bulk\cite{Weatherup2015,Anderson2017a}. Defect-free and epitaxial monoatomic layer of graphene has long been produced on SiC(0001) forming a continuous membrane over the whole surface including surface steps\cite{Forbeaux1998}.
With these advances, modifying the electronic properties of graphene has been investigated either by depositing inert metals\cite{ren2010,Forster2012} and metal-oxides\cite{Wang2008} or by intercalating between the graphene and the metal substrate\cite{Premlal2009,Schumacher2014,Voloshina2014,Huttmann2017} or between graphene and the SiC buffer-layer\cite{Sung2017}. Intercalation of metal donors or molecular acceptors into graphite is an old topic that culminated in recipes that enable control of the superstructures (staging phenomena), electrical conduction, superconductivity and even electrical energy storage in batteries (i.e., CF and CLi$_6$)\cite{Dresselhaus1981}. Thus, intercalation with magnetic metals is a route to modify interfacial magneto-electronic properties with potential applications in spintronics. Here, we report on the magnetic properties of intercalated Eu atoms between graphene and the SiC buffer-layer by employing synchrotron X-ray magnetic circular dichroism (XMCD). We also, report on the chemical stability of the buried Eu layer as the sample is exposed to air over a period of months. Recently, intercalation of Eu between Ir substrate and graphene (prepared by CVD) reveals that the structure and magnetic properties of the intercalated Eu depend on the coverage which does not seem to affect the electronic structure\cite{Schumacher2014}. However, a recent study shows that Eu intercalation between graphene and the SiC buffer layer modifies the $\pi-$band of graphene significantly\cite{Sung2017}. We note that besides the different substrates, the intercalated phases formed in SiC\cite{Sung2017} are of higher coverages than those reported on graphene/Ir metal\cite{Schumacher2014}.
\section*{Samples and Methods}
The substrate used in our studies, 6H-SiC(0001) purchased from Cree, Inc., is graphitized in ultra-high vacuum (UHV, $P \approx1\cdot10^{-10}$ Torr) by direct current heating of the sample to $\sim$1300 C (measured with an infrared pyrometer). Figure\ \ref{fig:STM}b shows a graphene layer with distinct 6$\times$6 superstructure commonly observed with graphene on SiC\cite{Forbeaux1998}. Metal intercalation is achieved by initial deposition of nominal several monolayers of Eu metal on a SiC supported graphene (see Fig.\ \ref{fig:STM}c) followed by annealing, leading to two competing processes namely, intercalation/diffusion of metal atoms through the graphene sheet and atom desorption from the graphene surface into the vacuum (see illustration in Fig.\ \ref{fig:STM}a). Slow step-wise annealing up to the metal desorption temperature provides conditions preferred for intercalation. After complete atom desorption, STM images show an undamaged graphene surface but with bright spots due to Eu clusters that are situated at the vertices of the 6$\times$6 superstructure (Fig.\ \ref{fig:STM}d-e, Fig.\ \ref{fig:STMclose}, and in the SI\cite{SI-EU}). The high-resolution STM images (Fig.\ \ref{fig:STMclose}) confirm that clusters are formed beneath the graphene, and that the cluster superstructure is rotated 30$^{\circ}$ with respect to the graphene.
Under further prolonged annealing up to 1200 C, Eu atoms de-intercalate and the initial graphene interface can be restored. This indicates that the density of an intercalated metal can be controlled in intercalation/de-intercalation cycling.
We note that lower annealing temperatures has been reported in the Ref.\ \cite{Sung2017} (120 C), confining the Eu diffusion between graphene and buffer layer, whereas annealing at 300 C shifts the Eu between the buffer and SiC and transforms the graphene to a bilayer. The annealing temperatures are higher in the current study resulting in a self-organized network of clusters of $\sim$25 atoms separated by at least 1.8 nm which behave independently in their magnetic response.
\begin{figure}
\centering \includegraphics [width = 76 mm] {G-Eu-STM_v4.pdf}
\caption{(Color online) (a) Schematic illustration of intercalation after deposition of Eu metal on graphene. During the annealing process, some atoms penetrate through the graphene and intercalate and some just evaporate. (b) STM image of a pristine graphene on a SiC(0001) surface showing the well established diffuse 6$\times$6 superlattice. (c) deposited Eu islands on graphene before intercalation. (d) Eu intercalated under graphene forming 2-3 nm clusters. (e) Eu clusters seem to randomly occupy the vertices on the superstructure grid.}
\label{fig:STM}
\end{figure}
The location of the intercalated metal whether between graphene and buffer layer or between buffer layer and SiC is an outstanding question. The metal position depends on the preparation conditions and dramatically affects the properties of the intercalated system. The use of high temperatures in the current study ($\sim$800 C) desorbs most of the deposited Eu and generates the cluster phase. Other phases are possible in the system for lower annealing temperatures. A similar cluster phase has also been observed for intercalated Au in graphene on SiC achieved at relatively high temperatures $\sim$700 C\cite{NarayananNair2016}. The Au cluster position is also defined by the 6$\times$6 supercell with average separation between the clusters $\sim$2.2 nm. Moreover, this study suggests that, the Au or Eu formed clusters between the buffer layer and graphene, not only explain the preference of nucleation to be at the vertices of the 6$\times$6 supercell but also that the cluster phase is a more general phenomenon of metal intercalation into graphene-SiC.
\begin{figure}
\centering \includegraphics [width = 76 mm] {G-Eu-STM_close_v2.pdf}
\caption{(Color online) (a) STM image of a higher density cluster region showing that the graphene can still be seen on top of the clusters. The superstructure is rotated 30$^{\circ}$ with respect to the graphene lattice. (b) Enlarged and enhancedA region of the pristine graphene from Fig.\ \ref{fig:STM} showing that the superstructure from the buffer layer-SiC interface (solid diamond) is rotated 30$^{\circ}$ with respect to the graphene (dashed line). }
\label{fig:STMclose}
\end{figure}
XMCD measurements are performed at the 4-ID-C beamline at the Advanced Photon Source (Argonne National Laboratory) in a chamber equipped with a high magnetic field ($<7$ T) produced by a split-coil superconducting magnet. Field dependence of the XMCD spectra are collected in helicity-switching mode in external magnetic fields applied parallel to the incident x-ray wave vector at energies that cover the Eu $M_4$ (1158 eV) and $M_5$ (1127 eV) binding energies. Measurements of x-ray absorption spectroscopy (XAS) signals are collected by total electron yield (TEY). For data analysis and normalization, the individual XAS, $\mu_+$ and $\mu_-$, are normalized by their respective monitors to compensate for incident-beam intensity variations. For the initial background subtraction, the XAS ($\mu_+$ and $\mu_-$) has a flat value subtracted such that the lowest energy (i.e. sufficiently far from the edge) is at 0 intensity, removing both background and offsets due to the beam. The total XAS ($\mu_++\mu_-$) is then scaled by a factor such that its maximum intensity is 1. That scale factor is then used to also scale the individual ($\mu_+$ and $\mu_-$) XAS. The XMCD signal is obtained from the difference between two XAS spectra of the left- and right-handed helicities, $\mu_+$ and $\mu_-$. More details on data reduction is provided elsewhere\cite{Anderson2017}. We note that our intercalated samples are removed from the ultra-high vacuum chamber and transported in air for the XMCD experiments. As we discuss below and in the SI\cite{SI-EU}, we have also tested the samples after exposure of 9 months in air.
\section*{Results and Discussion}
\begin{figure}\centering \includegraphics [width = 76 mm] {EuInterc-Eu2O3-TripleGraph_v3.pdf}
\caption{(Color online) The XAS, XMCD, and total XAS of the intercalated Eu (left) and \ce{Eu2O3} (right) at $B=5$ T and $T=15$ K. The total XAS signals for intercalated Eu and \ce{Eu2O3} are consistent with Eu$^{2+}$ and Eu$^{3+}$, respectively. }
\label{fig:EuXAS}
\end{figure}
Figure\ \ref{fig:EuXAS} shows the XAS, XMCD, and total XAS at the Eu $M_4$ and $M_5$ edges at $T =15$ K and $B = 5$ T for intercalated Eu (left) and for \ce{Eu2O3} (right). We measure \ce{Eu2O3} as a control to monitor possible oxidation of our sample as it is exposed to air. Each of the three signals shows a significant contrast between the two samples. Figures \ref{fig:EuXAS}a and \ref{fig:EuXAS}d show the XAS of the intercalated Eu and \ce{Eu2O3} with the latter exhibiting noticeable splitting of the $M_5$ peak, which has been documented as corresponding to Eu$^{3+}$ \cite{Thole1985a,Mizumaki2005}. However, the intercalated Eu has a very prominent difference between the $\mu_+$ and $\mu_-$ while the \ce{Eu2O3} has almost none. This leads to a strong XMCD signal for the intercalated (Fig. \ref{fig:EuXAS}b) but to nearly flat XMCD signal for the oxide (Fig. \ref{fig:EuXAS}e). The zero XMCD signal for \ce{Eu2O3} is expected for the non-magnetic Eu$^{3+}$ where $L=S=3$ and a total moment $J=0$\cite{Concas2011}.
The XMCD of the intercalated Eu enables to quantitatively determine the orbital, $\langle L_Z \rangle$, and spin, $\langle S_Z \rangle$, contributions to the total moment, $\langle J_Z \rangle$, of Eu$^{2+}$ via sum rules derived by Carra \textit{et. al}.\cite{Carra1993} as follows:
\begin{equation}
\langle L_Z \rangle=\frac{2(p+q)}{r}n_{\textsc{\tiny \emph{H}}}
\label{eq:LZ}
\end{equation}
and
\begin{equation}
\langle S_Z \rangle=\frac{2p-3q}{2r}n_{\textsc{\tiny \emph{H}}}-3\langle T_Z \rangle \approx \frac{2p-3q}{2r}n_{\textsc{\tiny \emph{H}}}
\label{eq:SZ}
\end{equation}
where $p=\int_{M_5}\mu_+-\mu_-$, $q=\int_{M_4}\mu_+-\mu_-$, $r=\int_{M_4+M_5}(\mu_++\mu_-)$, and $n_{\textsc{\tiny \emph{H}}}$ is the number of electron holes in the valence shell ($n_{\textsc{\tiny \emph{H}}}=7$ for Eu$^{2+}$) (it should be noted that our definition for $q$ differs from the $q$ used in Ref.\ \citen{Schumacher2014}). In Eq. \ref{eq:SZ}, the $\langle T_Z \rangle$ term vanishes due to the zero orbital moment. We note that a strong spin moment, $\langle S_Z \rangle$, and nearly zero orbital moment, $\langle L_Z \rangle$, are consistent with Hund's rules for Eu$^{2+}$ ($L=0 \mbox{; } S=J=7/2$)\cite{Schumacher2014,Carra1993,Crocombette1996,Wu1994} and thus $\langle S_Z \rangle=\langle J_Z \rangle$. Figure\ \ref{fig:EuField} shows moment calculations at $T = 15$ K as a function of magnetic field from +5 T to -5 T. Scans are conducted at both 20$^{\circ}$ and 90$^{\circ}$ angle between the magnetic field direction and the surface showing nearly paramagnetic-like behavior with no evidence of magnetic anisotropy. The dependence of the moment on magnetic field shown in Fig. \ref{fig:EuField} is similar in shape to the Brillouin function (solid line) but with smaller moment than that expected for paramagnetic Eu$^{2+}$. That the magnetic moment does not saturate at finite fields is another indication of no strong collective behavior of intercalated Eu clusters under graphene.
The fact that the magnetic moment $\langle J_Z \rangle$ is well below its saturation value 7$\mu_B$, at high field and at the low temperature $T =15$ K, is puzzling.
\begin{figure}\centering \includegraphics [width = 76 mm] {Eu-FieldDep_v2.pdf}
\caption{(Color online) The magnetic field dependence of the $\langle J_Z \rangle$ (triangles) and $\langle L_Z \rangle$ (square and circle) of intercalated Eu at $T = 15$ K. To check for anisotropy, measurements were conducted at incident beam angles of 20$^{\circ}$ (blue) and 90$^{\circ}$ (green). The $\langle L_Z \rangle$ components are nearly 0, which is consistent with Hund's rules for Eu$^{2+}$. The calculated Brillouin function for Eu$^{2+}$ at $T=15 K$ is also included for comparison as a smooth solid line.}
\label{fig:EuField}
\end{figure}
We emphasize that the XMCD unequivocally determines the electronic configuration of the intercalant as Eu$^{2+}$, as expected for metal Eu but also for ferromagnetic EuO. Indeed, previous $M_4-M_5$ XAS measurements of Eu metal and EuO are almost indistinguishable due to the $d-f$ core levels, involved in the transitions, that are hardly influenced by the specific chemistry of the element\cite{Thole1985a}. However, detailed comparison of our XMCD with that of thin films EuO indicate differences that point to the fact that the intercalated Eu is in its metallic state. Also, the magnetic ground states of the metal and oxide are distinct at low temperatures. Whereas EuO is ferromagnetic at T$_C$ $\approx 67 $ K\cite{Wachter1972,Lettieri2003} with finite hysteresis\cite{Lettieri2003,Wackerlin2015}, Eu metal undergoes an incommensurate helical magnetic structure at T$_N\approx 91 $ K\cite{Nereson1964,Jensen1991}. As shown in Fig.\ \ref{fig:EuField} there is no evidence of magnetic moment saturation or anisotropy that is expected from a ferromagnet ruling out the possibility that the intercalated Eu is an oxide (i.e., EuO). Another possibility is that intercalated Eu under graphene adopts the $(\sqrt{3}\times\sqrt{3})$R$30^\circ$ superstructure as an intercalated Eu in highly pyrolytic graphite (HOPG) crystals, namely, \ce{C6Eu}\cite{Suematsu1983}. However, magnetization and specific heat of \ce{C6Eu}
indicate it becomes antiferromagnetic (AFM) at about 40 K. This scenario can also be discarded since AFM systems do not yield XMCD signals, and we do observe a strong XMCD signal below 40 K in our samples.
To further explore the magnetic properties of the intercalated Eu nano-clusters, we have collected XMCD spectra at the $M_5$ regime (from 1120 to 1140 eV) at various temperatures and at fixed $B = 5$ T. As discussed in the SI\cite{SI-EU}, because $\langle L_Z \rangle =0$ for Eu$^{2+}$, measuring the XMCD on either the M$_5$ or M$_4$ is sufficient to determine the magnetic moment. Figure \ref{fig:EuTemp} shows the temperature dependence of $\langle J_Z \rangle$ from the XMCD spectra for the M$_5$ as a function of temperature, with characteristic increase common to a paramagnetic system. However, the 1/$\langle J_Z \rangle$ of the same data shows two distinct regions that overlaid by linear fits (dashed lines) intersect at $T^* \approx 90 $ K. We note that $T^*$ is very close to the the N{\`e}el temperature, $T_N$, of bulk metallic Eu at 91 K (vertical dashed line in Fig.\ \ref{fig:EuTemp})\cite{Nereson1964,Jensen1991}. As mentioned previously, the Curie temperature of EuO is at $T_c \approx 67$ K, which is substantially lower than the anomaly observed in our temperature dependence\cite{Wachter1972,Lettieri2003}. This is yet another indication that the intercalated Eu-clusters under graphene are likely in their metallic structure. In the SI we propose three scenarios of possible layers underneath graphene that may also explain the finite clustering size $2.5$ nm in diameter.
\begin{figure}\centering \includegraphics [width = 76 mm] {Eu_1XT_v4.pdf}
\caption{(Color online) Temperature dependence of the total moment $\langle J_Z \rangle$ and 1/$\langle J_Z \rangle$ for Intercalated Eu at $B = 5$ T. Bulk Eu has a transition to helical structure at 91 K, which is indicated by the vertical dashed line. The two dashed lines are linear fits below and above two temperature regions with intersection at $\approx 90$ K.}
\label{fig:EuTemp}
\end{figure}
\section*{Conclusions}
In conclusion, we have succeeded to intercalate Eu under epitaxial graphene on SiC buffer layer. Our XMCD results show the electronic configuration of the intercalant is that of Eu$^{2+}$ likely in its metallic state or as a Eu-silicide\cite{Averyanov2016}. Our STM images show that the Eu forms relatively uniform nano-clusters of approximately 2.5 nm in diameter, and although the clusters are randomly distributed they preferably nucleate at the vertices of the 6$\times$6 super structure of graphene on SiC which act as nucleation centers. We argue that unlike intercalated \ce{C6Eu}, the Eu under graphene forms clusters that likely conform to the square unit cell of metallic Eu and that, due to the incommensurabilty between graphene and the Eu, the clusters are limited in size. The temperature dependence of $\langle J_Z \rangle$ at fixed magnetic field $B=5$ T is consistent with the paramagnetic behavior displayed in the magnetic field dependence at $T=15$ K, namely, no anisotropy or hysteresis effects are observed. Although Eu is a highly oxidizing metal in air, the epitaxial graphene layer formed on SiC is practically defect free that protects the intercalated Eu against oxidation under atmospheric conditions over periods of months.
\section{Acknowledgments}
Ames Laboratory is operated by Iowa State University by support from the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-AC02-07CH11358. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, is supported by the U.S. DOE under Contract No. DE-AC02-06CH11357.
\bibliographystyle{apsrev4-2}
|
2,869,038,154,918 | arxiv | \section{}
\section{Suspensions and equivalent fluid}
The particles are dispersed in a mixture of water (74\% w/w), glycerol (25\% w/w) and
polyethylene oxide with a molar weight of \unit{300}\kilo\gram\per\mole\ (PEO, 1\% w/w, from Sigma Aldrich). The water/glycerol mixture has a shear viscosity of $\ensuremath{\eta_0}\xspace=1.9\,{\rm mPa.s}$, a surface tension $\gamma=68 \pm 2\,{\rm mN.m^{-1}}$ and a density $\rho=1059 \pm 3\,{\rm kg\,m^{-3}}$. The polystyrene particles of density $\rho \simeq 1057 \pm 3 \,{\rm kg.m^{-3}}$ are neutrally buoyant in the mixture over the timescale of an experiment. The volume fraction is defined as the ratio of the volume of particles to the total volume, $\phi=V_g/V_{tot}$ and is varied in the range $0\%$ to $40\%$.
\medskip
The equivalent fluid to a given suspension of volume fraction \ensuremath{\phi_\text{eq}}\xspace is defined as the water-glycerol-PEO mixture
with the same PEO content and a water-to-glycerol weight ratio chosen so that its shear viscosity
is equal to that of the suspension.
The composition of the equivalent fluids used in the present study is summarized in Table~\ref{tab:liqeq}.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\ensuremath{\phi_\text{eq}}\xspace (\%) & Water (\%) & Glycerol (\%) & PEO300 (\%) \\
\hline
\textbf{0} & \textbf{74} & \textbf{25} & \textbf{1} \\
\hline
10 & 69 & 30 & 1 \\
\hline
20 & 59 & 40 & 1 \\
\hline
30 & 47 & 52 & 1 \\
\hline
40 & 33 & 66 & 1 \\
\hline
\end{tabular}
\caption{Mass composition of the equivalent liquids.
The first line describes the interstitial fluid in the suspension.
}
\label{tab:liqeq}
\end{table}
\section{Videos}
The snapshots in Fig.~\ref{fig:timeline} are extracted from three videos available
in supplemental materials:
\begin{itemize}
\item Interstitial\_fluid.avi;
\item 20\micro\meter\_40\%.avi;
\item 140\micro\meter\_40\%.avi.
\end{itemize}
The videos are slowed down 1000 times.
The nozzle at the top of the image is \unit{2.75}\milli\meter\ wide.
\medskip
\section{Contour detection and processing}
The image processing used to extract the time evolution of the minimal diameter \ensuremath{h_\text{min}}\xspace is done in two steps.
First, the contour of the drop and the ligament is detected on each frame of the video using a thresholding method with ImageJ.
We obtain an array of points representing the 2D position of that contour.
In a second time, a custom-made Python routine translates the contour of the neck into the thickness profile $h(z,t)$.
Fig.~\ref{fig:contour} shows several thickness profiles at different time,
regularly spaced by $\Delta t = \unit{3}\milli\second$.
The neck width $\ensuremath{h_\text{min}}\xspace(t)$ is defined as the global minimum of $h(z,t)$ in the Newtonian
regime. In the viscoelastic regime, the wide and constant minimum of $h(z,t)$ defines $\ensuremath{h_\text{min}}\xspace(t)$.
By comparing the results of this automatic processing to the direct measurement of
\ensuremath{h_\text{min}}\xspace on the video, we find a maximum error of 2 pixels,
\textit{i.e.}, around \unit{10}\micro\meter.
\begin{figure}[h]
\centering
\includegraphics[width=.45\linewidth]{figSM}
\caption{Thickness profiles for the thinning of the interstitial fluid,
corresponding to
Fig.~1(a)
in the main article.
The time step between two profiles is constant and equals \unit{3}\milli\second.
The circles represent the neck width \ensuremath{h_\text{min}}\xspace in the Newtonian regime.
}
\label{fig:contour}
\end{figure}
\section{Reproducibility of the thinning experiments}
Achieving reproducibility can be a significant challenge when dealing with dense suspensions. However, since we considered dilute and moderate volume fraction ($\phi \leq 40\%$), the reproducibility of the thinning experiments is not an issue here. For instance,
Fig.~\ref{fig:repro} reports the thinning dynamic $h = f(t-\ensuremath{t_\text{c}}\xspace)$ for ten realizations of the same experiment,
in this case, the pinch-off of a suspension drop containing a solid fraction $\phi=40\%$ of \unit{140}\micro\meter\
particles.
The small variations observed between the different realizations can be understood since
the suspension remains dilute ($\phi \le 40\%$) and the particles small enough compared to the system.
The example presented here holds for other suspensions considered in this study and confirms the reproducibility of our experiments.
\begin{figure}[h]
\centering
\includegraphics[width=.45\linewidth]{figSM2}
\caption{Time evolution of the minimal diameter \ensuremath{h_\text{min}}\xspace for ten realizations of the same thinning experiments for a suspension with $\phi=40\%$ of
\unit{140}\micro\meter\ particles. Each color refer to a different realization.
}
\label{fig:repro}
\end{figure}
\end{document}
|
2,869,038,154,919 | arxiv | \section{Introduction}\label{sec:introduction}
\emph{Hybrid systems} exhibit both discrete \emph{jump}
and continuous \emph{flow} dynamics. Quality assurance of such systems
are of paramount importance due to the current ubiquity of
\emph{cyber-physical systems (CPS)} like cars, airplanes, and many
others.
For the formal verification approach to hybrid systems, the challenges are: 1) to incorporate
flow-dynamics; and 2) to do so at the lowest possible cost, so that the
existing discrete framework smoothly transfers to hybrid situations. A large
body of existing work uses \emph{differential equations} explicitly in the
syntax; see the discussion of related work below.
In~\cite{Suenaga2011}, instead, an alternative approach of
\emph{nonstandard static analysis}---combining \emph{static analysis}
and \emph{nonstandard analysis}---is proposed. Its basic idea is
to introduce a constant
$\ensuremath{\mathtt{d\hspace{-.05em}t}}$ for an \emph{infinitesimal} (i.e.\ infinitely small) value, and
\emph{turn flow into jump}. With $\ensuremath{\mathtt{d\hspace{-.05em}t}}$, the continuous operation of
integration can be represented by a while-loop, to which
existing discrete techniques such as Hoare-style program logics readily
apply. For a rigorous mathematical development they employ
\emph{nonstandard analysis (NSA)} beautifully formalized by
Robinson~\cite{Robinson1966}.
Concretely, in~\cite{Suenaga2011} they took the common combination of a
$\textsc{While}$-language and a Hoare
logic (e.g.\ in the textbook~\cite{Winskel1993}); and added a constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$
to obtain a modeling and verification framework for hybrid systems. Its
components are called $\While^{\dt}$ and $\Hoare^{\dt}$.
The soundness of $\Hoare^{\dt}$ is proved
against denotational semantics defined in the language
of NSA.
Subsequently in the \emph{nonstandard static analysis} program: in~\cite{Hasuo2012} they presented a prototype
automatic theorem prover for $\Hoare^{\dt}$;
and in~\cite{Suenaga2013}
they applied the same idea to stream processing systems, realizing
a verification framework for \emph{signal processing} as in Simulink.
Underlying these technical developments is the idea of so-called \emph{sectionwise execution}.
Although this paper does not rely explicitly on it, it is still useful
for laying out the ``operational'' intuition of nonstandard static analysis.
See the following example.
\vspace*{1em}
\noindent
\begin{minipage}{.77\textwidth}
\begin{myexpl}\label{example:elapsedTime}
Let $c_{\mathsf{elapse}}$ be the
program on the right.
The value of $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ is infinitesimal; therefore the
$\mathtt{while}$ loop will not terminate within finitely many steps.
Nevertheless it is somehow intuitive to expect that after an ``execution'' of
this program, the value of $t$
should be infinitesimally close to $1$ and larger than it.
\end{myexpl}
\end{minipage}
\hfill
\begin{math}
\begin{array}{l}
t := 0\;;\quad
\\
\mathtt{while}\; t\le 1\; \mathtt{do}
\\
\quad t:=t+\ensuremath{\mathtt{d\hspace{-.05em}t}}
\end{array}
\end{math}
\begin{wrapfigure}{r}{7em}
\begin{math}
\begin{array}{l}
t := 0\;;\quad
\\
\mathtt{while}\; t\le 1\; \mathtt{do}
\\
\quad t:=t+\frac{1}{i+1}
\end{array}
\end{math}
\end{wrapfigure}
One possible way of thinking is to imagine \emph{sectionwise execution}. For each
natural number $i$ we consider the \emph{$i$-th section} of the program
$c_{\mathsf{elapse}}$, denoted by $c_{\mathsf{elapse}}\sect{i}$
and shown on the right.
Concretely, $c_{\mathsf{elapse}}\sect{i}$ is obtained by replacing the
infinitesimal $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ in $c_{\mathsf{elapse}}$ with $\frac{1}{i+1}$.
Informally $c_{\mathsf{elapse}}\sect{i}$ is the ``$i$-th approximation'' of
the original $c_{\mathsf{elapse}}$.
A section $c_{\mathsf{elapse}}\sect{i}$ does terminate within
finite steps and
yields $1+\frac{1}{i+1}$ as the value of $t$.
Now we collect the outcomes of sectionwise executions and obtain
a sequence
\begin{equation}\label{equation:sequenceThatIsOnePlusDt}
\small
\begin{array}{l}
(\,1+1, \;
1+\frac{1}{2},\;
1+\frac{1}{3},\;
\dotsc,\;
1+\frac{1}{i},\;
\dotsc\,
)
\end{array}
\end{equation}
which is thought of as a progressive approximation of the actual
outcome of the original program $c_{\mathsf{elapse}}$. Indeed, in the
language of NSA, the sequence~(\ref{equation:sequenceThatIsOnePlusDt}) represents a \emph{hyperreal number}
$r$ that is infinitesimally close to $1$.
\vspace*{.0em}
We note that
a program in $\While^{\dt}$ is \emph{not} intended to be executed: the program $c_{\mathsf{elapse}}$
does not terminate.
It is however an advantage of
\emph{static} approaches to verification and analysis, that programs need not be executed to prove their
correctness. Instead well-defined mathematical semantics suffices. This
is what we do here as well as in~\cite{Suenaga2011,Hasuo2012,Suenaga2013}, with the denotational
semantics of $\While^{\dt}$ exemplified in Example~\ref{example:elapsedTime}.
\vspace*{.2em}
\noindent
\textbf{Our Contribution}
\quad
In the previous work~\cite{Suenaga2011,Hasuo2012,Suenaga2013} \emph{invariant
discovery} has been a big obstacle in scalability of the proposed
verification techniques---as
is usual in deductive verification. The current work, as a first step
towards scalability of the approach,
extends \emph{abstract interpretation}~\cite{Cousot1977} with
infinitesimals. The abstract interpretation methodology is known for
its ample applicability (it is employed in model checking as
well as in many deductive verification frameworks) and scalability (the static
analyzer Astr\'{e}e~\cite{Cousot2005} has been successfully used e.g.\
for Airbus's flight control system).
Our theoretical contribution includes: the theory of \emph{nonstandard
abstract interpretation} where (standard) abstract domains are ``$*$-transformed,'' in a rigorous NSA sense, to
the abstract domains for hyperreals; their soundness in over-approximating
semantics of $\While^{\dt}$ programs and hybrid system modeling by them; and introduction of the notion of
\emph{uniform} widening operators. With the latter, inductive
approximation is guaranteed to terminate within finitely many
steps---even after extension to the nonstandard setting. We show that
many known widening operators, if not all, are indeed uniform.
Although we focus on the domain of convex polyhedra in this paper, it
is also possible to extend other abstract domains like
ellipsoids~\cite{Feret2004}
in the same way.
These theoretical results form a basis of our prototype
implementation,\footnote{The prototype is available on-line: \href{http://www-mmm.is.s.u-tokyo.ac.jp/~kkido/}{http://www-mmm.is.s.u-tokyo.ac.jp/\~{}kkido/}} that successfully analyzes: \emph{water-level monitor},
a common example of piecewise-linear hybrid dynamics; and also
\emph{thermostat} that is beyond piecewise-linear.
The prototype deals with the constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ as a truly infinitesimal number using computer algebra system.
\vspace*{.2em}
\noindent
\textbf{Related Work}
\quad
There has been a lot of research work for verification of hybrid systems
and it
has led to quite a few system verification tools, including
HyTech~\cite{Henzinger1997},
PHAVer~\cite{Frehse05},
SpaceEx~\cite{Frehse11},
HySAT/iSAT~\cite{Franzle2007},
Flow*~\cite{ChenAS13} and
KeYmaera~\cite{PlatzerQ08}. All these rely on ODEs (or the explicit
solutions of them) for expressing continuous dynamics, much like
\emph{hybrid automata}~\cite{Alur1992} do.
Our nonstandard static analysis approach is completely different from
those in the following point: we do not use ODEs at all, and model
hybrid systems as an imperative program with an infinitesimal constant.
It enables us to apply static methodologies for discrete systems as they are. For example, in HyTech and PHAVer, convex polyhedra is used to
over-approximate the reachable sets. They need, however, some special
techniques such as linear phase-portrait~\cite{Henzinger95}, to reduce the dynamics into
piecewise linear one. Our framework does not need such and usual
abstract interpretation works as it is.
There are many other works we rely on, such as
those on
abstract interpretation, nonstandard analysis, etc. These are discussed
later when they become relevant.
\vspace*{.2em}
\noindent
\textbf{Organization}
\quad
In~\S{}\ref{sec:exampleOfAnalysis} we start with the water-level monitor example and present how our nonstandard abstract interpretation framework works.
Then we go on to its theoretical foundations.
In~\S{}\ref{sec:preliminaries} we review
preliminaries on: abstract
interpretation; nonstandard analysis; and the modeling language $\While^{\dt}$
from~\cite{Suenaga2011}.
In~\S{}\ref{sec:NSAI} we extend the theory of abstract
interpretation with infinitesimals and build the theory of nonstandard
abstract
interpretation.
Its theorems include soundness of approximation, and
termination guaranteed by (the $*$-transform of) a \emph{uniform} widening
operator.
In~\S{}\ref{sec:implementation}
we present our prototype implementation and the experiment results with it.
Most proofs are deferred to Appendix~\ref{appendix:omittedProofs}.
\section{Leading Example: Verification of Water-Level Monitor}\label{sec:exampleOfAnalysis}
We shall start with an example of verification and let it exemplify how our
framework---that extends abstract interpretation with infinitesimals,
and handles continuous as well as discrete dynamics---works. We use the
well-known example of the water-level monitor~\cite{Alur1992}.
In the current section, in particular, we will first revisit how the usual
abstract interpretation workflow (without extension) would work, using
a
discretized variant of the problem. Our emphasis is on the fact that
our extended framework works just in the same manner: without any
explicit ODEs or any additional theoretical infrastructure for ODEs;
but only adding a constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$.
\begin{wrapfigure}[4]{r}{0pt}
\raisebox{-13mm}[0pt][0mm]{\includegraphics[width=.18\textwidth]{Figures/waterLevel.jpg}}
\end{wrapfigure}
The concrete problem is as follows.
See the figure on the right. A water tank has a
constant drain ($2$~cm per second). When the water level $x$ gets lower
than $5$~cm the
switch is turned on, which eventually makes the pump work but only after
a time lag of two seconds. While the pump is working, the water level
$x$
rises by $1$~cm per second. Once $x$ reaches $10$~cm the switch is
turned off, which will shut down the pump but again after a time lag of two seconds.
Our goal is the \emph{reachability analysis} of this hybrid dynamics,
that is, to see the water level $x$ remains in a certain ``safe'' range (we will see
that the range is $1\le x \le 12$).
\begin{wrapfigure}[13]{r}{0.5\hsize}
\vspace{-1em}
\begingroup
\fontsize{7pt}{8pt}\selectfont
\begin{verbatim}
(*Water-Level Monitor*)
l := 0; x := 1; p := 1; s := 0;
dt' := 0.2;
while true do {
if p = 1 then x := x + dt'
else x := x - 2 * dt';
if (x <= 5 && p = 0) then s := 1
else {if (x >= 10 && p = 1)
then s := 1
else s := 0
};
if s = 1 then l := l + dt'
else skip;
if s = 1 && l >= 2
then {p := 1 - p; s := 0; l := 0}
else skip
}
\end{verbatim}
\endgroup
\vspace{-1.7em}
\caption{Discretized water-level monitor}
\label{fig:whileCodeCaseStudy}
\end{wrapfigure}
\subsection{Analysis by (Standard) Abstract Interpretation, as a Precursor}\label{subsec:waterLevel0.2}
Let us first revisit the usual workflow in reachability analyses by
abstract interpretation.
We will use the \emph{discretized} model of the water-level monitor
in Fig.~\ref{fig:whileCodeCaseStudy}, where each iteration of its unique
loop amounts to the lapse of $\ensuremath{\mathtt{d\hspace{-.05em}t}}'=0.2$ seconds. The model in
Fig.~\ref{fig:whileCodeCaseStudy} is
an imperative program
with while loops, a typical subject of analyses by abstract
interpretation.
More specifically:
$x$ is the water level, $l$ is the counter for the time lag,
$p$ stands for the state of the pump
($p=0$ if the pump is off, and $p=1$ if on)
and $s$ is for ``signals,''
meaning $s=1$ if the pump has not yet responded to a signal from the
switch (such as, when the switch is on but the pump is not on yet).
The first step in the usual abstract interpretation workflow is
to fix \emph{concrete} and \emph{abstract domains}. Here in~\S\ref{subsec:waterLevel0.2} we will use the followings.
\begin{itemize}
\item \textbf{The concrete domain: $\bigl(\mathcal P({\mathbb{R}}^{2})\bigr)^{4}$.}
We have two numerical variables $l,x$ and two Boolean ones $p,s$
in Fig.~\ref{fig:whileCodeCaseStudy},
therefore a canonical concrete domain would be $\mathcal P(\mathbb{B}^{2}\times
\mathbb{R}^{2})$. We have the powerset operation $\mathcal P$ in it since we
are now interested in the \emph{reachable} set of memory states.
However, for a better fit with our abstract domain (namely
convex polyhedra), we shall use the set
$\bigl(\mathcal P({\mathbb{R}}^{2})\bigr)^{4}$ that is isomorphic to the above set
$\mathcal P(\mathbb{B}^{2}\times
\mathbb{R}^{2})$
\item \textbf{The abstract domain: $(\mathbb{CP}_{2})^{4}$.}
We use the domain of \emph{convex polyhedra}~\cite{Cousot1978},
one of the most commonly-used abstract domains.
Recall that a convex polyhedron is a subset of a Euclidean space
characterized by a finite conjunction of linear inequalities.
Specifically, we
let $\mathbb{CP}_{2}$, the set of 2-dimensional convex polyhedra,
approximate the set $\mathcal P(\mathbb{R}^{2})$. Therefore, as an abstract
domain for the program in Fig.~\ref{fig:whileCodeCaseStudy},
we
take $(\mathbb{CP}_{2})^{4}$ (that approximates
$\bigl(\mathcal P({\mathbb{R}}^{2})\bigr)^{4}$).
\end{itemize}
The next step in the workflow is to \emph{over-approximate} the set of
memory states that are reachable by the program in
Fig.~\ref{fig:whileCodeCaseStudy}---this is a subset of the concrete domain
$\bigl(\mathcal P({\mathbb{R}}^{2})\bigr)^{4}$---using the abstract domain
$(\mathbb{CP}_{2})^{4}$. Since the desired set can be thought of as a least
fixed point, this over-approximation procedure involves: 1)
\emph{abstract execution} of the program in $(\mathbb{CP}_{2})^{4}$ (that is
straightforward, see e.g.~\cite{Cousot1978}); and 2) acceleration of
least fixed-point computation in $(\mathbb{CP}_{2})^{4}$ via suitable use of a
\emph{widening operator}. For convex polyhedra several
widening operators are well-known. We shall use here $\nabla_M$, so-called the
\emph{widening up
to $M$} operator from \cite{Halbwachs1993, Halbwachs1997}. One big
reason for this choice is the \emph{uniformity} of the operator (a
notion we introduce later in~\S{}\ref{subsec:uniformWidening}), among
others. The set $M$ of linear constraints is a parameter for this
widening operator; we fix it as usual, collecting the linear constraints
that occur in the program in question. That is, $M=\{x\leq 5, x\geq 5,
x\leq 10, x\geq 10, l\leq 2, l\geq 2\}$.
This over-approximation procedure is depicted in the \emph{iteration
sequence} in Fig.~\ref{fig:iterseq}. Let us look at some of its
details. The graph $0$ represents the initial memory state (before
the first iteration), where the pump is on and the water level $x$ is precisely
$1$. After one iteration the water level will be incremented by $1\times
\ensuremath{\mathtt{d\hspace{-.05em}t}}'=0.2$~cm; as usual in abstract interpretation, however, at this moment we invoke the widening
operator
$\nabla_M$, and the next ``abstract reachable set'' is $x\in [1,5]$
instead of $x\in [1,1.2]$. Here the
upper bound $5$ comes from
the constraint
$x\le 5$ that is in the parameter $M$ of the widening operator
$\nabla_{M}$.
This
results in the graph 1 in Fig.~\ref{fig:iterseq}.
In the iteration sequence (Fig.~\ref{fig:iterseq}) the four
polyhedra (in four different colors) gradually grow: in the graph 2 the water
level $x$ can be $10$~cm so in the graph 3 appears a green polyhedron
(meaning that a
signal is sent from the switch to the pump); after the graphs 3 and 9 we
\emph{delay} widening, a heuristic commonly employed in abstract
interpretation~\cite{Cousot1981}. In the end, in the graph 12 we have a
prefixed point (meaning that the polyhedra do not grow any further). There
we can see, from the range of $x$ spanned by the polyhedra, that the
water level never reaches beyond $0.6 \leq x \leq 12.2$.
\subsection{Analysis by \emph{Nonstandard Abstract
Interpretation}}\label{subsec:waterLevelDt}
In the above ``standard'' scenario, we approximated the
dynamics of the water level by discretizing the continuous notion of
time ($\ensuremath{\mathtt{d\hspace{-.05em}t}}'=0.2$). While this made the usual abstract interpretation
workflow go around,
there is a price to pay---the analysis result is not
\emph{precise}. Specifically, the reachable region thus over-approximated is
$0.6 \leq x \leq 12.2$, while the real reachable region is $1\leq x\leq
12$.\footnote{There are also examples in which
discretization even leads to \emph{unsound} analysis results.}
\begin{wrapfigure}[13]{r}{0.5\hsize}
\vspace{-1em}
\begingroup
\fontsize{7pt}{8pt}\selectfont
\begin{verbatim}
(*Water-Level Monitor*)
l := 0; x := 1; p := 1; s := 0;
while true do {
if p = 1 then x := x + dt
else x := x - 2 * dt;
if (x <= 5 && p = 0) then s := 1
else {if (x >= 10 && p = 1)
then s := 1
else s := 0
};
if s = 1 then l := l + dt
else skip;
if s = 1 && l >= 2
then {p := 1 - p; s := 0; l := 0}
else skip
}
\end{verbatim}
\endgroup
\vspace{-1.7em}
\caption{Water-level monitor in $\While^{\dt}$}
\label{fig:whileDtCodeCaseStudy}
\end{wrapfigure}
Obviously we can ``tighten up'' the analysis by making the value $\ensuremath{\mathtt{d\hspace{-.05em}t}}'$ smaller.
Even better, we can leave the expression $\ensuremath{\mathtt{d\hspace{-.05em}t}}'$ in
Fig.~\ref{fig:whileCodeCaseStudy} as a variable, and imagine the ``limit'' of
analysis results when the value of $\ensuremath{\mathtt{d\hspace{-.05em}t}}'$ tends to $0$. However here is
a question: what is that ``limit,'' in mathematically rigorous terms?
Taking $\ensuremath{\mathtt{d\hspace{-.05em}t}}'=0$ obviously does not work: do so in
Fig.~\ref{fig:whileCodeCaseStudy}
and we have no dynamics whatsoever. The value of $\ensuremath{\mathtt{d\hspace{-.05em}t}}'$ must be strictly positive.
Our contribution is an extension of abstract interpretation that answers
the last question. In our framework, the same (hybrid) dynamics of
the water-level monitor is modeled
by a program in Fig.~\ref{fig:whileDtCodeCaseStudy}. Here the expression
$\ensuremath{\mathtt{d\hspace{-.05em}t}}$
is a new constant that stands for a \emph{positive} and \emph{infinitesimal}
(i.e.\ infinitely small) value. Therefore the modeling is not an
approximation by discretization; it is an \emph{exact} modeling.
It is important to notice that the program in
Fig.~\ref{fig:whileDtCodeCaseStudy} is the same as the one in
Fig.~\ref{fig:whileCodeCaseStudy}, except that now $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ is some strange
constant,
while $\ensuremath{\mathtt{d\hspace{-.05em}t}}'$ in Fig.~\ref{fig:whileCodeCaseStudy} stood for a real
number (namely $0.2$). This difference, however, does not prevent us
from applying the \emph{static}, \emph{symbolic} and \emph{syntax-based} analysis by abstract
interpretation. We can follow exactly the same path as
in~\S{}\ref{subsec:waterLevel0.2}---taking the abstract domain of convex
polyhedra, executing the program in Fig.~\ref{fig:whileDtCodeCaseStudy}
on it, applying the widening operator $\nabla_M$, and forming an
iteration sequence much like in Fig.~\ref{fig:iterseq}---and this leads
to the analysis result $1-2\ensuremath{\mathtt{d\hspace{-.05em}t}} \leq x \leq 12+\ensuremath{\mathtt{d\hspace{-.05em}t}}$. Since $\ensuremath{\mathtt{d\hspace{-.05em}t}}$
is an infinitesimal number, the last result is practically as good as
$1\le x \le 12$. We have a prototype implementation that automates
this analysis (\S{}\ref{sec:implementation}).
What remains to be answered is the legitimacy of this extended abstract
interpretation framework. Is the outcome $1-2\ensuremath{\mathtt{d\hspace{-.05em}t}} \leq x \leq 12+\ensuremath{\mathtt{d\hspace{-.05em}t}}$
\emph{sound}, in the sense that it indeed over-approximates the true
reachable
set? Even before that, what do we mean by the ``true
reachable
set'' of the program in Fig.~\ref{fig:whileDtCodeCaseStudy},
with an exotic infinitesimal constant like $\ensuremath{\mathtt{d\hspace{-.05em}t}}$? Moreover, are
iteration sequences via the widening operator $\nabla_M$ guaranteed to
terminate
within finitely many steps, as is the case in the standard
framework~\cite{Halbwachs1993, Halbwachs1997}?
The rest of the paper is mostly devoted to (answering positively to) the
last questions. In it we use Robinson's \emph{nonstandard analysis
(NSA)}~\cite{Robinson1966} and give infinitesimal numbers---clearly
such do not exist in the set of (standard) real numbers---a status as first-class citizens. The program in
Fig.~\ref{fig:whileDtCodeCaseStudy} is in fact in the programming (or
rather \emph{modeling}) language $\While^{\dt}$ from~\cite{Suenaga2011,Hasuo2012};
and its semantics can be understood in the line of
Example~\ref{example:elapsedTime}. It turns out that the theory of
NSA---in particular its celebrated result of the \emph{transfer
principle}---allows us to ``transfer'' meta results from the standard
abstract interpretation to our extension. That is, what is true in the
world of standard reals (soundness, termination, etc.) is also true in
that of \emph{hyperreals}.
\begin{figure}[tb]
\begin{tabular}{lll}
\begin{minipage}[t]{.32\textwidth}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/0.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/1.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/2.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/3.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/4.pdf}
\end{minipage}
\begin{minipage}[t]{.32\textwidth}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/5.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/6.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/7.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/8.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/9.pdf}
\end{minipage}
\begin{minipage}[t]{.32\textwidth}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/10.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/11.pdf}
\includegraphics[width=\textwidth]{Figures/watertank_graphs_new/12.pdf}
\end{minipage}
\end{tabular}
\caption{An iteration sequence for the water-level monitor example.\newline
To save space, here we depict
an element of $(\mathbb{CP}_{2})^{4}$---i.e.\ a quadruple of convex
polyhedra---on the same plane $\mathbb{R}^{2}$. The four convex polyhedra come in different colors:
those in blue, green, red and yellow correspond to the values
$(p,s)=(1,0), (1,1), (0,0)$ and $(0,1)$ of the Boolean variables,
respectively.
}
\label{fig:iterseq}
\end{figure}
\section{Preliminaries}\label{sec:preliminaries}
In~\S{}\ref{sec:NSAI} we will present our \emph{soundness} and
\emph{termination} results as a ``metatheory'' that justifies the
workflow described in~\S{}\ref{subsec:waterLevelDt};
in this section we recall some preliminaries that are needed for those theoretical developments.
First, the general theory of abstract interpretation is introduced in~\S{}\ref{subsec:abstinterp} and the specific domain of convex polyhedra is presented in~\S{}\ref{subsec:convexpoly}.
Next, some basic notions in nonstandard analysis are explained in~\S{}\ref{subsec:preliminariesNSA}.
Finally, in~\S{}\ref{subsec:whiledt}, the modeling language $\While^{\dt}$ from~\cite{Suenaga2011} and its (denotational) collecting semantics based on nonstandard analysis are presented.
\subsection{Abstract Interpretation}\label{subsec:abstinterp}
\emph{Abstract interpretation}~\cite{Cousot1978} is a well-established
technique in static analysis. We make a brief review of its basic
theory; it is
mostly for the purpose of fixing notations.
The goal of abstract interpretation is to over-approximate a \emph{concrete semantics} defined on an \emph{concrete domain} by an \emph{abstract semantics} on an \emph{abstract domain}.
We assume that the concrete semantics is defined as a least fixed point on the concrete domain.
The following proposition guarantee the over-approximation of the least fixed point in the concrete domain by a prefixed point in the abstract domain.
In the proposition, the order $\sqsubseteq$ on the domain $L$ is extended to the order on $L\rightarrow L$ pointwisely.
And the \emph{least fixed point relative to ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}$}, denoted by ${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}F$, is the
least among the fixed points of $F$ above ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}$; by the cpo
structure of $L$ and the continuity of $F$, it is given by $\bigsqcup_{n\in\mathbb{N}}F^{n}{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}$.
Note that we are using the concretization-based framework described in~\cite{Cousot1992a}.
\begin{myprop}\label{prop:concretization}
Let $(L, \sqsubseteq)$ be a cpo;
$F:L\rightarrow L$ be a continuous function;
and ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\in L$ be such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\sqsubseteq F({\ooalign{{$\bot$}\crcr{\hss{-}\hss}}})$.
Let $(\overline{L}, \overline{\sqsubseteq})$ be a preorder;
$\gamma: \overline{L} \rightarrow L$ be a function (it is called \emph{concretization}) such that
$\overline{a}\mathrel{\overline{\sqsubseteq}}\overline{b} \Rightarrow \gamma(\overline{a})\sqsubseteq \gamma(\overline{b})$ for all $\overline{a}, \overline{b} \in \overline{L}$; and
$\overline{F}:\overline{L}\rightarrow\overline{L}$ be a monotone function such that $F\circ\gamma\sqsubseteq\gamma\circ\overline{F}$.
Assume further that $\overline{x}\in\overline{L}$ is a prefixed point of $\overline{F}$
(i.e.\ $\overline{F}(\overline{x})\mathrel{\overline\sqsubseteq}\overline{x}$)
such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\mathrel{\sqsubseteq}\gamma(\overline{x})$.
Then $\overline{x}$ over-approximates ${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}F$, that is,
${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}F\sqsubseteq \gamma(\overline{x})$.
\qed
\end{myprop}
In~\S{}\ref{subsec:waterLevel0.2} where we analyzed the discretized water-level monitor, the set ${\mathcal P}(\mathbb{R}^n)$ of subsets of memory states is used as a concrete domain $L$; and the \emph{domain of convex polyhedra} is used as an abstract domain $\overline{L}$. The interpretations $F$ and $\overline{F}$ on each domains are defined in a standard manner.
Towards the goal of obtaining $\overline{x}$ in Prop.~\ref{prop:concretization}, (i.e.\ finding a prefixed point in the abstract domain), the following notion of
\emph{widening} is used (often together with \emph{narrowing} that we will not
be using).
Note that in the following definition and proposition, the domain $(L, \sqsubseteq)$ is the abstract domain, corresponding to $(\overline{L}, \overline{\sqsubseteq})$ in Prop.~\ref{prop:concretization}.
\begin{mydef}[widening operator]\label{def:widen}
Let $(L, \sqsubseteq)$ be a preorder.
A function $\nabla: L \times L \rightarrow L$ is said to be a \emph{widening operator} if the following two conditions hold.
\begin{itemize}
\item (\emph{Covering}) For any $x, y \in L$, $x \sqsubseteq x \nabla y$ and $y \sqsubseteq x \nabla y$.
\item (\emph{Termination}) For any ascending chain $\langle x_i
\rangle \in L^\mathbb{N}$, the chain $\langle y_i \rangle \in
L^\mathbb{N}$ defined by
$y_0 = x_0$
and
$y_{i+1} = y_{i} \nabla x_{i+1}$ for each $i \in \mathbb{N}$
is ultimately stationary.
\end{itemize}
\end{mydef}
\noindent A widening operator on a fixed abstract domain $\overline{L}$
is not at all unique. In this paper we will discuss three widening operators previously introduced for $\mathbb{CP}_{n}$.
The use of widening is as in the following proposition: the covering
condition ensures that the outcome is a prefixed point; and the
procedure terminates thanks to the
termination condition.
\begin{myprop}[convergence of iteration sequences]\label{prop:widen}
Let $(L, \sqsubseteq)$ be a preorder; $F: L\rightarrow L$ be a monotone function; ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}} \in L$ be such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}} \sqsubseteq F({\ooalign{{$\bot$}\crcr{\hss{-}\hss}}})$; $\nabla : L\times L \rightarrow L$ be a widening operator; and $\langle X_i \rangle_{i\in\mathbb{N}} \in L^{\mathbb{N}}$ be the infinite sequence defined by
$$
\small
\begin{array}{c}
X_0 = {\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\enspace; \qquad
\text{and, for each $i\in\mathbb{N}$,}\quad
X_{i+1} =
\begin{cases}
X_{i} & (\text{if}\; F(X_i) \sqsubseteq X_{i})\\
X_{i} \nabla F(X_{i}) & (otherwise)
\end{cases}
\end{array}
$$
Then the sequence $\langle X_i \rangle_{i\in\mathbb{N}}$ is increasing and
ultimately
stationary; moreover its limit $\bigsqcup_{i\in\mathbb{N}}X_{n}$ is a prefixed
point of $F$ such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\sqsubseteq\bigsqcup_{i\in\mathbb{N}}X_{n}$.
\qed
\end{myprop}
\subsection{The Domain of Convex Polyhedra}\label{subsec:convexpoly}
The \emph{domain of convex polyhedra}, introduced in~\cite{Cousot1978},
is one of the most commonly used relational numerical abstract domains.
\begin{mydef}[domain of convex polyhedra $\mathbb{CP}_{n}$]\label{def:convexPoly}
An $n$-dimensional \emph{convex polyhedron} is the intersection of
finitely many (closed) affine half-spaces.
We denote the set of convex polyhedra in $\mathbb{R} ^n$ by $\mathbb{CP}_n$.
Its preorder $\sqsubseteq$ is given by the inclusion order (actually it is a partial order).
The concretization function $\gamma_{\mathbb{CP}_n} : \mathbb{CP}_{n}\rightarrow{\mathcal P}(\mathbb{R}^n)$ is defined in an obvious manner.
\end{mydef}
\auxproof{
\begin{mydef}
A \emph{generator system} is a triple $(L, R, T)$ of finite sets of
vectors.
Here
$L$
is the set of \emph{lines}, $R$ is the set of \emph{rays} and $T$ is the
set of \emph{points}.
A generator system $G=(\{\vec{l_1}, \cdots, \vec{l_l}\}, \{\vec{r_1},
\cdots, \vec{r_r}\}, \{\vec{t_1}, \cdots, \vec{t_t}\})$ induces a
convex polyhedron ${\sf gen}(G)$ in the following way, where
$\mathbb{R}_{\geq 0}$ is the set of nonnegative reals.
$$
{\sf gen}(G)=\biggl\{\sum_{i=1}^l\lambda_i\vec{l_i} + \sum_{i=1}^r\rho_i\vec{r_i} + \sum_{i=1}^t\pi_i\vec{t_i} \,\biggr|\, \lambda_i \in \mathbb{R}, \rho_i \in \mathbb{R}_{\geq 0}, \pi_i \in \mathbb{R}_{\geq 0}, \sum_{i=1}^t \pi_i = 1 \biggr\}$$
\end{mydef}
}
We will be studying three widening operators on $\mathbb{CP}_{n}$.
They are namely:
the \emph{standard widening operator} $\nabla_{S}$~\cite{Halbwachs1979};\footnote{The name ``standard'' is
confusing with the distinction between \emph{standard} and
\emph{nonstandard} entities in NSA. The use of ``standard'' in the
former sense is scarce in
this paper.}
the \emph{widening operator $\nabla_M$ up to $M$}~\cite{Halbwachs1993, Halbwachs1997}; and
the \emph{precise widening operator} $\nabla_{N}$~\cite{Bagnara2005}.
We briefly describe the former two; the definition of the last is
omitted for the lack of space. In the following definitions, the function ${\sf con}$ maps a set of linear constraints (called a \emph{constraint system}) to the convex polyhedron induced by the conjunction of those linear constraints.
\begin{mydef}[standard widening $\nabla_{S}$]\label{def:stdwiden}
Let $P_{1}, P_{2}\in\mathbb{CP}_{n}$; and $C_{1}$ and $C_{2}$ be constraints
system that induce $P_{1}$ and $P_{2}$, respectively.
\emph{The standard widening operator} $\nabla_S : \mathbb{CP}_n \times \mathbb{CP}_n
\rightarrow \mathbb{CP}_n$ is defined by
\scalebox{0.87}{\parbox{1.1\linewidth}{
$$P_1 \nabla_S P_2 := \begin{cases}
P_2 & \text{if\;}P_1 = \emptyset\\
{\sf con}\Biggl(
\begin{minipage}[c]{8cm}
$\{\varphi \in C_1 \mid
\text{$C_{2}$ implies $\varphi$, i.e.\
$\varphi$ is everywhere true in $P_{2}$}\}$\\
$\cup\bigl\{\psi \in C_2 \,\bigr|\, \exists \varphi \in C_1.\, P_1 = {\sf con}(C_1[\psi/\varphi])\bigr\}$
\end{minipage}\Biggr)& \text{otherwise}.
\end{cases}$$
}}
\end{mydef}
\noindent Intuitively $P_1 \nabla_S P_2$ is represented by the set of those linear
constraints of $P_1$ which are satisfied by every point of $P_2$.
The following second widening operator $\nabla_{M}$ refines $\nabla_{S}$. This is
what we use in our implementation. Here $M$ is a parameter.
\begin{mydef}[widening up to $M$, $\nabla_{M}$]\label{def:widenupto}
Let ${P}_1, {P}_2 \in \mathbb{CP}_n$, and $M$ be a (given) finite set of linear inequalities.
The \emph{widening operator up to $M$} is defined by
$${P}_1 \nabla_M {P}_2 \;:=\; \bigl(\mathcal{P}_1 \nabla_S {P}_2\bigr)\; \cap\; {\sf con}\bigl(\{\varphi \in M \mid {P}_i\subseteq {\sf con}(\{\varphi\}){\rm \ for\ }i=1,2 \}\bigr)\enspace.$$
\end{mydef}
\noindent
The parameter $M$ is usually taken to be the set of linear
inequalities that occur in the program under analysis.
\subsection{Nonstandard Analysis}\label{subsec:preliminariesNSA}
Here we list a minimal set of necessary definitions and results in
nonstandard analysis (NSA)~\cite{Robinson1966}.
Some further details
can be found in Appendix~\ref{appendix:NSAPrimer};
fully-fledged and accessible expositions of NSA are
found e.g.\ in~\cite{Hurd1985, Goldblatt1998}.
The following notions will play important roles.
\begin{itemize}
\item
\emph{Hyperreals} that extends reals by infinitesimals, infinites,
etc.;
\item
The \emph{transfer principle}, a celebrated result in NSA that states
that
reals and hyperreals share ``the same properties'';
\item
The first-order language $\mathscr{L}_{\baseSet}$ that specifies formulas in which syntax, precisely,
are preserved by the transfer principle; and finally
\item
The semantical construct of \emph{superstructure} for
interpreting $\mathscr{L}_{\baseSet}$-formulas.
\end{itemize}
What is of paramount importance is the transfer principle; in order to
formulate it in a mathematically rigorous manner,
the two last items (the language $\mathscr{L}_{\baseSet}$ on the syntactic side, and
superstructures on the semantical side) are used.
The first-order language $\mathscr{L}_{\baseSet}$ is essentially that of set theory and has
two predicates $=$ and $\in$. The \emph{superstructure} $V(X)$ is
then a semantical ``universe'' for such formulas, constructed from the
base set $X$: concretely $V(X)$ is the union of $X$, $\mathcal P(X)$,
$\mathcal P(X\cup \mathcal P(X))$, and so on. Finally, when we take $X=\mathbb{R}$ then the
set $^{\ast}\hspace{-.15em}{X}=^{\ast}\hspace{-.15em}{\mathbb{R}}$ is that of \emph{hyperreals}; and the
transfer principle claims that $A$ holds for reals if and only if
$^{\ast}\hspace{-.15em}{A}$---a formula essentially the same as $A$---holds for
hyperreals.
Its precise
statement is:
\begin{mylem}[the transfer principle]\label{lemma:transferPrinciplePreview}
For any closed formula $A$ in $\mathscr{L}_{\baseSet}$, the following are equivalent.
\begin{itemize}
\item The formula $A$ is valid in the superstructure $V(X)$.
\item The \emph{*-transform} $^{\ast}\hspace{-.15em}{A}$ of $A$---this is a formula in the
language $\mathscr{L}_{\hyper{\baseSet}}$---is valid in the superstructure $V(^{\ast}\hspace{-.15em}{X})$.
\end{itemize}
\end{mylem}
The transfer principle guarantees that we can employ the same abstract
interpretation
framework, for reals and hyperreals alike---\emph{literally} the same, in the sense that we express the framework
in the
language $\mathscr{L}_{\R}$. Concretely, various
constructions
and meta results (such as soundness and termination) in abstract
interpretation will be expressed as $\mathscr{L}_{\R}$-formulas, and since they are valid
in $V(\mathbb{R})$, they are valid in the ``nonstandard universe''
$V(^{\ast}\hspace{-.15em}{\mathbb{R}})$ too, by the transfer principle.
\paragraph{Hyperreals}
We fix an \emph{index set} $I=\mathbb{N}$, and an \emph{ultrafilter} $\mathcal{F}\subseteq\mathcal P(I)$ that extends
the cofinite filter $\mathcal{F}_{\mathrm{c}}:=\{S\subseteq I\mid
I\setminus S \text{ is finite}\}$. Its properties to be noted: 1)
for any $S\subseteq I$, exactly one of $S$ and $I\setminus S$
belongs to $\mathcal{F}$; 2) if $S$ is \emph{cofinite} (i.e.\ $I\setminus
S$ is finite), then $S$ belongs to $\mathcal{F}$.
\begin{mydef}[hyperreal
$r\in^{\ast}\hspace{-.15em}{\mathbb{R}}$]\label{definition:hypernumber}
We define the set
$^{\ast}\hspace{-.15em}{\mathbb{R}}$ of \emph{hyperreal numbers} (or \emph{hyperreals})
by
\begin{math}
^{\ast}\hspace{-.15em}{\mathbb{R}}:=\mathbb{R}^{I}/{\sim_{\mathcal{F}}}
\end{math}. It is therefore the set of infinite sequences on $\mathbb{R}$ modulo the
following equivalence $\sim_{\mathcal{F}}$: we have
$(a_{0},a_{1},\dotsc)\sim_{\mathcal{F}} (a'_{0},a'_{1},\dotsc)$ if
\begin{equation}\label{equation:defOfSimFilt}
\{i\in I\mid a_{i}=a'_{i}\}\in\mathcal{F}\enspace,
\quad\text{for which we say ``$d_{i}=d'_{i}$ for almost every $i$.''}
\end{equation}
A \emph{hypernatural} $n\in^{\ast}\hspace{-.15em}{\mathbb{N}}$ is defined similarly.
\end{mydef}
\noindent
It follows that:
two sequences $(a_{i})_{i}$ and $(a'_{i})_{i}$ that
coincide except for finitely many indices $i$ represent the
same hyperreal. The predicates besides $=$ (such as $<$) are defined in the same
way.
A notable consequence is the existence of infinite numbers in the set of hyperreals and hypernaturals:
$\omega:=[(1, 2, 3, \dotsc)]$ is a positive infinite since it is larger than any positive real $r = [(r, r, \dotsc)]$ ($i > r$ for almost every $i\in\mathbb{N}$).
In addition, the set of hyperreals includes infinitesimal numbers: a hyperreal
$\omega^{-1}:=[\,(1,\frac{1}{2},\frac{1}{3},\dotsc)\,]$ is positive
($0<\omega^{-1}$) but is smaller than any (standard) positive real
$r$
\paragraph{Superstructure}
A \emph{superstructure} is a ``universe,'' constructed step by
step
from
a certain base set $X$ (whose typical examples are
$\mathbb{R}$ and $^{\ast}\hspace{-.15em}{\mathbb{R}}$). We assume
$\mathbb{N}\subseteq X$.
\begin{mydef}[superstructure]\label{definition:superstructure}
A \emph{superstructure} $V(X)$ over $X$ is
defined by $ V(X):=\bigcup_{n\in\mathbb{N}}V_{n}(X)$, where
$V_{0}(X):=X$ and
$V_{n+1}(X) := V_{n}(X) \cup \mathcal P (V_{n}(X))$.
\end{mydef}
The superstructure $V(X)$ might seem to be a closure of $X$ only under powersets, but it accommodates many set-forming operations.
For example, ordered pairs $\rtuple{a,b}$ and tuples $\rtuple{a_{1},\dotsc,a_{m}}$ are
defined in $V(X)$ as is usually done in set theory, e.g.\
$\rtuple{a,b}:=\{\{a\},\{a,b\}\}$
The function space $a\to b$ is
thought of as a collection of special binary relations (i.e.\ $a\to
b\subseteq \mathcal P (a\times b))$, hence is in $V(X)$.
\paragraph{The First-Order Language $\mathscr{L}_{\baseSet}$}
We use the following
first-order language $\mathscr{L}_{\baseSet}$, defined for each choice of the base set
$X$ like $\mathbb{R}$ and $^{\ast}\hspace{-.15em}{\mathbb{R}}$.
\begin{mydef}[the language $\mathscr{L}_{\baseSet}$]\label{definition:languageLX}
\emph{Terms} in $\mathscr{L}_{\baseSet}$ consist of: variables $x,y,x_{1},x_{2},\dotsc$;
and a constant $a$ for each entity $a\in V(X)$.
\emph{Formulas} in $\mathscr{L}_{\baseSet}$ are constructed as follows.
\begin{itemize}
\item The predicate symbols are $=$ and $\in$; both are binary. The
\emph{atomic formulas} are of the form $s=t$ or $s\in t$ (where $s$
and $t$ are terms).
\item We allow Boolean combinations of formulas. We use the
symbols
$\land, \lor,\lnot$ and $\Rightarrow$.
\item Given a formula $A$, a variable $x$ and a term $s$, the expressions
$\forall x\in s.\, A$ and $\exists x\in s.\, A$
are formulas.
\end{itemize}
\end{mydef}
\noindent Note that
quantifiers always come with a bound $s$.
The language $\mathscr{L}_{\baseSet}$ depends on the choice of $X$ (it
determines the set of constants).
We shall also use
the following syntax sugars in $\mathscr{L}_{\baseSet}$, as is common in set theory and NSA.
\begin{displaymath}\footnotesize
\begin{array}{ll}
\rtuple{s,t} & \text{pair}
\qquad \qquad \qquad
\rtuple{s_{1},\dotsc,s_{m}} \quad \text{tuple}
\qquad \qquad \qquad
s\times t \quad \text{direct product}
\\
s\subseteq t &\text{inclusion, short for $\forall x\in s. \,x\in t$}
\\
s(t) &\text{function application; short for $x$ such that
$\rtuple{t,x}\in s$}
\\
s\mathbin{\circ} t &\text{function composition, $(s\mathbin{\circ} t)(x)=s(t(x))$}
\\
s\le t&\text{inequality in $\mathbb{N}$; short for $\rtuple{s,t}\in{\le}$
where ${\le}\subseteq\mathbb{N}^{2}$}
\end{array}
\end{displaymath}
\begin{mydef}[semantics of $\mathscr{L}_{\baseSet}$]\label{definition:semanticsOfLX}
We interpret $\mathscr{L}_{\baseSet}$ in the superstructure $V(X)$ in the obvious way.
Let $A$ be a closed formula; we say $A$ is \emph{valid}
if $A$ is true in $V(X)$.
\end{mydef}
\paragraph{The $*$-Transform and the Transfer Principle}
As we mentioned the transfer principle says that a closed formula $A$
in the language $\mathscr{L}_{\baseSet}$ is valid in $V(X)$ if and only if $^{\ast}\hspace{-.15em}{A}$ in
$\mathscr{L}_{\hyper{\baseSet}}$ is valid in $V(^{\ast}\hspace{-.15em}{X})$.
We shall describe how we syntactically transform
$A$ in $\mathscr{L}_{\baseSet}$ into
$^{\ast}\hspace{-.15em}{A}$ in $\mathscr{L}_{\hyper{\baseSet}}$.
For that purpose, in particular in translating constants in $\mathscr{L}_{\baseSet}$ (for entities in
$V(X)$) to $\mathscr{L}_{\hyper{\baseSet}}$, we will need the following \emph{semantical} translation.
The so-called \emph{ultrapower construction} yields a canonical map
\begin{equation}\label{equation:*TransferMap}\footnotesize
\begin{array}{c}
^{\ast}\hspace{-.15em}{(\place)}
\;:\;
V(X) \longrightarrow V(^{\ast}\hspace{-.15em}{X})\enspace,
\qquad
a \longmapsto ^{\ast}\hspace{-.15em}{a}\enspace
\end{array}
\end{equation}
that is called the \emph{*-transform}.
It is a map from the universe $V(X)$ of standard entities
to $V(^{\ast}\hspace{-.15em}{X})$ of nonstandard entities. The details of its
construction are in Appendix~\ref{appendix:NSAPrimer} or in ~\cite{Hurd1985}.
The above map $^{\ast}\hspace{-.15em}{(\place)}\colon V(X)\to V(^{\ast}\hspace{-.15em}{X})$ becomes a \emph{monomorphism}, a notion in
NSA. Most notably it will satisfy
the \emph{transfer principle} (Lem.~\ref{lemma:transferPrinciple}).
\begin{mydef}[*-transform of formulas]\label{definition:starTransformOfFormulas}
Let $A$ be a formula in $\mathscr{L}_{\baseSet}$. The \emph{*-transform} of $A$, denoted
by $^{\ast}\hspace{-.15em}{A}$, is
a formula in $\mathscr{L}_{\hyper{\baseSet}}$ obtained by replacing each constant $a$ occurring
in $A$ with the constant $^{\ast}\hspace{-.15em}{a}$ that designates the element $^{\ast}\hspace{-.15em}{a}\in
V(^{\ast}\hspace{-.15em}{X})$.
\end{mydef}
\begin{mylem}[the transfer principle]\label{lemma:transferPrinciple}
For any closed formula $A$ in $\mathscr{L}_{\baseSet}$, $A$ is valid (in $V(X)$) if and
only if $^{\ast}\hspace{-.15em}{A}$ is valid (in $V(^{\ast}\hspace{-.15em}{X})$). \qed
\end{mylem}
We can prove, for instance, the following proposition using the transfer
principle (the proof is in Appendix~\ref{appendix:omittedProofs}).
This proposition has a practical implication: our implementation relies
on it
in simplifying formulas including the infinitesimal constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$.
\begin{myprop}\label{prop:dtToQE}
Let $A$ be an $\mathscr{L}_{\R}$-formula with a unique free variable $x$; to
emphasize it we write $A(x)$ for $A$.
Then the validity of the formula
$$\exists r \in \mathbb{R}. \left(0<r \wedge \forall x \in \mathbb{R}. \left(0<x<r \Rightarrow A \left(x\right)\right)\right)$$
(in $V(\mathbb{R})$) implies the validity of $^{\ast}\hspace{-.15em} A (\ensuremath{\mathtt{d\hspace{-.05em}t}})$ in $V(^{\ast}\hspace{-.15em}{\mathbb{R}})$.
\qed
\end{myprop}
\subsection{The Modeling Language $\While^{\dt}$}\label{subsec:whiledt}
$\While^{\dt}$, a modeling language for hybrid systems based on NSA, is introduced in \cite{Suenaga2011}.
It is an augmentation of a usual imperative language (such as ${\mathbf{IMP}}$ in~\cite{Winskel1993}) with a constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ that expresses an infinitesimal number.
\begin{mydef
Let $\mathbf{Var}$ be the set of variables.
The syntax of $\While^{\dt}$ is as follows:
\begin{eqnarray*}\footnotesize
\begin{array}[tb]{rl}
\mathbf{AExp} \ni a ::=& x \mid r \mid a_1 \mathrel{\mathtt{aop}} a_2 \mid \ensuremath{\mathtt{d\hspace{-.05em}t}}\\
&\text{\; where\; } x \in \mathbf{Var}, r \in \mathbb{R} \text{\; and\; } \mathrel{\mathtt{aop}} \in \{+, -, \cdot, / \}\\
\mathbf{BExp} \ni b ::=& \mathtt{true} \mid \mathtt{false} \mid b_1 \wedge b_2 \mid \neg b \mid a_1 < a_2 \\
\mathbf{Cmd} \ni c ::=& \mathtt{skip} \mid x:=a \mid c_1;c_2 \mid \mathtt{if}\; b\; \mathtt{then}\; c_1\; \mathtt{else}\; c_2 \mid \mathtt{while}\; b\; \mathtt{do}\; c.
\end{array}
\end{eqnarray*}
An expression $a \in \mathbf{AExp}$ is an \emph{arithmetic expression}, $b \in \mathbf{BExp}$ is a \emph{Boolean expression} and $c \in \mathbf{Cmd}$ is a \emph{command}.
\end{mydef}
\begin{wrapfigure}[9]{r}{0.46\hsize}
\vspace{-1em}
\begingroup
\fontsize{7pt}{8pt}\selectfont
\begin{verbatim}
(*Thermostat*)
x := 22; p := 0;
while true do {
if p = 0 then x := x - 3 * x * dt
else x := x + 3 * (30 - x) * dt;
if x >= 22 then p := 0
else {if x <= 18 then p := 1
else skip
}
}
\end{verbatim}
\endgroup
\vspace{-1.7em}
\caption{Thermostat in $\While^{\dt}$}
\label{fig:thermostat}
\end{wrapfigure}
As we explained in~\S{}\ref{sec:introduction}, the infinitesimal constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ enables us to model not only discrete dynamics but also continuous dynamics without explicit ODEs.
For example, the water-level monitor is modeled as a $\While^{\dt}$ program shown in Fig.~\ref{fig:whileDtCodeCaseStudy}.
As another example, the thermostat can be modeled as the program on the right.
One can see that the continuous dynamics modeled in this example is beyond piecewise-linear.
Even dynamics defined by nonlinear ODEs can be modeled in $\While^{\dt}$ in
the same manner. To go further to accommodate an arbitrary hybrid automaton we must
properly deal with \emph{nondeterminism}, a feature currently lacking in
$\While^{\dt}$. Although we expect that to be not hard, precise comparision
between $\While^{\dt}$ and hybrid automata in expressivity is future work.
In the usual, standard abstract interpretation (without $\ensuremath{\mathtt{d\hspace{-.05em}t}}$), a
command $c$ is assigned its \emph{collecting semantics}
${\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R}) \rightarrow {\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})$ (see e.g.~\cite{Cousot1977}). This is semantics by reachable sets of memory states, as the concrete semantics.
Presence of $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ in the syntax of $\While^{\dt}$ calls for an
infinitesimal number in the picture.
The first thing to try would be to replace $\mathbb{R}$ with $^{\ast}\hspace{-.15em}\mathbb{R}$, and
let $\While^{\dt}$ commands interpreted as functions of the type
${\mathcal P}(\mathbf{Var}\rightarrow^{\ast}\hspace{-.15em}{\mathbb{R}}) \rightarrow
{\mathcal P}(\mathbf{Var}\rightarrow^{\ast}\hspace{-.15em}{\mathbb{R}})$. This however is not suited for the
purpose of interpreting recursion in presence of $\ensuremath{\mathtt{d\hspace{-.05em}t}}$.\footnote{If we interpret commands
as functions $
{\mathcal P}(\mathbf{Var}\rightarrow^{\ast}\hspace{-.15em}{\mathbb{R}})
\to
{\mathcal P}(\mathbf{Var}\rightarrow^{\ast}\hspace{-.15em}{\mathbb{R}})$, the interpretation
$\sem{\mathtt{while}\; x<10 \;\mathtt{do}\; x:= x+\ensuremath{\mathtt{d\hspace{-.05em}t}}}\{(x\mapsto 0)\}$ by a least fixed point
will be $\{x\mapsto r \mid \exists n \in \mathbb{N}.\; r = n*\ensuremath{\mathtt{d\hspace{-.05em}t}}\}$, not
$\{x\mapsto r \mid \exists n \in ^{\ast}\hspace{-.15em}\mathbb{N}.\; r = n*\ensuremath{\mathtt{d\hspace{-.05em}t}} \wedge
r\leq10\}$ as we expect. The problem is that
\emph{internality}---an ``well-behavedness'' notion in
NSA---is not preserved in such a modeling.} We rely instead on our theory of
\emph{hyperdomains} that is used in~\cite{Suenaga2013} and
described in Appendix~\ref{appendix:domainTheoryTransferred}
; see the
interpretation of while loops in Table~\ref{table:densem}. This calls
for the interpretation of commands to be of the type $^{\ast}\hspace{-.15em}{\bigl(\,
{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R}) \rightarrow {\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})
\,\bigr)}$, a subset of
$^{\ast}\hspace{-.15em}{
{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})} \rightarrow ^{\ast}\hspace{-.15em}{{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})
}$. The last type will be used in the following definition.
\begin{mydef
\label{def:whiledtsem}
\emph{Collecting semantics} for $\While^{\dt}$,
in Table~\ref{table:densem}, has the following
types where $\mathbb{B}$ is $\{\text{t\hspace{-0.1em}t}, \text{ff}\}$:
$\sem{a}\colon ^{\ast}\hspace{-.15em}(\mathbf{Var}\rightarrow\mathbb{R}) \rightarrow ^{\ast}\hspace{-.15em}\mathbb{R} $ for
$a\in\mathbf{AExp}$;
$\sem{b}\colon ^{\ast}\hspace{-.15em}(\mathbf{Var}\rightarrow\mathbb{R}) \rightarrow \mathbb{B}$ for
$b\in\mathbf{BExp}$; and
$\sem{c}\colon ^{\ast}\hspace{-.15em}{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R}) \rightarrow
^{\ast}\hspace{-.15em}{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})$ for $c\in\mathbf{Cmd}$.
\end{mydef}
In~\cite{Suenaga2011} and in~\S{}\ref{sec:introduction}, the
semantics of a while loop is defined using the idea of
sectionwise execution, instead of as a least fixed point. This is not
suited for employing abstract
interpretation---the latter is after all for computing least fixed points.
The collecting semantics in Def.~\ref{def:whiledtsem}
(Table~\ref{table:densem}) does use least fixed
points; it is based on the alternative $\While^{\dt}$ semantics
introduced in~\cite{Kido2013} (it will also appear in the forthcoming full version
of~\cite{Suenaga2011, Hasuo2012}). The equivalence of the two semantics
is established in~\cite{Kido2013}.
\begin{table}[tbp]
\centering
\scalebox{0.8}{\parbox{1.1\linewidth}{%
\begin{align*}
& \sem{x}\boldsymbol{\sigma} := \boldsymbol{\sigma}(x) \text{\;for each\;}x\in\mathbf{Var} &&\sem{\mathtt{true}}\boldsymbol{\sigma} := {\text{t\hspace{-0.1em}t}} \\
& \sem{r}\boldsymbol{\sigma} := r \text{\;for each\;}r \in\mathbb{R} &&\sem{\mathtt{false}}\boldsymbol{\sigma} := {\text{ff}} \\
& \sem{a_1 \mathrel{\mathtt{aop}} a_2}\boldsymbol{\sigma} := \sem{a_1} \mathrel{\mathtt{aop}} \sem{a_2} &&\sem{b_1 \wedge b_2}\boldsymbol{\sigma} := \sem{b_1} \wedge \sem{b_2} \\
& \sem{\ensuremath{\mathtt{d\hspace{-.05em}t}}}\boldsymbol{\sigma} := \textstyle[(1, \frac{1}{2}, \frac{1}{3}, \cdots)] &&\sem{\neg b}\boldsymbol{\sigma} := \neg(\sem{b}\boldsymbol{\sigma})
\end{align*}
\vspace{-2.5em}
\begin{align*}
&\sem{\mathtt{skip}}\mathbf{S} := \mathbf{S} \\
&\sem{x := a}\mathbf{S} := \{\boldsymbol{\sigma}[\sem{a}\boldsymbol{\sigma}/x] \mid \boldsymbol{\sigma} \in \mathbf{S}\} \\
&\sem{c_1; c_2}\mathbf{S} := \sem{c_2}(\sem{c_1}\mathbf{S}) \\
&\sem{\mathtt{if}\; b \;\mathtt{then}\; c_1 \;\mathtt{else}\; c_2}\mathbf{S} :=
\begin{minipage}[c]{8cm}
$\{\sem{c_1}\boldsymbol{\sigma} \mid \boldsymbol{\sigma}\in \mathbf{S},\; \sem{b}\boldsymbol{\sigma} = \text{t\hspace{-0.1em}t} \} \\ \cup \{\sem{c_2}\boldsymbol{\sigma} \mid \boldsymbol{\sigma}\in \mathbf{S},\; \sem{b}\boldsymbol{\sigma} = \text{ff} \}$
\end{minipage}\\
&\sem{\mathtt{while}\; b \;\mathtt{do}\; c} := {\rm lfp}\bigl(^{\ast}\hspace{-.15em}\Phi\left(\sem{b}\right)\left(\sem{c}\right)\bigr) \\
&\qquad\text{where\;}\Phi: \left(\mathbf{St} \rightarrow \mathbb{B}\cup\{\bot\}\right) \rightarrow \bigl({\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\rightarrow {\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\bigr) \rightarrow \\
&\qquad\qquad\quad \Bigl(\bigl({\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\rightarrow{\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\bigr) \rightarrow \bigl({\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\rightarrow{\mathcal P}\left(\mathbf{Var}\rightarrow\mathbb{R}\right)\bigr)\Bigr)\\
&\qquad\text{\;is\;defined\;by\;}
\Phi(f)(g) = \lambda\psi.\;\lambda S.\;
S\cup\psi\{(g(\sigma))\mid \sigma\in S,\; f(\sigma) = \text{t\hspace{-0.1em}t}\} \cup \{\sigma\mid\sigma\in S, \; f(\sigma) = \text{ff}\}.
\end{align*}
}}
\caption{$\While^{\dt}$ collecting semantics}
\label{table:densem}
\end{table}
In the rest of the paper we restrict the set of variables $\mathbf{Var}$ to be
finite. This as\-sump\-tion---a realistic one when we focus on the program
to be analyzed---makes our NSA framework much simpler.
Therefore
${\mathcal P}(\mathbf{Var}\rightarrow \mathbb{R})$ and $^{\ast}\hspace{-.15em}{\mathcal P}(\mathbf{Var}\rightarrow\mathbb{R})$ are equal to ${\mathcal P}(\mathbb{R}^n)$ and $^{\ast}\hspace{-.15em}{\mathcal P}(\mathbb{R}^n)$ for some $n\in\mathbb{N}$ respectively; we prefer the latter notations in what follows.
\section{Abstract Interpretation Augmented with Infinitesimals}
\label{sec:NSAI}
In the current section are our main theoretical contributions---a
metatheory of \emph{nonstandard abstract interpretation} that justifies
the workflow in~\S{}\ref{subsec:waterLevelDt}.
(Standard) abstract interpretation infrastructure such as Prop.~\ref{prop:concretization} and Prop.~\ref{prop:widen} is not applicable to $\While^{\dt}$ programs.
since $^{\ast}\hspace{-.15em}{\mathcal P}(\mathbb{R}^n)$ is not a cpo.\footnote{One can see that the ascending chain defined by $X_n := \{k*\ensuremath{\mathtt{d\hspace{-.05em}t}} \mid 0\leq k \leq n\}$ does not have the supremum in $^{\ast}\hspace{-.15em}{\mathcal P}(\mathbb{R}^n)$ since $\{k*\ensuremath{\mathtt{d\hspace{-.05em}t}} \mid k\in\mathbb{N}\}$ is not \emph{internal} (see Appendix~\ref{appendix:NSAPrimer})
.}
Thus, building on the theoretical foundations in the above, we now extend the abstract interpretation framework for the
analysis of $\While^{\dt}$ programs (and the hybrid systems modeled
thereby). We introduce an \emph{abstract hyperdomain} over $^{\ast}\hspace{-.15em}{\mathbb{R}}$ as the transfer of
the
(standard, over $\mathbb{R}$) domain of convex polyhedra. We then interpret $\While^{\dt}$
programs in them, and transfer the three widening operators mentioned in~\S{}\ref{subsec:abstinterp} to the
nonstandard setting. We classify them into \emph{uniform} ones---for
which termination is guaranteed even in the nonstandard setting---and
non-uniform ones.
The main theorems are Thm.~\ref{thm:hyperconcretization} and Thm.~\ref{thm:newunifwidenwithinfinitesimal}, for soundness (in place of Prop.~\ref{prop:concretization}) and termination (in place of Prop.~\ref{prop:widen}) respectively.
\subsection{The Domain of Convex Polyhedra over Hyperreals}
\label{subsec:abstractDomainOverHyperreals}
We extend convex polyhedra to the current nonstandard setting.
\begin{mydef}[convex polyhedra over $^{\ast}\hspace{-.15em}{\mathbb{R}}$]\label{def:hyperConvexPoly}
A \emph{convex polyhedron} on $(^{\ast}\hspace{-.15em}\mathbb{R})^n$ is an intersection of
finite number of affine half-spaces on $(^{\ast}\hspace{-.15em}\mathbb{R})^n$, that is, the
set of points $\mathbf{x}\in(^{\ast}\hspace{-.15em}\mathbb{R})^n$ that satisfy a certain
finite set of linear inequalities.
The set of all convex polyhedra on $(^{\ast}\hspace{-.15em}\mathbb{R})^n$ is denoted by
$\mathbb{CP}_n^{^{\ast}\hspace{-.15em}{\mathbb{R}}}$.
\end{mydef}
\begin{myprop}\label{prop:hyperconvexpoly}
The set $\mathbb{CP}_n^{^{\ast}\hspace{-.15em}{\mathbb{R}}}$ of all convex polyhedra over $(^{\ast}\hspace{-.15em}{\mathbb{R}})^{n}$
is a (proper) subset of
$^{\ast}\hspace{-.15em}\mathbb{CP}_n$, the $*$-transform of the (standard) domain of convex polyhedra over $\mathbb{R}^n$.
\qed
\end{myprop}
What lies in the difference between the two sets
$\mathbb{CP}_n^{^{\ast}\hspace{-.15em}{\mathbb{R}}}\subsetneq^{\ast}\hspace{-.15em}\mathbb{CP}_n$ is, for example, a disk as a
subset of $\mathbb{R}^{2}$ (hence of $^{\ast}\hspace{-.15em}{\mathbb{R}}^{2}$). In $^{\ast}\hspace{-.15em}\mathbb{CP}_2$ one can use a constraint system whose number of linear constraints is a hypernatural number $m\in^{\ast}\hspace{-.15em}{\mathbb{N}}$; using e.g.\
$m=\omega=[(0,1,2,\dotsc)]$
allows us to approximate a disk with progressive precision.
In the following development of nonstandard abstract interpretation, we
will use $^{\ast}\hspace{-.15em}\mathbb{CP}_n$ as an abstract domain since it allows transfer
of properties
of $\mathbb{CP}_n$.
We note, however, that
our over-approximation of
the interpretation $\sem{c}$ of a loop-free $\While^{\dt}$ program $c$
is always given in $\mathbb{CP}_n^{^{\ast}\hspace{-.15em}{\mathbb{R}}}$, i.e.\ with finitely many linear inequalities.
\subsection{Theory of Nonstandard Abstract Interpretation}
\label{subsec:theoryOfHyperAbstractInterp}
Our goal is to over-approximate the collecting semantics for $\While^{\dt}$ programs (Table~\ref{table:densem}) on convex polyhedra over $^{\ast}\hspace{-.15em}\mathbb{R}$.
As we mentioned at the beginning of this section, however, abstract interpretation infrastructure cannot be applied since $^{\ast}\hspace{-.15em}{\mathcal P}(\mathbb{R}^n)$ is not a cpo.
Fortunately it turns out that we can rely on the \emph{$*$-transform}
(\S{}\ref{subsec:preliminariesNSA}) of
the theory in~\S{}\ref{subsec:abstinterp}, where it suffices to impose the cpo structure only on ${\mathcal P}(\mathbb{R})$ and the \emph{$*$-continuity}---instead of the (standard)
continuity---on
the function
$\sem{c}$.
This theoretical framework of \emph{nonstandard abstract
interpretation}, which we shall describe here, is an extension of the
\emph{transferred domain theory} studied
in~\cite{BeauxisM11,Suenaga2013}. Part of the latter is found also in Appendix~\ref{appendix:domainTheoryTransferred}.
\begin{mythm}\label{thm:hyperconcretization}
Let $(L, \sqsubseteq)$ be a cpo;
$F:^{\ast}\hspace{-.15em}{L}\rightarrow ^{\ast}\hspace{-.15em}{L}$ be a *-continuous function;
and ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\in^{\ast}\hspace{-.15em}{L}$ be such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} F({\ooalign{{$\bot$}\crcr{\hss{-}\hss}}})$.
Let $(\overline{L}, \overline{\sqsubseteq})$ be a preorder;
$\gamma: \overline{L} \rightarrow L$ be a function such that $\overline{a}\mathrel{\overline{\sqsubseteq}}\overline{b} \Rightarrow \gamma(\overline{a})\sqsubseteq \gamma(\overline{b})$ for all $\overline{a}, \overline{b}\in\overline{L}$; and
$\overline{F}:^{\ast}\hspace{-.15em}{\overline{L}}\rightarrow^{\ast}\hspace{-.15em}{\overline{L}}$ be a *-continuous function that is monotone with respect to $^{\ast}\hspace{-.15em}{\overline\sqsubseteq}$ and satisfies $F\circ^{\ast}\hspace{-.15em}\gamma \mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} ^{\ast}\hspace{-.15em}\gamma\circ\overline{F}$.
Note that $(^{\ast}\hspace{-.15em}{\overline{L}}, ^{\ast}\hspace{-.15em}{\overline\sqsubseteq})$ is also a preorder.
Assume further that $\overline{x}\in^{\ast}\hspace{-.15em}{\overline{L}}$ is a prefixed point of $\overline{F}$
(i.e.\ $\overline{F}(\overline{x})\mathrel{^{\ast}\hspace{-.15em}{\overline\sqsubseteq}}\overline{x}$)
such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\mathrel{^{\ast}\hspace{-.15em}{\sqsubseteq}}^{\ast}\hspace{-.15em}{\gamma}(\overline{x})$.
Then $\overline{x}$ over-approximates ${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}F$, that is,
${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}F\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} ^{\ast}\hspace{-.15em}\gamma(\overline{x})$.
\qed
\end{mythm}
Our goal is
over-approximation of
the semantics of iteration of a loop-free $\While^{\dt}$ program $c$,
relying on Thm.~\ref{thm:hyperconcretization}.
Towards the goal, the next step
is to find a suitable
$\overline{F}: ^{\ast}\hspace{-.15em}{\overline{L}} \rightarrow ^{\ast}\hspace{-.15em}{\overline{L}}$
that ``stepwise approximates'' $F=\sem{c}$, the collecting semantics of $c$.
The next result implies that the $*$-transformation of
$\semcp{\place}$ (defined in a usual manner in standard abstract interpretation, as mentioned in~\S{}\ref{subsec:abstinterp}) can be used in
such $\overline{F}$.
\begin{myprop}
Let
$(L, \sqsubseteq), (\overline{L}, \overline{\sqsubseteq}), \gamma: \overline{L} \rightarrow L$ satisfy the hypotheses in Thm.~\ref{thm:hyperconcretization}.
Assume that a continuous function $F: L \rightarrow L$ is stepwise approximated by a monotone function
$\overline{F}: \overline{L}\rightarrow\overline{L}$, that is,
$F\circ\gamma \sqsubseteq \gamma\circ\overline{F}$.
Then the *-continuous function $^{\ast}\hspace{-.15em}{F}: ^{\ast}\hspace{-.15em}{L} \rightarrow ^{\ast}\hspace{-.15em}{L}$ is over-approximated by the monotone and internal function $^{\ast}\hspace{-.15em}{\overline{F}}: ^{\ast}\hspace{-.15em}{\overline{L}}\rightarrow ^{\ast}\hspace{-.15em}{\overline{L}}$, i.e. $^{\ast}\hspace{-.15em}{F} \circ^{\ast}\hspace{-.15em}\gamma \mathrel{^{\ast}\hspace{-.15em}{\sqsubseteq}} ^{\ast}\hspace{-.15em}{\gamma}\circ^{\ast}\hspace{-.15em}{\overline{F}}$.
\qed
\end{myprop}
We summarize what we observed so far on nonstandard abstract
interpretation by instantiating the abstract domain to $^{\ast}\hspace{-.15em}{\mathbb{CP}_{n}}$.
In the following $\sem{c}$ is from Def.~\ref{def:whiledtsem}.
\begin{mycor}[soundness of nonstandard abstract interpretation on $^{\ast}\hspace{-.15em}{\mathbb{CP}_{n}}$]
\label{cor:soundnessOfHyperAbstractDomains}
Let $c$ be a loop-free $\While^{\dt}$ command;
and let ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\in^{\ast}\hspace{-.15em}{({\mathcal P}(\mathbb{R}^{n}))}$
and $\overline{x} \in ^{\ast}\hspace{-.15em}{\mathbb{CP}_n}$ be such that $(^{\ast}\hspace{-.15em}{\semcp{c}})(\overline{x})
\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq}
\overline{x}$ and
${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq}^{\ast}\hspace{-.15em}{\gamma_{\mathbb{CP}_n}}(\overline{x})$.
Then we have
${\rm lfp}_{{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}}\sem{c}\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq}^{\ast}\hspace{-.15em}{\gamma_{\mathbb{CP}_n}}(\overline{x})$.
\qed
\end{mycor}
\subsection{Hyperwidening and Uniform Widening Operators}
\label{subsec:uniformWidening}
Towards our goal of using Thm.~\ref{thm:hyperconcretization}, the last
remaining step is to find a prefixed point $\overline{x}$,
i.e.\
$\overline{F}(\overline{x})\mathrel{^{\ast}\hspace{-.15em}{\overline{\sqsubseteq}}}\overline{x}$.
This is where widening operators are standardly used; see~\S{}\ref{subsec:abstinterp}.
We can try $*$-transforming a (standard) notion---a strategy that we have used repeatedly in the current section. This
yields the following result, that has a problem that is discussed shortly.
\begin{mythm}
\label{thm:widenwithinf}
Let $(L, \sqsubseteq)$ be a preorder and $\nabla: L\times L\rightarrow L$ be a widening operator on $L$.
Let $F: ^{\ast}\hspace{-.15em}{L} \rightarrow^{\ast}\hspace{-.15em}{L}$
be a monotone and internal function;
and ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\in ^{\ast}\hspace{-.15em}{L}$
be such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}} \mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} F({\ooalign{{$\bot$}\crcr{\hss{-}\hss}}})$.
The iteration \emph{hyper}-sequence
$\langle X_{i}\rangle_{i\in^{\ast}\hspace{-.15em}\mathbb{N}}$---indexed by hypernaturals
$i\in^{\ast}\hspace{-.15em}{\mathbb{N}}$---that is defined by
$$
\footnotesize
\begin{array}{c}
X_0 = {\ooalign{{$\bot$}\crcr{\hss{-}\hss}}},\quad X_{i+1} = \begin{cases}
X_{i} & ({\text if }\; F(X_i) \mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} X_{i})\\
X_{i} ^{\ast}\hspace{-.15em}\nabla F(X_{i}) & (otherwise)
\end{cases} {\rm for \; all}\; i \in ^{\ast}\hspace{-.15em}\mathbb{N}
\end{array}
$$
reaches its limit within some hypernatural number of steps and the limit
$\bigsqcup_{i\in\mathbb{N}}X_i$ is a prefixed point of $F$ such that ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}
\mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} \bigsqcup_{i\in\mathbb{N}}X_i$. \qed
\end{mythm}
The problem of Thm.~\ref{thm:widenwithinf} is that the \emph{finite-step
convergence} of iteration sequences for the original widening operator (described in Prop.~\ref{prop:widen})
is now transferred to \emph{hyper\-finite-step convergence}.
This is not desired. All the entities from NSA that we have used so far
are constructs in denotational semantics---whose only role is to ensure
soundness of verification methodologies\footnote{Recall that $\While^{\dt}$
is a \emph{modeling} language and we do not execute them.} and on which
we never actually operate---and therefore their
infinite/infinitesimal nature has been not a problem. In contrast,
computation of the iteration hypersequence $\langle
X_{i}\rangle_{i\in^{\ast}\hspace{-.15em}\mathbb{N}}$ is what we actually compute to
over-approximate program semantics; and therefore its termination
guarantee within $i\in^{\ast}\hspace{-.15em}{\mathbb{N}}$ steps (Thm.~\ref{thm:widenwithinf})
is of no use.
As a remedy we introduce a new notion of \emph{uniformity} of the (standard) widening
operators. It strengthens the original termination condition
(Def.~\ref{def:widen})
by imposing a uniform bound $i$ for stability of arbitrary chains
$\langle x_i \rangle \in L^\mathbb{N}$. Logically the change means
replacing $\forall\exists$ by $\exists\forall$.
\begin{mydef}[uniform widening]\label{def:unifwiden}
Let $(L, \sqsubseteq)$ be a preorder.
A function $\nabla: L \times L \rightarrow L$ is said to be a \emph{uniform widening operator} if the following two conditions hold.
\begin{itemize}
\item (Covering) For any $x, y \in L$, $x \sqsubseteq x \nabla y$ and $y \sqsubseteq x \nabla y$.
\item (Uniform termination) Let $x_0 \in L$. There exists a
\emph{uniform bound} $i \in \mathbb{N}$ such that: for any ascending
chain $\langle x_k \rangle \in L^\mathbb{N}$ starting from $x_0$, there
exists $j \le i$ at which the chain $\langle y_k \rangle \in
L^\mathbb{N}$, defined by $y_0 = x_0$ and $y_{k+1} = y_{k} \nabla x_{k+1} {\rm\; for \; all}\; k \in \mathbb{N}$, stabilizes (i.e.\ $y_{j} = y_{j+1}$).
\end{itemize}
\end{mydef}
It is straightforward that uniform termination implies termination.
We investigate uniformity of some of the commonly-known widening operators on convex polyhedra.
\begin{mythm}\label{thm:uniformityOfKnownWidening}
Among the three widening operators in~\S{}\ref{subsec:abstinterp}, $\nabla_{S}$ (Def.~\ref{def:stdwiden}) and $\nabla_{M}$ (Def.~\ref{def:widenupto}) are uniform, but $\nabla_{N}$ (\cite{Bagnara2005}) is not.
\qed
\end{mythm}
For example, the widening operator $\nabla_{S}$ is uniform because once the first element $x_0$ of an iteration sequence is fixed, the length of the iteration sequence is at most the number of linear inequalities that define the convex polyhedra $x_0$.
However, $\nabla_{N}$ is not uniform because an iteration sequence can be arbitrarily long even if the first element of it is fixed,
The following theorem is a ``practical'' improvement of
Thm.~\ref{thm:widenwithinf};
its proof relies on instantiating the uniform bound $i$
in a suitable $\mathscr{L}_{\R}$-formula with a Skolem
constant, before transfer.
\begin{mythm}
\label{thm:newunifwidenwithinfinitesimal}
Let $(L, \sqsubseteq)$ be a preorder and $\nabla \in L \times L \rightarrow L$ be a uniform widening operator on $L$.
Let $F: ^{\ast}\hspace{-.15em}{L} \rightarrow^{\ast}\hspace{-.15em}{L}$
be a monotone and internal function;
and ${\ooalign{{$\bot$}\crcr{\hss{-}\hss}}}\in L$
be such that $^{\ast}\hspace{-.15em}{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}} \mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} F(^{\ast}\hspace{-.15em}{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}})$.
The iteration sequence
$\langle X_i\rangle_{i \in \mathbb{N}}$ defined by
$$
\footnotesize
\begin{array}{c}
X_0 = ^{\ast}\hspace{-.15em}{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}},\quad
X_{i+1} =
\begin{cases}
X_{i} & ({\text if }\; F(X_i) \mathrel{^{\ast}\hspace{-.15em}{\sqsubseteq}} X_{i})\\
X_{i} \mathrel{^{\ast}\hspace{-.15em}{\nabla}} F(X_{i}) & (otherwise)
\end{cases} \quad{\rm for \; all}\; i \in \mathbb{N}
\end{array}
$$
reaches its limit within some finite number of steps; and the limit
$\bigsqcup_{i\in\mathbb{N}}X_i$ is a prefixed point of $F$ such that
$^{\ast}\hspace{-.15em}{\ooalign{{$\bot$}\crcr{\hss{-}\hss}}} \mathrel{^{\ast}\hspace{-.15em}\sqsubseteq} \bigsqcup_{i\in\mathbb{N}}X_i$. \qed
\end{mythm}
Note that uniformity of $\nabla$ is a \emph{sufficient condition} for
the termination of nonstandard iteration sequences (by
$^{\ast}\hspace{-.15em}{\nabla}$); Thm.~\ref{thm:newunifwidenwithinfinitesimal} does not
prohibit other useful widening operators in the nonstandard setting.
Furthermore, there can be a useful (nonstandard) widening operator
except for the ones $^{\ast}\hspace{-.15em}\nabla$ that arise
via standard ones $\nabla$.
It is a direct consequence of Thm.~\ref{thm:newunifwidenwithinfinitesimal} and Thm.~\ref{thm:uniformityOfKnownWidening} that the analysis of $\While^{\dt}$ programs on $^{\ast}\hspace{-.15em}{\mathbb{CP}_n}$ is terminating with $\nabla_S$ or $\nabla_M$.
\section{Implementation and Experiments}\label{sec:implementation}
\subsection{Implementation}
We implemented a prototype tool for analysis of $\While^{\dt}$ programs.
The tool currently supports: $^{\ast}\hspace{-.15em}\mathbb{CP}_{n}$ as an abstract domain; and
$^{\ast}\hspace{-.15em}\nabla_{M}$, *-transformation of $\nabla_M$ in Def.~\ref{def:widenupto} as a widening operator.
Its input is
a $\While^{\dt}$ program.
It outputs a convex polyhedron that over-approximates the set of reachable memory states for each modes (or the values of discrete variables).
Our tool consists principally of the following two components:
1) an OCaml frontend for parsing, forming an iteration sequence and making the set $M$ for $^{\ast}\hspace{-.15em}\nabla_{M}$; and
2) a Mathematica backend for executing operations on convex polyhedra.
The two components are interconnected by a C++ program, via MathLink.
There are some libraries such as Parma Polyhedra Library~\cite{BagnaraHZ08SCP} that are commonly used to execute operations on convex polyhedra.
They cannot be used in our implementation because we have to handle the infinitesimal constant $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ as an truly infinitesimal value.
Instead we implemented Chernikova's algorithm~\cite{Chernikova1964, Chernikova1965, Chernikova1968, LeVerge1992} symbolically, using \emph{computer algebra system (CAS)} on Mathematica based on Prop.~\ref{prop:dtToQE}.
Prop.~\ref{prop:dtToQE} ensures that the transformation from $^{\ast}\hspace{-.15em} A (\ensuremath{\mathtt{d\hspace{-.05em}t}})$ \\to
$\exists r \in \mathbb{R}. \left(0<r \wedge \forall x \in \mathbb{R}. \left(0<x<r \Rightarrow A \left(x\right)\right)\right)$ does not violate the soundness of the analysis.
When we have to evaluate a formula including $\ensuremath{\mathtt{d\hspace{-.05em}t}}$, we instead resolve $\exists r \in \mathbb{R}. \left(0<r \wedge \forall x \in \mathbb{R}. \left(0<x<r \Rightarrow A \left(x\right)\right)\right)$ using CAS (e.g. quantifier elimination).
\subsection{Experiments}\label{subsec:experiments}
We analyzed two $\While^{\dt}$ programs---the water-level monitor (Fig.~\ref{fig:whileDtCodeCaseStudy}) and the thermostat
(Fig.~\ref{fig:thermostat})---with our prototype.
The experiments were on Apple MacBook Pro with 2.6 GHz Dual-core Intel Core i5 CPU and 8 GB memory and the execution times are the average of 10 runs.
\noindent
\textbf{Water-Level Monitor}
This is a piecewise-linear dynamics
and a typical example used in hybrid automata literature. Our tool
automates the analysis presented in~\S{}\ref{sec:exampleOfAnalysis};
the execution time was 22.151 sec.
\noindent
\textbf{Thermostat}
The dynamics of this example is beyond piecewise-linear.
The nonstandard abstract interpretation
successfully analyzes this example without explicit piece\-wise-linear
approximation. We believe this result witnesses a potential of our approach. We skip how it analyzes this example since the procedure is the same as the water-level monitor case.
Our tool executes in 2.259 sec.\ and outputs an approximation from which
we obtain an invariant $18-54*\ensuremath{\mathtt{d\hspace{-.05em}t}}\leq x \leq 22+24*\ensuremath{\mathtt{d\hspace{-.05em}t}}$.
\section{Conclusions and Future Work}
We presented an extended abstract interpretation framework in which
hybrid systems are \emph{exactly} modeled as programs with infinitesimals.
The logical infrastructure by \emph{nonstandard analysis} (in particular
the \emph{transfer principle}) establishes its
soundness.
Termination is also ensured for \emph{uniform} widening operators.
Our prototype analyzer automates the extended abstract interpretation on the domain of convex polyhedra.
Regrettably our current implementation is premature and does not
compare---in precision or scalability---with the state-of-art tools for
hybrid system reachability such as SpaceEx~\cite{Frehse11} and Flow*~\cite{ChenAS13}.
In fact the two examples in~\S{}\ref{subsec:experiments} are the only
ones that we have so far succeeded to analyze. For other
examples---especially nonlinear ones, to which our framework is
applicable in principle---the analysis results are too imprecise to be useful.
To improve
there are some possible directions of future work to enhance the precision and scalability.
Firstly, we could utilize trace partitioning~\cite{Mauborgne2005}, narrowing operators (the use of narrowing operators in the domain of convex polyhedra is indicated in~\cite[\S{3.4}]{Henriksen2007}) and other techniques that have been introduced to enhance the precision of the analysis.
Secondly,
we believe abstract domains such as
\emph{ellipsoids}~\cite{Feret2004}, or some new ones that are tailored
to nonlinear dynamics,
can improve our analyzer.
Finally,
the lack of scalability is mainly due to our current way of eliminating $\ensuremath{\mathtt{d\hspace{-.05em}t}}$ (namely via
Prop.~\ref{prop:dtToQE}): it relies on \emph{quantifier elimination
(QE)} that is highly expensive. A faster alternative is desired.
\bibliographystyle{splncs03}
|
2,869,038,154,920 | arxiv | \section{Introduction}
Recently, deep reinforcement learning has enabled neural network policies to achieve state-of-the-art performance on many high-dimensional control tasks, including Atari games (using pixels as inputs) \cite{Mnih2015, Mnih2016}, robot locomotion and manipulation \cite{Schulman2006, Levine2016, Lillicrap2016}, and even Go at the human grandmaster level \cite{Silver2016}.
In reinforcement learning (RL), agents learn to act by trial and error, gradually improving their performance at the task as learning progresses. Recent work in deep RL assumes that agents are free to explore \emph{any behavior} during learning, so long as it leads to performance improvement. In many realistic domains, however, it may be unacceptable to give an agent complete freedom. Consider, for example, an industrial robot arm learning to assemble a new product in a factory. Some behaviors could cause it to damage itself or the plant around it---or worse, take actions that are harmful to people working nearby. In domains like this, \emph{safe exploration} for RL agents is important \cite{Moldovan2012, Amodei2016}. A natural way to incorporate safety is via constraints.
A standard and well-studied formulation for reinforcement learning with constraints is the constrained Markov Decision Process (CMDP) framework \cite{Altman1999}, where agents must satisfy constraints on expectations of auxilliary costs. Although optimal policies for finite CMDPs with known models can be obtained by linear programming, methods for high-dimensional control are lacking.
Currently, policy search algorithms enjoy state-of-the-art performance on high-dimensional control tasks \cite{Mnih2016, Duan2016}. Heuristic algorithms for policy search in CMDPs have been proposed \cite{Uchibe2007}, and approaches based on primal-dual methods can be shown to converge to constraint-satisfying policies \cite{Chow2015}, but there is currently no approach for policy search in continuous CMDPs that guarantees every policy during learning will satisfy constraints. In this work, we propose the first such algorithm, allowing applications to constrained deep RL.
Driving our approach is a new theoretical result that bounds the difference between the rewards or costs of two different policies. This result, which is of independent interest, tightens known bounds for policy search using trust regions \cite{Kakade2002,Pirotta2013,Schulman2006}, and provides a tighter connection between the theory and practice of policy search for deep RL. Here, we use this result to derive a policy improvement step that guarantees both an increase in reward and satisfaction of constraints on other costs. This step forms the basis for our algorithm, \textit{Constrained Policy Optimization} (CPO), which computes an approximation to the theoretically-justified update.
In our experiments, we show that CPO can train neural network policies with thousands of parameters on high-dimensional simulated robot locomotion tasks to maximize rewards while successfully enforcing constraints.
\section{Related Work}
Safety has long been a topic of interest in RL research, and a comprehensive overview of safety in RL was given by \cite{Garcia2015}.
Safe policy search methods have been proposed in prior work. Uchibe and Doya \yrcite{Uchibe2007} gave a policy gradient algorithm that uses gradient projection to enforce active constraints, but this approach suffers from an inability to prevent a policy from becoming unsafe in the first place. Bou Ammar et al. \yrcite{BouAmmar2015} propose a theoretically-motivated policy gradient method for lifelong learning with safety constraints, but their method involves an expensive inner loop optimization of a semi-definite program, making it unsuited for the deep RL setting. Their method also assumes that safety constraints are linear in policy parameters, which is limiting. Chow et al. \yrcite{Chow2015} propose a primal-dual subgradient method for risk-constrained reinforcement learning which takes policy gradient steps on an objective that trades off return with risk, while simultaneously learning the trade-off coefficients (dual variables).
Some approaches specifically focus on application to the deep RL setting. Held et al. \yrcite{Held2017} study the problem for robotic manipulation, but the assumptions they make restrict the applicability of their methods. Lipton et al. \yrcite{Lipton2017} use an `intrinsic fear' heuristic, as opposed to constraints, to motivate agents to avoid rare but catastrophic events. Shalev-Shwartz et al. \yrcite{Shalev-Shwartz2016} avoid the problem of enforcing constraints on parametrized policies by decomposing `desires' from trajectory planning; the neural network policy learns desires for behavior, while the trajectory planning algorithm (which is not learned) selects final behavior and enforces safety constraints.
In contrast to prior work, our method is the first policy search algorithm for CMDPs that both 1) guarantees constraint satisfaction throughout training, and 2) works for arbitrary policy classes (including neural networks).
\section{Preliminaries}
A Markov decision process (MDP) is a tuple, ($S,A,R,P,\mu$), where $S$ is the set of states, $A$ is the set of actions, $R : S \times A \times S \to \Real{}$ is the reward function, $P : S \times A \times S \to [0,1]$ is the transition probability function (where $P(s'|s,a)$ is the probability of transitioning to state $s'$ given that the previous state was $s$ and the agent took action $a$ in $s$), and $\mu : S \to [0,1]$ is the starting state distribution. A stationary policy $\pi : S \to {\mathcal P}(A)$ is a map from states to probability distributions over actions, with $\pi(a|s)$ denoting the probability of selecting action $a$ in state $s$. We denote the set of all stationary policies by $\Pi$.
In reinforcement learning, we aim to select a policy $\pi$ which maximizes a performance measure, $J(\pi)$, which is typically taken to be the infinite horizon discounted total return, $J(\pi) \doteq \tauE{\pi}{\sum_{t=0}^{\infty} \gamma^t R(s_t, a_t, s_{t+1})}$. Here $\gamma \in [0,1)$ is the discount factor, $\tau$ denotes a trajectory ($\tau = (s_0, a_0, s_1, ...)$), and $\tau \sim \pi$ is shorthand for indicating that the distribution over trajectories depends on $\pi$: $s_0 \sim \mu$, $a_t \sim \pi(\cdot|s_t)$, $s_{t+1} \sim P(\cdot | s_t, a_t)$.
Letting $R(\tau)$ denote the discounted return of a trajectory, we express the on-policy value function as $V^{\pi} (s) \doteq {\mathrm E}_{\tau \sim \pi}[R(\tau) | s_0 = s]$ and the on-policy action-value function as $Q^{\pi} (s,a) \doteq {\mathrm E}_{\tau \sim \pi} [R(\tau) | s_0= s, a_0 = a]$. The advantage function is $A^{\pi} (s,a) \doteq Q^{\pi}(s,a) - V^{\pi}(s)$.
Also of interest is the discounted future state distribution, $d^{\pi}$, defined by $d^{\pi} (s) = (1-\gamma) \sum_{t=0}^{\infty} \gamma^t P(s_t = s | \pi)$. It allows us to compactly express the difference in performance between two policies $\pi', \pi$ as
\begin{equation}
J(\pi') - J(\pi) = \frac{1}{1-\gamma} \underset{\begin{subarray}{c} s \sim d^{\pi'} \\ a \sim \pi' \end{subarray}}{{\mathrm E}} \left[ A^{\pi} (s,a) \right], \label{connection}
\end{equation}
where by $a \sim \pi'$, we mean $a \sim \pi'(\cdot |s)$, with explicit notation dropped to reduce clutter. For proof of (\ref{connection}), see \cite{Kakade2002} or Section \ref{appendix} in the supplementary material.
\section{Constrained Markov Decision Processes}
A constrained Markov decision process (CMDP) is an MDP augmented with constraints that restrict the set of allowable policies for that MDP. Specifically, we augment the MDP with a set $C$ of auxiliary cost functions, $C_1, ..., C_m$ (with each one a function $C_i : S \times A \times S \to \Real{}$ mapping transition tuples to costs, like the usual reward), and limits $d_1, ..., d_m$. Let $J_{C_i} (\pi)$ denote the expected discounted return of policy $\pi$ with respect to cost function $C_i$: $J_{C_i} (\pi) = \tauE{\pi}{\sum_{t=0}^{\infty} \gamma^t C_i (s_t, a_t, s_{t+1})}$. The set of feasible stationary policies for a CMDP is then
\begin{equation*}
\Pi_{C} \doteq \left\{\pi \in \Pi \; : \; \forall i, J_{C_i} (\pi) \leq d_i\right\},
\end{equation*}
and the reinforcement learning problem in a CMDP is
\begin{equation*}
\pi^* = \arg\max_{ \pi \in \Pi_{C} } J(\pi).
\end{equation*}
The choice of optimizing only over stationary policies is justified: it has been shown that the set of all optimal policies for a CMDP includes stationary policies, under mild technical conditions. For a thorough review of CMDPs and CMDP theory, we refer the reader to \cite{Altman1999}.
We refer to $J_{C_i}$ as a \textit{constraint return}, or $C_i$-return for short. Lastly, we define on-policy value functions, action-value functions, and advantage functions for the auxiliary costs in analogy to $V^{\pi}$, $Q^{\pi}$, and $A^{\pi}$, with $C_i$ replacing $R$: respectively, we denote these by $V_{C_i}^{\pi}$, $Q_{C_i}^{\pi}$, and $A_{C_i}^{\pi}$.
\section{Constrained Policy Optimization}
For large or continuous MDPs, solving for the exact optimal policy is intractable due to the curse of dimensionality \cite{Sutton1998}. Policy search algorithms approach this problem by searching for the optimal policy within a set $\Pi_{\theta} \subseteq \Pi$ of parametrized policies with parameters $\theta$ (for example, neural networks of a fixed architecture). In local policy search \cite{Peters2008a}, the policy is iteratively updated by maximizing $J(\pi)$ over a local neighborhood of the most recent iterate $\pi_k$:
\begin{equation} \label{polopt}
\begin{aligned}
\pi_{k+1} = \arg \max_{\pi \in \Pi_{\theta}} \; & J(\pi) \\
\text{s.t. } & D (\pi, \pi_k) \leq \delta,
\end{aligned}
\end{equation}
where $D$ is some distance measure, and $\delta > 0$ is a step size. When the objective is estimated by linearizing around $\pi_k$ as $J(\pi_k) + g^T (\theta - \theta_k)$, $g$ is the policy gradient, and the standard policy gradient update is obtained by choosing $D(\pi, \pi_k)= \|\theta - \theta_k\|_2$ \cite{Schulman2006}.
In local policy search for CMDPs, we additionally require policy iterates to be feasible for the CMDP, so instead of optimizing over $\Pi_{\theta}$, we optimize over $\Pi_{\theta} \cap \Pi_C$:
\begin{equation}\label{cpolopt}
\begin{aligned}
\pi_{k+1} = \arg \max_{\pi \in \Pi_{\theta}} \; & J(\pi) \\
\text{s.t. } & J_{C_i} (\pi) \leq d_i \;\;\; i=1, ..., m\\
& D(\pi, \pi_k) \leq \delta.
\end{aligned}
\end{equation}
This update is difficult to implement in practice because it requires evaluation of the constraint functions to determine whether a proposed point $\pi$ is feasible. When using sampling to compute policy updates, as is typically done in high-dimensional control \cite{Duan2016}, this requires off-policy evaluation, which is known to be challenging \cite{Jiang2015}. In this work, we take a different approach, motivated by recent methods for trust region optimization \cite{Schulman2006}.
We develop a principled approximation to (\ref{cpolopt}) with a particular choice of $D$, where we replace the objective and constraints with \textit{surrogate} functions. The surrogates we choose are easy to estimate from samples collected on $\pi_k$, and are good local approximations for the objective and constraints. Our theoretical analysis shows that for our choices of surrogates, we can bound our update's worst-case performance and worst-case constraint violation with values that depend on a hyperparameter of the algorithm.
To prove the performance guarantees associated with our surrogates, we first prove new bounds on the difference in returns (or constraint returns) between two arbitrary stochastic policies in terms of an average divergence between them. We then show how our bounds permit a new analysis of trust region methods in general: specifically, we prove a worst-case performance degradation at each update. We conclude by motivating, presenting, and proving gurantees on our algorithm, Constrained Policy Optimization (CPO), a trust region method for CMDPs.
\subsection{Policy Performance Bounds}
In this section, we present the theoretical foundation for our approach---a new bound on the difference in returns between two arbitrary policies. This result, which is of independent interest, extends the works of \cite{Kakade2002}, \cite{Pirotta2013}, and \cite{Schulman2006}, providing tighter bounds. As we show later, it also relates the theoretical bounds for trust region policy improvement with the actual trust region algorithms that have been demonstrated to be successful in practice \cite{Duan2016}. In the context of constrained policy search, we later use our results to propose policy updates that both improve the expected return and satisfy constraints.
The following theorem connects the difference in returns (or constraint returns) between two arbitrary policies to an average divergence between them.
\begin{restatable}{theorem}{performancebound}
\label{maintheorem_0} For any function $f: S \to \Real{}$ and any policies $\pi'$ and $\pi$, define $\delta_f (s,a,s') \doteq R(s,a,s') + \gamma f(s') - f(s)$,
\begin{equation*}
\epsilon_f^{\pi'} \doteq \max_s \left| {\mathrm E}_{a \sim \pi', s'\sim P} [\delta_f (s,a,s')] \right|,
\end{equation*}
\begin{align*}
&L_{\pi,f} (\pi') \doteq \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \left(\frac{\pi'(a|s)}{\pi(a|s)} - 1 \right) \delta_f(s,a,s') \right], \text{ and }\label{surrogate_0}
\end{align*}
\begin{equation*}
\begin{aligned}
&D_{\pi,f}^{\pm} (\pi') \doteq \frac{L_{\pi,f} (\pi')}{1-\gamma} \pm \frac{2\gamma \epsilon_f^{\pi'}}{(1-\gamma)^2} \underset{s \sim d^{\pi}}{{\mathrm E}} \left[ D_{TV} (\pi'||\pi)[s] \right],
\end{aligned}
\end{equation*}
where $D_{TV}(\pi'||\pi)[s] = (1/2)\sum_a \left| \pi'(a|s) - \pi(a|s) \right|$ is the total variational divergence between action distributions at $s$. The following bounds hold:
\begin{equation}
D_{\pi,f}^{+} (\pi') \geq J(\pi') - J(\pi) \geq D_{\pi,f}^{-} (\pi'). \label{bound}
\end{equation}
Furthermore, the bounds are tight (when $\pi' = \pi$, all three expressions are identically zero).
\end{restatable}
Before proceeding, we connect this result to prior work. By bounding the expectation ${\mathrm E}_{s \sim d^{\pi}}\left[D_{TV} (\pi' || \pi)[s]\right]$ with $\max_s D_{TV} (\pi'||\pi)[s]$, picking $f = V^{\pi}$, and bounding $\epsilon_{V^{\pi}}^{\pi'}$ to get a second factor of $\max_s D_{TV} (\pi'||\pi)[s]$, we recover (up to assumption-dependent factors) the bounds given by Pirotta et~al. \yrcite{Pirotta2013} as Corollary 3.6, and by Schulman et~al. \yrcite{Schulman2006} as Theorem 1a.
The choice of $f = V^{\pi}$ allows a useful form of the lower bound, so we give it as a corollary.
\begin{corollary}
For any policies $\pi', \pi$, with $\epsilon^{\pi'} \doteq \max_s | {\mathrm E}_{a \sim \pi'} [A^{\pi} (s,a) ] |$, the following bound holds:
\begin{equation}
\begin{aligned}
&J(\pi') - J(\pi)\\
& \geq \frac{1}{1-\gamma} \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi' \end{subarray}}{{\mathrm E}} \left[ A^{\pi} (s,a) - \frac{2\gamma \epsilon^{\pi'}}{1-\gamma} D_{TV} (\pi'||\pi)[s] \right]. \label{bound2}
\end{aligned}
\end{equation}
\label{advantage-bound}
\end{corollary}
The bound (\ref{bound2}) should be compared with equation (\ref{connection}). The term $(1-\gamma)^{-1} {\mathrm E}_{s\sim d^{\pi}, a\sim \pi'}[A^{\pi}(s,a)]$ in (\ref{bound2}) is an approximation to $J(\pi') - J(\pi)$, using the state distribution $d^{\pi}$ instead of $d^{\pi'}$, which is known to equal $J(\pi') - J(\pi)$ to first order in the parameters of $\pi'$ on a neighborhood around $\pi$ \cite{Kakade2002}. The bound can therefore be viewed as describing the worst-case approximation error, and it justifies using the approximation as a \textit{surrogate} for $J(\pi') - J(\pi)$.
Equivalent expressions for the auxiliary costs, based on the upper bound, also follow immediately; we will later use them to make guarantees for the safety of CPO.
\begin{corollary}
For any policies $\pi', \pi$, and any cost function $C_i$, with $\epsilon^{\pi'}_{C_i} \doteq \max_s | {\mathrm E}_{a \sim \pi'} [A^{\pi}_{C_i} (s,a) ] |$, the following bound holds:
\begin{equation}
\begin{aligned}
&J_{C_i} (\pi') - J_{C_i} (\pi)\\
& \leq \frac{1}{1-\gamma} \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi' \end{subarray}}{{\mathrm E}} \left[ A^{\pi}_{C_i} (s,a) + \frac{2\gamma \epsilon^{\pi'}_{C_i}}{1-\gamma} D_{TV} (\pi'||\pi)[s] \right]. \label{bound3}
\end{aligned}
\end{equation}
\label{safety-advantage-bound}
\end{corollary}
The bounds we have given so far are in terms of the TV-divergence between policies, but trust region methods constrain the KL-divergence between policies, so bounds that connect performance to the KL-divergence are desirable. We make the connection through Pinsker's inequality \cite{Csiszar1981}: for arbitrary distributions $p,q$, the TV-divergence and KL-divergence are related by $D_{TV} (p||q) \leq \sqrt{D_{KL} (p||q) /2}$. Combining this with Jensen's inequality, we obtain
\begin{align}
\underset{s \sim d^{\pi}}{{\mathrm E}}\left[D_{TV}(\pi'||\pi)[s]\right] &\leq \underset{s \sim d^{\pi}}{{\mathrm E}}\left[\sqrt{\frac{1}{2} D_{KL}(\pi'||\pi)[s]}\right] \nonumber \\
&\leq \sqrt{\frac{1}{2} \underset{s \sim d^{\pi}}{{\mathrm E}}\left[D_{KL}(\pi'||\pi)[s]\right]} \label{tvkl}
\end{align}
From (\ref{tvkl}) we immediately obtain the following.
\begin{corollary} \label{klbound}
In bounds (\ref{bound}), (\ref{bound2}), and (\ref{bound3}), make the substitution
\begin{equation*}
\underset{s \sim d^{\pi}}{{\mathrm E}}\left[D_{TV}(\pi'||\pi)[s]\right] \to \sqrt{\frac{1}{2} \underset{s \sim d^{\pi}}{{\mathrm E}}\left[D_{KL}(\pi'||\pi)[s]\right]}.
\end{equation*}
The resulting bounds hold.
\end{corollary}
\subsection{Trust Region Methods}
Trust region algorithms for reinforcement learning \cite{Schulman2006, Schulman2016} have policy updates of the form
\begin{equation}
\begin{aligned}
\pi_{k+1} = \arg \max_{\pi \in \Pi_{\theta}} \; & \underset{\begin{subarray}{c}s \sim d^{\pi_k} \\ a \sim \pi\end{subarray}}{{\mathrm E}}\left[A^{\pi_k}(s,a)\right] \\
\text{s.t. } &\bar{D}_{KL} (\pi || \pi_k) \leq \delta,
\end{aligned} \label{trpo}
\end{equation}
where $\bar{D}_{KL}(\pi || \pi_k) = {\mathrm E}_{s\sim\pi_k}\left[D_{KL}(\pi||\pi_k)[s] \right]$, and $\delta > 0$ is the step size. The set $\{\pi_{\theta} \in \Pi_{\theta} : \bar{D}_{KL} (\pi||\pi_k) \leq \delta\}$ is called the \textit{trust region}.
The primary motivation for this update is that it is an approximation to optimizing the lower bound on policy performance given in (\ref{bound2}), which would guarantee monotonic performance improvements. This is important for optimizing neural network policies, which are known to suffer from performance collapse after bad updates \cite{Duan2016}. Despite the approximation, trust region steps usually give monotonic improvements \cite{Schulman2006, Duan2016} and have shown state-of-the-art performance in the deep RL setting \cite{Duan2016, Gu2017}, making the approach appealing for developing policy search methods for CMDPs.
Until now, the particular choice of trust region for (\ref{trpo}) was heuristically motivated; with (\ref{bound2}) and Corollary \ref{klbound}, we are able to show that it is principled and comes with a worst-case performance degradation guarantee that depends on $\delta$.
\begin{proposition}[Trust Region Update Performance] \label{trpoprop} Suppose $\pi_k$, $\pi_{k+1}$ are related by (\ref{trpo}), and that $\pi_k \in \Pi_{\theta}$. A lower bound on the policy performance difference between $\pi_k$ and $\pi_{k+1}$ is
\begin{equation}
J(\pi_{k+1}) - J(\pi_k) \geq \frac{ -\sqrt{2\delta} \gamma \epsilon^{\pi_{k+1}}}{(1 -\gamma)^2}, \label{trpobound}
\end{equation}
where $\epsilon^{\pi_{k+1}} = \max_s \left| {\mathrm E}_{a \sim \pi_{k+1}} \left[A^{\pi_k}(s,a)\right]\right|$.
\end{proposition}
\begin{proof}
$\pi_k$ is a feasible point of (\ref{trpo}) with objective value 0, so ${\mathrm E}_{s \sim d^{\pi_k}, a \sim \pi_{k+1}} \left[A^{\pi_k}(s,a)\right] \geq 0$. The rest follows by (\ref{bound2}) and Corollary \ref{klbound}, noting that (\ref{trpo}) bounds the average KL-divergence by $\delta$.
\end{proof}
This result is useful for two reasons: 1) it is of independent interest, as it helps tighten the connection between theory and practice for deep RL, and 2) the choice to develop CPO as a trust region method means that CPO inherits this performance guarantee.
\subsection{Trust Region Optimization for Constrained MDPs}
\textit{Constrained policy optimization} (CPO), which we present and justify in this section, is a policy search algorithm for CMDPs with updates that approximately solve (\ref{cpolopt}) with a particular choice of $D$. First, we describe a policy search update for CMDPs that alleviates the issue of off-policy evaluation, and comes with guarantees of monotonic performance improvement and constraint satisfaction. Then, because the theoretically guaranteed update will take too-small steps in practice, we propose CPO as a practical approximation based on trust region methods.
By corollaries \ref{advantage-bound}, \ref{safety-advantage-bound}, and \ref{klbound}, for appropriate coefficients $\alpha_k$, $\beta_k^i$ the update
\begin{equation*}\label{cpo_ancestor}
\begin{aligned}
& \pi_{k+1} = \arg \max_{\pi \in \Pi_{\theta}} \; \smallsurr{\pi_k}{\pi}{A^{\pi_k}(s,a)} - \alpha_k \sqrt{\bar{D}_{KL}(\pi || \pi_k)} \\
& \text{s.t. } J_{C_i} (\pi_k) + \smallsurr{\pi_k}{\pi}{\frac{A^{\pi_k}_{C_i} (s,a)}{1-\gamma}} + \beta_k^i \sqrt{\bar{D}_{KL}(\pi || \pi_k)} \leq d_i
\end{aligned}
\end{equation*}
is guaranteed to produce policies with monotonically nondecreasing returns that satisfy the original constraints. (Observe that the constraint here is on an upper bound for $J_{C_i}(\pi)$ by (\ref{bound3}).) The off-policy evaluation issue is alleviated, because both the objective and constraints involve expectations over state distributions $d^{\pi_k}$, which we presume to have samples from. Because the bounds are tight, the problem is always feasible (as long as $\pi_0$ is feasible). However, the penalties on policy divergence are quite steep for discount factors close to 1, so steps taken with this update might be small.
Inspired by trust region methods, we propose CPO, which uses a trust region instead of penalties on policy divergence to enable larger step sizes:
\begin{equation}\label{cpo}
\begin{aligned}
\pi_{k+1} = &\arg \max_{\pi \in \Pi_{\theta}} \; \smallsurr{\pi_k}{\pi}{A^{\pi_k}(s,a)} \\
\text{s.t. } & J_{C_i} (\pi_k) + \surrA{\pi_k}{\pi}{A^{\pi_k}_{C_i} (s,a)} \leq d_i \;\;\; \forall i\\
& \bar{D}_{KL} (\pi || \pi_k) \leq \delta.
\end{aligned}
\end{equation}
Because this is a trust region method, it inherits the performance guarantee of Proposition \ref{trpoprop}. Furthermore, by corollaries \ref{safety-advantage-bound} and \ref{klbound}, we have a performance guarantee for approximate satisfaction of constraints:
\begin{proposition}[CPO Update Worst-Case Constraint Violation] \label{cpobound} Suppose $\pi_k, \pi_{k+1}$ are related by (\ref{cpo}), and that $\Pi_{\theta}$ in (\ref{cpo}) is any set of policies with $\pi_k \in \Pi_{\theta}$. An upper bound on the $C_i$-return of $\pi_{k+1}$ is
\begin{equation*}
J_{C_i} (\pi_{k+1}) \leq d_i + \frac{\sqrt{2\delta} \gamma \epsilon^{\pi_{k+1}}_{C_i}}{(1 -\gamma)^2},
\end{equation*}
where $\epsilon^{\pi_{k+1}}_{C_i} = \max_s \left| {\mathrm E}_{a \sim \pi_{k+1}} \left[A^{\pi_k}_{C_i} (s,a)\right]\right|$.
\end{proposition}
\section{Practical Implementation}
In this section, we show how to implement an approximation to the update (\ref{cpo}) that can be efficiently computed, even when optimizing policies with thousands of parameters. To address the issue of approximation and sampling errors that arise in practice, as well as the potential violations described by Proposition \ref{cpobound}, we also propose to tighten the constraints by constraining upper bounds of the auxilliary costs, instead of the auxilliary costs themselves.
\subsection{Approximately Solving the CPO Update}
For policies with high-dimensional parameter spaces like neural networks, (\ref{cpo}) can be impractical to solve directly because of the computational cost. However, for small step sizes $\delta$, the objective and cost constraints are well-approximated by linearizing around $\pi_k$, and the KL-divergence constraint is well-approximated by second order expansion (at $\pi_k = \pi$, the KL-divergence and its gradient are both zero). Denoting the gradient of the objective as $g$, the gradient of constraint $i$ as $b_i$, the Hessian of the KL-divergence as $H$, and defining $c_i \doteq J_{C_i}(\pi_k) - d_i$, the approximation to (\ref{cpo}) is:
\begin{equation}
\begin{aligned}
\theta_{k+1} =\arg\max_{\theta} \;\;\;& g^T (\theta - \theta_k) \\
\text{s.t. } \;\;\;& c_i + b_i^T (\theta - \theta_k) \leq 0 \;\;\; i=1,...,m \\
& \frac{1}{2} (\theta - \theta_k)^T H (\theta - \theta_k) \leq \delta.
\end{aligned} \label{cpoapprox}
\end{equation}
Because the Fisher information matrix (FIM) $H$ is always positive semi-definite (and we will assume it to be positive-definite in what follows), this optimization problem is convex and, when feasible, can be solved efficiently using duality. (We reserve the case where it is not feasible for the next subsection.) With $B \doteq [b_1 , ..., b_m]$ and $c \doteq [c_1, ..., c_m]^T$, a dual to (\ref{cpoapprox}) can be expressed as
\begin{equation}
\begin{aligned}
\max_{\begin{subarray}{c} \lambda \geq 0 \\ \nu \succeq 0\end{subarray}} \frac{-1}{2\lambda} \left( g^T H^{-1} g - 2 r^T \nu + \nu^T S \nu\right) + \nu^T c - \frac{\lambda \delta}{2},
\end{aligned} \label{cpodual}
\end{equation}
where $r\doteq g^T H^{-1} B$, $S\doteq B^T H^{-1} B$. This is a convex program in $m+1$ variables; when the number of constraints is small by comparison to the dimension of $\theta$, this is much easier to solve than (\ref{cpoapprox}). If $\lambda^*, \nu^*$ are a solution to the dual, the solution to the primal is
\begin{equation}
\theta^* = \theta_k + \frac{1}{\lambda^*} H^{-1} \left( g - B \nu^*\right). \label{proposed}
\end{equation}
Our algorithm solves the dual for $\lambda^*, \nu^*$ and uses it to propose the policy update (\ref{proposed}). For the special case where there is only one constraint, we give an analytical solution in the supplementary material (Theorem \ref{thmlqclp}) which removes the need for an inner-loop optimization. Our experiments have only a single constraint, and make use of the analytical solution.
Because of approximation error, the proposed update may not satisfy the constraints in (\ref{cpo}); a backtracking line search is used to ensure surrogate constraint satisfaction. Also, for high-dimensional policies, it is impractically expensive to invert the FIM. This poses a challenge for computing $H^{-1}g$ and $H^{-1} b_i$, which appear in the dual. Like \cite{Schulman2006}, we approximately compute them using the conjugate gradient method.
\subsection{Feasibility}
Due to approximation errors, CPO may take a bad step and produce an infeasible iterate $\pi_k$. Sometimes (\ref{cpoapprox}) will still be feasible and CPO can automatically recover from its bad step, but for the infeasible case, a recovery method is necessary. In our experiments, where we only have one constraint, we recover by proposing an update to purely decrease the constraint value:
\begin{equation}
\theta^* = \theta_k - \sqrt{\frac{2\delta}{b^T H^{-1} b}} H^{-1} b. \label{infeasibleproposal}
\end{equation}
As before, this is followed by a line search. This approach is principled in that it uses the limiting search direction as the intersection of the trust region and the constraint region shrinks to zero. We give the pseudocode for our algorithm (for the single-constraint case) as Algorithm \ref{alg1}.
\begin{algorithm}[tb]
\caption{Constrained Policy Optimization}
\label{alg1}
\begin{algorithmic}
\STATE {\bfseries Input:} Initial policy $\pi_0 \in \Pi_{\theta}$ tolerance $\alpha$
\FOR{$k = 0,1,2,...$}
\STATE Sample a set of trajectories ${\mathcal D} = \{\tau\} \sim \pi_k = \pi(\theta_k)$
\STATE Form sample estimates $\hat{g}, \hat{b}, \hat{H}, \hat{c}$ with ${\mathcal D}$
\IF{approximate CPO is feasible}
\STATE Solve dual problem (\ref{cpodual}) for $\lambda^*_k, \nu^*_k$
\STATE Compute policy proposal $\theta^*$ with (\ref{proposed})
\ELSE
\STATE Compute recovery policy proposal $\theta^*$ with (\ref{infeasibleproposal})
\ENDIF
\STATE Obtain $\theta_{k+1}$ by backtracking linesearch to enforce satisfaction of sample estimates of constraints in (\ref{cpo})
\ENDFOR
\end{algorithmic}
\end{algorithm}
\subsection{Tightening Constraints via Cost Shaping}
Because of the various approximations between (\ref{cpolopt}) and our practical algorithm, it is important to build a factor of safety into the algorithm to minimize the chance of constraint violations. To this end, we choose to constrain upper bounds on the original constraints, $C_i^+$, instead of the original constraints themselves. We do this by cost shaping:
\begin{equation}
C_i^+ (s,a,s') = C_i (s,a,s') + \Delta_i (s,a,s'), \label{safetyfactor}
\end{equation}
where $\Delta_i : S \times A \times S \to \Realp{}$ correlates in some useful way with $C_i$.
In our experiments, where we have only one constraint, we partition states into \textit{safe states} and \textit{unsafe states}, and the agent suffers a safety cost of $1$ for being in an unsafe state. We choose $\Delta$ to be the probability of entering an unsafe state within a fixed time horizon, according to a learned model that is updated at each iteration. This choice confers the additional benefit of smoothing out sparse constraints.
\section{Connections to Prior Work}
Our method has similar policy updates to primal-dual methods like those proposed by Chow et al. \yrcite{Chow2015}, but crucially, we differ in computing the dual variables (the Lagrange multipliers for the constraints). In primal-dual optimization (PDO), dual variables are stateful and learned concurrently with the primal variables \cite{Boyd2003}. In a PDO algorithm for solving (\ref{cpolopt}), dual variables would be updated according to
\begin{equation}
\nu_{k+1} = \left(\nu_k + \alpha_k \left(J_C (\pi_k) - d\right)\right)_+, \label{pdodual}
\end{equation}
where $\alpha_k$ is a learning rate. In this approach, intermediary policies are not guaranteed to satisfy constraints---only the policy at convergence is. By contrast, CPO computes new dual variables from scratch at each update to exactly enforce constraints.
\begin{figure*}[t]
\centering
Returns:
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Point-Circle-Final-Return}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Ant-Long-Return-1}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.4cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Humanoid-Long-Return}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-PointGather-Return-1}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-AntGather-Revised-500-Return}
\end{subfigure}
\rule{\linewidth}{0.5pt}
Constraint values: (closer to the limit is better)
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Point-Circle-Final-Safety}
\caption{Point-Circle}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Ant-Long-Safety-1}
\caption{Ant-Circle}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-Humanoid-Long-Safety}
\caption{Humanoid-Circle}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-PointGather-Safety-1}
\caption{Point-Gather}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=\linewidth, trim={0.5cm, 0.5cm, 0.5cm, 0}, clip]{img/CPO-AntGather-Revised-500-Safety}
\caption{Ant-Gather}
\end{subfigure}
\caption[]{Average performance for CPO, PDO, and TRPO over several seeds (5 in the Point environments, 10 in all others); the $x$-axis is training iteration. CPO drives the constraint function almost directly to the limit in all experiments, while PDO frequently suffers from over- or under-correction. TRPO is included to verify that optimal unconstrained behaviors are infeasible for the constrained problem.}
\label{main_results}
\end{figure*}
\section{Experiments}
In our experiments, we aim to answer the following:
\begin{itemize}
\item Does CPO succeed at enforcing behavioral constraints when training neural network policies with thousands of parameters?
\item How does CPO compare with a baseline that uses primal-dual optimization? Does CPO behave better with respect to constraints?
\item How much does it help to constrain a cost upper bound (\ref{safetyfactor}), instead of directly constraining the cost?
\item What benefits are conferred by using constraints instead of fixed penalties?
\end{itemize}
We designed experiments that are easy to interpret and motivated by safety. We consider two tasks, and train multiple different agents (robots) for each task:
\begin{itemize}
\item \textbf{Circle}: The agent is rewarded for running in a wide circle, but is constrained to stay within a safe region smaller than the radius of the target circle.
\item \textbf{Gather}: The agent is rewarded for collecting green apples, and constrained to avoid red bombs.
\end{itemize}
For the Circle task, the exact geometry is illustrated in Figure \ref{geom} in the supplementary material. Note that there are no physical walls: the agent only interacts with boundaries through the constraint costs. The reward and constraint cost functions are described in supplementary material (Section \ref{envirosec}). In each of these tasks, we have only one constraint; we refer to it as $C$ and its upper bound from (\ref{safetyfactor}) as $C^+$.
We experiment with three different agents: a point-mass $(S \subseteq \Real{9}, A \subseteq \Real{2})$, a quadruped robot (called an `ant') $(S \subseteq \Real{32}, A \subseteq \Real{8})$, and a simple humanoid $(S \subseteq \Real{102}, A \subseteq \Real{10})$. We train all agent-task combinations except for Humanoid-Gather.
For all experiments, we use neural network policies with two hidden layers of size $(64,32)$. Our experiments are implemented in rllab \cite{Duan2016}.
\begin{figure}[t]
\centering
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{humanoid.png}
\caption{Humanoid-Circle}
\end{subfigure}%
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{pointgather.png}
\caption{Point-Gather}
\end{subfigure}
\caption[]{The Humanoid-Circle and Point-Gather environments. In Humanoid-Circle, the safe area is between the blue panels. }
\label{graphics}
\end{figure}
\subsection{Evaluating CPO and Comparison Analysis}
Learning curves for CPO and PDO are compiled in Figure \ref{main_results}. Note that we evaluate algorithm performance based on the $C^+$ return, instead of the $C$ return (except for in Point-Gather, where we did not use cost shaping due to that environment's short time horizon), because this is what the algorithm actually constrains in these experiments.
For our comparison, we implement PDO with (\ref{pdodual}) as the update rule for the dual variables, using a constant learning rate $\alpha$; details are available in supplementary material (Section \ref{pdoimplement}). We emphasize that in order for the comparison to be fair, we give PDO every advantage that is given to CPO, including equivalent trust region policy updates. To benchmark the environments, we also include TRPO (trust region policy optimization) \cite{Schulman2006}, a state-of-the-art \textit{unconstrained} reinforcement learning algorithm. The TRPO experiments show that optimal unconstrained behaviors for these environments are constraint-violating.
We find that CPO is successful at approximately enforcing constraints in all environments. In the simpler environments (Point-Circle and Point-Gather), CPO tracks the constraint return almost \textit{exactly} to the limit value.
By contrast, although PDO usually converges to constraint-satisfying policies in the end, it is not consistently constraint-satisfying throughout training (as expected). For example, see the spike in constraint value that it experiences in Ant-Circle. Additionally, PDO is sensitive to the initialization of the dual variable. By default, we initialize $\nu_0 = 0$, which exploits no prior knowledge about the environment and makes sense when the initial policies are feasible. However, it may seem appealing to set $\nu_0$ high, which would make PDO more conservative with respect to the constraint; PDO could then decrease $\nu$ as necessary after the fact. In the Point environments, we experiment with $\nu_0 = 1000$ and show that although this does assure constraint satisfaction, it also can substantially harm performance with respect to return. Furthermore, we argue that this is not adequate in general: after the dual variable decreases, the agent could learn a new behavior that increases the correct dual variable more quickly than PDO can attain it (as happens in Ant-Circle for PDO; observe that performance is approximately constraint-satisfying until the agent learns how to run at around iteration 350).
We find that CPO generally outperforms PDO on enforcing constraints, without compromising performance with respect to return. CPO quickly stabilizes the constraint return around to the limit value, while PDO is not consistently able to enforce constraints all throughout training.
\subsection{Ablation on Cost Shaping}
In Figure \ref{cs-results}, we compare performance of CPO with and without cost shaping in the constraint. Our metric for comparison is the $C$-return, the `true' constraint. The cost shaping does help, almost completely accounting for CPO's inherent approximation errors. However, CPO is nearly constraint-satisfying even without cost shaping.
\begin{figure}[t]
\centering
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{With-and-without-constraint-cost-shaping/CPO-Ant-Circle-Bonus-Comparison-Return.png}
\caption{Ant-Circle Return}
\end{subfigure}%
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{With-and-without-constraint-cost-shaping/CPO-AntGather-Bonus-Comparison-Revised-500-Return.png}
\caption{Ant-Gather Return}
\end{subfigure}%
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{With-and-without-constraint-cost-shaping/CPO-Ant-Circle-Bonus-Comparison-Safety.png}
\caption{Ant-Circle $C$-Return}
\end{subfigure}%
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{With-and-without-constraint-cost-shaping/CPO-AntGather-Bonus-Comparison-Revised-500-Safety.png}
\caption{Ant-Gather $C$-Return}
\end{subfigure}%
\caption[]{Using cost shaping (CS) in the constraint while optimizing generally improves the agent's adherence to the true constraint on $C$-return.}
\label{cs-results}
\end{figure}
\subsection{Constraint vs. Fixed Penalty}
\begin{figure}[t]
\centering
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{Fixed-Penalty-Comparison/CPO-FPO-Comparison-Ant-Circle-Return.png}
\caption{Ant-Circle Return}
\end{subfigure}%
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{Fixed-Penalty-Comparison/CPO-FPO-Comparison-Ant-Circle-Safety.png}
\caption{Ant-Circle $C^+$-Return}
\end{subfigure}%
\caption[]{Comparison between CPO and FPO (fixed penalty optimization) for various values of fixed penalty. }
\label{FPO}
\end{figure}
In Figure \ref{FPO}, we compare CPO to a fixed penalty method, where policies are learned using TRPO with rewards $R(s,a,s') + \lambda C^+ (s,a,s')$ for $\lambda \in \{1, 5, 50\}$.
We find that fixed penalty methods can be highly sensitive to the choice of penalty coefficient: in Ant-Circle, a penalty coefficient of 1 results in reward-maximizing policies that accumulate massive constraint costs, while a coefficient of 5 (less than an order of magnitude difference) results in cost-minimizing policies that never learn how to acquire any rewards. In contrast, CPO automatically picks penalty coefficients to attain the desired trade-off between reward and constraint cost.
\section{Discussion}
In this article, we showed that a particular optimization problem results in policy updates that are guaranteed to both improve return and satisfy constraints. This enabled the development of CPO, our policy search algorithm for CMDPs, which approximates the theoretically-guaranteed algorithm in a principled way. We demonstrated that CPO can train neural network policies with thousands of parameters on high-dimensional constrained control tasks, simultaneously maximizing reward and approximately satisfying constraints. Our work represents a step towards applying reinforcement learning in the real world, where constraints on agent behavior are sometimes necessary for the sake of safety.
\section*{Acknowledgements}
The authors would like to acknowledge Peter Chen, who independently and concurrently derived an equivalent policy improvement bound.
Joshua Achiam is supported by TRUST (Team for Research in Ubiquitous Secure Technology) which receives support from NSF (award number CCF-0424422). This project also received support from Berkeley Deep Drive and from Siemens.
\subsubsection{Environments} \label{envirosec}
In the Circle environments, the reward and cost functions are
\begin{align*}
R(s) &= \frac{v^T [-y, x]}{1+ \left| \|[x,y]\|_2 - d \right|}, \\
C(s) &= \pmb{1} \left[ |x| > x_{lim}\right],
\end{align*}
where $x,y$ are the coordinates in the plane, $v$ is the velocity, and $d, x_{lim}$ are environmental parameters. We set these parameters to be
\begin{center}
\begin{tabular}{c|ccc}
& Point-mass & Ant & Humanoid \\
\hline
$d$ & 15 & 10 & 10 \\
$x_{lim}$ & 2.5 & 3 & 2.5
\end{tabular}
\end{center}
In Point-Gather, the agent receives a reward of $+10$ for collecting an apple, and a cost of $1$ for collecting a bomb. Two apples and eight bombs spawn on the map at the start of each episode. In Ant-Gather, the reward and cost structure was the same, except that the agent also receives a reward of $-10$ for falling over (which results in the episode ending). Eight apples and eight bombs spawn on the map at the start of each episode.
\begin{figure}[t]
\centering
\begin{subfigure}{.22\textwidth}
\centering
\includegraphics[height=0.9\linewidth]{experiment_setup.png}
\end{subfigure}
\caption[]{In the Circle task, reward is maximized by moving along the green circle. The agent is not allowed to enter the blue regions, so its optimal constrained path follows the line segments $AD$ and $BC$. }
\label{geom}
\end{figure}
\subsubsection{Algorithm Parameters}
In all experiments, we use Gaussian policies with mean vectors given as the outputs of neural networks, and with variances that are separate learnable parameters. The policy networks for all experiments have two hidden layers of sizes $(64,32)$ with $\tanh$ activation functions.
We use GAE-$\lambda$ \cite{Schulman2016} to estimate the advantages and constraint advantages, with neural network value functions. The value functions have the same architecture and activation functions as the policy networks. We found that having different $\lambda^{GAE}$ values for the regular advantages and the constraint advantages worked best. We denote the $\lambda^{GAE}$ used for the constraint advantages as $\lambda_C^{GAE}$.
For the failure prediction networks $P_{\phi} (s \to U)$, we use neural networks with a single hidden layer of size $(32)$, with output of one sigmoid unit. At each iteration, the failure prediction network is updated by some number of gradient descent steps using the Adam update rule to minimize the prediction error. To reiterate, the failure prediction network is a model for the probability that the agent will, at some point in the next $T$ time steps, enter an unsafe state. The cost bonus was weighted by a coefficient $\alpha$, which was $1$ in all experiments except for Ant-Gather, where it was $0.01$. Because of the short time horizon, no cost bonus was used for Point-Gather.
For all experiments, we used a discount factor of $\gamma = 0.995$, a GAE-$\lambda$ for estimating the regular advantages of $\lambda^{GAE} = 0.95$, and a KL-divergence step size of $\delta_{KL} = 0.01$.
Experiment-specific parameters are as follows:
\begin{center}
\begin{tabular}{c|ccccc}
Parameter & Point-Circle & Ant-Circle & Humanoid-Circle & Point-Gather & Ant-Gather\\
\hline
Batch size & 50,000 & 100,000 & 50,000 & 50,000 & 100,000\\
Rollout length & 50-65 & 500 & 1000 & 15 & 500 \\
Maximum constraint value $d$ & 5 & 10 & 10 & 0.1 & 0.2 \\
Failure prediction horizon $T$ & 5 & 20 & 20 & (N/A) & 20\\
Failure predictor SGD steps per itr & 25 & 25 & 25 & (N/A) & 10 \\
Predictor coeff $\alpha$ & 1 & 1 & 1 & (N/A) & 0.01 \\
$\lambda_C^{GAE}$ & 1 & 0.5 & 0.5 & 1 & 0.5
\end{tabular}
\end{center}
Note that these same parameters were used for all algorithms.
We found that the Point environment was agnostic to $\lambda_C^{GAE}$, but for the higher-dimensional environments, it was necessary to set $\lambda_C^{GAE}$ to a value $<1$. Failing to discount the constraint advantages led to substantial overestimates of the constraint gradient magnitude, which led the algorithm to take unsafe steps. The choice $\lambda_C^{GAE} = 0.5$ was obtained by a hyperparameter search in $\{0.5,0.92,1\}$, but $0.92$ worked nearly as well.
\subsubsection{Primal-Dual Optimization Implementation} \label{pdoimplement}
Our primal-dual implementation is intended to be as close as possible to our CPO implementation. The key difference is that the dual variables for the constraints are stateful, learnable parameters, unlike in CPO where they are solved from scratch at each update.
The update equations for our PDO implementation are
\begin{align*}
\theta_{k+1} &= \theta_k + s^j \sqrt{\frac{2\delta}{ (g - \nu_k b)^T H^{-1} (g - \nu_k b)}} H^{-1} \left(g - \nu_k b\right) \\
\nu_{k+1} &= \left( \nu_k + \alpha \left( J_C (\pi_k) - d \right) \right)_+,
\end{align*}
where $s^j$ is from the backtracking line search ($s \in (0,1)$ and $j \in \{0,1,...,J\}$, where $J$ is the backtrack budget; this is the same line search as is used in CPO and TRPO), and $\alpha$ is a learning rate for the dual parameters. $\alpha$ is an important hyperparameter of the algorithm: if it is set to be too small, the dual variable won't update quickly enough to meaningfully enforce the constraint; if it is too high, the algorithm will overcorrect in response to constraint violations and behave too conservatively. We experimented with a relaxed learning rate, $\alpha = 0.001$, and an aggressive learning rate, $\alpha = 0.01$. The aggressive learning rate performed better in our experiments, so all of our reported results are for $\alpha = 0.01$.
Selecting the correct learning rate can be challenging; the need to do this is obviated by CPO.
\subsubsection{Preliminaries}
Our analysis will make extensive use of the discounted future state distribution, $d^{\pi}$, which is defined as
\begin{equation*}
d^{\pi} (s) = (1-\gamma) \sum_{t=0}^{\infty} \gamma^t P(s_t = s | \pi).
\end{equation*}
It allows us to express the expected discounted total reward compactly as
\begin{equation}
J(\pi) = \frac{1}{1-\gamma} \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ R(s,a,s') \right], \label{jpi}
\end{equation}
where by $a \sim \pi$, we mean $a \sim \pi(\cdot|s)$, and by $s' \sim P$, we mean $s' \sim P(\cdot|s,a)$. We drop the explicit notation for the sake of reducing clutter, but it should be clear from context that $a$ and $s'$ depend on $s$.
First, we examine some useful properties of $d^{\pi}$ that become apparent in vector form for finite state spaces. Let $p^t_{\pi} \in \Real{|S|}$ denote the vector with components $p^t_{\pi} (s) = P(s_t = s | \pi)$, and let $P_{\pi} \in \Real{|S|\times|S|}$ denote the transition matrix with components $P_{\pi} (s'|s) = \int da P(s'|s,a) \pi(a|s)$; then $p^t_{\pi} = P_{\pi} p^{t-1}_{\pi} = P^t_{\pi} \mu$ and
\begin{eqnarray}
d^{\pi} &=& (1-\gamma) \sum_{t=0}^{\infty} (\gamma P_{\pi})^t \mu \nonumber \\
&=& (1-\gamma) (I - \gamma P_{\pi})^{-1} \mu. \label{dpi}
\end{eqnarray}
This formulation helps us easily obtain the following lemma.
\begin{lemma} For any function $f : S \to \Real{}$ and any policy $\pi$,
\begin{equation}
(1-\gamma) \underset{s \sim \mu}{{\mathrm E}}\left[f(s)\right] + \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \gamma f(s') \right] - \underset{s \sim d^{\pi}}{{\mathrm E}}\left[f(s)\right] = 0.
\end{equation}
\end{lemma}
\begin{proof} Multiply both sides of (\ref{dpi}) by $(I - \gamma P_{\pi})$ and take the inner product with the vector $f \in \Real{|S|}$.
\end{proof}
Combining this with (\ref{jpi}), we obtain the following, for any function $f$ and any policy $\pi$:
\begin{equation}
J(\pi) = \underset{s \sim \mu}{{\mathrm E}}[f(s)] + \frac{1}{1-\gamma} \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[R(s,a,s') + \gamma f(s') - f(s)\right]. \label{jpif}
\end{equation}
This identity is nice for two reasons. First: if we pick $f$ to be an approximator of the value function $V^{\pi}$, then (\ref{jpif}) relates the true discounted return of the policy ($J(\pi)$) to the estimate of the policy return (${\mathrm E}_{s\sim \mu}[f(s)]$) and to the on-policy average TD-error of the approximator; this is aesthetically satisfying. Second: it shows that reward-shaping by $\gamma f(s') - f(s)$ has the effect of translating the total discounted return by ${\mathrm E}_{s\sim \mu} [f(s)]$, a fixed constant independent of policy; this illustrates the finding of Ng. et~al. \yrcite{Ng1999} that reward shaping by $\gamma f(s') + f(s)$ does not change the optimal policy.
It is also helpful to introduce an identity for the vector difference of the discounted future state visitation distributions on two different policies, $\pi'$ and $\pi$. Define the matrices $G \doteq (I - \gamma P_{\pi})^{-1}$, $\bar{G} \doteq (I - \gamma P_{\pi'})^{-1}$, and $\Delta = P_{\pi'} - P_{\pi}$. Then:
\begin{eqnarray*}
G^{-1} - \bar{G}^{-1} &=& (I - \gamma P_{\pi}) - (I - \gamma P_{\pi'}) \\
&=& \gamma \Delta;
\end{eqnarray*}
left-multiplying by $G$ and right-multiplying by $\bar{G}$, we obtain
\begin{equation*}
\bar{G} - G = \gamma \bar{G} \Delta G.
\end{equation*}
Thus
\begin{eqnarray}
d^{\pi'} - d^{\pi} &=& (1-\gamma) \left(\bar{G} - G\right) \mu \nonumber \\
&=& \gamma (1-\gamma) \bar{G} \Delta G \mu \nonumber \\
&=& \gamma \bar{G} \Delta d^{\pi}. \label{dpidiff}
\end{eqnarray}
For simplicity in what follows, we will only consider MDPs with finite state and action spaces, although our attention is on MDPs that are too large for tabular methods.
\subsubsection{Main Results}
In this section, we will derive and present the new policy improvement bound. We will begin with a lemma:
\begin{lemma}\label{policybound0} For any function $f: S \to \Real{}$ and any policies $\pi'$ and $\pi$, define
\begin{equation}
L_{\pi,f} (\pi') \doteq \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \left(\frac{\pi'(a|s)}{\pi(a|s)} - 1 \right) \left(R(s,a,s') + \gamma f(s') - f(s) \right)\right],\label{surrogate}
\end{equation}
and $\epsilon_f^{\pi'} \doteq \max_s \left| {\mathrm E}_{a \sim \pi', s'\sim P} [R(s,a,s') + \gamma f(s') - f(s)] \right|$. Then the following bounds hold:
\begin{align}
J(\pi') - J(\pi) \geq \frac{1}{1-\gamma}\left(L_{\pi,f} (\pi') - 2\epsilon_f^{\pi'} D_{TV} (d^{\pi'} || d^{\pi})\right), \label{bound0} \\
J(\pi') - J(\pi) \leq \frac{1}{1-\gamma}\left(L_{\pi,f} (\pi') + 2\epsilon_f^{\pi'} D_{TV} (d^{\pi'} || d^{\pi})\right),
\label{bound0b}
\end{align}
where $D_{TV}$ is the total variational divergence. Furthermore, the bounds are tight (when $\pi' = \pi$, the LHS and RHS are identically zero).
\end{lemma}
\begin{proof} First, for notational convenience, let $\delta_f (s,a,s') \doteq R(s,a,s') + \gamma f(s') - f(s)$. (The choice of $\delta$ to denote this quantity is intentionally suggestive---this bears a strong resemblance to a TD-error.) By (\ref{jpif}), we obtain the identity
\begin{equation*}
J(\pi') - J(\pi) = \frac{1}{1-\gamma} \left(\underset{\begin{subarray}{c} s \sim d^{\pi'} \\ a \sim \pi' \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \delta_f (s,a,s') \right] - \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \delta_f(s,a,s') \right].\right)
\end{equation*}
Now, we restrict our attention to the first term in this equation. Let $\bar{\delta}_f^{\pi'} \in \Real{|S|}$ denote the vector of components $\bar{\delta}_f^{\pi'} (s) = {\mathrm E}_{a \sim \pi', s' \sim P} [\delta_f(s,a,s') |s]$. Observe that
\begin{eqnarray*}
\begin{aligned}
\underset{\begin{subarray}{c} s \sim d^{\pi'} \\ a \sim \pi' \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \delta_f (s,a,s') \right] & = \left\langle d^{\pi'}, \bar{\delta}_f^{\pi'} \right\rangle\\
& = \left\langle d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle + \left\langle d^{\pi'} - d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle
\end{aligned}
\end{eqnarray*}
This term is then straightforwardly bounded by applying H\"{o}lder's inequality; for any $p,q \in [1, \infty]$ such that $1/p + 1/q = 1$, we have
\begin{equation*}
\left\langle d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle + \left\| d^{\pi'} - d^{\pi}\right\|_p \left\|\bar{\delta}_f^{\pi'} \right\|_q \geq \underset{\begin{subarray}{c} s \sim d^{\pi'} \\ a \sim \pi' \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \delta_f (s,a,s') \right] \geq \left\langle d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle - \left\| d^{\pi'} - d^{\pi}\right\|_p \left\|\bar{\delta}_f^{\pi'} \right\|_q.
\end{equation*}
The lower bound leads to (\ref{bound0}), and the upper bound leads to (\ref{bound0b}).
We choose $p=1$ and $q = \infty$; however, we believe that this step is very interesting, and different choices for dealing with the inner product $\left\langle d^{\pi'} - d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle$ may lead to novel and useful bounds.
With $\left\| d^{\pi'} - d^{\pi}\right\|_1 = 2 D_{TV} (d^{\pi'} || d^{\pi})$ and $\left\|\bar{\delta}_f^{\pi'} \right\|_{\infty} = \epsilon_f^{\pi'}$, the bounds are almost obtained. The last step is to observe that, by the importance sampling identity,
\begin{eqnarray*}
\left\langle d^{\pi}, \bar{\delta}_f^{\pi'} \right\rangle &=& \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi' \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[ \delta_f (s,a,s') \right] \\
&=& \underset{\begin{subarray}{c} s \sim d^{\pi} \\ a \sim \pi \\ s' \sim P\end{subarray}}{{\mathrm E}} \left[\left( \frac{\pi'(a|s)}{\pi(a|s)} \right)\delta_f (s,a,s') \right].
\end{eqnarray*}
After grouping terms, the bounds are obtained.
\end{proof}
This lemma makes use of many ideas that have been explored before; for the special case of $f = V^{\pi}$, this strategy (after bounding $D_{TV} (d^{\pi'} || d^{\pi})$) leads directly to some of the policy improvement bounds previously obtained by Pirotta et~al. and Schulman et~al. The form given here is slightly more general, however, because it allows for freedom in choosing $f$.
\begin{remark}
It is reasonable to ask if there is a choice of $f$ which maximizes the lower bound here. This turns out to trivially be $f = V^{\pi'}$. Observe that ${\mathrm E}_{s' \sim P} \left[\delta_{V^{\pi'}}(s,a,s') | s,a\right] = A^{\pi'} (s,a)$. For all states, ${\mathrm E}_{a \sim \pi'} [A^{\pi'} (s,a)] = 0$ (by the definition of $A^{\pi'}$), thus $\bar{\delta}^{\pi'}_{V^{\pi'}} = 0$ and $\epsilon_{V^{\pi'}}^{\pi'} = 0$. Also, $L_{\pi, V^{\pi'}} (\pi') = -{\mathrm E}_{s \sim d^{\pi}, a \sim \pi} \left[A^{\pi'}(s,a)\right]$; from (\ref{jpif}) with $f = V^{\pi'}$, we can see that this exactly equals $J(\pi') - J(\pi)$. Thus, for $f = V^{\pi'}$, we recover an exact equality. While this is not practically useful to us (because, when we want to optimize a lower bound with respect to $\pi'$, it is too expensive to evaluate $V^{\pi'}$ for each candidate to be practical), it provides insight: the penalty coefficient on the divergence captures information about the mismatch between $f$ and $V^{\pi'}$.
\end{remark}
Next, we are interested in bounding the divergence term, $\|d^{\pi'} - d^{\pi}\|_1$. We give the following lemma; to the best of our knowledge, this is a new result.
\begin{lemma}\label{divergencebound}
The divergence between discounted future state visitation distributions, $\|d^{\pi'} - d^{\pi}\|_1$, is bounded by an average divergence of the policies $\pi'$ and $\pi$:
\begin{equation}
\|d^{\pi'} - d^{\pi}\|_1 \leq \frac{2\gamma}{1-\gamma} \underset{s \sim d^{\pi}}{{\mathrm E}} \left[ D_{TV} (\pi' || \pi)[s]\right],
\end{equation}
%
where $D_{TV} (\pi'||\pi)[s] = (1/2) \sum_a |\pi'(a|s) - \pi(a|s)|$.
\end{lemma}
\begin{proof}
First, using (\ref{dpidiff}), we obtain
\begin{eqnarray*}
\|d^{\pi'} - d^{\pi}\|_1 &=& \gamma \|\bar{G} \Delta d^{\pi}\|_1 \\
&\leq & \gamma \|\bar{G}\|_1 \|\Delta d^{\pi}\|_1.
\end{eqnarray*}
$\|\bar{G}\|_1$ is bounded by:
\begin{equation*}
\|\bar{G}\|_1 = \|(I - \gamma P_{\pi'})^{-1}\|_1 \leq \sum_{t=0}^{\infty} \gamma^t \left\|P_{\pi'}\right\|_1^t = (1-\gamma)^{-1}
\end{equation*}
To conclude the lemma, we bound $\|\Delta d^{\pi}\|_1$.
\begin{eqnarray*}
\|\Delta d^{\pi}\|_1 &=& \sum_{s'} \left| \sum_s \Delta(s'|s) d^{\pi}(s) \right| \\
&\leq& \sum_{s,s'} \left| \Delta(s'|s)\right| d^{\pi}(s) \\
&=& \sum_{s,s'} \left| \sum_a P(s'|s,a) \left(\pi'(a|s) - \pi(a|s) \right)\right| d^{\pi}(s) \\
&\leq& \sum_{s,a,s'} P(s'|s,a) \left|\pi'(a|s) - \pi(a|s) \right| d^{\pi}(s) \\
&=& \sum_{s,a} \left|\pi'(a|s) - \pi(a|s) \right| d^{\pi}(s) \\
&=& 2 \underset{s \sim d^{\pi}}{{\mathrm E}} \left[ D_{TV} (\pi'||\pi)[s] \right].
\end{eqnarray*}
\end{proof}
The new policy improvement bound follows immediately.
\performancebound*
\begin{proof} Begin with the bounds from lemma \ref{policybound0} and bound the divergence $D_{TV} (d^{\pi'} || d^{\pi})$ by lemma \ref{divergencebound}.
\end{proof}
|
2,869,038,154,921 | arxiv | \section{Introduction}
The emission-line stars are spread over the entire HR digram. They concern young, not evolved stars or at the opposite evolved
stars and of course Main Sequence stars. They are or not massive stars.
For instance, among others, one can found TTauri stars, UV Ceti stars, flare stars, Mira stars, HBe/Ae stars,
B[e] stars, Be/Oe stars, Of stars, Supergiant stars, LBV, WR stars.
This document concerns mainly OeBeAe stars.
The OeBeAe stars have properties similar to OBA type star (see Evans contribution, this volume) and specific properties
due to their extreme nature.
The first Be star, $\gamma$ Cas was discovered by the father Secchi in 1867 (\cite[Secchi 1967]{secchi1867}).
\cite[Collins (1987)]{collins1987} wrote the first definition of Be stars:
``they are non-supergiant OBA-type stars that have displayed at least once in their spectrum emission lines (H$\alpha$).''
As example, see Fig.~\ref{fig1}.
Actually the emission-lines come from the circumstellar decretion disk (\cite[Struve 1931]{struve1931}) formed by episodic matter
ejections from the central star.
Are the matter ejections related to the rotation?
Be stars are very fast rotators, rotating close to the breakup velocity but often an additional mechanism is needed
for the star being able to eject the matter.
In the next sections, the rotation and other mechanisms (binarity, stellar evolution, magnetic fields, pulsations) are examined as well as the effects of the metallicity, the stellar
evolution, etc.
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.4in]{halphalines.ps}
\caption{Examples of H$\alpha$ emission-line in different LMC Be stars.
The shape depends among other parameter of the inclination angle.}
\label{fig1}
\end{center}
\end{figure}
\cite[Porter \& Rivinius (2003)]{Porter2003} presented different kind of emission-line stars with their known properties
and how they differ to the ``classical Be stars''.
Since 2003, with the improvements of the technology (interferometry, MOS, etc) and the efforts made on the modeling and theory,
the knowledge of Be stars was improved. The next sections will provide some very summarized informations (since 2004, more than 240
refeered articles were published dealing with Be stars) about these recent developments (see also contribution by Baade, this volume).
{\underline{\it -Techniques for detecting a Be star}}
Before studying Be stars, it is necessary to find/recognize them.
There are several ways to detect them.
The first possibility is to use the photometric techniques.
Combining different colour/colour or colour/magnitude diagrams would help to pre-classify the stars and
detect potential Be star candidates.
\cite[Keller et al. (1999)]{keller99} provide examples of CMD with given thresholds above which the Be stars could fall.
They also show that Be stars tend to form a redder sequence than normal B stars
\cite[Dachs et al. (1988)]{dachs1988} have shown that the infrared excess is related to the circumstellar disk of Be stars.
It is also linked to the H$\alpha$ equivalent width.
More recent studies found similar results in other wavelength domains such as in the infrared with the AKARI survey
(\cite[Ita et al. 2010]{ita2010}) or with SPITZER (\cite[Bonanos et al. 2010]{bonanos2010}).
However, this kind of study has some limits due to the intrinsic properties of Be stars (variability, change of phases Be-B).
\cite[McSwain \& Gies (2005)]{McSwaingies2005} in their photometric survey of Galactic open clusters found 63 \%
of the known Be stars due to the transience of the Be phenomenon.
Thus with the photometry the coupling of indices is necessary.
It is also necessary to disentangle the local reddening and the infrared excess
due to the circumstellar disk of Be stars.
Another way to find the emission-line stars, and the Be stars, is to do slitless spectroscopy and combine the spectroscopic diagnostic
with the photometry as presented in \cite[Martayan et al. (2010a)]{marta2010a}.
The slitless spectroscopy gives the advantage to do not be sensitive to the diffuse nebulosity and the spectra are not contaminated
by other nebular lines.
Another possibility for finding Be stars is to study the behaviour and the variation of lightcurves.
The Be phenomenon is transient, as a consequence the star could appear as a Be star or as a B star when the disk was blown up.
Moreover, Be stars show often outbursts.
These characteristics help to find this kind of stars using the lightcurves variations as published by
\cite[Mennickent et al. (2002)]{mennickent2002} and \cite[Sabogal et al. (2005)]{sabogal2005}.
{\underline{\it -The Surveys}}
Be stars with the techniques mentionned above can be found mainly in open clusters and in fields.
In the Galaxy, there are different surveys, let us mention, the photometric survey by
\cite[McSwain \& Gies (2005)]{McSwaingies2005} and the slitless spectroscopic survey by
\cite[Mathew et al. (2008)]{Mathew2008}.
In other galaxies such as the Magellanic Clouds but also in the Milky Way, there are the following surveys
among others:
a slitless spectroscopic survey of the SMC by \cite[Meyssonnier \& Azzopardi (1993)]{MA1993},
another in the SMC/LMC/MW by \cite[Martayan et al. (2010a)]{Marta2010a}, \cite[Martayan et al. (2008)]{Marta2008}
(see Fig.~\ref{fig4}).
\begin{figure}[h]
\begin{center}
\includegraphics[width=3.4in]{figngc6611.ps}
\caption{Example of slitless spectroscopy in the NGC6611 open cluster by \cite[Martayan et al. (2008)]{Marta2008}.
The stars appear as spectra.}
\label{fig4}
\end{center}
\end{figure}
And recently, with the VLT-FLAMES, some spectroscopic surveys were and are performed in different galaxies
(\cite[Evans et al. 2005]{evans2005}, \cite[Martayan et al. 2006a]{Marta2006a},
\cite[Martayan et al. 2007a]{Marta2007a}).
With the polarimetry,
some Be stars were found in the SMC/LMC (\cite[Wisniewski et al. 2007]{wis2007},
\cite[Wisniewski \& Bjorkman 2006]{wis2006}),
and with FORS2 in IC1613 (\cite[Bresolin et al. 2007]{bresolin2007}), which correspond to the farthest Be stars detected.
\section{Mid/long term monitoring of Be stars, V/R variations}
{\underline{\it -Monitoring of Be stars}}
Why long-term monitorings are needed?
Due to the transience of the Be phenomenon for having a chance to detect a Be star, a mid or long term
monitoring of B/Be stars is needed. Indeed, for the follow-up of the variability of the star
related to outbursts and/or pulsations, or binarity, or modifications of the CS disk, etc, a monitoring is needed.
Surveys such as the photometric ones of MACHO or OGLE are useful in that sense
but also the spectroscopic surveys.
Here the amateur astronomers have a role to play. With the new efficient, good quality, low cost
spectrograph that can be mounted in amateur telescopes, see contribution by Blanchard et al. (this volume), some useful good quality
observations can be performed.
Moreover, without long-term monitoring (more than 10 years of spectroscopy)
the results concerning the Be star $\mu$ Cen and the predictions
of the outbursts dates could not have been obtained.
For all details about this pioneer study, see \cite[Rivinius et al. (1998a)]{rivi1998a},
\cite[Rivinius et al. (1998b)]{rivi1998b}.
This kind of study could thanks to the databases be performed more easily now, let us cite the
Be Star Spectra database http://basebe.obspm.fr/basebe/
(\cite[Neiner et al. 2007]{neiner2007}, contribution by De Batz et al., this volume).
{\underline{\it -The V/R variation of the emission lines}}
A long-term spectroscopic monitoring of Be stars can show in certain cases variations
of the V and R peaks of the emission-lines. This variation is now understood and is related to global
one-armed oscillations in the circumstellar disk.
The case of the Be star $\zeta$ Tau (see \cite[Stefl et al. 2009]{stefl2009}) is representative
of that phenomenon with a V/R cycle of about 1400 days without correlation with the period of 133 days
of the companion. Fig.~\ref{fig3b} from \cite[Stefl et al. (2009)]{stefl2009} illustrates this phenomenon.
The (polarimetric, spectroscopic, interferometric) observations are properly reproduced by a modeling
of a one-armed spiral viscous disk as described in
\cite[Carciofi et al. (2009)]{carciofi2009}, see also contributions by Carciofi and Stee, this volume.
\begin{figure}[h]
\begin{center}
\includegraphics[width=5in]{figstefl.ps}
\caption{Example of V/R variations in the Be star $\zeta$ Tau. Figure adapted from \cite[Stefl et al. (2009)]{stefl2009}.}
\label{fig3b}
\end{center}
\end{figure}
\section{Stellar winds}
The diagnostic lines are mostly present in the UV and optical but also several of them are
in the infrared. Let us mention the lines of HeI, HeII, Si IV, C IV, N V, Br$\gamma$, etc.
For more details, see \cite[Martins et al. (2010)]{martins2010},
\cite[Mokiem et al. (2006)]{mokiem2006}, and Henrichs et al., this volume.
The structure of the wind differs among B stars.
For the normal B stars, the wind is spherical, while for Be stars, we can expect a radiative bi-polar wind
mainly due to the Von Zeipel effect. The poles in case of fast rotation becomes brighter than
the equator, it explains this asymetry.
In case of Be stars, one can also expect an additional mechanical wind
at the equator that creates the CS disk. In such case the needed value of the $\Omega/\Omega$$_{c}$
ratio (angular velocity to critical angular velocity ratio) for ejecting matter has to be determined (see Sect.\ \ref{rotvel}).
Fortunately, recent interferometric facilities (such as the VLTI-AMBER), allowed to measure the shape
of the CS environment of Be stars and both winds were found.
It is for example the case of the Be stars $\alpha$ Arae, Achernar
(\cite[Meilland et al. 2007]{meilland2007}, \cite[Kervella et al. 2006]{kervella2006}).
\section{Rotational velocities}
\label{rotvel}
Be stars are known to be very fast rotator but how fast are they rotating?
Fig.~\ref{figdeforme} shows effect of the rotation on the shape of the stars
for different $\Omega/\Omega$$_{c}$ in the Roche model, rigid rotation approach.
Next sections provide some clues about this point.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3.4in]{figdeformeBe.ps}
\caption{Example of star shape depending on the $\Omega/\Omega$$_{c}$ ratio in the Roche model and rigid rotation
approach. Due to the Von Zeipel effect there is a temperature gradient from the poles to the equator.
Courtesy by Y. Fr\'emat, see also \cite[Fr\'emat et al. (2005)]{fremat2005}.}
\label{figdeforme}
\end{center}
\end{figure}
{\underline{\it -From the interferometry}}
The interferometric observations show that all observed Be stars are found flattened
revealing high $\Omega/\Omega$$_{c}$ratios.
The flattening of Achernar was found equal to 1.56 $\pm$0.05 by \cite[Domiciano de Souza et al. (2003)]{domi2003}.
However, this value could also be due to remaining small disk indicating that Achernar is not rotating
at the breakup velocity but close to it.
{\underline{\it -Line saturation effect at high $\Omega/\Omega$$_{c}$}}
\cite[Townsend et al. (2004)]{townsend2004} showed that the HeI 4471 \AA \ line has a saturation effect at high
$\Omega/\Omega$$_{c}$, typically above 80\%. This implies that the Vsini measurements based on the use of 1 He line
are limited in case of fast rotating stars, typically Be stars and possibly Bn stars.
Using several He lines as well as metallic lines, the saturation effect occurs again but at higher $\Omega/\Omega$$_{c}$
($\sim$90-95\%), see \cite[Fr\'emat et al. (2005)]{fremat2005}.
This explains the discrepancy between the measurements of Be stars Vsini between
\cite[Hunter et al. (2008)]{hunter2008} and \cite[Martayan et al. (2007a)]{Marta2007a}.
Fig.~\ref{fig6} shows the measurements differences from \cite[Hunter et al. (2008)]{hunter2008} and the saturation curves at $\Omega/\Omega$$_{c}$=70 and 80 \% from
\cite[McSwain et al. (2008)]{mcswain2008}. It clearly shows that the Be star Vsini measurements done by
\cite[Hunter et al. (2008)]{hunter2008} are saturated and the measurements by \cite[Martayan et al. (2007a)]{Marta2007a}
saturate at $\sim$90\%.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=3.4in]{figsatureBe.ps}
\caption{Vsini measuremnts of B and Be stars by \cite[Hunter et al. (2008)]{hunter2008} compared to those
of \cite[Martayan et al. 2007a]{Marta2007a}. The dotted curve corresponds to $\Omega/\Omega$$_{c}$=70\% and the solid curve
to $\Omega/\Omega$$_{c}$=80\% (line saturation effect with 1 He line measurement).
The curves come from \cite[McSwain et al. (2008)]{mcswain2008}.}
\label{fig6}
\end{center}
\end{figure}
{\underline{\it -Fast rotation effects correction}}
Accordingly to the previous section, in case of fast rotators, one has to take into account the fast rotation effects
(flattening, gravitational darkening). These effects introduce underestimates of the Vsini, and it implies too that
the spectral type/luminosity classes are modified from the late to earlier spectral type and from evolved to less
evolved luminosity classes. The codes BRUCE by \cite[Townsend et al. (2004)]{townsend2004} and
FASTROT by \cite[Fr\'emat et al. (2005)]{fremat2005} are able to properly correct the fundamental parameters of the fast
rotating stars.
{\underline{\it -The veiling effect}}
Another important effect in case of Be stars is the veiling effect due to the circumstellar disk
that affects the continuum level and thus the lines depth. Not taking this effect into account will imply
bad measurements of the fundamental parameters but also of the chemical abundances.
For taking this effect into account, \cite[Ballereau et al. (1995)]{ballereau1995} proposed a method based on the He4471 emission-line EW
measurements.
Other possibility is to prefer the bluest lines when spectrum fitting is performed but also to determine the disk influence
by fitting the Spectral Energy Distribution. The new spectrocopic facility, the VLT-XSHOOTER that covers the near UV to the K-band
simultaneously should help in that approach.
Alternatively, there is the method consisting to fit the Balmer discontinuity as described by \cite[Zorec et al. (2009)]{zorec2009}
{\underline{\it -Metallicity effects}}
\cite[Keller (2004)]{keller2004}, \cite[Martayan et al. (2006a)]{marta2006a}, \cite[Martayan et al. (2007a)]{marta2007a},
and \cite[Hunter et al. (2008)]{hunter2008} found that SMC OB stars rotate faster than LMC OB stars, which rotate faster than
their Galactic counterparts.
In the SMC, the mass-loss of OB stars by radiatively driven winds was found lower than in the Galaxy by
\cite[Bouret et al. (2003)]{bouret2003} and \cite[Vink (2007)]{vink2007}.
Consequently, the stars should loose less angular momentum and could rotate faster (\cite[Maeder \& Meynet 2001]{MM2001}).
The comparisons between SMC Be stars Vsini fairly agree with the theoretical models by \cite[Ekstr\"om et al. (2008)]{ekstrom2008}.
The SMC Be stars are found to be rotating very close to the critical velocity.
{\underline{\it -ZAMS rotational velocities}}
The ZAMS rotational velocities of SMC, LMC, and Milky Way (MW) intermediate-mass Be stars were determined
by \cite[Martayan et al. (2007a)]{marta2007a}.
They found a metallicity effect on the ZAMS rotational velocities. At lower metallicity Be stars rotate faster
since their birth. This could be related to an opacity effect, for an identical $\Omega/\Omega$$_{c}$,
at lower metallicitty, the radii are smaller thus the stars can rotate faster.
The comparison of the SMC ZAMS rotational velocities of intermediate-mass and massive Be stars with the theoretical tracks by
\cite[Ekstr\"om et al. (2008)]{ekstrom2008} are also in a fair agreement.
{\underline{\it -Number of Be stars vs. the metallicity and redshift}}
As a consequence of faster rotational velocities at lower metallicity, one can expect to find more Be-type stars in low-metallicity
environments.
Using the ESO-WFI in its slitless mode (\cite[Baade et al. 1999]{baade1999}), \cite[Martayan et al. (2010a)]{marta2010a} found 3 to
5 times more Be stars in SMC open clusters than in the Galactic ones (\cite[McSwain \& Gies 2005]{McSwaingies2005}),
see Fig.~\ref{fig8}.
This quantified the results by \cite[Maeder et al. (1999)]{maeder1999} and \cite[Wisniewski et al. (2008)]{wis2008}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=2.5in]{ratioBestarswithZ.ps}
\caption{Ratio of Be to B stars by spectral categories in the SMC and MW. Figure adapted
from \cite[Martayan et al. (2010a)]{marta2010a}.Red dashed bars are for the SMC, white bars for the MW.}
\label{fig8}
\end{center}
\end{figure}
Another consequence of this increase of the Be stars number with decreasing metallicity is when a survey of OBA-type
populations is performed, one should find an increased number/ratio of OeBeAe stars in the sample.
This was exactly the case of the survey by \cite[Bresolin et al. (2007)]{bresolin2007} who found 6 Be stars in the 6 MS B stars
observed in the low metallicity galaxy IC1613.
Actually at lower metallicity than the SMC, and probably at higher redshift, the stars could rotate again faster,
then one can expect to have more Be stars but also
that the Be phenomenon is extended to other categories of stars.
{\underline{\it -Chemical abundances}}
The measurement of chemical abundances is very important for testing the stellar evolution models with fast rotation, including the
rotational mixing due to the rotation. Moreover, it is also important for distinguishing the stars following a chemical homogeneous evolution than the usual
evolution (\cite[Maeder et al. 1987]{maeder1987}, \cite[Yoon et al. 2006]{yoon2006}).
It is also of interest to know them for the pulsating stars of different metallicity environments.
\cite[Lennon et al. (2005)]{lennon2005} found 2 Be stars without N enhancement, while the theoretical models using the
rotational mixing expect a N enhancement and a C depletion.
Dunstall et al. (this volume) and Peters (this volume) also found no N enrichment in Be stars.
Does it mean that the rotational mixing theory is not able to explain the measurements?
Not necessarily, \cite[Porter (1999)]{porter1999} found that due to the temperature gradient in Be stars, the ions are displaced
from the poles to the equator. This could occur to the N, implying that the corresponding lines become weaker and the
measurements biased.
\section{Stellar pulsations}
Be stars lie in $\beta$ Cep and SPB regions, thus p and g modes are expected to be found. The $\kappa$ mechanism is at the origin of
the observed pulsations due to the iron bump.
A beating of non radial pulsations combined to the fast rotation of Be stars could be at the origin of the matter ejection.
According to \cite[Townsend et al. (2004)]{townsend2004} an $\Omega/\Omega$$_{c}$=95\% is needed for launching the matter in orbit, while
\cite[Cranmer (2009)]{cranmer2009} shows that depending on the $\Omega/\Omega$$_{c}$ value the ejected matter will be part of the wind
or will form a disk.
\cite[Rivinius et al. (1998a)]{rivi1998a} have shown that the combination
fast rotation + beating of non radial pulsations is able to reproduce and predict the outbursts in the Be star $\mu$ Cen.
More recently, with the new photometric space missions MOST, COROT, KEPLER, a large number of pulsational frequencies
(g and p modes) were found in several Be stars.
\cite[Huat et al. (2009)]{huat2009}, in the Be star HD49330, have found that the pulsational frequencies
but also their amplitude change between the quiescence phase, the pre burst phase, and during the burst phase.
Is it related to the subsurface convection zones in OB stars? For more details, see Cantiello contribution (this volume).
\begin{figure}[h!]
\begin{center}
\includegraphics[width=4.5in]{BepulsationsHuat.ps}
\caption{Evolution of pulsational frequencies in the Be star HD49330 from COROT data during the different pre, during, and
after an outburst phases. The figure comes from \cite[Huat et al. (2009)]{huat2009}.}
\label{fig9}
\end{center}
\end{figure}
\section{Spectropolarimetry: magnetic fields and disks}
The spectropolarimetry is a useful tool for finding the Be stars due to their disks but also for detecting the magnetic fields.
Recent spectropolarimetric facilities such as ESPADONS, NARVAL, allow the search of magnetic fields in Be stars but up to now,
there is only 1 Be star in which a magnetic field was found: $\omega$ Ori (\cite[Neiner et al. 2003]{neiner2003}).
The presence of magnetic field ($\Omega$ transport from the core to the surface or magnetic reconnection)
combined to the fast rotation could also be an additional mechanism for ejecting matter and create the Be star CS disk.
\section{Circumstellar disks}
Huge progresses were performed thanks to the interferometry (VLTI, CHARA, NPOI) and the multi-wavelengths studies.
The main result is that the disk is rotating in a Keplerian way around its central star,
see for example \cite[Meilland et al. (2007)]{meilland2007}.
In addition, the thermal structure of the CS disk is also better known (see Tycner contribution, this volume).
\section{Stellar formation and evolution, Gamma Ray Bursts}
What kind of objects is at the origin of Be stars and their fast rotation? Could they be related to the Herbig Ae/Be objects?
Are the fast B rotator Bn stars at the Be star origin, while their number is roughly equal to the number of Be stars in the MW?
What is the IMF of the Be stars? Does it differ from the normal B stars?
It seems that the open cluster density has no effect on the appearance of Be stars (\cite[McSwain \& Gies 2005]{McSwaingies2005},
\cite[Martayan et al. 2010a]{marta2010a}).
Taking into account the fast rotation effects, \cite[Zorec et al. (2005)]{zorec2005} derived the evolutionary status of Be stars in
the MW (see Fig.~\ref{fig10}). Their location/life in the MS strongly depends on their mass and on the evolution
of the $\Omega/\Omega$$_{c}$ ratio.
In the LMC, the diagram of Be stars appearance is similar than in the MW but in the SMC, at the opposite of the MW, massive Be stars
and Oe stars appear or are still existing in the second part of the MS (\cite[Martayan et al. 2007a]{marta2007a}).
These differences are also consistent with the $\Omega/\Omega$$_{c}$ evolution at different metallicities.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=2.5in]{stellarevolution.ps}
\caption{Evolutionary diagram in the MS of the Be stars in the MW by \cite[Zorec et al. (2005)]{zorec2005}.}
\label{fig10}
\end{center}
\end{figure}
However, the life of a Be star is not finished at the end of the MS. It seems that the more massive SMC Be and Oe stars could play a
role in the explanation of the type 2 GRBs (also called Long Gamma Ray Bursts). \cite[Martayan et al. (2010b)]{marta2010b}
were able to reproduce the number and ratio of LGRBs using the populations of SMC massive Be and Oe stars accross
their stellar evolution.
The type 3 GRBs (without SN counterpart) could be explained in a binary scenario involving Be-binary stars
(\cite[Tutukov \& Fedorova 2007]{tutu2007}).
\section{Binarity}
While 60 to 70\% of O stars are known to be binaries (see \cite[Sana et al. 2009]{sana2009} and his contribution in this volume),
the question remains opened for the B and Be stars.
The last study by \cite[Oudmaijer et al. (2010)]{oud2010} using the VLT-NACO AO facility found that:
29 to 35 $\pm$8\% of the nearby B stars are binaries and that 30 to 33 $\pm$8\% of the nearby Be stars are binaries.
They scanned the separations larger than 20 AUs with a flux contrast ratio of 10 corresponding to M companions.
Scanning smaller scales needs the interferometry. About 40 \% of all Be stars observed with the VLTI and CHARA
were found to have a companion.
However, certain type of Be stars are known to be binaries such as the Be-X rays (see for example \cite[Coe et al. 2008]{coe2008}).
The $\gamma$ Cas-like stars have possible companions and have magnetic field activity in their CS environment (\cite[Smith et al. 2006]{smith2006}).
About the Be-binary disruptions scenario when the companion has exploded in SN, one should find that Be stars are runaway stars or see SN
remnants. In both cases, this is not observed for the large majority of Be stars.
The chemical abundances could help to find a potential previous interaction with a companion.
In that case, one can expect abundances anomalies.
\section{Conclusion}
To better understand the properties of these stars, it is necessary to do multiwavelength studies and multi-techniques studies
with the use for example of: the VLT-XSHOOTER, IR and X-rays satellites, MOS such as the VLT-FLAMES, the interferometry,
the AO facilities, and if possible simultaneous observations.
In the future, it will be possible to do similar studies than today but in very far galaxies, to scan other metallicity ranges, etc,
thanks to the ELTs and their instruments but also to ALMA, SKA, etc.
As shown, the Be phenomenon could be not restricted to the late O, B, and early A stars and could
also not be restricted to the MS, specially in lower metallicity environments.
Some of the questions about them and their CS environment are now solved but some others are still
opened and the definition of Be stars should be revised.
Finally, one can propose this new definition of the Be phenomenon:
{\it This is a star with innate or acquired very fast rotation, which combined to other mechanism such as non-radial pulsations beating
leads to episodic matter ejections creating a CS decretion disk or envelope.}
This definition implies that the stars are not restricted to the MS, not restricted to B-type, and from the CS material the
emission lines come.
|
2,869,038,154,922 | arxiv | \section{Introduction}
Mathematical models involving
partial differential equations (PDEs) depending on a set of parameters are ubiquitous in applied sciences and engineering. These input parameters are defined to characterize, e.g., material properties, loads, boundary/initial conditions, source terms, or geometrical features. High-fidelity simulations based on full-order models (FOMs), like the finite element method (FEM), entail huge computational costs in terms of CPU time and memory, if a large number of degrees of freedom is required. Complexity is amplified whenever interested in going beyond a single direct simulation, such as in the multi-query contexts of optimization, parameter estimation and uncertainty quantification.
To face these problems, several strategies to build reduced order models (ROMs) have been developed over the years, aiming at computing reliable solutions to parametrized PDEs at a greatly reduced cost.
A large class of ROMs relies on a projection-based approach, which aims at approximating the unknown state quantities as a linear superimposition of basis functions; these latter then span a subspace which the governing equations are projected onto
\cite{benner2015survey, benner2017model}. Among these, the reduced basis (RB) method \cite{quarteroni2016reduced, hesthaven2016certified} is a powerful and widely used technique, characterized by a splitting of the reduction procedure into an expensive, parameter-independent offline phase (however performed once and for all) and an efficient, parameter-dependent online phase. Its efficiency mainly relies on two crucial assumptions:
\begin{enumerate}
\item the solution manifold is low-dimensional, so that the FOM solutions can be approximated as a linear combination of few reduced modes with a small error;
\item the online stage is completely independent of the high-fidelity dimension \cite{farhat20205}.
\end{enumerate}
Assumption 1 concerns the approximability of the solution set and is associated with the slow decay of the Kolmogorov $N$-width \cite{pinkus2012n}. However, for physical phenomena characterized by a slow $N$-width decay, such as those featuring coherent structures that propagate over time \cite{fresca2020deep}, the manifold spanned by all the possible solutions is not of small dimension, so that ROMs relying on linear (global) subspaces might be inefficient. Alternative strategies to overcome this bottleneck can be, e.g., local RB methods \cite{amsallem2012nonlinear, pagani2018numerical, vlachas2021local}, or nonlinear approximation techniques, mainly based on deep learning architectures, see, e.g., \cite{lee2020model, kim2020efficient, fresca2021comprehensive, fresca2021pod, franco2021deep}.
Assumption 2 is automatically verified in linear, affinely parametrized problems \cite{quarteroni2016reduced}, but cannot be fulfilled when dealing with nonlinear problems, as the online assembling of the reduced operators requires to reconstruct the high-fidelity ones. To overcome this issue, a further level of approximation, or {\em hyper-reduction}, must be introduced. State-of-the-art methods, such as the empirical interpolation method (EIM) \cite{barrault2004empirical}, the discrete empirical interpolation method (DEIM) \cite{chaturantabut2010nonlinear}, its variant matrix DEIM \cite{negri2015efficient}, the missing point estimation \cite{astrid2008missing} and the Gauss-Newton with approximated tensors (GNAT) \cite{carlberg2011efficient}, aim at recovering an affine expansion of the nonlinear operators by computing only a few entries of the nonlinear terms. EIM, DEIM and GNAT can be seen as {\em approximate-then-project} techniques, since operator approximation is performed at the level of FOM quantities, prior to the projection stage. On the other hand, {\em project-then-approximate} strategies have also been introduced, aiming at approximating directly ROM operators, such as the reduced nonlinear residual and its Jacobian. An option in this sense is represented by the so-called Energy Conserving Sampling and Weighting (ECSW) technique \cite{farhat2015structure}. See. e.g., \cite{farhat20205} for a detailed review. \\
Although extensively applied in many applications, spanning from fluid flows models to cardiac mechanics \cite{drohmann2012reduced, amsallem2012nonlinear, tiso2013discrete, radermacher2016pod, ghavamian2017pod, bonomi2017reduced}, these strategies are code-intrusive and, more importantly, might impact on the overall efficiency of the ROM approximation in complex applications. Very often, when dealing with highly nonlinear problems expensive hyper-reduction strategies are required if aiming at preserving the physical constraints at the ROM level, that is, if ROMs are built consistently with the FOM through a projection-based strategy. For instance, a large number of DEIM basis vectors are required to ensure the convergence of the reduced Newton systems arising from the linearization of the nonlinear hyper-ROM when dealing with highly nonlinear elastodynamics problems \cite{cicci2021cardiacDEIM}, even if few basis functions are required to approximate the state solution in a low-dimensional subspace. An alternative formulation of DEIM in a finite element (FE) framework, known as unassembled DEIM \cite{tiso2013modified}, has been proposed to preserve the sparsity of the problem, while in \cite{peherstorfer2014localized} a localized DEIM selecting smaller local subspace for the approximation of the nonlinear term is presented.
Semi-intrusive strategies, avoiding the construction of a ROM through a Galerkin projection, have been recently proposed exploiting surrogate models to determine the RB approximation. For instance, neural networks (NNs) or Gaussian process (GP) regression can be employed to learn the map between the input parameters and the reduced-basis expansion coefficients in a non-intrusive way \cite{hesthaven2018non,guo2018reduced,guo2019data,swischuk2019projection}. An approximation of the nonlinear terms arising in projection-based ROMs is obtained in \cite{gao2020non} through deep NNs (DNNs) that exploit the projection of FOM solutions.
NNs have also also been recently applied in the context of {\em operator inference} for (parametrized) differential equations, combining ideas from classical model reduction with data-driven learning. For instance, the design of NNs able to accurately represent linear/nonlinear operators, mapping input functions to output functions, has been proposed recently in \cite{lu2021learning}; based on the universal ap\-pro\-xi\-ma\-tion theorem of operators \cite{chen1995universal}, a general deep learning framework, called DeepONet, has been introduced to learn continuous operators, such as solution operators of PDEs, using DNNs; see also \cite{wang2021learning}. In \cite{peherstorfer2016data} a non-intrusive projection-based ROM for parametrized time-dependent PDEs including low-order polynomial nonlinear terms is considered, inferring an approximation of the reduced operators directly from data of the FOM. Finally, the obtained low-dimensional system is solved -- in this case, the learning task consist in the solution to a least squares problem; see also \cite{benner2020operator}. Projection-based ROMs and machine learning have been fused in \cite{qian2019transform} aiming at the approximation of linear and quadratic ROM operators, focusing on the solution to a large class of fluid dynamics applications. Similarly, in \cite{bai2021non} a non-intrusive technique, exploiting machine learning regression algorithms, is proposed for the approximation of ROM operators related to projection-based methods for the solution of parametrized PDEs. Finally, \cite{bhattacharya2021model} combines principal component analysis-based model reduction with a NNs for approximation, in a purely data-driven fashion, of infinite-dimensional solution maps, such as the solution operator for time-dependent PDEs. \\
In this paper, we develop a novel semi-intrusive, deep learning-enhanced hyper-reduced order modeling strategy, which hereon we refer to as Deep-HyROMnet, by leveraging a Galerkin-RB method for solution dimensionality reduction and DNNs to perform hyper-reduction. Since the efficiency of the nonlinear ROM hinges upon the cost-effective approximation of the projections of the (discrete) reduced residual operator and its Jacobian (when an implicit numerical scheme is employed), the key idea is to overcome the computational bottleneck associated with classical, intrusive hyper-reduction techniques, e.g. DEIM, by relying on DNNs to approximate otherwise expensive reduced nonlinear operators at a greatly reduced cost. Unlike data-driven-based methods, for which the predicted output is not guaranteed to satisfy the underlying PDE, our method is physics-based, as it computes the ROM solution by actually solving the reduced nonlinear systems by means of Newton method, thus exploit the physics of the problem. A further benefit of the method proposed lies on the fact that the inputs given to the NN are low-dimensional arrays, so that the overwhelming training times and costs that may be required by even moderately large FOM dimensions can be avoided. We point out that Deep-HyROMnet aims at efficiently approximate the nonlinear operators given by the composition of the reduced solution operator, that maps the input parameter vector and time to the corresponding ROM solution, and the reduced residual/Jacobian operator, that maps the ROM solution to the reduced residual/Jacobian evaluated on the ROM solution. To the best of our knowledge, this is the first method of its kind. We apply the novel methodology to the solution of problems in nonlinear solid mechanics, with particular focus on parametrized nonlinear elastodynamics and complex (e.g., exponential nonlinear) constitutive relations of the material undergoing large de\-for\-ma\-tions, showing that Deep-HyROMnet outperforms the DEIM-based ROM in terms of computational speed-up in the online stage, still achieving accurate results. \\
The paper is structured as follows. We recall the formulation of the RB method for nonlinear unsteady parametrized PDEs in Section~\ref{sec:RBM}, relying on POD for the construction of the reduced subspace and on DEIM as hyper-reduction technique. Deep-HyROMnet and the DNN architecture employed to perform reduced operator approximation are detailed in Section~\ref{sec:Deep-HyROMnet}. The numerical performances of the method are assessed in Section~\ref{sec:tests} on several benchmark problems related with nonlinear elastodynamics. Finally, conclusions and future perspective are reported in Section~\ref{sec:conclusion}.
\section{Projection-based ROMs: the reduced basis method}\label{sec:RBM}
Our goal is to pursue an efficient solution to nonlinear unsteady PDE problems depending on a set of input parameters, which can be written in abstract form as follows: given an input parameter vector $\bm\mu\in\mathcal{P}$, $\forall t\in(0,T]$, find $\mathbf{u}(t;\bm \mu)\in V$ such that
\begin{equation} \label{eq:motion}
R(\mathbf{u}(t;\bm \mu),t;\bm\mu)=0 \quad \text{in } V',
\end{equation}
where the parameter space $\mathcal{P}\subset\mathbb{R}^P$ is a compact set and $H^1(\Omega)^m \subseteq V \subseteq H_0^1(\Omega)^m$ is a suitable functional space, depending on the boundary conditions at hand, whereas $\Omega \subset \mathbb{R}^d$ is a bounded domain in $d$ dimensions, $d = 1, 2, 3$. In particular, we are interested in vector problems ($m=3$) set in $d=3$ dimensions. The role of the parameter vector $\bm\mu$ depends on the particular application at hand; in the case of nonlinear elastodynamics, $\bm\mu$ is related to the coefficients of the constitutive relation, the material properties and the boundary conditions.
By performing discretization in space and time, we end up with a fully-discrete nonlinear system
\begin{equation}\label{eq:residual}
\mathbf{R}(\mathbf{u}_h^n(\bm\mu),t^n;\bm\mu) = \mathbf{0} \quad \text{in } \mathbb{R}^{N_h},
\end{equation}
at each time step $t^n$, $n=1,\dots,N_t$, which can be solved by means of the Newton method: given $\bm\mu\in\mathcal{P}$ and an initial guess $\mathbf{u}_h^{n,(0)}(\bm\mu)$, for $k\geq0$, find $\mathbf{\delta u}_h^{n,(k)}(\bm\mu)\in\mathbb{R}^{N_h}$ such that
\begin{equation}\label{eq:motionFOM} \left\{ \begin{array}{llll}
\mathbf{J}(\mathbf{u}_h^{n,(k)}(\bm\mu),t^{n};\bm\mu)\mathbf{\delta u}_h^{n,(k)}(\bm\mu) = - \mathbf{R}(\mathbf{u}_h^{n,(k)}(\bm\mu),t^{n};\bm\mu) \\
\mathbf{u}_h^{n,(k+1)}(\bm\mu) = \mathbf{u}_h^{n,(k)}(\bm\mu) + \mathbf{\delta u}_h^{n,(k)}(\bm\mu)
\end{array} \right.
\end{equation}
until suitable stopping criteria are fulfilled. Here, $\mathbf{u}_h^{n,(k)}(\bm\mu)$ represents the solution vector for a fixed parameter $\bm\mu$ computed at time step $t^{n}$ and Newton iteration $k$, while $\mathbf{R}\in\mathbb{R}^{N_h}$ and $\mathbf{J}\in\mathbb{R}^{N_h\times N_h}$ denote the residual vector and the corresponding Jacobian matrix, respectively. We refer to (\ref{eq:motionFOM}) as the high-fidelity, full-order model (FOM) for problem (\ref{eq:motion}). In particular, we rely on a Galerkin-finite element method (FEM) for space approximation, and consider implicit finite difference schemes for time discretization, i.e.,
\begin{align*}
\partial_t\mathbf{u}_h(t^{n}) \approx \frac{\mathbf{u}_h^{n}-\mathbf{u}_h^{n-1}}{\Delta t}, &&
\partial_t^2\mathbf{u}_h(t^{n}) \approx \frac{\mathbf{u}_h^{n}-2\mathbf{u}_h^{n-1}+\mathbf{u}_h^{n-2}}{\Delta t^2},
\end{align*}
which do not require restrictions on $\Delta t$ \cite{quarteroni2013numerical}.
The high-fidelity dimension $N_h$ is determined by the underlying mesh and the chosen FE polynomial order and can be extremely big whenever high accuracy is required for the problem at hand.
To reduce the FOM numerical complexity, we introduce a projection-based reduced-order model (ROM), by relying on the reduced basis (RB) method \cite{quarteroni2016reduced}. The idea of the RB method is to suitably select $N\ll N_h$ vectors of $\mathbb{R}^{N_h}$, forming the so-called RB matrix $\mathbf{V}\in\mathbb{R}^{N_h\times N}$, and to generate a reduced problem by performing a Galerkin projection of the FOM onto the subspace $V_N=\text{Col}(\mathbf{V})\subset\mathbb{R}^{N_h}$ generated by these vectors. This method relies on the assumption that the reduced-order approximation can be expressed as a linear combination of few, problem-dependent, basis functions, that is
\begin{equation*}
\mathbf{Vu}_N^{n}(\bm\mu) \approx \mathbf{u}_h^{n}(\bm\mu),
\end{equation*}
for $n=\,\dots,N_t$, where $\mathbf{u}_N^{n}(\bm\mu)\in\mathbb{R}^N$ denotes the vector of the ROM degrees of freedom at time $t^n\geq0$. The latter is obtained by imposing that the projection of the FOM residual computed on the ROM solution is orthogonal to the trial subspace (in the case of a Galerkin projection): given $\bm\mu\in\mathcal{P}$, at each time $t^{n}$, for $n=1,\dots,N_t$, we seek $\mathbf{u}_N^{n}(\bm\mu)\in\mathbb{R}^{N}$ such that
\begin{equation*}
\mathbf{V}^T\mathbf{R}(\mathbf{Vu}_N^{n}(\bm\mu),t^{n};\bm\mu) = \mathbf{0}.
\end{equation*}
From now on, we will denote the reduced residual $\mathbf{V}^T\mathbf{R}$ and the corresponding Jacobian $\mathbf{V}^T\mathbf{JV}$ as $\mathbf{R}_N$ and $\mathbf{J}_N$, respectively. Then, the associated reduced Newton problem at time $t^{n}$ reads: given $\mathbf{u}_N^{n,(0)}(\bm\mu)$, for $k\geq0$, find $\mathbf{\delta u}_N^{n,(k)}(\bm\mu)\in\mathbb{R}^{N}$ such that
\begin{equation}\label{eq:motionROM} \left\{ \begin{array}{llll}
\mathbf{J}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^{n};\bm\mu)\mathbf{\delta u}_N^{n,(k)}(\bm\mu) = - \mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^{n};\bm\mu), \\
\mathbf{u}_N^{n,(k+1)}(\bm\mu) = \mathbf{u}_N^{n,(k)}(\bm\mu) + \mathbf{\delta u}_N^{n,(k)}(\bm\mu),
\end{array} \right.
\end{equation}
until a suitable stopping criterion is fulfilled.
\subsection{Solution-space reduction: proper orthogonal decomposition}
In this section we provide an overview of the proper orthogonal decomposition (POD) technique used to compute the reduced basis $\mathbf{V}$ through the so-called method of snapshots \cite{POD,benner2015survey}.
Let
\begin{equation*}
\mathcal{M}_{u_h} = \{\mathbf{u}_h^n(\bm\mu)\in\mathbb{R}^{N_h}~\lvert ~\bm\mu\in\mathcal{P}, ~n=1,\dots,N_t \}
\end{equation*}
be the (discrete) solution manifold identified by the image of $\mathbf{u}_h$, that is, the set of all the PDE solutions for $\bm\mu$ varying in the parameter space and $t^n$ in the partition of the time interval. Our goal is to approximate $\mathcal{M}_{u_h}$ with a reduced linear manifold, the {\itshape trial manifold}
\begin{equation*}
\mathcal{M}_{u_N}^{lin} = \{\mathbf{Vu}_N^n(\bm\mu)~\lvert ~\mathbf{u}_N^n(\bm\mu)\in\mathbb{R}^N, ~\bm\mu\in\mathcal{P},~n=1,\dots,N_t \}.
\end{equation*}
To do this, given $n_s<N_h$ sampled instances of $\bm\mu\in\mathcal{P}$, we define the snapshots matrix
\begin{equation*}
\mathbf{S} = \left[ \mathbf{u}_h(t^1;\bm\mu_1)~|~\dots~|~\mathbf{u}_h(t^{N_t};\bm\mu_1)~|~\dots~|~\mathbf{u}_h(t^1;\bm\mu_{n_s})~|~\dots~|~\mathbf{u}_h(t^{N_t};\bm\mu_{n_s}) \right]\in\mathbb{R}^{N_h\times n_s}
\end{equation*}
which contains the FOM solutions $\mathbf{u}_h(t^n;\bm\mu_\ell)$ as its columns. Sampling can be performed, e.g., through a latin hypercube sampling design, as well as through suitable low-discrepancy points sets.
The POD basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ spanning the subspace $V_N$ is obtained by performing the singular value decomposition of $\mathbf{S}$,
\begin{equation*}
\mathbf{S}=\mathbf{U\Sigma Z}^T,
\end{equation*}
and then collecting the first $N$ columns of $\mathbf{U}$, corresponding to the $N$ largest singular values stored in the diagonal matrix $\bm\Sigma = \text{diag}(\sigma_1,\dots,\sigma_{r})\in\mathbb{R}^{n_s\times n_s}$, with $\sigma_1\geq\dots\geq\sigma_{r}\geq0$ and $r\leq N_h\land n_s$ being the rank of $\mathbf{S}$. The columns of the matrices $\mathbf{U}\in\mathbb{R}^{N_h\times n_s}$ and $\mathbf{Z}\in\mathbb{R}^{n_s\times N_h}$ correspond to the left and the right singular vectors of $\mathbf{S}$, respectively. This yields an orthonormal basis that, among all $N$-dimensional orthonormal basis $\mathbf{W}\in\mathcal{V}_N$, minimizes the least square error of snapshot reconstruction
\begin{align*}
&\lVert \mathbf{S} - \mathbf{V}\mathbf{V}^T\mathbf{S} \rVert_2^2 = \underset{\mathbf{W}\in\mathcal{V}_N}{\min} \lVert \mathbf{S} - \mathbf{W}\mathbf{W}^T\mathbf{S} \rVert_2^2 = \sigma_{N+1}^2, \\
&\lVert \mathbf{S} - \mathbf{V}\mathbf{V}^T\mathbf{S} \rVert_F^2 = \underset{\mathbf{W}\in\mathcal{V}_N}{\min} \lVert \mathbf{S} - \mathbf{W}\mathbf{W}^T\mathbf{S} \rVert_F^2 = \sum_{i=N+1}^r \sigma_i^2,
\end{align*}
where $\lVert \cdot \rVert_2$ and $\lVert \cdot \rVert_F$ are the Euclidean norm and the Frobenius norm, respectively. Hence, singular values' decay directly impacts on the size $N$, usually computed as the minimum integer satisfying \vspace{-0.1cm}
\begin{equation}\label{eq:POD-dim}
RIC(N) = \frac{\sum_{\ell=1}^{N}\sigma_i^2}{\sum_{\ell=1}^{r}\sigma_i^2} \geq 1-\varepsilon_{POD}^2 \vspace{-0.1cm}
\end{equation}
for a given tolerance $\varepsilon_{POD}>0$. The POD technique constructs a low-dimensional optimal subspace of $\mathbb{R}^{N_h}$ retaining as much as possible of the snapshots relative information content ($RIC$), provided that a sufficiently rich set of snapshots has been chosen. We summarize the POD technique in Algorithm \ref{alg:POD}.
\begin{algorithm}\caption{Proper orthogonal decomposition (POD)}\label{alg:POD}
INPUT: $\mathbf{S}\in\mathbb{R}^{N_h\times n_s}$\\
OUTPUT: $\mathbf{V}\in\mathbb{R}^{N_h\times N}$\\ \vspace{-1.1em}
\begin{algorithmic}[1]
\STATE Perform SVD of $\mathbf{S}$, i.e., $\mathbf{S=U\Sigma Z}^T$
\STATE Select basis dimension $N$ as the minimum integer fulfilling condition (\ref{eq:POD-dim})
\STATE Construct $\mathbf{V}$ collecting the first $N$ columns of $\mathbf{U}$
\end{algorithmic}
\end{algorithm}
For the sake of efficiency, in our work we rely on a non-deterministic version of POD, exploiting the so-called randomized-SVD, see Algorithm \ref{alg:randSVD}. Randomization offers, in fact, a powerful tool for performing low-rank matrix approximation, especially when dealing with massive datasets. The randomized approach usually beats its classical competitors in terms of computational speed-up, accuracy and robustness \cite{halko2011finding}.
The key idea of randomized SVD is to split the task of computing an approximated singular value decomposition of a given matrix into a first random stage, and a second deterministic one. The former exploits random sampling to construct a low-dimensional subspace that captures most of the action of the input matrix; the latter is meant to restrict the given matrix to this subspace and then manipulate the associated reduced matrix with classical deterministic algorithms, to obtain the desired low-rank approximations. This randomized approach is convenient when the snapshots matrix is high-dimensional, i.e. when $N_h$ and $n_s$ are large. In fact, finding the first $k$ dominant singular-values for a dense input matrix of dimension $N_h\times n_s$, requires $O(N_hn_s\log(k))$ floating-point operations for a randomized algorithm, in contrast with $O(N_hn_sk)$ flops for a classical one.
\begin{algorithm}\caption{Randomized-SVD}\label{alg:randSVD}
INPUT: $\mathbf{S\in\mathbb{R}}^{m\times n}$, target rank $k\in\mathbb{N}$\\
OUTPUT: $\mathbf{U\Sigma Z}^T\approx\mathbf{S}$\\ \textbf{\textit{stage 1}}
\begin{algorithmic}[1]
\STATE Generate a Gaussian matrix $\bm\Theta\in\mathbb{R}^{N_h\times k}$
\STATE Compute $\mathbf{Q}\in\mathbb{R}^{N_h\times k}$ whose columns form an orthonormal basis for the range of $\mathbf{S}\bm\Theta$ and such that
\begin{equation*}
\lVert \mathbf{S} - \mathbf{QQ}^T\mathbf{S}\rVert_2 \leq \min_{rank(\mathbf{X})\leq k} \lVert \mathbf{S} - \mathbf{X}\rVert_2,
\end{equation*}
e.g., using the QR factorization.
\end{algorithmic}
\textbf{\textit{stage 2}}
\begin{algorithmic}[1]
\STATE Form $\widetilde{\mathbf{S}}=\mathbf{Q}^T\mathbf{S}\in\mathbb{R}^{k\times n}$
\STATE Compute SVD of $\widetilde{\mathbf{S}}=\widetilde{\mathbf{U}}\bm\Sigma \mathbf{Z}^T$
\STATE Set $\mathbf{U}=\mathbf{Q}\widetilde{\mathbf{U}}$
\end{algorithmic}
\end{algorithm}
\begin{remark}
Note that Algorithm~\ref{alg:randSVD} can be also adapted to solve the following problem: given a target error tolerance $\varepsilon>0$, find $k=k(\varepsilon)$ and $\mathbf{Q}\in\mathbb{R}^{N_h\times k}$ satisfying
$\lVert \mathbf{S} - \mathbf{QQ}^T\mathbf{S}\rVert_2 \leq \varepsilon$.
\end{remark}
\subsection{Hyper-reduction: the discrete empirical interpolation method}\label{sec:DEIM}
In the case of parametrized PDEs featuring nonaffine dependence on the parameter and/or nonlinear (high-order polynomial or nonpolynomial) dependence on the field variable, a further level of reduction, known as {\itshape hyper-reduction}, must be introduced \cite{grepl2007efficient,negri2015efficient}. Note that if nonlinearities only include quadratic (or, at most, cubic) terms and do not feature any parameter dependence, assembling of nonlinear terms in the ROM can be performed by projection of the corresponding FOM quantities, once and for all \cite{gobat2021reduced}.
For the case at hand, the residual $\mathbf{R}_N$ and the Jacobian $\mathbf{J}_N$ appearing in the reduced Newton system (\ref{eq:motionROM}) depend on the solution at the previous iteration and, therefore, must be computed at each step $k\geq0$. It follows that, for any new instance of the parameter $\bm\mu$, we need to assemble the high-dimensional FOM-arrays before projecting them onto the reduced subspace, entailing a computational complexity which is still of order $N_h$. To setup an efficient offline-online computational splitting, an approximation of the nonlinear operators that is independent of the high-fidelity dimension is required.
Several techniques have been employed to provide this further level of approximation \cite{barrault2004empirical, chaturantabut2010nonlinear, astrid2008missing, carlberg2011efficient, farhat2015structure}; among these, DEIM has been successfully applied to stationary or quasi-static nonlinear mechanical problems \cite{bonomi2017reduced,ghavamian2017pod}. Its key idea is to replace the nonlinear arrays in (\ref{eq:motionROM}) with a collateral reduced basis expansion, computed through an inexpensive interpolation procedure. In this framework, the high-dimensional residual $\mathbf{R}(\bm\mu)$ is projected onto a reduced subspace of dimension $m<N_h$ spanned by a basis $\bm\Phi_\mathcal{R}\in\mathbb{R}^{N_h\times m}$ \vspace{-0.05cm}
\begin{equation*}
\mathbf{R}(\bm\mu) \approx \bm\Phi_\mathcal{R}\mathbf{r}(\bm\mu), \vspace{-0.05cm}
\end{equation*}
where $\mathbf{r}(\bm\mu)\in\mathbb{R}^m$ is the vector of the unknown amplitudes. The matrix $\bm\Phi_\mathcal{R}$ can be precomputed offline by performing POD on a set of high-fidelity residuals collected when solving (\ref{eq:motionROM}) for $n_s'$ training input parameters \vspace{-0.05cm}
\begin{equation*}
\mathbf{S}_{\bm\rho} =\left[\mathbf{R}(\mathbf{Vu}_N^{n,(k)}(\bm\mu_\ell),t^{n};\bm\mu_\ell)), k\geq0\right]_{n=1,\dots,N_t}^{\ell=1,\dots,n_s'}. \vspace{-0.05cm}
\end{equation*}
The unknown parameter-dependent coefficient $\mathbf{r}(\bm\mu)$ is obtained online by collocating the approximation at the $m$ components selected by a greedy procedure, that is \vspace{-0.05cm}
\begin{equation*}
\mathbf{P}^T\mathbf{R}(\bm\mu) \approx \mathbf{P}^T\bm\Phi_\mathcal{R}\mathbf{r}(\bm\mu) \implies \mathbf{r}(\bm\mu) = (\mathbf{P}^T\bm\Phi_\mathcal{R})^{-1}\mathbf{P}^T\mathbf{R}(\bm\mu), \vspace{-0.05cm}
\end{equation*}
where $\mathbf{P}\in\mathbb{R}^{N_h\times m}$ is the boolean matrix associated with the interpolation constraints. We thus define the hyper-reduced residual vector as \vspace{-0.05cm}
\begin{equation*}
\mathbf{R}_{N,m}(\bm\mu) := \mathbf{V}^T\bm\Phi_\mathcal{R}(\mathbf{P}^T\bm\Phi_\mathcal{R})^{-1}\mathbf{P}^T\mathbf{R}(\bm\mu) \approx \mathbf{V}^T\mathbf{R}(\bm\mu). \vspace{-0.05cm}
\end{equation*}
To avoid confusion, we recall that $\mathbf{R}_N = \mathbf{V}^T\mathbf{R}$, so that $\mathbf{R}_{N,m}\approx\mathbf{R}_N$. Finally, the associated Jacobian approximation $\mathbf{J}_{N,m}(\bm\mu)$ can be computed as the derivative of $\mathbf{R}_{N,m}(\bm\mu)$ with respect to the reduced displacement, obtaining \vspace{-0.05cm}
\begin{align*}
\mathbf{J}_{N,m}(\bm\mu) = \mathbf{V}^T{\bm\Phi_\mathcal{R}}(\mathbf{P}^T{\bm\Phi_\mathcal{R}})^{-1} \mathbf{P}^T\mathbf{J}(\bm\mu)\mathbf{V}, \vspace{-0.05cm}
\end{align*}
or by relying on the so-called matrix DEIM (MDEIM) algorithm \cite{negri2015efficient}, as done in \cite{bonomi2017reduced,manzoni2018reduced}.
However, the application of DEIM in this setting can be rather inefficient, especially when turning to complex problem which require a high number of residual basis, thus interpolation points, to ensure the convergence of the hyper-reduced Newton system
\begin{equation*}
\left\{ \begin{array}{llll}
\mathbf{J}_{N,m}(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^{n};\bm\mu)\mathbf{\delta u}_N^{n,(k)}(\bm\mu) = - \mathbf{R}_{N,m}(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^{n};\bm\mu) \\
\mathbf{u}_N^{n,(k+1)}(\bm\mu) = \mathbf{u}_N^{n,(k)}(\bm\mu) + \mathbf{\delta u}_N^{n,(k)}(\bm\mu).
\end{array} \right.
\end{equation*}
In fact, the $m$ points selected by the DEIM algorithm correspond to a subset of nodes of the computational mesh, which, together with the neighboring nodes (i.e. those sharing the same cell), form the so-called \textit{reduced mesh}, see, e.g., the sketch reported in Figure~\ref{fig:redMesh}. Since the entries of any FE-vector are associated with the degrees of freedom (dofs) of the problem, $\mathbf{P}^T\mathbf{R}(\bm\mu)$ is computed by integrating the residual only on the quadrature points belonging to the reduced mesh, which, nevertheless, can be rather dense.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.475]{redMesh.png}
\caption{Sketch of a reduced mesh for an hexahedral computational grid in a two-dimensional case. The red dots represent the points selected by the DEIM algorithm and, together with the blue ones, represent the vertices of the elements (light blue) of the reduced mesh.}
\label{fig:redMesh}
\end{figure}
A modification of the DEIM algorithm, the so-called unassembled DEIM (UDEIM), has been proposed in \cite{tiso2013discrete} to exploit the sparsity of the problem and minimize the number of element function calls. However, a high number of nonlinear function evaluations is still required when the number of magic points is sufficiently big. Indeed, DEIM-based affine approximations are effective, in terms of computational costs, provided that few entries of the nonlinear terms can be cheaply computed; however, this situation does not occur neither for dynamical systems arising from the linearization of a nonlinear system around a steady state, nor when dealing with global nonpolynomial nonlinearities.
In this paper, we propose an alternative technique to perform hyper-reduction, which is independent of the underlying mesh and relies on a deep neural network architecture to approximate reduced residual vectors and Jacobian matrices. The introduction of a surrogate model to perform operator approximation is justified by the fact that, often, most of the CPU time needed online for each new parameter instance is required by DEIM for assembling arrays such as residual vectors or corresponding Jacobian matrices on the reduced mesh.
\section{Operator approximation: a deep learning-based technique (Deep-HyROMnet)}\label{sec:Deep-HyROMnet}
To recover the offline-online efficiency of the RB method, overcoming the need to assemble the nonlinear arrays onto the computational mesh as in the case of the DEIM, we present a novel projection-based method which relies on DNNs for the approximation of the nonlinear terms. We refer to this strategy as to a \textit{hyper-reduced order model enhanced by deep neural networks} (Deep-HyROMnet). Our goal is the efficient numerical approximation of the whole sets
\begin{align*}
\mathcal{M}_{R_N} &= \{\mathbf{R}_N(\mathbf{Vu}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)\in\mathbb{R}^{N} ~\lvert ~n=1,\dots,N_t, ~k\geq0, ~\bm\mu\in\mathcal{P} \}, \medskip \\
\mathcal{M}_{J_N} &= \{\mathbf{J}_N(\mathbf{Vu}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)\in\mathbb{R}^{N\times N} ~\lvert ~n=1,\dots,N_t, ~k\geq0, ~\bm\mu\in\mathcal{P} \},
\end{align*}
which we refer to as the \textit{reduced residual manifold} and \textit{reduced Jacobian manifold}, respectively, in a way that depends only on the ROM dimension $N$ and on the number of parameters $P$. To achieve this task, we employ the DNN architecture developed in \cite{fresca2021comprehensive} for the DL-ROM techniques. It is worthy to note that, except for the approximation error of the reduced nonlinear operators, the proposed Deep-HyROMnet approach is a physics-based method and that the computed solution satisfies the nonlinear equation of the problem under investigation, up to a further approximation of ROM residual and Jacobian arrays -- thus, similarly to what happened for a POD-Galerkin-DEIM ROM. The main idea of the deep learning-based operator approximation approach that replaces the DEIM in our new Deep-HyROMnet strategy, is to learn the following input-to-residual and input-to-Jacobian maps, respectively:
\begin{align*}
\bm\rho_N\colon&(\bm\mu,t^n,k)\longmapsto {\bm\rho}_N(\bm\mu,t^n,k) \approx \mathbf{R}_N(\mathbf{Vu}_N^{n,(k)}(\bm\mu),t^n;\bm\mu),\\
\bm\iota_N\colon&(\bm\mu,t^n,k)\longmapsto {\bm\iota}_N(\bm\mu,t^n,k) \approx \mathbf{J}_N(\mathbf{Vu}_N^{n,(k)}(\bm\mu),t^n;\bm\mu),
\end{align*}
provided $(\bm\mu,t^n,k)\in\mathcal{P}\times\{t^1,\dots,t^{N_t}\}\times\mathbb{N}^+$, and to finally replace the linear system in (\ref{eq:motionROM}) with
\begin{equation*}
{\bm\iota}_N(\bm\mu,t^n,k)\mathbf{\delta u}^{n,(k)}(\bm\mu) = - {\bm\rho}_N(\bm\mu,t^n,k).
\end{equation*}
Hence, Deep-HyROMnet aims at approximating the residual vector and the Jacobian matrix obtained after their projection onto the reduced space of dimension $N\ll N_h$. Indeed, performing POD-Galerkin on the solution space allows to severely reduce the problem dimension from $N_h$ to $N$ and, hence, to ease the learning task with respect the reconstruction of the full-order $\mathbf{R}$ and $\mathbf{J}$.
\begin{remark}
As an alternative to Newton iterative scheme, we can rely on Broyden's method \cite{Broyden}, which belongs to the class of quasi-Newton methods. This allows to avoid the computation of the Jacobian matrix at each iteration $k\geq0$ by relying on rank-one updates, based on residuals computed at previous iterations. However, we are able to compute
Jacobian matrices
very efficiently using automatic differentiation (AD), so that the computational burden is the assembling of residual vectors. For this reason, in this paper we will focus on the Newton method only, that is, the solution of problem (\ref{eq:motionROM}).
\end{remark}
To summarize, in the case of the Newton approach, we end up with the following reduced problem: given $\bm\mu\in\mathcal{P}$ and, for $n=1,\dots,N_t$, the initial guess $\mathbf{u}_N^{n,(0)}(\bm\mu) = \mathbf{u}_N^{n-1}(\bm\mu)$, find $\delta\mathbf{u}_N^{n,(k)}\in\mathbb{R}^N$ such that, for $k\geq0$,
\begin{equation}\label{eq:motionDeep} \left\{ \begin{array}{llll}
\bm\iota_N(\bm\mu,t^{n},k) \delta\mathbf{u}_N^{n,(k)}(\bm\mu) = - \bm\rho_N(\bm\mu,t^{n},k), \\
\mathbf{u}_N^{n,(k+1)}(\bm\mu) = \mathbf{u}_N^{n,(k)}(\bm\mu) + \delta\mathbf{u}_N^{n,(k)}(\bm\mu),
\end{array} \right.
\end{equation}
until $\lVert\bm\rho_N(\bm\mu,t^n,k)\rVert_2 / \lVert\bm\rho_N(\bm\mu,t^n,0)\rVert_2 < \varepsilon$, where $\varepsilon>0$ is a given tolerance. In Algorithms~\ref{alg:Deep-HyROMnet_offline} and \ref{alg:Deep-HyROMnet_online}, we report a summary of the offline and online stages of Deep-HyROMnet, respectively.
\begin{algorithm}[H]
\caption{Deep-HyROMnet for nonlinear time-dependent problems. Offline stage.}
\label{alg:Deep-HyROMnet_offline}
INPUT: $\bm\mu_\ell$, for $\ell=1,\dots,n_s$, and $\bm\mu_{\ell'}$, for $\ell'=1,\dots,n_s'$\\
OUTPUT: $\mathbf{V}\in\mathbb{R}^{N_h\times N}$
\begin{algorithmic}[1]
\STATE \textbf{for} $\ell=1,\dots,n_s$ \textbf{do}
\STATE $\quad$ \textbf{for} $n=1,\dots,N_t$ \textbf{do}
\STATE $\quad\quad$ \textbf{for} $k\geq0$ \textbf{until convergence do}
\STATE $\quad\quad\quad$ Assemble and solve problem (\ref{eq:motionFOM})
\STATE $\quad\quad\quad$ Collect $\mathbf{S}_u \leftarrow\mathbf{S}_u\cup \left[\mathbf{u}_h^{n,(k)}(\bm\mu_\ell)\right]$ column-wise
\STATE Construct $\mathbf{V}=\text{POD}(\mathbf{S}_u, \varepsilon_{POD})$ (see Algorithm~\ref{alg:POD})
\STATE \textbf{for} $\ell'=1,\dots,n_s'$\textbf{ do}
\STATE $\quad$ \textbf{for} $n=1,\dots,N_t$ \textbf{do}
\STATE $\quad\quad$ \textbf{for} $k\geq0$ \textbf{until convergence do}
\STATE $\quad\quad\quad$ Assemble and solve reduced problem (\ref{eq:motionROM})
\STATE $\quad\quad\quad$ Collect $\mathbf{S}_{\bm\rho} \leftarrow\mathbf{S}_{\bm\rho}\cup \left[\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu_{\ell'}),t^{n};\bm\mu_{\ell'})\right]$ column-wise
\STATE $\quad\quad\quad$ Collect $\mathbf{S}_{\bm\iota} \leftarrow\mathbf{S}_{\bm\iota}\cup \left[\mathbf{J}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu_{\ell'}),t^{n};\bm\mu_{\ell'})\right]$ column-wise
\STATE Train the DNNs (see Algorithm~\ref{alg:DLROM1})
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[H]
\caption{Deep-HyROMnet for nonlinear time-dependent problems. Online stage.}
\label{alg:Deep-HyROMnet_online}
INPUT: $\bm\mu\in\mathcal{P}$\\
OUTPUT: $\mathbf{Vu}_N^n(\bm\mu)\in\mathbb{R}^{N_h}$, for $n=1,\dots,N_t$\\ \vspace{-1.1em}
\begin{algorithmic}[1]
\STATE \textbf{for} $n=0,\dots,N_t-1$ \textbf{do}
\STATE $\quad$ \textbf{for }$k\geq0$ \textbf{until convergence do}
\STATE $\quad\quad$ Compute $\bm\rho_N(\bm\mu,t^n,k)$ and $\bm\iota_N(\bm\mu,t^n,k)$ (see Algorithm~\ref{alg:DLROM2})
\STATE $\quad\quad$ Solve hyper-reduced problem (\ref{eq:motionDeep})
\STATE Recover $\mathbf{Vu}_N^n(\bm\mu)$, for $n=1,\dots,N_t$
\end{algorithmic}
\end{algorithm}
\subsection{DL-ROM-based neural network}\label{sec:DLROM}
For the sake of generality, we will focus on the DNN-based approximation of the reduced residual vector only, that is
\begin{equation*}
{\bm\rho}_N(\bm\mu,t^n,k) \approx\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)\in\mathbb{R}^N.
\end{equation*}
In fact, by relying on a suitable transformation, we can easily write the Jacobian matrix as a vector of dimension $N^2$ and apply the same procedure described in the following for the residual vector also in the case of the Jacobian matrix. In particular, we define the transformation
\begin{equation*}
vec\colon\mathbb{R}^{N\times N}\rightarrow\mathbb{R}^{N^2}, \quad vec(\mathbf{J}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)) = \mathbf{j}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu),
\end{equation*}
which consists in stacking the columns of $\mathbf{J}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)$ in a vector on which is then applied the DL-ROM technique, thus obtaining
\begin{equation*}
\widetilde{\bm\iota}_N(\bm\mu,t^n,k)\approx \mathbf{j}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)\in\mathbb{R}^{N^2}.
\end{equation*}
Finally, we revert the $vec$ operation, so that ${\bm\iota}_N(\bm\mu,t^n,k) = vec^{-1}(\widetilde{\bm\iota}_N(\bm\mu,t^n,k))$.
We thus aim at efficiently approximating the whole set $\mathcal{M}_{R_N}$
by means of the reduced residual trial manifold, defined as
\begin{equation*}
\mathcal{M}_{\rho_N} = \{ {\bm\rho}_N(\bm\mu,t^n,k) ~|~\bm\mu\in\mathcal{P}, ~n=1,\dots,N_t, ~k\geq0\}\subset\mathbb{R}^{N}.
\end{equation*}
The DL-ROM approximation of the ROM residual $\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)$ takes the form
\begin{equation*}
\bm\rho_N(\bm\mu,t^n,k) = \widetilde{\mathbf{R}}_N(\bm\mu,t^n,k;\bm\theta_{DF},\bm\theta_{D}) = \mathbf{f}^D_N(\bm\phi_q^{DF}(\bm\mu,t^n,k;\bm\theta_{DF});\bm\theta_{D})
\end{equation*}
where
\begin{itemize}
\item $\bm\phi_q^{DF}(\cdot~;\bm\theta_{DF})\colon\mathbb{R}^{P+2}\rightarrow\mathbb{R}^q$ such that
\begin{equation*}
\bm\phi_q^{DF}(\bm\mu,t^n,k;\bm\theta_{DF}) = \mathbf{R}_q(\bm\mu,t^n,k;\bm\theta_{DF})
\end{equation*} is a deep feedforward neural network (DFNN), consisting in the subsequent composition of a nonlinear activation function, applied to a linear transformation of the input, multiple times. Here, $\bm\theta_{DF}$ denotes the vector of parameters of the DFNN, collecting all the corresponding weights and biases of each layer and $q$ is as close as possible to the input size $P+2$;
\item $\mathbf{f}^D_N(\cdot~;\bm\theta_{D})\colon\mathbb{R}^q\rightarrow\mathbb{R}^N$ such that
\begin{equation*}
\mathbf{f}^D_N(\mathbf{R}_q(\bm\mu,t^n,k;\bm\theta_{DF});\bm\theta_{D}) = \widetilde{\mathbf{R}}_N(\bm\mu,t^n,k;\bm\theta_{DF},\bm\theta_{D})
\end{equation*}
is the decoder function of a convolutional autoencoder (CAE), obtained as the composition of several layers (some of which are convolutional), depending upon a vector $\bm\theta_{D}$ collecting all the corresponding weights and biases.
\end{itemize}
The encoder function of the CAE is exploited, during the training stage only, to map the reduced residual $\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu)$ associated to $(\bm\mu,t^n,k)$ onto a low-dimensional representation
\begin{equation*}
\mathbf{f}^E_q(\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu);\bm\theta_{E}) = \widetilde{\mathbf{R}}_q(\bm\mu,t^n,k;\bm\theta_{E}),
\end{equation*}
where $\mathbf{f}^E_q(\cdot~;\bm\theta_{E})\colon\mathbb{R}^N\rightarrow\mathbb{R}^q$ denotes the encoder function, depending upon a vector $\bm\theta_{E}$ of parameters.
\begin{remark}\label{rmk:zero-padded}
We point out that the input of the encoder function, that is, the reduced residual vector $ \mathbf{R}_N$, is reshaped into a square matrix by rewriting its elements in row-major order, thus obtaining $\mathbf{R}_N^{reshape}\in\mathbb{R}^{\sqrt{N}\times \sqrt{N}}$. If $N$ is not a square, the input $\mathbf{R}_N$ is zero-padded as explained in \cite{Goodfellow-et-al-2016}, and the additional elements are subsequently discarded.
\end{remark}
Regarding the prediction of the reduced residual for new unseen instances of the inputs, given $\bm\mu_{test}\in\mathcal{P}$, computing the DL-ROM approximation of $\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu_{test})$, for any possible $n=1,\dots,N_t$ and $k\geq0$, corresponds to the testing stage of a DFNN and of the decoder function of a convolutional AE; thus, at testing time, we discard the encoder function. The architecture used during the training stage is reported in Figure~\ref{fig:DNN}, whereas, during the testing phase, the encoder function is discarded.
\begin{figure}
\centering
\includegraphics[width=0.95\textwidth]{architecture.png}
\caption{DNN architecture used during the training phase for the reduced residual vector.}
\label{fig:DNN}
\end{figure}
Let us define the reduced residual snapshots matrix $\mathbf{S}_{\bm\rho}\in\mathbb{R}^{N\times N_{train}}$, with $N_{train}=n_s'N_tN_k$, as
\begin{equation*}
\mathbf{S}_{\bm\rho} = \left[\mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu),t^n;\bm\mu_\ell)\right]_{\ell=1,\dots,n_s',n=1,\dots,N_t,k\geq0},
\end{equation*}
that is, the matrix collecting column-wise ROM residuals computed for $n_s'$ sampled parameters $\bm\mu_\ell\in\mathcal{P}$, at different time instances $t^1,\dots,t^{N_t}$ and for each Newton iteration $k\geq0$, and the parameter matrix $\mathbf{M}\in\mathbb{R}^{(P+2)\times N_{train}}$ of the corresponding triples
\begin{equation*}
\mathbf{M} = \left[\left(\bm\mu_\ell,t^n,k\right)\right]_{\ell=1,\dots,n_s',n=1,\dots,N_t,k\geq0}.
\end{equation*}
The training stage consists in solving the following optimization problem in the weights variable $\bm\theta = (\bm\theta_{E},\bm\theta_{DF},\bm\theta_{D})$:
\begin{equation*}
\mathcal{J}(\bm\theta) = \dfrac{1}{N_{train}}\sum_{\ell=1}^{n_s'}\sum_{n=1}^{N_t}\sum_{k=0}^{N_k}\mathcal{L}(\bm\mu_\ell,t^n,k;\bm\theta) \rightarrow \underset{\bm\theta}{\min}
\end{equation*}
where
\begin{equation}\label{eq:loss}
\begin{aligned}
\mathcal{L}(\bm\mu_\ell,t^n,k;\bm\theta) = &\dfrac{\omega_h}{2} \lVert \mathbf{R}_N(\mathbf{V}\mathbf{u}_N^{n,(k)}(\bm\mu_\ell),t^n;\bm\mu_\ell) - \widetilde{\mathbf{R}}_N(\bm\mu_\ell,t^n,k;\bm\theta_{DF},\bm\theta_{D}) \rVert^2 \\
& + \dfrac{1-\omega_h}{2} \lVert \widetilde{\mathbf{R}}_q(\bm\mu_\ell,t^n,k;\bm\theta_{E}) - \mathbf{R}_q(\bm\mu_\ell,t^n,k;\bm\theta_{DF}) \rVert^2,
\end{aligned}
\end{equation}
with $\omega_h\in[0,1]$. The loss function (\ref{eq:loss}) combines the reconstruction error, i.e. the error between the ROM residual and the DL-ROM approximation, and the error between the intrinsic coordinates and the output of the encoder. The training stage of the DNN involved in Deep-HyROMnet is detailed in Algorithm~\ref{alg:DLROM1}; in particular, we denote by $\alpha$ the training-validation splitting fraction, by $\eta$ the starting learning rate, by $N_b$ the batch size, by $n_b = (1-\alpha)N_{train}/N_b$ the number of minibatches and by $N_e$ the maximum number of epochs. The testing stage of the DNN is detailed in Algorithm~\ref{alg:DLROM2}. See, e.g., \cite{fresca2021comprehensive, fresca2021pod} for further details.
\begin{algorithm}
\caption{Training stage for the DNN, based on Algorithm~1 of \cite{fresca2021comprehensive}}
\label{alg:DLROM1}
INPUT: $\mathbf{M}\in\mathbb{R}^{(P+2)\times N_{train}}$, $\mathbf{S}_{\bm\rho}\in\mathbb{R}^{N\times N_{train}}$, $\alpha$, $\eta$, $N_b$, $n_b$, $N_e$, early-stopping criterion \\
OUTPUT: $\bm\theta^* = (\bm\theta_E^*,\bm\theta_{DF}^*,\bm\theta_D^*)$ (optimal)
\begin{algorithmic}[1]
\STATE Randomly shuffle $\mathbf{M}$ and $\mathbf{S}$
\STATE Split data in $\mathbf{M} = \left[\mathbf{M}^{train},\mathbf{M}^{val}\right]$ and $\mathbf{S}_{\bm\rho} = \left[\mathbf{S}_{\bm\rho}^{train},\mathbf{S}_{\bm\rho}^{val}\right]$ (according to $\alpha$)
\STATE Normalize $\mathbf{M}$ and $\mathbf{S}$ according to (\ref{eq:standardization})
\STATE Randomly initialize $\bm\theta^0 = (\bm\theta_E^0,\bm\theta_{DF}^0,\bm\theta_D^0)$
\STATE $n_e = 0$
\STATE \textbf{while} $\lnot$early-stopping \textbf{and} $n_e\leq N_e$ \textbf{do}
\STATE $\quad$ \textbf{for} $k=1,\dots,n_b$ \textbf{do}
\STATE $\quad\quad$ Sample a minibatch $(\mathbf{M}^{batch},\mathbf{S}^{batch})\subset(\mathbf{M}^{train},\mathbf{S}^{train})$
\STATE\label{state:start} $\quad\quad$ $\mathbf{S}^{batch} = reshape(\mathbf{S}^{batch})$
\STATE $\quad\quad$ $\widetilde{\mathbf{S}}^{batch}_q(\bm\theta_E^{n_bn_e+k}) = \mathbf{f}^E_q(\mathbf{S}^{batch};\bm\theta_E^{n_bn_e+k})$
\STATE $\quad\quad$ $\mathbf{S}^{batch}_q(\bm\theta_{DF}^{n_bn_e+k}) = \bm\phi^{DF}_q(\mathbf{M}^{batch};\bm\theta_{DF}^{n_bn_e+k})$
\STATE $\quad\quad$ $\widetilde{\mathbf{S}}^{batch}_N(\bm\theta_{DF}^{n_bn_e+k},\bm\theta_D^{n_bn_e+k}) = \mathbf{f}^D_N(\mathbf{S}^{batch}_q(\bm\theta_{DF}^{n_bn_e+k});\bm\theta_D^{n_bn_e+k})$
\STATE\label{state:end} $\quad\quad$ $\widetilde{\mathbf{S}}^{batch}_N = reshape(\widetilde{\mathbf{S}}^{batch}_N)$
\STATE $\quad\quad$ Accumulate loss (\ref{eq:loss}) on $(\mathbf{M}^{batch},\mathbf{S}^{batch})$ and compute $\hat{\nabla}_{\bm\theta}\mathcal{J}$
\STATE $\quad\quad$ $\bm\theta^{n_bn_e+k+1} = \text{ADAM}(\eta,\hat{\nabla}_{\bm\theta}\mathcal{J},\bm\theta^{n_bn_e+k})$
\STATE $\quad$ Repeat instructions \ref{state:start}-\ref{state:end} on $(\mathbf{M}^{val},\mathbf{S}^{val})$ to evaluate early-stopping criterion
\STATE $\quad$ $n_e = n_e+1$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Testing stage for the DNN, based on Algorithm~2 of \cite{fresca2021comprehensive}}
\label{alg:DLROM2}
INPUT: $(\bm\mu,t^n,k)\in\mathcal{P}\times\{t^1,\dots,t^{N_t}\}\times\mathbb{N}^+$, $(\bm\theta_{DF}^*,\bm\theta_D^*)$ (optimal)\\
OUTPUT: $\widetilde{\mathbf{S}}_N$ (i.e. $\bm\rho_N(\bm\mu,t^n,k)$ or $\bm\iota_N(\bm\mu,t^n,k)$)
\begin{algorithmic}[1]
\STATE $\mathbf{S}_q(\bm\theta_{DF}^*) = \bm\phi^{DF}_q(\bm\mu,t^n,k;\bm\theta_{DF}^*)$
\STATE $\widetilde{\mathbf{S}}_N(\bm\theta_{DF}^*,\bm\theta_D^*) = \mathbf{f}^D_N(\mathbf{S}_q(\bm\theta_{DF}^*);\bm\theta_D^*)$
\STATE $\widetilde{\mathbf{S}}_N = reshape(\widetilde{\mathbf{S}}_N)$
\end{algorithmic}
\end{algorithm}
\begin{remark}
Differently from the scaling techniques used in \cite{fresca2021comprehensive,fresca2021pod}, which are based on a min-max procedure, we standardize the input and output of the DNN as follows. After splitting the data into training and validation sets according to a user-defined training-validation splitting fraction, $\mathbf{M} = \left[\mathbf{M}^{train},\mathbf{M}^{val}\right]$ and $\mathbf{S}_{\bm\rho} = \left[\mathbf{S}_{\bm\rho}^{train},\mathbf{S}_{\bm\rho}^{val}\right]$, we define for each row of the training set the corresponding mean and standard deviation
\begin{equation*}
M_{mean}^i = \dfrac{1}{N_{train}} \sum_{j=1}^{N_{train}} M_{ij}^{train}, \quad M_{sd}^i = \sqrt{\dfrac{1}{N_{train}-1} \sum_{j=1}^{N_{train}} (M_{ij}^{train}-M_{mean}^i)^2},
\end{equation*}
so that parameters are normalized by applying the following transformation
\begin{equation}\label{eq:standardization}
M_{ij}^{train} \mapsto \dfrac{M_{ij}^{train}-M_{mean}^i}{M_{sd}^i}, \quad i=1,\dots,P+2, \quad j=1,\dots,N_{train}
\end{equation}
that is, each feature of the training parameter matrix is standardized. The same procedure is applied to the training snapshots matrix $\mathbf{S}_{\bm\rho}^{train}$ by replacing $M^i_{*}$ with $S^i_{*}$, where $*\in\{mean,sd\}$ respectively. Transformation (\ref{eq:standardization}) is applied to the validation and testing sets as well, but considering the mean and the standard deviation computed over the training set. In order to rescale the reconstructed solution to the original values, we apply the inverse transformation.
\end{remark}
\section{Numerical results}\label{sec:tests}
In this Section, we investigate the performances of Deep-HyROMnet on different applications related to the parametrized nonlinear time-dependent PDE problems, focusing on structural mechanics. In particular, we consider {\em (i)} a series of structural tests on a rectangular beam, with different loading conditions and a simple nonlinear constitutive law, and then {\em (ii)} a test case on an idealized left ventricle geometry, simulating cardiac contraction. In the following subsection we formulate both these problems in the framework of nonlinear elastodynamics.
\subsection{Nonlinear elastodynamics} \label{sec:nonlinear_elastodynamics}
Let us consider a continuum body $\mathcal{B}$ embedded in a three-dimensional Euclidean space at a given time $t>0$. Let $\Omega_0$ be the reference configuration, which we assume to coincide with the initial configuration, and be $\mathbf{X}\in\Omega_0$ a generic point. The motion of the body $\mathcal{B}$ is given by \vspace{-0.1cm}
\begin{equation*}
\chi(\mathbf{X},t;\bm\mu) = \mathbf{x} \quad \forall t>0, \vspace{-0.1cm}
\end{equation*}
which maps the material position $\mathbf{X}$ in the reference configuration $\Omega_0$ to the spatial position $\mathbf{x}$ in the deformed or current configuration $\Omega_t$ for all times $t>0$. A motion $\chi$ of a body $\mathcal{B}$ will change the body's shape, position and/or orientation. For a given parameter vector $\bm\mu\in\mathcal{P}$, the displacement vector field \vspace{-0.1cm}
\begin{equation*}
\mathbf{u}(\mathbf{X},t;\bm\mu) = \chi(\mathbf{X},t;\bm\mu) - \mathbf{X} \vspace{-0.1cm}
\end{equation*}
relates the position $\mathbf{X}$ of a particle in the reference configuration to its position $\mathbf{x}$ in the current configuration at time $t>0$. A crucial quantity in nonlinear mechanics is the deformation gradient \vspace{-0.1cm}
\begin{equation*}
\mathbf{F}(\mathbf{X},t;\bm\mu) = \frac{\partial \chi(\mathbf{X},t;\bm\mu)}{\partial \mathbf{X}} = \mathbf{I} + \nabla_0 \mathbf{u}(\mathbf{X},t;\bm\mu), \vspace{-0.1cm}
\end{equation*}
which characterizes changes of material elements during motion. The change in volume between the reference and the current configurations at time $t>0$ is given by $J(\mathbf{X},t;\bm\mu) = \det\mathbf{F}(\mathbf{X},t;\bm\mu)>0$. Common measures of strain are the right Cauchy-Green strain tensor and the Green-Lagrange strain tensor, that are defined as \vspace{-0.1cm}
\begin{equation}\label{eq:strain}
\mathbf{C} = \mathbf{F}^T\mathbf{F}, \qquad \mathbf{E} = \frac{1}{2}(\mathbf{C}-\mathbf{I}), \vspace{-0.1cm}
\end{equation}
respectively. The equation of motion for a continuous medium is given by the conservation of mass and the balance of the linear momentum,
in material coordinates, reads as follows:
\begin{equation*}
\rho_0\partial_t^2\mathbf{u}(\mathbf{X},t;\bm\mu) - \nabla_0\cdot\mathbf{P}(\mathbf{F}(\mathbf{X},t;\bm\mu)) = \mathbf{b}_0(\mathbf{X},t;\bm\mu), \qquad \mathbf{X}\in\Omega_0,~t>0
\end{equation*}
where $\rho_0$ is the density of the body, $\mathbf{P(F)}$ is the first Piola-Kirchhoff stress tensor and $\mathbf{b}_0$ is an external body force. Proper boundary and initial conditions must be specified to ensure the well-posedness of the problem. In addition, we need a constitutive equation for $\mathbf{P}$, that is, a stress-strain relationship describing the material behavior. Here, we consider hyperelastic materials, for which the existence of a strain density function $\mathcal{W}\colon Lin^+\rightarrow\mathbb{R}$ such that
\begin{equation*}
\mathbf{P(F)} = \frac{\partial\mathcal{W}(\mathbf{F})}{\partial\mathbf{F}}
\end{equation*}
is postulated. Note that, since $\mathbf{F}$ depends on the displacement $\mathbf{u}$, we can equivalently write $\mathbf{P(F)}$ or $\mathbf{P}(\mathbf{u})$. The strong formulation of a general initial boundary-valued problem in elastodynamics thus reads as follows: given a body force $\mathbf{b}_0 = \mathbf{b}_0(\mathbf{X},t;\bm\mu)$, a prescribed displacement $\bar{\mathbf{u}} = \bar{\mathbf{u}}(\mathbf{X},t;\bm\mu)$ and surface traction $\bar{\mathbf{T}} = \bar{\mathbf{T}}(\mathbf{X},t,\mathbf{N};\bm\mu)$, find the unknown displacement field $\mathbf{u}(\bm\mu)\colon\Omega_0\times(0,T]\rightarrow\mathbb{R}^3$ so that
\begin{align}\label{eq:strong-IBVP}
\left\{ \begin{array}{lllr}
\rho_0\partial^2_t {\mathbf{u}}(\mathbf{X},t;\bm\mu) - \nabla_0\cdot\mathbf{P}(\mathbf{u}(\mathbf{X},t;\bm\mu)) = \mathbf{b}_0(\mathbf{X},t;\bm\mu) && \text{in } & \Omega_0\times(0,T]\\
\mathbf{u}(\mathbf{X},t;\bm\mu) = \bar{\mathbf{u}}(\mathbf{X},t;\bm\mu) && \text{on } & \Gamma_0^{D}\times(0,T]\\
\mathbf{P}(\mathbf{u}(\mathbf{X},t;\bm\mu))\mathbf{N} = \bar{\mathbf{T}}(\mathbf{X},t,\mathbf{N};\bm\mu) && \text{on } & \Gamma_0^{N}\times(0,T]\\
\mathbf{P}(\mathbf{u}(\mathbf{X},t;\bm\mu))\mathbf{N} + \alpha \mathbf{u}(\mathbf{X},t;\bm\mu) + \beta \partial_t\mathbf{u}(\mathbf{X},t;\bm\mu) = \mathbf{0} && \text{on } & \Gamma_0^{R}\times(0,T]\\
\mathbf{u}(\mathbf{X},0;\bm\mu) = \mathbf{u}_0(\mathbf{X};\bm\mu),~~ \partial_t{\mathbf{u}}(\mathbf{X},0;\bm\mu) = \dot{\mathbf{u}}_0(\mathbf{X};\bm\mu) && \text{in } & \Omega_0\times\{0\}
\end{array} \right.
\end{align}
where $\mathbf{N}$ is the outer normal unit vector and $\alpha,\beta\in\mathbb{R}$. The boundary of the reference domain is divided such that $\Gamma_0^D\cup\Gamma_0^N\cup\Gamma_0^R = \Gamma$, with $\Gamma_0^i\cap\Gamma_0^j=\emptyset$ for $i,j\in\{D,N,R\}$. The corresponding variational form can we written as: $\forall t\in(0,T]$, find the unknown displacement field $\mathbf{u}(t;\bm\mu)\in V$ such that
\begin{align}\label{eq:weak-IBVP}
\langle R(\mathbf{u}(t;\bm\mu),t;\bm\mu),\bm\eta\rangle &:= \int_{\Omega_0} \rho_0\partial_t^2{\mathbf{u}}(t;\bm\mu) \cdot{\bm\eta}d\Omega + \int_{\Omega_0} \mathbf{P}(\mathbf{u}(t;\bm\mu)):\nabla{\bm\eta}d\Omega \nonumber \\
&+ \int_{\Gamma_0^R} \left(\alpha\mathbf{u}(t;\bm\mu)+\beta\partial_t\mathbf{u}(t;\bm\mu)\right)\cdot{\bm\eta}d\Gamma - \int_{\Gamma_N} \bar{\mathbf{T}}(t,\mathbf{N};\bm\mu)\cdot{\bm\eta}d\Gamma - \int_{\Omega_0} \mathbf{b}_0(t;\bm\mu)\cdot{\bm\eta}d\Omega = 0\\
\int_{\Omega_0} \mathbf{u}(0;\bm\mu) \cdot{\bm\eta}d\Omega &= \int_{\Omega_0} \mathbf{u}_0(\bm\mu) \cdot{\bm\eta}d\Omega \nonumber, \qquad \int_{\Omega_0} \partial_t\mathbf{u}(0;\bm\mu) \cdot{\bm\eta}d\Omega = \int_{\Omega_0} \dot{\mathbf{u}}_0(\bm\mu) \cdot{\bm\eta}d\Omega \nonumber
\end{align}
for any test function $\bm\eta$, where $V=V(\Omega_0)$ denotes a suitable Hilbert space on the reference configuration $\Omega_0\in\mathbb{R}^3$ and $V'$ its dual. This equation is inherently nonlinear and additional source of nonlinearity is introduced in the material law, i.e. when using a nonlinear $\mathcal{W} =\mathcal{W}(\mathbf{F})$, which is often the case of engineering applications.
For the sake of simplicity, in all test cases, we neglect the body forces $\mathbf{b}_0(\bm\mu)$ and consider zero initial conditions $\mathbf{u}_0(\bm\mu) = \dot{\mathbf{u}}_0(\bm\mu) = \mathbf{0}$. Regarding boundary conditions, we consider $\bar{\mathbf{u}}(\bm\mu) = \mathbf{0}$ on the Dirichlet boundary $\Gamma_0^D$ and always assume $\alpha=\beta=0$, so that we actually impose homogeneous Neumann conditions on $\Gamma_0^R$. Finally, the traction vector is given by
\begin{equation*}
\bar{\mathbf{T}}(\mathbf{X},t,\mathbf{N};\bm\mu) = -\mathbf{g}(t;\bm\mu)J\mathbf{F}^{-T}\mathbf{N},
\end{equation*}
where $\mathbf{g}(t;\bm\mu)$ represents an external load and will be specified according to the application at hand.
The residual in (\ref{eq:residual}) is given by
\begin{equation*}
\begin{aligned}
\mathbf{R}(\mathbf{u}_h^{n}(\bm\mu), t^{n};\bm\mu) &:= \left(\dfrac{\rho_0}{\Delta t^2}\mathcal{M} + \dfrac{1}{\Delta t}\mathcal{F}_{\beta}^{int} + \mathcal{F}_{\alpha}^{int}\right)\mathbf{u}_h^{n}(\bm\mu) + \mathcal{S}(\mathbf{u}_h^{n}(\bm\mu)) - \left(\dfrac{2\rho_0}{\Delta t^2}\mathcal{M} + \dfrac{1}{\Delta t}\mathcal{F}_{\beta}^{int}\right)\mathbf{u}_h^{n-1}(\bm\mu) \\ & + \dfrac{\rho_0}{\Delta t^2}\mathcal{M}\mathbf{u}_h^{n-2}(\bm\mu) - \mathcal{F}^{ext,n}(\bm\mu),
\end{aligned}
\end{equation*}
for $n=1,\dots,N_t$, where $\mathbf{u}_h^0(\bm\mu)$ and $\mathbf{u}_h^{-1}(\bm\mu)$ are known for the initial condition, and
\begin{align*}
&[\mathcal{M}]_{ij} = \int_{\Omega_0} \bm\varphi_j\cdot\bm\varphi_id\Omega,
\qquad
[\mathcal{F}_{\beta}^{int}]_{ij} = \int_{\Gamma_0^R} \beta~\bm\varphi_j\cdot\bm\varphi_i d\Gamma, \qquad
[\mathcal{F}_{\alpha}^{int}]_{ij} = \int_{\Gamma_0^R} \alpha~\bm\varphi_j\cdot\bm\varphi_i d\Gamma,\\
&[\mathcal{S}(\mathbf{u}_h^n(\bm\mu))]_{i} = \int_{\Omega_0} \mathbf{P}(\mathbf{u}_h^n(\bm\mu))\colon \nabla{\bm\varphi_i}d\Omega,
\qquad
[\mathcal{F}^{ext,n}(\bm\mu)]_{i} = \int_{\Gamma_0^N} \bar{\mathbf{T}}^n(\mathbf{N};\bm\mu)\cdot{\bm\varphi_i}d\Gamma + \int_{\Omega_0} \mathbf{b}_0^n(\bm\mu)\cdot{\bm\varphi_i}d\Omega,
\end{align*}
for all $i,j=1,\dots,N_h$, being $\{\bm\varphi_i\}_{i=1}^{N_h}$ a basis for the finite element (FE) space.
\newpage
As a measure of accuracy of the reduced approximations with respect to the FOM solution, we consider time-averaged $L^2$-errors of the displacement vector, that are defined as follows: \vspace{-0.2cm}
\begin{equation}\label{eq:error}
\begin{aligned}
\epsilon_{abs}(\bm\mu) &= \frac{1}{N_t}\sum_{n=1}^{N_t} \lVert \mathbf{u}_h(\cdot,t^n;\bm\mu) - \mathbf{Vu}_N(\cdot,t^n;\bm\mu)\rVert_{2}, \vspace{-0.1cm}\\
\epsilon_{rel}(\bm\mu) &= \frac{1}{N_t}\sum_{n=1}^{N_t} \frac{\lVert \mathbf{u}_h(\cdot,t^n;\bm\mu) - \mathbf{Vu}_N(\cdot,t^n;\bm\mu)\rVert_{2}}{\lVert \mathbf{u}_h(\cdot,t^n;\bm\mu)\rVert_{2}}.
\end{aligned}
\end{equation}
The CPU time ratio, that is the ratio between FOM and ROM computational times, is used to measure efficiency, since it represents the speed-up offered by the ROM with respect to the FOM. The code is implemented in Python in our software package pyfe$^\text{x}$, a Python binding with the in-house Finite Element library \texttt{life$^\texttt{x}$} (\url{https://lifex.gitlab.io/lifex}), a high-performance C++ library based on the \texttt{deal.II} (\url{https://www.dealii.org}) Finite Element core \cite{dealII92}. Computations have been performed on a PC desktop computer with 3.70GHz Intel Core i5-9600K CPU and 16GB RAM.
\subsection{Deformation of a clamped rectangular beam}\label{sec:DB}
The first series of test cases represents a typical structural mechanical problem, with reference geometry $\bar\Omega_0 = [0,10^{-2}]\times[0,10^{-3}]\times[0,10^{-3}]$ m$^3$, reported in Figure~\ref{fig:DB_mesh}.
\begin{figure}[b!]
\centering
\includegraphics[width=0.795\textwidth]{DB_geometry.png}
\caption{Rectangular beam geometry (left) and computational grid (right).}
\label{fig:DB_mesh}
\end{figure}
For the continuum body $\mathcal{B}$ under investigation, we consider a nearly-incompressible neo-Hookean material, which is characterized by the following strain density energy function \vspace{-0.1cm}
\begin{equation*}
\mathcal{W}(\mathbf{F}) = \frac{G}{2}(\mathcal{I}_1 - 3) + \frac{K}{4}( (J-1)^2 + \ln^2(J) ), \vspace{-0.1cm}
\end{equation*}
where $G>0$ is the shear modulus, $\mathcal{I}_1 = J^{-\frac{2}{3}}\det(\mathbf{C})$ and the latter term is needed to enforce in\-com\-pres\-si\-bi\-li\-ty, being the bulk modulus $K>0$ the penalization factor. This choice leads to the following first Piola-Kirchhoff stress tensor, characterized by a nonpolynomial nonlinearity, \vspace{-0.1cm}
\begin{equation*}
\mathbf{P}(\mathbf{F}) = GJ^{-\frac{2}{3}}\left(\mathbf{F} - \frac{1}{3}\mathcal{I}_1\mathbf{F}^T\right) + \frac{K}{2}J\left(J-1+\frac{1}{J}\ln(J)\right)\mathbf{F}^T. \vspace{-0.1cm}
\end{equation*}
The beam is clamped at the left-hand side, that is, Dirichlet boundary conditions are imposed on the left face $x=0$, whilst a pressure load changing with the deformed surface orientation is applied to the entire bottom face $z=0$ (i.e. $\Gamma_0^N$). Homogeneous Neumann conditions are applied on the remaining boundaries (i.e. $\Gamma_0^R$ with $\alpha=\beta=0$). As possible functions for the external load $\mathbf{g}(t;\bm\mu)$, we choose
\begin{enumerate}
\item a linear function $\mathbf{g}(t;\bm\mu) = \widetilde{p}~t/T$;
\item a triangular or hat function $\mathbf{g}(t;\bm\mu) = \widetilde{p}~\left(2t~\chi(t)_{\left(0,\frac{T}{2}\right]} + 2(T-t)~\chi(t)_{\left(\frac{T}{2},T\right]}\right)$;
\item a step function $\mathbf{g}(t;\bm\mu) = \widetilde{p}~\chi(t)_{\left(0,\frac{T}{3}\right]}$, so that the presence of the inertial term is not negligible.
\end{enumerate}
Here, $\widetilde{p}>0$ is a parameter controlling the maximum load. The FOM is built on a hexahedral mesh with $640$ elements and $1025$ vertices, resulting in a high-fidelity dimension $N_h=3075$ (since $\mathbb{Q}_1$-FE are employed). The resulting computational mesh in the reference configuration is reported in Figure~\ref{fig:DB_mesh}.
\newpage
The following sections are organized as follows: first, we analyze the accuracy and the efficiency of the ROM without hyper-reduction with respect to the POD tolerance $\varepsilon_{POD}$, thus resulting in reduced subspaces of different dimensions $N\in\mathbb{N}$. Then, for a fixed basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$, POD-Galerkin-DEIM approximation capabilities are investigated for different sizes of the reduced mesh, associated with different values of the tolerance $\varepsilon_{DEIM}$ for the computation of the residual basis $\bm\Phi_{\mathcal{R}}\in\mathbb{R}^{N_h\times m}$. Finally, the performances of Deep-HyROMnet are assessed and compared to those of DEIM-based hyper-ROMs.
\subsubsection{Test case 1: linear function for the pressure load}
Let us consider the parametrized linear function
\begin{equation*}
\mathbf{g}(t;\bm\mu) = \widetilde{p}~t/T,
\end{equation*}
for the pressure load, describing a situation in which a structure is progressively loaded. We choose a time interval $t\in[0,0.25]$~s and employ a uniform time step $\Delta t = 5\cdot10^{-3}$~s for the time discretization scheme, resulting in a total number of $50$ time iterations. As parameters, we consider:
\begin{itemize}
\item the shear modulus $G\in[0.5\cdot10^4,1.5\cdot10^4]$~Pa;
\item the bulk modulus $K\in[2.5\cdot10^4,7.5\cdot10^4]$~Pa;
\item the external load parameter $\widetilde{p}\in[2,6]$~Pa.
\end{itemize}
Given a training set of $n_s=50$ points generated from the three-dimensional parameter space $\mathcal{P}$ through latin hypercube sampling (LHS), we compute the reduced basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ using the POD method with tolerance \vspace{-0.2cm}
\begin{equation*}
\varepsilon_{POD}\in\{10^{-3}, 5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\}. \vspace{-0.2cm}
\end{equation*}
The corresponding reduced dimensions are $N=3$, $4$, $5$, $6$, $8$, $9$ and $15$, respectively. In Figure~\ref{fig:DB_ramp_svd_uh} we show the singular values of the snapshot matrix related to the FOM displacement $\mathbf{u}_h$, where a rapid decay of the plotted quantity means that a small number of RB functions are needed to correctly approximate the high-fidelity solution.
\begin{figure}[b!]
\centering
\includegraphics[width=0.45\textwidth]{DB_ramp_svd_uh.png}
\caption{Test case 1. Decay of the singular values of the FOM solution snapshots matrix.}
\label{fig:DB_ramp_svd_uh}
\end{figure}
The average relative error $\epsilon_{rel}$ between the FOM and the POD-Galerkin ROM solutions computed over a testing set of 50 randomly chosen parameters, different from the ones used to compute the solution snapshots, is reported in Figure~\ref{fig:DB_ramp_ROM}, together with the CPU time ratio. The approximation error decreases up to an order of magnitude when reducing the POD tolerance $\varepsilon_{POD}$ from $10^{-3}$ to $10^{-6}$, corresponding to an increase of the RB dimension from $N=3$ to $N=15$. Despite being the RB space low-dimensional, the computational speed-up achieved by the reduced model is negligible. This is due to the fact that the ROM still depends on the FOM dimension $N_h$ during the online stage. For this reason, we need to rely on suitable hyper-reduction techniques.
\begin{figure}[t!]
\vspace{-0.2cm}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_ramp_ROM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_ramp_ROM_speedup.png}
\end{subfigure}
\caption{Test case 1. Average over 50 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of ROM without hyper-reduction.}
\label{fig:DB_ramp_ROM}
\end{figure}
For the construction of both hyper-reduced models (POD-Galerkin-DEIM and Deep-HyROMnet), we need first to compute snapshots from the ROM solutions for given parameter values and time instants, in order to build either the DEIM basis $\bm\Phi_\mathcal{R}$ or train the DNNs $\bm\rho_N$ and $\bm\iota_N$. To this goal, we choose a POD-Galerkin ROM with dimension $N=4$, being it a good balance between accuracy and computational effort for the test case at hand, and perform ROM simulations for a given set of $n_s'=200$ parameter samples to collect residual and Jacobian data.
In order to investigate the impact of hyper-reduction onto the ROM solution reconstruction error, we compute the DEIM basis $\mathbf{\bm\Phi}_{\mathcal{R}}$ for the approximation of the residual using the POD method with different tolerances, that are
\begin{equation*}
\varepsilon_{DEIM}\in\{10^{-3}, 5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\},
\end{equation*}
corresponding to $m=22$, $25$, $30$, $33$, $39$, $43$, $51$, respectively. Larger POD tolerances were not sufficient to ensure the convergence of Newton method for all considered combinations of parameters, so that higher speed-ups cannot be achieved by decreasing the basis dimension $m$.
\begin{figure}[b!]
\vspace{-0.2cm}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_ramp_N4_DEIM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_ramp_N4_DEIM_speedup.png}
\end{subfigure}
\caption{Test case 1. Average over 50 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of POD-Galerkin-DEIM with $N=4$.}
\label{fig:DB_ramp_DEIM_N4}
\end{figure}
The average relative error $\epsilon_{rel}$ is evaluated over the testing set and plotted in Figure~\ref{fig:DB_ramp_DEIM_N4}, as well as the CPU time ratio. To compute the high-fidelity solutions, $26$~s are required in average, while a POD-Galerkin-DEIM ROM, with $N=4$ and $m=22$, requires only $2.4$~s, thus yielding a speed-up of $\times11$ compared to the FOM.
Data related to the performances of the POD-Galerkin-DEIM method for $N=4$ and different values of $m$ are shown in Table~\ref{tab:DB_ramp}. The number of elements of the reduced mesh represents a small percentage of the one forming the original grid, so that the cost related to the residual assembling is remarkably alleviated. Nonetheless, it is obvious that the main computational bottleneck is the construction of the reduced system at each Newton iteration, and in particular the assembling of the residual vector on the reduced mesh, which requires between $78\%$ and $88\%$ of the total (online) CPU time. In particular, almost $90\%$ of this computational time is demanded for assembling the residual $\mathbf{R}(\mathbf{Vu}_N^{n,(k)}(\bm\mu),t^{n};\bm\mu)$ on the reduced mesh, while computing the associated Jacobian matrix using the automatic differentiation tool takes less than $1\%$.
\begin{table}[h!]
\centering
\begin{tabular}{|l||c|c|c|}
\hline
POD tolerance $\varepsilon_{DEIM}$ & $5\cdot10^{-4}$ & $5\cdot10^{-5}$ & $5\cdot10^{-6}$\\
\hline
DEIM interpolation dofs $m$ & $25$ & $33$ & $43$ \\
\hline
Reduced mesh elements (total: $640$) & $86$ & $115$ & $168$ \\
\hline
\hline
Online CPU time & $2.8$~s & $3.6$~s & $4.3$~s \\
$\quad\circ$ system construction $[*]$ & $78\%$ & $83\%$ & $88\%$\\
$\quad\circ$ system solution & $0.16\%$ & $0.13\%$ & $0.09\%$\\
\hline
\hline
$[*]$ System construction for each Newton iteration & $0.02$~s & $0.02$~s & $0.03$~s \\
$\quad\circ$ residual assembling & $89\%$ & $87\%$ & $88\%$\\
$\quad\circ$ Jacobian computing through AD & $0.6\%$ & $0.4\%$ & $0.5\%$\\
\hline
\hline
Computational speed-up & $\times$9.4 & $\times$7.3 & $\times$6.0 \\
\hline
Time-averaged $L^2(\Omega_0)$-absolute error & $3\cdot10^{-5}$ & $2\cdot10^{-5}$ & $2\cdot10^{-5}$\\
\hline
Time-averaged $L^2(\Omega_0)$-relative error & $8\cdot10^{-3}$ & $5\cdot10^{-3}$ & $5\cdot10^{-3}$ \\
\hline
\end{tabular}
\caption{Test case 1. Computational data related to POD-Galerkin-DEIM with $N=4$ and different values of $m$.}
\label{tab:DB_ramp}
\end{table}
Finally, we analyze the performances of Deep-HyROMnet and compare them in terms of both accuracy and efficiency with POD-Galerkin-DEIM ROMs. The average of the absolute error $\epsilon_{abs}$, the relative error $\epsilon_{rel}$ and the CPU time ratio are reported in Table~\ref{tab:DB_ramp_N4_hyper-ROM}. In terms of efficiency, the DNN-based ROMs outperform the DEIM-based hyper-ROMs substantially, being almost $100$ times faster than POD-Galerkin-DEIM ROM with $m=22$, whist achieving the same accuracy. In particular, Deep-HyROMnet is able to compute the reduced solutions in less than 0.03~s, thus yielding an overall speed-up of order $\mathcal{O}(10^3)$ compared to the FOM.
\begin{table}[b!]
\centering
\vspace{0.15cm}
\begin{tabular}{|l||c|c|c|}
\hline
& DEIM ($m=$22) & DEIM ($m=$30) & Deep-HyROMnet\\
\hline
\hline
Computational speed-up & $\times$11 & $\times$8 & $\times$1012\\
\hline
Avg. CPU time & 2~s & 3~s & 0.026~s\\
\hline
Time-avg. $L^2(\Omega_0)$-absolute error & $7.4\cdot10^{-5}$ & $1.9\cdot10^{-5}$ & $7.7\cdot10^{-5}$\\
\hline
Time-avg. $L^2(\Omega_0)$-relative error & $9.7\cdot10^{-3}$ & $5.0\cdot10^{-3}$ & $8.3\cdot10^{-3}$\\
\hline
\end{tabular}
\caption{Test case 1. Computational data related to POD-Galerkin-DEIM ROMs and Deep-HyROMnet, for $N=4$.}
\label{tab:DB_ramp_N4_hyper-ROM}
\end{table}
The evolution of the $L^2(\Omega_0)$-absolute error, averaged over the testing parameters, is reported in Figure~\ref{fig:DB_ramp_N4_hyper-ROM_err_abs} for all of the hyper-ROMs considered.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{DB_ramp_N4_hyperROM_err_abs.png}
\caption{Test case 1. Evolution in time of the average $L^2(\Omega_0)$-absolute error for $N=4$ computed using POD-Galerkin-DEIM and Deep-HyROMnet.}
\label{fig:DB_ramp_N4_hyper-ROM_err_abs}
\end{figure}
The final accuracy of the hyper-ROMs equals that of the ROM without hyper-reduction, i.e. $\epsilon_{rel}\approx10^{-2}$, meaning that the projection error dominates over the nonlinear operators approximation error. The difference between the FOM and Deep-HyROMnet solutions at time $T=0.25$~s is shown in Figure \ref{fig:DB_ramp_N4_DeepHyROMnet_mu10_mu13_error} in two scenarios.
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\textwidth]{DB_ramp_N4_DeepHyROMnet_mu10_mu13_error.png}
\caption{Test case 1. FOM (wireframe) and Deep-HyROMnet (colored) solutions at time $T=0.25$~s for $\bm\mu = [1.3225\cdot10^4~\text{Pa}, 3.9875\cdot10^4~\text{Pa}, 3.43~\text{Pa}]$ (left) and $\bm\mu = [0.6625\cdot10^4~\text{Pa}, 5.8625\cdot10^4~\text{Pa}, 4.89~\text{Pa}]$ (right).}
\label{fig:DB_ramp_N4_DeepHyROMnet_mu10_mu13_error}
\end{figure}
In order to increase the accuracy of the reduced solution, we should consider higher values of the RB dimension $N$. As a matter of fact, by increasing the RB dimension, the task of the DNNs becomes more complex, meaning that more training samples and a larger size of the neural networks themselves may be required. In Table~\ref{tab:DB_ramp_N8_hyper-ROM} are reported the computational data associated with POD-Galerkin-DEIM and Deep-HyROMnet hyper-ROMs when $N=8$, but the same number of training snapshots and the same DNN architectures of previous case (i.e. $N=4$) are employed. We observe that POD-Galerkin-DEIM is able to provide more accurate approximations of the high-fidelity solution by increasing the size $m$ of the residual basis, albeit reducing the online speed-up with respect to the FOM. On the other hand, in the context of multi-query problems, such as uncertainty quantification or optimization, where thousands of queries to the parameter-to-solution map are required, it is of paramount importance to decrease the CPU time needed for the solution of the reduced problem at each new instance of the input parameter vector.
\begin{table}[h]
\centering
\begin{tabular}{|l||c|c|c|}
\hline
& DEIM ($m=$29) & DEIM ($m=$51) & Deep-HyROMnet\\
\hline
\hline
Computational speed-up & $\times$8 & $\times$5 & $\times$949\\
\hline
Avg. CPU time & 3~s & 5~s & 0.027~s\\
\hline
Time-avg. $L^2(\Omega_0)$-absolute error & $2.6\cdot10^{-5}$ & $3.9\cdot10^{-6}$ & $9.0\cdot10^{-5}$\\
\hline
Time-avg. $L^2(\Omega_0)$-relative error & $1.1\cdot10^{-2}$ & $6.0\cdot10^{-4}$ & $8.1\cdot10^{-3}$\\
\hline
\end{tabular}
\caption{Test case 1. Computational data related to POD-Galerkin-DEIM ROMs and Deep-HyROMnet, for $N=8$.}
\label{tab:DB_ramp_N8_hyper-ROM}
\vspace{-0.2cm}
\end{table}
\subsubsection{Test case 2: hat function for the pressure load}
Let us now consider a piecewise linear pressure load given by the following hat function
\begin{equation*}
\mathbf{g}(t;\bm\mu) = \widetilde{p}~\left(2t~\chi(t)_{\left(0,\frac{T}{2}\right]} + 2(T-t)~\chi(t)_{\left(\frac{T}{2},T\right]}\right),
\end{equation*}
describing the case in which a structure is increasingly loaded until a maximum pressure is reached, and then linearly unloaded in order to recover the initial resting state. For the case at hand, we choose $t\in[0,0.35]$~s and $\Delta t = 5\cdot10^{-3}$~s, resulting in a total number of $70$ time steps. As parameter, we consider the external load parameter $\widetilde{p}\in[2,12]$ Pa. The shear modulus $G$ and the bulk modulus $K$ are fixed to the values $10^4$ Pa and $5\cdot10^4$ Pa, respectively. Let us consider a training set of $n_s=50$ points generated from the one-dimensional parameter space $\mathcal{P}=[2,12]$ Pa through LHS and build the RB basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ with $N=4$, corresponding to $\varepsilon_{POD}=10^{-4}$. The singular values of the solution snapshots matrix are reported in Figure~\ref{fig:DB_hat_svd_uh}.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{DB_hat_svd_uh.png}
\caption{Test case 2. Decay of the singular values of the FOM solution.}
\label{fig:DB_hat_svd_uh}
\end{figure}
Given the Galerkin-ROM nonlinear data collected for $n_s'=300$ sampled parameters, the DEIM residual basis $\mathbf{\bm\Phi}_{\mathcal{R}}$ is computed using the POD method with tolerance \vspace{-0.1cm}
\begin{equation*}
\varepsilon_{DEIM}\in\{10^{-3}, 5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\}, \vspace{-0.1cm}
\end{equation*}
corresponding to $m=14$, $16$, $22$, $24$, $29$, $31$, $37$, respectively. Tolerances $\varepsilon_{DEIM}$ larger than the values reported above were not sufficient to ensure convergence of Newton method for all the considered parameters.
\begin{figure}[h]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_hat_N4_DEIM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_hat_N4_DEIM_speedup.png}
\end{subfigure}
\caption{Test case 2. Average over 50 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of POD-Galerkin-DEIM with $N=4$.}
\label{fig:DB_hat_DEIM_N4}
\end{figure}
The relative error $\epsilon_{rel}$, evaluated over a testing set of $50$ parameters, is about $10^{-2}$ when using $m=14$ residual basis, and can be further reduced of one order of magnitude when increasing the DEIM dimension to $m=29$, albeit highly decreasing the CPU time ratio, as shown in Figure~\ref{fig:DB_hat_DEIM_N4}.
Table \ref{tab:DB_hat_hyper-ROM_N4} shows the comparison between POD-Galerkin-DEIM (with $m=14$ and $m=29$) and Deep-HyROMnet hyper-reduced models on a testing set of 50 parameter instances. As observed in the previous test case, Deep-HyROMnet is able to achieve good results in terms of accuracy, comparable with the fastest DEIM-based model ($m=14$), at a greatly reduced cost. Also in this case, the speed-up achieved by our DNN-based hyper-ROM is of order $\mathcal{O}(10^3)$ with respect to the FOM, since less than $0.04$~s are needed to compute the reduced solution for each new instance of the parameter, against a time of about $40$~s required by the FOM, and of $3$~s required by POD-Galerkin-DEIM.
\begin{table}[t]
\centering
\begin{tabular}{|l||c|c|c|}
\hline
& DEIM ($m=14$) & DEIM ($m=29$) & Deep-HyROMnet \\
\hline
\hline
Computational speed-up & $\times$14 & $\times$9 & $\times$1153\\
\hline
Avg. CPU time & 3~s & 5~s & 0.035~s\\
\hline
Time-avg. $L^2(\Omega_0)$-absolute error & $9.0\cdot10^{-5}$ & $6.8\cdot10^{-6}$ & $2.0\cdot10^{-4}$\\
\hline
Time-avg. $L^2(\Omega_0)$-relative error & $1.5\cdot10^{-2}$ & $1.4\cdot10^{-3}$ & $1.7\cdot10^{-2}$\\
\hline
\end{tabular}
\caption{Test case 2. Computational data related to POD-Galerkin-DEIM ROMs and Deep-HyROMnet, for $N=4$.}
\label{tab:DB_hat_hyper-ROM_N4}
\end{table}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth]{DB_hat_N4_hyperROM_err_abs.png}
\caption{Test case 2. Evolution in time of the average $L^2(\Omega_0)$-absolute error for $N=4$ computed using POD-Galerkin-DEIM and Deep-HyROMnet.}
\label{fig:DB_hat_hyper-ROM_N4_err_abs}
\end{figure}
The evolution in time of the average $L^2(\Omega_0)$-absolute error for DEIM and Deep-HyROMnet models is shown in Figure~\ref{fig:DB_hat_hyper-ROM_N4_err_abs}.
The accuracy obtained using Deep-HyROMnet, although slightly lower than the ones achieved using a DEIM-based approximation, is satisfying in all the considered scenarios. Figure \ref{fig:DB_hat_N4_DeepHyROMnet_mu36_error} shows the FOM and the Deep-HyROMnet displacements at different time instances obtained for a given testing parameter.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{DB_hat_N4_DeepHyROMnet_mu36_error.png}
\caption{Test case 2. FOM (wireframe) and Deep-HyROMnet (colored) solutions computed at different times for $\bm\mu = [10.7375~\text{Pa}]$.}
\label{fig:DB_hat_N4_DeepHyROMnet_mu36_error}
\end{figure}
\subsubsection{Test case 3: step function for the pressure load}
As last test case for the beam geometry, we consider a pressure load acting on the bottom surface area for only a third of the whole simulation time, that is
\begin{equation*}
\mathbf{g}(t;\bm\mu) = \widetilde{p}~\chi(t)_{\left(0,\frac{T}{3}\right]},
\end{equation*}
such that the resulting deformation features oscillations. This case is of particular interest in nonlinear elastodynamics, since the inertial term cannot be neglected, as it has a crucial impact on the deformation of the object. For the case at hand, we choose $t\in[0,0.27]$~s and a uniform time step $\Delta t = 3.6\cdot10^{-3}$~s, resulting in a total number of $75$ time iterations. For what concerns the input parameters, we vary the external load $\widetilde{p}\in[2,12]$ Pa and consider $G=10^4$ Pa and $K=5\cdot10^4$ Pa fixed.
We build the reduced basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ from a training set of $n_s=50$ FOM solutions using $\varepsilon_{POD}=10^{-3}$, thus obtaining a reduced dimension of $N=4$, and perform POD-ROM simulations for a given set of $n_s'=300$ parameter samples to collect the nonlinear terms data necessary for the construction of both POD-Galerkin-DEIM and Deep-HyROMnet models. The DEIM basis $\mathbf{\bm\Phi}_{\mathcal{R}}$ for the approximation of the residual is computed by performing POD on the associated snapshots matrix with tolerance
\begin{equation*}
\varepsilon_{DEIM}\in\{10^{-3}, 5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\},
\end{equation*}
where $\varepsilon_{DEIM}=10^{-3}$ is the larger POD tolerance that allows to guarantee the convergence of the reduced Newton algorithm for all testing parameters. The corresponding number basis for $\mathbf{R}$ is $m=18$, $20$, $27$, $30$, $38$, $40$, $50$, respectively. The results regarding the average relative error $\epsilon_{rel}$ and the computational speed-up, evaluated over 50 instances of the parameter, are shown in Figure \ref{fig:DB_step_DEIM_N4}.
\begin{figure}[b!]
\vspace{-0.2cm}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_step_N4_DEIM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{DB_step_N4_DEIM_speedup.png}
\end{subfigure}
\caption{Test case 3. Average over 50 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of POD-Galerkin-DEIM with $N=4$.}
\label{fig:DB_step_DEIM_N4}
\end{figure}
Like for the previous test cases, we compare POD-Galerkin-DEIM and Deep-HyROMnet ROMs, with respect to the displacement error and the CPU time ratio.
\begin{table}[h]
\centering
\begin{tabular}{|l||c|c|c|}
\hline
& DEIM ($m=18$) & DEIM ($m=38$) & Deep-HyROMnet \\
\hline
\hline
Computational speed-up & $\times$12 & $\times$6 & $\times$1350\\
\hline
Avg. CPU time & 4~s & 8~s & 0.038~s\\
\hline
Time-avg. $L^2(\Omega_0)$-absolute error & $2.4\cdot10^{-3}$ & $1.3\cdot10^{-4}$ & $4.8\cdot10^{-4}$\\
\hline
Time-avg. $L^2(\Omega_0)$-relative error & $6.7\cdot10^{-1}$ & $2.4\cdot10^{-2}$ & $1.0\cdot10^{-1}$\\
\hline
\end{tabular}
\caption{Test case 3. Computational data related to POD-Galerkin-DEIM ROMs and Deep-HyROMnet, for $N=4$.}
\label{tab:DB_step_N4_hyper-ROM}
\end{table}
As reported in Table~\ref{tab:DB_step_N4_hyper-ROM}, Deep-HyROMnet outperforms DEIM substantially in terms of efficiency also in this case when handling the nonlinear terms. Indeed, Deep-HyROMnet yields a ROM that is more than $1000$ times faster than the FOM (this latter requiring $51$~s in average to be solved), still providing satisfactory results in terms of accuracy.
Figure~\ref{fig:DB_step_N4_DeepHyROMnet_mu1_mu13} represent the Deep-HyROMnet solution at different time instants for two different values of the parameter and show that the hyper-ROM is able to correctly capture the nonlinear behavior of the continuum body also when the inertial term cannot be neglected.
\begin{figure}[h!]
\centering
\begin{subfigure}{0.85\textwidth}
\centering
\includegraphics[width=\textwidth]{DB_step_N4_DeepHyROMnet_mu1.png}
\caption{$\bm\mu = [5.1125~\text{Pa}]$}
\end{subfigure}
\begin{subfigure}{0.85\textwidth}
\centering
\includegraphics[width=\textwidth]{DB_step_N4_DeepHyROMnet_mu13.png}
\caption{$\bm\mu = [11.4625~\text{Pa}]$}
\end{subfigure}
\caption{Test case 3. FOM (wireframe) and Deep-HyROMnet (colored) solutions computed at different times.}
\label{fig:DB_step_N4_DeepHyROMnet_mu1_mu13}
\end{figure}
\subsection{Passive inflation and active contraction of an idealized left ventricle}
The second problem we are interested in is the inflation and contraction of a prolate spheroid geometry representing an idealized left ventricle (see Figure~\ref{fig:Prolate_BCs}) where the boundaries $\Gamma_0^R$, $\Gamma_0^N$ and $\Gamma_0^D$ represent the epicardium, the endocardium and the base of a left ventricle, respectively, the latter being the artificial boundary resulting from truncation of the heart below the valves in a short axis plane.
\begin{figure}
\centering
\includegraphics[width=0.725\textwidth]{ProlateC_geometry.png}
\caption{Passive inflation and active contraction of an idealized left ventricle. Idealized truncated ellipsoid geometry (left) and computational grid (right).} \label{fig:Prolate_BCs}
\end{figure}
We consider transversely isotropic material properties for the myocardial tissue, adopting a nearly-incompressible for\-mu\-la\-tion of the constitutive law proposed in \cite{guccione1995finite}, whose strain-energy density function is given by
\begin{equation*}
\mathcal{W}(\mathbf{F}) = \frac{C}{2}(e^{Q(\mathbf{F})} - 1),
\end{equation*}
with the following form for $Q$ to describe three-dimensional transverse isotropy with respect to the fiber coordinate system,
\begin{equation*}
Q = b_{f} E_{ff}^2 + b_{s} E_{ss}^2 + b_{n} E_{nn}^2 + b_{fs}(E_{fs}^2 + E_{sf}^2) + b_{fn}(E_{fn}^2 + E_{nf}^2) + b_{sn}(E_{sn}^2 + E_{ns}^2).
\end{equation*}
Here, $E_{ij}$, $i,j\in\{f,s,n\}$, are the components of the Green-Lagrange strain tensor (\ref{eq:strain}), the material constant $C>0$ scales the stresses and the coefficients $b_f$, $b_s$, $b_n$ are related to the material stiffness in the fiber, sheet and transverse directions, respectively. This leads to a (passive) first Piola-Kirchhoff stress tensor characterized by exponential nonlinearity. In order to enforce the incompressibility constraint, we consider an additional term $\mathcal{W}_{vol}(J)$ in the definition of the strain energy density function, which must grow as the deformation deviates from being isochoric. A common choice for $\mathcal{W}_{vol}$ is a convex function with null slope in $J=1$, e.g.,
\begin{equation*}
\mathcal{W}_{vol}(J) = \frac{K}{4}( (J-1)^2 + \ln^2(J) ),
\end{equation*}
where the penalization factor is the bulk modulus $K>0$. Furthermore, to reproduce the typical twisting motion of the ventricular systole, we need to take into account a varying fiber distribution and contractile forces. The fiber direction is computed using the rule-based method proposed in \cite{rossi2014thermodynamically}, which depends on parameter angles $\bm\alpha^{epi}$ and $\bm\alpha^{endo}$. Active contraction is modeled through the active stress approach \cite{ambrosi2012active}, so that we add to the passive first Piola-Kirchoff stress tensor a time-dependent active tension, which is assumed to act only in the fiber direction
\begin{equation*}
\mathbf{P} = \left(\frac{\partial\mathcal{W}(\mathbf{F})}{\partial\mathbf{F}} + \frac{\partial\mathcal{W}_{vol}(J)}{\partial\mathbf{F}}\right) + T_a(t)(\mathbf{Ff}_0\otimes\mathbf{f}_0),
\end{equation*}
where $\mathbf{f}_0\in\mathbb{R}^3$ denotes the reference unit vector in the fiber direction and $T_a$ is a parametrized function that surrogates the active generation forces. In our case, since we are modeling only the systolic contraction, we define
\begin{equation*}
T_a(t) = \widetilde{T}_a~t/T, \quad t\in(0,T),
\end{equation*}
with $\widetilde{T}_a>0$. To model blood pressure inside the chamber we assume a linearly increasing external load
\begin{equation*}
\mathbf{g}(t;\bm\mu) = \hat{p}~t/T, \quad t\in(0,T).
\end{equation*}
Since we want to assess the performance of Deep-HyROMnet to reduce the myocardium contraction, we consider as unknown parameters those related to the active components of the strain energy function:
\begin{itemize}
\item the maximum value of the active tension $\widetilde{T}_a\in[49.5\cdot10^3,70.5\cdot10^3]$ Pa, and
\item the fiber angles $\bm\alpha^{epi}\in[-105.5,-74.5]^\circ$ and $\bm\alpha^{endo}\in[74.5,105.5]^\circ$.
\end{itemize}
All other parameters are fixed to the reference values taken from \cite{land2015verification}, namely $b_{f}=8$, $b_{s}=b_{n}=b_{sn}=2$, $b_{fs}=b_{fn}=4$, $C=2\cdot10^3$~Pa, $K=50\cdot10^3$~Pa and $\widetilde{p}=15\cdot10^3$. Regarding the time discretization, we choose $t\in[0,0.25]$~s and a uniform time step $\Delta t = 5\cdot10^{-3}$~s, resulting in a total number of $50$ time iterations. The FOM is built on a hexahedral mesh with $4804$ elements and $6455$ vertices, depicted in Figure \ref{fig:Prolate_BCs}, corresponding to a high-fidelity dimension $N_h=19365$, since $\mathbb{Q}_1$-FE (that is, linear FE on a hexahedral mesh) are used. In this case, the FOM requires almost $360$~s to compute the solution dynamics for each parameter instance.
Given $n_s=50$ points obtained by sampling the parameter space $\mathcal{P}$, we construct the corresponding solution snapshots matrix $\mathbf{S}_u$ and compute the reduced basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ using the POD method with tolerance
\begin{equation*}
\varepsilon_{POD}\in\{10^{-3}, 5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\}.
\end{equation*}
From Figure~\ref{fig:ProlateC_svd_uh}, we observe a slower decay of the singular values of $\mathbf{S}_u$ with respect to the structural problems of Section~\ref{sec:DB}. In fact, we obtain larger reduced basis dimensions $N=16$, $22$, $39$, $50$, $87$, $109$ and $178$, respectively.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{ProlateC_svd_uh.png}
\caption{Passive inflation and active contraction of an idealized left ventricle. Decay of the singular values of the FOM solution snapshots matrix.}
\label{fig:ProlateC_svd_uh} \vspace{-0.1cm}
\end{figure}
The error and the CPU speed-ups averaged over a testing set of 20 parameters are both shown in Figure \ref{fig:ProlateC_ROM}, as functions of the POD tolerance $\varepsilon_{POD}$.
\begin{figure}[b!]
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ProlateC_ROM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ProlateC_ROM_speedup.png}
\end{subfigure}
\caption{Passive inflation and active contraction of an idealized left ventricle. Average over 20 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of ROM without hyper-reduction.}
\label{fig:ProlateC_ROM}
\end{figure}
As already discussed, the speed-up achieved by the ROM is negligible, since at each Newton iteration without hyper-reduction the ROM still depends on the high-fidelity dimension $N_h$. For what concerns the approximation error, we observe a reduction of almost two orders of magnitude when going from $N=16$ to $N=178$. \\
Given the reduced basis $\mathbf{V}\in\mathbb{R}^{N_h\times N}$ with $N=16$, we construct the POD-Galerkin-DEIM ap\-proxima\-tion by considering $n_s'=200$ parameter samples. Figure~\ref{fig:ProlateC_N16_svd_RN} shows the decay of the singular values of $\mathbf{S}_R$, that is, the snapshots matrix of the residual vectors $\mathbf{R}(\mathbf{Vu}_N^{n,(k)}(\bm\mu_{\ell'}),t^n;\bm\mu_{\ell'})$. We observe that the reported curve decreases very slowly, so that we expect that a large number of basis functions is required to correctly approximate the nonlinear operators.
\begin{figure}[t!]
\centering
\includegraphics[width=0.45\textwidth]{ProlateC_N16_svd_RN.png}
\caption{Passive inflation and active contraction of an idealized left ventricle. Decay of the singular values of the ROM residual snapshots matrix.}
\label{fig:ProlateC_N16_svd_RN}
\vspace{-0.25cm}
\end{figure}
In fact, by computing $\bm\Phi_{\mathcal{R}}\in\mathbb{R}^{N_h\times m}$ using the following POD tolerances:
\begin{equation*}
\varepsilon_{DEIM}\in\{5\cdot10^{-4}, 10^{-4}, 5\cdot10^{-5}, 10^{-5}, 5\cdot10^{-6}, 10^{-6}\},
\end{equation*}
we obtain $m=303$, $456$, $543$, $776$, $902$ and $1233$, respectively. Higher values of $\varepsilon_{DEIM}$ (related to hopefully smaller dimensions $m$) were not sufficient to guarantee the convergence of the reduced Newton problem for all the parameter combinations considered. The average relative error over a set of 20 parameters and the computational speed-up are both reported in Figure~\ref{fig:ProlateC_DEIM_N16}. In particular, we observe that the relative error is between $4\cdot10^{-3}$ and $8\cdot10^{-3}$, as we could expect from the projection error reported in Figure~\ref{fig:ProlateC_ROM}, that is, POD-Galerkin-DEIM is able to achieve the same accuracy of the ROM without hyper-reduction.
\begin{figure}
\centering
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ProlateC_N16_DEIM_error.png}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{ProlateC_N16_DEIM_speedup.png}
\end{subfigure}
\caption{Passive inflation and active contraction of an idealized left ventricle. Average over 20 testing parameters of relative error $\epsilon_{rel}$ (left) and average speed-up (right) of POD-Galerkin-DEIM with $N=16$.}
\label{fig:ProlateC_DEIM_N16}
\vspace{-0.3cm}
\end{figure}
The data reported in Table~\ref{tab:ProlateC} leads to the same conclusions regarding the computational bottleneck of the DEIM technique as those reported in Table~\ref{tab:DB_ramp}. In fact, assembling the residual on the reduced mesh requires around $85\%$ of the online CPU time, thus undermining the hyper-ROM efficiency.
\begin{table}
\centering
\begin{tabular}{|l||c|c|c|}
\hline
POD tolerance $\varepsilon_{DEIM}$ & $5\cdot10^{-4}$ & $5\cdot10^{-5}$ & $5\cdot10^{-6}$\\
\hline
DEIM interpolation dofs $m$ & $303$ & $543$ & $902$ \\
\hline
Reduced mesh elements (total: $4804$) & $914$ & $1345$ & $1855$ \\
\hline
\hline
Online CPU time & $58$~s & $81$~s & $110$~s \\
$\quad\circ$ system construction $[*]$ & $89\%$ & $93\%$ & $94\%$\\
$\quad\circ$ system solution & $0.01\%$ & $0.01\%$ & $0.01\%$\\
\hline
\hline
$[*]$ System construction for each Newton iteration & $0.4$~s & $0.6$~s & $0.9$~s \\
$\quad\circ$ residual assembling & $94\%$ & $94\%$ & $94\%$\\
$\quad\circ$ Jacobian computing through AD & $0.24\%$ & $0.24\%$ & $0.26\%$\\
\hline
\hline
Computational speed-up & $\times$6.2 & $\times$4.5 & $\times$3.3 \\
\hline
Time-averaged $L^2(\Omega_0)$-absolute error & $1\cdot10^{-3}$ & $7\cdot10^{-4}$ & $6\cdot10^{-4}$\\
\hline
Time-averaged $L^2(\Omega_0)$-relative error & $8\cdot10^{-3}$ & $5\cdot10^{-3}$ & $5\cdot10^{-3}$\\
\hline
\end{tabular}
\caption{Passive inflation and active contraction of an idealized left ventricle. Computational data related to POD-Galerkin-DEIM with $N=16$ and different values of $m$.}
\label{tab:ProlateC}
\end{table}
Finally, Table \ref{tab:ProlateC_N16_hyper-ROM} reports the computational data of POD-Galerkin-DEIM ROMs obtained for a number of magic points equals to $m=303$ and $m=543$, and of the Deep-HyROMnet, clearly showing that the latter outperforms the classical reduction strategy regarding the computational speed-up.
In fact, Deep-HyROMnet is able to approximate the solution dynamics in $0.1$~s, that is even faster than real-time, while a POD-Galerkin-DEIM ROM requires $1$~min in average, where the final simulation time $T$ is set equal to $0.25$~s. Although the Deep-HyROMnet error is one order of magnitude higher than the one evaluated by a DEIM-based hyper-ROM (see Figure~\ref{fig:ProlateC_N16_hyper-ROM_err_abs}), the results are satisfactory in terms of accuracy. In Figures~\ref{fig:ProlateC_N16_DeepHyROMnet_mu1_mu10_mu19} the FOM and the DNN-based hyper-ROM displacements at time $T=0.25$~s are reported for three different values of the parameters, together with the error between the high-fidelity and the reduced solutions.
\begin{table}[h!]
\centering
\begin{tabular}{|l||c|c|c|}
\hline
& DEIM ($m=303$) & DEIM ($m=543$) & Deep-HyROMnet \\
\hline
\hline
Speed-up & $\times$6 & $\times$5 & $\times$3554 \\
\hline
Avg. CPU time & 58~s & 75~s & 0.1~s\\
\hline
mean$_{\bm\mu}$ $\epsilon_{abs}(\bm\mu)$ & $1.3\cdot10^{-3}$ & $6.6\cdot10^{-4}$ & $1.5\cdot10^{-2}$ \\
\hline
mean$_{\bm\mu}$ $\epsilon_{rel}(\bm\mu)$ & $7.5\cdot10^{-3}$ & $5.0\cdot10^{-3}$ & $6.4\cdot10^{-2}$ \\
\hline
\end{tabular}
\caption{Passive inflation and active contraction of an idealized left ventricle. Computational data related to DEIM-based and DNN-based hyper-ROMs, for $N=16$.}
\label{tab:ProlateC_N16_hyper-ROM}
\end{table}
\begin{figure}[b!]
\centering
\includegraphics[width=0.925\textwidth]{ProlateC_N16_hyperROM_err_abs.png}
\caption{Passive inflation and active contraction of an idealized left ventricle. Evolution in time of the average $L^2(\Omega_0)$-absolute error computed using DEIM-based and DNN-based hyper-ROMs, for $N=16$.}
\label{fig:ProlateC_N16_hyper-ROM_err_abs}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\textwidth]{ProlateC_N16_DeepHyROMnet_mu1_mu10_mu19.png}
\caption{Passive inflation and active contraction of an idealized left ventricle. FOM (wireframe) and Deep-HyROMnet (colored) displacements (frontal view on top, lateral view in the middle) and corresponding difference (bottom) at time $T=0.25$~s for $\bm\mu = [61942.5~\text{Pa},-77.5225^\circ, 87.9075^\circ]$ (left), $\bm\mu = [59737.5~\text{Pa},-102.3225^\circ, 91.1625^\circ]$ (center) and $\bm\mu = [50497.5~\text{Pa},-100.9275^\circ, 80.0025^\circ]$ (right).}
\label{fig:ProlateC_N16_DeepHyROMnet_mu1_mu10_mu19}
\end{figure}
To conclude, we repeat that the approximation of the reduced nonlinear operators with Deep-HyROMnet does not depend directly on the high-fidelity dimension $N_h$, but rather on reduced basis dimension $N$.
To test its performances using a higher FOM dimension, we address the solution to the problem described in this Section, however considering a finer hexahedral mesh with $9964$ elements and $13025$ vertices, thus obtaining $N_h=39075$ as FOM dimension. In this case, about 13 minutes are required to compute the high-fidelity dynamics. On the other hand, a reduced basis of dimension $N=16$ is computed for $\varepsilon_{POD}=10^{-3}$; the computational data, averaged over a testing set of $20$ parameter samples, are reported in Table \ref{tab:ProlateC_hyper-ROM_fine}. Almost unexpectedly, the online CPU time required by Deep-HyROMnet doubles as we double $N_h$. This may be due to the higher time required to perform matrix-vector multiplication for the reconstruction of the reduced solutions $\mathbf{Vu}_N^n(\bm\mu)$, for $n=1,\dots,N_t$. Further analysis should be performed to investigate this issue. Nonetheless, is it worth saying that the overall computational speed-up of Deep-HyROMnet increases as the FOM dimension $N_h$ grows, while the number $N$ of reduced basis function remains small, so that reduced solutions can be computed extremely fast. For what concerns the approximation accuracy of the hyper-ROMs with respect to the associate FOMs, we obtain almost the same results, showing that Deep-HyROMnet is able to deal with higher high-fidelity dimensions.
\begin{table}
\centering
\begin{tabular}{|l||c|c|}
\hline
& \multicolumn{2}{c|}{Deep-HyROMnet} \\
\hline
\hline
$N_h$ & $19365$ & $39075$ \\
\hline
FOM time & 5~min 54~s & 13~min 01~s \\
\hline
$N$ & \multicolumn{2}{c|}{$16$} \\
\hline
Speed-up & $\times$3554 & $\times$3886 \\
\hline
Avg. CPU time & 0.1~s & 0.2~s \\
\hline
mean$_{\bm\mu}$ $\epsilon_{abs}(\bm\mu)$ & $1.5\cdot10^{-2}$ & $2.7\cdot10^{-2}$ \\
\hline
mean$_{\bm\mu}$ $\epsilon_{rel}(\bm\mu)$ & $6.4\cdot10^{-2}$ & $8.3\cdot10^{-2}$ \\
\hline
\end{tabular}
\caption{Passive inflation and active contraction of an idealized left ventricle. Computational data related to Deep-HyROMnet for $N_h=19365$ and $N_h=39075$.}
\label{tab:ProlateC_hyper-ROM_fine}
\end{table}
\section{Conclusions}\label{sec:conclusion}
In this work we have addressed the solution to the parametrized elastodynamics equation, correlated with nonlinear constitutive law, by means of a new projection-based reduced order model (ROM), developed to accurately capture the state solution dynamics at a reduced computational cost with respect to full-order models (FOMs) providing expensive high-fidelity approximations.
We focused on Galerkin-reduced basis (RB) methods, characterized by a projection of the differential problem onto a low-dimensional subspace built, e.g., by performing proper orthogonal decomposition (POD) on a set of FOM solutions, and by the splitting of the reduction procedure into a costly offline phase and an inexpensive online phase. Numerical experiments showed that, despite their highly nonlinear nature, elastodynamics problems can be reduced by exploiting projection-based strategies in an effective way, with POD-Galerkin ROMs achieving very good accuracy even in presence of a handful of basis functions. However, when dealing with nonlinear problems, a further level of approximation is required to make the online stage independent of the high-fidelity dimension.
Hyper-reduction techniques, such as the discrete empirical interpolation method (DEIM), are necessary to efficiently handle the nonlinear operators. However, a serious issue is represented by the assembling (albeit onto a reduced mesh) of the approximated nonlinear operators in this framework. This observation suggested the idea of relying on surrogate models to perform operator approximation, overcoming the need to assemble the nonlinear terms onto the computational mesh.
Pursuing this strategy, we have proposed a new projection-based, deep learning-based ROM, \textit{Deep-HyROMnet}, which combines the Galerkin-RB approach with deep neural networks (DNNs) to assemble the reduced Newton system in an efficient way, thus avoiding the computational burden entailed by classical hyper-reduction strategies. This approach allows to rely on physics-based (thus, consistent) ROMs retaining the underlying structure of the physical model, as DNNs are employed only for the approximation of the reduced nonlinear operators, so that the problem displacement at each time instance is computed by solving the reduced nonlinear system. Regarding the offline cost of this hybrid reduction strategy, we point out that:
\begin{itemize}
\item FOM solutions are required only for the construction of the reduced basis functions;
\item since the nonlinear operators are collected during Newton iterations at each time step, a smaller number of ROM simulations with respect to purely data-driven approaches is sufficient for training the DNNs;
\item being the training data low-dimensional, we can avoid the overwhelming training times and costs that would be required by the DNN if FOM arrays were used.
\end{itemize}
Deep-HyROMnet has been successfully applied in a nonlinear solid mechanics context, showing remarkable improvement in terms of online CPU time with respect to POD-Galerkin-DEIM ROMs. Our goal in future works is to apply the developed strategy to other classes of nonlinear problems for which traditional hyper-reduction techniques represent a computational bottleneck.
|
2,869,038,154,923 | arxiv | \section{Introduction} \label{sec_introduction}
\revision{Distributed learning is a branch of machine learning in which large datasets are stored on several client devices, and a central server controls the learning process \cite{NIPS2010_abea47ba,MLSYS2019_c74d97b0}.
In this setting, the local data is never shared or transferred across clients, or shared with the central server, and the actual optimization can be performed either by the central server or separately on each client \cite{NIPS2006_77ee3bc5,45187}.
In the first case, learning is performed by the central server requesting (stochastic) gradient information from each client and aggregating this information within its optimization procedure \cite{10.1109/ALLERTON.2019.8919791,JMLR:v11:teo10a}.
From a mathematical optimization perspective, distributed learning presents several challenges including communication overheads \cite{46622,Dai2021} and asynchronicity \cite{10.1109/ALLERTON.2019.8919791}.
This may be further complicated in the case of federated learning, where the client datasets are assumed to be heterogeneous \cite{wang2021field}.}
Here, we consider the case of \revision{distributed} learning in the presence of adversarial attacks; the reliance on many clients and network communication makes this a more relevant concern than in traditional learning.
Specifically, we consider the case of Byzantine adversaries, which alter the gradient information sent from (potentially multiple) clients to the server, for example through data poisoning \cite{Biggio2012,chen2017targeted}, where individual clients' datasets are altered, or model update poisoning \cite{bhagoji2019analyzing}, where the information sent to the central server is directly corrupted.
The goal of adversarial attacks can be untargeted, attempting to decrease the model's accuracy overall \cite{Biggio2012}, or targeted, aiming to change the model's behavior on only a specific subset of inputs \cite{chen2017targeted,bhagoji2019analyzing}.
A comprehensive taxonomy of adversarial attacks is given in \cite[Section 5]{kairouz2021advances}.
In the presence of adversarial attacks, standard optimization methods such as stochastic gradient descent (SGD) can fail \cite{blanchard2017byzantinetolerant}, and so the development of optimization algorithms which are Byzantine resilient is of critical importance to the success of \revision{distributed} learning.
Where the individual clients' datasets are i.i.d., algorithms such as SGD can be \revision{made} Byzantine resilient using robust averaging techniques.
In this paper, we introduce a simplified convergence analysis of Byzantine Resilient SGD (BRSGD) \cite{blanchard2017byzantinetolerant} for nonconvex learning problems.
Our analysis covers the same range of approaches for aggregating gradient information from i.i.d.~client datasets, but uses more standard smoothness assumptions on the objective function and the non-corrupted stochastic gradients.
As a trade-off, we show convergence to first-order optimality in expectation, rather than almost surely as in \cite{blanchard2017byzantinetolerant}.
\revision{We also show a convergence rate of $\bigO(1/K^{(1-p)/2})$ when the learning rate sequence is chosen as $\bigO(k^{-p})$ for $p\in(1/2,1)$, which was not provided in \cite{blanchard2017byzantinetolerant}.}
\paragraph{Related Work}
A general analysis of SGD in a federated learning setting \revision{(which generalizes distributed learning)} for nonconvex objectives can be found in \cite{li2020unified}.
In the case of Byzantine resilient SGD for i.i.d.~client datasets, several different robust averaging techniques have been proposed.
These include the geometric median \cite{Chen2017,xie2018generalized,pillutla2019robust}, coordinate-wise medians and trimmed means \cite{pmlr-v80-yin18a,pmlr-v97-yin19a}, neighborhood-based averaging \cite{blanchard2017byzantinetolerant}, iterative filtering \cite{su2019securing,pmlr-v97-yin19a} and combinations of multiple such approaches \cite{pmlr-v80-mhamdi18a}.
The types of convergence theory of these different methods varies.
Convergence only for strongly convex objectives is considered in \cite{Chen2017,pillutla2019robust,pmlr-v80-yin18a}, and in \cite{su2019securing} for the full population loss (rather than the empirical loss).
For nonconvex objectives, high probability convergence to approximate first- and second-order optimal points is given in \cite{pmlr-v80-yin18a} and \cite{pmlr-v97-yin19a} respectively.
Alternatively, \cite{blanchard2017byzantinetolerant} considers BRSGD applied to nonconvex objectives with decreasing learning rates, which proves almost sure convergence to stationary points rather than to a neighborhood, similar to the standard SGD setting \cite{bottou2018optimization}.
This analysis applies to a generic robust aggregator satisfying specific assumptions.
Different aggregations methods which satisfy this assumption have been proposed in \cite{blanchard2017byzantinetolerant,xie2018generalized,pmlr-v80-mhamdi18a}.
We conclude by noting that \cite{Xie2019,data2020byzantineresilient,pmlr-v139-data21a} consider the case of Byzantine resilient federated learning in the case of non-i.i.d.~datasets and have developed specific robust aggregators suited for this setting (with associated convergence theory).
This is a more difficult problem, made visible for instance by \cite{data2020byzantineresilient} requiring a maximum of 25\% of clients be corrupted (rather than approximately 50\% for the i.i.d.~case).
\paragraph{Contributions}
In this paper, we present a simplified convergence result for BRSGD.
Unlike the original analysis in \cite{blanchard2017byzantinetolerant} (which was based on the analysis of SGD in \cite{Bottou98}), we use standard smoothness assumptions on the objective, closer in spirit to the standard analysis of SGD \cite{bottou2018optimization}.
For the (non-corrupted) stochastic gradient estimates, we use a general expected smoothness assumption based on \cite{khaled2020}.
Under these conditions, we prove the convergence of BRSGD to stationary points in expectation: we note that this is weaker than the almost-sure result from \cite{blanchard2017byzantinetolerant}, coming from our more standard problem assumptions and simpler analysis. Our result and proof technique has some similarities to \cite[Lemma 4.3]{sebbouh2021sure}, which shows almost sure convergence of standard SGD under general expected smoothness conditions on stochastic gradients.
As described above, since BRSGD in \cite{blanchard2017byzantinetolerant} is a generic framework, our results are applicable to any aggregation function satisfying the same assumptions, including all those in \cite{blanchard2017byzantinetolerant,xie2018generalized,pmlr-v80-mhamdi18a}.
\paragraph{Structure}
We begin by describing the general model of \revision{distributed} learning with Byzantine adversaries and describe the BRSGD method in \secref{sec_model}.
Our new convergence analysis is given in \secref{sec_convergence}.
\revision{The corresponding convergence rates are shown in \secref{sec_complexity}}, and we conclude in \secref{sec_conclusion}.
\paragraph{Notation}
We use $\|\cdot\|$ to be the Euclidean norm and $\langle \cdot, \cdot \rangle$ to be the corresponding inner product on $\R^d$, and let $[m]:=\{1,\ldots,m\}$.
\section{Problem \& Byzantine Resilient SGD} \label{sec_model}
We begin by describing the Byzantine adversarial model problem and the Byzantine resilient SGD algorithm from \cite{blanchard2017byzantinetolerant}.
\subsection{Byzantine Adversarial Model Problem}
In the \revision{distributed} learning problem, our data is \revision{split} across $m$ nodes (or devices) while our model is centralized \cite{kairouz2021advances}. For our $i^{\text{th}}$ node, $\{((x_i)_j, (y_i)_j)\}_{j=1}^{n_i}$ is our dataset, where $n_i$ is the number of data-points stored at that node. We will assume that each element is drawn from a distribution that is common across all nodes, $((x_i)_j, (y_i)_j) \sim \Omega$. As this distribution is theoretical, we replace our unknown distribution with the known empirical distribution $\Omega'_i$, where we select each element of our dataset with probability $\frac{1}{n_i}$.
Having been given this dataset, from a space of functions parametrized by $w \in \mathbb{R}^d$, we wish to find a model function $f$ for which $f(x) \approx y$, for all $i \in [m]$ and $(x,y) \sim \Omega'_i$. Hence, we look for a function $f(\cdot;w)$ that minimizes the average empirical risk across our $m$ nodes:
\begin{equation}
\min_{w \in \mathbb{R}^d} F(w) := \frac{1}{m} \sum^m_{i=1} F_i(w), \label{prob:2}
\end{equation}
where $F_i$ is the empirical risk of the model $f(w)$ on the $i^{\text{th}}$ node. Specifically:
\begin{equation}
F_i(w) := \frac{1}{n_i} \sum^{n_i}_{j=1} l((y_i)_j,f((x_i)_j;w)).
\end{equation}
where $l(\cdot,\cdot)$ is a loss function that quantifies the difference between the two values. Note that both the loss function and the model remain constant across the nodes.
We solve \eqref{prob:2} with iterative methods converging to a neighbourhood of stationary points of our problems.
These iterative methods involve each node sending an estimation of the gradient of our function at the current point to the central server. We model component failure or corruption by setting some of our nodes to be Byzantine adversaries \cite{blanchard2017byzantinetolerant}, who may send arbitrary values to the central server.
In our problem, of our $m$ nodes, $q$ will be Byzantine adversaries as defined below. Typically we require $q < m/2$, however there are slight differences in some methods.
\begin{definition}[Byzantine Adversaries]
Let $\{(g_1)_k, (g_2)_k, ..., (g_m)_k\}$ be the set of correct local gradient estimators calculated by each node for the $k^{\text{th}}$ iteration of an algorithm. If $q$ out of $m$ vectors are Byzantine adversaries, then, the set of correct vectors at every iteration are partially replaced by vectors $\{(\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\}$, according to:
\begin{equation}
(\Tilde{g}_i)_k :=
\begin{cases}
(B_i)_k,& \text{if}\ \text{the $i^{\text{th}}$ node is Byzantine,}\\
(g_i)_k, & \text{otherwise.}
\end{cases}
\end{equation}
The indices of the adversaries may change across iterations and the value of the Byzantine gradient, $(B_i)_k$, may be a function dependent on $\{(g_1)_k, (g_2)_k, ..., (g_m)_k\}$, the current value of the model $x_k$, the current step-size $\alpha_k$, or any previous information.
\end{definition}
In the taxonomy of \cite[Section 5]{kairouz2021advances}, this framework corresponds to dynamic, white box adversaries using within-update collusion, where the adversary functions via data poisoning or model update poisoning.
\subsection{BRSGD Algorithm}
We now outline the Byzantine Resilient SGD (BRSGD) algorithm framework for solving \eqref{prob:2} in the presence of Byzantine adversaries.
In iteration $k$ of this method, initially from \cite{blanchard2017byzantinetolerant}, the central server initially collects the (possibly corrupted) local gradient information $(\Tilde{g}_i)_k \in \R^d$ from each node $i \in [m]$.
It then aggregates these $m$ vectors to produce a final gradient estimate $A_k\in\R^d$ using some aggregation function $\text{\sc Agg}\left((\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\right)$: we will later give specific requirements on $\text{\sc Agg}$.
We then take a gradient descent-type step with a pre-specified learning rate $\alpha_k>0$.
The full BRSGD method is given in \algref{alg_brsgd}.
\begin{algorithm}[tb]
\begin{algorithmic}[1]
\Require Initial point $w_0 \in \mathbb{R}^d$, sequence of learning rates $\{\alpha_k\}^{\infty}_{k=0}$ with $\alpha_k > 0$ for all $k$, gradient aggregation function $\text{\sc Agg}: \underbrace{\R^d \times \cdots \times \R^d}_{\text{$m$ times}} \to \R^d$.
\vspace{0.5em}
\For{$k=0,1,2,\ldots$}
\For{machines $i = 1,2,...,m$ in parallel}
\If{$i$ is a non-Byzantine node}
\State Compute local stochastic gradient\ $(\Tilde{g}_i)_k = (g_i)_k$, where $(g_i)_k \in \mathbb{R}^d$ is the (stochastic) gradient estimate at node $i$.
\Else
\State Set $(\Tilde{g}_i)_k = (B_i)_k$, where $(B_i)_k \in \mathbb{R}^d$ is the attack gradient.
\EndIf
\EndFor
\State Aggregate received gradient information: $A_k = \text{\sc Agg}\left((\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\right)$.
\State Set $w_{k+1} = w_k - \alpha_k A_k$.
\EndFor
\end{algorithmic}
\caption{Byzantine Resilient Stochastic Gradient Descent (BRSGD) \cite{blanchard2017byzantinetolerant}}
\label{alg_brsgd}
\end{algorithm}
Since the non-corrupted local gradient estimators $(g_i)_k$ can be stochastic gradient estimates, we also outline our formal stochastic approach to this problem.
We introduce a probability space, $(\mathbb{P},\mathcal{F},\Omega)$, with an associated filtration $(\mathcal{F}_k)_{k\in\mathbb{N}}$ to model our gradient information as an adapted process on a filtration, based on framework from \cite{paquette2018}.
Our sample space will be $\Omega' := \Omega_1' \times \Omega_2' \times... \times \Omega_m'$. Let the random variable modelling the sample drawn at the $i^{th}$ node for the $k^{th}$ iterate be $(\zeta_k)_i := (\zeta_k)_i (\omega)$.
The value of the gradient estimator at the $i^{th}$ node for the $k^{th}$ iterate will be modelled by $g_i(w_k) := G(w_k,(\zeta_k)_i)$, where $G(w,\zeta)$ is the random variable modelling gradient updates at point $w$ and it will be denoted $G(w)$.
Since our datasets are assumed to be i.i.d., our function $G(w_k, \zeta)$ is not dependent on the node, and hence, we have the following property:
\begin{equation}
\mathbb{E}[g_1(w_k)] = \mathbb{E}[g_2(w_k)] = \cdots = \mathbb{E}[g_m(w_k)].
\end{equation}
Our filtration $\mathcal{F}_k$ will be the $\sigma$-algebra generated by our previous random variables, i.e. $\mathcal{F}_k = \sigma \left(\bigcup^{k-1}_{i=0} \bigcup^{m}_{j=1} \sigma((\zeta_i)_j) \right)$ and let $\mathcal{F} := \bigcup^{\infty}_{n=1} \mathcal{F}_k$.
Therefore, both our aggregated and individual gradient estimates are an adapted processes on the filtration.
Furthermore, the iterates of our algorithm will have the property:
\begin{equation}
\mathbb{E}_k[w_{k}] = w_{k},
\end{equation}
where we denote the conditional expectation of a random variable with respect to filtration as $\mathbb{E}_k[\cdot] := \mathbb{E}[\cdot|\mathcal{F}_k]$.
\paragraph{Choice of aggregation function}
For our convergence theory to hold, the aggregation function $\text{\sc Agg}$ must satisfy the following assumption, from \cite{blanchard2017byzantinetolerant}.
\begin{assumption}[\revision{$\alpha$}-Byzantine Resilience] \label{ass_resilient_agg}
Let $A_k := \text{\sc Agg}\left((\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\right)$.
Then there exists a constant \revision{$\alpha\in[0,\pi/2)$} such that for all $k$ we have
\begin{enumerate}
\item \revision{$\langle \mathbb{E}_k[A_k], \nabla F(w_k) \rangle \geq (1 - \sin(\alpha)) \|\nabla F(w_k)\|^2$}, and
\item For $r=2,3,4$, $\mathbb{E}_k[\|A_k\|^r]$ bounded above by a linear combination of terms: \linebreak $\mathbb{E}_k[\|G(w_k)\|^{r_1}] \cdot \mathbb{E}_k[\|G(w_k) \|^{r_2}]\cdot ... \cdot \mathbb{E}_k[\|G(w_k)\|^{r_n}]$ with $r_1 + r_2 + ... + r_n = r$.
\end{enumerate}
\end{assumption}
\revision{
\begin{remark}
We note that while \cite{blanchard2017byzantinetolerant} requires $r=2,3,4$ in \assref{ass_resilient_agg}(2), our convergence result only requires $r=2$.
In fact, our result requires only the weaker condition
\begin{align}
\mathbb{E}_k[\|A_k\|^2] \leq E \mathbb{E}_k[\|G(w_k)\|^2], \label{eq_resilient_agg_weaker}
\end{align}
for some constant $E$.
The $r=2$ case of \assref{ass_resilient_agg}(2) implies \eqref{eq_resilient_agg_weaker} via
\begin{align}
\mathbb{E}_k[\|A_k\|^2] \leq C_1 \mathbb{E}_k[\|G(w_k)\|^2] + C_2 \mathbb{E}_k[\|G(w_k)\|]^2 \leq (C_1+C_2) \mathbb{E}_k[\|G(w_k)\|^2],
\end{align}
for some constants $C_1,C_2>0$, and where the second inequality follows from Jensen's inequality for conditional expectations.
\end{remark}
}
The works \cite{blanchard2017byzantinetolerant,xie2018generalized,pmlr-v80-mhamdi18a} give multiple examples of suitable functions $\text{\sc Agg}$ which satisfy \assref{ass_resilient_agg}, where \revision{$\alpha$} typically depends on $\text{\sc Agg}$, the number of clients $m$ and the number of corrupted clients $q$.
\revision{Specifically, the $\text{\sc Agg}$ functions proposed in these works are:
\begin{itemize}
\item Krum \cite{blanchard2017byzantinetolerant}, which returns the vector from $\{(\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\}$ that minimizes the distance between it and its nearest neighbours;
\item Variations of the median \cite{xie2018generalized}, including the marginal (i.e.~component-wise) median, the geometric median
\begin{align}
\text{\sc Agg}\left((\Tilde{g}_1)_k, (\Tilde{g}_2)_k, ..., (\Tilde{g}_m)_k\right) = \argmin_{g\in\R^d} \sum_{i=1}^{m} \|g-(\Tilde{g}_i)_k\|,
\end{align}
and the `mean-around-median', the (component-wise) mean of the closest vectors to the marginal median;
\item Bulyan \cite{pmlr-v80-mhamdi18a}, which uses an existing $\text{\sc Agg}$ function such as the above to generate a set of candidate gradients and then apply a `mean-around-median' aggregation to that set.
\end{itemize}
}
\subsection{Existing Convergence Theory}
We now describe the underlying assumptions and state the existing convergence theory for BRSGD (\algref{alg_brsgd}) as given in \cite{blanchard2017byzantinetolerant}.
The requirements on the objective \eqref{prob:2} are based on \cite{Bottou98}:
\begin{assumption} \label{ass_old_smoothness}
The objective function $F$ is $C^3$, bounded below, and there exist constants $D,\epsilon \geq 0$ and $0 \leq \beta < \frac{\pi}{2} $ such that, for all $w \in \mathbb{R}^d$ with $\|w\|^2 \geq D$, we have:
\begin{equation}
\|\nabla F(w)\| \geq \epsilon, \quad \text{and} \quad \frac{\langle w, \nabla \revision{F}(w) \rangle}{\|w\| \cdot \|\nabla \revision{F}(w)\|} \geq \cos(\beta). \label{eq_convex_eventually}
\end{equation}
\end{assumption}
Next, the requirement on the stochastic gradients is given by the following two assumptions.
\begin{assumption} \label{ass_old_gradients1}
The gradient estimator $G(w_k)$ is unbiased, $\mathbb{E}_k[G(w_k)] = \nabla F(w_k)$, and for all $r \in \{2,3,4\}$, there exist non-negative constants $A_r$ and $B_r$ such that \linebreak $\mathbb{E}_k[\|G(w_k)\|^r] \leq A_r + B_r \|w_k\|^r$.
\end{assumption}
We are now able to state the main convergence result from \cite{blanchard2017byzantinetolerant}.
\begin{theorem}[Proposition 2, \cite{blanchard2017byzantinetolerant}] \label{thm_old_convergence}
Suppose that Assumptions \ref{ass_resilient_agg}, \ref{ass_old_smoothness} and \ref{ass_old_gradients1} hold, with $\alpha + \beta < \pi/2$ (with $\alpha$ from \revision{\assref{ass_resilient_agg}} and $\beta$ from \assref{ass_old_smoothness}), \revision{and
\begin{align}
\eta(m,q) \sqrt{\mathbb{E}[\|G(w,\zeta)-\grad F(w)\|^2]} \leq \sin(\alpha)\|\grad F(w)\|, \label{eq_eta_condition}
\end{align}
where $m > 2q+2$ and
\begin{align}
\eta(m,q) := \sqrt{2\left(m-q+\frac{q(m-q-2) + q^2(m-q-1)}{m-2q-2}\right)}.
\end{align}
}
\revision{If} we run \algref{alg_brsgd} with learning rates satisfying $\sum_{k=0}^{\infty} \alpha_k = \infty$ and $\sum_{k=0}^{\infty} \alpha_k^2 < \infty$.
Then $\lim_{k\to\infty} \|\grad F(w_k)\| = 0$ almost surely.
\end{theorem}
We conclude by noting the technical condition $\alpha+\beta < \pi/2$, relating the errors in the stochastic gradients and the smoothness of the objective.
We do not need a condition like this in our analysis.
\revision{The condition \eqref{eq_eta_condition} relates the stochastic gradients to the aggregation function (provided there are not too many adversaries). Although our analysis does not require this condition explicitly, similar conditions are required to show Byzantine resilience of aggregation functions such as Krum \cite{blanchard2017byzantinetolerant} and the geometric median \cite{xie2018generalized}.}
\section{New Convergence Analysis} \label{sec_convergence}
We now present our new, simplified analysis of BRSGD.
Compared to \thmref{thm_old_convergence}, our result uses simpler and more common assumptions on both the objective $F$ and the stochastic gradient estimator $G(w,\zeta)$, and a specific choice of learning rate.
As a result, the conclusion is weaker: we get convergence to stationary points in expectation rather than almost surely.
It has the same requirements on the aggregating function $\text{\sc Agg}$.
In particular, we note that \thmref{thm_old_convergence} requires \eqref{eq_convex_eventually}, essentially that $F$ is `convex enough' outside a certain bounded region.
This has the downside, for example, of excluding the model space of neural networks with soft-max activation functions, a common model space in distributed learning, as the activation function has flat asymptotes \cite{Bottou98}.
\begin{assumption} \label{ass_new_smoothness}
Each term in the objective function $F$ is $L$-smooth (i.e.~continuously differentiable and $\grad F$ is $L$-Lipschitz continuous) and bounded below by some $F_{\text{low}}$.
\end{assumption}
The key implication of $F$ being $L$-smooth is the upper bound
\begin{equation}
F(y) \leq F(x) + \langle \nabla F(x),y-x \rangle + \frac{L}{2}\|y-x\|^2, \qquad \forall x,y\in\R^d. \label{eq_lipschitz}
\end{equation}
For our stochastic gradient estimator, we will use a version of the expected smoothness property \cite[Assumption 3.2]{khaled2020}.
We note that \cite{khaled2020} shows that variants of SGD including minibatching, importance sampling, gradient combinations and their combinations all satisfy expected smoothness.
\begin{assumption}(Expected Smoothness) \label{ass_new_gradients}
For all $k$, the gradient estimator $G(w_k)$ is unbiased:
\begin{equation}
\mathbb{E}_k[G(w_k)] = \nabla F(w_k), \label{ass:1}
\end{equation}
and there exist non-negative constants $A$, $B$ and $C$ (independent of $k$), such that:
\begin{equation}
\mathbb{E}_k[\|G(w_k)\|^2] \leq 2A(F(w_k) -F_{\text{low}}) + B\|\nabla F(w_k)\|^2 + C. \label{ass:2}
\end{equation}
\end{assumption}
To prove our result, we will need the below technical lemma, a generalization of \cite[Lemma 2]{khaled2020}, which corresponds to the case of $a_k \equiv a$ for all $k$.
\begin{lemma} \label{lem_weighting_sequence}
Let $\{r_k\}_{n \in \mathbb{N}}$, $\{d_k\}_{n \in \mathbb{N}}$ be positive-valued sequences, $\{a_k\}_{n \in \mathbb{N} \cup \{-1\}}$ be a non-increasing, positive-valued sequence and $L$, $A$ and $C$ be positive constants such that, for all $k \in \mathbb{N}$,
\begin{equation}
\frac{a_k}{2} r_k \leq (1+a_k^2 L A)d_k -d_{k+1} + \frac{L C a^2_k}{2}. \label{lem4:1}
\end{equation}
Then, the following identity holds:
\begin{equation}
\min_{0 \leq k \leq K-1} r_k \leq \frac{2 d_0}{a_{-1} K W_{K-1}} + \frac{L C}{K W_{K-1}} \sum^{K-1}_{k=0} a_k W_k, \label{lem4:4}
\end{equation}
where $W_k$ is a weighting sequence defined by $W_{-1} = 1$ and $W_k = W_{k-1} \frac{a_k}{a_{k-1} (1+L A a_k^2)}$ for $k\geq 0$.
\end{lemma}
\begin{proof}
We begin by taking \eqref{lem4:1} and multiplying through by $\frac{W_k}{a_k}$ for each $k \in \mathbb{N}$. Thus,
\begin{equation}
\frac{1}{2} r_k W_k \leq (1+a_k^2 L A)d_k \frac{W_k}{a_k} - d_{k+1} \frac{W_k}{a_k} + \frac{L C a_k W_k}{2}.
\end{equation}
Noting that $\frac{W_k (1 + a_k^2 L A)}{a_k} = \frac{W_{k-1}}{a_{k-1}}$, we simplify:
\begin{equation}
\frac{1}{2} r_k W_k \leq d_k \frac{W_{k-1}}{a_{k-1}} - d_{k+1} \frac{W_k}{a_k} + \frac{L C a_k W_k}{2}.
\end{equation}
This allows us to apply a telescoping sum, adding together the first $K-1$ inequalities we receive:
\begin{equation}
\frac{1}{2} \sum^{K-1}_{k=0} r_k W_k \leq \frac{ d_0}{a_{-1}} - \frac{W_{K-1} d_K}{a_{K-1}} + \frac{L C}{2} \sum^{K-1}_{k=0} a_k W_k.
\end{equation}
Let $\hat{W} := \sum^{K-1}_{k=0} W_k$ and divide through by $\hat{W}$ on both sides:
\begin{align}
\frac{1}{2 \hat{W}} \sum^{K-1}_{k=0} r_k W_k &\leq \frac{ d_0}{a_{-1} \hat{W}} - \frac{W_{K-1} d_K}{a_{K-1} \hat{W}} + \frac{L C}{2 \hat{W}} \sum^{K-1}_{k=0} a_k W_k, \\
\frac{1}{2 \hat{W}} \sum^{K-1}_{k=0} r_k W_k + \frac{W_{K-1} d_K}{a_{K-1} \hat{W}} &\leq \frac{d_0}{a_{-1} \hat{W}} + \frac{L C}{2 \hat{W}} \sum^{K-1}_{k=0} a_k W_k.
\end{align}
To simplify further we note:
\begin{equation}
\frac{1}{2} \min_{0 \leq k \leq K-1} r_k = \frac{1}{2 \hat{W}} \left(\min_{0 \leq k \leq K-1} r_k\right) \sum^{K-1}_{k=0} W_k \leq \frac{1}{2 \hat{W}} \sum^{K-1}_{k=0} r_k W_k + \frac{w_{K-1} d_K}{a_{K-1} \hat{W}}, \label{lem4:2}
\end{equation}
where \eqref{lem4:2} holds as $W_{K-1}$, $d_K$, $a_{K-1}$ and $\hat{W}$ are all strictly positive. Hence:
\begin{equation}
\frac{1}{2} \min_{0 \leq k \leq K-1} r_k \leq \frac{d_0}{a_{-1} \hat{W}} + \frac{L C}{2 \hat{W}} \sum^{K-1}_{k=0} a_k W_k \label{lem4:22}.
\end{equation}
We now note $\frac{a_k}{a_{k-1}} \leq 1$, as $\{a_k\}_{n \in \mathbb{N} \cup \{-1\}}$ is non-increasing. Furthermore, $\frac{1}{1+ L A a_k^2} \leq 1$, as $L,A\ \text{and}\ a_k > 0$. Therefore:
\begin{equation}
W_{k} = W_{k-1} \frac{a_k}{a_{k-1}} \frac{1}{(1+L A a_k^2)} \leq W_{k-1},
\end{equation}
and hence, $\{W_n\}_{\{n \in \mathbb{N} \cup \{-1\} \}}$ is non-increasing. Using this, we note:
\begin{equation}
\frac{1}{\hat{W}} = \frac{1}{\sum^{K-1}_{k=0} W_k} \leq \frac{1}{\sum^{K-1}_{k=0} \min_{0 \leq k \leq K-1} W_k} = \frac{1}{K W_{K-1}}. \label{lem4:3}
\end{equation}
Substituting \eqref{lem4:3} into the right-hand side of \eqref{lem4:22} and multiplying by 2 provides our result.
\end{proof}
We are now ready to prove our main result.
\begin{theorem} \label{thm_main_convergence}
Suppose that Assumptions~\ref{ass_resilient_agg}, \ref{ass_new_smoothness} and \ref{ass_new_gradients} hold.
If we run \algref{alg_brsgd} with learning rate sequence that satisfies: $\{\alpha_k\}_{k \in \mathbb{N}}$ is non-increasing, $\alpha_0 \leq \frac{1 - \sin (\alpha)}{L B'}$, $\sum_{k=0}^{\infty} \alpha_k^2 < \infty$, and $\lim_{k \to \infty} \frac{1}{k \alpha_{k-1}} = 0$
(where $B' = B \revision{E}$; $L$, $B$ and \revision{$E$} are defined in Assumptions~\ref{ass_new_smoothness} and \ref{ass_new_gradients}, and \revision{\eqref{eq_resilient_agg_weaker}---a consequence of \assref{ass_resilient_agg}(2)---respectively, and $\alpha$ comes from \assref{ass_resilient_agg}(1)}), then
\begin{equation}
\lim_{K \to \infty} \left(\min_{0\leq k \leq K-1} \mathbb{E}[\|\nabla F (w_k)\|]\right) = 0.
\end{equation}
\end{theorem}
\begin{proof}
We begin by applying \eqref{eq_lipschitz} and simplifying using $w_{k+1} = w_k - \alpha_k A_k$:
\begin{align}
F(w_{k+1}) &\leq F(w_k) + \langle \nabla F(w_k), w_{k+1}-w_k \rangle + \frac{L}{2} \|w_{k+1}-w_k\|^2, \\
&= F(w_k) - \alpha_k \langle \nabla F(w_k) , A_k \rangle + \frac{L \alpha_k^2}{2} \|A_k\|^2.
\end{align}
We then take the expectation of the above conditioned on $\mathcal{F}_k$ from our filtration:
\begin{align}
\mathbb{E}_k[F(w_{k+1})] \leq F(w_k) - \alpha_k \langle \nabla F(w_k) , \mathbb{E}_k[A_k] \rangle + \frac{L \alpha_k^2}{2} \mathbb{E}_k[\|A_k\|^2].
\end{align}
To simplify, we apply \revision{\assref{ass_resilient_agg}(2)} to get:
\begin{equation}
(1 - \sin(\alpha))\|\nabla F(w_k)\|^2 \leq \langle \nabla F(w_k),\mathbb{E}_k[A_k] \rangle, \label{thm52:1}
\end{equation}
and, for some constant $\revision{E}$,
\begin{equation}
\mathbb{E}_k [\|A_k\|^2] \leq \revision{E} \mathbb{E}_k [\|G(w_k)\|^2]. \label{thm52:2}
\end{equation}
Applying \eqref{thm52:1} and \eqref{thm52:2}, and subtracting $F_{\text{low}}$ from both sides, we simplify:
\begin{align}
\mathbb{E}_k[F(w_{k+1})] - F_{\text{low}} &\leq F(w_k) - F_{\text{low}} - \alpha_k \langle \nabla F(w_k) , \mathbb{E}_k[A_k] \rangle + \frac{L \alpha_k^2}{2} \mathbb{E}_k[\|A_k\|^2],\\
&\leq F(w_k) - F_{\text{low}} - \alpha_k (1 - \sin(\alpha))\|\nabla F(w_k)\|^2 \\
&\quad + \frac{L \alpha_k^2}{2} \revision{E} \mathbb{E}_k [\|G(w_k)\|^2].
\end{align}
We now simplify using \assref{ass_new_gradients}:
\begin{equation}
\revision{E} \mathbb{E}_k [\|G(w_k)\|^2] \leq \revision{E} \left(2A(F(w_k) - F_{\text{low}}) + B \|\nabla F(w_k)\|^2 + C \right).
\end{equation}
Collecting terms, we will rewrite the upper bound for our final term as:
\begin{equation}
\revision{E} \mathbb{E}_k [\|G(w_k)\|^2 \leq 2A'(F(w_k) - F_{\text{low}}) + B' \|\nabla F(w_k)\|^2 + C'.
\end{equation}
Thus:
\begin{align}
\mathbb{E}_k[F(w_{k+1})] - F_{\text{low}} &\leq (1+L \alpha_k^2 A')(F(w_k) - F_{\text{low}}) \\
&\quad - \left(\alpha_k \left(1 - \sin(\alpha)\right) -\frac{L \alpha_k^2 B'}{2}\right)\|\nabla F(w_k)\|^2 + \frac{L \alpha_k^2 C'}{2}.
\end{align}
Taking total expectations and using the Tower Property:
\begin{align}
\mathbb{E}[F(w_{k+1}) - F_{\text{low}}] &\leq (1 + L \alpha_k^2 A') \mathbb{E}[(F(w_k)-F_{\text{low}})] \\
&\quad- \left(\alpha_k \left(1 - \sin(\alpha)\right) -\frac{L \alpha_k^2 B'}{2}\right) \mathbb{E}[\|\nabla F(w_k)\|^2] + \frac{L \alpha_k^2 C'}{2}. \label{thm52:11}
\end{align}
Defining $\delta_{k} := \mathbb{E}[F(w_{k}) - F_{\text{low}}]$, \eqref{thm52:11} becomes:
\begin{equation}
\alpha_k\left(1 - \sin(\alpha) - \frac{L B' \alpha_k}{2}\right) \mathbb{E}[\|\nabla F(w_k)\|^2] \leq (1 + L \alpha_k^2 A') \delta_k - \delta_{k+1} + \frac{L \alpha_k^2 C'}{2}.
\end{equation}
Furthermore, our requirements on \revision{the learning rate sequence $\alpha_k$} guarantee $1-\sin(\alpha) - \frac{LB'\alpha_k}{2} \geq \frac{1-\sin(\alpha)}{2}$, hence we define $r_k := (1-\sin(\alpha)) \mathbb{E}[\|\nabla F(w_k)\|^2]$ and get:
\begin{equation}
\frac{\alpha_k}{2} r_k \leq (1 + L \alpha_k^2 A') \delta_k - \delta_{k+1} + \frac{L \alpha_k^2 C'}{2}.
\end{equation}
We now apply \lemref{lem_weighting_sequence} to our problem using the following weighting sequence:
\begin{equation}
W_k :=
\begin{cases}
1,& \text{if}\ k = -1,\\
W_{k-1} \frac{\alpha_k}{\alpha_{k-1} (1+LA' \alpha_k^2)}, & \text{otherwise.}
\end{cases}
\end{equation}
Hence:
\begin{equation}
\min_{0 \leq k \leq K-1} r_k \leq \frac{2 \delta_0}{\alpha_{-1} K W_{K-1}} + \frac{LC'}{K W_{K-1}} \sum^{K-1}_{k=0} \alpha_k W_k, \label{eq_after_lemma}
\end{equation}
where we define $\alpha_{-1} := \alpha_0$ for convenience.
We now show $\lim_{K \to \infty} \left( \min_{0 \leq k \leq K-1} r_k \right) = 0$. To do this, we return to the definition of our weighting sequence and note:
\begin{equation}
W_{k} = \frac{ \alpha_k}{\alpha_{-1}} \prod^{k}_{j=0} \frac{1}{1 + L A' \alpha_j^2}.
\end{equation}
Hence:
\begin{align}
\min_{0 \leq k \leq K-1} r_k &\leq \frac{2 \delta_0}{\alpha_{-1}} \left( \frac{\alpha_{-1}}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \right) \nonumber \\
&\quad + L C' \left( \frac{\alpha_{-1}}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \right) \sum^{K-1}_{k=0} \left( \frac{\alpha_{k}^2}{\alpha_{-1}} \prod^{k}_{j=0} \frac{1}{1 + L A' \alpha_j^2} \right). \label{eq_tmp1}
\end{align}
In order to simplify, we will bound a pair of infinite products and a summation. We will first show that:
\begin{equation}
\prod^{\infty}_{k=0} (1 + L A' \alpha_k^2) < \infty.
\end{equation}
Recall that, for a sequence of positive real numbers $\{y_n\}_{n \in \mathbb{N}}$, $\prod^{\infty}_{k=0} y_k$ converges if and only if $\sum^{\infty}_{k=0} \log(y_k)$ converges. From our specification of $\alpha_k$, we know $\sum^{\infty}_{k=0} \alpha_k^2 < \infty$. Let:
\begin{equation}
Q := \sum^{\infty}_{k=0} \alpha_k^2.
\end{equation}
When $y > -1$, $\log(1+y) \leq y$. Hence:
\begin{align}
\sum^{\infty}_{k=0} \log(1 + L A' \alpha_k^2) &\leq \sum^{\infty}_{k=0} L A' \alpha_k^2 \leq L A' Q. \label{eq_tmp_ref}
\end{align}
Therefore, $P := \prod^{\infty}_{k=0} (1 + L A' \alpha_k^2) < \infty$. Secondly, we note that, for all $k \in \mathbb{N}$:
\begin{equation}
\prod^{k}_{j=0} \frac{1}{1 + L A' \alpha_k^2}\leq 1, \label{eq_tmp2}
\end{equation}
as, $L A' \alpha_k^2 > 0$, for all $k \in \mathbb{N}$. This, in turn, allows us to simplify another summation. Specifically:
\begin{align}
\sum^{\infty}_{k=0} \left( \frac{\alpha_{k}^2}{\alpha_{-1}} \prod^{k}_{j=0} \frac{1}{1 + L A' \alpha_j^2} \right) &\leq \frac{1}{\alpha_{-1}} \sum^{\infty}_{k=0} \alpha_k^2 = \frac{Q}{\alpha_{-1}}.
\end{align}
We now take the limit of both sides of \eqref{eq_tmp1} and recover our result.
\begin{align}
\lim_{K \to \infty} &\left(\min_{0 \leq k \leq K-1} r_k\right) \nonumber \\
&\leq \lim_{K \to \infty} \left( \frac{2 \delta_0}{\alpha_{-1}} \left( \frac{\alpha_{-1}}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \right) \right) \\
&\quad+ \lim_{K \to \infty} \left(L C' \left( \frac{\alpha_{-1}}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \right) \sum^{K-1}_{k=0} \left( \frac{\alpha_k^2}{\alpha_{-1}} \prod^{k}_{j=0} \frac{1}{1 + \alpha_j^2 L A'} \right) \right), \\
&\leq \revision{2\delta_0 \left(\lim_{K\to\infty} \frac{1}{K\alpha_{K-1}}\right) \prod_{k=0}^{\infty}(1+LA' \alpha_k^2)} \nonumber \\
&\quad \revision{+ L C' \alpha_{-1} \left(\lim_{K\to\infty} \frac{1}{K\alpha_{K-1}}\right) \left(\prod_{k=0}^{\infty}(1+LA'\alpha_k^2)\right) \left(\sum^{\infty}_{k=0} \left( \frac{\alpha_{k}^2}{\alpha_{-1}} \prod^{k}_{j=0} \frac{1}{1 + L A' \alpha_j^2} \right)\right)}, \\
&= \revision{(2\delta_0 P + L C' P Q) \lim_{K\to\infty} \frac{1}{K\alpha_{K-1}}}, \label{eq_nearly_done}
\end{align}
\revision{where $P$ is defined after \eqref{eq_tmp_ref}}. By assumption, we have $\lim_{K \to \infty} \frac{1}{K \alpha_{K-1}} = 0 $, therefore:
\begin{equation}
\lim_{K \to \infty} \left(\min_{0 \leq k \leq K-1} (1-\sin(\alpha)) \mathbb{E}[\|\nabla F(w_k)\|^2]\right) = 0.
\end{equation}
Our result then follows from Jensen's inequality, $\mathbb{E}[\|\nabla F(w)\|]^2 \leq \mathbb{E}[\|\nabla F(w)\|^2]$.
\end{proof}
We note that our assumptions on the learning rate sequence $\{\alpha_k\}_k$ allows for sequences which decay as $\alpha_k \sim k^{-p}$ for any $p\in(1/2,1)$.
\revision{
\subsection{Iterate Selection} \label{sec_iterate_selection}
The above result shows that there is a subsequence of iterates which converges in expectation.
We now give a probabilistic procedure to select a single iterate with small expected gradient, inspired by the analysis in \cite[Theorem 6.1]{Lan2019}.
In this context, we assume that \algref{alg_brsgd} has been run for $K$ iterations, and we randomly select an iterate from $w_0,\ldots,w_{K-1}$ according to a specific probability distribution depending on the learning rate sequence $\alpha_k$.
However, compared to \cite[Corollary 6.1]{Lan2019}, the choice of $\alpha_k$ does not depend on $K$, and so \algref{alg_brsgd} can always be continued from its previous endpoint if the desired accuracy is not achieved.
This analysis requires only small modifications of \lemref{lem_weighting_sequence} and \thmref{thm_main_convergence}, which we present here.
\begin{lemma} \label{lem_weighting_sequence_random}
Suppose the assumptions of \lemref{lem_weighting_sequence} hold, including the definition of the weighting sequence $W_k$.
If, for any $K\geq 0$, we define the random variable $R_K\in\{0,\ldots,K-1\}$ by
\begin{align}
\mathbb{P}(R_K=k) := \frac{W_{k}}{\sum^{K-1}_{i=0} W_i}, \quad k=0,...,K-1, \label{eq_weighting_iterate}
\end{align}
then
\begin{equation}
\mathbb{E}[r_{R_K}] \leq \frac{2 d_0}{a_{-1} K W_{K-1}} + \frac{L C}{K W_{K-1}} \sum^{K-1}_{k=0} a_k W_k.
\end{equation}
\end{lemma}
\begin{proof}
The proof of this result is identical to that of \lemref{lem_weighting_sequence}, except that \eqref{lem4:2} is replaced by
\begin{align}
\frac{1}{2} \mathbb{E}[r_{R_K}] = \frac{1}{2 \hat{W}} \sum^{K-1}_{k=0} r_k W_k \leq \frac{1}{2 \hat{W}} \sum^{K-1}_{k=0} r_k W_k + \frac{w_{K-1} d_K}{a_{K-1} \hat{W}},
\end{align}
from which we conclude
\begin{align}
\frac{1}{2} \mathbb{E}[r_{R_K}] \leq \frac{d_0}{a_{-1} \hat{W}} + \frac{L C}{2 \hat{W}} \sum^{K-1}_{k=0} a_k W_k,
\end{align}
in place of \eqref{lem4:22}.
\end{proof}
\begin{corollary} \label{cor_convergence_random_iterate}
Suppose that the assumptions of \thmref{thm_main_convergence} hold, and for any $K\geq 0$ we define the random variable $R_K$ as per \eqref{eq_weighting_iterate}.
Then
\begin{equation}
\lim_{K \to \infty} \mathbb{E}[\|\nabla F (w_{R_K})\|] = 0.
\end{equation}
\end{corollary}
\begin{proof}
The proof of this result is identical to that of \thmref{thm_main_convergence}, but we replace \eqref{eq_after_lemma} with
\begin{align}
\mathbb{E}[r_{R_K}] \leq \frac{2 \delta_0}{\alpha_{-1} K W_{K-1}} + \frac{LC'}{K W_{K-1}} \sum^{K-1}_{k=0} \alpha_k W_k,
\end{align}
which follows from \lemref{lem_weighting_sequence_random}.
Hence instead of \eqref{eq_nearly_done} we reach
\begin{align}
\lim_{K \to \infty} \mathbb{E}[r_{R_K}] &\leq (2\delta_0 P + L C' P Q) \lim_{K\to\infty} \frac{1}{K\alpha_{K-1}} = 0,
\end{align}
from which the result follows by Jensen's inequality, $(1-\sin(\alpha)) \mathbb{E}[\|\nabla F(w_{R_K})\|]^2 \leq r_{R_K}$.
\end{proof}
}
\revision{
\section{Convergence Rate} \label{sec_complexity}
The previous section gives a convergence analysis for \algref{alg_brsgd} under general assumptions on the decreasing learning rate sequence $\alpha_k$.
We now specialize these results to give a convergence rate for the case $\alpha_k \sim k^{-p}$ for $p\in(1/2,1)$.
\begin{corollary} \label{cor_complexity}
Suppose the assumptions of \thmref{thm_main_convergence} hold, and the learning rate sequence is given by
\begin{align}
\alpha_k = \frac{1-\sin(\alpha)}{L B' (k+1)^p}, \qquad k=0,1,2,\ldots,
\end{align}
for some $p\in(1/2,1)$.
Then
\begin{align}
\min_{0 \leq k \leq K-1} \mathbb{E}[\|\grad F(w_k)\|] &\leq \left(\frac{e^{L A' Q_K} \left(2\delta_0 + L C' Q_K\right)}{(1-\sin(\alpha))K \alpha_{K-1}}\right)^{1/2}, \label{eq_complexity1}
\end{align}
where $Q_K := \sum_{k=0}^{K-1} \alpha_k^2$, $A':=AE$, $C':=CE$ (with $A$ and $C$ from \assref{ass_new_gradients} and $E$ from \eqref{eq_resilient_agg_weaker}), and $\delta_{0} := F(w_0)-F_{\text{low}}$.
In particular, this means that
\begin{align}
\min_{0 \leq k \leq K-1} \mathbb{E}[\|\grad F(w_k)\|] = \bigO\left(e^{A'/(2 L (B')^2)} \left(2\delta_0+\frac{C'}{L (B')^2}\right)^{1/2} (L B')^{1/2} \frac{1}{K^{(1-p)/2}}\right),
\end{align}
as $K\to\infty$ (considering the dependency in terms of $K$, $\delta_0$, $A'$, $B'$, $C'$, and $L$).
\end{corollary}
\begin{proof}
We continue from \eqref{eq_tmp1} in the proof of \thmref{thm_main_convergence}, where after applying \eqref{eq_tmp2} we have
\begin{align}
\min_{0 \leq k \leq K-1} r_k &\leq \frac{2 \delta_0}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) + \left( \frac{L C'}{K \alpha_{K-1}} \prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \right) \sum^{K-1}_{k=0} \alpha_{k}^2,
\end{align}
where $r_k := (1-\sin(\alpha)) \mathbb{E}[\|\nabla F(w_k)\|^2]$.
By the same reasoning leading to \eqref{eq_tmp_ref} we get
\begin{align}
\sum_{k=0}^{K-1} \log(1+LA'\alpha_k^2) \leq L A' Q_K,
\end{align}
and so $\prod^{K-1}_{k=0} (1 + L A' \alpha_k^2) \leq e^{L A' Q_K}$, giving
\begin{align}
\min_{0 \leq k \leq K-1} r_k &\leq \frac{e^{L A' Q_K} \left(2\delta_0 + L C' Q_K\right)}{K \alpha_{K-1}}.
\end{align}
Jensen's inequality gives $\mathbb{E}[\|\nabla F(w)\|]^2 \leq \mathbb{E}[\|\nabla F(w)\|^2]$, and we recover \eqref{eq_complexity1}.
Now since $\alpha_k = \alpha_0 (k+1)^{-p}$ we have
\begin{align}
Q_K \leq \alpha_0^2 + \int_{1}^{K-1} \alpha_0^2 k^{-2p} dk \leq \alpha_0^2 + \int_{1}^{K} \alpha_0^2 k^{-2p} dk = \frac{\alpha_0^2}{2p-1}\left(2p-2 + \frac{1}{K^{2p-1}}\right).
\end{align}
So for $K$ large, we have $Q_K = \bigO(\alpha_0^2) = \bigO\left(\frac{1}{L^2 (B')^2}\right)$ and $K \alpha_{K-1} = \bigO\left(\frac{K^{1-p}}{L B'}\right)$, which gives second result.
\end{proof}
\begin{remark}
Identical results to \corref{cor_complexity} hold when considering $\mathbb{E}[\|\grad F(w_{R_K})\|]$ instead of $\min_{k\leq K} \mathbb{E}[\|\grad F(w_k)\|]$, following the reasoning in \secref{sec_iterate_selection}.
\end{remark}
Our convergence rate result \corref{cor_complexity} essentially gives $\min_{k\leq K} \mathbb{E}[\|\grad F(w_k)\|] = \bigO\left(\frac{1}{K^{(1-p)/2}}\right)$.
In the best case, $p\to 1/2$, this convergence rate approaches $\bigO(K^{-1/4})$, or equivalently requiring $K = \bigO(\epsilon^{-4})$ to achieve optimality level $\mathbb{E}[\|\grad F(w_k)\|] \leq \epsilon$ (or alternatively $\mathbb{E}[\|\grad F(w_{R_K})\|] \leq \epsilon$).
This matches the standard worst-case complexity of stochastic gradient descent for nonconvex functions \cite{pmlr-v119-drori20a}.
}
\section{Conclusion} \label{sec_conclusion}
Having algorithms for \revision{distributed} learning in the presence of Byzantine adversaries is an important part of improving the utility of \revision{distributed} learning.
In this work we presented a simplified analysis of Byzantine resilient SGD (BRSGD) as developed in \cite{blanchard2017byzantinetolerant}, proving convergence in expectation \revision{and corresponding convergence rates} under more realistic assumptions on the objective function and the (non-corrupted) stochastic gradient estimators.
Since BRSGD is a generic algorithm, allowing the use of any aggregation function satisfying \assref{ass_resilient_agg}, our analysis applies to all the specific choices given in \cite{blanchard2017byzantinetolerant,xie2018generalized,pmlr-v80-mhamdi18a}.
Directions for future work include extending our analysis to more flexible learning rate regimes and other Byzantine resilient learning algorithms not based on the framework from \cite{blanchard2017byzantinetolerant}, such as those in \cite{pmlr-v80-yin18a,pmlr-v97-yin19a}.
\addcontentsline{toc}{section}{References}
\bibliographystyle{siam}
|
2,869,038,154,924 | arxiv | \section{Introduction}
Consider the following result.
\begin{theorem}
\label{thm:field-family}
Suppose that $\alpha \pm 8$ is squarefree, where $\alpha \in \mathbb{Z}$. Then the field $K_\alpha = \mathbb{Q}(\theta)$ where $\theta$ is a root of the irreducible polynomial $T^4 - 6T^2 - \alpha T - 3$ has ring of integers $\mathbb{Z}[\theta]$; in other words, $K_\alpha$ is a quartic monogenic field.
\end{theorem}
The discriminant of this polynomial, and hence the field $\mathbb{Q}(\theta)$, is $-27(\alpha-8)^2(\alpha+8)^2$. We do not doubt that monogenicity can be deduced by classical computations, but the novelty of this paper is our method: we discover this family of quartic fields as partial torsion fields (fields generated by a root of a division polynomial) of a particular family of elliptic curves, and deduce monogenicity by reference to reduction properties of the elliptic curve. In particular, we prove the following.
\begin{theorem}
\label{thm:ec-family}
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, such that some twist $E'$ of $E$ has a $4$-torsion point defined over $\mathbb{Q}$. Then the following are equivalent:
\begin{enumerate}
\item \label{item:red} $E'$ has reduction types $I^*_1$ and $I_1$ only;
\item \label{item:sqf} $E$ has $j$-invariant with squarefree denominator except a possible factor of 4.
\item \label{item:j} $E$ has $j$-invariant $j = \frac{(\alpha^2 - 48)^3}{(\alpha - 8)(\alpha + 8)}$, where $\alpha \in \mathbb{Z}$, $\alpha \pm 8$ are squarefree.
\end{enumerate}
Let $K_n$ be the field defined by adjoining the $x$-coordinate of an $n$-torsion point of $E$.
If any of the above hypotheses holds, then
$K_3$ is monogenic with a generator given by a root of $T^4 - 6T^2 - \alpha T - 3$. In particular, the field $K_3$ has discriminant $-27(\alpha-8)^2(\alpha+8)^2$.
\end{theorem}
Some examples of small values of $\alpha$ for which $K_3$ is monogenic are:
$$\pm2, \pm3, \pm5, \pm6, \pm7, \pm9, \pm11, \pm13, \pm14, \pm15, \pm18, \pm21, \pm22, \pm23, \pm25.$$
The methods used in the proof turn information about reduction properties of an elliptic curve into information about the index $[\mathcal{O}_{\mathbb{Q}(\theta)} : \mathbb{Z}[\theta]]$ where $\theta$ is a special value of an elliptic function (namely, a zero of a division polynomial). Theorem \ref{thm:ec-family} is meant primarily to showcase our methods. A more detailed analysis using these same methods can provide bounds and even exact formulae for the discriminants of partial torsion fields in general. This will be described in a follow-up paper by the second author.
In fact, Fleckinger and V\'erant studied the number fields of Theorem \ref{thm:field-family}, motivated by their status as partial torsion fields \cite{FV}. However, as they write, ``We note that the arithmetic of elliptic curves is not used once we have these polynomials.'' They describe a basis for the ring of integers in general (which is not a power basis), and show that they are quartic $S_4$ fields. See Section \ref{sec:ant}.
There is an abundance of literature on both monogenic number fields and number fields obtained by adjoining torsion points of elliptic curves. Monogenicity is rare: while our favourites, including quadratic fields, and the cyclotomic fields, are monogenic, it is known for example that almost all abelian extensions of $\mathbb{Q}$ with degree coprime to $6$ are non-monogenic \cite{Gras}. For an in-depth bibliography of monogenicity, see Narkiewicz \cite[pp. 79-81]{Nark} and the book of Ga\'al \cite{MR1896601}, and for fundamental algorithmic work, see Gy\H ory \cite{MR0437489}. We content ourselves here with listing a few recent works concerning monogenic quartic fields. In \cite{spear}, Spearman describes an infinite family of $A_4$ monogenic fields arising from $x^4+18x^2-4tx +t^2+81$ when $t(t^2+81)$ is squarefree. The $D_8$ fields are studied by Kable \cite{MR1688180} and Huard, Spearman, and Williams \cite{MR1321725}. While the pure quartic case is investigated by Funakura, who finds infinitely many monogenic fields \cite{MR0779772}. Fleckinger and V\'erant also have a monogenic family which appears to be $D_8$ \cite[(2)]{FV}. In \cite{tangras}, Gras and Tano\'e list necessary and sufficient conditions for certain biquadratic extensions of $\mathbb{Q}$ to be monogenic; Motoda constructs an infinite family \cite{MR2017249}. It is also known that infinitely many quartic cyclic fields are non-monogenic, by work of Motoda, Nakahara, Shah and Uehara \cite{MR2605782} and also Olajos \cite{MR2169516}. As for $S_4$ fields, little is known; however, B\'erczes, Evertse and Gy\H ory restrict the multiply monogenic orders in such fields \cite{MR3114010}. See the experimental data in Section \ref{sec:exp} for three more families of quartic fields which appear to be monogenic.
The field over which the $n$-torsion points of an elliptic curve are defined is often denoted $\mathbb{Q}(E[n])$ and plays a crucial role in the study of elliptic curves and their Galois representations. It is often referred to as a division field or a torsion field. For a survey, see \cite{MR1836119}. In general, the discriminants of such fields are not known, although there has been some work on their ramification \cite{MR2028514, MR3438391, MR3501021}. In the case when $n$ is prime the different has been computed \cite{KrausCali, Kraus}. In the case of $3$-division fields, generators, Galois groups and subfields have been very explicitly described \cite{BP2012}; see \cite{BP2016} for higher order. However, little similar work has been done on the subfields defined by division polynomials.
The Fueter polynomial we study arises from changing coordinates to the Fueter form of an elliptic curve: this choice has a history in explicit class field theory. Specifically, in \cite{ct87}, Cassou-Nogu\`{e}s and Taylor pursue Kronecker's Jugendtraum for certain ray class fields of imaginary quadratic fields. They study elliptic curves with complex multiplication and good reduction away from $2$. Let $K$ be an imaginary quadratic field with discriminant $d_K<-4$ and suppose $2$ splits in $K$. If $I$ is any $\mathcal{O}_K$ ideal, let $K(I)$ denote the ray class field of $K$ mod $I$. Now suppose $\xi$ is an odd $\mathcal{O}_K$ ideal. Cassou-Nogu\`{e}s and Taylor show that $\mathcal{O}_{K(4\xi)}$ is monogenic over $\mathcal{O}_{K(4)}$, using special values of the coordinates of the Fueter form.
Although the methods and the class of monogenic fields found in \cite{ct87} differ, we adopt their use of the Fueter form to access special values of an elliptic function.
It is remarkable that in the non-CM case, these special values still seem to offer some advantage in describing partial torsion fields explicitly, in the form of monogenic generators. Is it possible that these special values provide the best power basis for general partial torsion fields?
Our main method involves two ingredients: the algorithm of Gu\`ardia, Montes and Nart \cite{gmn12}, which computes $[\mathcal{O}_{\mathbb{Q}(\theta)}:\mathbb{Z}[\theta]]$; and the $p$-adic valuations of division polynomials (in particular, $T^4 - 6T^2 - \alpha T - 3$, the $3$-division polynomial in Fueter form), which are computed in detail in work of the third author \cite{s16}. A basic description of the Montes algorithm is to be found in Section \ref{sec:montes}. Briefly, the algorithm uses the Newton polygon to compute $v_p([\mathcal O_{\mathbb{Q}(\theta)}:\mathbb{Z}[\theta]])$ in terms of the number of lattice points on and under the polygon. The simplest case is a polygon which bounds no points, and this case corresponds to the $p$-adic valuation being 0. Thus, by picking $\alpha$ so that all the polygons are simple, we ensure that the corresponding field is monogenic.
It is possible to apply the Montes algorithm to the polynomial $T^4 - 6T^2 - \alpha T - 3$ directly, but the computations are rather involved. This would provide a proof of Theorem \ref{thm:field-family}, but it would not demonstrate the new methods dependent upon interpreting the polynomial as a division polynomial of an elliptic curve. In particular, the efficient choice of lift $\phi_i$ (see Section \ref{sec:montes}) is guided by the elliptic curve.
One can view this project as part of the study the discriminants of number fields associated with Latt\`es maps. Briefly, if $\psi\colon E \to E$ is an elliptic curve endomorphism and $\pi\colon E \to \mathbb P^1$ a finite covering, then a rational map $\phi\colon \mathbb P^1 \to \mathbb P^1$ is a Latt\`es map if $\pi \circ \psi = \phi \circ \pi$. For example, one may take $\psi(P) = [n]P$ and $\pi(x,y) = x$. The corresponding Latt\`es map has degree $n^2$, and it is from these maps that the division polynomials are derived (see Section \ref{sec:div-pol}).
The idea to compute the discriminants of number fields associated to Latt\`es maps is motivated by similar computations done for the power maps and Chebyshev polynomials. These three families of maps---Latt\`es, Chebyshev, and power---are postcritically finite. Consequently, if $f$ is a member of any one of these families, then the tower of number fields generated by $f^n(x) - c$ is unramified outside a finite set of primes \cite{ch12}. In some sense this simplifies the computation of the index as only finitely many primes need be analysed. In the case that $f$ is a Chebyshev or power map, the first author has used the Montes algorithm to compute the field discriminant precisely, and produced infinite towers of monogenic fields \cite{g14, g17}. In the case of the $n$-division polynomial, we need only consider the primes dividing $n$ and the discriminant of the curve. The shape of the Newton polygons tend to evolve predictably from one iterate to the next.
\subsection*{Acknowledgements} The authors are indebted to David Grant, \'{A}lvaro Lozano-Robledo and Joseph H. Silverman for helpful conversations.
\section{The Montes Algorithm} \label{sec:montes}
In this section we give a basic description of the Montes algorithm so that Theorem \ref{th:gmn} is understood. We refer more interested readers to \cite{gmn12} for the full details.
Let $\Phi \in \mathbb{Z}[x]$ be a monic irreducible polynomial whose root $\theta$ generates a number field $K$, and denote by $\mathcal{O}_K$ the ring of integers of $K$. Define $\operatorname{ind} \Phi = [ \mathcal{O}_K : \mathbb{Z}[\theta] ]$. Let $\operatorname{ind}_p \Phi = v_p(\operatorname{ind} \Phi)$ denote the $p$-adic valuation of $\operatorname{ind} \Phi$. The value $\operatorname{ind}_p\Phi$ may be computed as follows.
First, factor $\Phi$ modulo $p$ and write
\begin{align*}
\Phi(x) \equiv \phi_1(x)^{e_1} \cdots \phi_r(x)^{e_r} \pmod p,
\end{align*}
where the $\phi_i \in \mathbb{Z}[x]$ are monic lifts of the irreducible factors of $\Phi$ modulo $p$. The algorithm will terminate regardless of the choice of lifts, however this choice may simplify the computations significantly.
For each factor $\phi_i$, there is a unique expression
\begin{align*}
\Phi(x) = a_0(x) + a_1(x)\phi_i(x) + a_2(x)\phi_i(x)^2 + \cdots + a_s(x)\phi_i(x)^s,
\end{align*}
where the $a_j$ are integral polynomials satisfying $\deg a_j < \deg \phi_i$. This expression is called the \emph{$\phi_i$-development} of $\Phi$.
From the $\phi_i$-development, construct the \emph{$\phi_i$-Newton polygon} by taking the lower convex hull of the points
\begin{align} \label{eq:Newton poly}
\left\{\big(j,v_p(a_j(x))\big): 0 \le j \le s\right\},
\end{align}
where $v_p(a_j(x))$ is defined to be the minimal $p$-adic valuation of the coefficients of $a_j(x)$. Only the sides of negative slope are of import, and we call the set of sides of negative slope the \emph{$\phi_i$-polygon}. The set of lattice points under the $\phi_i$-polygon in the first quadrant carries important arithmetic data, and to keep track of these points, we define
\begin{align*}
\operatorname{ind}_{\phi_i}(\Phi) = (\deg \phi_i)\cdot \#\{ (x,y) \in \mathbb{N}^2: \text{ $(x,y)$ is on or under the $\phi_i$-polygon}\}.
\end{align*}
To each lattice point on the $\phi_i$-polygon, we attach a \emph{residual coefficient}
\begin{align*}
\operatornamewithlimits{res}(j) =
\begin{cases*}
\operatorname{red}(a_j(x)/p^{v_p(a_j(x))}) & if $\big(j,v_p(a_j(x))\big)$ is on the $\phi_i$-polygon\\
0 & otherwise,
\end{cases*}
\end{align*}
where $\operatorname{red}: \mathbb{Z}[x] \to \mathbb{F}_p[x]/(\phi_i(x))$ denotes the reduction map modulo $p$ and $\phi_i$.
For any side $S$ of the $\phi_i$-polygon, denote the left and right endpoints of $S$ by $(x_0,y_0)$ and $(x_1,y_1)$, respectively. We define the \emph{degree} of $S$ to be $\deg S = \gcd(y_1-y_0,x_1-x_0)$. In other words, $\deg S$ is equal to the number of segments into which the integral lattice divides $S$. We associate to $S$ a \emph{residual polynomial}
\begin{align*}
R_S(y) = \sum_{i=0}^{\deg S} \operatornamewithlimits{res}\left(x_0+i\frac{(x_1-x_0)}{\deg S}\right)y^i \in \mathbb{F}_p[x]/(\phi_i(x))[y].
\end{align*}
We note that $\operatornamewithlimits{res}(x_0)$ and $\operatornamewithlimits{res}(x_1)$ are necessarily non-zero, and in particular, it is always the case that $\deg S = \deg R_S$.
Finally, if $R_S$ is separable for each $S$ of the $\phi_i$-polygon, then $\Phi$ is \emph{$\phi_i$-regular}, and if $\Phi$ is $\phi_i$-regular for each factor $\phi_i$, then $\Phi$ is \emph{$p$-regular}.
\begin{theorem}[Theorem of the index] \label{th:gmn}
We have
\begin{align*}
\operatorname{ind}_p\Phi \ge \sum_{i=1}^r \operatorname{ind}_{\phi_i}(\Phi)
\end{align*}
with equality if $\Phi$ is $p$-regular.
\end{theorem}
\begin{proof}
See \cite[\S 4.4]{gmn12}.
\end{proof}
For our purposes, we need only the following simple corollary.
\begin{proposition}\label{prop:montes-for-us}
If $\Phi$ is monic, and $v_p(a_0) = 1$ for each $\phi_i$-development, then $\operatorname{ind}_{p} \Phi = 0$.
\end{proposition}
\begin{proof}
The Newton polygon for each $\phi_i$-development has exactly one side of negative slope to consider, running from $(0,1)$ to $(k_0, 0)$ for some $0 < k_0 \le s$. Therefore there are no points under or on the segment, and $\Phi$ is $p$-regular. The result follows from Theorem \ref{th:gmn}.
\end{proof}
\section{Fueter form and curves with a point of order $4$}
The goal of this section is to examine a particular one-parameter family of elliptic curves, namely a normal form for a curve with a rational point of order $4$ (although often called Tate's normal forms, such families of curves with rational $n$-torsion were known in the 19th century). This family was suggested by experimental data. In the next section we exhaustively analyse the valuations of special values of division polynomials for this family, describing all situations in which the Montes algorithm can be applied.
\subsection{Tate and Fueter forms}
\label{sec:tate-fueter-form}
Tate's normal form for an elliptic curve with a rational point of order $4$ is given by the Weierstrass form
\begin{equation}
\label{eqn:tate}
E: y^2 + (\alpha + 8 \beta) xy + \beta (\alpha + 8 \beta)^2 y = x^3 + \beta (\alpha + 8 \beta) x^2,
\end{equation}
where $\alpha, \beta \in \mathbb{Q}$. Though, by a change of coordinates, we may assume that $\alpha, \beta \in \mathbb{Z}$ and are coprime. Up to isomorphism, this is a one-parameter family of curves with $(0,0)$ being a point of order $4$. The invariants are:
\begin{equation}
\label{eqn:invariants}
\Delta = \beta^4(\alpha - 8\beta)(\alpha + 8 \beta)^7, \quad j = \frac{ (\alpha^2 - 48 \beta^2)^3 }{\beta^4(\alpha - 8\beta)(\alpha + 8\beta)}.
\end{equation}
Throughout the remainder of the paper, we will often use $a := \alpha + 8 \beta$ for ease of notation.
Appling the change of coordinates
\begin{equation}
\label{eqn:change-of-coord}
(x,y) = \left( \frac{a \beta}{T} - a\beta, \frac{1}{2} \left( \frac{ (a\beta)^{\frac32}T_1}{T^2} - \frac{a^2\beta}{T} \right) \right),
\end{equation}
one obtains
\[
T_1^2 = T\left(4T^2 + \frac{\alpha}{\beta} T + 4\right),
\]
which is known as a Fueter curve \cite{ct87}. The identity of the group is $(T,T_1) = (0,0)$, and the point $Q_0 := (1,\sqrt{a/\beta})=(1,\sqrt{8+\alpha/\beta})$ is a point of order $4$. Note that this change of coordinates is defined over a potentially quadratic extension $\mathbb{Q}(\sqrt{a\beta})$ but that the field of definition of the $x$-coordinate of a point is the same as the field of definition of the corresponding $T$ coordinate.
Suppose $p$ is a prime at which $E$ has bad reduction. If $p \mid a$ or $p \mid \beta$, then the singular point modulo $p$ on the Weierstrass curve, namely $(0,0)$, becomes $Q_0$ modulo $p$ on the Fueter curve. However, if $p \mid (\alpha-8\beta)$, then the singular point modulo $p$ on the Weierstrass curve, namely $(-2^5\beta^2,2^7\beta^3)$, becomes $(-1,0)$ modulo $p$ on the Fueter curve. Generally, when $p$ is an odd prime that divides $\alpha-8\beta$, a rational lift of the singular point will not necessarily exist.
\subsection{Division polynomials, Weierstrass and Fueter}
\label{sec:div-pol}
By definition, the $n$-th division polynomial $\Psi_n(x,y)$ for an elliptic curve $E$ in Weierstrass form
\[
E: y^2 + a_1xy + a_3 = x^3 + a_2x^2 + a_4x + a_6
\]
has the property that
\[
[n](x,y) = \left( \frac{\phi_n(x,y)}{\Psi_n(x,y)^2}, \frac{\omega_n(x,y)}{\Psi_n(x,y)^3} \right),
\]
where $\phi_n, \omega_n, \Psi_n$ are coprime polynomials. It can also be defined by stipulating that $\Psi_1(x,y) = 1, \Psi_2(x,y) = 2y + a_1x + a_3$ and for $n > 2$,
\[
\Psi_n(x,y) = \begin{dcases*}
n \sideset{}{'}\prod_{P \in E[n]\setminus\{\mathcal{O}\}} (x - x(P)) & $n$ is odd \\
\frac{n}{2} \Psi_2(x,y) \sideset{}{'}\prod_{P \in E[n]\setminus E[2]} (x - x(P)) & $n$ is even,
\end{dcases*}
\]
where the $'$ on the product indicates that we include only one of each pair $P$ and $-P$ in the product. In particular,
\begin{align*}
&\Psi_1=1,\\
&\Psi_2=2y+a_1x+a_3,\\
&\Psi_3=3x^4+b_2x^3+3b_4x^2+3b_6x+b_8,\\
&\Psi_4=\Psi_2(2x^6+b_2x^5+5b_4x^4+10b_6x^3+10b_8x^2+(b_2b_8-b_4b_6)x+(b_4b_8-b_6^2)).
\end{align*}
The odd division polynomials have degree $\frac{n^2-1}{2}$ in $x$. The $n$-th division polynomial has divisor $\sum_{P \in E[n]} (P) - n^2 (\mathcal{O})$. The group law of the elliptic curve manifests as a recurrence relation among the $\Psi_n$, $\omega_n$ and $\phi_n$; in particular, for $n \ge 3$,
\begin{equation}
\label{eqn:psi-rec}
\Psi_{2n-1} = \Psi_{n+1}\Psi_{n-1}^3 - \Psi_{n-2}\Psi_{n}^3, \quad
\Psi_{2n} \Psi_2 = \Psi_n \left( \Psi_{n+2}\Psi_{n-1}^2 - \Psi_{n-2}\Psi_{n+1}^2 \right).
\end{equation}
Therefore, having computed the first four division polynomials directly, we can obtain all the others recursively.
The discriminants of division polynomials have been computed by Verdure:
\begin{theorem}[{\cite[Theorem 1]{Verdure}}]
\label{thm:psi-disc}
\[
\operatorname{Disc}(\Psi_n) =
\begin{cases*}
(-1)^{\frac{n-1}{2}} n^{\frac{n^2-3}{2}} \Delta^{ \frac{n^4 - 4n^2 + 3}{24} } & $n$ odd \\
(-1)^{\frac{n-2}{2}} 16 n^{\frac{n^2-6}{2}} \Delta^{ \frac{n^4 - 10n^2 + 24}{24} } & $n$ even.
\end{cases*}
\]
\end{theorem}
In \cite{Fueter}, Fueter defined similar polynomials in $T$ and $T_1$ which we will call \emph{Fueter polynomials}. In particular, for a Fueter curve $T_1^2 = T(4T^2 + \frac{\alpha}{\beta}T + 4)$, one defines $F_1 = 1, F_2 = \frac{T_1}{\sqrt{T}}$, and for $n > 2$,
\[
F_n = \begin{dcases*}
\sideset{}{'}\prod_{P \in E[n]\setminus\{\mathcal{O}\}} (T - T(P)) & $n$ is odd \\
\frac{n}{2} F_2 \sideset{}{'}\prod_{P \in E[n]\setminus E[2]} (T - T(P)) & $n$ is even.
\end{dcases*}
\]
Here the above products are taken over the nontrivial $n$-torsion points with distinct $T$-coordinates. We also exclude the 2-torsion from the product when $n$ is even. The first few Fueter polynomials are:
\begin{align*}
&F_1=1,\\
&F_2=\frac{T_1}{\sqrt{T}},\\
&F_3=T^4-6T^2-\frac{\alpha}{\beta}T-3,\\
&F_4 = 2\frac{T_1}{\sqrt{T}}\left(T^6+\frac{\alpha}{\beta}T^5+10T^4-10T^2-\frac{\alpha}{\beta}T-2\right).
\end{align*}
Furthermore, they satisfy a recurrence relation:
\begin{align}
\label{eqn:f-rec}
F_{2n-1} &= (-1)^{n}(F_{n+1}F_{n-1}^3 - F_{n-2}F_{n}^3), \quad \notag \\
F_{2n} F_2 &= (-1)^nF_n \left( F_{n+2}F_{n-1}^2 - F_{n-2}F_{n+1}^2 \right).
\end{align}
Our Fueter polynomials for odd $n$ coincide with those defined by Cassou-Nogu\`{e}s and Taylor in \cite[\S IV.3]{ct87}. However, our even Fueter polynomials are distinct. In making our definition, we wished to preserve the recurrence relation.
One now observes that for odd $n$ (our primary interest), the polynomials $\Psi_n(x)$ and $F_n(T)$ define the same field extension. We will refer to this field extension as the \emph{$n$-th partial torsion field}. When $n$ is prime, it is the field of definition of the $x$-coordinate or $T$-coordinate of a single point of order $n$, which is generically of degree $(n^2-1)/2$.
Although we will only require the following proposition for odd $n$, we record the full relationship between the division polynomials of the Weierstrass and Fueter forms.
\begin{proposition}
\label{prop:psi-to-f}
Let $n$ be odd. Then
\[
\Psi_n = (-1)^\frac{n-1}{2} \left( \frac{a\beta}{T} \right)^{\frac{n^2-1}{2}} F_n,
\]
where $F_n$ is a monic polynomial in $T$ of degree $\frac{n^2-1}{2}$.
Let $n$ be even. Then
\[
\Psi_n = (-1)^\frac{n+2}{2} \left( \frac{a\beta}{T} \right)^{\frac{n^2-1}{2}} F_n,
\]
where $F_n=\frac{n}{2}\frac{T_1}{\sqrt{T}}f_n$ with $f_n$ a monic polynomial in $T$ of degree $\frac{n^2-4}{2}$.
\end{proposition}
\begin{proof}
Using the change of coordinates \eqref{eqn:change-of-coord}, we check the result directly for $n=1,2,3,4$.
Proceeding by induction, suppose we have the result for all $n<N$ and consider $\Psi_N$.
{\bf Case I: $N$ odd.} In this case, letting $N=2m+1$, we have by \eqref{eqn:psi-rec} that
$$\Psi_N=\Psi_{2m+1}=\Psi_{m+2}\Psi_m^3-\Psi_{m-1}\Psi_{m+1}^3.$$
Suppose $m$ is even. Then, using \eqref{eqn:f-rec} and the inductive hypothesis,
\begin{align*}
\Psi_N&=-\Big(\frac{a\beta}{T} \Big)^{\frac{(m+2)^2-1+3m^2-3}{2}}F_{m+2}F_m^3-\Big(\frac{a\beta}{T} \Big)^\frac{(m-1)^2-1+3(m+1)^2-3}{2}F_{m-1}F_{m+1}^3. \\
&=-\Big(\frac{a\beta}{T} \Big)^{\frac{(2m+1)^2-1}{2}}\left(F_{m+2}F_m^3 + F_{m-1}F_{m+1}^3\right) \\
&=-\Big(\frac{a\beta}{T} \Big)^{\frac{N^2-1}{2}}F_N.
\end{align*}
Keeping in mind the relationship $T_1^2=4T^3+\frac{\alpha}{\beta}T^2+4T$, we remark that $F_N$ is a polynomial in $T$.
Finally,
the leading term of $F_N(T)$ is determined by $F_{m-1}F_{m+1}^3$, which has degree $(N^2-1)/2$ and is monic.
An analogous computation yields the result if $m$ is odd.
{\bf Case II: $N$ even.} Letting $N=2m$, we have from \eqref{eqn:psi-rec} that
$$\Psi_2\Psi_N=\Psi_2\Psi_{2m}=\Psi_{m-1}^2\Psi_m\Psi_{m+2}-\Psi_{m-2}\Psi_m\Psi_{m+1}^2.$$
Again suppose $m$ is even. We have from \eqref{eqn:f-rec} and the inductive hypothesis that
\begin{align*}
\Psi_2\Psi_N&=-\Big(\frac{a\beta}{T} \Big)^{\frac{2(m-1)^2-2+m^2-1+(m+2)^2-1}{2}}F_{m-1}^2F_m F_{m+2}\\
&\quad\quad\quad+\Big(\frac{a\beta}{T} \Big)^{\frac{(m-2)^2-1+m^2-1+2(m+1)^2-2}{2}}F_{m-2} F_m F_{m+1}^2 \\
&=-\Big(\frac{a\beta}{T} \Big)^{\frac{(2m)^2+2}{2}}(F_{m-1}^2F_m F_{m+2} - F_{m-2} F_m F_{m+1}^2) \\
&=-\Big(\frac{a\beta}{T} \Big)^{\frac{N^2+2}{2}}F_2F_N.
\end{align*}
Dividing by $\Psi_2=\frac{(a\beta)^\frac{3}{2}T_1}{T^2}$ we obtain our desired expression. Note that
\begin{align*}
&F_{m-1}^2F_m F_{m+2} - F_{m-2} F_m F_{m+1}^2\\
&=\frac{T_1^2}{T}
\left(
\frac{m^2+2m}{4}F_{m-1}^2f_m f_{m+2} - \frac{m^2-2m}{4}f_{m-2} f_m F_{m+1}^2
\right).
\end{align*}
The quantity in the large parentheses is a polynomial in $T$, which, by induction, has leading term of degree $(N^2-4)/2$ with coefficient $m$. Finally, as before, if $m$ is odd, an analogous computation finishes the proof.
\end{proof}
We also record the discriminant of the odd Fueter polynomials.
\begin{proposition}\label{prop:f-disc}
For $n$ odd, we have
\[
\operatorname{Disc}(F_n) = (-1)^{\frac{n-1}{2}} n^{\frac{n^2-3}{2}} \left( \beta^{-2}(\alpha -8\beta)(\alpha + 8\beta) \right)^{ \frac{ n^4 - 4n^2 +3}{24} } .
\]
\end{proposition}
\begin{proof}
To compute the discriminant, we use Proposition \ref{prop:psi-to-f}. Let $d = (n^2-1)/2$, the degree of $\Psi_n$. Let $n$ be odd. Then,
\begin{align*}
\operatorname{Disc} F_n(T) &= (a\beta)^{-2d(d-1)} \operatorname{Disc} (a\beta)^d F_n(T) \\
&= (a\beta)^{-2d(d-1)} \operatorname{Disc} \left( \Psi_n\left( \frac{a\beta}{T} - a\beta \right) T^d \right) \\
&= (a\beta)^{-2d(d-1)} \operatorname{Disc} ( \Psi_n( a\beta T - a\beta ) ) \\
&= (a\beta)^{-d(d-1)} \operatorname{Disc} ( \Psi_n( T - a\beta ) ) \\
&= (a\beta)^{-d(d-1)} \operatorname{Disc} ( \Psi_n( T ) ).
\end{align*}
Next, we use the discriminant of $E$ \eqref{eqn:invariants} and Theorem \ref{thm:psi-disc}.
\end{proof}
\subsection{Tate's algorithm}
The purpose of this subsection is to give a full analysis of the reduction of the curve $E$ in Tate's Weierstrass form, via Tate's algorithm.
\begin{proposition}
\label{prop:tate-alg}
Let $p$ be an odd prime, $p \mid \Delta$. Let $\widetilde{E}$ denote the reduction of $E$ modulo $p$. Let $f$ denote the exponent of $p$ in the conductor of $E$. Let $c$ be the number of components in the special fiber over the minimal proper regular model of the curve over $\mathbb{Z}_p$. Then:
\begin{enumerate}
\item If $p \mid \beta$, then $f=1$, $c = 4 v_p(\beta)$, and $E$ has Kodaira type $I_{4v_p(\beta)}$. In this case, $E$ is in minimal Weierstrass form with respect to $p$, and the point $(0,0)$ has singular reduction.
\item If $p \mid (\alpha - 8 \beta)$, then $f = 1$ and $E$ has Kodaira type $I_{v_p(\alpha - 8\beta)}$. Furthermore,
\begin{enumerate}
\item If $p \equiv 1 \pmod 4$, then $c = v_p(\alpha - 8\beta)$.
\item If $p \equiv 3 \pmod 4$, then
\[
c = \begin{cases*}
1 & if $v_p(\alpha - 8\beta)$ is odd \\
2 & if $v_p(\alpha - 8 \beta)$ is even.
\end{cases*}
\]
\end{enumerate}
In these cases, $E$ is in minimal Weierstrass form with respect to $p$, and the point $(-2^5 \beta^2, 2^7 \beta^3)$ on $\widetilde{E}$ is singular.
\item If $p \mid (\alpha + 8 \beta)$, we let $w=\lfloor\frac{v_p(\alpha+8\beta)}{2}\rfloor$. Then
\begin{enumerate}
\item If $v_p(\alpha + 8 \beta)$ is odd, then $f=2$, $c=4$, and $E$ has Kodaira type $I^*_{v_p(\alpha + 8 \beta)}$.
\item If $v_p(\alpha + 8 \beta)$ is even, then $f=1$, $E$ has Kodaira type $I_{v_p(\alpha + 8 \beta)}$, and
\[
c = \begin{cases*}
v_p(\alpha + 8\beta) & if $\left( \frac{ \beta(\alpha + 8 \beta)p^{-2w} }{p} \right) = 1$ \\
2 & if $\left( \frac{ \beta(\alpha + 8 \beta)p^{-2w} }{p} \right) = -1$.
\end{cases*}
\]
\end{enumerate}
When $p \mid (\alpha+8\beta)$, $E$ is in minimal Weierstrass form with respect to $p$ after the change of coordinates $(x,y)=(p^{2w}x',p^{3w}y')$ and the point $(0,0)$ has singular reduction.
\end{enumerate}
\end{proposition}
\begin{proof}
We follow Tate's algorithm as described in \cite[IV \S 9]{ATAEC}.
{\bf Case I:} Suppose $p \mid \beta$. We apply Tate's algorithm and note that $p\nmid b_2=(\alpha+8\beta)^2+4\beta(\alpha+8\beta)$. Hence we have Kodaira type $I_{4v_p(\beta)}$ and $f=1$. Since $T^2-\alpha T$ splits completely over $\mathbb{Z}/p\mathbb{Z}$, $c=4v_p(\beta)$.
{\bf Case II:} Suppose $p \mid (\alpha -8\beta)$. In this case the singular point on the reduced curve is $(-2^5\beta^2,2^7\beta^3)$. Following Tate's algorithm, we make a change of coordinates $(x',y')=(x-2^5\beta^2,y+2^7\beta^3)$. Recall the notation $a=(\alpha+8\beta).$ For ease of notation we will write $x'$ as $x$ and $y'$ as $y$. We now have
\begin{align*}
E' &: y^2+axy+(2^8\beta^3+2^5\beta^2a+\beta a^2)y \\
&= x^3+(-3\cdot 2^5\beta^2+\beta a)x^2+(-2^6\beta^3a-2^7\beta^3a+3\cdot2^{10}\beta^4)x \\
& \quad +(-2^7\beta^4a^2+5\cdot2^{10}\beta^5a-3\cdot 2^{14}\beta^6).
\end{align*}
Continuing, we compute $b_2=a_1^2+4a_2$. Note $a\equiv 2^4\beta$ mod $p$. We have
\[
b_2=a^2+2^2(3x_1+\beta a)\equiv 2^8\beta^2-3\cdot2^7\beta^2+2^6\beta^2=-2^6\beta^2.
\]
This shows that $p\nmid b_2$ so that we have Kodaira type $I_{v_p({\alpha-8\beta})}$ and $f=1$. Continuing, we consider $T^2+aT+(3\cdot 2^5\beta^2 -\beta a)$ over $\mathbb{Z}/p\mathbb{Z}$. Reducing we have $T^2+2^4\beta T+5\cdot 2^4\beta^2$. Applying the quadratic formula, the roots are $-8u\beta \pm 4\beta \sqrt{-1}$. Thus the splitting field is $\mathbb{Z}/p\mathbb{Z}$ if and only if $p\equiv 1 \mod 4$. Hence $c=v_p(\alpha-8\beta)$ if $p\equiv 1 \mod 4$. Further, if $p\equiv 3 \mod 4$, then $c=1$ if $v_p(\alpha-8\beta)$ is odd and $c=2$ if $v_p(\alpha-8\beta)$ is even.
{\bf Case III:} Now assume $p \mid (\alpha+8\beta)$. Recall $w=\lfloor\frac{v_p(\alpha+8\beta)}{2}\rfloor$. We make the change of coordinates $(x,y)=(p^{2w}x',p^{3w}y')$. We have $a_1\mapsto a_1p^{-w}$, $a_2\mapsto a_2p^{-2w}$, and $a_3\mapsto a_3p^{-3w}$. Note $\Delta'=\Delta p^{-12w}$ so that $v_p(\Delta')=7v_p(\alpha-8\beta)-12w=v_p(\alpha-8\beta)$.
{\bf Part a:}
Suppose $v_p(\alpha+8\beta)$ is odd. Applying Tate's algorithm, we see $p \mid b_2'=(a_1p^{-w})^2+4a_2p^{-2w}$, $p^3 \mid b_8'=a_2a_3^2p^{-8w}$, and $p^3 \mid b_6'=a_3^2p^{-6w}$. Hence we consider $T^3-a_2p^{-v}T^2$ over $\mathbb{Z}/p\mathbb{Z}$. This polynomial has a double root at $T=0$ and a simple root at $T=a_2p^{-2w}$. Thus we have Kodaira type $I^*_{v_p(\alpha+8\beta)}$ and $f=2$. Following the subprocedure to step 7, we find $c=4$.
{\bf Part b:} Suppose $v_p(\alpha+8\beta)$ is even. Applying Tate's algorithm, we see that $p\nmid b_2'=(a_1p^{-w})^2-4a_2p^{-2w}$. Hence we have Kodaira type $I_{v_p(\alpha+8\beta)}$ and $f=1$. Considering $T^2-\beta(\alpha+8\beta)p^{-2w}$ over $\mathbb{Z}/p\mathbb{Z}$, we see that if $\left(\frac{\beta(\alpha+8\beta)p^{-2w}}{p}\right)=1$, then $c=v_p(\alpha+8\beta)$. Conversely, if $\left(\frac{\beta(\alpha+8\beta)p^{-2w}}{p}\right)=-1$ then $c=2$.
\end{proof}
Care must be taken when $E$ has bad reduction at 2. When $2 \mid \beta$, the results and proof used above can be applied by replacing $p$ with 2. When $2 \mid (\alpha+8\beta)$ we see $2 \mid \alpha$ and hence $2 \mid \alpha-8\beta$.
\begin{proposition}
\label{prop:tate-alg2}
Let the notation be as before and recall, $a=\alpha+8\beta$.
\begin{enumerate}
\item If $v_2(a)=1$, then $E$ has Kodaira type $I_1^*$, $f=3$, and $c=4$. In this case, $E$ is in minimal Weierstrass form with respect to 2 and the point $(0,0)$ has singular reduction.
\item If $v_2(a)=2$, then $E$ has Kodaira type $III$.
\item If $v_2(a)$ is odd and greater than 1, the $E$ has Kodaira type $I^*_{v_2(a)}$.
\item If $v_2(a)=4$ and $\frac{\beta a+4a-16}{32}$ is odd, then $E$ has Kodaira type $I_0^*$.
\item If $v_2(a)=4$ and $\frac{\beta a+4a-16}{32}$ is even, then we have two subcases.
\begin{enumerate}
\item If $\frac{\beta a^2}{2^8}\equiv 1$ mod 4, then $E$ has Kodaira type $I_2^*$.
\item If $\frac{\beta a^2}{2^8}\equiv 3$ mod 4, then $E$ has Kodaira type $I_3^*$.
\end{enumerate}
\item If $v_2(a)>4$ is even, we have several subcases:
\begin{enumerate}
\item If $\frac{\beta a+4a-16}{32}$ is odd, then we have Kodaira type $I_{v_2(a)-4}^*$.
\item If $\frac{\beta a+4a-16}{32}$ is even, we have further subcases:
\begin{enumerate}
\item If $v_2(a)=6$, we have Kodaira type $III^*$.
\item If $v_2(a)=8$, then $E$ is nonsingular at 2.
\item If $v_2(a)\geq 10$, we have Kodaira type $I_{v_2(a)-8}$.
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{proposition}
\begin{proof}
We follow Tate's algorithm as described in \cite[IV \S 9]{ATAEC}.
{\bf Case I: $v_2(a)=1$.} Applying Tate's algorithm, we see $2 \mid b_2$, $4 \mid a_6$, $8 \mid b_8$, and $8 \mid b_6$. Thus we consider $$P(T)=T^3+\frac{\beta a}{2}T^2=T^2\left(T+\frac{\beta a}{2}\right).$$
We see $P(T)$ has a simple root and a double root modulo 2. Hence we have Kodaira type $I_n^*$ and $f=v_2(\Delta)-4-n$. To determine $n$ and $c$ we consider the polynomial
$$Y^2+\frac{\beta a^2}{4}Y.$$
This polynomial has distinct roots in $\mathbb{Z}/2\mathbb{Z}$. Hence $n=1$ and $c=4$. Noting $v_2(\Delta)=8$, the result follows.
{\bf Case II: $v_2(a)>1$.} We define $w=\lfloor \frac{v_2(a)}{2}\rfloor$ and we make the change of coordinates $(x,y)=(2^{2w}x',2^{3w}y')$. For ease of notation we will write $x$ and $y$ for $x'$ and $y'$.
{\bf Case II-A: $v_2(a)=2$.} Then $8\nmid b_8$ and we have type $III$.
{\bf Case II-B: $v_2(a)$ odd.}
If $v_2(a)$ is odd we consider $P(T)\equiv T^2(T+1)$ mod 2. When the subprocedure to step 7 terminates, we are left with type $I^*_{v_2(a)}$.
{\bf Case II-C: $v_2(a)=4$.} In step 6 we change coordinates to obtain
$$y^2 +\left(\frac{a}{2^w}+2\right)xy+\frac{\beta a^2}{2^{3w}}y=x^3+\left(\frac{\beta a}{2^{2w}}+\frac{a}{2^w}-1\right)x^2+\frac{\beta a^2}{2^{3w}}x.$$
We consider $$P(T)=T^3+\frac{\beta a+2^wa-2^{2w}}{2^{2w+1}}T^2+\frac{\beta a^2}{2^{3w+2}}T.$$
If $\frac{\beta a+2^wa-2^{2w}}{2^{2w+1}}$ is odd, then we have type $I_0^*$. If $\frac{\beta a+2^wa-2^{2w}}{2^{2w+1}}$ is even, we change coordinates setting $x=x'+2$ and again abuse notation by letting $x=x'$. Our curve becomes
\begin{align*}
y^2 &+\left(\frac{a}{2^3}+2\right)xy+\left(\frac{\beta a^2}{2^{3w}}+\frac{a}{2^{w-1}}+4\right)y \\
&=x^3+\left(\frac{\beta a+2^wa-2^{2w}+6\cdot 2^{2w}}{2^{2w}}\right)x^2+\left(\frac{\beta a^2}{2^{3w}}+\frac{\beta a +2^wa-2^{2w}}{2^{2w}}+12\right)x.
\end{align*}
Following the subprocedure to step 7, we obtain the desired result.
{\bf Case II-D: $v_2(a)>4$ even and $\frac{\beta a+4a-16}{32}$ odd.} Then $P(T)\equiv T^2(T+1)$ mod 2. Following the subprocedure to step 7, we find we have type $I_{v_2(a)-4}$.
{\bf Case II-E: $v_2(a)>4$ even and $\frac{\beta a+4a-16}{32}$ even.} Then $P(T)$ has a triple root.
{\bf Case II-E-i: $v_2(a)=6$ and $\frac{\beta a+4a-16}{32}$ even.} Then $16\nmid a_4=\frac{\beta a^2}{2^{3w}}$ so we have type $III^*$.
{\bf Case II-E-ii: $v_2(a) > 6$ even and $\frac{\beta a+4a-16}{32}$ even.} Then our Weierstrass equation was not minimal. We make the change of coordinates $(x,y)=(4x',8y')$ to obtain
$$y^2 +\left(\frac{a}{2^{w+1}}+1\right)xy+\frac{\beta a^2}{2^{3w+3}}y=x^3+\frac{\beta a+2^wa-2^{2w}}{2^{2w+2}}x^2+\frac{\beta a^2}{2^{3w+4}}x.$$
{\bf Case II-E-ii-a: $v_2(a)=8$ and $\frac{\beta a+4a-16}{32}$ even.} One checks that if $v_2(a)=8$, our curve is nonsingular at 2.
{\bf Case II-E-ii-b: $v_2(a)>8$ even and $\frac{\beta a+4a-16}{32}$ even.} We have type $I_{v_2(a)-8}$.
\end{proof}
\section{Valuation of Division Polynomials}
The purpose of this section is to determine the valuation of $F_n$ evaluated at the singular point. This is done by reference to the valuations of $\Psi_n$ at the singular point, and the change of variables of Proposition \ref{prop:psi-to-f}. To obtain the valuations of $\Psi_n$, we demonstrate two methods. The first is to apply the results of \cite{s16}, which give explicit valuations based on the reduction data of Proposition \ref{prop:tate-alg}. The second is a hands-on approach using the recurrence relations for division polynomials, which is possible in simpler cases. We consider only odd primes.
\subsection{Odd primes dividing $\alpha - 8 \beta$}
Recall that, when $p \mid (\alpha - 8\beta)$, the singular point modulo $p$ is $(-2^5\beta^2, 2^7\beta^3)$.
\begin{proposition}
\label{prop:val-minus}
Suppose $p \mid (\alpha - 8\beta)$. Let $Q$ be a point of $E(\overline{\mathbb{Q}})$ which is singular modulo $p$, and satisfies $x(Q) = -2^5\beta^2$. Let $Q'$ be the image of $Q$ under the change of coordinates to Fueter form.
Suppose that $n$ is odd. Then,
\[
v_p( F_n(Q')) = v_p(\Psi_n(Q)) =
v_p(\alpha - 8 \beta) \frac{n^2-1}{8}.
\]
\end{proposition}
To prove Proposition \ref{prop:val-minus}, we begin with a lemma.
\begin{lemma} \label{lem:2Q}
Suppose $p \mid (\alpha - 8\beta)$ and let $Q$ be as above. Then, $[2]Q$ does not reduce to the singular point mod $p$.
\end{lemma}
\begin{proof} Recall $a=\alpha+8\beta$. We compute
\begin{align*}
x([2]Q)&=\frac{2^{20}\beta^8-b_42^{10}\beta^4+b_62^6\beta^2-b_8}{-2^{17}\beta^6+b_22^{10}\beta^4-b_42^6\beta^2+b_6}\\
&=\frac{2^{20}\beta^6-2^{10}a^3\beta^3+2^6a^4\beta^2-a^5\beta}{-2^{17}\beta^4+2^{10}a^2\beta^2+2^{12}a\beta^3-2^6a^3\beta+a^4}.
\end{align*}
We divide the numerator and denominator by $a-16\beta=\alpha-8\beta$ to obtain
\[
\frac{-a^4\beta+3\cdot2^4a^3\beta^2-2^8a^2\beta^3-2^{12}a\beta^4-2^{16}\beta^5}{a^3-3\cdot2^4a^2\beta+2^8a\beta^2+2^{13}\beta^3}.
\]
Reducing mod $p$ we obtain
\[
x([2]Q)\equiv -2^4\beta^2.
\]
Thus $[2]Q$ does not reduce to the singular point.
\end{proof}
Following \cite{s16}, we define, for any integers $a, \ell$ such that $\ell \neq 0$, the sequence
\begin{equation}
\label{eqn:rn}
R_n(a,\ell) = \left\lfloor \frac{ n^2\widehat{a}(\ell-\widehat{a}) }{2\ell} \right\rfloor - \left\lfloor \frac{ \widehat{na}(\ell - \widehat{na}) }{2\ell} \right\rfloor ,
\end{equation}
where $\widehat{x}$ denotes the least non-negative residue of $x$ modulo $\ell$. Theorem 9.3 of \cite{s16} gives the valuations of the sequence of division polynomials, evaluated at a point of multiplicative reduction, in terms of such sequences. We apply this to our specific situation here.
In particular, we will encounter the sequence $R_n(1,2)$, which begins from $n=1$ as follows:
\[
0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, \ldots
\]
The odd terms of the sequence have a simple closed form.
\begin{lemma}
\label{lem:triang}
For $n$ odd, $R_n(1,2) = \frac{n^2-1}{4}$.
\end{lemma}
\begin{proof}
For $n$ odd, we have $\widehat{a}=\widehat{na}=1$ in \eqref{eqn:rn}. Therefore,
\[
R_n(1,2) = \left\lfloor \frac{ n^2 }{4} \right\rfloor - \left\lfloor \frac{ 1 }{4} \right\rfloor
= \left\lfloor \frac{ n^2 }{4} \right\rfloor = \frac{n^2-1}{4}.
\]
\end{proof}
\begin{proposition}
\label{prop:unified}
Suppose $p \mid (\alpha - 8\beta)$ and let $Q$ be as above. Let $K$ be the potentially quadratic extension of $\mathbb{Q}$ so that $Q\in E(K)$ and let $L$ be an unramified, potentially quadratic extension of $K$ such that $E$ has split multiplicative reduction over $L$ (which exists by Proposition \ref{prop:tate-alg}). Let $v_p'$ be a lift of $v_p$ to $L$.
Let $n>0$ and suppose $4 \nmid n$. Then $v_p' = 2v_p$ if and only if $v_p(\alpha - 8\beta)$ is odd; otherwise $v_p' = v_p$.
We have
$$v_p'(\Psi_n(Q))=\frac{v_p'(\alpha-8\beta)}{2}R_n(1,2).$$
If furthermore $n$ is odd, then
$$v_p(\Psi_n(Q)) = v_p(\alpha - 8\beta)\frac{n^2-1}{8}.$$
\end{proposition}
\begin{proof}
One can compute that $K$ is the quadratic extension obtained by adjoining
\begin{align*}
&\sqrt{\alpha^4-2^5\alpha^3\beta-2^7\alpha^2\beta^2+5\cdot 2^{11}\alpha\beta^3-15\cdot 2^{12}\beta^4} \\
&= \sqrt{\alpha -8 \beta}\sqrt{ \alpha^3 - 24\alpha^2\beta - 320 \alpha \beta^2 +7680 \beta}.
\end{align*}
We also have
\[
\alpha^3 - 24\alpha^2\beta - 320 \alpha \beta^2 + 7680 \beta \equiv 2^{12}\beta^3 \pmod{\alpha - 8\beta}.
\]
Therefore, since $p$ is odd, divides $(\alpha - 8\beta)$, and is coprime to $\beta$, we have that the extension $K$ is ramified at $p$ if and only if $v_p(\alpha - 8\beta)$ is odd. Hence, $v_p' = 2 v_p$ if and only if $v_p(\alpha - 8\beta)$ is odd; otherwise $v_p' = v_p$.
The group of components over $L$ is isomorphic to $\mathbb{Z}/v_p'(\alpha-8\beta)\mathbb{Z}$ since we have split multiplicative reduction. The component containing $Q$ has additive order exactly 2 by Lemma \ref{lem:2Q}. Thus it may be identified with $v_p'(\alpha-8\beta)/2$.
Hence, in the language of \cite{s16}, $\ell_Q=v_p'(\alpha-8\beta)$ and $a_Q=v_p'(\alpha-8\beta)/2$. Applying \cite[Theorem 9.3]{s16}, we find that
$$v_p'(\Psi_n(Q))=R_n(v_p'(\alpha-8\beta)/2,v_p'(\alpha-8\beta)).$$
By \cite[Proposition 8.2(iv)]{s16},
$$v_p'(\Psi_n(Q))=\frac{v_p'(\alpha-8\beta)}{2}R_n(1,2). $$
For odd $n$, $\Psi_n(x)$ is a polynomial in $x$ alone and therefore $\Psi_n(Q) \in \mathbb{Q}$. Accordingly, by Lemma \ref{lem:triang}, we obtain the given statement.
\end{proof}
Proposition \ref{prop:val-minus} follows from Propositions \ref{prop:unified} and \ref{prop:psi-to-f} (recall that $\alpha, \beta$ are coprime integers).
\subsection{Odd primes dividing $\alpha + 8 \beta$ or $\beta$}
In this case, we apply the recurrence relation for the division polynomial to obtain valuations.
\begin{proposition}
\label{prop:val-others}
Suppose $p \mid \beta$ or $p \mid (\alpha + 8 \beta)$ (these cases are mutually exclusive). Then $(0,0)$ is a point of order $4$ and has singular reduction on $\widetilde{E}$; the corresponding point in Fueter form has $T=1$. Suppose that $n$ is odd.
If $p \mid \beta$, then
\[
v_p(\Psi_n(0)) = \frac{3n^2 -3}{8} v_p(\beta), \quad
v_p(F_n(1)) = -\frac{n^2 -1}{8} v_p(\beta).
\]
If $p \mid (\alpha + 8\beta)$, then
\[
v_p(\Psi_n(0)) = \frac{5n^2 - 5}{8} v_p(\alpha + 8 \beta), \quad
v_p(F_n(1)) = \frac{n^2 -1}{8} v_p(\alpha + 8\beta).
\]
\end{proposition}
\begin{proof}
We will proceed by induction. Recall $a=\alpha+8\beta$. For the base cases we have $\Psi_1(0)=1$, $\Psi_2(x,y)=2y+ax+a^2$ so $\Psi_2(0)=a^2$. Further, $\Psi_3=3x^4+b_2x^3+3b_4x^2+3b_6x+b_8=3x^4+(a^2+4\beta a)x^3+3\beta a^3x^2+3\beta^2a^4x+\beta^3a^5$. Hence $\Psi_3(0)=\beta^3a^5$. We have $\Psi_4=\Psi_2(2x^6+b_2x^5+5b_4x^4+10b_6x^3+10b_8x^2+(b_2b_8-b_4b_6)x+(b_4b_8-b_6^2))$. Evaluating at 0 we obtain $\Psi_4(0)=\Psi_2(0)(b_4b_8-b_6^2)=\Psi_2(0)(\beta^4a^8-\beta^4a^8)=0$.
First we prove if $4 \mid n$, $\Psi_n(0)=0$. Suppose we have the result for all $n<N$ and suppose $4 \mid N$. Let $N=2m$, so that $m$ is even. Then
$$\Psi_2\Psi_N=\Psi_2\Psi_{2m}=\Psi_{m-1}^2\Psi_m\Psi_{m+2}-\Psi_{m-2}\Psi_m\Psi_{m+1}^2.$$
Now either $4 \mid m$ or $4 \mid m-2$ and $4 \mid m+2$. Hence the result follows by induction.
Now suppose that $v_p(\Psi_n(0))=v_p(a)\frac{5n^2-5}{8}+v_p(\beta)\frac{3n^2-3}{8}$ for all $n<N$. Suppose $N$ is odd, and write $N=2m+1$. We have
$$\Psi_N=\Psi_{2m+1}=\Psi_{m+2}\Psi_m^3-\Psi_{m-1}\Psi_{m+1}^3.$$
Suppose first that $m$ is even. Then either $m$ or $m+2$ is divisible by 4. Hence
\begin{align*}
v_p(\Psi_N(0))&=v_p(\Psi_{m-1}(0))+3v_p(\Psi_{m+1}(0))\\
&=v_p(a)\frac{5(m-1)^2-5}{8}+v_p(\beta)\frac{3(m-1)^2-3}{8}\\
&\quad\quad\quad\quad+v_p(a)3\frac{5(m+1)^2-5}{8}+v_p(\beta)3\frac{3(m+1)^2-3}{8}\\
&=v_p(a)\frac{5(2m+1)^2-5}{8}+v_p(\beta)\frac{3(2m+1)^2-3}{8}.
\end{align*}
Likewise, if $m$ is odd, either $m-1$ or $m+1$ is divisible by 4. Hence
\begin{align*}
v_p(\Psi_N(0))&=v_p(\Psi_{m+2}(0))+3v_p(\Psi_{m}(0))\\
&=v_p(a)\frac{5(m+2)^2-5}{8}+v_p(\beta)\frac{3(m+2)^2-3}{8}\\
&\quad\quad\quad\quad+v_p(a)3\frac{5m^2-5}{8}+v_p(\beta)3\frac{3m^2-3}{8}\\
&=v_p(a)\frac{5(2m+1)^2-5}{8}+v_p(\beta)\frac{3(2m+1)^2-3}{8}.
\end{align*}
This gives the stated results for $\Psi_n$. For $F_n$, we use the change of coordinates between Weierstrass and Fueter form and Proposition \ref{prop:psi-to-f}.
\end{proof}
\section{Proof of the Main Theorem}
\begin{proof}[Proof of Theorem \ref{thm:ec-family}]
Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$, and suppose a twist $E'$ has a rational $4$-torsion point, hence can be put into Tate normal form as in \eqref{eqn:tate} with $\alpha, \beta \in \mathbb{Z}$ coprime. The $j$-invariant of the elliptic curve is invariant under twisting. In Tate normal form, the discriminant and $j$-invariant are of the form
\[
\Delta = \beta^4(\alpha - 8\beta)(\alpha + 8\beta)^7, \quad j = \frac{ (\alpha^2 - 48 \beta^2)^3 }{ \beta^4 (\alpha - 8\beta)(\alpha + 8\beta) }, \quad \alpha,\beta \in \mathbb{Z}.
\]
Therefore $E'$ has good reduction modulo $p$ unless $p \mid \beta(\alpha - 8\beta)(\alpha + 8\beta)$.
We now show that conditions \eqref{item:red}, \eqref{item:sqf} and \eqref{item:j} of the statement are equivalent. Under condition \eqref{item:red}, we have $\beta=1$ by Proposition \ref{prop:tate-alg}. In this case, requirements \eqref{item:sqf} and \eqref{item:j} are evidently equivalent. For odd primes, Proposition \ref{prop:tate-alg} implies that $p^2$ may not divide $\alpha \pm 8$. For $p=2$, Proposition \ref{prop:tate-alg2} implies that $v_2(\alpha + 8) = 0$ or $1$. This implies $v_2(\alpha - 8) = 0$ or $1$ also, and we have demonstrated condition \eqref{item:j}. Hence \eqref{item:red} implies \eqref{item:sqf} and \eqref{item:j}.
Conversely, if condition \eqref{item:sqf} holds, we apply Propositions \ref{prop:tate-alg} and \ref{prop:tate-alg2} to conclude that \eqref{item:red} holds. Thus we have demonstrated all the conditions are equivalent.
The field $K_\alpha$ generated by the $x$-coordinate of a single point of order $3$ is invariant under the twist. Therefore we now assume $E$ itself has a rational $4$-torsion point.
Change coordinates so that $E$ is in Tate normal form and Fueter form as in Section \ref{sec:tate-fueter-form} with $\alpha \in \mathbb{Z}$ and $\beta=1$. We then find that the partial $3$-torsion field is generated by the $3$-division Fueter polynomial, $F_3(T) = T^4 - 6T^2 - \alpha T - 3$. Let $\theta$ be a root of this polynomial, and let $K = \mathbb{Q}(\theta)$.
Under the equivalent conditions of the theorem, the polynomial $F_3(T)$ is irreducible, as observed in \cite[Proposition 2.10]{FV}, so $K$ is a quartic field.
We apply the Montes algorithm. It calls for examining the polynomial $F_3$ developed around any lift of a repeated irreducible factor modulo $p$; each such situation may contribute a factor to the index $[\mathcal{O}_K: \mathbb{Z}[\theta]]$. If no such non-trivial factors appear, we can conclude $\mathcal{O}_K = \mathbb{Z}[\theta]$.
We will show prime-by-prime that the only repeated factors are linear of the form $T-T_0$ and that $v_p(F_3(T_0)) = 1$.
{\bf Case I: $p=2$.} Modulo $2$, the polynomial $F_3$ becomes $T^4 - \alpha T - 1$. If $\alpha$ is odd, this is irreducible with no repeated roots. If $\alpha$ is even, then the repeated root is $1$, so we develop $F_3$ around $T-1$, obtaining a constant term of $-\alpha - 8$, which we have assumed to be squarefree. Therefore in this case $v_2(F_3(1)) = 1$.
{\bf Case II: $p=3$.}
Modulo $3$, the polynomial $F_3$ becomes $T^4 - \alpha T$, and $\alpha$ is a repeated root. If $3$ divides $\alpha$, then a lift of this root is $0$, and $v_3(F_3(0)) = 1$. If $\alpha \equiv 1 \pmod 3$, then $4$ is a lift, and $v_3(F_3(4)) = 1$. Else $-4$ is a lift of $\alpha$, and $v_3(F_3(-4)) = 1$.
{\bf Case III: $p \ge 5$.}
Now, suppose $F_3$ has a repeated irreducible factor modulo an odd prime $p$. The roots of $F_3$ are the four $x$-coordinates of non-trivial $3$-torsion; this means that reduction modulo $p$ fails to be injective on $E[3]$. This occurs if and only if $E$ has bad reduction at $p$, or $p=3$.
Suppose $p \ge 5$ is a prime of bad reduction, and suppose $Q$ is a point on $E$ having singular reduction modulo $p$. Specifically, if $p \mid \alpha + 8$, take $Q = (0,0)$. If $p \mid \alpha - 8$, take $x(Q) = -2^5$. Then, the only repeated root of $F_3$ modulo $p$ is $T(Q)$ (since the failure of injectivity under reduction must take the form of $3$-torsion points mapping to the singular point, as the map to the non-singular part has torsion-free kernel). Then, using the fact that $\alpha \pm 8$ are not divisible by $p^2$, we learn from Propositions \ref{prop:val-minus} and \ref{prop:val-others} that $v_p(F_3(T(Q))) = 1$.
In each case, we find that $v_p(F_3(T_0))=1$ where $T_0$ is the repeated root. Therefore the associated Newton polygon starts at height $1$ on the $y$-axis. Hence, the polygon cannot pass through any lattice points and cannot contain any lattice points, and the polygon has only one segment, as in Proposition \ref{prop:montes-for-us}. Therefore it is $p$-regular. By the Montes algorithm, this implies that the index $[\mathcal{O}_K: \mathbb{Z}[\theta]]$ is not divisible by $p$.
As we have verified that the index $[\mathcal{O}_K : \mathbb{Z}[\theta]]$ is not divisible by any prime, we conclude that $\mathcal{O}_K = \mathbb{Z}[\theta]$.
\end{proof}
Theorem \ref{thm:field-family} follows immediately.
\section{Algebraic number theory of the family $T^4 - 6T^2 - \alpha T - 3$}\label{sec:ant}
Let $\theta$ be a root of $T^4 - 6T^2 - \alpha T - 3$. Consider the field $K_\alpha = \mathbb{Q}(\theta)$.
This family of number fields was studied by Fleckinger and V\'erant \cite{FV}. Let $\alpha \ge 9$, $\alpha \in \mathbb{Z}$, and $\alpha \neq 24$. Then Fleckinger and V\'erant showed that $K_\alpha$ is an $S_4$ quartic field with two real embeddings \cite[Proposition 2.10]{FV}. They give an explicit basis for the ring of integers in general \cite[Proposition 2.11]{FV}, but it is not a power basis and they do not mention monogenicity. Finally, they remark that when $3 \mid \alpha$, then $1 + \frac{\alpha}{3} \theta + 2\theta^2$ is a unit. In fact, they point out that there are no other parametrized units in this field.
Experimentally, we observed surprisingly small regulators and surprisingly large class groups for these fields; the existence of a simple parametrized unit is a possible explanation.
\section{A related family}
Fleckinger and V\'erant also study the family of quartic fields given by $T^4 + \frac{\alpha}{2} T^3 + 6 T^2 + \frac{\alpha}{2} T + 1$ of discriminant $-4\left( \left(\frac{\alpha}{2}\right)^2 - 16 \right)^3$, which they observe arise from a point of order four on a Fueter model \cite{FV}. The authors prove that this family is monogenic whenever $(\alpha/2)^2 - 16$ is odd and squarefree, and $\alpha \ge 12$ \cite[Corollary 1.4]{FV}. This appears to be a $D_8$ family. We leave it as an open question whether the methods of this paper may apply to this family.
\section{Experimental Data}
\label{sec:exp}
As part of our exploration, we took a survey of elliptic curves to determine the prevalence of monogenic fields, using Sage Mathematics Software \cite{Sage} and pari/GP \cite{PARI}. Up to isogeny, there are 11575 curves of conductor less than 10000 whose 3-division field is monogenic. The torsion points of many curves share the same field of definition, and in all, these 11575 curves yield 1026 unique fields. In particular, the following families of fields are prevalent.
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l|l}
\text{Polynomial} & \text{Discriminant}\\
\hline
$T^4 - 6s T^2 - t T - 3s^2$ & $-3^3(t^2 - 64s^3)^2$ \\
$T^4 - T^3 - 3s T^2 - (4t + 3s^2)T + t$ & $-3^3(16t^2 + (24s^2 + 12s + 1)t + (9s^4 + s^3))^2$\\
$T^4 - 2T^3 - 6s T^2 - (2t+6s^2)T + t$ & $-2^4 3^3 (t^2 + (6s^2 + 6s + 1)t + (9s^4 + 2s^3))^2$
\end{tabular}
\end{center}
In the table above $T$ is the indeterminate, while $s,t \in \mathbb{Z}$ parametrize the family. Each of these quartic field families appears to be $S_4$ monogenic under appropriate conditions on the discriminant and the parameters.
|
2,869,038,154,925 | arxiv | \section{Introduction}
\label{sec:intro}
The upcoming construction and commissioning of SKA Phase 1 will bring with it a slew of blind HI surveys to be carried out by precursor facilities. While many of these surveys will be shallow or medium depth, wide area surveys, there are several ultra deep single pointing and small field surveys that aim to probe HI galaxies out to unprecedented redshifts.
Stacking has become a key tool for HI astronomers in recent years as measurements of the evolution of HI density with redshift have been attempted \citep{Lah+2007,Delhaize+2013,Rhee+2013}, and low mass and HI-deficient galaxies have been studied at low redshift \citep[e.g.][]{Fabello+2011,Fabello+2011b,Fabello+2012}. As surveys push to increasingly high redshifts, stacking will become an evermore invaluable tool in the attempt to study normal HI galaxies out to a redshift of order unity and beyond.
As surveys become deeper, both in terms of their sensitivity and redshift range, confusion becomes an increasing concern. Longer integration times mean surveys are sensitive to less massive galaxies, but this also means that background emission makes up a larger fraction of the signal detected. Probing HI at higher redshift causes an increasingly large number of objects to be contained in an individual beam width, as the physical size of the beam grows with redshift and therefore encloses more volume. When undetected target objects are stacked this low level emission from the surrounding galaxies will also be coadded. Eventually, when the survey data is deep enough, this confused emission will contribute a significant fraction of the final stacked spectrum and create a bias in the results. The scale of this bias should be estimated so that it can be anticipated and potentially corrected for.
A small number of measurements and predictions of confusion have been made, that are applicable to very deep HI surveys. \citet{Duffy+2008} made predictions for potential FAST (Five hundred metre Aperture Spherical Telescope) surveys, using a similar approach to that used here, but assumed a uniform universe (i.e. neglected the correlation function). As we shall show this leads to an order of magnitude underestimation of the signal due to confusion. \citet{Delhaize+2013} took a different approach by estimating the contribution of confusion in a stack that was known to be heavily confused, based on the optical parameters of the galaxies in the field. This provides a means to interpret a stacked spectrum with confusion, but could also be used to predict the amount of confusion. However, as this would require the specific (optical) input catalogue, and we intend to produce a general tool to assess a generic survey's confusion, this approach will not be discussed in detail in this paper.
In this paper we make use of the currently available HI correlation function (CF) and measurements of the mean ($z=0$) HI density to predict how much HI mass will be contained in a stacked spectrum, in addition to that of the intended targets. This is intended to be a universal tool which can be used to calculate a realistic, but computationally cheap, estimate of the impact of confusion on any HI survey. Section \ref{sec:surveys} briefly outlines the upcoming surveys for which predictions will be made. Section \ref{sec:model} describes how the analytic model is derived, as well as its caveats and limitations. In section \ref{sec:results} we present our results and their implications are discussed in section \ref{sec:discuss}.
\section{Deep Surveys}
\label{sec:surveys}
In the coming years a host of new HI galaxy surveys will begin as part of the precursors to the SKA. In \citet{Jones+2015} we assessed the impact of confusion on shallow and medium depth surveys, whereas this paper focuses on the three deepest of upcoming surveys, LADUMA (Looking At the Distant Universe with MeerKAT), CHILES (COSMOS HI Large Extragalactic Survey) and DINGO UDEEP (Deep Investigation of Neutral Gas Origins - Ultra Deep). We also briefly discuss FAST in a more general sense as the specifics of the surveys it will perform have yet to be determined.
LADUMA \citep{Holwerda+2012} intends to integrate a single pointing with MeerKAT for 5,000 hours. This makes the total field of view of the survey simply the primary beam of a single dish, 0.9 deg$^{2}$ at $z=0$. MeerKAT will have a maximum baseline of 8 km, potentially allowing the synthesised beams to reach down to sizes of $\sim$10 arcsec. The bandwidth of the survey will in theory permit detections of HI sources out to a redshift of $\sim$1.5.
CHILES \citep{Fernandez+2013}, which recently began taking data with the VLA (Very Large Array), is also a single pointing survey, with an integration time of 1,000 hours. Due to the longer baselines of the VLA, the minimum synthesised beam is 5 arcsec across, while the larger dishes reduce the field of view to 0.25 deg$^{2}$ at $z=0$. The narrower bandwidth that CHILES adopts (compared to LADUMA), sets its maximum possible redshift for HI detection at 0.45.
Unlike the two deepest planned pathfinder surveys DINGO UDEEP \citep{Meyer2009,Duffy+2012c} will not be a single pointing. ASKAP (Australian Square Kilometre Array Pathfinder) will survey 60 deg$^{2}$ over the redshift range 0.1-0.43. The survey is intended to be 5,000 hours, and should detect tens of thousands of HI sources. However, due to the computational demands of forming multiple beams (from ASKAP's phased array feeds) and correlating all the signals over this wide bandwidth, it is not yet certain whether ASKAP will achieve a resolution of 10 or 30 arcsec for this survey.
FAST is a single-dish telescope (the only one in this list) currently under construction in China. The 305 m Arecibo observatory in Puerto Rico is the only existing telescope of a comparable size and design. However, unlike Arecibo's fixed reflector, FAST's segmented 500 m primary reflector will be deformable, and the instrument platform will be movable, allowing for zenith angles up to 40$^{\circ}$, which is double the sky area observable from Arecibo. While FAST is observing, a 300 m segment of the reflector will be deformed into a parabola \citep{Nan2006}, giving it a resolution of approximately 3 arcmin for 21 cm radiation, compared to almost 4 arcmin for Arecibo. FAST's larger area will produce greater sensitivity than Arecibo, while its proposed 19 feed horn array (compared to the 7 horn Arecibo L-band Feed Array, or ALFA) will increase its survey speed to a factor of a few faster than Arecibo. Assuming that FAST's feed array has a system temperature of 31 K (as does ALFA) the figure of merit (FoM), which effectively measures a telescope's sensitivity divided by the time taken to map a given area, is 37 for FAST, compared to 4.6 for ALFA on Arecibo (on a scale where 1 pixel with a system temperature of 25 K on Arecibo has a FoM of 1). Although the exact surveys that FAST will carry out have yet to be defined, it has been suggested \citep[e.g.][]{Duffy+2008} that it might probe HI galaxies out to a redshift of $\sim$0.5.
In addition to these upcoming ultra deep surveys, we will reference the two currently available large area, blind HI surveys, ALFALFA (Arecibo Legacy Fast ALFA) and HIPASS (HI Parkes All Sky Survey). The ALFALFA survey \citep{Giovanelli+2005} covers approximately 6,900 deg$^{2}$, with a mean source density of 4 deg$^{-2}$ and a mean redshift of 0.03. The HI properties and functions used throughout this paper \citep{Martin+2010,Papastergis+2013} were derived from the $\alpha$.40 catalogue \citep{Haynes+2011}, which covers 40\% of the nominal sky area. HIPASS \citep{Barnes+2001,Meyer+2004} covers approximately a hemisphere of sky area, but is less deep than ALFALFA, with a mean redshift of 0.01 and a mean source density of 0.2 deg$^{-2}$.
\section{Determining the Confusion in a Stack}
\label{sec:model}
In order to assess how confused a stacked spectrum is, it is necessary to calculate the relative contributions from the target objects versus those they are confused with. The signal due to confusion is found from the total HI mass there is (on average) in a given stack, in addition to that of the target objects. If this mass is negligible in comparison to the mass of the target sources, then clearly it is not a concern. However, if it is comparable in mass, then the spectral profile of this confused emission is also of interest, as this will determine how it alters the appearance of the stacked spectrum in practice. The following subsections outline how each of these quantities can be calculated.
\subsection{Confused Mass in a Stack}
\label{sec:mass_in_stack}
When creating a stack, the angular (or physical) size of the `postage stamps' (or `cut outs') must be chosen. This defines a scale on the sky, and the smallest it can meaningfully be is the size of the beam (or synthesised beam, for interferometers); which is what we shall assume. For simplicity, the fact that the final maps will be made up of pixels is ignored, and the `cut outs' are assumed to be circular. The same analysis could be done with square `cut outs', but given the other uncertainties (see section \ref{sec:lims}) this factor of order unity is unimportant.
Next, a velocity range in the spectrum must be chosen in which the relevant signal is believed to reside. The broadest HI galaxy velocity widths are around 600 km/s, so with an accurate input redshift, a velocity slice of $\pm$300 \kms \ is a conservative choice, and is what will be used here. The results are less sensitive to this choice than might be expected, because the correlation function (CF) causes the signal to be strongly peaked around zero relative velocity.
Together these dimensions define a cylinder in redshift space that is centred on the target being stacked. The amount of HI mass, in addition to the central source, that is within this volume (on average) determines the strength of the confusion signal in the final stacked spectrum, and we will refer to it as the ``confused mass".\footnote{We will also use the phrase ``confused sources" throughout this paper to mean the sources that a target object is confused with, not including the target object itself.}
In order to calculate the mean confused mass in a stack, two things must be known: the expected number of HI galaxies residing in the cylinder surrounding the target object, and the mean HI mass of an HI-selected galaxy. The first of these can be calculated from the CF, and the second by the integral of the HI mass function (HIMF).
The CF is the excess probability (above random) of two sources being separated by a given distance, here denoted by $\xi(\kappa,\beta)$, where $\kappa$ is the separation perpendicular to the line of sight, and $\beta$ is the separation along it. In general it is not symmetric with respect to $\kappa$ and $\beta$, as distance along the line of sight is usually determined from redshifts, and so peculiar velocities alter the derived separations. Although these distortions along the line of sight are not physical, in the sense that the galaxies may not be separated by the distances calculated, they are directly applicable to this scenario as the depth of the cylinder is also a pseudo-distance (a velocity divided by the Hubble constant). Thus, we make use of the 2D CF for HI sources, as calculated in \citet{Papastergis+2013}, and for convenience, will use the simple analytic fit from \citet{Jones+2015} to approximate it:
\begin{equation}
\xi(\kappa,\beta) = \left( \frac{1}{r_{0}}\sqrt{\frac{\kappa^{2}}{a^{2}} + \frac{\beta^{2}}{b^{2}}} \right)^{\gamma},
\label{eqn:2dcf}
\end{equation}
where $ab = 1$, $r_{0} = 9.05$ Mpc, $a = 0.641$, and $\gamma = -1.13$.
Integrating $1+\xi$ over the cylinder defined by the choice of `postage stamp' size and velocity range, and multiplying by the mean HI source number density, gives the expected number of additional HI sources within the volume. Finally, multiplying by the mean HI mass of an HI source \citep{Martin+2010}, returns the total mass in these sources within the beam on average, $M_{\mathrm{conf}}$.
\begin{eqnarray}\nonumber
M_{\mathrm{conf}} & = & 4 \mathrm{\pi} \Omega_{\mathrm{HI}} \rho_{\mathrm{c}} \int_{0}^{\beta_{\mathrm{sep}}} \int_{0}^{\kappa_{\mathrm{sep}}} \kappa \left[ 1+\xi(\kappa,\beta) \right] \mathrm{d}\kappa \mathrm{d}\beta \\
& = & 2 \mathrm{\pi} \Omega_{\mathrm{HI}} \rho_{\mathrm{c}} a \left[ \frac{\beta_{\mathrm{sep}} \kappa^{2}_{\mathrm{sep}}}{b a^{2}} + I \right],
\label{eqn:model}
\end{eqnarray}
where
\begin{eqnarray}\nonumber
I &=& \frac{2\frac{\beta_{\mathrm{sep}}}{b} \left( \frac{\kappa_{\mathrm{sep}}}{a} \right)^{\gamma+2}(\gamma+3)}{(\gamma+2)(\gamma+3) r_{0}^{\gamma}} \\ &&\left[ _{2}F_{1}\left( \frac{1}{2},-\frac{\gamma}{2}-1;\frac{3}{2};-\frac{a^{2} \beta^{2}_{\mathrm{sep}}}{b^{2} \kappa^{2}_{\mathrm{sep}}} \right) -2\left( \frac{\beta_{\mathrm{sep}}}{b} \right)^{\gamma+3} \right]
\end{eqnarray}
and $\mathrm{_{2}F_{1}}$ is the Gaussian hypergeometric function, $\beta_{\mathrm{sep}}$ is the velocity half range, in this case $300/70$ Mpc, $\kappa_{\mathrm{sep}}$ is the physical radius of the beam in Mpc at the distance of the target object, and $\Omega_{\mathrm{HI}}$ is the background density of HI ($\rho_{\mathrm{HI}}$) relative to the critical density ($\rho_{\mathrm{c}}$) in \Msol$\,\mathrm{Mpc}^{-3}$ (equivalent to the mean source number density times the mean source mass). We adopt $\Omega_{\mathrm{HI}} = 4.3 \times 10^{-3}$, as found by \citet{Martin+2010}. Refer to \citet{Jones+2015} for the full details of the fit to $\xi(\kappa,\beta)$ and how to evaluate its integral.
The above equation for the confused mass is independent of the shape of the HIMF, because the quantity of HI in a given volume only depends on its integral. However the variance of the confused mass is dependent on the shape of the HIMF. This can be understood by considering where most of the HI mass in the Universe resides, which at present is in $M_{*}$ galaxies. If the faint-end slope was steeper and most of the HI mass resided in highly abundant dwarf galaxies, then the variance in the confused mass would be small (ignoring the environmental dependence that would likely be present in such a universe) as the Poisson noise in the number counts within the cylinder would be low. Alternatively if the faint-end slope were to be very flat and the knee mass very high, then although the integral could be identical, most of the HI mass would be contained in exceptionally rare, highly massive systems. As a result the Poisson noise associated with the counts of such galaxies would be very large, leading to high variance in the confused mass.
\subsection{Spectral Profile of Confusion}
\label{sec:spec_prof}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{spec_profile.eps}
\caption{The solid black line shows a simulated stack of the average spectral profile contributed by confused sources only (target sources have been removed), in a stack at $z=0.029$ for a survey with a beam size of 15.5 arcmin (at $z=0$), intended to mimic the \citet{Delhaize+2013} experiment with HIPASS. The green dashed line shows the double Gaussian fit to the black profile, while the red dotted and blue dash-dot lines show the two separate components of the fit.}
\label{fig:spec_prof}
\end{figure}
If all the additional mass in the cylinder was uniformly distributed in velocity space then it would not pose a problem to deriving physical properties from the stacked spectrum, as the confusion signal would just represent a DC shift in the baseline. However, if the confusion signal is peaked around the central frequency, then it can contribute an unknown amount to the final flux, or worse, make up all of the flux and give a false positive (in the event that the central sources are not detected even in the stacked spectrum).
The spectral shape of the confusion signal (which we will refer to as the ``confusion profile") can be calculated using a similar method to that in section \ref{sec:mass_in_stack}, which reveals it takes a double Gaussian form. However, this method neglects the velocity widths of each galaxy contributing to the confusion signal. Therefore, we have estimated the confusion profile using mock stacks (see section \ref{sec:stack_sims}), shown by the solid black line in figure \ref{fig:spec_prof}. The inclusion of velocity widths broadens the confusion profile, however it maintains a double Gaussian shape (see figure \ref{fig:spec_prof}). The two components arise from the peak in the CF at zero velocity separation, and the uncertainty in the input catalogue of target redshifts. For the latter we assume a Gaussian distribution centred on zero with a width of 35 \kms, as found by \citet{Toribio+2011}.
Here it should be reiterated that figure \ref{fig:spec_prof} includes only the stacked emission of the confused sources, with emission from the target galaxies removed. The profile is well fit by a double Gaussian with a narrow and a broad component, which highlights that caution must be used when interpreting heavily confused stacks, as this profile shape is similar to what might be expected for a stack detection on top of confusion noise, not just from confusion alone.
\subsection{Mock Stacks}
\label{sec:stack_sims}
In order to help assess our findings and potential strategies to mitigate confusion, we make use of simulated HI stacks. Our approach is similar to that of \citet{Maddox+2013}, which used the template HI profiles of \citet{Saintonge+2007}, however our mock stacks are intentionally noiseless and the masses and velocity widths are drawn randomly from a fit to the $\alpha$.40 mass-width function \citep{Papastergis+2015,Jones+2015}.
When simulating the signal from confusion, galaxy masses and widths are drawn from the mass-width function. A lower HI mass bound (of $10^{6.2}$ \Msol, the lowest that ALFALFA can measure the HIMF to) must be set, and only masses greater than this are selected. However, the results are insensitive to this bound as most of the HI mass in the Universe is contained in much more massive systems. The number of confused sources to be included (around each target object) is chosen from a Poisson distribution with an expectation equal to $M_{\mathrm{conf}}/\bar{M}_{\mathrm{HI}}$, where $\bar{M}_{\mathrm{HI}}$ is the mean HI mass of a galaxy, and $M_{\mathrm{conf}}$ is the confused mass as calculated in equation \ref{eqn:model}. The galaxy masses and widths are then drawn from the mass-width function and are placed at angular and velocity separations (away from the central target) drawn from the 2D CF (equation \ref{eqn:2dcf}). Finally, the profiles are added to the stack at the appropriate frequencies (the angular information is ignored except when non-uniform beam weightings are consider in section \ref{sec:beam_weight}).
To simulate the contribution of the target objects, we make the assumption that all the targets have the same mass and then draw only the velocity width (for the relevant mass) from the mass-width function. A redshift error is added to the profile, drawn from a Gaussian of width 35 \kms, and then it is added to the stacked spectrum. All stacked targets are assumed to be the same mass for simplicity and generality, however information about the mass distribution of targets, which might be available when modelling a particular survey, would be straightforward to incorporate. This assumption has no impact on the amount of confused mass we calculate, but could alter ratio of confusion to target signals.
\subsection{Modelling Limitations}
\label{sec:lims}
The model and simulation methods described above have a number of caveats and shortcomings which are outlined in this section. A general note is that this methodology only applies to the average values present in a large stack. This will require on the order of 1,000 spectra in a given stack, such that extreme cases and small number statistics are not dominant.
\subsubsection{Redshift Evolution}
\label{sec:red_evo}
Although there is some evidence for $z$-dependence of $\Omega_{\mathrm{HI}}$ from stacking, damped Lyman-$\alpha$ observations and HI intensity mapping experiments \citep[e.g.][]{Rao+2006,Lah+2007,Prochaska+Wolfe2009,Chang+2010,Freudling+2011,Delhaize+2013,Rhee+2013,Hoppmann+2015}, there is no observational data describing how the shape of the HIMF may evolve, or how the HI CF evolves. Due to these limitations we choose to display our results for two separate assumptions: constant $\Omega_{\mathrm{HI}}$, and $\rho_{\mathrm{HI}} \propto (1+z)^{3}$, with both using the $z=0$ CF throughout. The first case will likely under predict the confused mass at high redshift as the observations indicate a factor of $\sim$2 increase in $\Omega_{\mathrm{HI}}$ by $z=1$, while the second case actually appears to overestimate the increase of HI density with redshift. Thus, barring a major shift in the HI CF, we expect the true value to lie between these two cases.
\subsubsection{Sharp Edges \& Point Sources}
\label{sec:edges_points}
The response of the telescope beam is assumed to be a step-function. When stacking based on `cut outs' from a uniform survey map, where the shape of the beam response has already been accounted for, this is the simplest choice. In section \ref{sec:beam_weight} we discuss the possibility of using a different weighting as a way to reduce confusion.
When simulating stacks to verify the analytic results and test mitigation strategies (section \ref{sec:mit_strat}), the confused sources, which in reality would be galaxies with their own velocity widths and spatial patterns, are modelled with realistic HI profile shapes \citep{Saintonge+2007} in frequency space, but as point sources on the sky. Given the simplistic weighting of the beam, modelling sources as points (spatially) is sufficient. However, as the finite velocity widths inevitably broaden the profile of any confusion signal (see figure \ref{fig:spec_prof}), it might be expected that the total mass within a $\pm$300 \kms \ window might differ from the value derived via equation \ref{eqn:model}. This has been explicitly checked for in our mock stacks, and while the spectral profile of the confusion signal becomes broader, it maintains a double Gaussian shape and the total confused mass is consistent with the analytic model.
\subsubsection{Redshift Error Distribution}
In order to stack non-detections an input (presumably) optical catalogue of positions and redshifts must be used. When calculating the profile of the confusion signal a Gaussian distribution with a width of 35 \kms \ is assumed to represent the deviations between the HI and optical redshifts. In practice the scale of this dispersion is dependent on the quality of the spectra in the input catalogue. \citet{Maddox+2013} found a smaller dispersion between SDSS and ALFALFA when only including the highest S/N ALFALFA detections, while \citet{Delhaize+2013} quoted the uncertainty in their input redshifts as 85 \kms. Although the value we chose to adopt changes the width of our resulting profile, it does not alter the qualitative results.
For a particular survey there may be more knowledge about how these redshifts differ from each other which, when available, should be used instead. Alternatively, the bias from confusion could be estimated by calculating the cross correlation function between the HI and optical catalogue when possible, and use this in place of equation \ref{eqn:2dcf}.
\subsubsection{Model Uncertainties and Variance}
\label{sec:mod_var}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{conf_mass_variance.eps}
\caption{The fractional uncertainty in the confused mass (standard deviation divided by mean value) estimated by simulating 100 stacks, each of 1,000 targets, at each redshift and beam size. The bold lines indicate those mocks which assume $\Omega_{\mathrm{HI}}$ is constant, and the standard weight lines are for a $\rho_{\mathrm{HI}} \propto (1+z)^3$ model. The solid (red) lines represent a beam size of 30 arcsec, and the dashed (green) lines a beam size of 10 arcsec. These are estimates of the 1-$\sigma$ fractional uncertainties in the red (second from top) and green (second from bottom) solid lines in figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo}.}
\label{fig:conf_mass_var}
\end{figure}
There is an error associated with our choice of the parametric forms used to fit both the 2D CF (equation \ref{eqn:2dcf}) and the mass-width function \citep[see ][]{Jones+2015}, as well as the exact ranges we chose to fit them over. As this is a single choice that involves the judgement of the individual performing the fit, it is very difficult to estimate a quantitative error for. Thus, rather than quoting an error we have chosen to a) demonstrate that the model we use is consistent with both the number counts and the observed rate of confusion between detections in the ALFALFA data set \citep{Jones+2015}, and b) present arguments (sections \ref{sec:red_evo} and \ref{sec:discuss}) that the two extremes which we adopt for any redshift evolution, likely bracket the true value.
The above concerns aside, there is still another uncertainty that is important. Equation \ref{eqn:model} gives the confused mass that is present \emph{on average} in a stacked spectrum. As alluded to previously (section \ref{sec:mass_in_stack}) the variance of this quantity depends on the shape of the HIMF, and the more top-heavy it is, the higher the variance in $M_{\mathrm{conf}}$. In addition to the shape of the HIMF, the variance of $M_{\mathrm{conf}}$ also depends on the number of spectra being stacked, the angular size of the `cut outs', and the redshift of the stack. To estimate the scale of the variance we ran 100 realisations of stacks of 1,000 targets at redshifts 0.1 to 1.4 (in increments of 0.1), for two beam sizes, 10" and 30" (at $z=0$). Figure \ref{fig:conf_mass_var} shows the fractional uncertainties (standard deviation divided by the mean) in the confused mass calculated from these realisations. While the uncertainty for the mock stacks with a 30 arcsec beam quickly (by $z \sim 0.3$) drop to less that 10\%, for the stacks using a 10 arcsec beam the uncertainty starts off at almost 100\% and does not fall to 10\% until between a redshift of 0.5 and 1 (depending on the assumed evolution of $\Omega_{\mathrm{HI}}$). This indicates that accounting for confusion in a statistical way will be difficult for surveys with small beam sizes, as the variance in any individual stack will be so large. However, as is shown below confusion will turn out to be only a minor concern for surveys achieving beam sizes of 10 arcsec.
\section{Results}
\label{sec:results}
Before proceeding with predictions for upcoming surveys the CF model was tested against an existing study of HI stacking in a highly confused regime by \citet{Delhaize+2013}. In that paper HIPASS non-detections were stacked based on Two-Degree-Field Galaxy Redshift Survey (2dFGRS) positions and redshifts. The mean redshift of their sample was 0.029, and the stacked spectrum has a mass of $3 \times 10^{9}\,h_{70}^{-2}\,$\Msol \ between velocities $\pm300$ \kms. They also estimated that each source was confused with three others (on average), which increased the effective luminosity of the stacked sample by a factor of 2.5. Assuming a constant mass-light-ratio, this means that the contribution of confusion to the stack was approximately $1.8 \times 10^{9}\,h_{70}^{-2}\,$\Msol. A higher redshift sample of targeted follow-up was also stacked, giving a mean mass of $1.4 \times 10^{10}\,h_{70}^{-2}\,$\Msol \ at a mean redshift of 0.096, of which $1.1 \times 10^{10}\,h_{70}^{-2}\,$\Msol \ was estimated to be due to confusion.
Using our framework (and assuming constant $\Omega_{\mathrm{HI}}$) to estimate the confused mass in a stack at a redshift of 0.029 in HIPASS data returns a value of $1.9 \times 10^{9}\,$\Msol \ for a beam size of 15.5', and $3.3 \times 10^{9}\,$\Msol \ for a beam size of 21.9'. The Parkes telescope beam size is 15.5' for a wavelength of 21 cm, but the weighting used in \citet{Delhaize+2013} produces an effective beam size of 21.9' (and 21.2' for the higher $z$ sample). We quote results for both beam sizes as our model does not incorporate the beam profile weighting they assume. For the higher redshift sample we estimate a confused mass of between $1.3$ and $2.0 \times 10^{10}\,h_{70}^{-2}\,$\Msol \ for beam sizes 15.5' and 21.2' respectively. Both of these results appear approximately consistent, although the exact confidence is not possible to assess (see section \ref{sec:discuss}).
The confused mass present, on average, in a stack made from a generic survey at a given redshift, was estimated based on the integral of the 2D CF over the telescope beam and $\pm300$ \kms \ in redshift space (see section \ref{sec:mass_in_stack}). Figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo} show the results for various telescope resolutions, each solid line represents a different angular resolution: 5, 10, 20, 30 arcsec and 3 arcmin (at $z=0$), from bottom to top. The dashed lines represent the confused mass that would be present if the Universe were perfectly uniform, and the faint dotted lines are the results obtained using the projected CF \citep{Papastergis+2013}, which removes the difference in the physical and velocity directions. The two figures are identical except that figure \ref{fig:conf_mass_const_Omega} assumes $\Omega_{\mathrm{HI}}$ does not change from its value at $z=0$, while figure \ref{fig:conf_mass_evo} assumes $\rho_{\mathrm{HI}}$ grows like $(1+z)^{3}$.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{mass_in_stack_general_const_Omega.eps}
\caption{The predicted average HI mass due to sources of confusion in a stacked spectrum as a function of the redshift (distance) of the stack, assuming $\Omega_{\mathrm{HI}}$ is fixed at its zero redshift value. The line styles indicate the method used to generate the estimate, with solid lines representing the 2D correlation function, dotted lines the projected (or 1D) correlation function, and dashed lines assume the Universe is uniform in HI. The blue, green, orange, red and black lines use beam sizes of 5, 10, 20, 30 arcsec, and 3 arcmin (at $z=0$) respectively, or equivalently smallest to largest beam going bottom to top.}
\label{fig:conf_mass_const_Omega}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{mass_in_stack_general_evo.eps}
\caption{Identical to the figure above except $\rho_{\mathrm{HI}}$ increases in proportion to $(1+z)^{3}$ from its zero redshift value.}
\label{fig:conf_mass_evo}
\end{figure*}
Below we outline the results relevant to each upcoming survey. Wherever a value is quoted for the constant $\Omega_{\mathrm{HI}}$ case, the $\rho_{\mathrm{HI}} \propto (1+z)^{3}$ value will immediately follow in parentheses (if different at the stated precision).
\subsection{CHILES}
CHILES has a resolution of 5 arcsec, so the solid blue (lowest) lines in figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo} are the appropriate estimates of the confusion in stacked CHILES data. Even at the maximum redshift (0.45) the confused mass within one synthesised beam, and $\pm$300 \kms, will still only be $\sim10^{7}$ \Msol, indicating that CHILES will have no major concerns due to confusion when stacking sources. However, as CHILES will spatially resolve almost all the sources it detects, a more appropriate measure of the confused mass can be derived by choosing a constant physical scale for the `postage stamp' cut out of a galaxy in a stack. To be overly conservative we choose 100 kpc (diameter), which gives a confused mass of $1 \times 10^{8}$ \Msol \ at $z=0$, which increases to $1.6 \times 10^{8}$ \Msol \ ($3.1 \times 10^{8}$ \Msol) at $z=0.45$. In other words, CHILES would only encounter non-negligible amounts of confusion bias if very low mass objects (presumably at lower redshift) were to be stacked, which seems unlikely given the that CHILES is a pencil beam survey.
\subsection{LADUMA}
For LADUMA the angular size of the minimum synthesised beam is still not set, however as MeerKAT's maximum baseline will be smaller than the VLA's B-configuration baseline, here we assume LADUMA will have a resolution of 10 arcsec. This is represented by the solid green (second lowest) line in figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo}. As mentioned above, in reality the confused mass is unlikely to ever drop much below $10^{8}$ \Msol \ even at low redshifts, as the physical size of the sources (rather than the size of the beam) will determine the `postage stamp' size.
Again this indicates that LADUMA will be safe from the impact of confusion when stacking sources significantly more massive than $10^{8}$ \Msol, at least up to intermediate redshifts. By the outer edge of LADUMA's bandpass ($z=1.45$) the mass in confusion will have risen to $1.8 \times 10^{9}$ \Msol \ ($5.4 \times 10^{9}$ \Msol), potentially large enough to influence the stacking of $M_{*}$ galaxies.
However, if LADUMA were to be unable to achieve its intended synthesised beam size, then things would look quite different. The orange (third lowest) lines show the case for a 20 arcsec beam, which at the outermost redshift (1.45) would contain over $5 \times 10^{9}$ \Msol \ ($1.5 \times 10^{10}$ \Msol) of confused HI, and even by $z \sim 0.5$ would contain $5 \times 10^{8}$ \Msol \ ($1 \times 10^{9}$ \Msol). Preliminary estimates of LADUMA's detection capability (A. Baker, private communication) suggest that at $z \sim 0.5$ targets down to masses of $3 \times 10^{8}$ \Msol \ might be detectable via stacking, and by the outer edge of the survey this will have increased to $3 \times 10^{9}$ \Msol. In both cases, if LADUMA were to have a beam size of 20 arcsec rather than 10, then these stacks would contain more mass in confused HI than in the target objects. While this may not prevent progress via stacking, it would add a strong additional bias and a new level of complexity to the process that would require careful consideration, compared to if the survey achieves its target resolution.
\subsection{DINGO UDEEP}
Similarly to CHILES, if ASKAP is able to achieve 10 arcsec resolution then the stacking capabilities of DINGO UDEEP will be limited by the physical size of objects, rather than the survey's angular resolution, throughout most of its redshift range (0.1-0.43). Whereas, if only a 30 arcsec resolution can be achieved then, as the red (second highest) line in figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo} shows, the confused mass will soon rise well above $10^{8}$ \Msol, complicating the interpretation of any stacks of objects of comparable mass. Although stacking of objects above $10^{9}$ \Msol \ should still be relatively unimpeded, as the confused mass does not reach $10^{9}$ \Msol \ until $z \sim 0.4$ and, as will be discussed in section \ref{sec:mit_strat}, the confusion signal can be effectively removed until it becomes comparable to the target signal.
Using the \citet{Jones+2015} expression for a general survey detection limit and assuming an order of magnitude improvement from stacking, we estimate that DINGO UDEEP will be capable of detecting an object with an HI mass of $3 \times 10^{8}$ \Msol \ via stacking at $z=0.2$, but at that redshift the predicted confused mass is $1.7 \times 10^{8}$ \Msol \ ($2.4 \times 10^{8}$ \Msol) for a 30 arcsec beam. At $z=0.4$ the situation is slightly worse, with the confused mass becoming $6.2 \times 10^{8}$ \Msol \ ($1.1 \times 10^{9}$ \Msol) and the mass detectable via stacking being $1 \times 10^{9}$ \Msol.
\subsection{FAST}
Unlike the other telescopes discussed here FAST is a single dish, and thus will have a much poorer resolution. The black solid (highest) line in figures \ref{fig:conf_mass_const_Omega} \& \ref{fig:conf_mass_evo} shows the expected confused mass for a FAST based survey, which rises above $10^{9}$ \Msol \ by a redshift of $\sim$0.1 and by 0.4 (0.3) even the most HI massive galaxies will be severely impacted by confusion. FAST's vast collecting area will mean it might be capable of directly detecting HI galaxies in a survey out to $z=0.2$ or greater, and would certainly by capable of doing so via stacking, but regardless of how these sources might be detected they will still be subject to considerable bias from confusion.
\section{Discussion}
\label{sec:discuss}
The approximate agreement shown between the estimates of the confused mass from stacks of Parkes data \citep{Delhaize+2013} and our model is an encouraging validation. However, both our predictions are somewhat higher than the estimates from that paper. The significance of this is difficult to assess as the values quoted from \citet{Delhaize+2013} are not given with errors at the relevant stage of their calculation. The simplest potential explanation might be that this is variance between the average value and two particular examples, however using similar multiple realisations of mocks stacks to those in section \ref{sec:mod_var} it is clear that this cannot be the explanation, as we measure only a standard deviation of approximately a percent between equivalent simulated stacks.
If this offset is real then the reason for it is uncertain; one possibility is that this model uses the HI auto-correlation function, whereas the HI-optical cross-correlation function might be the most appropriate. As shown in \citet{Papastergis+2013} the correlation function of SDSS blue galaxies is almost indistinguishable from that of an HI population, but HI-rich galaxies are much less likely to be found in regions with high densities of red galaxies. Therefore, an input sample that contains any red galaxies will have less confused HI mass around those targets than would targets based on an HI selected sample. Another possible explanation is that the assumption of a constant HI mass to light ratio across the target and confused sources might not be valid at the level of the discrepancy.
Assuming that the upcoming interferometric HI surveys can achieve their desired beam sizes they should have minimal amounts of confusion when making stacks throughout most of their bandpass ranges. However, due to its beam size, confusion is considerably more worrisome for FAST. \citet{Duffy+2008} estimated the contribution of confusion to a FAST survey and found that even for very long integration times (over 15 hours) it would not be a concern until beyond a redshift of 0.5. The dashed black (highest) line in figure \ref{fig:conf_mass_evo} shows the confused mass calculated assuming a uniform universe for a FAST sized beam (3 arcmin). This is equivalent to how the confused mass was defined by \citet{Duffy+2008}, but our value of $\Omega_{\mathrm{HI}}$ is 16 percent larger. As can be seen here the inclusion of the CF (solid black line), compared to assuming a uniform background, increases the confused mass by more than an order of magnitude (over the relevant redshift range). This will severely limit FAST's ability to probe HI galaxies much beyond $z=0.1$, which reiterates the conclusion of \citet{Jones+2015}, that future blind HI surveys with single dish telescopes should focus on the nearby universe where their larger beam sizes are a strength rather than a hindrance.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{sim_stack_beam3_z01.eps}
\includegraphics[width=\columnwidth]{sim_stack_beam3_z02.eps}
\caption{Simulated noiseless stacks of 1,000 galaxies of HI mass $3 \times 10^{9}$ \Msol, showing the contributions from the target galaxies (dashed green lines) and from confusion (red dash-dot lines). Both plots assume a zero redshift beam size of 3 arcmin (as expected for FAST). The left plot is for a stack at $z=0.1$, and the right at $z=0.2$. Both assume $\rho_{\mathrm{HI}} \propto (1+z)^{3}$.}
\label{fig:FAST_stack}
\end{figure*}
To show how confusion may affect a stack of data from a FAST survey we have simulated two stacks of galaxies with target objects of $3 \times 10^{9}$ \Msol, at redshifts 0.1 and 0.2 (see figure \ref{fig:FAST_stack}). If the total signal is (incorrectly) assumed to be made up of two Gaussian components, a broad one due to confusion and a narrow one due to the target signal, the mean target masses are found to be 3.3 and $4.4 \times 10^{9}$ \Msol \ respectively at $z = 0.1$ and 0.2. The excess signal that is incorporated into the narrow Gaussian component originates from the fact that the confusion profile is itself a double Gaussian, and is therefore not adequately subtracted by the broad component alone. In fact the overestimation would be worse, but some of the target signal is clipped (by the $\pm 300$ \kms \ boundary), and some is incorporated into the broad Gaussian along with the confusion signal.
A major uncertainty in our predictions is redshift evolution, which due to the current lack of data is inadequately modelled. We argued in section \ref{sec:red_evo} that the two cases presented for the evolution of HI density likely bracket the true evolution in that quantity, however the impact of the change in the HI CF is more difficult assess. \cite{Hartley+2010} find that the correlation length of blue galaxies in the UKIDSS Ultra Deep Survey increases by approximately a factor of 2 going from $z=0$ to 1.5. At $z=0$ blue galaxies and HI-rich galaxies are proxies for each other. Therefore, it is reasonable to assume that the HI CF would also be raised with increasing redshift, meaning the curves shown in figure \ref{fig:conf_mass_const_Omega} would represent lower limits on the confused mass in stacks.
The two models of the evolution of $\rho_{\mathrm{HI}}$ unsurprisingly give similar results at low redshift, but start to diverge at larger redshift, leaving LADUMA with the most uncertain measure of confused mass. The shape of the confused mass versus redshift curve for the constant $\Omega_{\mathrm{HI}}$ model (figure \ref{fig:conf_mass_const_Omega}) is qualitatively similar to the shape of a model detection limit for an HI survey. \citet{Fabello+2011} found that an order of magnitude below the detection limit is the most that could be gained by stacking, before non-Gaussian noise became dominant (although \citet{Delhaize+2013} indicates that deeper stacks might be possible with very well characterised noise). Therefore, assuming that at all redshifts there are sufficient stacking targets available that are approximately an order of magnitude below the detection limit, we arrive at the somewhat counter intuitive result that the ratio of the mean mass of these targets to the confused mass in their stack, is almost independent of redshift.\footnote{Note that this may appear to be in conflict with the \citet{Delhaize+2013} experiment, however that is because their two datasets have very different integration times, allowing them to probe lower masses than would otherwise be possible in their higher redshift sample, and thus making the stack more confused.} It should be noted however that this will break down at the lowest redshifts because, as stated previously, in practice the physical size of galaxies will prevent the confused mass ever dropping much below $10^{8}$ \Msol. In the case where $\rho_{\mathrm{HI}}$ increases with the Universe's decreasing volume (figure \ref{fig:conf_mass_evo}) the confused mass rises much more steeply with redshift, producing much more severe confusion at higher $z$. While this might seem like the most conservative model to use, the currently available data (\citet{Rhee+2013} and references within) indicate that $\rho_{\mathrm{HI}}$ does not increase this quickly with redshift.
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{sim_stack_beam10_z04.eps}
\includegraphics[width=\columnwidth]{sim_stack_beam30_z04.eps}
\caption{Simulated noiseless stacks of 1,000 galaxies of HI mass $10^{9}$ \Msol, showing the contributions from the target galaxies (dashed green lines) and from confusion (red dash-dot lines). Both plots are for a stack at $z=0.4$ and assume that $\rho_{\mathrm{HI}} \propto (1+z)^{3}$, but the left has a zero redshift beam size of 10 arcsec, while the right has a 30 arcsec beam. }
\label{fig:DINGO_stack}
\end{figure*}
Regardless of which evolution model is assumed to be correct, the results show that for surveys like LADUMA and DINGO UDEEP, where the synthesised beam size is not yet fixed, there is much to be gained in terms of the stacking performance by pushing to a lower beam size (in this case 10 arcsec). The difference in confused mass between a beam size of 10 and 30 arcsec is approximately an order of magnitude. For DINGO UDEEP a 30 arcsec beam would mean that a large fraction of the mass in stacks (probing the lowest possible HI masses) will be contributed by confusion, at all redshifts; whereas with a 10 arcsec beam the contribution would be almost negligible. For LADUMA there is little option but to use a $\sim$10 arcsec beam if stacking is going to be a viable option. Even with a 20 arcsec beam the smallest masses that are in theory detectable via stacking would likely always be below the level of the confused mass, but with a 10 arcsec beam this would not be the case until the very largest redshifts.
Figure \ref{fig:DINGO_stack} shows the contributions of confusion in two simulated stacks at approximately the outer edge of DINGO UDEEP's bandpass ($z=0.4$) for beam sizes of 10 and 30 arcsec (at $z=0$). The target galaxies have HI masses of $10^{9}$ \Msol, the lowest that will likely be detectable via stacking with this survey at $z=0.4$. While the 30 arcsec beam introduces $1.1 \times 10^{9}$ \Msol \ of confusion, the 10 arcsec beam only introduces $1.5 \times 10^{8}$ \Msol. In this case naively splitting the resulting total signal into two Gaussian components gives a mean target mass of 1.3 and $1.0 \times 10^{9}$ \Msol, for the 30 and 10 arcsec beams respectively.
For regimes where the confused mass in a stack is comparable to the anticipated mass of the targets, the spectral profile calculated in section \ref{sec:spec_prof} indicates that caution must be used. The profile of confusion alone appears to be well fit by a double Gaussian, where the two components arise from the width of the velocity space CF and the distribution of redshift uncertainties in the input catalogue. This is precisely the profile that might be expected from a stack detection with a small amount of confusion, a narrow Gaussian (presumed from the target objects) superimposed on a broader Gaussian (presumed to be from confusion). Thus, in a severe case it is possible that confusion alone could be misidentified as a detection and confusion. In a more moderate case, where there is a real detection, it is desirable to minimise the amount of confused mass and to understand how much it still contributes to the final stack. Strategies to accomplish this are discussed below.
\subsection{Mitigation Strategies}
\label{sec:mit_strat}
In any stacking experiment where a significant contribution from confusion is anticipated (not limited to the surveys discussed here), there are two approaches that can be taken to improve the outcome: either strategies to remove confused mass can be implemented, or the amount of the final signal that is contributed by confusion can be estimated.
As a first approximation the model presented in this paper can be used to predict how much confusion there is in a stack, however there are a number of situations where this might give a poor estimate. For example, a stack with a small number of targets, at high redshift, or with an input catalogue of galaxies not selected for HI content. In such cases other strategies might be necessary. One approach could be to explore the properties of such stacks in a simulation, another is to attempt to mitigate the impact of confusion when extracting the final parameters from a stack, which is the approach we discuss below.
\subsubsection{Double Gaussian Decomposition}
As figure \ref{fig:spec_prof} shows, a large fraction of the signal from confusion is expected to be in a broad Gaussian component, whereas most of the target emission should be in a narrow component. Although there is also a narrow component to the confusion profile, removing the broad component will help to alleviate much of the confusion.
This approach was tested by simulating the confusion in a stack using representative HI line profile shapes \citep{Saintonge+2007}, positions from the CF, and assuming the $z=0$ value of HI density \citep{Martin+2010}. The narrow Gaussian component of the total profile was found to reproduce the mean target mass well (within $\sim$10\%) in the cases where the confused mass was less that about 2/3 of the target mass, although results were marginally worse for more massive, broader targets. Presumably the portion of the target signal that is excluded from the narrow Gaussian is approximately made up for by the inclusion of some of the narrow component of confusion. However, when the confused mass becomes almost as large as the target mass, the narrow Gaussian integral begins to diverge from the mean target mass.
Thus, this straightforward method is very successful for stacks with low levels of confusion, but cannot adequately separate target signal and confusion when the confusion is more severe.
\subsubsection{Beam Weighting}
\label{sec:beam_weight}
In the regime where the telescope beam (or synthesised beam) is considerably larger than the target source, the weighting of the pixels can be tapered away from the target. This will have little impact on the target flux (presumably concentrated in the central pixel), but will give lower weight to the surrounding confusion signal.
This approach was tested using mock stacks, as before. For stacks with physical beam sizes of 100-600 kpc, assuming 4 pixels across a beam width and a Gaussian weighting scheme, the confused mass was reduced by approximately 25-30\% compared to a uniform weighting. However, for larger beam sizes there are diminishing returns as in addition to the target, many of the confused objects also lie within the central pixel.
\subsubsection{Inclusion and Exclusion of Confused Targets}
Possibly the most obvious solution to confusion is to simply excluded the stacking targets that are likely to be heavily confused. Most of the HI mass in the Universe is contained in $M_{*}$ systems, which are likely to be visible in the optical input catalogue. Targets that are in close proximity to $M_{*}$ galaxies (provided they have optical redshifts) could in principle be removed from the input catalogue. As most of the HI mass is contained in these galaxies, this would remove most of the confused mass from the stack.
This approach has some promise for the cases where low mass galaxies are being stacked at low redshift and $M_{*}$ galaxies are uncommon, but for higher redshifts where the beam sizes become larger, many targets are confused, often multiple times \citep{Delhaize+2013,Jones+2015}. Thus, it becomes impractical to remove them.
The approach taken by \citet{Delhaize+2013} was to include such targets, but to note the presence of likely sources of confusion. By assuming a constant HI mass to light ratio they were able to estimate the fraction of the stack mass that was contributed by confusion. As shown in section \ref{sec:results} our results are roughly consistent with their findings. This procedure could be taken further by using HI scaling relations with stellar mass or disc size to improve the estimate of the confused galaxy masses \citep{Toribio+2011,Huang+2012}.
\subsubsection{Exclusion in Velocity Space}
Weighting the beam cuts confusion by eliminating sources spatially, but this can also be done in velocity space. \citet{Fabello+2011} used the Tully-Fisher relation (TFR) to remove the section of the spectrum containing the intended target, in order to estimate the rms noise in the rest of the spectrum. The same method could be used to stack just the region of the spectrum that is likely to contain emission from the target galaxy, thereby removing additional sources in front or behind the target that would otherwise contribute to a stack made with a conservative $\pm 300$ \kms \ cut.
This method was simulated as before, but with each contributing spectrum cut off at $\pm (W_{TF}/2 + \sigma_{\mathrm{input}})$ away from the target redshift. Where $W_{TF}$ is the target's simulated velocity width ($W_{50}$) with 0.2 dex of scatter introduced (designed to emulate the TFR), and $\sigma_{\mathrm{input}}$ is the standard deviation of the redshift uncertainty in the input catalogue (35 \kms). This gave approximately a 60\% reduction in confused mass when stacking targets in the mass range $10^{8}$ - $10^{9}$ \Msol, and a 45\% reduction for targets in the range $10^{9}$ - $10^{10}$. However, it also typically removed 30-35\% of the flux from to the target objects, with the higher mass stack more effected.
\section{Conclusions}
We created a model to predict the average amount of HI mass contributed by confused sources to a stack from a generic survey. The analytic expression of our model (equation \ref{eqn:model}) is derived in the general case, allowing for different beam sizes, velocity ranges, HI background densities or fits to the CF to be used to make quick estimates of the amount of confusion in any HI survey. This model, based on the ALFALFA correlation function, shows agreement with estimates of the confusion present in stacks of Parkes data \citep{Delhaize+2013}, and predicts approximately an order of magnitude more confused HI than found from assuming a uniform universe \citep{Duffy+2008}.
The largest uncertainty in the predictions comes from our relative ignorance of the redshift evolution of HI-rich galaxies. However, we argued that the true values likely fall between the two idealised cases presented here, and that the smaller of the two is in fact a lower limit.
The results for upcoming SKA precursors surveys, like LADUMA and DINGO UDEEP, reveal that it would be highly advantageous if these surveys could achieve their initially intended resolutions (10 arcsec), as any resolution substantially poorer than this would lead to stacks that are dominated by confusion, rather than their target objects.
Confusion was the most concerning for FAST; its larger (single dish) beam size results in the mass in confusion rapidly overtaking even that of $M_{*}$ galaxies, as redshift increases. This will prevent a FAST based blind HI survey from probing individual galaxies much beyond $z=0.1$ with either stacking or direct detections. Similarly to the findings of our previous work \citep{Jones+2015} this indicates that single dish telescopes should focus their HI galaxy studies on the local Universe.
When simulating stacks with a large component of confusion we had limited success in implementing mitigation strategies. Weighting pixels in a Gaussian pattern reduced the confused mass by about 30\%, but is only suitable when one pixel is larger than the angular extent of the targets. Using the TFR to exclude regions of the spectrum beyond the target's emission was even more successful at removing unwanted confusion, however it also removed around 30\% of the target emission. Simply decomposing the total spectrum into broad and narrow Gaussian components was very successful at estimating the mean target mass with even moderate levels of confusion, despite it not being an accurate model of the profile shape of targets combined with confusion. However, when the confused mass approached that of the targets, the results began to diverge from the true values. Thus in the event of of heavily confused stack, the best approach will likely be not to try to exclude sources of confusion, but to use optical data or simulations to model and account for their HI properties.
\section*{Acknowledgements}
The authors acknowledge the work of the entire ALFALFA collaboration in observing, flagging, and extracting the catalogue of galaxies that this paper makes use of. The ALFALFA team at Cornell is supported by NSF grants AST-0607007 and AST-1107390 to RG and MPH and by grants from the Brinson Foundation. EP is supported by a NOVA postdoctoral fellowship at the Kapteyn Institute. MGJ would like to thank both Kelley Hess and Andrew Baker for helpful discussions concerning CHILES and LADUMA.
|
2,869,038,154,926 | arxiv | \section{Introduction}
Over the last 20 years, there is a dramatic rise in the use and misuse of opioids in the United States, including misuse of prescription opioids, resurgence in heroin use and increase in abuse of illicit synthetic opioids such as fentanyl, which led to the current opioid epidemic and caused a rising number of overdose deaths\cite{skolnick2018opioid}.
According to the Centers for Disease Control and Prevention, the rate of opioid overdose deaths keeps rising from 1999 to 2018, posing a major threat to public health\cite{hedegaard2020drug}.
Opioid overdose happens when an excessive amount of opioid agonists work on the \update{$\mu$-opioid receptor (MOR)} in the brain, resulting in respiratory depression and eventually death\cite{schiller2019opioid}.
To reverse opioid overdoses, naloxone as shown Figure \ref{Naloxone}, an antagonist to the MOR, is used as the most common antidote, usually in the nasal formulation so as to efficiently bypass the blood brain barrier (BBB) and exert an immediate effect\cite{skolnick2018opioid}.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.4]{amia_template/pics/Naloxone.pdf}
\caption{Structure and Physicochemical Properties of Naloxone. \cite{wishart2018drugbank}}
\label{Naloxone}
\end{figure}
However, naloxone can be distributed away from the brain rapidly, leading to a brief period of pharmacodynamic action, which is possibly caused by its limited BBB permeability\cite{clarke2005naloxone}. Considering that opioid agonists have a longer half-life, there is a risk of inadequate response or re-narcotization after a single dose of naloxone, especially in patients who have taken large doses or long-acting opioid formulations\cite{rzasa2018naloxone}. Administration of repeated doses of naloxone may be necessary if respiratory depression recurs\cite{wermeling2013response}. \update{At the same time, the price of naloxone nasal spray is high \cite{gupta2016rising}.
Other attempts to lengthen the time for reversing opioid overdose, such as combining naloxone with other opioid antagonists, have failed \cite{krieter2019pharmacokinetic}. }
For all these reasons, there is a demand for more effective opioid antagonists with enhanced brain retention ability, which corresponds to high BBB permeability.
Nevertheless, developing new drugs costs 2.6 billion dollars on average, and can take more than 10 years \cite{chan2019advancing}.
Drug discovery for lead compounds, i.e., promising drug candidates, requires iterative organic synthesis and screening assays, with a failure rate higher than 90\% \cite{hughes2011principles}. Recently, the increase in the amount of chemical and biomedical data has encouraged the use of `data-hungry' machine learning algorithms such as deep learning to generate and optimize molecules, which significantly accelerates the drug discovery process by reducing resources spent on wet-lab synthesis and characterization of bad lead compounds\cite{chen2018rise, elton2019deep}.
By representing molecules as simplified molecular-input line-entry system (SMILES) strings, the generation of potential drug molecules can be treated as a sequence generation problem. With reinforcement learning (RL)\cite{sutton2018reinforcement}, the generation can be biased towards molecules with certain desired properties. For example, Popova et al \update{used RL to train a molecule generator} to generate novel compound libraries with a desired physicochemical or biological property \cite{popova2018deep}.
For naloxone, it targets the central nervous system (CNS). Several physicochemical factors underlie permeation through the BBB for CNS drugs\cite{mikitsh2014pathways}. For instance, CNS active drugs tend to have smaller molecular weight (MW). Molecules with MW less than 500 can undergo significant free diffusion and when MW increases from 200 to 450, BBB permeability \update{decreases} 100-fold. Besides, CNS drugs must have sufficient lipophilicity (measured by the partition coefficient between octanol and water, logP) to cross the hydrophobic phospholipid bilayer of cell membranes. One example of increased BBB permeability with higher logP is that heroin (logP=2.3) exhibits much higher brain uptake than morphine (logP=0.99). Besides, solubility (measured by logS) is also an important property because successful nasal products like naloxone nasal spray usually require the active ingredient to be highly soluble\cite{wermeling2013response}.
The driving question, in this study, is whether there can be molecules with both sufficient opioid antagonistic effect (i.e., a higher negative logarithm of the experimental half maximal inhibitory concentration, pIC50) and enhanced brain retention ability (i.e., a smaller MW and a higher logP) while maintaining high solubility (i.e., a higher logS). \update{Given that the number of drug-like molecules is estimated to be between $10^{30}$ and $10^{60}$, routine virtual screening on existing compound libraries can not guarantee finding molecules with multiple desired properties and exhaustive searching in the huge chemical space can be prohibitively expensive\cite{popova2018deep}.} Therefore, a multi-objective deep reinforcement learning (DRL) framework is used for the discovery of better opioid antagonists.
\section{Methods}
\subsection{A Deep Reinforcement Learning Framework}
Our framework consists of three major components: 1) a generative model based on an RNN model that can generate SMILES strings; 2) a predictive model that predicts the properties of interest for a given SMILES string; and 3) an RL engine which biases the generative model towards generating SMILE strings with desired properties, the values of which are predicted by the predictive model.
Inspired by Popova et al\cite{popova2018deep}, we first train the generative model on a large corpus ($\sim$1.9M) of real-world compound SMILES strings to learn the syntax of SMILES so that the generative model is able to generate valid SMILES strings. The learned weights provide a good initialization for the generative model during the RL stage. Second, we train the predictive model which contains a predictive sub-model for every property of interest. In this paper, we built three sub-models for pIC50, logP and logS respectively. There is no sub-model for MW since it can be directly calculated. With the learned predictive model and well-initialized generative model, we use an RL algorithm, REINFORCE \cite{williams1992simple}, to further train the generative model in an end-to-end fashion such that the generated SMILES strings can have the desired properties. Figure \ref{fig:pipeline} depicts an overview of the DRL framework.
\begin{figure}[b]
\centering
\includegraphics[scale=0.4]{amia_template/pics/Methodology.pdf}
\caption{Overview of the Deep Reinforcement Learning Framework. First, the generative model samples SMILES strings whose properties are predicted by the predictive model; the RL engine then combines all properties of each sampled SMILES into a reward as feedback to train the generative model to generate SMILES with desired properties.}
\label{fig:pipeline}
\end{figure}
\textbf{Reinforcement Learning. }
Reinforcement learning refers to the problem of learning an optimal decision-making policy to acquire the maximal amount of rewards in a sequential decision scenario \cite{sutton2018reinforcement}. Given the previously generated SMILES characters $\bm{s}_t=\{c_i\}_{i=0}^t$ where $c_0$ is the start token, the stochastic policy (i.e., the generative model) $\pi_\theta(a|\bm{s}_t)$ samples the next SMILES character $a$ as its action. Then, a reward $r_{t+1}$ is provided by the reward function $R(\bm{s}_t, a)$ and the state $\bm{s}_t$ will be updated to $\bm{s}_{t+1}=\{c_0, \cdots, c_t, a\}$. This generation process repeats until the generative model samples a termination token or reaches a predefined maximum length of a SMILES string. RL seeks an optimal policy $\pi^*_\theta$ to maximize the expected cumulative future rewards (i.e., return)
\begin{equation}
\pi^*_\theta = \arg\max_{\pi_\theta\in \mathcal{G}} \E_{\eta \sim\pi_\theta}\bigg[\sum_{t=0}^{T} \gamma^t r_{t+1}\bigg],
\label{eq:rl}
\end{equation}
where $\mathcal{G}$ is the set of all candidates policies $\pi_\theta$, and $\eta$ is a sampled roll-out (i.e., a SMILES string) of length $T$ from $\pi_\theta$. $\gamma$ is the discount factor and usually $\gamma < 1$. Solving equation \eqref{eq:rl} is a difficult problem, and many algorithms have been proposed in past decades \cite{williams1992simple,schulman2017proximal,sutton2018reinforcement}. Here, we use a classical RL algorithm called REINFORCE.
\textbf{REINFORCE. }
REINFORCE belongs to a family of RL algorithms called policy gradient methods \cite{sutton2018reinforcement} which estimates the gradient of certain performance measures of a decision-making policy and inputs the gradient into a stochastic gradient ascent algorithm to improve the policy toward higher total rewards.
Formally, if $G_t=\sum_{k=t}^T \gamma^{k-t}r_{t+1}$,
REINFORCE seeks to maximize
$$\mathcal{J}(\theta) = \E\big[\log \pi_\theta(a_t|\bm{s}_t)G_t\big],$$
where $\mathcal{J}(\theta)$ captures the expected return under the distribution of all possible state and action sequences. Note that in REINFORCE, the policy is probabilistic, i.e., $\pi_\theta(a_t|s_t)$ is the probability of taking action $a_t$ in state $s_t$. Hence, the gradient of the objective function $\mathcal{J}(\theta)$ can be written as follows
$$\nabla\mathcal{J}(\theta)=\E\big[\nabla_\theta \log \pi_\theta(a_t|s_t)G_t\big].$$
However, computing $\nabla\mathcal{J}(\theta)$ is non-trivial due to the high dimensionality of the space of possible state and action sequences. REINFORCE addresses this problem by using Monte Carlo sampling and approximating the gradient by
\begin{equation}
\nabla\mathcal{J}(\theta)\approx \frac{1}{K}\sum_{i=1}^K \sum_{t=0}^{T_i}\nabla_\theta \log \pi_\theta(a_t^i|\bm{s}_t^i)G_t^i.
\label{eq:grad_approx}
\end{equation}
At each iteration, REINFORCE samples $K$ roll-outs $\{\bm{s}_t^i, a_t^i, r^i_{t+1}\}_{i=1}^K$ from the current policy $\pi_\theta$ (i.e., the generative model), which are used to estimate $\nabla\mathcal{J}(\theta)$ using Equation \eqref{eq:grad_approx}. Then, parameters of the policy $\pi_\theta$ can be updated as
\begin{equation}
\theta' = \theta + \alpha \nabla\mathcal{J}(\theta),
\end{equation}
where $\alpha$ is the learning rate.
Due to the high variance in the sampling process, training can be unstable. To address this, a baseline reward $b_t^i$ is often estimated and subtracted from $G_t$. Hence, the gradient becomes
\begin{equation}
\nabla\mathcal{J}(\theta)\approx \frac{1}{K}\sum_{i=1}^K \sum_{t=0}^{T_i}\nabla_\theta \log \pi_\theta(a_t^i|\bm{s}_t^i)(G_t^i-b^i_t)\big].
\label{eq:reinforce_bs}
\end{equation}
Thus, REINFORCE can learn the parameters of the generative model in an end-to-end fashion by using backpropagation. During training, actions leading to higher total rewards $G_t$ will be reinforced through increasing $\log \pi_\theta$; while actions resulting in lower total reward will be suppressed by decreasing $\log \pi_\theta$.
\textbf{The Multi-Objective Reward Function. }
RL algorithms require a properly defined reward function. In this paper, we aim to learn a generative model that is able to generate SMILES strings with multiple desired properties: 1) smaller MW; 2) higher logP; 3) higher logS and 4) higher pIC50. Note that we only build predictors for logP, logS and pIC50, while MW is computed directly from the SMILES string. Therefore, we introduce a multi-objective reward function which is a weighted sum of these properties
\begin{equation}
r(\bm{s}_T)=\sum_{p\in \mathcal{P}} w_p r_p(\bm{s}_T),
\label{eq:weighted_rwd}
\end{equation}
where $\mathcal{P}=\{\text{MW, logP, logS, pIC50}\}$, $w_p$ is the weight assigned to each property and $r(\bm{s}_T)$ is the predicted property value for the generated SMILES string $\bm{s}_T$. Besides, we assign a negative weight to MW to convert the minimization task to maximization.
In addition, to ensure the validity of most generated SMILES strings, we regularize the generative model by penalizing the model when it generates an invalid SMILES string, with a negative reward $r_p$. Hence, we define the reward function $R(\bm{s}_t, a)$ as follows:
\begin{align}
R(\bm{s}_t, a) = \left\{ \begin{array}{ll}
0 & \text{if } t<T \\
r(\bm{s}_T) & \text{if } t=T \text{ and } \bm{s}_t\text{ is valid} \\
r_p & \text{otherwise}\\
\end{array} \right.
\label{eq:rwd_fun}
\end{align}
The reward is only provided at the $T$th (last) step of the generation. REINFORCE uses this reward function to learn a customized generative model that is capable of generating valid SMILES strings with multiple desired properties.
\subsection{Data Collection }
In order to train the generative model, we set up a SMILES-strings corpus with 1,870,310 unique compounds, which is retrieved from ChEMBL25 database\cite{gaulton2017chembl}. Note that all SMILES strings here are canonicalized, which means that they are uniquely mapped to compounds.
To train the predictive model, we acquire logS and logP data from the literature\cite{sorkun2019aqsoldb, popova2018deep} and removed the duplicates. IC50 data against MOR (ChEMBL ID: 4354) are retrieved from ChEMBL25 database\cite{gaulton2017chembl}. We only include compounds with explicit IC50 values at the same scale. If the IC50 value is low, then the corresponding compound is highly potent against its target since only a very little amount of the compound can inhibit the target. We take their negative logarithm to get the pIC50 dataset. A high pIC50 means that the compound has high potency.
Basic statistics for the three datasets are summarized in Table~\ref{number of unique compounds}.
\begin{table}[b]
\centering
\caption{Statistics for the Datasets in the Predictive Model}
\label{number of unique compounds}
\begin{tabular}{l|llll}
\hline
Property & Min & Max & Median & Count \\
\hline
logP & -5.1 & 11.3 & 2.0 & 14,152 \\
logS & -13.2 & 2.1 & -2.6 & 9,981 \\
pIC50 & 1.8 & 10.2 & 6.1 &915 \\
\hline
\end{tabular}
\end{table}
\subsection{Model Architecture}
\textbf{The Generative Model. }
To accelerate the training of the generative model in the RL stage, we first train the generative model to learn the syntactical rules for constructing SMILES strings.
At each time step, the generative model takes a current prefix string of a training instance (i.e., a SMILES string), and predicts the probability distribution of the next character (Figure~\ref{fig:Generative_Model}(a)). A cross-entropy loss is calculated at each step and parameters of the model are updated through back propagation. By treating each step as a multi-class classification problem, we fit the generative model to existing SMILES strings such that the model can generate valid SMILES strings. Importantly, the learned weights later serve as a good initialization for the generative model and expedite the training in the RL stage.
\begin{figure}[h!]
\centering
\includegraphics[width=0.99\linewidth]{amia_template/pics/Generative_Model.pdf}
\vspace{-.1in}
\caption{The Generative Model. (a) Model architecture. (b) Illustration of the generation of a SMILES string.}
\label{fig:Generative_Model}
\end{figure}
The generative model is a recurrent neural network \cite{lecun2015deep} which is capable of generating SMILES strings. Essentially, a SMILES string is composed of a sequence of characters $\bm{s}=\{c_1, \cdots, c_T\}$ from a vocabulary $V$ and $c_t\in V$ for all $t\in [1,T]$. To facilitate the generation process, we append a start token and a termination token to the head and tail of $\bm{s}$, respectively. As shown in Figure~\ref{fig:Generative_Model}(b), at each time step $t\in[0, T]$, the input to the generative model is a character $c_t$ ($c_0$ is the start token). The generative model first uses an embedding layer to convert the categorical character $c_t$ into an embedding vector of continuous scalars, which is then processed by a recurrent layer to update its hidden state. Finally, a dense layer and a softmax layer are used to map the hidden state to a probability distribution of the next possible character, from which we sample the next character. By repeating this process until a termination token is sampled, the generative model generates a complete SMILES string.
\textbf{The Predictive Model. }
The predictive model consists of multiple property predictors, each for one property of interest. Here, we consider three properties, namely, logP, logS and pIC50. Property prediction is essentially a regression task where we aim to map a SMILES string to a scalar value. All property predictors are RNN-based models with the same architecture. Figure \ref{Predictive_Model_DL} illustrates the predictive model architecture. First, a SMILES string is passes through an embedding layer, converting each character in the SMILES string into an embedding vector. Second, the recurrent layer sequentially processes the embedding vectors and constructs a temporal feature vector for the input SMILES string. Last, three consecutive dense layers are used to map the feature vector to a property value. The network is trained with a mean squared error (MSE) loss.
However, a large number of training examples are often required for deep learning models, like RNN, before they can achieve superior performance. For cases where only limited training examples are available, Support Vector Machines \cite{cortes1995support} and Random Forests \cite{liaw2002classification} are often more competitive. Hence,
for each property, we compare three different models: Support Vector Machines \cite{cortes1995support}, Random Forests \cite{liaw2002classification} and the proposed RNN-based model, and select the classifier with the smallest MSE.
We find Random Forest works best for pIC50 prediction; while the RNN-based model works best for logS and logP prediction.
This can be due to the fact that we only have a small number of data points ($\sim$1k) for pIC50. In contrast, $\sim$14k and $\sim$10k training examples are available for logP and logS, respectively.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{amia_template/pics/Predictive_Model_DL.pdf}
\vspace{-.2in}
\caption{The Architecture of the Predictive Model.}
\label{Predictive_Model_DL}
\end{figure}
\subsection{Implementation Details.}
In Equation \eqref{eq:rwd_fun}, we set the penalty of invalid SMILES to $r_p=\hat{r}_\mu - \hat{r}_\sigma$, where $\hat{r}_\mu$ and $\hat{r}_\sigma$ are the average and standard deviation of the weighted sum of rewards (i.e., $r(\bm{s}_T)$ in Equation \eqref{eq:weighted_rwd}) of the SMILES strings sampled from the initialized generative model (prior to the RL training stage). Importantly, we empirically find that the RL algorithm is sensitive to the $r_p$ and our setting of $r_p$ strikes a good balance between validity and desired properties. We assign an equal weight of $0.25$ to logP, logS and pIC50, and $-0.25$ to MW.
During the RL training stage, the learning rate and maximum length of SMILES strings are set to $10^{-5}$ and $200$, respectively. The generative model is trained for 240 episodes. In each episode, we sample 200 SMILES strings with a batch size of 10.
The size of the dictionary is 58 with 56 distinct SMILES tokens plus a start token and a termination token. Each token is embedded into a 512-dimensional vector.
Traditional RNN models like GRU \cite{cho2014gru} and LSTM \cite{Hochreiter-Schmidhuber-NC97} are inferior in memorizing and counting thus often fail to capture algorithmic patterns in sequences \cite{joulin2015inferring}. Meanwhile, the construction of SMILES strings needs to follow certain algorithmic rules such as atom valence constraints and bracket opening-closure. Thus, here we use Stack-augmented GRU (StackGRU) \cite{joulin2015inferring} as the recurrent layer in the generative model. The stack width, stack depth and hidden size of StackRNN are 256, 200, 512, respectively.
For the predictive model, an ordinary single-layer GRU with hidden size 512 is used as the recurrent layer. The following three dense layers have dimension sizes 128, 32, and 1 respectively, the first two of which are followed by a ReLU layer and a batch-norm layer \cite{ioffe2015batch}. The deep learning models are implemented with PyTorch \cite{paszke2019pytorch}. RDKit \cite{landrum2006rdkit} is used for validating and visualizing the molecules. The Random Forest model for predicting pIC50 has $100$ trees and is implemented with Scikit-Learn \cite{pedregosa2011scikit}.
\section{Results}
\subsection{Evaluation of the DRL Framework}
\textbf{GRU vs StackGRU.}
We first compare the GRU and StackGRU in learning the syntax of SMILES strings. Specifically, we train two generative models (see Figure \ref{fig:Generative_Model}(a): one with GRU as the recurrent layer; the other with StackGRU) on the training corpus, and sample 10k SMILES strings from each model.
By using syntactical and chemical validity check functions from RDKit, we calculate the percentage of syntactically and chemically valid compounds.
We also measure the novelty of generated compounds by calculating the percentage of non-overlapping compounds between the sample and the training corpus. Furthermore, by examining the percentage of non-duplicates within the sample, we quantify the uniqueness of the generated sample.
\begin{table}[b]
\centering
\caption{Performance Comparison between GRU and StackGRU}
\label{performance of generative model}
\begin{tabular}{l|llll}
\hline
Configuration & Syntactical Validity (\%) & Chemical Validity (\%) & Novelty (\%) & Uniqueness (\%) \\
\hline
GRU & 75.87 & 60.74 & 99.32 &99.99 \\
StackGRU & 87.29 & 77.40 & 98.92 & 99.97 \\
\hline
\end{tabular}
\end{table}
As can be seen from Table~\ref{performance of generative model}, both syntactical validity and chemical validity are relatively low when using a standard GRU compared to the StackGRU. With StackGRU, syntactical validity increased to 87.29\% and chemical validity increased to 77.40\%, which demonstrates that StackGRU is better at learning the SMILES syntax. Both novelty and uniqueness are close to 100\%, which indicates that the generative model is able to generate novel and unique SMILES strings, and does not just memorize training examples.
\textbf{Property Prediction.}
For the predictive model, we plot the predicted value vs true value in Figure ~\ref{Prediction_Performance}.
\begin{figure}[h]
\centering
\includegraphics[scale=0.55]{amia_template/pics/Prediction_Performance.pdf}
\caption{Predicted Value vs True Value from the Predictive Model.}
\label{Prediction_Performance}
\end{figure}
Indicated by a high correlation coefficient, i.e., $R^2$ and low rooted mean squared error (RMSE), the predictors for logP and logS have high precision and accuracy. However, the sub-predictive model for pIC50 has relatively poor performance, which can be caused by the insufficient data points in the pIC50 dataset. We also apply the predictive model to Naloxone. The predicted values for logP and logS of Naloxone are 1.94 and -2.67, respectively, which align reasonably well with the reported properties of Naloxone (logP=1.47 and logS=-1.8) \cite{wishart2018drugbank}.
\textbf{Molecule Generation. }
Given that our goal is to bias the generation of molecules towards higher logP, logS, pIC50 and smaller MW, we sample 10k SMILES strings after the generative model is trained using REINFORCE for 0, 80, 160 and 240 episodes, respectively.
Figure~\ref{performance_4multi_RL} shows the distribution of each property value for chemically valid strings from the samples. The pIC50 of the generated molecules are biased toward higher values, albeit not significantly. The distribution of logS is significantly shifted to the right and the distribution of MW is also significantly shifted to the left, which means molecules with higher logS and smaller MW are more likely to be generated over training episodes. However, generated molecules tend to have lower logP values, indicated by the left-shifted distribution of logP. This phenomenon is probably because logP and logS are contradictory by nature. When there are more hydrophilic groups, higher logS and lower logP are expected and \textit{vice versa} when there are more hydrophobic groups. Overall, our DRL framework is able to bias the properties of generated molecules.
\begin{figure}[h]
\centering
\includegraphics[width=0.9\linewidth]{amia_template/pics/performance_4multi_RL.pdf}
\caption{Shifted Distribution of Targeted Properties during RL Training. }
\label{performance_4multi_RL}
\end{figure}
Table~\ref{performance of multiRL model} summarizes the syntactical validity, chemical validity, novelty and uniqueness of the generated samples.
\begin{table}[b]
\centering
\caption{Evaluation of Generated Samples during RL Training}
\label{performance of multiRL model}
\begin{tabular}{l|llll}
\hline
Episode & Syntactical Validity (\%) & Chemical Validity (\%) & Novelty (\%) & Uniqueness (\%) \\
\hline
0th & 87.29 & 77.40 & 98.92 & 99.97 \\
80th & 91.92 & 89.60 & 95.58 & 98.50 \\
160th & 96.77 & 95.90 & 93.79 & 65.52 \\
240th & 99.40 & 99.16 & 96.46 & 21.51 \\
\hline
\end{tabular}
\end{table}
One major issue with RL in \textit{de novo} drug design in previous studies is the reduced validity\cite{popova2018deep}. Here, by incorporating penalty for invalid SMILES strings in the reward function, our DRL framework generates SMILES strings approaching 100\% validity when the training episodes increase. Besides, the novelty of the generated samples is also high. Uniqueness is decreasing as the number of episodes goes up since as the DRL training episodes increase, the generative model tends to converge to the distribution of a smaller number of SMILES strings with the desired properties.
\subsection{Identification of Potential Lead Compounds}
From the RL-trained generative model at the 160th episode, we sample 10k SMILES strings and then use the following criteria to filter out molecules with: 1) logP $>1.94$, 2) logS $> -2.67$, 3) pIC50 $>6$ and 4) MW $< 327.37$. The reason why we choose the 160th episode is that it enables balanced performance with regard to the validity, novelty, uniqueness and the ability to shift property distribution. The cutoff values used in logP, logS and MW are the predicted/calculated values for naloxone in our DRL framework. Note that our goal is to discover molecules with sufficient inhibitory activity against MOR and optimal logP, logS and MW values to ensure prolonged brain retention ability. We set the cutoff value of pIC50 at 6 despite the predicted pIC50 for naloxone by our predictive model is 6.93 since pIC50 $\geq 6$ corresponds to active compounds\cite{popova2018deep}.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\linewidth]{amia_template/pics/structures_generated.pdf}
\caption{Novel Molecules Generated in the DRL Framework. (a) At the 160th Episode. (b) At the 0th Episode.}
\label{generated_sructures}
\end{figure}
We filtered out six novel SMILES strings from the 10k-size sample, as the identified potential lead compounds. Figure~\ref{generated_sructures} (a) shows their structures drawn by RDKit, which are simple by direct visual checking. We also calculated their synthetic accessibility score (SAS)\cite{ertl2009estimation}. SAS can range between 1 and 10. A high SAS, usually above 6, corresponds to high molecule complexity and increased synthesis difficulty. For the six molecules, their SAS values range from 1.40 to 3.12, indicating that our DRL framework generates highly feasible molecules.
To further demonstrate the usefulness of the DRL framework, we also sample 10k SMILES strings at the 0th episode (i.e., without RL training) and filter with the same criteria. Three SMILES strings are filtered out. Figure~\ref{generated_sructures} (b) shows their structures and predicted properties. Despite having the expected properties, their structures are very complex with high SAS values, which indicates that the molecules generated without RL training are much less feasible.
\section{Conclusion and Discussion}
Recent years have seen an increasing use of deep learning in drug discovery with the rise of the `big data' era\cite{chen2018rise}. Linear representations of molecules, such as the SMILES strings, are broadly used in ligand-based drug discovery studies\cite{lipinski2019advances}. By linking molecules to end points like physicochemical properties, inhibitory activity, BBB permeability, etc, end-to-end drug discovery and development is now becoming a reality \cite{ekins2019exploiting}.
For example, in 2019, Insilico Medicine succeeded in using deep learning to design new lead compounds for discoidin domain receptor 1 (DDR1) kinase inhibitors from scratch in just 21 days\cite{zhavoronkov2019deep}.
In this study, we aim to discover better opioid antagonists to help to combat the opioid epidemic, where the most common antidote for opioid overdose, naloxone, has limited blood brain barrier permeability.
To accelerate the discovery for better opioid antogonists candidates, we implement a multi-objective DRL framework. The framework is able to identify valid, novel and feasible molecules with sufficient opioid antagonistic activity and enhanced brain retention ability.
More importantly, the proposed multi-objective DRL framework has great potential in accelerating drug discovery, which is a multi-property optimization task \textit{per se}. For instance, effective and safe drugs need to exhibit a fine-tuned combination of pharmacokinetic and pharmacodynamic properties, such as high potency, affinity and selectivity against the drug target as well as optimal absorption, distribution, metabolism, excretion and toxicity (ADMET)\cite{ferreira2019admet}. Our study shows that with well-designed reward functions, the multi-objective DRL framework can be customized to generate molecules with optimal properties from different drug development aspects.
\bibliographystyle{unsrt}
|
2,869,038,154,927 | arxiv | \section{Introduction}
The elicitation of requirements is one of the key phases in the requirements engineering process \cite{paetsch2003requirements}. In this phase, requirements from all relevant stakeholders must be considered. However, in situations where key aspects of the system to be developed are still unclear, conflicting requirements can be elicited from different stakeholders \cite{van2000requirements}. It is important to create a shared understanding of the desired system between stakeholders to solve such conflicts and to keep them from threatening the project's success. Videos present one possible medium of creating shared understanding. In these so-called vision videos, concrete visions of the functionality of a system are presented. Stakeholders watching the videos can detect, discuss and resolve misalignments of their mental models \cite{karras2020representing}.
Once vision videos are created, stakeholders usually attend a conjoined meeting in which a vision video is shown and discussed \cite{brill_videos_2010, Schneider2017}. However, stakeholders can not always attend such meetings in person. They can be distributed among different locations, time zones or even continents. In these cases, vision videos need to be watched and discussed online. Online meetings make it easier for requirements engineers to collaborate with distributed stakeholders since no travel arrangements have to be made. Nevertheless, finding a time that is suitable for every stakeholder can be a challenging or sometimes even impossible task. This problem only increases in importance with larger numbers of relevant stakeholders, for example when eliciting requirements from a crowd~\cite{groen2015towards, todoran2013cloud}. Discussing vision videos with only a subset of all relevant stakeholders introduces the risk of missing out on valuable ideas, especially those that build upon suggestions of others.
The overall goal of this research is to \textit{find a viable method to conduct vision video meetings in online settings}. Hence, we investigate synchronous and asynchronous viewings of vision videos in online settings. Synchronous viewings are the closest adaption of traditional vision video meetings. We look to find advantages and disadvantages of their use in an online context. Asynchronous viewings can be used when it is impossible to find a suitable time for a conjoined meeting. Therefore, we seek to explore the applicability of distributed viewing of vision videos to requirements engineering processes with crowds of stakeholders. The online setting allows many more people, or a crowd, to watch vision videos and provide feedback on the visions presented.
In this paper, we introduce concepts for synchronous and asynchronous viewings. The two methods are tested in a preliminary experiment. Our results indicate that vision videos can still lead to meaningful discussions regardless of the online setting. Furthermore, we find asynchronous viewings to facilitate the use of vision videos for crowds of stakeholders. The approach enables large numbers of stakeholders to generate ideas and express concerns about illustrated visions without inhibitions.
The rest of this paper is structured as follows: Section \ref{sec:rw} presents related work. We introduce our methodology including a preliminary experiment in section \ref{sec:methodology}. Results of the experiment are laid out in section \ref{sec:results} and discussed in section \ref{sec:disc}. The paper is concluded in section \ref{sec:concl}.
\section{Related Work}
\label{sec:rw}
Videos have been used in requirements engineering for many years.
Creighton et al.~\cite{creighton_software_2006} proposed ``software cinema'', a scenario video creation methodology.
Such videos show workflows of customers which are not implemented yet. Xu et al. \cite{xu_user_2013} also examined scenario videos in the scope of software development and applied them in the context of a case study.
Brill et al.~\cite{brill_videos_2010} have listed opportunities for using videos in different phases of requirements engineering.
They have conducted a study to investigate whether video creation is effective and efficient in comparison to use cases.
Gathering feedback on videos from a crowd in online settings has already been applied before the Covid-19 pandemic.
Keimel et al.~\cite{keimel_qualitycrowd_2012} have proposed a web application to collect opinions from a crowd regarding video quality.
For more complex feedback, Lasecki et al.~\cite{lasecki_glance_2014} developed a video coding tool with a conversation interaction paradigm between user and system. A user asks questions in natural language and receives answers from crowd workers. Crowd opinions are aggregated as final feedback.
Crowd opinions can be gathered from a group or from multiple sessions with individuals. In the field of psychology, several researchers studied the decision-making mechanisms of groups.
Gruenfeld et al.~\cite{gruenfeld_group_1996} have tested whether group decision making is dependent on the degree of homogeneity within the group.
Toma and Butera~\cite{toma_hidden_2009} provided selected information to influence a group's decision making.
Clart et al.~\cite{clark_comparing_2015} have conducted a game to compare decision making of groups and individuals. Their results indicate that groups perform better than individuals with regard to decision rationality.
In the engineering domain, Ezin et al.~\cite{ezin2019} have built a video recommendation system for individuals based on group preferences.
All the literature listed indicates that group decision making is a complex process that seems to depend on group members, external information, and individual opinions.
During the pandemic, Karras et al.~\cite{karras2020using} used vision videos in a virtual focus group of five participants.
Based on their experiences, they formulated recommendations for the use of vision videos in virtual focus groups.
In our work, we use vision videos for the evaluation of different visions. We collect feedback from potential stakeholders on different system visions in online meetings. The stakeholders are asked to choose one of the presented visions and provide reasons for their decision. Our efforts focus on the viewing of vision videos online in group settings or individually. We examine whether differences between these settings can be observed.
\section{Methodology}
\label{sec:methodology}
The goal of this research is to \textit{find a viable method to conduct vision video meetings in online settings}. For this purpose we designed two different methods, namely synchronous and asynchronous viewing.
\subsection{Synchronous Viewing}
\label{sec:sync}
Synchronous viewing of vision videos in online settings aims directly at simulating the experience of traditional meetings. For this purpose, stakeholders join a video conferencing tool like Zoom\footnote{\url{https://zoom.us/meetings}} or BigBlueButton\footnote{\url{https://demo.bigbluebutton.org/gl}}. Here, they can use a microphone to speak to other attendees. Webcams can also be used to enable stakeholders to see one another. The mentioned video conferencing tools allow users to share videos within them. Vision videos can be shown and controlled by the meeting's organizers. The control elements change the video for all attendees simultaneously. This means that pauses, rewinds or jumps to specific parts of the video are visible for all participants. Meeting organizers can therefore control the focus of the discussion. Video sections that are unclear can be revisited and discussed explicitly by using these controls.
\subsection{Asynchronous Viewing}
\label{sec:async}
A different method of watching vision videos in online settings is to do so asynchronously. Asynchronous viewing of videos means that each stakeholder watches the video by themselves. There is no fixed time at which all stakeholders have to watch simultaneously. Instead, a deadline can be given. In this method, all control over the video lies with the stakeholder. Asynchronous viewing requires additional means to record stakeholder’s ideas or concerns.
\subsection{Goal and Research Questions}
\label{sec:study}
According to the goal definition template by Wohlin et al.~\cite{wohlin_experimentation_2012}, we formulated our research goal as follows:
\begin{framed}\noindent
\textbf{Goal definition:} \newline \textbf{We analyze} different methods of viewing vision videos
\newline \textbf{for the purpose of} finding a viable method
\newline \textbf{with respect to} the feedback elicited
\newline \textbf{from the viewpoint of} stakeholders
\newline \textbf{in the context of} an online experiment setting.
\end{framed}
Based on our research goal, we formulated the following research questions:
\begin{itemize}
\item \textbf{RQ1:} What are the advantages and disadvantages of viewing vision videos synchronously?
\item \textbf{RQ2:} What are the advantages and disadvantages of viewing vision videos asynchronously?
\end{itemize}
\begin{figure*}[!h]
\centering
\includegraphics[scale=0.58]{Figure1.pdf}
\caption{Experiment design}
\label{fig:Experiment Design}
\end{figure*}
\subsection{Hypotheses}
Based on the two methods of synchronous and asynchronous viewing of vision videos, we conducted an experiment with two sets of participants: a group (synchronous viewing) and individuals (asynchronous viewing).
Therefore, we analyzed the following hypotheses:
\newline \textbf{H1\textsubscript{0}}: There is no difference between the sets of participants regarding a change in the choice of variants after the discussion.
\newline \textbf{H2\textsubscript{0}}: There is no difference between the sets of participants regarding the amount of feedback given by single participants.
The respective alternative hypotheses {H1\textsubscript{1}} and {H2\textsubscript{1}} state the opposite of the null hypotheses, namely that there is a difference between the sets of participants.
Both hypotheses aim at possibly existing differences in the two online settings considered. Based on these differences we formulate advantages, disadvantages and potential use cases.
\subsection{Experiment Design}
\paragraph{Material}
In our experiment, we used a video that we divided into two thematic sections by pausing it. At the beginning of the first video section, there was an introduction to familiarize viewers with the topic of shopping in rural areas and the potential challenges associated with it. After this introduction, the first video section showed three possible options for ordering products: by taking a picture, by pressing a button, or by automatic measurement. The second section of the video includes three delivery options for products by receiving the package from a neighbour, by drone delivery, or by dropping the package in the trunk of a car. The videos used in our experiment were produced and already used in the context of the paper ``Refining Vision Videos'' by Schneider et al.\cite{schneider_refining_2019}.
At the end of the experiment, participants were asked to complete a LimeSurvey\footnote{\url{https://www.limesurvey.org/de/}} questionnaire. The questionnaire contained demographic questions and questions about the experiment.
\paragraph{Participant Selection}
All 16 participants of our experiment took part voluntarily. The participants were between 21 and 33 years old (M = 25.8, SD = 4.1). Three of the participants were female and thirteen male.
Ten of the participants are students; six are currently employed or have been employed in the past. Fourteen of the participants are studying subjects in STEM\footnote{STEM (Science Technology Engineering Mathematics)} fields or are currently working or have worked in STEM fields in the past.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.46]{Participants_Demographics2.pdf}
\caption{How often have you been involved in software development processes?}
\label{fig: Demographics}
\end{figure}
Figure \ref{fig: Demographics} shows the distribution of responses to the question \textit{How often have you been involved in software development processes?}. On a likert scale labeled from 1 (never before) to 5 (very often), two participants stated that they had never been involved in a software development process before.
\paragraph{Experiment Procedure}
At the beginning of the planning phase of the experiment participants signed up for appointments in an online calendar. Subsequently, a date was set for the group session on which at least six participants had registered. The other participants were assigned to the group of individual sessions.
Prior to the experiment, a consent form and an overview were sent to the participants by e-mail. Six of the participants took part in a group session, the other ten participants took part individually. The experiment was conducted virtually in BigBlueButton.
Our design is divided into two segments. Figure \ref{fig:Experiment Design} shows the sequence of sections for the individuals and the group of six participants.
At the beginning of our experiment the participants saw the thematic intro section and the three order variants (labeled A, B and C). After watching the videos, participants voted on which variant they would choose via a short poll started by the experimenter.
Then the group participants discussed their choices among each other. The individuals were asked about pros and cons of each variant by the interviewer. After this exchange, participants voted again on which order variant they would choose via a short poll started by the experimenter. This sequence of video watching, voting and an exchange was repeated in the second segment regarding the delivery variant videos (labeled 1, 2 and 3).
At the end of the experiment, the participants were asked to complete an online questionnaire.
\paragraph{Data Analysis Procedures}
In our experiment, we collected data in three different ways. We have documented the participants' votes of the variants, a total of four votes for each participant. In addition, we analyzed the audio recordings of the sessions to be able to count the mentioned pros and cons for each order and delivery variant. The last data source we used are the results of the LimeSurvey questionnaire. Regarding the results of the questionnaire, we cleaned the data by removing incomplete response data-sets.\\
\textit{Analyzing the Choice of Variants}: To examine our data regarding the first hypothesis, we analyzed the results of two questions of the online questionnaire in a descriptive way. Within the scope of these questions, we asked the participants whether they had changed their opinion regarding the choice of variants after the discussion and if so, why. \\
\textit{Analyzing the Amount of Arguments}: To answer our second research question, we listened to the audio recordings of the sessions to count the arguments for each variant. Based on the results of the counting, we calculated the average arguments per participant for both groups. \\
\textit{Analyzing Additional Results}: For the analysis of the additional results (\textit{Video Quality, Changes of Opinion, Chosen Variants}) the results of the online questionnaire were used. The respective average of the data, the median and the standard deviation were calculated for each subsection.
\section{Results}
\label{sec:results}
Analyzing the data as described leads to the following results:
\subsection{Choice of Variants}
We asked our participants to choose one of the shown variants four times. The exit survey included a question regarding whether or not participants had changed their choices. We also asked about the reasons for these changes.
Four out of six participants (66\%) of the group session indicated that their choices had changed. When asked for the reasons behind these changes, they mentioned misunderstandings being cleared up and critical questions being answered in the discussion with other attendees.
As for the asynchronous viewers partaking in the individual sessions, six out of ten test subjects (60\%) changed their opinion. Three of them added that the additional time they spent thinking about the advantages and disadvantages of individual variants was the reason for these changes. Another participant mentioned that they needed time to adjust to their role as a stakeholder for a system concerning rural infrastructure. Based on these results, we reject \textbf{H1$_0$} and accept \textbf{H1$_1$}.
\subsection{Arguments}
During the experiment we counted the arguments in favor and against individual variants mentioned by participants.
For example, the advantages ``minimum cost" and ``flexibility" were stated by participants regarding order variant A in which a picture of the product is taken using a smartphone. An example for a mentioned disadvantage of the delivery by drone was that ``It could be problematic if I were not at home when the drone comes".
The group attending the synchronous meeting gave a total of 24 arguments. With six participants taking part in the meeting, this means that each participant mentioned four arguments on average. Furthermore, we can calculate the average amount of arguments for each variant shown in the video per participant. With six variants present, this results in 0.66 arguments per variant and participant.
For the ten individual test subjects partaking in asynchronous viewings, we counted a total of 151 arguments. This means that an average of 15.1 arguments per participant was given. As the same video was shown, the average number of arguments per variant and participant can be calculated in the same manner resulting in a value of 2.5.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.98]{Figure3.pdf}
\caption{Average number of arguments per variant and participant}
\label{fig: NumberOfArguments}
\end{figure}
Figure~\ref{fig: NumberOfArguments} shows that the average number of arguments per variant and participant from those taking part in the asynchronous viewings (dark green) is far higher than the corresponding data from the synchronous viewing (light green).
This difference represents an increased amount of feedback. Therefore, we can not accept \textbf{H2$_0$}. Instead, we accept \textbf{H2$_1$}.
Based on the given arguments, we have also collected new requirements from participants.
One participant suggested scanning a QR-Code as an alternative to ordering by taking a picture (variant A), because the product can be recognized precisely.
While criticizing the missing recipient on drone delivery (variant 2), one participant proposed that the landing time and place should be arranged before delivery.
Apart from the requirements presented, numerous enhanced or new requirements were stated by the participants.
Based on the results of our preliminary experiment, we infer that listing the advantages and disadvantages can support the elicitation phase.
\subsection{Additional Results}
\label{addtional-result}
In addition to the results of the experiment, we asked participants further questions in an exit survey on their experiences during the experiment. We collect their answers to gather information about the validity of our results.
\paragraph{Video Quality}
One question of the exit survey asked about participant's agreement with a statement expressing that the video's quality was good enough to understand its contents. With an answer of 1 indicating full disagreement and a rating of 5 indicating full agreement, participants gave an average rating of 4.13 (M = 4, SD = 0.96). Notably, 75\% of participants agreed with the statement. Only a single test subject disagreed.
\paragraph{Changes of Opinion}
The survey included a question regarding participant's self-reported tendencies to change their opinions after exchanges with others. With an answer of 1 meaning that they were very unlikely to change their opinion, while 5 indicated a high probability, participants' answers provided an average of 2.56 (M = 2.5, SD = 0.96).
\paragraph{Chosen Variants}
Finally, we asked participants whether they were satisfied with the variants chosen in the experiment. Again, ratings of 1 and 5 indicated strong dissatisfaction and strong satisfaction respectively. The participants of our experiment answered with an average of 3.25 (M = 4, SD = 0.93). Notably, the strongest ratings of 1 and 5 were not chosen by any test subjects.
\section{Discussion}
\label{sec:disc}
Based on observations made by researchers during the preliminary experiment and the collected results, we gathered insights about our research questions.
The results of our preliminary experiment reveal insights on our research questions.
\subsection{Pros and Cons of Synchronous Viewing (RQ1)}
We trialed synchronous viewings of vision videos in online settings as described with a group of six participants. Based on our observations and the results of the survey we found the following advantages and disadvantages:
\paragraph{Advantages}
Viewing vision videos in a synchronous manner resembles traditional meetings most closely. Processes that worked for vision video meetings in person can be adopted for synchronous online meetings with minimal adjustments.
Synchronous viewings of vision videos mean that stakeholders attend the same meeting. They can directly discuss the contents of the shown video. This means that misaligned mental models can quickly be identified and cleared up. The exchanges can also result in new ideas. Stakeholders can express their opinions and suggest solutions to open issues. They can also expand on suggestions of other attendees which they would not have considered otherwise. At the end of the meeting, stakeholders should agree on a concrete vision of the system. The importance of discussions with other stakeholders is underlined by the acceptance of \textbf{H1$_1$}.
Furthermore, synchronous viewings provide a clear time frame for requirements engineers.
Planned meetings provide clear start and end times. A well structured meeting should lead to satisfying results within the set time frame. This makes it easier for requirements engineers to foresee when they can expect the results of the vision video viewings.
\paragraph{Disadvantages}
One of the major difficulties when organizing synchronous meetings with many attendees is finding a time that is suitable for everyone. Vision videos can be watched with only a subset of all relevant stakeholders. However, this introduces the risk of missing out on ideas, especially ideas that build upon suggestions of others.
Moreover, conjoined meetings often are of a fixed length. This leads to discussions having to be cut short in order to maintain the meeting's schedule.
Another disadvantage is the fact that synchronous viewings also bear the risk of not gathering all possible arguments from shy stakeholders. In meetings with many attendees, some participants are likely to be more active than others. Other stakeholders may be reluctant to express new ideas or disagree with their peers. Stakeholders can resign to just repeating what other attendees mentioned instead of thinking about new ideas themselves.
\subsection{Pros and Cons of Asynchronous Viewing (RQ2)}
As for asynchronous viewings, we tested our method with ten test subjects. We found the following advantages and disadvantages:
\paragraph{Advantages}
One of the main advantages of asynchronous viewings of vision videos is the fact that stakeholders can freely choose a time at which they watch the video. Requirements engineers do not need to find a single time that is suitable for all stakeholders. However, it is advisable to set a larger time frame in which the video should be watched.
Another benefit is that stakeholders have full control over the video during these asynchronous viewings. They can move freely along the video’s time line and explore its content at their own pace. This also allows them to re-watch individual sequences or even the whole video. Stakeholders can record their own ideas and concerns without inhibition. Our acceptance of \textbf{H2$_1$} emphasizes this advantage.
Asynchronous viewings of vision videos are a way to detect conflicts within the visions of the video’s creators and stakeholders. The method requires minimal effort from requirements engineers as they do not need to be present while the video is watched. Conflicts can then be addressed in meetings that should be shorter than completely synchronous viewings. Providing stakeholders with the video before a meeting also means that they have more time to prepare thoughtful suggestions. This could benefit the requirements engineering process.
\paragraph{Disadvantages}
The most important drawback of asynchronous viewing is that stakeholders miss out on the possibility to directly discuss suggestions and concerns with their peers. They can not respond to issues raised by others or expand on their ideas.
The missing discussion also means that conflicts in the visions of stakeholders can not be resolved immediately. While a single stakeholder can raise their concerns with the video’s creators and start a discussion with them, they are unable to confer with other stakeholders. This means that conflicting mental models between such stakeholders would need to be resolved in a separate manner.
Another disadvantage of asynchronous viewing is the fact that stakeholders need to record their thoughts themselves. This means that a way to record stakeholders’ comments is needed. Such a collaborative system could for example work through annotations that viewers can add to relevant video sections.
Additionally, stakeholders might not understand the video’s contents correctly. Such misunderstandings can not be rectified immediately when the video is watched asynchronously. Instead, another discussion, at least via e-mail, is required.
\subsection{Research Goal}
Both methods of watching vision videos presented in this paper have major advantages and disadvantages. It is not possible to objectively decide which method is superior.
Ultimately, the two presented methods are best suited for different use cases. Synchronous viewings most closely simulate the traditional use of vision videos. They fulfill the same purpose and should develop similar results. Asynchronous viewings can support the requirements elicitation process when it is hard to find a time at which all stakeholders can attend the same meeting.
A combination of the two methods could present the most suitable method for using vision videos in online settings. By sharing the video with stakeholders they can prepare themselves for the conjoined meeting. Misalignments between the vision presented in the video and a stakeholder's mental model can be detected before the meeting, meaning that they can point to the relevant video sections. Additionally, they can also develop ideas expanding on the video's contents or decide on any shown variants. This way, the synchronous meeting can be reduced to a compact discussion between stakeholders which in turn should lead to more ideas being discussed in a shorter amount of time.
The most important use case for asynchronous meetings is the elicitation of requirements from crowds. When dealing with large numbers of stakeholders, it is often impossible to organize and conduct a synchronous meeting. Discussions between individual stakeholders are also impossible due to the scale of the requirements engineering process. Letting crowds of stakeholders watch vision videos asynchronously enables each individual stakeholder to contribute their ideas and suggestions. A system recording these comments is still necessary but can also include automatic aggregations of the given data. This way, requirements engineers can easily gather high quality data from a crowd of stakeholders.
\subsection{Threats to Validity}
There are a number of limitations to our results. In this section, we discuss these threats to validity according to the four types defined by Wohlin et al.~\cite{wohlin_experimentation_2012}.
\paragraph{Construct Validity}
The construct validity of our results is threatened by the fact that the answers to our questionnaire heavily rely on self reporting. Aspects like the tendency to change one's opinion after discussions with peers were not measured by any objective metrics. Instead, participants simply reported on a likert scale.
Another threat is that during individual sessions, participants may not understand the video's content, so that they cannot make immediate decisions after watching (see figure~\ref{fig:Experiment Design}). To mitigate this threat, the researcher asked the participant to recall the variants first. If they still have difficulty making a decision, the interviewer provided necessary descriptions in key words.
Moreover, bad video quality can lead to contents not being shown clearly. This may lead to requirements being misunderstood. To check whether participants perceived video quality to be an issue, we asked them question about video quality. As results in section~\ref{addtional-result} show, most participants rated the video quality positively. This indicates that this threat did not exist in the context of our experiment.
\paragraph{Internal Validity}
The results of our experiment could have been effected by participants' fatigue or boredom. To minimize this threat to validity we kept the execution time of our experiment as short as possible. Not a single participant took longer than 30 minutes.
Another threat to the internal validity of our results is that the outcome of discussions between participants is not objective. Arguments given for or against a particular variant shown in the vision video are solely based on the opinion of the participants. Repeating the experiment with a different set of test subjects might lead to a different outcome. However, we do not expect a change to the observed advantages and disadvantages of each method.
\paragraph{Conclusion Validity}
A major threat to the conclusion validity of our results is the small sample size. Our preliminary experiment only included 16 participants. We still found a number of advantages and disadvantages of both methods presented in this paper.
Moreover, for the results of participants taking part in the individual interviews, we counted all arguments mentioned. We did not differentiate between already mentioned points and completely new notions. Nevertheless, arguments that are expressed multiple times still present meaningful feedback. Concerns shared by a number of stakeholders can be seen as more important than those only mentioned once.
\paragraph{External Validity}
Our selection of participants also threatens the external validity of this research. Most of them study or work in STEM related fields. As for the attendees of our group session simulating synchronous viewing of vision videos, we purposefully selected participants who did not know each other before the experiment. This ensured that participants did not have an inclination to respond to particular other participants who they had already discussed other topics with.
Furthermore, most participants of our experiment revealed that they had been involved in software development processes before taking part. They are more likely to be familiar with the role of stakeholders. Conducting the experiment with participants less familiar with development processes might lead to different discussions. However, we do not expect this to have impacted the usefulness of our methods.
\section{Conclusion}
\label{sec:concl}
Vision videos can be used as a way to create a shared understanding between stakeholders. The video itself can be used as the foundation for discussions on the design of a new system. However, the use of vision videos in times of social distancing has not been fully explored in scientific research.
In this paper, we present concepts for two different methods of using vision videos in online settings. We expand on the advantages and disadvantages of both synchronous and asynchronous viewing of videos and conduct a preliminary experiment. We observed participants watching a vision video in a between-groups design. One group watched the video synchronously, while the other group watched in individual sessions.
Our results indicate clear advantages of both methods. Synchronous viewings most closely simulate traditional vision video meetings where all relevant stakeholders meet in person to watch the video. Asynchronous viewings enable vision videos to be used with large numbers of stakeholders, when the sheer number of participants would prohibit a conjoined meeting due to scheduling constraints. Using vision videos and asynchronous viewing, requirements can be elicited from crowds. Both methods facilitate distributed teams to use vision videos without having to meet in person and therefore without major travel costs.
The experiment presented in this paper is only a preliminary one. A larger experiment with practitioners would lead to more refined results. We also seek to evaluate whether our concepts are transferable to in-person uses of vision videos. Watching the video asynchronously before a shorter synchronous meeting could reduce the time required of stakeholders.
The use of our concepts for crowds of stakeholders should also be investigated further. Finally, a collaborative system to collect the feedback from stakeholders who watch the vision video asynchronously is yet to be developed.
\section*{Acknowledgment}
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant No.: 289386339, project ViViUse.
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